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## Material Information

Title:
Cascaded lattice Boltzmann methods based on central moments for thermal convection, multiphase flows and complex fluids
Creator:
Ouderji, Farzaneh Hajabdollahi
Place of Publication:
Denver, CO
Publisher:
Publication Date:
Language:
English

## Thesis/Dissertation Information

Degree:
Doctorate ( Doctor of philosophy)
Degree Grantor:
Degree Divisions:
College of Engineering, Design, and Computing, CU Denver
Degree Disciplines:
Engineering and applied science
Committee Chair:
Welch, Samuel
Committee Members:
Premnath, Kannan
Jenkins, Peter
Biringen, Sedat
Lee, Taehun

Abstract:

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Auraria Library
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Copyright Farzaneh Hajabdollahi Ouderji. Permission granted to University of Colorado Denver to digitize and display this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.

Full Text
CASCADED LATTICE BOLTZMANN METHODS BASED ON CENTRAL MOMENTS FOR THERMAL CONVECTION, MULTIPHASE FLOWS AND COMPLEX FLUIDS
by
FARZANEH HAJABDOLLAHI OUDERJI BS, Bahonar University of Kerman (BUK), 2009 MS, Ferdowsi University of Mashhad, 2011
A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Engineering and Applied Science Program
2019

ii
This thesis for the Doctor of Philosophy degree by Farzaneh Hajabdollahi Ouderji has been approved for the Engineering and Applied Science Program by
Samuel Welch, Chair Kannan Premnath, Advisor Peter Jenkins Sedat Biringen Taehun Lee
Date: May 18, 2019

Ill
Hajabdollahi Ouderji, Farzaneh (PhD, Engineering and Applied Sciences)
CASCADED LATTICE BOLTZMANN METHODS BASED ON CENTRAL MOMENTS FOR
THERMAL CONVECTION, MULTIPHASE FLOWS AND COMPLEX FLUIDS
Thesis directed by Assistant Professor Kannan Premnath
ABSTRACT
Lattice Boltzmann (LB) methods are kinetic schemes based on stream-and-collide procedures for the evolution of particle distribution functions, and are of much interest to the computational fluid dynamics community due to the locality of their algorithmic steps and other numerical features. Among the more recent developments in this area is the cascaded LB formulation, which is based on central moments and multiple relaxation times that was originally constructed for athermal, single-phase or single-fluid flows, and is promising due to its natural ability to impose Galilean invariance and its enhanced numerical properties such as improved numerical stability when compared to the other collision models. In this dissertation, we advance its state-of-the-art by proposing several new schemes based on cascaded LB approach with improved convergence and/or accuracy or numerical stability, with a common theme involving the use of double (or more) distribution functions that evolve under the relaxation of various central moments during the collision steps for the computation of various multi-physics fluid dynamics applications, including heat transfer and multiphase systems. Specifically, we present and analyze the following: (i) convergence acceleration by preconditioning of the cascaded LB method with additional Galilean invariant (GI) corrections to improve accuracy by reducing aliasing errors due to the discreteness and symmetries of the lattice for steady state flow simulations, (ii) implementation strategies for body forces and source terms in the simulations of fluid motion and scalar transport using cascaded LB schemes via operator splitting, (iii) cascaded LB methods for thermal convective flows in three-dimensions and cylindrical coordinates with axial symmetry, (iv) a kinetic approach for computing the vorticity fields locally in a double

distribution functions based LB formulation without relying on finite differences for spatial derivatives, and (v) cascaded LB schemes for incompressible, two-phase flows and interface capturing at high density ratios based on phase-field models, including extensions for representing surfactant effects. Numerical validation against a variety of complex flow benchmark problems demonstrate the accuracy and effectiveness of the approach for each of the new developments. We show significant improvements in convergence acceleration by preconditioning and accuracy by GI corrections, and improvements in stability with the use of cascaded LB formulations when compared to other collision models for simulations of thermal convective flows and multiphase systems. Overall, the cascaded LB schemes are found to be versatile and numerically robust for a variety of complex flows with attendant multi-physics effects.
The form and content of this abstract are approved. I recommend its publication.
Approved: Kannan Premnath

V
DEDICATION
This dissertation is dedicated to my family, and especially my sister, for their understanding, motivation and patience during the challenges of graduate school and life. I am truly thankful for having all of you in my life. I would to also dedicate this dissertation to my major Professor Prof. Kannan Premnath for his vital contributions to my professional development and for his numerous help during my graduate study. This work would not have been possible without his immense expertise that he shared and his ablest guidance.

ACKNOWLEDGEMENTS
I am grateful to Prof. Kannan Premnath for his constant support and guidance, and for providing an excellent opportunity for my academic and professional development. I would like to thank Profs. Samuel Welch, Peter Jenkins, Sedat Biringen and Taehun Lee for agreeing to serve on my dissertation committee and for their comments and suggestions.
My Ph.D. research program was financially supported by the startup funds of Prof. Premnath and the US National Science Foundation (NSF) under Grant CBET-f7056630, and I gratefully acknowledge their support.

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I INTRODUCTION 1
1.1 Background and Motivation.................................................. 1
1.2 Single-Relaxation-Time (SRT) Lattice Boltzmann Method...................... 3
1.3 Cascaded Lattice Boltzmann Method based on Central Moments................ 6
1.4 Research Goals and Original Contributions of This Dissertation............ 14
1.4.1 Preconditioned cascaded LB scheme for convergence acceleration with improved Galilean invariance................................................ 15
1.4.2 Symmetrized operator splitting formulations force/source implementations
in cascaded LB methods for flow and scalar transport............... 16
1.4.3 Three-dimensional cascaded LB method for thermal convective flows.. 17
1.4.4 Axisymmetric central moment LB schemes for flows with heat transfer including swirling effects.................................................. 18
1.4.5 Local vorticity computation approach using double distribution functions
based LB techniques for flow and scalar transport ................. 19
1.4.6 Cascaded LB methods for two-phase flows with interface capturing based
on a phase-field model including surfactant effects ................ 20
1.5 Organization of This Dissertation......................................... 21
II GALILEAN INVARIANT PRECONDITIONED CENTRAL MOMENT LATTICE BOLTZ-
MANN METHOD WITHOUT CUBIC VELOCITY ERRORS FOR EFFICIENT STEADY FLOW SIMULATIONS 23
2.1 Introduction.............................................................. 23
2.2 Preconditioned Cascaded Central Moment Lattice Boltzmann Method: Non-Galilean
Invariant Formulation..................................................... 28

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2.3 Derivation of Non-Galilean Invariant Spurious Terms in the Preconditioned Cascaded Central Moment LB Method: Chap man-Enskog Analysis..................... 30
2.4 Derivation of Corrections via Extended Moment Equilibria for Elimination of Cubic Velocity errors in Preconditioned Macroscopic Equations.................. 38
2.5 Galilean Invariant Preconditioned Cascaded Central Moment LBM without Cubic
Velocity Errors on a Standard Lattice.......................................... 44
2.6 Numerical Results ............................................................ 47
2.6.1 Lid-driven Cavity Flow ................................................. 47
2.6.2 Laminar Flow over a Square Cylinder..................................... 48
2.6.3 Backward-Facing Step Flow............................................... 55
2.6.4 Hartmann Flow........................................................... 56
2.6.5 Four-rolls Mill Flow Problem: Comparison between GI Corrected and Uncorrected Preconditioned Cascaded LBM .............................. 58
2.7 Summary and Conclusions....................................................... 63
III SYMMETRIZED OPERATOR SPLIT SCHEMES FOR FORCE AND SOURCE MODELING IN CASCADED LATTICE BOLTZMANN METHODS FOR FLOW AND SCALAR TRANSPORT 67
3.1 Introduction.................................................................. 67
3.2 Operator Splitting Methods.................................................... 70
3.3 Strang Splitting of Lattice Boltzmann Method Including Body Forces ........... 73
3.4 Body Force Scheme for 2D Cascaded LB Method for Fluid Flow via Strang Splitting 75
3.5 Extension of the Symmetrized Operator Split Implementation for Cascaded LB
Method for Passive Scalar Transport Including Sources.......................... 84
3.6 Results and Discussion........................................................ 88
3.6.1 Poiseuille Flow......................................................... 89
3.6.2 Hartmann Flow
91

ix
3.6.3 Womersley Flow........................................................ 92
3.6.4 Flow through a Square Duct .......................................... 96
3.6.5 Four-rolls Mill Flow Problem.......................................... 96
3.6.6 Thermal Couette Flow with Viscous Heat Dissipation...................100
3.7 Summary and Conclusions.....................................................103
IV CENTRAL MOMENTS-BASED CASCADED LATTICE BOLTZMANN METHOD FOR THERMAL CONVECTIVE FLOWS IN THREE-DIMENSIONS 104
4.1 Introduction................................................................104
4.2 Three-dimensional Cascaded LBE for Thermal Convective Flows Using D3Q15
Lattice.....................................................................107
4.2.1 3D Cascaded LB Model for Fluid Flow..................................107
4.2.2 3D Cascaded LB Model for Transport of Temperature Field...............108
4.3 Results and Discussion......................................................119
4.4 Summary and Conclusions.....................................................127
V CASCADED LATTICE BOLTZMANN METHOD BASED ON CENTRAL MOMENTS FOR
AXISYMMETRIC THERMAL FLOWS INCLUDING SWIRLING EFFECTS 128
5.1 Introduction................................................................128
5.2 Cascaded LBM for Axisymmetric Thermal Convective Flows: Swirling Effects . . . 132
5.2.1 Governing equations for thermal flows in cylindrical coordinates with axial
symmetry..............................................................132
5.2.2 Cascaded LB scheme for axial and radial velocity fields: operator splitting
for mass and momentum source terms..........................135
5.2.3 Cascaded LB scheme for azimuthal velocity field: operator splitting for
source term...........................................................141
5.2.4 Cascaded LB scheme for temperature field : operator splitting for source term 145
5.3 Results and Discussion......................................................147

X
5.3.1 Taylor-Couette flow ..........................................................148
5.3.2 Natural convection in an annulus between two coaxial vertical cylinders . . . 150
5.3.3 Swirling flow in a lid-driven cylindrical container...........................156
5.3.4 Mixed convection in a slender vertical annulus between two coaxial cylinders 161
5.3.5 Melt flow and convection during Czochralski crystal growth in a rotating
cylindrical crucible .........................................................164
5.3.6 Comparison of single relaxation time and cascaded LB models for axisym-
metric flow simulations ......................................................168
5.4 Summary and Conclusions.........................................................171
VI LOCAL VORTICITY COMPUTATION IN DOUBLE DISTRIBUTION FUNCTIONS BASED LATTICE BOLTZMANN MODELS FOR FLOW AND SCALAR TRANSPORT 173
6.1 Introduction........................................................................173
6.2 MRT-LBE for Fluid Motion............................................................177
6.2.1 Moment relationships for the symmetric velocity gradient tensor: Chapman-
Enskog Analysis ..............................................................180
6.3 MRT-LBE for Transport of a Passive Scalar .......................................183
6.3.1 Moment relationships for the scalar gradient vector and skew-symmetric
velocity gradient tensor: Chapman-Enskog Analysis ............................185
6.4 Derivation of local expressions for the complete velocity gradient tensor and vor-
ticity field.........................................................................188
6.5 Results and discussion..............................................................190
6.5.1 Poiseuille flow...............................................................190
6.5.2 Four-rolls mill flow..........................................................191
6.5.3 Womersley flow................................................................194
6.6 Summary and Conclusions.............................................................196
VIEASCADED LATTICE BOLTZMANN METHOD FOR PHASE-FIELD MODELING OF IN-

XI
COMPRESSIBLE MULTIPHASE FLOWS 199
7.1 Introduction.............................................................199
7.2 Governing Macroscopic Equations: Interface Capturing and Two-Phase Fluid Motion 203
7.3 Modified Continuous Boltzmann Equation for Two-Phase Flows and Central Moments of Equilibria and Sources..............................................206
7.3.1 Continuous Central Moments of Equilibria and Sources of MCBE......208
7.4 Cascaded LB Method for Solution of Two-Phase Fluid Motion ...............211
7.5 Cascaded LB Method for Solution of Phase-Field based Interfacial Dynamics .... 217
7.6 Results and Discussion...................................................220
7.6.1 Evolution of a circular interface in imposed shear flow...........221
7.6.2 Laplace-Young relation of a static drop............................221
7.6.3 Rayleigh-Taylor instability........................................225
7.6.4 Falling drop under gravity.........................................227
7.6.5 Buoyancy-driven rising bubble .....................................228
7.6.6 Impact of a drop on a thin liquid layer............................231
7.7 Comparative study of numerical stability of different collision models...234
7.8 Summary and Conclusions..................................................238
VIHUMMARY, CONCLUSIONS AND OUTLOOK 240
BIBLIOGRAPHY 247
APPENDIX
A 265
1.1 Strang Splitting Implementation of Body Forces in 3D Central Moment LB Method 265
B 271

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2.1 Structure of the 3D Central Moment-based Collision Kernel for Fluid Flow using a
D3Q15 Lattice .....................................................................271
C 275
3.1 Source Terms for the 3D Cascaded LBE for Scalar Field using D3Q15 Lattice . . . 275
D 277
4.1 3D Cascaded LB Model for Transport of Temperature Field using a D3Q7 Lattice . 277
E 281
5.1 Relation between non-equilibrium moments and spatial derivatives of components
of moment equilibria for D2Q9 lattice ............................................281
F 283
6.1 Phase-field modeling of effect of soluble surfactant dynamics on two-phase flows . . . 283
6.1.1 Introduction ...............................................................283
6.1.2 Surfactant concentration evolution equation.......285
6.1.3 Interfacial force in the presence of surfactants via a geometric approach for
the two-fluid motion........................................................287
6.1.4 Cascaded LB method for the solution of the surfactant concentration field . . 288
6.1.5 Numerical results...........................................................291

Xlll
LIST OF TABLES
2.1 Comparison between the global relative errors in the computed solutions for the
velocity field using the GI corrected preconditioned cascaded LB scheme and the uncorrected preconditioned cascaded LB scheme for the four-rolls mill flow problem at Re = 20, uo = 0.045 and a grid resolution of 251 x 251.................. 62
3.1 Relative error between the numerical results obtained using the 2D symmetrized
operator split cascaded LB source scheme for a passive scalar transport and the analytical solution for the simulation of the thermal Couette flow at various Eckert numbers Ec..............................................................102
4.1 Qualitative comparison of key flow and thermal characteristics in natural convection in a cubic cavity in the symmetry plane (z=0.5) between the 3D cascaded
LBM and the reference benchmark results obtained using a NSE solver [1]........126
5.1 Grid convergence study given in terms of the average Nusselt number Nu for Ra =
104 for natural convection in a cylindrical annulus computed using axisymmetric cascaded LB schemes.....................................................154
5.2 Comparison of the average Nusselt number Nu for different Ra for natural convection in a cylindrical annulus computed using axisymmetric cascaded LB schemes
with other reference numerical solutions and new results for Ra = 106 and 107. . . . 156
5.3 Comparison of the mean equivalent thermal conductivity at the inner cylinder in a slender vertical cylindrical annulus during mixed convection for Re = 100, Pr =
0.7, R^ = 2, rj = 10 at different values of a..................................164
5.4 Comparison of the maximum value of the stream function ipmax computed using
the axisymmetric cascaded LB schemes with reference numerical solutions for the Wheelerâ€™s benchmark problem....................................................168

XIV
LIST OF FIGURES
1.1 Two-dimensional, nine-velocity (D2Q9) lattice.................................. 5
2.1 Comparison of the computed horizontal velocity u/Up and vertical velocity v/Up
profiles along the geometric centerlines of the cavity using the Galilean invariant preconditioned cascaded central moment LBM with the benchmark results of [2] (symbols) for Re=3200 and 5000 and 7 = 0.1...................................... 49
2.2 Convergence histories of the GI preconditioned cascaded central moment LBM and
the standard cascaded LBM (7 = 1) for lid-driven cavity flow for Re=3200........ 49
2.3 Schematic representation of the flow over a square cylinder in a 2D channel..... 50
2.4 Stream function contours for flow over a square cylinder for four different Reynolds
numbers; Re=l, Re=15 and Re=30 using the GI preconditioned cascaded central moment LBM with 7 = 0.5.................................................. 51
2.5 Comparison of the computed velocity profiles along and across the square cylinder along its centerline for both the horizontal u and vertical v velocity components obtained using the GI preconditioned cascaded central moment LBM with 7 =
0.5 for Re = 100 with benchmark results obtained using the Gas Kinetic Scheme (GKS) [3]................................................................ 53
2.6 Comparison of the computed Reynolds number dependence of the recirculating
wake length Lr on the left (a) and the Reynolds number dependence of the drag coefficient Cd on the right with (b) benchmark correlation (Eq. (2.63)) and GKS-based numerical results [3] respectively................................. 54
2.7 Convergence histories of the GI preconditioned cascaded central moment LBM and
the standard cascaded LBM (7 = 1) for flow over the square cylinder for Re=30. . . 54
2.8 Schematic representation of the flow over a backward-facing step in a 2D channel. . 55

XV
2.9 Streamline contours for flow over a backward-facing step at (a) Re = fOO, (b) Re = 300, (c) Re = 500, (d) Re = 800 computed using the GI preconditioned cascaded central moment LBM with 7 = 0.3................................................ 57
2.f0 Comparison of the reattachment length as a function of the Reynolds number Re computed using the GI preconditioned cascaded central moment LBM with 7 =
0.3 (symbols) against the benchmark results of [4].............................. 57
2.11 Comparison of the computed velocity profile using the preconditioned GI cascaded central moment LBM (7 = 0.1) with the analytical solution for Hartmann flow for various Ha at Ma = 0.02. The lines indicate analytical results, and the symbols are
the solutions obtained by the GI preconditioned cascaded LBM.................... 58
2.12 Steady state streamline patterns for the four-rolls mill flow problem at uq = 0.045
and Re = 20 computed using the GI corrected preconditioned cascaded LB scheme with 251 x 251 grid nodes and 7 = 0.3................................... 60
2.13 Comparison of the computed and analytical vertical velocity profiles uy(x) at y =
7T for the four-rolls mill flow problem at Re = 20 obtained using the GI corrected preconditioned cascaded LB scheme with 251x251 grid nodes, uq = 0.45 and 7 = 0.3. 60
2.14 Distribution of the diagonal strain rate component dxux = â€”dyuy for the four-rolls
mill flow problem with uo = 0.045............................................... 61
3.1 Comparison of the computed velocity profiles using the 2D symmetrized operator split cascaded LB forcing scheme with the analytical solution for Poiseuille flow for body force magnitudes of 10-7 and 10-8. The lines indicate the analytical results,
and the symbols are the solutions obtained by our present numerical scheme......90
3.2 Grid convergence for 2D Poiseuille flow with a constant Reynolds number Re =
100 and relaxation time r = 0.55 computed using the 2D symmetrized operator cascaded LB forcing scheme.............................................. 91

XVI
3.3 Comparison of the computed velocity profiles using the 2D symmetrized operator split cascaded LB forcing scheme with the analytical solution for Hartmann flow for Hartmann numbers Ha of 3 and fO. The lines indicate the analytical results,
and the symbols are the solutions obtained by our present numerical scheme....93
3.4 Comparison of computed and analytical velocity profiles at different instants within a time period of pulsatile flow at two different Womersley numbers of Wo = 4
Wo = 40.7. Here, lines represent the analytical solution and symbols refer to the numerical results obtained using the 2D symmetrized operator split cascaded LB forcing scheme................................................................ 95
3.5 Comparison of the computed velocity profiles using the 3D symmetrized opera-
tor split cascaded LB forcing scheme and the analytical solution, for flow through a square duct in presence of a body force magnitude of Fx = 40-7 for different values of y. Here, lines represent the analytical solution and symbols refer to the results obtained using the present numerical scheme........................... 97
3.6 Comparison of the computed and analytical vertical velocity profiles uy(x) at y =
7T for the four-rolls mill flow problem at Uo = 0.04, v = 0.0044 and N = 96. Here, line represents the analytical solution and the symbol refers to the numerical results obtained using the 2D symmetrized operator split cascaded LB forcing scheme. 99
3.7 Streamlines (a) computed using the 2D symmetrized operator split cascaded LB forcing scheme and (b) obtained using the analytical solution for the four-rolls mill
flow problem at uo = 0.04, v = 0.0044 and N = 96.............................. 99
3.8 Grid convergence for the four-rolls mill flow problem at uo = 0.04, v = 0.0044 computed using the 2D symmetrized operator split cascaded LB forcing scheme
under the convective scaling..................................................400

xvii
3.9 Comparison between numerical results of the temperature profile computed using the 2D symmetrized operator split cascaded LB source scheme for a passive scalar transport and the analytical solution for the thermal Couette flow for various values of the Eckert number Ec. Here, lines represent the analytical solution and symbols refer to the results obtained using the present numerical scheme..............102
4.1 Geometric configuration for the physical model of the 3D cubic cavity and the coordinate system, model of 3D cavity with coordinate system........................121
4.2 Comparison of the temperature (top) and velocity profiles (bottom) for Rayleigh
number Ra = 105 on the symmetry center plane x â€” z; symbols â€ o â€ denote the reference benchmark solutions [1], and line â€ â€” â€ by present work............123
4.3 Projections of streamlines in natural convection in a 3D cavity computed using 3D
cascaded LBM on different center planes at Rayleigh number Ra = 104 (left) and Ra = 105 (right). Top row: y â€” z plane, Middle row x â€” z plane Bottom raw: y â€” x plane.................................................................................124
4.4 Temperature distribution in natural convection in a 3D cavity computed using 3D
cascaded LBM on different center planes at Rayleigh numbers Ra = 104 (left) and Ra = 105 (right). Top row: y â€” z plane, Middle row x â€” z plane Bottom row: y â€” x plane.................................................................................125
5.1 Comparison between the analytical velocity profile (solid lines) and the cascaded LB solution (symbols) for the Taylor-Couette flow between two circular cylinders
at an angular velocity ratio k = 0.1 and for various values of the radius ratio /?. . . 149
5.2 Evaluation of order of accuracy for Taylor-Couette flow with a constant Reynolds
number Re = 5, radius ratio fi = 1/3 and relaxation time r = 0.6 at different grid resolutions computed using the axisymmetric cascaded LB scheme.......................150
5.3 Schematic illustration of the geometry and boundary conditions for natural convection in a vertical annulus..............................................................151

XV111
5.4 Streamlines and isotherms for the natural convection between two co-axial vertical cylinders at Pr = 0.7 and (a,d) Ra = 103, (b,e) Ra = 104 and (c,f) Ra = 105 computed using cascaded LB schemes. Top row presents streamlines and the bottom
row the isotherms...............................................................153
5.5 Streamlines and isotherms for the natural convection between two co-axial verti-
cal cylinders at Pr = 0.7 and (a,c) Ra = 106, (b,d) Ra = 107 computed using cascaded LB schemes. Top row presents streamlines and the bottom row the isotherms. Grid resolution used is 300 x 300............................ 155
5.6 Schematic of swirling flow in a confined cylinder driven by a rotating top lid.157
5.7 Computed streamline patterns in the meridian plane due to swirling flow in a con-
fined cylinder driven by a rotating lid at various aspect ratios and Reynolds numbers using the axisymmetric cascaded LB sachems: (a) Ra = 1.5 and Re = 990,
(b) Ra = 1-5 and Re = 1290 (c)Ra = 2.5 and Re = 1010 and (d)RA = 2.5 and
Re = 2200....................................................................... 159
5.8 Dimensionless axial velocity profile uz/u0 as a function of the dimensional axial distance z/H for (a) Ra = 1.5 and Re = 990, (b) Ra = 1.5 and Re = 1290
(c) Ra = 2.5 and Re = 1010 and (d)R,A = 2.5 and Re = 2200: Comparison between axisymmetric cascaded LB scheme predictions and NS-based solver results ([5]) . . . 160
5.9 Schematic of the arrangement for mixed convection in a slender cylindrical annulus
with inner lateral wall rotation................................................161
5.10 Contours of (a) azimuthal velocity, (b) temperature, (c) vorticity, and (d) streamlines for mixed convection in a slender cylindrical annulus for three different values
of a computed using the axisymmetric cascaded LB schemes........................163
5.11 Geometric arrangement of melt flow and convection during Czochralski crustal
growth in a rotating crucibleâ€”Wheelerâ€™s benchmark problem.......................165

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5.12 Streamlines (upper row) and isotherms (bottom row) corresponding to two cases of the Wheelerâ€™s benchmark problem of melt flow and convection during Czochralshi crystal growth: Rex = 100, Rec = â€”25 (left) and Rex = 1000, Rec = â€”250 (right). . 167
5.13 Comparison of the maximum Reynolds number for numerical stability of single
relaxation time (SRT) and cascaded LB methods for simulation of the shear driven swirling flow in a confined cylinder at different grid resolutions...............170
6.1 Comparison of the computed profiles of the vorticity field and the analytical solution in a Poiseuille flow for different values of the centerline velocity Umax = 0.01,0.03,0.05, and 0.08 obtained by varying the fluid viscosity at a fixed body force Fx = 3 x 10-6. Here, the lines represent the analytical solution and symbols
refer to the numerical results obtained by the present DDF MRT-LB scheme.........191
6.2 Comparison of the spatial distribution of the computed vorticity field with the an-
alytical solution in a four-rolls mill flow within a square domain of size 2tt x 2tt for Re = 40. The surface plot on the left corresponds to the numerical results obtained by the present DDF MRT-LB scheme and that on the right is based on the analytical solution..............................................................193
6.3 Comparison of computed profiles of the vorticity field and the analytical solution in a four-rolls mill flow along various horizontal sections at y = 0, 7t/4, 7t/2, 7t, 57t/4.
Here, the lines represent the analytical solution and symbols refer to the numerical results obtained by the present DDF MRT-LB scheme................................193
6.4 Evaluation of the order of accuracy of the present DDF MRT-LB scheme for vor-
ticity computation in the four-rolls mill flow problem with a constant kinematic viscosity v = 0.00218 at different grid resolutions..............................195

XX
6.5 Comparison of computed profiles of the vorticity field and the analytical solution in a pulsatile flow in a channel (i.e., Womersley flow) at different instants within
a time period for two different Womersley numbers of Wo = 4.0 and Wo = 7.0.
Here, lines represent the analytical solution and the symbols refer to the numerical results obtained using the present DDF MRT-LB scheme................................496
7.4 Snapshots of the interface under an imposed shear flow with an initially circular
shape computed by the cascaded LB method.........................................222
7.2 Surface contours of the pressure distribution of a single static drop of radius R =
30 at different density ratios pa/pb with surface tension u=lx 40-3 in a periodic square domain.............................................................223
7.3 Comparison of the computed pressure differences (symbols) obtained using the cascaded LB method against the analytical predictions using the Laplace-Young relation for various values of the drop curvature 1/R with surface tension a =
5 x fO-3, f x fO-3, f x f0-4.....................................................224
7.4 Snapshots of simulation of Rayleigh-Taylor instability at At = 0.5 and (a) Re =
256 and (b) Re = 3000............................................................ 226
7.5 Time evolution of the positions of the bubble front and the spike tip for Rayleigh-
Taylor instability at At = 0.5 and (a) Re = 256 and (b) Re = 3000................ 227
7.6 Evolution of a deforming drop falling under gravity for various values of the Ohne-
sorge number Oh of 0.3, 0.7 and f.O at a fixed Eotvos number Eo = 43 shown at time instants T = 0, 2.04, 3.05, 4.07, 5.09, 6.44, 7.43, 8.44, and 9.46 (from top to bottom)...................................................................229
7.7 Evolution of the interface of a buoyancy-driven rising bubble at Re = 35 and (a)
Eo = 40, (b) Eo = 50, (c) Eo = 425............................................... 232
7.8 Time history of the non-dimensional center of mass of a buoyancy-driven rising
bubble at Re = 35 and Eo = 425
233

XXI
7.9 Evolution of the splashing of a drop on a thin film at We = 8000 and pa/Pb =
fOOO for (a) Re = 20 (b)Re = fOO...............................................235
7.10 Evolution of the interface of an oscillating liquid cylinder starting from an initial elliptic shape configuration with semi-major axis a = 25 and semi-minor axis b =
15; surface tension parameter k = 0.1, kinematic viscosity va = Rb = 0.01 and density ratio pa/pb = 100........................................... 236
7.11 Comparison of the ratios of the minimum achievable viscosities for single relaxation time (SRT) and cascaded LB formulations allowing numerically stable simu-
lations of an oscillating liquid cylinder with surface tension parameter k = 0.01 at different density ratios.............................................................237
6.1 Comparison of computed drop migration velocity under imposed constant surfac-
tant concentration gradient in the simulation of Youngâ€™s problem (solid lines) with the analytical solution for the terminal velocity (dashed lines) for surface tension sensitivities 6.2 Snapshots of the evolution of a migrating drop under imposed constant surfactant
concentration gradient in the simulation of Youngâ€™s problem for surface tension sensitivities 6.3 Comparison of computed (symbols) and analytical (liner) equilibrium profiles for surfactant concentration tp for a planar interface. Simulations are performed for the imposed surfactant concentrations in the bulk fluids ipb = 0.0001, ipb = 0.0002
and tpb = 0.0003.................................................................. 296

1
CHAPTER I INTRODUCTION
1.1 Background and Motivation
Complex fluid motion arises widely in a number of engineering applications, geophysical contexts and biochemical systems. Of particular interests are those that involve thermal convective flows in configurations involving heat transfer and/or interfacial flows in multiphase or multi-fluid systems. Often the presence of nonlinearity and multiple scales inherent in such multiphysics situations make it quite challenging to investigate such systems. Simulations based on computational methods enable fundamental studies of the fluid mechanics and as predictive tools for engineering design of configurations that handle fluid flows. Thus, the development of accurate, efficient and robust numerical methods plays a pivotal role in computational fluid dynamics (CFD). Classical numerical techniques such as the finite difference, finite volume and finite element methods have traditionally been used to perform discretization of the partial differential equations such as the Navier-Stokes (NS) equations that represent the fluid motion including any attendant multiphysics effects. From a different perspective, more recently, the lattice Boltzmann (LB) method has demonstrated to be a very effective alternate numerical technique to simulate a variety of complex fluid flow systems [6, 7, 8].
The lattice Boltzmann method is a mesoscopic method based on local conservation and discrete symmetry principles, and may be derived as a special discretization of the Boltzmann equation [9]. Hence, it can be regarded as a kinetic scheme. Algorithmically, it involves the streaming of the particle distribution functions as a perfect shift advection step along the lattice directions and followed by a local collision step as a relaxation process towards an equilibria, and accompanied by special strategies for the implementation of impressed forces. The hydrodynamic fields char-

2
acterizing the fluid motion are then obtained via the various kinetic moments of the evolving distribution functions and its consistency to the NS equations may be established by a Chapman-Enskog expansion or Taylor series expansions under appropriate scaling between the discrete space step and time step. As such, the LB method has been applied for the computation of a wide range of complex flows including turbulence, multiphase and multicomponent flows, particulate flows and microflows (see e.g., [fO, ff, 12]).
The LB method has the following advantages. Its streaming step is linear and exact and all nonlinearity is modeled locally in the collision step; by contrast, the convective term in the NS equation is nonlinear and nonlocal. The pressure field is obtained locally LB methods, circumventing the need for the solution of the time consuming elliptic Poisson equation as in traditional methods. The exact-advection in the streaming step combined with the collision step based on a relaxation model leads to a second order accurate method with relatively low numerical dissipation. The kinetic model for the collision step can be tailored to introduce additional physics as necessary and its additional degree of freedom can be tuned to improve numerical stability. Various boundary conditions for complex geometries can be represented using relatively simple rules for the particle populations. Finally, the locality of the method makes it amenable for almost ideal implementation on parallel computers for large scale flow simulations.
During the last decade or so, the LB method has undergone a number of refinements, especially with the development of more sophisticated collision models as well as in adopting advanced strategies from CFD to further improve its numerical stability, accuracy and efficiency. As such, the collision step, which represents various physics associated with the fluid motion including the momentum diffusion as a relaxation process, plays a main role in the numerical stability of the method. Among the earliest collision models is the single relaxation time (SRT) model [13], which, while being popular due to its simplicity, is susceptible to numerical instability at relatively high Reynolds numbers. A significant improvement is achieved by the multiple relaxation time model (MRT) [14] in which different raw moments relax at different rates. More re-

3
cently, further enhancement in stability was made possible by the introduction of a cascaded LB method, which is a multi-parametric scheme that is based on considering relaxation in terms of central moments, which are formulated by shifting the particle velocity by the local fluid velocity [15]. The significant advantages of such more advanced collision models were numerically demonstrated more recently (see e.g., [16, 17]).
Central moments based cascaded LB method has so far mainly been developed and applied for athermal flows (i.e., without heat transfer effects) and single-phase flows. In particular, development of new cascaded LB schemes for thermal convective flows in different spatial dimensions and two-phase flows with interfacial dynamics would significantly improve the state-of-the-art in the LB formulations and expand their scope for various complex fluid dynamics applications. In addition, strategies for convergence acceleration and for improvements in the accuracy of the cascaded LB method, and the construction of novel simpler and efficient approaches to handle body forces/sources for such methods would make further contributions of significant interest to the field. These are among the various topics of our research endeavors presented in this dissertation. Before we outline the specific original research goals and contributions of this dissertation later in this chapter, in what follows, we will first present some details of the LB method, including the cascaded formulation, which would serve as a basis for further improvements.
1.2 Single-Relaxation-Time (SRT) Lattice Boltzmann Method
First, we will briefly discuss the simplest version of the LB method, i.e., the single-relaxation time (SRT) approach, which will be used as a basis for comparison. A typical lattice in two-dimensions (2D) is the two-dimensional, nine velocity (D2Q9) lattice, which is illustrated in Fig. 1.1,

4
and components of the particle velocities are given by The particle velocity ea may be written as
/
(0,0) a = 0
(Â±1,0),(0,Â±1) a = 1,- â–  â€¢ ,4 (f.f)
(Â±1,Â±1) a = 5, â€¢ â€¢ â€¢ ,8
The lattice represents the discrete characteristic particle directions ea, where a = 0,1, 2, â€¢ â€¢ â€¢ ,8 along which the discrete particle distribution function fa advects (i.e., streams) and then undergo a scattering process (i.e., collision). According to the SRT LB method, the distribution function relaxes to its corresponding equilibrium distribution function ffq at a single relaxation time r. Thus, the SRT LB method can be written as
fa(x + ea,t+ 1) = fa{x, t) -- [fa{x, t) ~ /Â«9(X, t)} , (f .2)
T
where the equilibrium distribution function ffq is related to the local fluid density p and u velocity fields, and is given by
fa = wap
1 +
u
+
(ea â€¢ uf
2 ci
u u
2c2e
(f.3)
Here, wa are the weighting factors for different particle directions and given by wa are given by Wo = 4/9, wa = f/9, where a = f, 2, 3, 4 and wa = 1 /36, where a = 5, 6, 7, 8 and P = ^2fa, pu = ^2faea, p = c2sp. (4.4)
a a
This method simulates the fluid motion, with the kinematic viscosity v given by v = c2(t â€” f/2). R may be noted that the SRT LB method is prone to numerical instabilities when it is required to simulate fluid flows with relatively low viscosities, i.e., when r â€”> f/2 and it does not possess any additional degrees of freedom to enhance its numerical properties to address such issues. R may be noted that various boundary formulations to represent the no-slip, free slip, open flow

5
FIGURE 1.1: Two-dimensional, nine-velocity (D2Q9) lattice.

6
boundary conditions, among others, in terms of their effect on the kinetic variables, i.e., the distribution function have been devised. For example, the no-slip condition can be represented by means of the so-called half-way bounce back scheme and its more sophisticated versions based on interpolations for curved walls. See Ref. [18] for details.
1.3 Cascaded Lattice Boltzmann Method based on Central Moments
We will now discuss a significantly improved version of the LB scheme based on central moments, which is referred to as the cascaded LB method [15], including the forcing terms [19]. It provides a context to this dissertation research, where we will further improve its efficiency and numerical accuracy, extend its capabilities, and broaden its scope of applications, in particular, to those related to thermo-fluid dynamics and complex fluids with interfaces. Generally, the cascaded LB method represents collision process via the relaxation of various central moments to their corresponding equilibria. Here, the central moments are defined as the sum over all particle directions of the products of the distribution function weighted with the monomials of the components of the particle velocity shifted by the fluid velocity (see below). The use of central moments naturally imposes Galilean invariance of those moment components independently supported by a given lattice (but not for those unsupported higher order moments that alias with the lower order ones due to the discreteness and symmetries of the lattice). In this regard, for the purpose of illustration, again the standard D2Q9 lattice is employed. As the cascaded LBM is a moment based approach, we start with a set of the following nine nonorthogonal basis vectors obtained

7
using the monomials eâ„¢xeâ„¢y in a ascending order:
\$o = (1,1,1,1,1,1, l,!,!)1,
!=ex = (0,1, 0, -1, 0,1, -1, -1, l)f,
\$2 = ey = (0, 0,1, 0, -1,1,1, -1, -l)f, (1.5)
\$3 = e^, o e^, T ey o e^, \$4 = ex 0 0 \$5 = qx o e^,
^6 â€” ^x o ex o ey, \$7 â€” e^, o sy o e^, â€” ex o sx o sy o e^,
where f is the transpose operator. In the above, for any two ^-dimensional vectors (q = 9 here) a and b, we define the elementwise vector multiplication by a o b. That is, (a o b)a = aaba, where a represents a component of the vector and the implicit summation convention is not assumed here. To facilitate the presentation in the following, we also define a standard scalar inner product of any two such vectors as (a, b). That is, (a, b) = ^a=oaÂ«^Â«- By applying the Gram-Schmidt orthogonalization method they result into the following equivalent set of orthogonal basis vectors:
Ko = \$0, Ki = \$1, K2 = \$2,
K3 = 33>3 â€” 43>o, K4 = \$4, K5 = \$5,
Kg = â€”3\$g + 2\$2, K7 = â€”3\$7 + 2\$i, KÂ§ = 9\$8 â€” 6\$3 + 4\$o-Collecting them as an orthogonal matrix K, we get
K
[K0, K1; K2, K3, K4, K5, Kg, Kr, K8]
(1.6)
See [15, 20, 19], which enumerates the details of the matrix K. Now we define the continuous equilibrium central moments as [15]:
II;
M
xmyn
fOO /*oo
fM(ix-ux)m(ty-uy)ndtxdty.
(1.7)
' â€” CO J â€” CO
where fM is the local Maxwell-Boltzmann distribution in the continuous particle velocity space
Â£ = (Â£x,Â£y) and is given by: fM = fM(p,u,Â£) = Aexp
(t-uf
2c{
. Here, p is the density

8
and u = (ux,uy) is the macroscopic fluid velocity. By evaluating IIâ€ž in the increasing order of moments for the D2Q9 lattice we obtain
nÂ£l = p, nf=o, n^ = o, n% = c2p, n^ = c2sP, n^ = o, = o, nJJj, = o, u^yy = cAsp.
Similarly, we define the central moments of the sources of order (m + n) due to a force field F =
(Fx,Fy) as
/OO POO
/ ^fF(,Cx ~ Ux)m(Cy ~ uy)nd^xd^y, (1-8)
OO J â€” CO
where AfF is the change in the distribution function due to force fields. Again we can evaluate Eq. (1.8) as
r,
'F
fF â€” p
L X â€” 1 XI
TF â€” F
L y ~ ry>
r
F
o, r
F
yy
o, r
F
xy
0,
rF = c2f
x xxy ^s-1- Vi
rF = c2f
xyy us x x ,
r
F
xxyy
o.
Based on the above continuous central moments and using the trapezoidal rule for evaluating the source term to retain second order accuracy, the elements of the cascaded LBE can be formulated as follows [15, 19]:
fa(x + ea,t + 1) = fa(x, t) + wa{x, t) + i [Sa(x, t) + Sa(x + ea,t + 1)] (1.9)
In the above equation, the collision term can be modeled by
Q^Q^(f,g) = (K-g)Â«, (1.10)
where f = (/o, /i, /a,..., /s)^ is the vector of distribution functions and g = (<7o, <7i,#2, â€¢ â€¢ â€¢ , is the vector of the unknown change in different moments supported by the lattice under collision that is given later. The discrete form of the source term Sa in the cascaded LBE represents the influence of the force field (Fx, Fy) in the velocity space that is given as: S = (So, Si, S2, â–  â–  â– , Sg)^. As Eq. (1.9) is semi-implicit, by using the standard variable transformation by He et al. [21, 22],

9
and, in particular, by He et al. [23] /Â« = /Â« â€” ^Sa, so that the implicitness is removed and a second-order accuracy is maintained, we get
Ja(x + ea,t + 1) = 7Â«(Â®, t) + t) + Sa(x, t). (1.11)
In order to determine the structure of the cascaded collision operator and the source terms in the presence of general spatially and/or temporally variable body forces, we define the following set of central moments of order (m + n), respectively, as
f K \ ( f \
l%Xrnyn Ja
Ke (7xrnyn a Sa
\ Pxmyn / \ fa j
(eax ~ ux)m(eay ~ uy)n.
(1.12)
By equating the discrete central moments for both the distribution functions and source terms
with the corresponding continuous central moments
â€”
x"Ly,L
TTAi /t
(Tgiriyn
r
F
xmyn i
we get
-eg
-eg
-eg
0,
Ke<1
<^XX
c2sP,
--eq
^yy
c2sP,
vreg
<^xy
^eq ftxxy
o,
^eq
K>xyy
'--eq
l^xxyy
c4sp.
(1.13)
(TO â€” 0, Ox â€” Fxi &y â€” Fyi @xx â€” 0, @yy â€” 0) &xy â€” 0, &xxy â€” 0, &xyy â€” 0, 0Xxyy â€” 0.
(1.14)
Here, we apply the effect of the variable body forces only on the first order central moments of the sources to obtain consistent macroscopic equations [19]. Similarly, for the transformed central moments we have
^eg
Â«0
â€”eg 1 ^ â€”eg Fx â€” 2^xâ€™ ^y â€” â€” 1F 2ry> ^eg o â€”e(i 2 ^eq
P, ^xx Csp, ^yy dsP> f^xy
-eq cs â€ž â„¢xxy 2 â€”eg _ ^xyy c2 p â€”eg _ 4 2 hi ^xxyy ^sP-
0,
(1.15)
On the other hand, the actual calculations in the cascaded formulation are carried out in the terms of the raw moments. Hence, we define the following set of raw moments (designated here

10
with the prime (') symbol), which will be used later:
( F \ ( f \
r\>rgm yn Ja
keq' â€™hxrn yn = E req Ja
(7 rglTlyn a Sa
^ / y K xmyn y \ fa j
m n ^ax^ay
(1.16)
We can readily convert the central moments into a combination of the raw moments by using the binomial theorem.
Thus, we first summarize the following set of raw moments for the source terms for the D2Q9 lattice:
cl 2 Fxux -c' â€˜2Fyuy cl Fxuy+Fyux
U XX -y2 1 uyy y2 1 u xy y2 1
Qxxy â€” Eg//,. I â€˜^â€˜FxUxUy, ^xyy â€” FxUy I 2FyUyUx,
O'xxyy â€” 2 I f // ,â–  Il y | 2 f 'yllyll r.
Next, as an intermediate step, the source moments projected to the orthogonal moment space nip = (Kg,S) are obtained as
mf =0, m\ = Fx, mf = Fy, ms3 = 6F-u, m% = 2(Fxux - FyUy), ms5 = (Fxuy + Fyux), fhs6 = (1- 3ul) Fy - 6Fxuxuy, mf = (l - 3u^) Fx - 6Fyuyux,
= 3 [(6u2y - 2) Fxux + (6u2x - 2) .
Equivalently, the above equations can be written in a matrix form as K S = (mg, mf, mf,..., m|)t. Now, inverting the above equation, we can obtain explicit expressions for Sa in terms of the gen-

11
eral variable body force F and fluid velocity u in the particle velocity space as
50
51
52
53
54
55
s6
S7
Ss
^ [â€”m3 +
h |6"*! k |6"â€˜2
â€” [â€”6m 36 L
â€” [â€”6m 36 1
k |6"â€˜!
â€” [â€”6m 36 L
â€” [â€”6m 36 L
^ |6"â€˜!
(1.17a)
- 2mg], (1.17b)
- 2mg], (1.17c)
â–  - 2fhi\, (1.17d)
~ 2mg], (1.17e)
â€” 3m,g â€” 3m-7 + m|] , (1-17f)
1,5 â€” 3m,g + 3m-7 + m|], (l-17g)
1,5 + 3m,g + 3m-7 + m|], (1.17h)
+ 3m,g â€” 3m-7 + m|] . (1-17i)
Then, to determine the structure of the cascaded collision operator in the presence of forcing terms we start from the lowest-order nonconservative (i.e., second order) moments, and perform relaxation of various central moments to their corresponding equilibria, each carried out at different relaxation times. See Refs. [15, 19] for details. Thus, the change of different moments under

12
cascaded collision can be summarized as
90
h
91
95
96
97 9s
9i=h = 0,
J 2 p(u2x + u2y) fj ^ 1 / /
12 ) 3^ ' /y \^xx ' ftyy) 2 ^xx ' Â®yy)
UJ4
T
P(UI - U2y) ^ ^ l ,
\KXX 17yy) \'~7XX (7yy)
7
W5 ( pUxUy cJ 1
4 l 7
^xy 2 *^Â£2/ f â€™
W6
S 2pUxUy + HXXy 2UxKXy UyKxx n(Txxyf 0uy('>)93 + 74) ^Uxg5,
U)7 I 2â€” _ â€” -
,, 4 â€˜2â€˜pUXUy KXyy â€˜I'UyKj.y U X K yy
- 2^(3^3 -74) - 2uyg5,
= ~y\\p + Â°OpU2xU2y â€”
9'
+ 4 UXUyKxy
r^/
xxyy 2"Ux!7Xyy â€˜^â€˜â€˜^y^xxy 4~ ^x:^yy 4~ ^y^xx
1_,
â€” â€œO',
2" xxyy r 293 2 Uy(3g3 + <74) 2^x^93 9i)
- iuxuyg5 - 2uyg6 - 2uxg7.
(1.18a)
(1.18b)
(1.18c)
(1.18d)
(1.18e)
(1.18f)
(1.18g)
For ease of implementation, the cascaded lattice Boltzmann equation with forcing term can then be presented in terms of following collision and streaming steps, respectively:
laix, t) = 7a(*> t) + + Sa(x, t) (1.19)
7a(Â® + ea,i+l) =7a(Â®,i). (1-20)
where, is obtained using Eq. (7.41) and Eqs. (1.18a)-(1.18g). Finally, the explicit form of the

13
post-collision distribution function in the velocity space can be written as follows:
70 = 7o + [go - 4G7 - <78)] + s0,
71 = 7i + [5o+5i-53+54 + 2(<77-<78)]+'S'i,
= I2 + [go +92 -93 -9a + 2(^6 -gs)] + S2,
I3 = 73 + [go-gi-93+94-2(g7 + g8)}+ s3,
li = li + [go-g2-93-94-^{g3 + g3)]+sA,
75 = 75 + [go + gi+ 92 + 2g3 + 95-96-97 + gÂ»\ + s5,
7o = 76 + [go-gi+g2 + 2g3-g5-ge+g7+g8] + S6,
77 = 77 + [<7o -91-92 + +93 +95+96 +97 + gs] + S7,
7s = 7s + [go +91-92 + 293-95+96-g7 + 93] + s8.
The hydrodynamic variables are then obtained as
P = J2 7Â«> Pu = Yj7Â«ea + \Fâ€™ 'P = Â°2sPâ€¢ (L21)
a a
It can be shown via the standard multiscale Chapman-Enskog expansion that the solution of the above cascaded LB method represents the weakly compressible Navier-Stokes equations (NSE) (see e.g., [19])
dtp + V â€¢ (pu) = 0, (1.22a)
dt (pu) + V â€¢ (puu) = â€” Vp + V â€¢ [pv (2S â€” IV â€¢ u) + pÂ£l V â€¢ u] + F, (1.22b)
where S = |(Vm + (VÂ«)t) and I are the strain rate tensor and identity tensor, respectively, and F = (Fx,Fy). The transport coefficients of the fluid motion, such as the kinematic bulk viscosity ( and kinematic shear viscosity v are related to the relaxation times of the second order moments via
C = (1/3) (l/w3 - 1/2), v = (1/3) (1/wg â€” 1/2) , /? = 4, 5
and the relaxation times for higher order moments uip, where /3 = 6, 7, 8 can be independently adjusted to improve numerical stability. See Ref. [17] for a comparison of the cascaded LB method

14
against LB methods based on other collision models for standard benchmark problems involving the solution of the 2D NSE.
1.4 Research Goals and Original Contributions of This Dissertation
The research detailed in this dissertation presents several new innovations that advances the state-of-the-art in the lattice Boltzmann method based on central moments and multiple relaxation times for flow simulations with attendant additional multiphysics effects. The overall thrust is to improve the efficiency, i.e., the convergence accceleration, and accuracy of the cascaded LB method for computation of steady state flows, to expand its ability to handle force/source terms via a time-splitting approach, and to develop new cascaded LB schemes for simulation of thermal convective flows in multiple dimensions as well as for two-phase flows in conjunction with the capturing of interfaces. In addition, for the first time, we propose a new strategy in the LB method that enable local vorticity computation without requiring the finite differencing for the spatial derivatives in simulations of flows with a passive scalar transport. A central theme in our research is the use of two (or more) distribution functions that evolve using unified advanced LB formulations based on multiple relaxation times and central moments for various applications that involve the computation of fluid motion in conjunction with the transport of a scalar or capturing of interfaces, such as in heat transfer and multiphase flow problems, respectively. The specific goals of this dissertation are to develop, analyze, and perform numerical studies of the following:
â€¢ Preconditioned cascaded LB scheme for convergence acceleration with improved Galilean invariance
â€¢ Symmetrized operator splitting formulations force/source implementations in cascaded LB methods for flow and scalar transport

15
â€¢ Three-dimensional cascaded LB method for thermal convective flows
â€¢ Axisymmetric central moment LB schemes for flows with heat transfer including swirling effects
â€¢ Local vorticity computation approach using double distribution functions based LB techniques for flow and scalar transport
â€¢ Cascaded LB methods for two-phase flows with interface capturing based on a phase-field model including surfactant effects
In the following, we will briefly expand on the details of the above research goals and expected outcomes.
1.4.1 Preconditioned cascaded LB scheme for convergence acceleration with improved Galilean invariance
Lattice Boltzmann models used for the computation of fluid flows represented by the Navier-Stokes (NS) equations on standard lattices can lead to non Galilean invariant (GI) viscous stress involving cubic velocity errors. This arises from the dependence of their third order diagonal moments on the first order moments for standard lattices, i.e., the aliasing effects due to the discreteness and symmetry of the lattice. Strategies have recently been introduced to restore GI without such errors using a modified collision operator involving either corrections to the relaxation times or to the moment equilibria. Convergence acceleration in the simulation of steady flows can be achieved by solving the preconditioned NS equations, which contain a preconditioning parameter that can be used to tune the effective sound speed, and thereby alleviating the numerical stiffness.
In this research contribution, we will present a GI formulation of the preconditioned cascaded

16
central moment LB method used to solve the preconditioned NS equations, which is free of cubic velocity errors on a standard lattice, for steady flows. A Chapman-Enskog analysis is used to reveal the structure of the spurious non-GI defect terms and it is demonstrated that the anisotropy of the resulting viscous stress is dependent on the preconditioning parameter, in addition to the fluid velocity. It is shown that partial correction to eliminate the cubic velocity defects is achieved by scaling the cubic velocity terms in the off-diagonal third-order moment equilibria with the square of the preconditioning parameter. Furthermore, we develop additional corrections based on the extended moment equilibria involving gradient terms with coefficients dependent locally on the fluid velocity and the preconditioning parameter. Such parameter dependent corrections eliminate the remaining truncation errors arising from the degeneracy of the diagonal third-order moments and fully restores GI without cubic defects for the preconditioned LB scheme on a standard lattice.
Several conclusions will be drawn from the analysis of the structure of the non-GI errors and the associated corrections, with particular emphasis on their dependence on the preconditioning parameter. The new GI preconditioned central moment LB method will be validated for a number of complex flow benchmark problems and its effectiveness to achieve convergence acceleration and improvement in accuracy will be demonstrated.
1.4.2 Symmetrized operator splitting formulations force/source implementations in cascaded LB methods for flow and scalar transport
Fluid motion are generally driven/influenced by local body forces, and similarly the transport of a scalar field, such as temperature, by the local (heat) sources. We will present operator split forcing schemes exploiting a symmetrization principle, i.e. Strang splitting, for cascaded LB methods in two- and three-dimensions for fluid flows with impressed local forces. We will also derive analogous scheme for the passive scalar transport represented by a convection-diffusion equa-

17
tion with a source term in a novel cascaded LB formulation. They are based on symmetric applications of the split solutions of the changes on the scalar field/fluid momentum due to the sources/forces over half time steps before and after the collision step. The latter step is effectively represented in terms of the post-collision change of moments at zeroth and first orders, respectively, to represent the effect of the sources on the scalar transport and forces on the fluid flow.
Such symmetrized operator split cascaded LB schemes are consistent with the second-order Strang splitting and naturally avoid any discrete effects due to forces/sources by appropriately projecting their effects for higher order moments. All the force/source implementation steps are performed only in the moment space and they do not require formulations as extra terms and their additional transformations to the velocity space. These result in particularly simpler and efficient schemes to incorporate forces/sources in the cascaded LB methods unlike those considered previously. We will demonstrate the validity and accuracy, as well as the second-order convergence rate of the symmetrized operator split forcing/source schemes for the cascaded LB methods based on a numerical study of various benchmark problems in 2D and 3D for fluid flow problems with body forces and scalar transport with sources.
1.4.3 Three-dimensional cascaded LB method for thermal convective flows
Fluid motion driven by thermal effects, such as that due to buoyancy in differentially heated three-dimensional (3D) enclosures, arise in several natural settings and engineering applications.
It is represented by the solutions of the Navier-Stokes equations (NSE) in conjunction with the thermal energy transport equation represented as a convection-diffusion equation (CDE) for the temperature field. In this research, we will develop new 3D lattice Boltzmann (LB) methods based on central moments and using multiple relaxation times for the three-dimensional, fifteen velocity (D3Q15) lattice, as well as its subset, i.e., the three-dimensional, seven velocity (D3Q7)

18
lattice to solve the 3D CDE for the temperature field in a double distribution function framework. Their collision operators lead to a cascaded structure involving higher order terms. In this approach, the fluid motion is solved by another 3D cascaded LB model from prior work. Owing to the differences in the number of collision invariants to represent the dynamics of flow and the transport of the temperature field, the structure of the collision operator for the 3D cascaded LB formulation for the CDE will be shown to be markedly different from that for the NSE. We will validate the new 3D cascaded LB schemes for thermal convective flows for natural convection of air driven thermally on two vertically opposite faces in a cubic cavity enclosure at different Rayleigh numbers against prior numerical benchmark solutions.
1.4.4 Axisymmetric central moment LB schemes for flows with heat transfer including swirling effects
Fluid motion in cylindrical coordinates with axial symmetry that is driven by rotational effects and/or thermal buoyancy effects arise widely in a number of technological applications and geophysical contexts. Computational effort for such problems can be significantly reduced if axial symmetry can be exploited; in such cases the system of equations can be reduced to set of quasi-two-dimensional (2D) problems in the meridian plane, where the simulations can be performed for broader ranges of the parameter spaces more efficiently. We will present a cascaded LB approach to simulate thermal convective flows, which are driven by buoyancy forces and/or swirling effects, in a cylindrical coordinate system with axial symmetry. In this regard, the dynamics of the axial and radial momentum components along with the pressure are represented by means of the 2D Navier-Stokes equations with geometric mass and momentum source terms in the pseudo Cartesian form, while the evolutions of the azimuthal momentum and the temperature field are each modeled by an advection-diffusion type equation with appropriate local source terms.
Based on these, cascaded LB schemes involving three distribution functions will be formulated

19
to solve for the fluid motion in the meridian plane using a D2Q9 lattice, and to solve for the azimuthal momentum and the temperature field each using a D2Q5 lattice. The geometric mass and momentum source terms for the flow fields and the energy source term for the temperature field are included using a new symmetric operator splitting technique, via pre-collision and postcollision source steps around the cascaded collision step for each distribution function. These will result in a particularly simple and compact formulation to directly represent the effect of various geometric source terms consistently in terms of changes in the appropriate zeroth and first order moments. We wifi validate this new axisymmetric cascaded LB approach by means of simulations of several complex buoyancy-driven thermal flows and including rotational effects in cylindrical geometries and comparisons of the computed results against prior benchmark numerical results.
In addition, we will demonstrate significant improvements in numerical stability with the use of the cascaded LB formulation when compared to other collision models for axisymmetric flow simulations.
1.4.5 Local vorticity computation approach using double distribution functions based LB techniques for flow and scalar transport
Computation of vorticity, or the skew-symmetric velocity gradient tensor, in conjunction with the strain rate tensor plays an important role in fluid mechanics in the classification of flows, in identifying vortical structures and in the modeling of various complex flows. For the simulation of flows accompanied by the advection-diffusion transport of a scalar field, double distribution functions (DDF) based LB methods, involving a pair of lattice Boltzmann equations (LBEs) are commonly used. We will present a new local vorticity computation approach, i.e., without involving finite differences, by introducing an intensional anisotropy of the scalar flux in the third order, off-diagonal moment equilibria of the LBE for the scalar field, and then combining the second order non-equilibrium components of both the LBEs. As such, any pair of lattice sets in the DDF formulation that can independently support the third order off-diagonal moments would en-

20
able local determination of the complete flow kinematics, with the LBEs for the fluid motion and the transport of the passive scalar providing the necessary moment relationships to respectively determine the symmetric and skew-symmetric components of the velocity gradient tensor.
As an illustration of our approach, we will formulate a DDF-LB model for local vorticity computation using a pair of multiple relaxation times (MRT) based collision approaches on D2Q9 lattices, where the necessary moment relationships to determine the velocity gradient tensor and the vorticity will be established via a Chapman-Enskog analysis. We will present a numerical study that validates the predicted vorticity fields against the analytical solutions with good accuracy and second order convergence.
1.4.6 Cascaded LB methods for two-phase flows with interface capturing based on a phase-held model including surfactant effects
Simulation of multiphase flows, which are ubiquitous in nature and engineering applications, require coupled capturing or tracking of the interfaces in conjunction with the fluid motion often occurring at multiple scales. In this contribution, we will present unified cascaded LB methods for the solution of the incompressible two-phase flows at high density ratios and for the capturing of the interfacial dynamics. Based on a modified continuous Boltzmann equation (MCBE) for two-phase flows, where a kinetic transformation to the distribution function involving the pressure field is introduced to reduce the associated numerical stiffness at high density gradients, a central moment cascaded LB formulation for fluid motion will be constructed. In this LB scheme, the collision step is prescribed by the relaxation of various central moments to their equilibria that are reformulated in terms of the pressure field obtained via matching to the continuous equilibria based on the transformed Maxwell distribution. Furthermore, the differential treatments for the effects of the source term representing the change due to the pressure field and of the source term due to the interfacial tension force and body forces appearing in the MCBE on dif-

21
ferent moments are consistently accounted for in this cascaded LB solver that computes the pressure and velocity fields. In addition, another cascaded LB scheme will be developed to solve for the interfacial dynamics represented by a phase field model based on the conservative Allen-Cahn equation that evolves interfaces by advection and under the competing effects due to a diffusion term and an interfacial sharpening or phase segregation flux term. The latter is introduced into the cascaded LB scheme via a modification to the moment equilibria.
The use of central moments natural maintains the Galilean invariance of the all moments independently supported by the chosen lattice and improves the numerical stability of the resulting unified cascaded LB formulation for two-phase flows. Based on numerical simulations of a variety of two-phase flow benchmark problems, we will validate the new approach and demonstrate improvements in numerical stability. Furthermore, for surfactant-laden interfacial flows, where the presence of surfactant can be used to tune the interfacial dynamics, another cascaded LB scheme that solves its concentration field based on a phase-field model will be constructed. The resulting approach will be validated for self-propulsion of a drop under an imposed linear gradient of surfactant distribution and equilibrium profile of surfactant concentration under adsorption effects.
1.5 Organization of This Dissertation
Based on the above introduction and research goals, this dissertation is organized into a number of chapters, each addressing a particular area of investigation. Thus, each chapter will include a literature review, mathematical formulations and numerical scheme associated with the specific topic being addressed, which will then be followed by results and discussion, and a summary of the main findings. As required, additional details supporting the discussion are given in various appendices. In Chapter 2, we present a new preconditioned formulation of the cascaded LB method for efficient simulations, which is further improved for accuracy by a modification to preserve its Galilean invariance without cubic velocity errors. A simpler strategy based

22
on symmetrized operator splitting for including local body forces in fluid motion and sources in passive scalar transport using the cascaded LB method is elaborated in Chapter 3. Central moments based cascaded LB schemes for thermal convective flows in three-dimensions and axisym-metric coordinates are discussed in Chapters 4 and 5, respectively. Chapter 6 discusses a new LB method for local vorticity computation via double distribution functions based approach for flow and scalar transport. Then, new cascaded LB schemes for two-phase flows, based on a phase field model are presented in Chapter 7. The modeling of attendant surfactant effects in two-phase flows using a cascaded LB approach is discussed in Appendix F. Finally, Chapter 8 summarizes this dissertation research, its main conclusions and outlook for future research investigations in this area.

23
CHAPTER II
GALILEAN INVARIANT PRECONDITIONED CENTRAL MOMENT LATTICE BOLTZMANN METHOD WITHOUT CUBIC VELOCITY ERRORS FOR EFFICIENT STEADY FLOW SIMULATIONS
2.1 Introduction
The lattice Boltzmann (LB) method has now been established as a powerful kinetic scheme based computational fluid dynamics approach ([24], [7], [18]). It is a mesoscopic method based on local conservation and discrete symmetry principles, and may be derived as a special discretization of the Boltzmann equation. During the last decade, many efforts were made to further improve its numerical stability, accuracy and efficiency. In particular, sophisticated collision models based on multiple relaxation times and involving raw moments, central moments or cumulants, and entropic formulations have significantly expanded the capabilities of the LB method. The significant achievements of these developments and their applications to a variety of complex flow problems have been discussed, for example, in [11, 12, 25, 26, 27, 28, 29, 17, 30, 31, 32, 33].
There exist various additional aspects in the LB approach that require further attention and present scope for improvements. In particular, the finiteness of the lattice can introduce certain truncation errors that manifest as non-Galilean invariant viscous stress, i.e. fluid velocity dependent viscosity. This lack of Galilean invariance (GI) arises due to the fact that the diagonal terms in the third-order moments are not independently supported by the standard tensor-product lattices (i.e. D2Q9 and D3Q27). More specifically, for example,
eocx fa â€” C-axfa â€”
Here, and in the following, the primed quantities denote raw moments. In other words, there is a degeneracy of the third-order diagonal (longitudinal) moments that results in a deviation

24
between the emergent macroscopic equations derived by the Chapman-Enskog expansion and the Navier-Stokes (NS) equations. Such cubic-velocity truncation errors are grid independent and persist in finer grids especially under high shear and flow velocity. Moreover, such emergent anisotropic viscous stress may have a negative impact on numerical stability as a result of a negative dependence of the emergent viscosity on the fluid velocity. In order to overcome this shortcoming, various attempts have been made.
One possibility is to consider a lattice with a larger particle velocity set, such as the D2QI7 lattice in two-dimensions [34], which was pursued after [35] pointed out nonlinear, cubic-velocity deviations of the emergent equations of the LB models with standard lattice sets from the NS equations. This involved the use of higher order velocity terms in the equilibrium distribution. However, [36] showed that the specific equilibria adopted in [34] does not fully eliminate the cubic-velocity errors. Moreover, the use of non-standard lattice stencils with larger number of particle velocities increases the computational cost and propensity of the numerical instability at grid scales. On the other hand, it was shown more recently by various others ([37], [36], [38]) that partial corrections to the GI errors on the standard lattice (i.e. D2Q9 lattice) may be achieved by adopting special forms of the off-diagonal, third-order moments in the equilibria. That is,
K.xxy â€” CgpUy + fV/ly. ^xyy â€” gs fV/ ,â–  T p^x^y
Here, cs is the speed of sound and the particular choices of the cubic-velocity terms that are underlined are crucial to partially restore GI for the above identified moments. Here, we also point out that the above forms of the off-diagonal, third-order raw moment equilibria that allow such partial GI corrections naturally arise in the central moment LB formulations, when the equilibrium central moment components are set to zero and and then rewritten in terms of their corresponding raw moments. However, since K^qxx = klx and = Ky1 due to the degeneracy of the third-order longitudinal moments, which is inherent to the standard tensor-product lattices, additional corrections are required to restore GI completely free of cubic-velocity errors. In this regard, in order to compensate the terms which violate GI on standard lattices, LB schemes with

25
single relaxation time models were augmented with finite difference expressions [39, 40, 41]. On the other hand, more recently, [42] introduced small intentional anisotropies into a matrix collision operator that corrects the anisotropy in the resulting viscous stress tensor thereby addressing the above mentioned issue. In addition, independently, [43] introduced additional corrections involving velocity gradients to the equilibria that achieved equivalent results. These two studies provided strategies to represent the Navier-Stokes equations in LB models on standard lattices completely free of cubic-velocity errors. In addition, [44] presented finite difference based corrections to the method proposed in [45] to reduce the resulting spurious velocity dependent viscosity effects on standard lattices.
While the LB schemes have found applications to a wide range of fluid flow problems, there has also been considerable interest to an important class of problems related to low Reynolds number steady state flows. They include analysis and design optimization of a variety of Stokes flows through capillaries, porous media flows, heat transfer problems under stationary conditions, and since the LB methods are explicit marching in nature, efficient solution techniques need to devised to accelerate their convergence (see e.g. [46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58,
59, 60]). A recent review of the literature in the LB approach for such problems can be found in [58, 59]. Generally, multigrid and preconditioning techniques can be devised to improve the steady state convergence of the LB scheme. A comparison of a multigrid LB formulation with the conventional solvers showed significant improvement in efficiency [55]. At low Mach numbers, the convergence can be further accelerated by means of preconditioning for both the traditional single grid LB methods [49, 52, 53, 58, 59] and multigrid LB scheme [60]. The present chapter addresses a further refinement to the LB techniques for steady state flows, viz., improving the accuracy of the acceleration strategy based on the preconditioned LB formulation without GI cubic velocity and parameter dependent errors.
Thus, it is clear that another aspect of the LB method, similar to certain schemes based on the classical CFD, is its slow convergence to steady state at low Mach numbers. In such conditions,

26
there is a relatively large disparity between the sound speed and the convection speed of the fluid motion resulting in higher eigenvalue stiffness and larger number of iterations for convergence. This stiffness can be alleviated and convergence can be significantly improved by preconditioning. Reference [49] presented a preconditioned LB method based on a single relaxation time model by modifying the equilibrium distribution function by using a preconditioning parameter. Then, [52] and [53] presented preconditioned LB formulations based on multiple relaxation times. More recently, [58] presented a preconditioned scheme for the central moment based cascaded LB method [45] in the presence of forcing terms [49] and demonstrated significant convergence acceleration.
In general, such preconditioned LB schemes are intended to solve the preconditioned NS equations, which can be written as ([61], [62])
where p*, S and F are the pressure, strain rate tensor and the impressed force, respectively. Here, 7 is the preconditioning parameter, which can be used to tune the pseudo-sound speed, thereby alleviating the eigenvalue stiffness and improving convergence acceleration (e.g. [58]). However, the existing LB models for the preconditioned NS equations are not Galilean invariant and are expected to involve both velocity- and parameter-dependent anisotropic form of the viscous stress tensor. Development of the Galilean invariant preconditioned central moment based LB method without cubic-velocity defects and parameter free truncation errors for steady flow simulations is the main focus of this study. It may be noted that the preconditioned NS equations may be considered as a specific example of what may be called as the generalized NS equations containing a free parameter. In the present case, such a parameter is imposed by numerics due to preconditioning. On the other hand, such generalized NS equations arise in other contexts such as in the simulation of the fluid saturated variable porous media flows represented by the Brinkman-Forchheimer-Darcy equation. In such cases, the free parameter appearing in the generalized NS
f + v.oÂ») = o.
(2.1a)
(2.1b)

27
equations is imposed by physics, viz., the porosity. Thus, our present investigation on the development of the Galilean invariant LB models for the preconditioned NS equations on standard lattices without cubic-velocity and parameter dependent errors also has wider implications in other contexts.
In order to first identify such truncation errors, in this chapter we first perform a Chapman-Enskog analysis of the preconditioned central moment LB formulation and isolate various cubic-velocity and parameter dependent errors at various moment orders. It will be seen that the anisotropy of the stress tensor depends not just on the cubic-velocity terms (like in the previous studies), but also on the preconditioning parameter 7. Furthermore, we will also demonstrate that even to achieve partial corrections for the GI defects on the standard lattice, the cubic velocity terms appearing in the off-diagonal components of the third-order moment equilibria need to be appropriately scaled by 7 (e.g. Kxxy = c2spv,y + pu2uy/j2). In general, the various truncation error terms that arise due to the degeneracy of the third-order diagonal elements will be seen to have complex dependence on both the velocities and the preconditioning parameter. Once such GI defect terms are identified, new corrections are derived for the preconditioned central moment LB formulation based on the extended moment equilibria. This results in a GI central moment LB method for the preconditioned NS equations without cubic-velocity and parameter based defects on standard lattices. The present scheme is targeted towards efficient and accurate low Reynolds number steady state laminar flows by a preconditioned LB formulation without the discrete cubic velocity and parameter dependent effects via corrections to the moment equilibria.
This chapter is organized as follows. In the next section (Sec. 2.2), our previous central moment based preconditioned LBM with forcing terms on the D2Q9 lattice is summarized first. Section 3 performs a more refined analysis based on the Chapman-Enskog expansion and identifies various cubic-velocity and parameter dependent GI defect errors. Then, Sec 2.4 derives new corrections based on the extended moment equilibria and Sec. 2.5 presents a GI preconditioned central moment LB method free of cubic-velocity and parameter dependent errors. Numerical results are

28
presented in Sec. 2.6, which compares our numerical results for a variety of benchmark problems, including the lid-driven cavity flow, flow over a square cylinder, backward facing step flow, the Hartmann flow and the four-roll mills flow problem for the purpose of validation. In addition, convergence acceleration due to preconditioning and improvement in accuracy due to the GI corrected LB scheme are also illustrated. Finally, the main findings of this chapter are summarized in Sec. 2.7.
2.2 Preconditioned Cascaded Central Moment Lattice Boltzmann Method: Non-Galilean Invariant Formulation
In our previous work, we presented a modified cascaded central moment lattice Boltzmann method (LBM) with forcing terms for the computation of preconditioned NS equations [58]. However, this preconditioned LBM formulation is not Galilean invariant on standard lattices. This is because it results in grid-independent cubic-velocity errors that are sensitive to the preconditioning parameter. In fact, the derivation of the precise expression for the non-GI truncation errors will be derived in the next section. It may be noted that all other prior preconditioned LB schemes are also not Galilean invariant. However, the choice of central moments here partially corrects parts of the cubic velocity defects in the off-diagonal third order moments naturally (Sec. 2.3) and simplifies derivation of the correction terms to completely restore GI free of cubic velocity errors on standard lattice (Sec. 2.4). Here, we summarize our previous preconditioned central moment LB model setting the stage for further development in the following.
The preconditioned cascaded central moment LBM with forcing terms may be written as [58]
fa(x,t) = fa(x,t) + (K â€¢ g)â€ž + Sa(x,t), (2.2a)
fa(x + ea,t + 1) = fa(x,t), (2.2b)
where a variable transformation fa = fa â€” \Sa is introduced to maintain second order accuracy in the presence of forcing terms. In the above, K is the orthogonal transformation matrix and

g is the collision operator. In order to list the expressions for the collision kernel for the standard two-dimensional, nine particle velocity (D2Q9) lattice, we first define various sets of raw moments as follows on which it is based:
29
( 7 \ ( f \
'lxrnyn Ja
Ke<1' 'lxrnyn = E req Ja
(7xm yU a Sa
^ / \ I7xrnyri y \ fa j
m n ^ax^ay'
(2.3)
The preconditioned collision kernel set for the orthogonal moment basis using the preconditioning parameter 7 can be written as [58]
9a
97
UJ 7
T
9 3
go â€” 91 â€” 92 â€” 0,
- ^3 f 2 | Pj*lWy) W I 7' U
12 | 3^ ' 7 V^xx ' â„¢yy) 2^uxx ' uyy) f >
g4 = f _ tw
j (Eikik ^yy) 2 (Tkik Â®yy) [ i
WB I PUXUy _ ^ _ 17
i/5 4 1 -y f â€™
2Uy + â€” 2Uxnxy â€” UyKxx â€” 2Â°'xxy } â€” 2Uy(3g3 + 74) â€” 2ux95,
T I7xyy 2 UyKxy U X K yy ^ ^ 94^
4 1 ~
as = X HP + 3Pâ€œxÂ«y -
,2â€ž,2
r^/
"K.
â€˜lUxfc'jXyy ^^y^xxy ^x^yy ^11^.
xxyy
y^xx
+ 4:UXUyKxy
~ Wxxyy] - 2h - lu2y(3g3 + g4) - \u2x{3g3 - g4)
-4uxuyg5 - 2uyg6 - 2uxg7.
(2.4)
For further details, and including the choice of the collision matrix K and source raw moments axmyn, see [58]. This scheme results in a tunable pseudo-sound speed c*s = 7cs, where cs = ~^5x/5t, and the emergent viscosity v is given by v = 7(7^- â€” 7), (3 = 4, 5. While this scheme is intended to simulate the preconditioned NS equations given in Eq. (I), as will be shown via a consistency analysis based on the Chapman-Enskog expansion in the next section that it leads to velocity-and preconditioning parameter-dependent non-GI truncation errors. In particular, it will be seen that the components of the non-equilibrium parts of the second order moments, which

30
contribute to the viscous stress tensor, depends on cubic velocity truncation errors and modulated by the preconditioning parameter 7.
2.3 Derivation of Non-Galilean Invariant Spurious Terms in the Preconditioned Cascaded Central Moment LB Method: Chapman-Enskog Analysis
In order to facilitate the Chapman-Enskog analysis, the central moment LB formulation can be equivalently rewritten in terms of a collision process involving relaxation to a generalized equilibria in the lattice or rest frame of reference [58]. This strategy is considered in this chapter to further investigate the structure of the cubic velocity non-GI truncation errors for our preconditioned LB method. In this regard, it is convenient to define the non-orthogonal transformation matrix T which is the basis to obtain the orthogonal collision matrix K used in the previous section and on which the subsequent analysis follows:
=2 ) "ocyi
e2 ) ^ay/ 1 \&otx &a,y)
e2 ) |e2 e2 )1
(2.5)
where the usual bra-ket notation is used to represent the raw and column vectors in the q-dimensional space (q = 9) for the D2Q9 lattice. Then, the relation between the various sets of the raw moments and their corresponding states in the velocity space can be defined via this nominal, non-orthogonal transformation matrix T as
m = Tf, m = Tf, meq = TfeÂ«, S = TS,
(2.6)
where
f = (/o, /1, /2, â€¢ â€¢ â€¢, fs)j, f = (/o, fi, h, â–  â–  â– , /s)1,
ceq ___ ( Q _ (a c C C \t
1 â€” WO 5 Jl W2 5 â€¢ â€¢ â€¢ W8 ) > ^ â€” (wO) *->2) â€¢ â€¢ â€¢ 5 &8)

31
are the various quantities in the velocity space, and
m (jTiQj TTli^ 7712; * * * ; Tilg^ ( ^0 â€™ ^x > ^yi ^xx ~b ^yyi ^xx ^yy ^xyi ^xxyi ^xyy> ^xxyy ) ) (2.7a)
m (t71q; 771 \, 7712 j * * * ; ^8) ^^0 â€™ ^x â€™ ^y "> ^xx ~b ^yy ^xx ^yy â€™ ^xy â€™ ^xxy> ^xyy â€™ ^xxyy^j â€™ (2.7b)
â€” ea / â€” eq â€” eq â€” eg â€” ea\f /-~'eg/ â€¢-'-eg/ â€¢-'-eg/ â€¢-'-eg/ , â€¢-'-eg/ -~-â‚¬<7/ ---â‚¬ 111 â€” (^0 > ^1 ) ^2 ) * * * ) ^8 ) â€” ( ^0 ) ^y i ^a?a? H- ^yy j ^a?a? ^yy ? FXy ? TvXxy ? 71
xyy,
-eq1
T^xxyy
(2.7c)
S ( 8); *\$*1; *\$2; â€¢ â€¢ â€¢ ; *\$8 ) ( U"q , Ug,, Gy, G xx X @yy; & xx Â® yy> ^xyi Â®xxy i ^xyy^xxyy ) (^*^*-0
are the corresponding states in the moment space.
To facilitate the Chapman-Enskog analysis, we can rewrite the preconditioned LB model presented in Eq. (2.2a) and Eq. (2.2b) in terms of the raw moment space given in Eq. (2.6) as ([19], [58])
f (x + ea5t,t + Â£t) - f (apt) = T 1 -A (m - meq) +T 1 where the diagonal relaxation time matrix A is defined as
A = diag(0,0,0,w3,W4,W5,w6, w7,w8).
â– 4a1s
(2.8)
(2.9)
The preconditioned raw moments of the equilibrium distribution and source terms can be represented as
â€”eq â€”eq â€”eq
Kq = p, Kx = puxj /V = puy,
and
~eq: _ i â€ž, y y _ i â€ž, y y _
I^XX â€” 3 p t , rvyy 3' ' y â€™ ~ y â€™
â€”eq I ,
7y>XXy â€” 3 P^y â€œb
pU2xUy -~~eq' 1 .. pUXUy
.,2 j 7>xyy â€” ^P^x .,2
7 7
Kxkyy = T,P+ \p(ul + UD + puWy.
Gâ€ž
= o, ^ Fx u X y 1 Gy - Ev. 7 â€™
2F'x'ux 2FyUy rr _ Fx
yy
2 i u xy
FXUy+FyUX
I2
Â®xxy â€” FyUx ~b ^FxVjXVjy^ Â®xyy â€” FxUy Hâ€œ~ 2FyUyUXj
(2.10)
Oxxyy â€” ^{FxUxUy Hâ€œ FyUyUx^.
(2.11)

32
The following comments are in order here. Up to the second order moments, the above expressions coincide with those presented in our previous work [58]). In other words, UiUj terms in the moment equilibria are preconditioned by 7, while the first and second order moment terms, i.e. Fi and FiUj are preconditioned by 7 and 72, respectively. As a first new element towards a LB scheme with an improved GI, we precondition the third-order moment equilibria terms ryu2 terms by y2 (see the terms inside boxes in Eq. (2.10)). This partially restores GI without cubic velocity defects for the preconditioned LB model for the off-diagonal components of the third-order moments. In fact, as will be shown later in this section, in order to remove the spurious cross-velocity derivative terms appearing in the equivalent macroscopic equations of our preconditioned LB scheme (e.g. uxuydxuy and uyuxdyux), such a scaling of the cubic velocity terms in the third order moment equilibria is essential. Then, applying the standard Chapman-Enskog multiscale expansion to Eq. (2.8), i.e.
00 m = ^erarli(ra), (2.12)
n=0
dt = n=0 (2.13)
where e is a small bookkeeping perturbation parameter, and also using a Taylor expansion to simplify the streaming operator in Eq. (2.8), i.e.
FI
f(* + eae,t + e) = ^ fâ€”(dt + ea â€¢ V)f(agt). (2.14)
n=0
After converting all the resulting terms into the moment space using Eq. (2.6), we get the following moment equations at consecutive order in e:
O(e0) : m(Â°) = me 0{(l) : {c)t0 + Et9t)a(Â°) = -AmW + S, (2.15b)
9tlm(Â°) + (<9to + Eidi) 1 â€” = -Am(2) (2.15c)
where Ej = T(ejl)T l,i e {x,y}. The relevant components of the first-order O(e) equations Eq. (2.15b), i.e. up to the second order in moment space needed for deriving the preconditioned

33
macroscopic hydrodynamics equations are given as
dt0p + dx(pux) + dy(pUy) = 0,
dt0 (pUx) + dx + dy ^ pu^'y) = ^,
aâ€ž (pÂ»â€ž) + a, (^) + a, (7 + =i) = 7,
Â»â€ž ( Ip ++ a, (!pÂ», + + a, (>, + ^
â€” â€”f j -L
â€” UJ3IU3 -\~ 2 ;
a0 (!^) + , - ^) + a, (-Â§p% + ^)
-uj4m\ +
(1) . 2 (FXUX~FyUy)
Qt ( PUX Uy
) + 9x {\pUy + PUy2V j + 9y {\pUx + â€” U/5/U5 i- 2
72
(2.16a)
(2.16b)
(2.16c)
(2.16d)
(2.16e)
(2.16f)
Similarly, the leading order moment equations at 0(e2) can be obtained from Eq. (2.15c) as
dtlp = 0,
(2.17a)
dtl (pux) +dx \{l~ |w3) + \ (l - |w4) m'P + dy (l - |w5)
= 0,
(2.17b)
9*i (puy) + dx (l - ^w5) + 9y i (! - ^w3) m31} - \(l - \uA) m.
(i)
0.
(2.17c)
In the above equations, the second-order, non-equilibrium moments fn^\ fh^ and (corresponding to, Px^ +Pyl\ Px^ â€” Pyl^ and â€™Kxy\ respectively) are unknowns. Ideally, they should only be related to the strain rate tensor components to recover the correct physics related to the viscous stress. However, as will be should below, on the standard D2Q9 lattice there will be non-GI contributions dependent on the preconditioning parameter 7. In what follows, fh^\ and will be obtained from Eq. (2.16d), Eq. (2.16e) and Eq. (2.16f), respectively.

34
Now, from Eq. (2.16d), the non-equilibrium moment fh^ can be written as
p(v?x+v?y)
-(1) 1
m\' = -rr
* W3
-at0[lp + ^^)-9,[lPu, + ^)-ay[lPuy + epâ€˜
+
â€˜2(F'xUX~\~ FyUy)
72
(2.18)
In order to simplify Eq. (2.18) further, one needs to obtain expressions, in particular, for dt0 ( Ty-),
9t0 ( p), dx (pP1) and dy (pp). It follows from Eq. (2.16b) that
dto(pux) = -i;dxp ~ dx â€”21 - d.
pv,3_
f pUXUy\ Fc
+ â€”.
Rearranging dt0 ( ) as
<-) ~ i ~^y l I
3 V 7 ) \ 7 / 7
9 (2.19)
Using Eq. (2.19) and Eq. (2.16a) to replace the time derivative in the first and second terms respectively, on the right hand side of the above equation, we get.
9,
'to
pul
7
2ux
7
pUXUy
7
I Fx
~r 7
-\dxp-dx (ppj -dy(
+ V [dx (PUX) + dy (pUy)] â€¢
(2.20)
Similarly, we may write
9,
'to
pul
T
2 uv
-\dyP-8y -8X (
pUXUy
7
+u
7
+ -f \-dx (pUx> + dv 7%)] â€¢
(2.21)
Thus, the time derivative can be replaced with the spatial derivative. Also , it readily follows that
dx [ 'N 7^ O Â£ 7-, H w ss to = -yf dx{pux
-0y (Puluy\ { i2 ) = â€”^dyipUy
72
2pUXUy
9XUy,
^2 9yUX .
(2.22a)
(2.22b)
Rearranging Eq. (2.20) and simplifying it further by retaining all cubic velocity terms and neglecting all others higher order terms in velocity (e.g. fifth order and higher) we get
-dto ( ^
pdxP + pdX(Pul) + IpdyUy + 2-p^dyUX ~ ^
~Ydx{pux) - Ydy(puy)-
(2.23)

35
Similarly, it follows from Eq. (2.21) that
-dt0 ( ^ ) =
ifrdyP + ^dy(pul) + ^dxux
+
2 fmx'Uy
72
dx'Ui
I 2FyUy T Â«,2
â€” -Z~dX(pUX) - ^dy(pUy).
(2.24)
Now, to obtain an expression for fn^\ we group all the higher order terms given in Eqs. (2.22a), (2.22b), (2.23)
and (2.24). It follows that owing to the choice of the off-diagonal third-order equilibrium mo-
ments with the cubic velocity terms scaled by y2 (i.e. 'nexxy = \puy + pu*Zv, kxyy = \pux + pu*2V)
at the outset following Eq. (2.9) earlier, all the cross-derivative spurious terms, i.e.â€”2puxuydxuy
and â€”2puxUydyux cancel. Then, simplifying the grouping of all the remaining higher order terms
in Eq. (2.22a), Eq. (2.22b), Eq. (2.23) and Eq. (2.24) and retaining all cubic velocity terms and
neglecting terms of negligible higher orders and after considerable rearrangement, we get
-dt0[^)-dt0(p-^)-dx
( PUxV-l l 72
-
pAjuy
3y{uxdxp + Uydyp) ^2 (,FxUX + FyUy) + p
+ P

ur
dxu7
A _ i ) ,,2 , f A _ I
72 7 J y ' l 72 7
UZ
OyUy.
(2.25)
By substituting the above equation (Eq. (2.25)) in Eq. (2.18) and using dt0p = â€”dx(pux) â€” dy(puy) from Eq. (2.16a) to further simplify the resulting expressions, we finally get the form of the non-equilibrium moment as
7)
m =
+dyUy) + 3^ (y - l) (UXdXP + UydyP) +
dxux
JL
L03
+ A
W3

^-Fu2y+U-l)u2
OyUy.
(2.26)
Similarly, using Eq. (2.16e) and following analogous procedure as above for and using Eq. (2.16f) for after considerable algebraic manipulations and simplifications we get the expressions for

36
the remaining non-equilibrium second-order moments as
and
m
(i)
"zuSÂ®xUx Auy) E 3w4 (7 1 ^ (uxdxp uydyp) +
_p_
OJ4.
1 OJ4.
â€” - ) u7
7 71 x
1 1 I 1/2
T ~ 7 J uy
d7u7
J_ _ i I ,,2 , f J_ _ 1
72 7 y u"y ' l 72 7
â– u;
dyUy,
(2.27)
m
(i)
+ + 3^7 (7 â€œ X) (Ux9yp + %9xP) +
7 (72 - 7) PÂ«x% (dxUx + 9y%) â€¢
(2.28)
The first terms, which are underlined, in the right hand sides of Eq. (2.26), Eq. (2.27) and Eq. (2.28) are associated with the required flow physics related to the components of the viscous stress tensor. All the remaining terms in these equations are non-Galilean invariant terms for the preconditioned LB scheme. These spurious terms arise because the diagonal third-order moments Kxxx and nt/ly are not supported by the standard D2Q9 lattice. However, such discrete effects are not observed in the C-E analysis of the continuous Boltzmann equation. In order to eliminate the non-GI error terms by other means in the next section on the standard lattice, we explicitly identify the various non-GI terms in the components of the second-order non-equilibrium moments as
El
IPâ€” [ ~ ~ 1 ) (Ux8xp + Uydyp), ouj3 \7
EL = â€”
yu
P_
UJ3
OJ3
â€” - y2 _|_ (â€” u2
^2 ^ j ax + ^2 ^ j u,y
4 - -) 7+(4 - A Â«2
^ 'T / y V 7 7
d7u7
dyUy.
(2.29a)
(2.29b)
eL 2 --4
yp 3W4 \7 J
Ea _p_ (---
gu W4 AT 7
U,yUy^J, 1 1
72 7 ) uy
d7u7
UJ 4
â€œI 4l--)Ul+(^2--)u2
7^ 7 j y v 'â– y*
OyUy,
(2.30a)
(2.30b)

37
and
Egp â€” ^ 1 ) (uxdyp + Uydxp),
p ( 1 1
P
E ---- ( o ) UXUydXUX ~\~ ( o ) UXUydyUâ€˜
1 1
w5 \7 7/ w5 V7 7
Then, we can rewrite the non-equilibrium second-order moments
(2.31a)
(2.31b)
m
Â£(!)' +Â£(!)' -,VXX 1 ,vyy 2 P m (uxUx 3 w3 + dyUy) T Egp + EgU, (2.32)
say - ,vxx yy 2P (f) (oxux 6UJ4 ~ OyUy) + Egp + EgU, (2.33)
â€” (1) -M y m5 = KxJ = p (a (uXUy 3W5 T 9yUX ) + El + El. (2.34)
Some interesting observations can be made from the above analysis: (*) when the LB scheme is preconditioned, i.e. 7 / 1, non-GI terms persist in terms of velocity and density gradients for all the second-order non-equilibrium moments, including the off-diagonal moment (m^ = kI ), unlike that for the simulation of the standard NS equations (i.e. with 7 = 1). However, the non-GI cubic velocity contributions in vanish for incompressible flow (V â€¢ u = 0), i.e. Epu =
0 . (ii). In general the prefactors appearing in the non-GI terms for the diagonal components, i.e. in ml and ml exhibit dramatically different behaviour for the asymptotic limit cases: No preconditioning case (7 â€”>â€¢ 1): ~ 3,^77 â€œ7) ~ 0; strong preconditioning case (7 â€”>â€¢
0): ~ 77, ^77 â€” yj ~ 77. Thus, due to the complicated structure of the truncation
errors and their dependence on 7, the non-GI terms in the diagonal moment components modify significantly as 7 varies due to preconditioning, (in) when 7 = 1, i.e. when our preconditioned LB scheme reverts to the solution of the standard NS equations, Egp = Epp = Epp = 0, Epu = 77 (u^dxUx +UydyUy), Epu = (uldxux â€” UydyUy), and Epu = 0. That is, the non-GI terms become identical to the results reported by [42] and [43].

38
2.4 Derivation of Corrections via Extended Moment Equilibria for Elimination of Cubic Velocity errors in Preconditioned Macroscopic Equations
In order to effectively eliminate the non-GI error terms given in Eq. (2.29a)-(2.31b) that appear in the non-equilibrium moments fh^\ fhand in the previous section (see Eqs. (2.32)-(2.34)) arising due to the third-order diagonal equilibrium moments (kxxx and nt/yy) not being independently supported by the D2Q9 lattice, we consider an approach based on the extended moment equilibria. In other words, we extended the second-order moment equilibria by including extra gradient terms with unknown coefficients as follows:
â€” eq( 0) m0 0
â€” eq( 0) m4 0
â€” eq( 0) m2 0
â€” eq( 0) m3 â€” eq( 1) m3
â€” eq( 0) m4 + St â€” eq( 1) m4 =
peq(Â°) J 5 â€” eq( 1) m5
â€” eq( 0) m6 0
â€” eq( 0) m7 0
â€” eq( 0) m8 0
-^eq
<^x
-^eq
, ^eq <^xx \ nyy
-^eq
^xy
-^eq
6xdxUx I Gydyiiy I Axdxp I Xydyp
{eq = + St m^(1) = Kxl - Kyi + 5t 0^.dXUX - OydyUy + \\dxp ~ Xydyp |(2 *3 5)
OxdXUX | Gydylly | A x9xp | Xydyp 0 0 0
In other words, the corrections to the second-order moments are given by
^xxy
â€”~eq
K>xyy
'--eq
l^xxyy
0
0
0
^eq(l) = (Q3gxUx _|_ Q^dyUy) + (Ai\dXp + A ydyp),
-eg(i) = (QAdxUx _ Q^dyUy) + (AAxdXp - Xydyp),
m7{l) = (Â°t9xUx + OydyUy) + (A ldxp + A 5ydyp),
(2.36a)
(2.36b)
(2.36c)
where the coefficients 9X, 9y, Xx and Xqy, where j = 3, 4, 5 are to be determined from a modified Chapman-Enskog analysis so that the non-GI cubic velocity terms are effectively removed from the emergent preconditioned macroscopic moment equations.

39
We now apply a Chapman-Enskog (C-E) expansion by taking into account the modified equilibria which is now given as rheq = where is the moment equilibria pre-
sented in the previous section and me9W is the correction to this equilibria. As a result, the C-E expansion given as Eq. (2.12) and Eq. (2.13) are now replaced with
m = me9(-0) + em69*-1-1 + em*-1-1 + e2m<-2-) + â€¢ â€¢ â€¢ , dt = dt0 + eSq + e2<9q + â€¢ â€¢ â€¢ . (2.37)
Then, by using a Taylor expansion given in Eq. (2.14) for the streaming operator in Eq. (2b) along with above modified C-E expansion Eq. (2.37), we get the following hierarchy of moment equations at different orders in e:
O(e0) : m(Â°) = m69,
Oie1) : (dt0 + Eidi)mW = -A [m^ - m^1)] + S,
(2.38a)
(2.38b)
0(e2) : <9tlm(0) + (dto + E*<%)
l-5A
m
(-1-) + (dt0 + Ej9j) |Ameq^ = -Am(2( (2.38c)
where Ej = T(ejl)T_1 and i e {x,y} . The relevant 0(e) equations for the first order moments are given in Eqs. (2.16a)-(2.16c). However, the equations of the second order moments are now modified due to the presence of the extended moment equilibria m69^1) in Eq. (2.38b) which are now given by (instead of Eqs. (2.16d)-(2.16f))
Â»â€ž (h>++ at ( Ipu, + yyd + ad i ^

= + oj3me3q{1) + .
9,
'to
fp{ul-u2y)
) + 9X (|pux - + dy (~lpuy + E&bL^
,(1)
aqm) + W4m4 +
eq{ 1) , 2 (Fxux-FyUy)
a,. (^) + (><â€ž + *\$*) + + yyJ)
= + w5m5?(1) + F*uy+Fvu*.
(2.39a)
(2.39b)
(2.39c)
Similarly, the leading order moment equations of 0(e2) which are modified by m69^1) as shown in

40
Eq. (2.38c) are obtained as (instead of Eqs. (2.17a)-(2.17c))
dtlp = 0,
dtl {(nix) +dx \{l~ |w3) \ (l - |w4)
m
(i)
+ 9y
dx
l, . I 1, .
4W3m3 + jW4m4
+ dv
1 -^eq(l)
2W5%
(1 - Â±w5) fn^
= 0,
<9*1 (puy) + <9* (l - |w5)
a
1, .
2W5%
+ <9y + <%/
I (1 - ^3) mÂ£1} - |(1 - ^4)
m
(i)
+
+
1, . 1, ,
4W3 m3 â€” 4w4m4
0.
(2.40a)
(2.40b)
(2.40c)
The non-equilibrium moment m3^ is now obtained from Eq. (2.39a) as
m31} = -b-
3 W3
-^to 3P +
p(Â«2+M2)
7
- ( IPUX + ^ ~ dy (j PUy + â€œ
T
2{FXUX-\-FyUy)
72
1 â€” eq'(l)
+ m3 b
(2.41)
All the terms within the square brackets in the above equation exactly corresponds to Eq. (2.32). Hence, Eq. (2.41) reduces to
^3 1 T? 3 1 â„¢ e<4l)
.(i) _ 2p
3w3
ml = ~ â€” (dxux + dyUy) + Egp + Egu + m3 ,
(2.42)
where the non-GI error terms and Egu are given in Eqs. (2.29a) and (2.29b), respectively, and the extended moment equilibrium m39('1^ in Eq. (2.36a). Similarly, the non-equilibrium moment m4^ is obtained from Eq. (2.39b) and using Eq. (2.33) for simplification, and for using Eqs. (2.39c) and (2.34), we finally get
mA
4^ â€” ~ Â®vuy) + Egp + Egu + m^1^,
m
(i)
â€” ~3^(Â®xUy + dyv,x) + E5gp + E5gu +
m
eq{l)
(2.43)
(2.44)
Here, the non-GI error terms Egp and Egu are given in Eq. (2.30a) and Eq. (2.30b), respectively, and the correction equilibrium moment m49('1'> in Eq. (2.36b). Likewise, Egp and Egu are obtained from Eqs. (2.31a) and (2.31b) respectively and is presented in Eq. (2.36c).
Now, in order to obtain the preconditioned moment system for the conserved moments, we combine 0{e) equations Eqs. (2.16a)-(2.16c) with ex Eq. (2.40a)-(2.40c) for the corresponding equa-

41
tions at 0(e2), and using dt = ft0 + eft^, we get
dtp + dx(pux) + dy(pUy) = 0,
dt (pux) + dx p + + ft, ^pu^Uy j =
__ r f)
j t(JX
i (i - f) ft1â€™ + i (i -1)
m
(i)
- eft.
â‚¬Cjx
W3^eg(i) + ^ ~eftl)
Xm3
- eft.
H1 - f)
wsw^ft1)"
2 'U5
ft (p^) + 9, + 9, (Ip + f - 634(1 - f-)
-eft.
m
eq(l)
H1 - Â¥) ^31} - h (X â€œ f)m
(1)
e9x
WB^eft1) 2 'U5
- eft.
Xâ„¢3
eft1) _ W3-eg(l) 4 m4
(2.45a)
(2.45b)
(2.45c)
Our goal is to show that the above equations (Eq. (2.45a)-(2.45c)) is consistent with the preconditioned NS equations (Eq. (1)) presented in Sec. 2.1 without the identified truncation errors, i.e without involving the non-GI cubic velocity defects. Now, in order to relate the moment corrections m^1^, fheqt'l'> and fheqt'l'> appearing in the equilibria with the non-GI error terms, with a view to eliminate them, consider the right hand side of Eq. (2.45b) (i.e.the ^-momentum equation) and substitute for fh^\ m^ and from Eq. (2.42), Eq. (2.43) and Eq. (2.44), respectively, which becomes
â€” y- + eftc
+ 3 ( J3 2 J P(dXUX + dylly) + 3 ( 2 J P(dXUX dylly)
3 l UJ4
+efty
3 ( WK 2 ) P(dXUy + dyux)
-Â«V [i (1 - f) {Elâ€ž + Â£5â€ž} + 1 (1 - f) {El + }] - cd, [(1 - f) {El + Â£{â€ž}]
â‚¬C)X
lfheqW , 1 ^efti)
2 ra3 ' 2 ,nA
- eft,
m
eft 1)
(2.46)
The first two lines in the above equations correspond to the physics, while the third line corresponds to the spurious non-GI terms arising from discrete lattice effects and the fourth line are related to equilibrium corrections.
In order to eliminate the cubic velocity truncation errors, it follows that the third and fourth

42
lines in the above equation (Eq. (2.46)) sum to zero. This yields
(i - y ) {E3gp + E3gv] + mf(1) = 0, (2.47a)
(l - y ) {E% + EAgv] + mf(1) = 0, (2.47b)
(i - y ) Kc + + ^9(1) = Â°- (2-47c)
The above equations Eqs. (2.47a)-(2.47c), represent the key constraint relations between the non-GI error terms and the moment equilibria correction terms to obtain a preconditioned cascaded central moment LB model without cubic velocity defects.
Further analysis shows that these constraints hold identically for the y-momentum as well (Eq. 2.45c)). Now considering Eq. (2.47a) and using Eq. (2.29a) and (2.29b) for Egp and Egu, respectively, the extend moment equilibrium is given as
-eg(i) = (q3qxUx + e3dyuy) + (A 3xdxp + A 3ydyp),
where the coefficients obtained after matching are given by
n3 _ _ ( j_____1
\)P
4 1 1 â€ž2 i
~ ~( ) UT +
e3 =
y
Ul3
2 J P

â€” t )u:
J_ _ i ) ,,2 i f J_ _ 1
72 7 J ' l 72 7
U
\3 _ _2 / J_____1
Nc 3 l w3 2
V - 1 ) Ux,
\3 2 / _\_
Ay 3 w3
1 U,,
(2.48a)
(2.48b)
(2.48c)
(2.48d)
Similarly, from Eq. (2.30a),(2.30b), (2.36b) and (2.47b), we can obtain the coefficient of me^l\ and from Eq. (2.3fa), Eq. (2.31b), Eq. (2.36c) and Eq. (2.47c), those for can be deter-
mined. The results read as follows:
m
eg(l)
(eAxdxux - QydyUy) + (A%dxp - Xpyp),

43
where
and
where

94 =
y
\)P
\ â€” -) ux â€”
7 1 x

â€” 7 ) u:
OJ4
\)P

u
\4 _ _2 / J_________1
a:e ~ 3 I w4 2
7-1 tt
\4 _ _2 / J_ Ay 3 1 w4
1 U.
â– y,
m,
eq(l)
(19ldxux + OydyUy) + (A^p + A^p),
(2.49a)
(2.49b)
(2.49c)
(2.49d)
â€œ \) P - 7) uxuy, (2.50a)
^ - 5) P â€œ 7) (2.50b)
= â€œI - 5) P (7 â€œ x) %> (2-50c)
= "3 (Â± - 5) 0 (l - l) â€œ*â€¢ (2.50d)
Note that, as a special case, when 7 = 1, i.e. the LB model is used to solve the standard NS equations without preconditioning, then 9% = â€” 3p(^ â€” ^)ux, 9y = â€” 3p(^ â€” ^)Uy, 94 = â€” 3p(^ â€”
t;)ux, 94 = â€”3p(^- â€” ^)Uy, and all the remaining coefficient go to zero. In such a case, these moment corrections to the equilibria become identical to the GI corrections presented by [43] and equivalent to the alternative GI formulation without cubic velocity errors introduced by [42].
Finally, using the above extended moment equilibria (m^1', fh^1'1 and m^1^) and the expression for the non-equilibrium moments (fh^\ fh^ and m^) from Eq. (2.42)-Eq. (2.44) along with the constraint relations, i.e. Eqs. (2.47a)-(2.47c) in Eqs. (2.45a)-(2.45c), we get
dtp + V â€¢ j = 0,,
dtjx + V-(^) + * 1 eU 1 â€” (2 dxja - V â€¢ j) + V â€¢ j
V 7 / 7 [ 7 7 J
+ 0y (9Xjy ~\~ 9yjx) Fx 7
(2.51)
(2.52)

44
dtjy + V â€¢ (^j = -dyV- + dx
7
(9xjy + dyjx
+ cb

+ 7,
7
(2.53)
where p* by
^p is the pressure, j = pw, and the bulk and shear viscosities are, respectively given
tf3 = l (-~l
3 y W3 2
d4 = â€” f â€”â€” â€” â€”
3 \ W4 2
3 \ W5 2
(2.54)
Thus, Eqs. (2.51)-(2.53) are consistent with the preconditioned NS equations given in Eqs. (2.1a)-(2.1b) without cubic velocity defects in GI due to the use of the extended moment equilibria presented earlier.
2.5 Galilean Invariant Preconditioned Cascaded Central Moment LBM without Cubic Velocity Errors on a Standard Lattice
The cascaded central moment LBM with forcing term presented in Eqs. (2.2a), (2.2b), (2.3) and (4) modify to enforce GI without cubic velocity errors as follows. Equations Eq. (2.2a), Eq. (2.2b) and Eq. (2.3) remains the same as before and the collision kernel given in Eq. (4) is modified to account for the extended moment equilibria in the second order moments as well as corrections to the third-order equilibrium moments. The change of moments <73, p4 and <75 for the second order components follow by augmenting the corresponding moment equilibria with the extended moment equilibria incorporating the GI corrections identified in the previous section. On the other hand, owing to the cascaded structure of the collision kernel, the GI corrections to the third order moment changes % and <77, which depend on the lower order moment changes, for the preconditioned central moment LB scheme need to be constructed carefully. They are obtained by prescribing the relaxation of the third order central moment components to their corresponding central moment equilibria. Following the derivation given in [19], they can then be represented as -6%p3 - 2%p4 - 8Â«xp5 - 4p6 = W6 \th?xy - nxxy\ and -6uxg3 + 2uxgÂ± - 8uyg5 - 4p7 = ^i\^Xyy â€” Kxyy\, where kx:xy and kxyy are the third order central moment components, and Kxqxy

45
and 'Kxyy, respectively, are their equilibria. Rewriting these central moment relaxations in terms of the relaxations of the raw moment components of the third and lower orders via the binomial theorem, it follows that
9 6 9i
We
4
UJj
T
xxy yxxy) <^â€˜yxfaXy yxy) yy(.yxx yxx)
- uy ( 0^3 + 7:94 ) - 2uxfa,
1
iyxyy yxyy) ^^yi^xy yxy) yxfayy yyy) yx ( 093 ^9A ) 2v,ygÂ§.
2 2"
Now, using the components of the preconditioned raw moment equilibria, including those for the third order equilibrium moments with the GI corrections from Eq. (2.40), the final expressions for the change in moments for the collision kernel fa and fa can be derived. Thus, the modified preconditioned collision kernel with the GI corrections reads
9o = 0, fa = 0, fa = 0,
93 = ff {|P + P(Ul + y2y)h ~ (Kxx + Kyy) â€œ \(axx + ayy) +
(9xdxux + 9%dyUy)5t + (A3xdxp + Aydyp)5t\ ,
94 4 ^p('Ua, yy)/^1 {yxx yyy) 2^xx Â®yy)
{9%dxux - d*dyUy)5t + (AAxdxp - \*dyp)6t\ ,
95 = x \puxyyh - Kxy - \axy + (93dxux + d^dyUy)5t + (Ahxdxp + A^p)^}, 9a = ~4 ^ â€” pUxUy + HXXy â€” 2UxKXy â€” â€” 2Uy(^93 + 9a) ~ 2uxgÂ§,
97 = X | (f - ^2) PUxV2y + Kxyy - 2 UyKXy - UXKyy ^ ~ ^(3^ ~ fa) ~ 2Uyfa,
9& ~4~ "I gp T 3pUxUy yXXyy â€˜^â€˜yx!7Xyy â€˜^â€˜^y^'xxy T yxyyy T yy!7>xx
} - 2p3 - \u2y{3g3 + fa) - \u2x{3g3 - fa)
-Auxuyfa - 2Uyfa - 2uxfa.
+AuxUyRxy
where the various coefficients 9X, 9y, \x and Ay where j = 3,4 and 5 are given in Eqs. (2.48b)-(2.48d), and (2.49b)-(2.49d) and (2.50a)-(2.50d). The GI corrections are identified by means of the underlined terms in the cascaded collision kernel terms in the above equation.
It may be noted that other GI preconditioned LB schemes without cubic velocity errors can be

46
constructed from our results in the previous section. For example, a non-orthogonal moment based multiple relaxation time LB method readily follows from the analysis presented before.
The spatial gradients for the velocity components and the density appearing in the extended moment equilibria can be calculated using isotropic finite difference schemes. Alternatively, the diagonal strain rate components dxux and dyuy can be locally obtained from non-equilibrium moments as follows, which is used in our simulation studies presented in the next section. From Eqs. (2.42) and (2.47a) and rearranging, one may write the resulting expression as follows:
-cidxux - c2dyuy = m3^ - ep. (2.55)
Similarly, from Eq. (2.43) and Eq. (2.47b), it follows that
-cidxux + c2dyUy = ) - ep, (2.56)
where the coefficients c\, c2, c\ and c\ and the parameters ep and ep are defined as
Cl = â€” + P _3w3 + P, Cl = 3^7 + P1 P, (2.57)
C2 = P, C2 = 3^1 + Here,
P-y = â€” \ (Ajul + BjUy) , Qy â€” g (AyUy + BjV,G)
P-y = ~\ (AyU^ â€” B-yUy) 1 Qj = ~ 2 (AyUy â€” ByUx
where A7 = , By = , C7 = ( y - l) and
Gp = â€” Cj (uxdxp + Uydyp) , Gp = {^xdxp Uydyp)
Solving Eqs. (2.55) and (2.56) for dxux and dyuy , we get
dxux
dyv,y
c2{m(z) -ep) + c2(mp -ep) +ci(m!ip
/ [-cic2 - cic2],
eP
/ [-C1C2 - cic2] â€¢
(2.59)
(2.60a)
(2.60b)

47
Here, the density gradients appearing in ep and ep Eqs. (2.59) may be computed using a isotropic finite difference scheme. In Eqs. (2.60a) and (2.60b), all the coefficients involving 7 need to be computed only once before the start of computations for efficient implementation; quantities such as v% and Uy appearing in the factors P, Q, P and Q above need to be reused rather than perform the product calculations for every occurrence. A comparison of the computational costs for the uncorrected preconditioned LB scheme and the GI corrected preconditioned formulation is presented for a benchmark case study on the four-rolls mill flow problem at the end of the numerical results section (see Sec. 2.6.5), which also demonstrates a quantitative improvement in accuracy achieved with correction. The non-equilibrium moments and required in Eqs. (2.60a) and Eqs. (2.60b) are obtained as
-(i)
m ' =
^ 'Sfiax A &ay)fa
2 p(ul + U.
7+
7
m
(i)
2 _ 2 w _ P(ul uy)
/ ^ay/Jcz
7
(2.61a)
(2.61b)
2.6 Numerical Results
We will now present the validation of our new Galilean invariant preconditioned cascaded central moment LBM by making comparisons against prior numerical solutions for various complex flow benchmark problems. These include the lid-driven cavity flow, flow over a square cylinder, backward-facing step flow, the Hartmann flow and the four-roll mills flow problem. In addition, we will also demonstrate the convergence acceleration achieved using our preconditioning LB model for some of the benchmark flow problems.
2.6.1 Lid-driven Cavity Flow
As the first test problem, the GI preconditioned central moment LB model is applied for the simulation of steady, two-dimensional flow within a square cavity driven by the motion of the

48
top lid. This is one of the classical internal flow benchmark problems with complex flow structures. The numerical simulations are computed at two different Reynolds numbers of 3200 and 5000, which are resolved by computational meshes with a resolution of 400 x 400. To implement the moving top wall at a velocity Up, the standard momentum augmented half-way bounce back scheme is considered. In order to validate the numerical simulation results obtained with our GI preconditioned LB scheme, the computed dimensionless horizontal and vertical velocity profiles along the vertical and horizontal centerlines, respectively, for Reynolds number Re = 3200 and 5000 and preconditioning parameter 7 = 0.1, are presented with benchmark solutions of [2] in Fig. 2.1. The Mach number Ma considered in the simulations is 0.05. It is clear that the velocity profiles for all the cases agree very well with the prior numerical data. Next, we investigate how the steady state convergence histories are influenced by the use of our new preconditioned formulation for this benchmark problem. Figure 2.2 presents the convergence histories for Re = 3200 obtained by varying the preconditioning parameter 7. Here 7 = 1 corresponds to results without preconditioning. Obviously, the use of preconditioning accelerates the steady state convergence by at least one order of magnitude. For example, it can be seen that when compared to the case without preconditioning (7 = 1), the preconditioned GI cascaded LBM with 7 = 0.05, is at least 15 times faster.
2.6.2 Laminar Flow over a Square Cylinder
Next, in order to validate our preconditioned LB formulation for an external complex flow example, a two dimensional laminar flow over a square cylinder in a channel is studied. The geometry details and the set up of the flow problem is provided in Fig. 2.3. A fully developed velocity profile is considered at the inlet, and at the outlet, a convective boundary condition is used which is given by
dfUi + UmaxdxUi â€” 0
(2.62)

49
(c) Re=5000 (d) Re=5000
FIGURE 2.1: Comparison of the computed horizontal velocity u/Up and vertical velocity v/Up profiles along the geometric centerlines of the cavity using the Galilean invariant preconditioned cascaded central moment LBM with the benchmark results of [2] (symbols) for Re=3200 and 5000 and 7 = 0.1.
FIGURE 2.2: Convergence histories of the GI preconditioned cascaded central moment LBM and the standard cascaded LBM (7 = 1) for lid-driven cavity flow for Re=3200.

where Umax is the maximum velocity of the inflow profile. Computations were performed using L = 50D, H = 8D and L\ = 12.5D, where D is side of square the cylinder, L and H are the total length and width of computation domain, respectively and the location of square cylinder from entrance is defined by L\. In order to visualize the general complex features and patterns of the
Li
L
FIGURE 2.3: Schematic representation of the flow over a square cylinder in a 2D channel.
flow, the streamlines plots at four different Reynolds numbers Re = 1, Re = 15, Re = 30 and Re = 200 are presented in Fig. 2.4. In Fig. 2.4(a), as it may be expected, at a low Reynolds number, Re = 1, where the fluid velocity is relatively very slow and on the other hand, the viscosity is large, the fluid flow is creeping and symmetric without separation. However, with increasing Reynolds number an adverse pressure gradient is established which leads to the flow separation from the surface and a vortex pair regime is formed (Fig. 2.4(b)). As the Reynolds number is further increased further to Re = 30, the size of the recirculation zone increases; besides the flow is still steady and symmetric about, the horizontal centerline (Fig. 2.4(c)). These general features and flow patterns are consistent with the prior benchmark results (e.g. [63], [3]).
Then, we present the velocity profiles along the centerline at different sections at Re = 100 with a mesh resolution of 1000 x 320. Figure. 2.5 illustrates the horizontal and vertical components of the velocity profiles of u and v, respectively. By comparing the present results against the benchmark numerical results obtained using the Gas Kinetic scheme (GKS) [3], a good agreement. between the computational results is observed. An important, global feature of the flow over

51
(a) Re=l (b) Re=15
(c) Re=30
FIGURE 2.4: Stream function contours for flow over a square cylinder for four different Reynolds numbers; Re=l, Re=15 and Re=30 using the GI preconditioned cascaded central moment LBM with 7 = 0.5.

52
a cylinder is the length of the recirculating flow pattern formed behind the cylinder. Quantitative characterization of this wake length Lr and its dependence on the Reynolds number Re is a key element in the validation of numerical scheme. A widely used empirical correlation for the wake length Lr as a linear function of the Reynolds number is given by [63]
^ ^ -0.065 + 0.0554Ae, for 5 < Re < 60. (2.63)
As illustrated in Fig. 2.6a, the computed results for the wake length Lr obtained using the GI preconditioned cascaded central moment LBM are in very good agreement with the empirical correlation presented in Eq. (2.63). As may be expected, for the steady 2D flow over a square cylinder which, at relatively low Re is characterized by symmetry, the lift force is zero and, as a result, a main quantity of interest is the drag force or the drag coefficient Cd in dimensionless form whose magnitude varies significantly with Re. We use the standard momentum exchange method to compute the drag force on the square cylinder in our preconditioned LB formulation.
A comparison of the computed drag coefficient Cd obtained using our GI preconditioned LB scheme with the GKS scheme [3] based benchmark results is presented in Fig. 2.6b. It can be observed that the obtained results agree well with the benchmark solutions. Next, we analyze the influence of the precondition parameter 7 in our formulation on the steady state convergence of this complex flow problem. Figure 2.7 presents the convergence histories for Re = 30. It can be seen that when compared to the usual cascaded LBM without preconditioning (7 = 1), the preconditioned formulation (e.g. for 7 < 0.1) is able converge to the steady state significantly faster, with the residual error being reduced to the machine round off error by a factor of least 15 times more rapidly. Thus, the GI preconditioned cascaded central moments LBM exhibits significant convergence acceleration for complex flows.

53
(c) (d)
FIGURE 2.5: Comparison of the computed velocity profiles along and across the square cylinder along its centerline for both the horizontal u and vertical v velocity components obtained using the GI preconditioned cascaded central moment LBM with 7 = 0.5 for Re = 100 with benchmark results obtained using the Gas Kinetic Scheme (GKS) [3].

54
(a) (b)
FIGURE 2.6: Comparison of the computed Reynolds number dependence of the recirculating wake length Lr on the left (a) and the Reynolds number dependence of the drag coefficient Cd on the right with (b) benchmark correlation (Eq. (2.63)) and GKS-based numerical results [3] respectively.
FIGURE 2.7: Convergence histories of the GI preconditioned cascaded central moment LBM and the standard cascaded LBM (7 = 1) for flow over the square cylinder for Re=30.

55
2.6.3 Backward-Facing Step Flow
As the third flow benchmark flow problem involving complex separation and reattachment effects, we consider a two-dimensional laminar flow over a backward facing step, which is computed using the GI preconditioned central moment LBM. The geometry and boundary conditions for the simulation are shown in Fig. 2.8. For a step of height h, the flow entry is placed at L\ = 10h behind the step and the exit is located L-2 = 30h downstream of the step, and the channel height is defined as H = 2h. In this simulation, the number of nodes in resolving the step flow is defined by considering h = 94. At the entrance, a parabolic profile, and, at the outlet, a convective boundary condition are imposed, and, finally, the half-way bounce-back scheme is utilized for the no-slip boundary condition at the walls. The computational results are then presented for Reynolds numbers up to 800, where the Reynolds number is defined as Re = 2hl^ax. Here, Umax is the maximum speed at the inlet channel. For the purpose of investigating the flow behavior
Xr
* L 2=30h *
FIGURE 2.8: Schematic representation of the flow over a backward-facing step in a 2D channel.
in the vicinity of the step, the distributions of streamlines are plotted at four different Reynolds numbers in Fig. 2.9. Initially, a primary recirculation zone is created downstream of the step at Re = 100 (Fig. 2.9(a)). However, it can be seen from Fig. 2.9(a) to Fig. 2.9(d) that the Reynolds number has a remarkable effect on the structure recirculation regimes and the length of this zone is seen to increase by increasing the Reynolds number. Furthermore, a second recirculation zone occurs along the top wall at the higher Reynolds number of Re = 500 which becomes more vis-

56
ible at Re = 800. All these observed flow pattern are consistent with prior benchmark results.
In order to more precisely determine the quantitative effect of the Reynolds number on the reattachment length in the primary recirculation zone, our computed results based on the GI preconditioned cascaded central moment LBM for different Reynolds numbers are computed with the numerical results of [4], which are presented in Fig. 2.10. It can be observed that the agreement between the predictions based on our GI preconditioned LB scheme and the benchmark results is excellent. Moreover, it can be clearly seen that by increasing the Reynolds number, the reattachment length increased, consistent with prior observations.
2.6.4 Hartmann Flow
In this section, in order to validate our preconditioned scheme for a problem involving a body force, the Hartmann flow of an incompressible fluid bounded by two parallel plates is studied. An external uniform magnetic field Bz = Bo is applied perpendicular to the plates. Since the body force varies spatially arising due to the interaction of the flow velocity and the induced magnetic field, i.e. the Lorentz force, it represents appropriate test problem for the present study. In our preconditioned LB model, the moments of the source terms at different orders are preconditioned differently to correctly recover the macroscopic with variable body forces. The relationship between the external magnetic field Bq and an induced magnetic field Bx(z) across the channel is
given by Bx(z) = ^
sinh( Ha) L
where F& and L are driving force due to imposed pressure gradient and the half channel width, respectively, and Ha is the Hartmann number,which measures the ratio of the Lorentz force to viscous force.
The Lorentz force component is then defined as Fmx = Bq^j^. In consequence, the effective variable body force component is defined as Fx = Ff,+Fmx. The analytical solution for the Hartmann
flow has the following velocity profile ux{z) = /^coth(Ha)
cosh(Haf)
COSll(Ha)
, where rj is the
magnetic resistivity given by rj = Bq2L2/(Ra2u). Figure 2.11 presents comparisons of the com-

57
>
(a) Re=100
>1
(d) Re=800
FIGURE 2.9: Streamline contours for flow over a backward-facing step at (a) Re = 100, (b) Re = 300, (c) Re = 500, (d) Re = 800 computed using the GI preconditioned cascaded central moment LBM with 7 = 0.3.
FIGURE 2.10: Comparison of the reattachment length as a function of the Reynolds number Re computed using the GI preconditioned cascaded central moment LBM with 7 = 0.3 (symbols) against the benchmark results of [4].

58
puted velocity profiles using the GI preconditioned cascaded LBM with 7 = 0.1 and Mach number Ma = 0.02 against the exact, solution for various values of Ha. It can be observed that the GI preconditioned cascaded central moment LBM is able to reproduce the benchmark solution very well. In particular, as Ha is increased, the resulting higher magnitudes of the Lorent.z force causes significant flattering of the velocity profiles and this effect of Ha on the velocity profiles is represented by our preconditioned is model with very good accuracy.
2.5
2
S 1.5
3
1
0.5
z
x 10'
FIGURE 2.11: Comparison of the computed velocity profile using the preconditioned GI cascaded central moment LBM (7 = 0.1) with the analytical solution for Hartmann flow for various Ha at Ma = 0.02. The lines indicate analytical results, and the symbols are the solutions obtained by the GI preconditioned cascaded LBM.
2.6.5 Four-rolls Mill Flow Problem: Comparison between GI Corrected and Uncorrected Preconditioned Cascaded LBM
As seen in Sec. 2.3, the GI errors for the LBM on the standard, tensor product lattices, such as the D2Q9 lattice, are generally related to the strain rates in the principal directions (dxux and dyUy). Hence, in order to compare the GI corrected formulation (Sec. 2.5), which is constructed to eliminate such errors, with the uncorrected formulation (Sec. 2.2), we consider the four-rolls mill flow problem, which is characterized by local extensional/compression strain rates (i.e. dxux / 0,dyUy / 0), and for which a well-defined analytical solution is available. It is a

59
modified form of the classical Taylor-Green vortex flow driven by a local body force, whose components are given by
Fx(x,y) = 2uuo sin a; sin y, Fy(x,y) = 2uuo cos x cos y
in a periodic square domain of side length 27r (0 < x,y < 2ir), resulting in a steady vortical motion in the form of an array of counterrotating vortices. Here, v and uo are the kinematic viscosity and the velocity scale, respectively, and a unit reference density is considered. The analytical solution of the velocity field, which follows from a simplification of the Navier-Stokes equations impressed by the above body force, reads
ux(x, y) = uo sin x sin y, Fy(x, y) = uq cos a; cos y.
Clearly, the local flow field is subjected to local diagonal strain rates, i.e. dxux = â€”dyuy = uocosxsiny, and, as a result, the uncorrected LB scheme induces additional GI errors, which should be annihilated by the corrected LB method; and thus, the difference in the global flow fields against the analytical solution under a suitable norm in each case can be quantitatively studied and compared.
We performed computations on a square domain resolved by 251 x 251 grid nodes with a velocity scale uo = 0.045 for a Reynolds number Re = uqL/u, where L = 2tt, of 20. Figure 2.12 shows the streamline patterns at the steady state computed using the GI corrected preconditioned LB scheme (7 = 0.3), which manifest as a set of counterrotating vortices. The computed velocity profile uy(x,y = tt) obtained using the GI corrected LB scheme along the horizontal centerline of the domain presented in Fig. 2.13 are compared against the analytical solution given above, which show good agreement.
Furthermore, Fig. 2.14 presents a surface plot of the diagonal strain rate component dxux, which is seen to have a significant local variation, due to which quantitative differences in the solutions between the GI corrected and uncorrected preconditioned LB schemes can be expected, which will now be demonstrated in the following.

60
FIGURE 2.12: Steady state streamline patterns for the four-rolls mill flow problem at uq = 0.045 and Re = 20 computed using the GI corrected preconditioned cascaded LB scheme with 251 x 251 grid nodes and 7 = 0.3.
FIGURE 2.13: Comparison of the computed and analytical vertical velocity profiles uy(x) at y = 7T for the four-rolls mill flow problem at Re = 20 obtained using the GI corrected preconditioned cascaded LB scheme with 251 x 251 grid nodes, uq = 0.45 and 7 = 0.3.

61
FIGURE 2.14: Distribution of the diagonal strain rate component dxux = â€”dyuy for the four-rolls mill flow problem with uq = 0.045.

62
In order to make a quantitative comparison between the solutions obtained using the two differ-
ent LB methods, we first define the global relative errors for the velocity field 11 GRE^fz112 and 11 GRE^Z 112 between the components of the solution obtained using the GI corrected preconditioned LB scheme (i.e. (uc,vc)) and the analytical solution (i.e. (ua,va)) under a discrete Â£2 norm; and similarly 11GRE^112 and 11GRE^ 112 between the uncorrected preconditioned LB scheme (i.e. (Uuc, Vuc)) and the analytical solution. These are written as follows:
IIgre:
GI 11
u II2
'E(wc - Ua)2
EÂ«2
IIgre:
GI 1
'Uvc-vay
E v*
llpp-p II EKe Ua)2 llppp || /E(V'UC Va)2
11 â€lb v â€”inâ€”â€™ 11 "lb v â€”â€”â€™
where the summations in the above are carried out for the whole computational domain. Table 2.1 presents the above global relative errors for the velocity field components for both the preconditioned cascaded LB formulations for different values of the preconditioning parameter (7 = 0.2,0.3,0.4 and 0.5). It can be seen that significant improvements in accuracy is achieved by the GI corrected preconditioned LB scheme. In particular, the errors relative to the analytical solution are reduced by about a factor of two with the corrected preconditioned LB scheme for the conditions considered for the computation of this problem. Such improvements are consistent
with the fact that the corrected LB scheme eliminates the additional GI errors arising in this flow subjected to the local variations of the diagonal (compression/extension) strain rates, which are present in the uncorrected LB scheme.
TABLE 2.1: Comparison between the global relative errors in the computed solutions for the velocity field using the GI corrected preconditioned cascaded LB scheme and the uncorrected preconditioned cascaded LB scheme for the four-rolls mill flow problem at Re = 20, uq = 0.045 and a grid resolution of 251 x 251.
7 GI corrected u error ||GREf||2 Uncorrected u error ||GREu||2 GI corrected v error GRE(f7 2 Uncorrected v error ||GREâ€ž||2
0.2 3.386 x 10â€œ3 6.662 x 10-3 3.377 x 10â€œ3 6.665 x 10-3
0.3 1.850 x 10-3 4.104 x 10-3 1.854 x 10-3 4.126 x 10-3
0.4 1.384 x 10-3 2.851 x 10-3 1.389 x 10-3 2.865 x 10-3
0.5 1.135 x 10-3 2.113 x 10-3 1.140 x 10-3 2.123 x 10-3

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2.7 Summary and Conclusions
Lattice Boltzmann schemes on standard tensor product lattices can result in cubic-velocity errors in Galilean invariance (GI) as the third-order diagonal moments are not independently supported and degenerates to the first-order moments. Recent investigations have presented corrections to the collision operator to yield schemes free of these errors for the representation of the standard Navier-Stokes (NS) equations. Convergence acceleration of simulations of steady state flows can be achieved by solving the preconditioned NS equations involving a preconditioning parameter 7 to tune the pseudo-sound speed thereby alleviating the numerical stiffness. In our prior chapter, we devised a modified central moment based cascaded LBM to represent such preconditioned NS equations, which may be referred to as a specific example of an extended or generalized NS equa-

64
tions containing a free parameter, here the preconditioning parameter 7. In this chapter, we have presented a new preconditioned central moment based cascaded LB scheme that eliminates such non-GI cubic-velocity and parameter dependent errors for the simulation of steady state flows.
A detailed analysis based on the Chapman-Enskog expansion reveals the structure of the non-GI truncation errors that appear in the second-order non-equilibrium moment components, which are related to the viscous stress. Subsequently, we prescribe an extended second-order moment equilibria that restores GI free of cubic-velocity errors for the preconditioned LB model on the standard D2Q9 lattice. The following are among the main findings arising from our analysis:
â€¢ In general, the use of central moments in a LB scheme provides a natural setting to partially restore GI for the third-order off-diagonal moments. In particular, by setting the third-order central moment equilibria of the off-diagonal components to zero (e.g. l^xxy = 0), one naturally arrives at the precise forms of the corresponding raw moment equilibria (e.g. ^xxy = c2spv,y + pu2uy) that restores GI of such components in the representation
of the standard NS equations. On the other hand, in the preconditioned LB scheme, the cubic-velocity terms appearing in the third-order, off-diagonal moment equilibria needs to be scaled by y2 (e.g. Kxxy = c2puy + pu2uy/y2) to fully eliminate the spurious cubic-velocity cross-derivative terms (e.g. uxUydyUx, UyUxdxUy) appearing in the derivation of the preconditioned macroscopic equations.
â€¢ In order to effectively eliminate the non-GI, diagonal velocity gradient terms (e.g. u2dxux), the second-order, diagonal moment equilibria needs additional corrections in both the velocity and density gradients when 7 / 1, which are prescribed via extended moment equilibria. The velocity gradients can be locally and efficiently obtained using the nonequilibrium second order moment components; on the other hand, the density gradients can be computed using a finite-difference approximation.

Unlike that for the standard NS equations, the representation of the preconditioned NS

65
equations using a LB scheme results in additional, non-GI, cross-coupling velocity terms (e.g. Uydxux), which are also eliminated by our Gl-corrected preconditioned LB scheme.
â€¢ For the second-order, off-diagonal moment equilibria, additional gradient velocity correction terms are needed to restore GI for these components when 7 / 1. However, for incompressible flows (V â€¢ u = 0), they vanish regardless of the value of 7. Such a situation is unique to the representation of the preconditioned NS equations using LB schemes, as the non-GI corrections are generally restricted only to the diagonal components of the second-order equilibria for the representation of the standard NS equations.
â€¢ In general, the prefactors in GI defect terms exhibit dramatically different behaviors for
the asymptotic limit cases: For example, 7â€”^1 (No preconditioning):^^ â€” ~ 3 and
7 â€”> 0 (Strong preconditioning):^^ â€” yj ~
â€¢ When 7 = 1, i.e. when the present LB model is used to simulate flows represented by standard NS equations as a special case, all our results for the GI defect terms and corrections become identical with those derived by [42] and [43].
â€¢ Finally, the results of our present analysis can be extended to three-dimensions (e.g. D3Q27 lattice) and other collision models for the simulation of the preconditioned NS equations.
In addition, we have presented numerical validation of our new GI preconditioned LB scheme based on central moments against several complex flow benchmark problems including the lid-driven cavity flow, flow over a square cylinder, the backward facing step flow, the Hartmann flow and the four-roll mills flor problem. Comparison against prior numerical solutions show good agreement for the modified preconditioned scheme. In addition, it is demonstrated that our GI corrected preconditioned cascaded LB scheme results in significant convergence acceleration of complex flow simulations, and a quantitative improvement in accuracy when compared to the

66
uncorrected preconditioned LB scheme. Finally, it may be noted that our analysis of non-GI aspects for the preconditioned LB scheme has implications for LB schemes for other situations such as the porous media flows. For example, there is a formal analogy between the preconditioned NS equations and the Brinkman-Forchheimer-Darcy equations, where the porosity serves as a free parameter (e.g. [64, 65]). LB models constructed for such flows (e.g. [66]) can be further improved by the approach presented in this chapter.

67
CHAPTER III
SYMMETRIZED OPERATOR SPLIT SCHEMES FOR FORCE AND SOURCE MODELING IN CASCADED LATTICE BOLTZMANN METHODS FOR FLOW AND SCALAR TRANSPORT
3.1 Introduction
The lattice Boltzmann (LB) method is now a well established alternative numerical technique to computational fluid dynamics (CFD) problems. The collision step, which represents various physics associated with the fluid motion including the momentum diffusion as a relaxation process, plays a main role in the numerical stability of the method. Among the earliest collision models is the single relaxation time (SRT) model [67], which, while being popular due to its simplicity, is susceptible to numerical instability at relatively high Reynolds numbers. A significant improvement is achieved by the multiple relaxation time model (MRT) [14] in which different raw moments relax at different rates. More recently, further enhancement in stability was made possible by the introduction of a cascaded LB method, which is a multi-parametric scheme that is based on considering relaxation in terms of central moments, which are formulated by shifting the particle velocity by the local fluid velocity [15, 19]. The significant advantages of such more advanced collision models were numerically demonstrated more recently [43]. A strategy to accelerate the convergence of the cascaded LB method has also been devised and studied [58], which has been further extended with improved Galilean invariance properties [68].
Another aspect of the LB schemes, which is particularly important in applications, is the implementation strategy to represent the various impressed body forces, which can either arise within the fluids or imposed externally. Some examples include the local surface tension and phase segregation forces in multiphase fluid systems, Lorentz forces in magnetohydrodynamics, gravity and

68
Coriolis forces. In general, such body forces can be spatially varying and/or time dependent. Due to the kinetic nature of the LB method, special considerations are necessary and various forcing schemes have been introduced over the years ([69], [23], [70], [71], [72], [73]). In particular, the investigation by [71] highlighted the discrete effects arising in prior LB forcing schemes via the second order moments in the momentum flux tensor, and provided a consistent source term that avoids such spurious effects when used with the SRT collision model. This was further generalized to the MRT model by including source terms in the moment space in both two-dimensions (2D) and three-dimensions (3D) ([74], [75], [7]).
In the case of the cascaded LB method, the first consistent forcing scheme based on the central moments was presented by [19]. By taking the source term proposed by [76] as the starting point, they devised a forcing formulation without discrete effects, which was also shown to be a further generalization of that presented by [71] to the cascaded LB scheme under appropriate limits. Later, [77] constructed another type of forcing scheme for the cascaded LB method based on the exact difference method [73]. More recently, [78], [79] and [80] presented other variants of forcing schemes for LB methods based on central moments. While all these forcing schemes differ from one another due to the variations in the kinetic models for the source term, a common element among them is the presence of extra source terms or changes to the equilibria, which are usually taken together with the collision relaxation terms as part of the collision step. This generally involves computing source moments at different orders and transforming them back to the velocity space, which entails additional computational effort.
Based on the consideration that the LB schemes are generally fluid flow, i.e. Navier-Stokes (NS), solvers, and by avoiding the kinetic aspects for the implementation of the impressed forces, simpler and more efficient strategies can be constructed. The numerical framework for this is the operator splitting approach widely used to efficiently solve ordinary and partial differential equations arising in various applications including CFD ([81], [82]). The basis idea is to split the problem into a set of simpler subproblems and then devise a strategy that alternates between solv-

69
ing such simpler problems in certain sequence, which then approximate the solution to the full problem to a certain order of accuracy. Such operator splitting techniques are sometimes also referred to as the fractional step or time-splitting methods. Of particular importance is the Strang spliting [83], which achieves second-order accuracy by a symmetrized application of the solution method for one (or more) of the subproblems. The structure of the higher order splitting errors can be analyzed via the Taylor-Lie series [81] or using the Baker-Compbell-Hausdorff formulas [84]. From such a perspective, Dellar [85] presented a derivation of the lattice Boltzmann method based on Strang splitting with second order accuracy and interpreted both unsplit and time-split forcing schemes based on this approach. In particular, a uncoupled spin-step to implement body force in a SRT LB model introduced earlier by Salmon [86] was shown to be consistent with the Strang splitting. Furthermore, it was also extended to the MRT-LB models ([85], [87]).
In this chapter, our goal is to construct efficient body force implementation schemes based on the symmetrized operator (Strang) splitting for the cascaded LB methods. The lattice symmetry and the use of central moments naturally impose Galilean invariance for the chosen set of independent moments basis. The symmetric application of the separate body force steps in two half time steps in the cascaded LB formulation provides a second order accuracy. Unlike the unsplit forcing schemes presented earlier for the cascaded LB method [19], our approach does not require either the computation of various source moments at different orders or an extra transformation step to convert them back to velocity space. In essence, the operator-split forcing scheme involves one half application of the force before collision and the other half force step after collision. The latter step will be seen to lead to unique expressions for the post-collision change of first order moments in the cascaded collision operator. The precise structure of these expressions will be shown to depend on choice of the first order moment basis vectors associated with the type of lattice considered. In fact, we will present operator split forcing scheme for the cascaded LB method both in 2D and 3D for the computation of the fluid motion. In addition, in order to demonstrate the generality of our approach, we will extend it to represent the convective-diffusion equation

70
(CDE) with a source term, such as those arising in the convective thermal flows with internal heat generation. In this regard, a novel cascaded LB formulation for the solution of the CDE with source term using the Strang splitting will be constructed. Finally, we will present a numerical validation study of the symmetrized operator split forcing/source schemes for the cascaded LB method for fluid flow (i.e., the NS equations) and passive scalar transport (i.e., the CDE) and in different dimensions.
This chapter is organized as follows. In the next section (Sec. 2), we briefly review the various operator splitting approaches including the Strang splitting. Section 3 presents the general ideas behind the symmetrized operator splitting based forcing implementation in the LB method. Section 4 discusses the derivation and the algorithmic procedure of the symmetrized operator split forcing scheme for the 2D cascaded LB method for representing fluid flow subjected to local impressed forces. A corresponding 3D formulation is outlined in the Appendix A. Section 5 presents a symmetrized operator split approach source incorporation scheme for a 2D cascaded LB scheme for representing the convection-diffusion based transport of a passive scalar field with local sources. Numerical validation results of various symmetrized operator split forcing/source scheme are presented in Sec. 6. Finally, Sec. 7 summarizes our approach and presents the main conclusion arising from this work.
3.2 Operator Splitting Methods
We will now briefly review the various typical operator splitting methods, including the Strang splitting which will then be exploited to construct efficient second order accurate forcing schemes in the cascaded LB method. For the purpose of illustration, we will consider the numerical solutions of the following evolution problem:
dy
dt
Py + Qy, y(t) = y0 on [t, t + At]
(3.1)

71
where, for ease of presentation, P and Q are considered as linear operators. Nonlinear operators can be dealt with using Lie operator formalism [84]. Here, At is the time step. For reference, the unsplit solution yu of the full problem can be represented as
yU =eAt(P+Q)yo_ (3.2)
Now, a first order splitting scheme, which is sometimes known as the Lie-Trotter (LT) splitting or as the Godunov splitting scheme in the CFD literature, can be represented by means of the following steps, which compute solution to each subproblem involving P and Q separately:
du*
StepP: Solve â€”^ = Py*, y*{t' = t)=yo on [t, t + At] , (3.3a)
rill**
StepQ : Solve = Qy**, y**(t' = t) = y*{t + At) on[t,t + At], (3.3b)
Solution: yLT (t + At) = y** (t + At). (3.3c)
This solution of the Lie-Trotter splitting or the P-Q splitting scheme may be more compactly represented by means of the exponential operators as
yLT(t + At) = eAtQeAtPy0. (3.4)
The local error (Ei) incurred over a small time step At due to splitting when compared to the unsplit solution (Eq. (3.2)) can be estimated by means of a Lie-Taylor series (factored product expansions) as [81]
Ei,lt = yLT -yu = 7, [p> Q] yoAt2 + o(At3), (3.5)
where the symbol [â€¢, â€¢] represents the commutator, i.e., [X, Y] = XY â€” YX for any two operators X and Y. Then, the global error (Eg) over a time duration T or T/At number of steps is EgtLT = {T/At) â–  Ei}LT ~ O(At), which means that the Lie-Trotter scheme is first order accurate. This means that even if a higher order method is used to solve each subproblem (Step P and StepQ), the above splitting scheme is still overall first order accurate due to the decomposition error arising from the non-commuting operators, which is often the case in practice.

72
One possibility to improve the order of accuracy is to symmetrize the computation via taking the average of the two sequences of calculations, i.e. Step P - Step Q and Step Q - Step P results. Such an averaged scheme may be represented as [88]
yA= l(eAtPeAtQ+eAtQeAtR)?/o_ (3.6)
This approach introduces a local error relative to the unsplit solution (Eq. (3.2)), which can be written as [89]
Ei,a = yA - yu = R'At3 + 0(At4),
where
R, = _^([P,[P,Q]] + [Q,[Q,P]])2/o-
Hence, the global error becomes E9)A = (T/ At) â€¢ Ei>a ~ 0(At2). While this is theoretically interesting to gain an order of accuracy, it is computationally expensive as, for each time step, double the effort is required when compared to the previous scheme (P â€” Q splitting).
A more efficient strategy to achieve a global second order accuracy is to devise the Strang (S) splitting [83]. In this scheme, one of the operators (say P) is applied twice for a time step of length At/2, before and after the solution of the other subproblem (say, involving StepQ), which is solved for full step length of At. This may be represented as
StepP1/2: Solve ^-= Py*, y*(t' = t)=yo on [t, t + At/2], (3.7a)
StepQ: Solve = Qy**, y**{t' = t) = y*{t + A/2) on[t,t + At], (3.7b)
StepP1/2: Solve = Py***, y***(t1 = t) = y**(t + At) on [t, t + At/2p.7c)
Solution: ys(t + At) = y***(t + At/2). (3.7d)
This symmetric application of the operators in the P1/2 â€” Q â€” P1/2 scheme achieves second order accuracy, which may be deduced by first noting that the Strang splitting solution may be more compactly written in the exponential form as
ys(t + At) = eAt/2PeAtQeAt/2Py0.
(3.8)

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Its local error when compared to the unsplit solution (Eq. (3.2)) then follows via a Lie-Taylor series as [82]
Ei,s = ys -yu = RAf3 + 0(At4), (3.9)
where
R = ^([[P,Q],P]+2[[P,Q],Q])2/o. (3-10)
Then, the global error (Eg) over a time period T follows as = (T/Af) â€¢ EiyS 0(At2) and hence this scheme is second order accurate. An equally valid possibility to achieve a similar second order accuracy is to consider the Q1/2 â€” P â€” Q1/2 splitting, which is useful when StepP is more expensive to compute than StepQ. It may be noted that a similar scheme was independently devised by [90], who further analyzed and elaborated on its variants (see also [91]), and hence it is sometimes referred to as the Strang-Marchuk splitting scheme.
3.3 Strang Splitting of Lattice Boltzmann Method Including Body Forces
Lattice Boltzmann (LB) schemes are generally constructed to represent the evolution of the dynamics of the fluid motion represented by
dtp + V â€¢ (pu) = 0, (3.11a)
dt(pu) + V â€¢ (puu) = â€”VP + V â€¢ nv + F, (3.11b)
where p and u are the fluid density and velocity, respectively, P is the pressure and nv is the viscous stress tensor. Here, F represents the effect of the local impressed body forces, which can vary spatially and may be time dependent, i.e. for e.g. in 2D, F = (Fx,Fy) where Fx = Fx(x,t) and Fy = Fy(x,t) . An efficient approach to solve the above fluid flow equation in the LB framework is to solve the Eqs. (3.11a) and (3.11b), but without the body force F using the usual stream and collide procedure (subproblem A) and then separately solve dt(pu) = F as a forcing step (subproblem B) and subsequently combined appropriately in a certain sequence to yield a second order accurate scheme. This can be achieved via symmetrization of the operator

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splitting of the one of the subproblems over two half time steps. Dellar [85] performed a derivation and analysis of the LB method via Strang splitting, which will be used as formal starting point to construct efficient operator split forcing schemes for the cascaded LB method in the subsequent sections.
In the following, S, C and F are used to denote the operators used to perform the streaming step, collision step and the forcing step, respectively. For a lattice containing a = 0,1,2,... b directions, the collision and streaming steps can be represented as
StepC: i(x,t +At) = Ci(x,t) = i(x,t)+ K-g, (3.12a)
StepS: fa(x,t +At) = Sfa(x,t) = fa(x - eaAt,t). (3.12b)
Here, f = (/o, /i, /2 â€¢ â€¢ â€¢ fbV is a vector of size (6 + 1) representing the distribution functions, where f is the transpose operator, g = (<7o, <7i, <72 â€¢ â€¢ â€¢ <7is the vector representing the change of different moments under collision, and K is the transformation matrix of the cascaded LB method that maps changes in moments back to changes in the distribution functions, which are specified later.
It may be noted that C and S operators represent the split solution operators of the discrete analog of dtfa = and dtfa + ^a-^7fa = 0, respectively, of the discrete velocity Boltzmann equation dtfa + â€¢ Vfa = Q,a, whose emergent behavior represents the NS equations given in Eq. (3.11a)
and Eq. (3.11b), but without F. Then, the forcing step separately solves the following:
Step F : J^(pw) = F. (3.13)
One possibility to combine the above split steps to effectively achieve second order accuracy is to perform a symmetric application of the forcing steps over two half time steps, before and after the collision step, which is akin to the spin steps for the force presented by Salmon ([86]):
fa{x, t + At) = S F1/2 C Fl/2fa{x, t)
(3.14)

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where F1/2 represents performing the solution of Eq. (3.13) over time step of length At/2. Ref. [85] showed that this achieves second order accuracy similar to the Strang splitting extended to three operators: f'a(x,t + At) = C1/2F1/2 S F1/2C1/2f'a(x, t), where the two are related by f'a = C1/2Fi!2fa . Since the momentum is conserved during collisions, a second order scheme with Eq. (3.14) can be obtained by pu = J2a f Â«eÂ« = /Â«eÂ«)- We will adopt the above
strategy in our derivation of the symmetrized operator split forcing scheme for the cascaded LB method in the subsequent sections. Similar approach was recently adopted for the MRT LB models (e.g., [87]). In addition, Schiller [92] proposed a variant of the Strang splitting of forcing steps around streaming and collisions, where the half collision step is valid for the regime involving the relaxation time being much greater than the time step. Also, Dellar [42] showed that the Crank-Nicolson solution of the moment equations for combined collisions and time-independent forcing obtained by Strang splitting is equivalent to Kupershtokhâ€™s exact difference method [73].
3.4 Body Force Scheme for 2D Cascaded LB Method for Fluid Flow via Strang Splitting
We will consider a 2D cascaded LB formulation for a two-dimensional, nine velocity (D2Q9) lattice. The components of the particle velocities are then represented by the following vectors using the standard Diracâ€™s bra-ket notation:
Ie*) = (0,1,0, -1,0,1, -1, -1, l)f, (3.15a)
\ey) = (0,0,1,0, â€”1,1,1, â€”1, â€”l)f. (3.15b)
Their components for any particle velocity direction a (where a = 0,..., 8) are referred to as eax and eayi respectively. Furthermore, we need the following 9-dimensional vector:
|1) = (1,1,1,1,1,1,1,1,1)+. (3.16)
The zeroth moment is the Euclidean inner product of this vector with the distribution function.
We then consider the following specific set of orthogonal basis vectors used in the collision term

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of the cascaded LB method (e.g., [19]):
|iL0) = |l), \K\) = \ex), \K2) = \ey), |iL3> = 3 |e2 + e2} - 4 |1>,
\KA) = \e2x - e2y), \K5) = \exey), \K&) = -3\e2xey) + 2\ey),
\Kj) = â€”3 \exe2) + 2 le^), |Kg) = 9 |e2e2) â€” 6 |e2 + e2) + 4 |1). (3-17)
In the above, symbol such as \e2ey) = \exexey) represents a vector resulting from the elementwise vector multiplication (Hadamard product) of the sequence of vectors le^}, le^) and \ey). By combining the above 9 vectors, we then obtain the following orthogonal matrix
K = [\K0), \K\), |K2) , |K3), |K4), |K5), |K6), |K7) , |K8)] . (3.18)
Here, K maps changes of moments under collisions back to changes in the distribution functions. In order to determine the structure of the cascaded collision operator, we first define the following set of central moments of the distribution functions and its equilibria of order (m+n), respectively, as
\ K>Xrnyn = Â£ ( f \
keq 'lxrnyn / a y faq )
(6ax 1l>x) (,6-ay ^y)
(3.19)
By equating the discrete central moments of the equilibrium distribution function with the corresponding continuous central moments based on the local Maxwellian ([15], [19], [20]), we get
"-eg ""eg n ""eg n "-eg o -~-eg 2
/s)q â€” pj h'jX â€” 0, h'jy â€” 0, Hxx â€” CsP? ^yy â€” cspj
Kxy = 0, nScy = 0, KeXyy = 0, KxLyy = C4sp. (3.20)
where c2 = 1/3 with cs being the sound speed. This is set by applying the usual lattice units, i.e. Ax = At = 1 or the particle speed c = Ax/At = 1, and because c2 = c2/3 for the athermal LB scheme used in this work (see e.g. [18]). On the other hand, the actual computations in the cascaded formulations are carried out in terms of raw moments, which are defined as (designated here with the (') symbol)
(â– ' \ K>Xrnyn / = Â£ (f \ Ja
" eg y Kxrnyn y a y faq ,
m n ^ax^ay'
(3.21)

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The collide and stream steps (C and S) of the 2D cascaded LB method can then be, respectively, written as [15, 19]
Step C : /Â« = /Â« + (K â€¢ g)Â«
StepS: fa(x,t) = fK(x - eaAt,t),
(3.22a)
(3.22b)
where /Â« represents the post-collision distribution function and g = (go, <7i, <72 â–  â€¢ â€¢ <7s)^ is the change of different moments under collisions, which is determined based on the relaxation of various central moments to their corresponding equilibria in a cascaded fashion [15]. Since the mass and momentum are collision invariants, <70 = <7i = <72 = 0. As a result, the cascaded structure starts from the non-conserved second order moments, and the corresponding components of the change of different moments under collisions are given by
h
94
95
96
97 9s
12 y 3 ^ ^ ^x '
^ â€™U'y) {/7X^ <7
W5
â€™yy
^ ypuxuy nxyj ,
UJq f o
^ j^2puxUy T nxxy 2uxnxy
(jjj f 9
- â€” <^2pUXUy + Hxyy ~ 2 UyH^ ~ U X K yy
2uxnxy uynxx| 2 uy(^93 +94) 2
- 7)UX(3g3 - 94) - 2'
2
^xxyy ^^x^xyy ^^y^xxy ^x^yy
j 1 0 2 2
^ | gP ^>pUxUy
+4uxuyKXy | â€” 2gs â€” (3<73 + <74) â€” -v%(3g3 â€” <74)
^uxuyg5 2uyga 2uxgj.
(3.23)
where U3, uj4,... ,ujs are the relaxation parameters. These relaxation steps lead to the following expressions for the bulk and shear viscosities, respectively, as ( = |(^ â€” ^)At and v = |(A- â€” |)At where j = 4, 5, and the pressure field P is obtained via an equation of state as P = |p.
After the streaming step, i.e., Eq. (3.22b), we obtain the output velocity field components (desig-

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nated with a superscript â€oâ€) as the first moment of fa:
PuÂ°x = EÂ®=o Ueax, puÂ°y = ELo /Â«eÂ«w (3-24)
We then introduce the effect of the body force F = (Fx, Fy) as a solution of the subproblem in Eq. (3.13). This is accomplished by performing two symmetric steps of half time steps of length At/2, one before and the other after the collision step. Both these steps incorporate the effect of forces directly into the moment space. Solving Eq. (3.13) for the first part of the symmetric sequence of step yields pux â€” puÂ° = Fx^ and puy â€” puÂ° = Fy^. Thus,
Pre-collision Forcing StepF1/2 : ux = - ^pux + ^-At^j , uy = - ^puy + . (3.25)
Then, we use this updated velocity field (ux,uy) in Eq. (3.23) to perform the cascaded relaxation collision step to determine the change of different moments under collisions, i.e. 'gp, fi =
3,4,..., 8. As a result of correctly projecting the effect of the forces in the various higher order moments, it naturally eliminates the discrete effects identified earlier [71] (see the discussion at the end of this section). Then, to implement the other part of the symmetrized force step with half step to solve Eq. (3.13) post collision, we set pvFx â€” ux = Fx^ and pv^ â€” uy = Fy^ , where (ux,uy) is the result of the target velocity field due to the forcing step after collision. Thus,
F F
Post-collision Forcing Step F1/2 : pvPx = pux + At, pvPy = puy + yAt. (3.26)
Note that this can also be rewritten in terms of the output velocity field uÂ° = (uÂ°, uÂ°) given in Eq. (3.24) by using Eq. (3.25) as
pFPx = puÂ°x + FxAt, pFPy = puÂ° + FyAt. (3.27)
A main issue here is how to effectively design the post-collision distribution function /Â« in the cascaded LB method so that Eq. (3.27) is precisely satisfied. Now, using /Â« = /Â« + (K â€¢ g)a and taking its first moments, we get
pUx = Pjafa&ax = Eq, fa&ax T Fp {Kp | e^,) C)p, (3.28a)
pUy â€” Pjafa&ay â€” P^ocfoc&oey T Fp {Kp\cy')Qp. (3.28b)

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Based on the orthogonal basis vectors |Kp) given in Eq. (3.17), it follows that
'Zfi{Kfi\ex)gfi = &ji, Yip{Kp\ey)gp = 6 g2. (3.29)
Using Eqs. (3.24) and (3.29) in Eqs. (3.28a) and (3.28b) we, get the desired velocity field as
puP = puÂ° + 6gi, pmPy = puÂ° + 6)72. (3.30)
Comparing the result of the target velocity field following the second half of the symmetrized forcing steps given in Eq. (3.27) with the change of moments based expressions in Eq. (3.30), we obtain
gi = ^-At, g2 = ^At. (3.31)
o o
Equation (3.31) represents an algorithmic result that effectively implements the effect of the post-collision forcing step over a duration of half time step following collision. This is a consequence of the momentum needing to change by FAt over a time step, and the normalization is implied by our choice of basis for the moments. Then, the above relation (Eq. (3.31)) for the post-collision change of first moments due to the force field {cj\ and <72) along with the change of different higher moments under collisions <7)3, where (3 = 3, 4,..., 8, given in Eq. (3.23) effectively provide the desired post-collision states of the distribution function /Â«. Expanding Eq. (3.22a),

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ii ThisthesisfortheDoctorofPhilosophydegreeby FarzanehHajabdollahiOuderji hasbeenapprovedforthe EngineeringandAppliedScienceProgram by SamuelWelch,Chair KannanPremnath,Advisor PeterJenkins SedatBiringen TaehunLee Date:May18,2019

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xi COMPRESSIBLEMULTIPHASEFLOWS199 7.1Introduction.........................................199 7.2GoverningMacroscopicEquations:InterfaceCapturingandTwo-PhaseFluidMotion203 7.3ModiedContinuousBoltzmannEquationforTwo-PhaseFlowsandCentralMomentsofEquilibriaandSources..............................206 7.3.1ContinuousCentralMomentsofEquilibriaandSourcesofMCBE.......208 7.4CascadedLBMethodforSolutionofTwo-PhaseFluidMotion............211 7.5CascadedLBMethodforSolutionofPhase-FieldbasedInterfacialDynamics....217 7.6ResultsandDiscussion..................................220 7.6.1Evolutionofacircularinterfaceinimposedshearow.............221 7.6.2Laplace-Youngrelationofastaticdrop.....................221 7.6.3Rayleigh-Taylorinstability.............................225 7.6.4Fallingdropundergravity............................227 7.6.5Buoyancy-drivenrisingbubble..........................228 7.6.6Impactofadroponathinliquidlayer.....................231 7.7Comparativestudyofnumericalstabilityofdierentcollisionmodels.........234 7.8SummaryandConclusions.................................238 VIII SUMMARY,CONCLUSIONSANDOUTLOOK240 BIBLIOGRAPHY 247 APPENDIX A 265 1.1StrangSplittingImplementationofBodyForcesin3DCentralMomentLBMethod265 B 271

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xiii LISTOFTABLES 2.1Comparisonbetweentheglobalrelativeerrorsinthecomputedsolutionsforthe velocityeldusingtheGIcorrectedpreconditionedcascadedLBschemeandthe uncorrectedpreconditionedcascadedLBschemeforthefour-rollsmillowproblematRe=20, u 0 =0 : 045andagridresolutionof251 251..............62 3.1Relativeerrorbetweenthenumericalresultsobtainedusingthe2Dsymmetrized operatorsplitcascadedLBsourceschemeforapassivescalartransportandthe analyticalsolutionforthesimulationofthethermalCouetteowatvariousEckert numbersEc..........................................102 4.1Qualitativecomparisonofkeyowandthermalcharacteristicsinnaturalconvectioninacubiccavityinthesymmetryplanez=0.5betweenthe3Dcascaded LBMandthereferencebenchmarkresultsobtainedusingaNSEsolver[1]......126 5.1GridconvergencestudygivenintermsoftheaverageNusseltnumber Nu for Ra = 10 4 fornaturalconvectioninacylindricalannuluscomputedusingaxisymmetric cascadedLBschemes....................................154 5.2ComparisonoftheaverageNusseltnumber Nu fordierent Ra fornaturalconvectioninacylindricalannuluscomputedusingaxisymmetriccascadedLBschemes withotherreferencenumericalsolutionsandnewresultsfor Ra =10 6 and10 7 ....156 5.3Comparisonofthemeanequivalentthermalconductivityattheinnercylinderin aslenderverticalcylindricalannulusduringmixedconvectionfor Re =100 ;Pr = 0 : 7 ;R io =2 ; =10atdierentvaluesof ........................164 5.4Comparisonofthemaximumvalueofthestreamfunction max computedusing theaxisymmetriccascadedLBschemeswithreferencenumericalsolutionsforthe Wheeler'sbenchmarkproblem...............................168

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xiv LISTOFFIGURES 1.1Two-dimensional,nine-velocityD2Q9lattice......................5 2.1Comparisonofthecomputedhorizontalvelocity u=U p andverticalvelocity v=U p prolesalongthegeometriccenterlinesofthecavityusingtheGalileaninvariant preconditionedcascadedcentralmomentLBMwiththebenchmarkresultsof[2] symbolsforRe=3200and5000and =0 : 1.......................49 2.2ConvergencehistoriesoftheGIpreconditionedcascadedcentralmomentLBMand thestandardcascadedLBM =1forlid-drivencavityowforRe=3200......49 2.3Schematicrepresentationoftheowoverasquarecylinderina2Dchannel......50 2.4StreamfunctioncontoursforowoverasquarecylinderforfourdierentReynolds numbers;Re=1,Re=15andRe=30usingtheGIpreconditionedcascadedcentral momentLBMwith =0 : 5.................................51 2.5Comparisonofthecomputedvelocityprolesalongandacrossthesquarecylinder alongitscenterlineforboththehorizontal u andvertical v velocitycomponents obtainedusingtheGIpreconditionedcascadedcentralmomentLBMwith = 0 : 5for Re =100withbenchmarkresultsobtainedusingtheGasKineticScheme GKS[3]...........................................53 2.6ComparisonofthecomputedReynoldsnumberdependenceoftherecirculating wakelength L r ontheleftaandtheReynoldsnumberdependenceofthedrag coecient C D ontherightwithbbenchmarkcorrelationEq..63andGKSbasednumericalresults[3]respectively..........................54 2.7ConvergencehistoriesoftheGIpreconditionedcascadedcentralmomentLBMand thestandardcascadedLBM =1forowoverthesquarecylinderforRe=30...54 2.8Schematicrepresentationoftheowoverabackward-facingstepina2Dchannel..55

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xv 2.9Streamlinecontoursforowoverabackward-facingstepataRe=100,bRe =300,cRe=500,dRe=800computedusingtheGIpreconditionedcascadedcentralmomentLBMwith =0 : 3.........................57 2.10ComparisonofthereattachmentlengthasafunctionoftheReynoldsnumber Re computedusingtheGIpreconditionedcascadedcentralmomentLBMwith = 0 : 3symbolsagainstthebenchmarkresultsof[4]....................57 2.11ComparisonofthecomputedvelocityproleusingthepreconditionedGIcascaded centralmomentLBM =0.1withtheanalyticalsolutionforHartmannowfor variousHaatMa=0 : 02.Thelinesindicateanalyticalresults,andthesymbolsare thesolutionsobtainedbytheGIpreconditionedcascadedLBM.............58 2.12Steadystatestreamlinepatternsforthefour-rollsmillowproblemat u 0 =0 : 045 andRe=20computedusingtheGIcorrectedpreconditionedcascadedLBscheme with251 251gridnodesand =0 : 3...........................60 2.13Comparisonofthecomputedandanalyticalverticalvelocityproles u y x at y = forthefour-rollsmillowproblematRe=20obtainedusingtheGIcorrected preconditionedcascadedLBschemewith251 251gridnodes, u 0 =0 : 45and =0 : 3.60 2.14Distributionofthediagonalstrainratecomponent @ x u x = )]TJ/F35 10.9091 Tf 8.485 0 Td [(@ y u y forthefour-rolls millowproblemwith u 0 =0 : 045.............................61 3.1Comparisonofthecomputedvelocityprolesusingthe2Dsymmetrizedoperator splitcascadedLBforcingschemewiththeanalyticalsolutionforPoiseuilleowfor bodyforcemagnitudesof10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(7 and10 )]TJ/F33 7.9701 Tf 6.586 0 Td [(8 .Thelinesindicatetheanalyticalresults, andthesymbolsarethesolutionsobtainedbyourpresentnumericalscheme.....90 3.2Gridconvergencefor2DPoiseuilleowwithaconstantReynoldsnumber Re = 100andrelaxationtime =0 : 55computedusingthe2Dsymmetrizedoperator cascadedLBforcingscheme.................................91

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xvi 3.3Comparisonofthecomputedvelocityprolesusingthe2Dsymmetrizedoperator splitcascadedLBforcingschemewiththeanalyticalsolutionforHartmannow forHartmannnumbersHaof3and10.Thelinesindicatetheanalyticalresults, andthesymbolsarethesolutionsobtainedbyourpresentnumericalscheme.....93 3.4Comparisonofcomputedandanalyticalvelocityprolesatdierentinstantswithin atimeperiodofpulsatileowattwodierentWomersleynumbersofWo=4 Wo=10.7.Here,linesrepresenttheanalyticalsolutionandsymbolsrefertothe numericalresultsobtainedusingthe2DsymmetrizedoperatorsplitcascadedLB forcingscheme........................................95 3.5Comparisonofthecomputedvelocityprolesusingthe3DsymmetrizedoperatorsplitcascadedLBforcingschemeandtheanalyticalsolution,forowthrough asquareductinpresenceofabodyforcemagnitudeof F x =10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(7 fordierent valuesof y .Here,linesrepresenttheanalyticalsolutionandsymbolsrefertothe resultsobtainedusingthepresentnumericalscheme...................97 3.6Comparisonofthecomputedandanalyticalverticalvelocityproles u y x at y = forthefour-rollsmillowproblemat u 0 =0 : 01, =0 : 0011and N =96.Here, linerepresentstheanalyticalsolutionandthesymbolreferstothenumericalresultsobtainedusingthe2DsymmetrizedoperatorsplitcascadedLBforcingscheme.99 3.7Streamlinesacomputedusingthe2DsymmetrizedoperatorsplitcascadedLB forcingschemeandbobtainedusingtheanalyticalsolutionforthefour-rollsmill owproblemat u 0 =0 : 01, =0 : 0011and N =96....................99 3.8Gridconvergenceforthefour-rollsmillowproblemat u 0 =0 : 01, =0 : 0011 computedusingthe2DsymmetrizedoperatorsplitcascadedLBforcingscheme undertheconvectivescaling................................100

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xvii 3.9Comparisonbetweennumericalresultsofthetemperatureprolecomputedusingthe2DsymmetrizedoperatorsplitcascadedLBsourceschemeforapassive scalartransportandtheanalyticalsolutionforthethermalCouetteowforvariousvaluesoftheEckertnumberEc.Here,linesrepresenttheanalyticalsolution andsymbolsrefertotheresultsobtainedusingthepresentnumericalscheme.....102 4.1Geometriccongurationforthephysicalmodelofthe3Dcubiccavityandthecoordinatesystem.modelof3Dcavitywithcoordinatesystem..............121 4.2ComparisonofthetemperaturetopandvelocityprolesbottomforRayleigh number Ra =10 5 onthesymmetrycenterplane x )]TJ/F35 10.9091 Tf 11.501 0 Td [(z ;symbols" "denotethe referencebenchmarksolutions[1],andline" )]TJ/F15 10.9091 Tf 10.909 0 Td [("bypresentwork...........123 4.3Projectionsofstreamlinesinnaturalconvectionina3Dcavitycomputedusing3D cascadedLBMondierentcenterplanesatRayleighnumber Ra =10 4 leftand Ra =10 5 right.Toprow: y )]TJ/F35 10.9091 Tf 10.604 0 Td [(z plane,Middlerow x )]TJ/F35 10.9091 Tf 10.603 0 Td [(z planeBottomraw: y )]TJ/F35 10.9091 Tf 10.604 0 Td [(x plane.............................................124 4.4Temperaturedistributioninnaturalconvectionina3Dcavitycomputedusing3D cascadedLBMondierentcenterplanesatRayleighnumbers Ra =10 4 leftand Ra =10 5 right.Toprow: y )]TJ/F35 10.9091 Tf 10.604 0 Td [(z plane,Middlerow x )]TJ/F35 10.9091 Tf 10.603 0 Td [(z planeBottomrow: y )]TJ/F35 10.9091 Tf 10.604 0 Td [(x plane.............................................125 5.1Comparisonbetweentheanalyticalvelocityprolesolidlinesandthecascaded LBsolutionsymbolsfortheTaylor-Couetteowbetweentwocircularcylinders atanangularvelocityratio =0 : 1andforvariousvaluesoftheradiusratio ...149 5.2EvaluationoforderofaccuracyforTaylor-CouetteowwithaconstantReynolds numberRe=5,radiusratio =1 = 3andrelaxationtime =0 : 6atdierentgrid resolutionscomputedusingtheaxisymmetriccascadedLBscheme...........150 5.3Schematicillustrationofthegeometryandboundaryconditionsfornaturalconvectioninaverticalannulus................................151

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xviii 5.4Streamlinesandisothermsforthenaturalconvectionbetweentwoco-axialvertical cylindersat Pr =0 : 7anda,d Ra =10 3 ; b,e Ra =10 4 andc,f Ra =10 5 computedusingcascadedLBschemes.Toprowpresentsstreamlinesandthebottom rowtheisotherms......................................153 5.5Streamlinesandisothermsforthenaturalconvectionbetweentwoco-axialverticalcylindersat Pr =0 : 7anda,c Ra =10 6 ,b,d Ra =10 7 computedusingcascadedLBschemes.Toprowpresentsstreamlinesandthebottomrowthe isotherms.Gridresolutionusedis300 300.......................155 5.6Schematicofswirlingowinaconnedcylinderdrivenbyarotatingtoplid.....157 5.7ComputedstreamlinepatternsinthemeridianplaneduetoswirlingowinaconnedcylinderdrivenbyarotatinglidatvariousaspectratiosandReynoldsnumbersusingtheaxisymmetriccascadedLBsachems:a R A =1 : 5and Re =990, b R A =1 : 5and Re =1290c R A =2 : 5and Re =1010andd R A =2 : 5and Re =2200..........................................159 5.8Dimensionlessaxialvelocityprole u z =u o asafunctionofthedimensionalaxial distance z=H fora R A =1 : 5and Re =990,b R A =1 : 5and Re =1290 c R A =2 : 5and Re =1010andd R A =2 : 5and Re =2200:Comparisonbetween axisymmetriccascadedLBschemepredictionsandNS-basedsolverresults[5]...160 5.9Schematicofthearrangementformixedconvectioninaslendercylindricalannulus withinnerlateralwallrotation...............................161 5.10Contoursofaazimuthalvelocity,btemperature,cvorticity,anddstreamlinesformixedconvectioninaslendercylindricalannulusforthreedierentvalues of computedusingtheaxisymmetriccascadedLBschemes..............163 5.11GeometricarrangementofmeltowandconvectionduringCzochralskicrustal growthinarotatingcrucible )]TJ/F15 10.9091 Tf 8.484 0 Td [(Wheeler'sbenchmarkproblem..............165

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xix 5.12Streamlinesupperrowandisothermsbottomrowcorrespondingtotwocasesof theWheeler'sbenchmarkproblemofmeltowandconvectionduringCzochralshi crystalgrowth: Re x =100, Re c = )]TJ/F15 10.9091 Tf 8.484 0 Td [(25leftand Re x =1000 ;Re c = )]TJ/F15 10.9091 Tf 8.485 0 Td [(250right..167 5.13ComparisonofthemaximumReynoldsnumberfornumericalstabilityofsingle relaxationtimeSRTandcascadedLBmethodsforsimulationofthesheardriven swirlingowinaconnedcylinderatdierentgridresolutions.............170 6.1ComparisonofthecomputedprolesofthevorticityeldandtheanalyticalsolutioninaPoiseuilleowfordierentvaluesofthecenterlinevelocity U max = 0 : 01 ; 0 : 03 ; 0 : 05,and0 : 08obtainedbyvaryingtheuidviscosityataxedbody force F x =3 10 )]TJ/F33 7.9701 Tf 6.586 0 Td [(6 .Here,thelinesrepresenttheanalyticalsolutionandsymbols refertothenumericalresultsobtainedbythepresentDDFMRT-LBscheme.....191 6.2Comparisonofthespatialdistributionofthecomputedvorticityeldwiththeanalyticalsolutioninafour-rollsmillowwithinasquaredomainofsize2 2 forRe=40.ThesurfaceplotontheleftcorrespondstothenumericalresultsobtainedbythepresentDDFMRT-LBschemeandthatontherightisbasedonthe analyticalsolution......................................193 6.3Comparisonofcomputedprolesofthevorticityeldandtheanalyticalsolution inafour-rollsmillowalongvarioushorizontalsectionsat y =0 ;= 4 ;= 2 ;; 5 = 4. Here,thelinesrepresenttheanalyticalsolutionandsymbolsrefertothenumerical resultsobtainedbythepresentDDFMRT-LBscheme..................193 6.4EvaluationoftheorderofaccuracyofthepresentDDFMRT-LBschemeforvorticitycomputationinthefour-rollsmillowproblemwithaconstantkinematic viscosity =0 : 00218atdierentgridresolutions.....................195

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xx 6.5Comparisonofcomputedprolesofthevorticityeldandtheanalyticalsolution inapulsatileowinachanneli.e.,Womersleyowatdierentinstantswithin atimeperiodfortwodierentWomersleynumbersofWo=4 : 0andWo=7 : 0. Here,linesrepresenttheanalyticalsolutionandthesymbolsrefertothenumerical resultsobtainedusingthepresentDDFMRT-LBscheme................196 7.1Snapshotsoftheinterfaceunderanimposedshearowwithaninitiallycircular shapecomputedbythecascadedLBmethod.......................222 7.2Surfacecontoursofthepressuredistributionofasinglestaticdropofradius R = 30atdierentdensityratios A = B withsurfacetension =1 10 )]TJ/F33 7.9701 Tf 6.586 0 Td [(3 inaperiodic squaredomain........................................223 7.3Comparisonofthecomputedpressuredierencessymbolsobtainedusingthe cascadedLBmethodagainsttheanalyticalpredictionsusingtheLaplace-Young relationforvariousvaluesofthedropcurvature1 =R withsurfacetension = 5 10 )]TJ/F33 7.9701 Tf 6.586 0 Td [(3 ; 1 10 )]TJ/F33 7.9701 Tf 6.586 0 Td [(3 ; 1 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(4 .................................224 7.4SnapshotsofsimulationofRayleigh-TaylorinstabilityatAt=0 : 5andaRe= 256andbRe=3000...................................226 7.5TimeevolutionofthepositionsofthebubblefrontandthespiketipforRayleighTaylorinstabilityatAt=0 : 5andaRe=256andbRe=3000...........227 7.6EvolutionofadeformingdropfallingundergravityforvariousvaluesoftheOhnesorgenumberOhof0.3,0.7and1.0ataxedEotvosnumberEo=43shownat timeinstants T =0 ; 2 : 04 ; 3 : 05 ; 4 : 07 ; 5 : 09 ; 6 : 11 ; 7 : 13 ; 8 : 14,and9 : 16fromtopto bottom............................................229 7.7Evolutionoftheinterfaceofabuoyancy-drivenrisingbubbleatRe=35anda Eo=10,bEo=50,cEo=125............................232 7.8Timehistoryofthenon-dimensionalcenterofmassofabuoyancy-drivenrising bubbleatRe=35andEo=125..............................233

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xxi 7.9EvolutionofthesplashingofadroponathinlmatWe=8000and A = B = 1000foraRe=20bRe=100.............................235 7.10Evolutionoftheinterfaceofanoscillatingliquidcylinderstartingfromaninitial ellipticshapecongurationwithsemi-majoraxis a =25andsemi-minoraxis b = 15;surfacetensionparameter~ =0 : 1,kinematicviscosity A = B =0 : 01and densityratio A = B =100.................................236 7.11ComparisonoftheratiosoftheminimumachievableviscositiesforsinglerelaxationtimeSRTandcascadedLBformulationsallowingnumericallystablesimulationsofanoscillatingliquidcylinderwithsurfacetensionparameter~ =0 : 01at dierentdensityratios....................................237 6.1ComparisonofcomputeddropmigrationvelocityunderimposedconstantsurfactantconcentrationgradientinthesimulationofYoung'sproblemsolidlineswith theanalyticalsolutionfortheterminalvelocitydashedlinesforsurfacetension sensitivities 0 =0 : 0048 ; 0 =0 : 0146and 0 =0 : 0244................293 6.2Snapshotsoftheevolutionofamigratingdropunderimposedconstantsurfactant concentrationgradientinthesimulationofYoung'sproblemforsurfacetension sensitivities 0 =0 : 0048 ; 0 =0 : 0146and 0 =0 : 0244................295 6.3Comparisonofcomputedsymbolsandanalyticallinerequilibriumprolesfor surfactantconcentration foraplanarinterface.Simulationsareperformedfor theimposedsurfactantconcentrationsinthebulkuids b =0 : 0001, b =0 : 0002 and b =0 : 0003.......................................296

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1 CHAPTERI INTRODUCTION 1.1BackgroundandMotivation Complexuidmotionariseswidelyinanumberofengineeringapplications,geophysicalcontexts andbiochemicalsystems.Ofparticularinterestsarethosethatinvolvethermalconvectiveows incongurationsinvolvingheattransferand/orinterfacialowsinmultiphaseormulti-uidsystems.Oftenthepresenceofnonlinearityandmultiplescalesinherentinsuchmultiphysicssituationsmakeitquitechallengingtoinvestigatesuchsystems.Simulationsbasedoncomputational methodsenablefundamentalstudiesoftheuidmechanicsandaspredictivetoolsforengineeringdesignofcongurationsthathandleuidows.Thus,thedevelopmentofaccurate,ecient androbustnumericalmethodsplaysapivotalroleincomputationaluiddynamicsCFD.Classicalnumericaltechniquessuchasthenitedierence,nitevolumeandniteelementmethods havetraditionallybeenusedtoperformdiscretizationofthepartialdierentialequationssuchas theNavier-StokesNSequationsthatrepresenttheuidmotionincludinganyattendantmultiphysicseects.Fromadierentperspective,morerecently,thelatticeBoltzmannLBmethod hasdemonstratedtobeaveryeectivealternatenumericaltechniquetosimulateavarietyof complexuidowsystems[6,7,8]. ThelatticeBoltzmannmethodisamesoscopicmethodbasedonlocalconservationanddiscrete symmetryprinciples,andmaybederivedasaspecialdiscretizationoftheBoltzmannequation[9]. Hence,itcanberegardedasakineticscheme.Algorithmically,itinvolvesthestreamingofthe particledistributionfunctionsasaperfectshiftadvectionstepalongthelatticedirectionsand followedbyalocalcollisionstepasarelaxationprocesstowardsanequilibria,andaccompanied byspecialstrategiesfortheimplementationofimpressedforces.Thehydrodynamiceldschar-

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4 andcomponentsoftheparticlevelocitiesaregivenbyTheparticlevelocity e maybewrittenas e = 8 > > > > > < > > > > > : ; 0 =0 1 ; 0 ; ; 1 =1 ; ; 4 1 ; 1 =5 ; ; 8 .1 Thelatticerepresentsthediscretecharacteristicparticledirections e ,where =0 ; 1 ; 2 ; ; 8 alongwhichthediscreteparticledistributionfunction f advectsi.e.,streamsandthenundergoascatteringprocessi.e.,collision.AccordingtotheSRTLBmethod,thedistribution functionrelaxestoitscorrespondingequilibriumdistributionfunction f eq atasinglerelaxation time .Thus,theSRTLBmethodcanbewrittenas f x + e ;t +1= f x ;t )]TJ/F15 10.9091 Tf 12.379 7.38 Td [(1 [ f x ;t )]TJ/F35 10.9091 Tf 10.909 0 Td [(f eq x ;t ] ; .2 wheretheequilibriumdistributionfunction f eq isrelatedtothelocaluiddensity and u velocityelds,andisgivenby f eq = w 1+ e u c 2 s + e u 2 2 c 4 s )]TJ/F52 10.9091 Tf 12.105 7.38 Td [(u u 2 c 2 s : .3 Here, w aretheweightingfactorsfordierentparticledirectionsandgivenby w aregivenby w 0 =4 = 9, w =1 = 9,where =1 ; 2 ; 3 ; 4and w =1 = 36,where =5 ; 6 ; 7 ; 8and c 2 s =1 = 3.Then, thehydrodynamicelds,i.e.,thedensityandvelocityareupdatedbytakingzerothandrstmoments,respectively,of f ,andthepressureeld p isobtainedfromdensityviaanequationof state: = X f ; u = X f e ;p = c 2 s : .4 Thismethodsimulatestheuidmotion,withthekinematicviscosity givenby = c 2 s )]TJ/F15 10.9091 Tf 11.057 0 Td [(1 = 2. ItmaybenotedthattheSRTLBmethodispronetonumericalinstabilitieswhenitisrequired tosimulateuidowswithrelativelylowviscosities,i.e.,when ! 1 = 2anditdoesnotpossess anyadditionaldegreesoffreedomtoenhanceitsnumericalpropertiestoaddresssuchissues.It maybenotedthatvariousboundaryformulationstorepresenttheno-slip,freeslip,openow

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5 FIGURE1.1:Two-dimensional,nine-velocityD2Q9lattice.

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7 usingthemonomials e m x e n y inaascendingorder: 0 = ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 y ; 1 = e x = ; 1 ; 0 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 y ; 2 = e y = ; 0 ; 1 ; 0 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 y ; .5 3 = e x e x + e y e y ; 4 = e x e x )]TJ/F25 10.9091 Tf 10.909 0 Td [(e y e y ; 5 = e x e y ; 6 = e x e x e y ; 7 = e x e y e y ; 8 = e x e x e y e y ; where y isthetransposeoperator.Intheabove,foranytwo q -dimensionalvectors q =9here a and b ,wedenetheelementwisevectormultiplicationby a b .Thatis, a b = a b , where representsacomponentofthevectorandtheimplicitsummationconventionisnotassumedhere.Tofacilitatethepresentationinthefollowing,wealsodeneastandardscalarinner productofanytwosuchvectorsas h a ; b i .Thatis, h a ; b i = P q =0 a b .ByapplyingtheGramSchmidtorthogonalizationmethodtheyresultintothefollowingequivalentsetoforthogonalbasisvectors: K 0 = 0 ; K 1 = 1 ; K 2 = 2 ; K 3 =3 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 0 ; K 4 = 4 ; K 5 = 5 ; K 6 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 6 +2 2 ; K 7 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 7 +2 1 ; K 8 =9 8 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 3 +4 0 : Collectingthemasanorthogonalmatrix K ,weget K =[ K 0 ; K 1 ; K 2 ; K 3 ; K 4 ; K 5 ; K 6 ; K 7 ; K 8 ] ; .6 See[15,20,19],whichenumeratesthedetailsofthematrix K .Nowwedenethecontinuous equilibriumcentralmomentsas[15]: b M x m y n = Z 1 Z 1 f M x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x m y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y n d x d y : .7 where f M isthelocalMaxwell-Boltzmanndistributioninthecontinuousparticlevelocityspace = x ; y andisgivenby: f M f M ; u ; = 2 c 2 s exp h )]TJ/F33 7.9701 Tf 9.68 5.374 Td [( )]TJ/F53 7.9701 Tf 6.586 0 Td [(u 2 2 c 2 s i : Here, isthedensity

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8 and u = u x ;u y isthemacroscopicuidvelocity.Byevaluating b M x m y n intheincreasingorderof momentsfortheD2Q9latticeweobtain b M 0 = ; b M x =0 ; b M y =0 ; b M xx = c 2 s ; b M yy = c 2 s ; b M xy =0 ; b M xxy =0 ; b M xyy =0 ; b M xxyy = c 4 s : Similarly,wedenethecentralmomentsofthesourcesoforder m + n duetoaforceeld F = F x ; F y as b )]TJ/F36 7.9701 Tf 6.818 4.504 Td [(F x m y n = Z 1 Z 1 f F x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x m y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y n d x d y ; .8 where f F isthechangeinthedistributionfunctionduetoforceelds.Againwecanevaluate Eq..8as b )]TJ/F36 7.9701 Tf 6.818 4.504 Td [(F 0 =0 ; b )]TJ/F36 7.9701 Tf 6.818 4.504 Td [(F x = F x ; b )]TJ/F36 7.9701 Tf 6.819 4.504 Td [(F y = F y ; b )]TJ/F36 7.9701 Tf 6.818 4.504 Td [(F xx =0 ; b )]TJ/F36 7.9701 Tf 6.818 4.504 Td [(F yy =0 ; b )]TJ/F36 7.9701 Tf 6.818 4.504 Td [(F xy =0 ; b )]TJ/F36 7.9701 Tf 6.818 4.505 Td [(F xxy = c 2 s F y ; b )]TJ/F36 7.9701 Tf 6.818 4.505 Td [(F xyy = c 2 s F x ; b )]TJ/F36 7.9701 Tf 6.818 4.505 Td [(F xxyy =0 : Basedontheabovecontinuouscentralmomentsandusingthetrapezoidalruleforevaluatingthe sourcetermtoretainsecondorderaccuracy,theelementsofthecascadedLBEcanbeformulated asfollows[15,19]: f x + e ;t +1= f x ;t + c x ;t + 1 2 [ S x ;t + S x + e ;t +1].9 Intheaboveequation,thecollisiontermcanbemodeledby c c f ; b g = K b g ; .10 where f = f 0 ;f 1 ;f 2 ;:::;f 8 y isthevectorofdistributionfunctionsand b g = b g 0 ; b g 1 ; b g 2 ;:::; b g 8 y isthevectoroftheunknownchangeindierentmomentssupportedbythelatticeundercollision thatisgivenlater.Thediscreteformofthesourceterm S inthecascadedLBErepresentsthe inuenceoftheforceeld F x ; F y inthevelocityspacethatisgivenas: S = S 0 ;S 1 ;S 2 ;:::;S 8 y : AsEq..9issemi-implicit,byusingthestandardvariabletransformationbyHeetal.[21,22],

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9 and,inparticular,byHeetal.[23] f = f )]TJ/F33 7.9701 Tf 13.043 4.295 Td [(1 2 S ,sothattheimplicitnessisremovedanda second-orderaccuracyismaintained,weget f x + e ;t +1= f x ;t + c x ;t + S x ;t : .11 Inordertodeterminethestructureofthecascadedcollisionoperatorandthesourcetermsinthe presenceofgeneralspatiallyand/ortemporallyvariablebodyforces,wedenethefollowingset ofcentralmomentsoforder m + n ,respectively,as 0 B B B B B B B B @ ^ x m y n ^ eq x m y n ^ x m y n ^ x m y n 1 C C C C C C C C A = X 0 B B B B B B B B @ f f eq S f 1 C C C C C C C C A e x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x m e y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y n : .12 Byequatingthediscretecentralmomentsforboththedistributionfunctionsandsourceterms withthecorrespondingcontinuouscentralmoments b eq x m y n = b M x m y n b x m y n = b )]TJ/F36 7.9701 Tf 6.818 3.959 Td [(F x m y n ,weget b eq 0 = ; b eq x =0 ; b eq y =0 ; b eq xx = c 2 s ; b eq yy = c 2 s ; b eq xy =0 ; b eq xxy =0 ; b eq xyy =0 ; b eq xxyy = c 4 s : .13 b 0 =0 ; b x = F x ; b y = F y ; b xx =0 ; b yy =0 ; b xy =0 ; b xxy =0 ; b xyy =0 ; b xxyy =0 : .14 Here,weapplytheeectofthevariablebodyforcesonlyontherstordercentralmomentsof thesourcestoobtainconsistentmacroscopicequations[19].Similarly,forthetransformedcentral momentswehave b eq 0 = ; b eq x = )]TJ/F33 7.9701 Tf 9.681 4.295 Td [(1 2 F x ; b eq y = )]TJ/F33 7.9701 Tf 9.681 4.295 Td [(1 2 F y ; b eq xx = c 2 s ; b eq yy = c 2 s ; b eq xy =0 ; b eq xxy = )]TJ/F36 7.9701 Tf 9.68 5.375 Td [(c 2 s 2 F y ; b eq xyy = )]TJ/F36 7.9701 Tf 9.681 5.375 Td [(c 2 s 2 F x ; b eq xxyy = c 4 s : .15 Ontheotherhand,theactualcalculationsinthecascadedformulationarecarriedoutinthe termsoftherawmoments.Hence,wedenethefollowingsetofrawmomentsdesignatedhere

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10 withtheprime 0 symbol,whichwillbeusedlater: 0 B B B B B B B B @ ^ 0 x m y n ^ eq 0 x m y n ^ 0 x m y n ^ 0 x m y n 1 C C C C C C C C A = X 0 B B B B B B B B @ f f eq S f 1 C C C C C C C C A e m x e n y .16 Wecanreadilyconvertthecentralmomentsintoacombinationoftherawmomentsbyusingthe binomialtheorem. Thus,werstsummarizethefollowingsetofrawmomentsforthesourcetermsfortheD2Q9 lattice: b 0 0 =0 ; b 0 x = F x ; b 0 y = F y ; b 0 xx = 2 F x u x 2 ; b 0 yy = 2 F y u y 2 ; b 0 xy = F x u y + F y u x 2 ; b 0 xxy = F y u 2 x +2 F x u x u y ; b 0 xyy = F x u 2 y +2 F y u y u x ; b 0 xxyy =2 F x u x u 2 y +2 F y u y u 2 x : Next,asanintermediatestep,thesourcemomentsprojectedtotheorthogonalmomentspace b m s = h K ; S i areobtainedas b m s 0 =0 ; b m s 1 = F x ; b m s 2 = F y ; b m s 3 =6 F u ; b m s 4 =2 F x u x )]TJ/F35 10.9091 Tf 10.909 0 Td [(F y u y ; b m s 5 = F x u y + F y u x ; b m s 6 = )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1 )]TJ/F15 10.9091 Tf 10.91 0 Td [(3 u 2 x F y )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 F x u x u y ; b m s 7 = )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 u 2 y F x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 F y u y u x ; b m s 8 =3 )]TJ/F15 10.9091 Tf 9.545 -8.836 Td [(6 u 2 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 F x u x + )]TJ/F15 10.9091 Tf 5 -8.836 Td [(6 u 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 F y u y : Equivalently,theaboveequationscanbewritteninamatrixformas K S = b m s 0 ; b m s 1 ; b m s 2 ;:::; b m s 8 y : Now,invertingtheaboveequation,wecanobtainexplicitexpressionsfor S intermsofthegen-

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11 eralvariablebodyforce F anduidvelocity u intheparticlevelocityspaceas S 0 = 1 9 [ )]TJ/F41 10.9091 Tf 10.244 0 Td [(b m s 3 + b m s 8 ] ; .17a S 1 = 1 36 [6 b m s 1 )]TJ/F41 10.9091 Tf 12.668 0 Td [(b m s 3 +9 b m s 4 +6 b m s 7 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b m s 8 ] ; .17b S 2 = 1 36 [6 b m s 2 )]TJ/F41 10.9091 Tf 12.668 0 Td [(b m s 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 b m s 4 +6 b m s 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b m s 8 ] ; .17c S 3 = 1 36 [ )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 b m s 1 )]TJ/F41 10.9091 Tf 12.668 0 Td [(b m s 3 +9 b m s 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 b m s 7 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b m s 8 ] ; .17d S 4 = 1 36 [ )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 b m s 2 )]TJ/F41 10.9091 Tf 12.668 0 Td [(b m s 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 b m s 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 b m s 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b m s 8 ] ; .17e S 5 = 1 36 [6 b m s 1 +6 b m s 2 +2 b m s 3 +9 b m s 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 b m s 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 b m s 7 + b m s 8 ] ; .17f S 6 = 1 36 [ )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 b m s 1 +6 b m s 2 +2 b m s 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 b m s 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 b m s 6 +3 b m s 7 + b m s 8 ] ; .17g S 7 = 1 36 [ )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 b m s 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 b m s 2 +2 b m s 3 +9 b m s 5 +3 b m s 6 +3 b m s 7 + b m s 8 ] ; .17h S 8 = 1 36 [6 b m s 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 b m s 2 +2 b m s 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 b m s 5 +3 b m s 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 b m s 7 + b m s 8 ] : .17i Then,todeterminethestructureofthecascadedcollisionoperatorinthepresenceofforcing termswestartfromthelowest-ordernonconservativei.e.,secondordermoments,andperform relaxationofvariouscentralmomentstotheircorrespondingequilibria,eachcarriedoutatdierentrelaxationtimes.SeeRefs.[15,19]fordetails.Thus,thechangeofdierentmomentsunder

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12 cascadedcollisioncanbesummarizedas b g 0 = b g 1 = b g 2 =0 ; .18a b g 3 = ! 3 12 2 3 + u 2 x + u 2 y )]TJ/F15 10.9091 Tf 10.909 0 Td [( b 0 xx + b 0 yy )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 2 b 0 xx + b 0 yy ; .18b b g 4 = ! 4 4 u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y )]TJ/F15 10.9091 Tf 10.909 0 Td [( b 0 xx )]TJ/F41 10.9091 Tf 11.021 2.182 Td [(b 0 yy )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 2 b 0 xx )]TJ/F41 10.9091 Tf 11.192 0 Td [(b 0 yy ; .18c b g 5 = ! 5 4 u x u y )]TJ/F41 10.9091 Tf 11.021 2.181 Td [(b 0 xy )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(1 2 b 0 xy ; .18d b g 6 = ! 6 4 2 u 2 x u y + b 0 xxy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x b 0 xy )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y b 0 xx )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 2 b xxy )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 2 u y b g 3 + b g 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x b g 5 ; .18e b g 7 = ! 7 4 2 u x u 2 y + b 0 xyy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y b 0 xy )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x b 0 yy )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(1 2 b xyy )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(1 2 u x b g 3 )]TJ/F41 10.9091 Tf 10.98 0 Td [(b g 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y b g 5 ; .18f b g 8 = ! 8 4 1 9 +3 u 2 x u 2 y )]TJ/F41 10.9091 Tf 10.909 12.109 Td [(h b 0 xxyy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x b 0 xyy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y b 0 xxy + u 2 x b 0 yy + u 2 y b 0 xx +4 u x u y b 0 xy i )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 2 b 0 xxyy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 3 )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 2 u 2 y b g 3 + b g 4 )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(1 2 u 2 x b g 3 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 u x u y b g 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y b g 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x b g 7 : .18g Foreaseofimplementation,thecascadedlatticeBoltzmannequationwithforcingtermcanthen bepresentedintermsoffollowingcollisionandstreamingsteps,respectively: e f x ;t = f x ;t + C + S x ;t .19 f x + e ;t +1= e f x ;t : .20 where, C isobtainedusingEq..41andEqs..18a{.18g.Finally,theexplicitformofthe

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13 post-collisiondistributionfunctioninthevelocityspacecanbewrittenasfollows: e f 0 = f 0 +[ b g 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 b g 3 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 8 ]+ S 0 ; e f 1 = f 1 +[ b g 0 + b g 1 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 3 + b g 4 +2 b g 7 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 8 ]+ S 1 ; e f 2 = f 2 +[ b g 0 + b g 2 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 3 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 4 +2 b g 6 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 8 ]+ S 2 ; e f 3 = f 3 +[ b g 0 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 1 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 3 + b g 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 7 + b g 8 ]+ S 3 ; e f 4 = f 4 +[ b g 0 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 2 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 3 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 6 + b g 8 ]+ S 4 ; e f 5 = f 5 +[ b g 0 + b g 1 + b g 2 +2 b g 3 + b g 5 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 6 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 7 + b g 8 ]+ S 5 ; e f 6 = f 6 +[ b g 0 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 1 + b g 2 +2 b g 3 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 5 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 6 + b g 7 + b g 8 ]+ S 6 ; e f 7 = f 7 +[ b g 0 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 1 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 2 +2 b g 3 + b g 5 + b g 6 + b g 7 + b g 8 ]+ S 7 ; e f 8 = f 8 +[ b g 0 + b g 1 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 2 +2 b g 3 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 5 + b g 6 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 7 + b g 8 ]+ S 8 : Thehydrodynamicvariablesarethenobtainedas = X f ; u = X f e + 1 2 F ;p = c 2 s : .21 ItcanbeshownviathestandardmultiscaleChapman-Enskogexpansionthatthesolutionofthe abovecascadedLBmethodrepresentstheweaklycompressibleNavier-StokesequationsNSE seee.g.,[19] @ t + r u =0 ; .22a @ t u + r uu = )]TJ/F55 10.9091 Tf 8.485 0 Td [(r p + r [ S )]TJ/F58 10.9091 Tf 10.909 0 Td [(I r u + I r u ]+ F ; .22b where S = 1 2 r u + r u y and I arethestrainratetensorandidentitytensor,respectively,and F = F x ;F y .Thetransportcoecientsoftheuidmotion,suchasthekinematicbulkviscosity andkinematicshearviscosity arerelatedtotherelaxationtimesofthesecondordermoments via = = 3 =! 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = 2 ; = = 3 =! )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = 2 ; =4 ; 5 andtherelaxationtimesforhigherordermoments ! ,where =6 ; 7 ; 8canbeindependentlyadjustedtoimprovenumericalstability.SeeRef.[17]foracomparisonofthecascadedLBmethod

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23 CHAPTERII GALILEANINVARIANTPRECONDITIONEDCENTRALMOMENT LATTICEBOLTZMANNMETHODWITHOUTCUBICVELOCITY ERRORSFOREFFICIENTSTEADYFLOWSIMULATIONS 2.1Introduction ThelatticeBoltzmannLBmethodhasnowbeenestablishedasapowerfulkineticschemebased computationaluiddynamicsapproach[24],[7],[18].Itisamesoscopicmethodbasedonlocal conservationanddiscretesymmetryprinciples,andmaybederivedasaspecialdiscretizationof theBoltzmannequation.Duringthelastdecade,manyeortsweremadetofurtherimproveits numericalstability,accuracyandeciency.Inparticular,sophisticatedcollisionmodelsbased onmultiplerelaxationtimesandinvolvingrawmoments,centralmomentsorcumulants,and entropicformulationshavesignicantlyexpandedthecapabilitiesoftheLBmethod.Thesignicantachievementsofthesedevelopmentsandtheirapplicationstoavarietyofcomplexow problemshavebeendiscussed,forexample,in[11,12,25,26,27,28,29,17,30,31,32,33]. ThereexistvariousadditionalaspectsintheLBapproachthatrequirefurtherattentionand presentscopeforimprovements.Inparticular,thenitenessofthelatticecanintroducecertain truncationerrorsthatmanifestasnon-Galileaninvariantviscousstress,i.e.uidvelocitydependentviscosity.ThislackofGalileaninvarianceGIarisesduetothefactthatthediagonalterms inthethird-ordermomentsarenotindependentlysupportedbythestandardtensor-productlatticesi.e.D2Q9andD3Q27.Morespecically,forexample, b 0 xxx = X e 3 x f = X e x f = b 0 x : Here,andinthefollowing,theprimedquantitiesdenoterawmoments.Inotherwords,there isadegeneracyofthethird-orderdiagonallongitudinalmomentsthatresultsinadeviation

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24 betweentheemergentmacroscopicequationsderivedbytheChapman-Enskogexpansionand theNavier-StokesNSequations.Suchcubic-velocitytruncationerrorsaregridindependent andpersistinnergridsespeciallyunderhighshearandowvelocity.Moreover,suchemergentanisotropicviscousstressmayhaveanegativeimpactonnumericalstabilityasaresultof anegativedependenceoftheemergentviscosityontheuidvelocity.Inordertoovercomethis shortcoming,variousattemptshavebeenmade. Onepossibilityistoconsideralatticewithalargerparticlevelocityset,suchastheD2Q17latticeintwo-dimensions[34],whichwaspursuedafter[35]pointedoutnonlinear,cubic-velocitydeviationsoftheemergentequationsoftheLBmodelswithstandardlatticesetsfromtheNSequations.Thisinvolvedtheuseofhigherordervelocitytermsintheequilibriumdistribution.However,[36]showedthatthespecicequilibriaadoptedin[34]doesnotfullyeliminatethecubicvelocityerrors.Moreover,theuseofnon-standardlatticestencilswithlargernumberofparticle velocitiesincreasesthecomputationalcostandpropensityofthenumericalinstabilityatgrid scales.Ontheotherhand,itwasshownmorerecentlybyvariousothers[37],[36],[38]that partialcorrectionstotheGIerrorsonthestandardlatticei.e.D2Q9latticemaybeachieved byadoptingspecialformsoftheo-diagonal,third-ordermomentsintheequilibria.Thatis, b eq 0 xxy = c 2 s u y + u 2 x u y ; b eq 0 xyy = c 2 s u x + u x u 2 y : Here, c s isthespeedofsoundandtheparticularchoicesofthecubic-velocitytermsthatareunderlinedarecrucialtopartiallyrestoreGIfortheaboveidentiedmoments.Here,wealsopoint outthattheaboveformsoftheo-diagonal,third-orderrawmomentequilibriathatallowsuch partialGIcorrectionsnaturallyariseinthecentralmomentLBformulations,whentheequilibriumcentralmomentcomponentsaresettozeroandandthenrewrittenintermsoftheircorrespondingrawmoments.However,since b eq 0 xxx = b eq 0 x and b eq 0 yyy = b eq 0 y duetothedegeneracyof thethird-orderlongitudinalmoments,whichisinherenttothestandardtensor-productlattices, additionalcorrectionsarerequiredtorestoreGIcompletelyfreeofcubic-velocityerrors.Inthis regard,inordertocompensatethetermswhichviolateGIonstandardlattices,LBschemeswith

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26 thereisarelativelylargedisparitybetweenthesoundspeedandtheconvectionspeedofthe uidmotionresultinginhighereigenvaluestinessandlargernumberofiterationsforconvergence.Thisstinesscanbealleviatedandconvergencecanbesignicantlyimprovedbypreconditioning.Reference[49]presentedapreconditionedLBmethodbasedonasinglerelaxationtime modelbymodifyingtheequilibriumdistributionfunctionbyusingapreconditioningparameter. Then,[52]and[53]presentedpreconditionedLBformulationsbasedonmultiplerelaxationtimes. Morerecently,[58]presentedapreconditionedschemeforthecentralmomentbasedcascadedLB method[15]inthepresenceofforcingterms[19]anddemonstratedsignicantconvergenceacceleration. Ingeneral,suchpreconditionedLBschemesareintendedtosolvethepreconditionedNSequations,whichcanbewrittenas[61],[62] @ @t + r u =0 ; .1a @ u @t + r uu = )]TJ/F15 10.9091 Tf 10.08 7.38 Td [(1 r p + 1 r S + F ; .1b where p , S and F arethepressure,strainratetensorandtheimpressedforce,respectively.Here, isthepreconditioningparameter,whichcanbeusedtotunethepseudo-soundspeed,thereby alleviatingtheeigenvaluestinessandimprovingconvergenceacceleratione.g.[58].However, theexistingLBmodelsforthepreconditionedNSequationsarenotGalileaninvariantandare expectedtoinvolvebothvelocity-andparameter-dependentanisotropicformoftheviscousstress tensor.DevelopmentoftheGalileaninvariantpreconditionedcentralmomentbasedLBmethod withoutcubic-velocitydefectsandparameterfreetruncationerrorsforsteadyowsimulationsis themainfocusofthisstudy.ItmaybenotedthatthepreconditionedNSequationsmaybeconsideredasaspecicexampleofwhatmaybecalledasthegeneralizedNSequationscontaining afreeparameter.Inthepresentcase,suchaparameterisimposedbynumericsduetopreconditioning.Ontheotherhand,suchgeneralizedNSequationsariseinothercontextssuchasin thesimulationoftheuidsaturatedvariableporousmediaowsrepresentedbytheBrinkmanForchheimer-Darcyequation.Insuchcases,thefreeparameterappearinginthegeneralizedNS

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27 equationsisimposedbyphysics,viz.,theporosity.Thus,ourpresentinvestigationonthedevelopmentoftheGalileaninvariantLBmodelsforthepreconditionedNSequationsonstandardlatticeswithoutcubic-velocityandparameterdependenterrorsalsohaswiderimplicationsinother contexts. Inordertorstidentifysuchtruncationerrors,inthischapterwerstperformaChapmanEnskoganalysisofthepreconditionedcentralmomentLBformulationandisolatevariouscubicvelocityandparameterdependenterrorsatvariousmomentorders.Itwillbeseenthattheanisotropy ofthestresstensordependsnotjustonthecubic-velocitytermslikeinthepreviousstudies, butalsoonthepreconditioningparameter .Furthermore,wewillalsodemonstratethateven toachievepartialcorrectionsfortheGIdefectsonthestandardlattice,thecubicvelocityterms appearingintheo-diagonalcomponentsofthethird-ordermomentequilibrianeedtobeappropriatelyscaledby e.g. b eq 0 xxy = c 2 s u y + u 2 x u y = 2 .Ingeneral,thevarioustruncationerror termsthatariseduetothedegeneracyofthethird-orderdiagonalelementswillbeseentohave complexdependenceonboththevelocitiesandthepreconditioningparameter.OncesuchGIdefecttermsareidentied,newcorrectionsarederivedforthepreconditionedcentralmomentLB formulationbasedontheextendedmomentequilibria.ThisresultsinaGIcentralmomentLB methodforthepreconditionedNSequationswithoutcubic-velocityandparameterbaseddefects onstandardlattices.ThepresentschemeistargetedtowardsecientandaccuratelowReynolds numbersteadystatelaminarowsbyapreconditionedLBformulationwithoutthediscretecubic velocityandparameterdependenteectsviacorrectionstothemomentequilibria. Thischapterisorganizedasfollows.InthenextsectionSec.2.2,ourpreviouscentralmoment basedpreconditionedLBMwithforcingtermsontheD2Q9latticeissummarizedrst.Section3 performsamorerenedanalysisbasedontheChapman-Enskogexpansionandidentiesvarious cubic-velocityandparameterdependentGIdefecterrors.Then,Sec2.4derivesnewcorrections basedontheextendedmomentequilibriaandSec.2.5presentsaGIpreconditionedcentralmomentLBmethodfreeofcubic-velocityandparameterdependenterrors.Numericalresultsare

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28 presentedinSec.2.6,whichcomparesournumericalresultsforavarietyofbenchmarkproblems, includingthelid-drivencavityow,owoverasquarecylinder,backwardfacingstepow,the Hartmannowandthefour-rollmillsowproblemforthepurposeofvalidation.Inaddition, convergenceaccelerationduetopreconditioningandimprovementinaccuracyduetotheGIcorrectedLBschemearealsoillustrated.Finally,themainndingsofthischapteraresummarized inSec.2.7. 2.2PreconditionedCascadedCentralMomentLatticeBoltzmannMethod:NonGalileanInvariantFormulation Inourpreviouswork,wepresentedamodiedcascadedcentralmomentlatticeBoltzmannmethod LBMwithforcingtermsforthecomputationofpreconditionedNSequations[58].However, thispreconditionedLBMformulationisnotGalileaninvariantonstandardlattices.Thisisbecauseitresultsingrid-independentcubic-velocityerrorsthataresensitivetothepreconditioning parameter.Infact,thederivationofthepreciseexpressionforthenon-GItruncationerrorswill bederivedinthenextsection.ItmaybenotedthatallotherpriorpreconditionedLBschemes arealsonotGalileaninvariant.However,thechoiceofcentralmomentsherepartiallycorrects partsofthecubicvelocitydefectsintheo-diagonalthirdordermomentsnaturallySec.2.3 andsimpliesderivationofthecorrectiontermstocompletelyrestoreGIfreeofcubicvelocity errorsonstandardlatticeSec.2.4.Here,wesummarizeourpreviouspreconditionedcentral momentLBmodelsettingthestageforfurtherdevelopmentinthefollowing. ThepreconditionedcascadedcentralmomentLBMwithforcingtermsmaybewrittenas[58] f f x ;t = f x ;t + K ^ g + S x ;t ; .2a f x + e ;t +1= f f x ;t ; .2b whereavariabletransformation f = f )]TJ/F33 7.9701 Tf 12.457 4.295 Td [(1 2 S isintroducedtomaintainsecondorderaccuracy inthepresenceofforcingterms.Intheabove, K istheorthogonaltransformationmatrixand

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29 ^ g isthecollisionoperator.Inordertolisttheexpressionsforthecollisionkernelforthestandardtwo-dimensional,nineparticlevelocityD2Q9lattice,werstdenevarioussetsofraw momentsasfollowsonwhichitisbased: 0 B B B B B B B B @ ^ 0 x m y n ^ eq 0 x m y n ^ 0 x m y n ^ 0 x m y n 1 C C C C C C C C A = X 0 B B B B B B B B @ f f eq S f 1 C C C C C C C C A e m x e n y : .3 Thepreconditionedcollisionkernelsetfortheorthogonalmomentbasisusingthepreconditioning parameter canbewrittenas[58] b g 0 = b g 1 = b g 2 =0 ; b g 3 = ! 3 12 n 2 3 + u 2 x + u 2 y )]TJ/F15 10.9091 Tf 10.909 0 Td [( b 0 xx + b 0 yy )]TJ/F33 7.9701 Tf 12.105 4.295 Td [(1 2 b 0 xx + b 0 yy o ; b g 4 = ! 4 4 n u 2 x )]TJ/F36 7.9701 Tf 6.587 0 Td [(u 2 y )]TJ/F15 10.9091 Tf 10.909 0 Td [( b 0 xx )]TJ/F41 10.9091 Tf 11.021 2.182 Td [(b 0 yy )]TJ/F33 7.9701 Tf 12.105 4.295 Td [(1 2 b 0 xx )]TJ/F41 10.9091 Tf 11.191 0 Td [(b 0 yy o ; b g 5 = ! 5 4 n u x u y )]TJ/F41 10.9091 Tf 11.022 2.182 Td [(b 0 xy )]TJ/F33 7.9701 Tf 12.104 4.295 Td [(1 2 b 0 xy o ; .4 b g 6 = ! 6 4 n 2 u 2 x u y + b 0 xxy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x b 0 xy )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y b 0 xx )]TJ/F33 7.9701 Tf 12.104 4.295 Td [(1 2 b xxy o )]TJ/F33 7.9701 Tf 12.105 4.295 Td [(1 2 u y b g 3 + b g 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x b g 5 ; b g 7 = ! 7 4 n 2 u x u 2 y + b 0 xyy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y b 0 xy )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x b 0 yy )]TJ/F33 7.9701 Tf 12.104 4.295 Td [(1 2 b xyy o )]TJ/F33 7.9701 Tf 12.104 4.295 Td [(1 2 u x b g 3 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y b g 5 ; b g 8 = ! 8 4 n 1 9 +3 u 2 x u 2 y )]TJ/F41 10.9091 Tf 10.909 12.109 Td [(h b 0 xxyy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x b 0 xyy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y b 0 xxy + u 2 x b 0 yy + u 2 y b 0 xx +4 u x u y b 0 xy i )]TJ/F33 7.9701 Tf 12.105 4.296 Td [(1 2 b 0 xxyy o )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 3 )]TJ/F33 7.9701 Tf 12.104 4.296 Td [(1 2 u 2 y b g 3 + b g 4 )]TJ/F33 7.9701 Tf 12.104 4.296 Td [(1 2 u 2 x b g 3 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 u x u y b g 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y b g 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x b g 7 : Forfurtherdetails,andincludingthechoiceofthecollisionmatrix K andsourcerawmoments ^ 0 x m y n ,see[58].Thisschemeresultsinatunablepseudo-soundspeed c s = c s ,where c s = 1 p 3 x = t ,andtheemergentviscosity isgivenby = 3 1 ! )]TJ/F33 7.9701 Tf 12.804 4.296 Td [(1 2 , =4 ; 5.Whilethisscheme isintendedtosimulatethepreconditionedNSequationsgiveninEq.,aswillbeshownviaa consistencyanalysisbasedontheChapman-Enskogexpansioninthenextsectionthatitleadsto velocity-andpreconditioningparameter-dependentnon-GItruncationerrors.Inparticular,itwill beseenthatthecomponentsofthenon-equilibriumpartsofthesecondordermoments,which

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30 contributetotheviscousstresstensor,dependsoncubicvelocitytruncationerrorsandmodulatedbythepreconditioningparameter . 2.3DerivationofNon-GalileanInvariantSpuriousTermsinthePreconditioned CascadedCentralMomentLBMethod:Chapman-EnskogAnalysis InordertofacilitatetheChapman-Enskoganalysis,thecentralmomentLBformulationcanbe equivalentlyrewrittenintermsofacollisionprocessinvolvingrelaxationtoageneralizedequilibriainthelatticeorrestframeofreference[58].Thisstrategyisconsideredinthischapterto furtherinvestigatethestructureofthecubicvelocitynon-GItruncationerrorsforourpreconditionedLBmethod.Inthisregard,itisconvenienttodenethenon-orthogonaltransformation matrix T whichisthebasistoobtaintheorthogonalcollisionmatrix K usedintheprevioussectionandonwhichthesubsequentanalysisfollows: T =[ e 0 ; j e x i ; j e y i ; j e 2 x + e 2 y i ; j e 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(e 2 y i ; j e x e y i ; j e 2 x e y i ; j e x e 2 y i ; j e 2 x e 2 y i ] ; .5 wheretheusualbra-ketnotationisusedtorepresenttherawandcolumnvectorsintheq-dimensional space q =9fortheD2Q9lattice.Then,therelationbetweenthevarioussetsoftherawmomentsandtheircorrespondingstatesinthevelocityspacecanbedenedviathisnominal,nonorthogonaltransformationmatrix T as b m = T f ; b m = T f ; b m eq = T f eq ; b S = T S ; .6 where f = )]TJ/F15 10.9091 Tf 7.349 -5.957 Td [( f 0 ; f 1 ; f 2 ;:::; f 8 y ; f = f 0 ;f 1 ;f 2 ;:::;f 8 y ; f eq = f eq 0 ;f eq 1 ;f eq 2 ;:::;f eq 8 y ; S = S 0 ;S 1 ;S 2 ;:::;S 8 y

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31 arethevariousquantitiesinthevelocityspace,and b m = )]TJ 6.759 -7.339 Td [(b m 0 ; b m 1 ; b m 2 ;:::; b m 8 y = b 0 0 ; b 0 x ; b 0 y ; b 0 xx + b 0 yy ; b 0 xx )]TJ/F41 10.9091 Tf 11.022 2.181 Td [(b 0 yy ; b 0 xy ; b 0 xxy ; b 0 xyy ; b 0 xxyy y ; .7a b m = b m 0 ; b m 1 ; b m 2 ;:::; b m 8 y = b 0 0 ; b 0 x ; b 0 y ; b 0 xx + b 0 yy ; b 0 xx )]TJ/F41 10.9091 Tf 11.021 0 Td [(b 0 yy ; b 0 xy ; b 0 xxy ; b 0 xyy ; b 0 xxyy y ; .7b b m eq = b m eq 0 ; b m eq 1 ; b m eq 2 ;:::; b m eq 8 y = b eq 0 0 ; b eq 0 x ; b eq 0 y ; b eq 0 xx + b eq 0 yy ; b eq 0 xx )]TJ/F41 10.9091 Tf 11.021 0 Td [(b eq 0 yy ; b eq 0 xy ; b eq 0 xxy ; b eq 0 xyy ; b eq 0 xxyy y ; .7c b S = b S 0 ; b S 1 ; b S 2 ;:::; b S 8 y = b 0 0 ; b 0 x ; b 0 y ; b 0 xx + b 0 yy ; b 0 xx )]TJ/F41 10.9091 Tf 11.191 0 Td [(b 0 yy ; b 0 xy ; b 0 xxy ; b 0 xyy b 0 xxyy y .7d arethecorrespondingstatesinthemomentspace. TofacilitatetheChapman-Enskoganalysis,wecanrewritethepreconditionedLBmodelpresentedinEq..2aandEq..2bintermsoftherawmomentspacegiveninEq..6as[19],[58] f x + e t ;t + t )]TJ/F25 10.9091 Tf 10.909 0 Td [(f x ;t = T )]TJ/F33 7.9701 Tf 6.587 0 Td [(1 h )]TJ/F15 10.9091 Tf 9.424 2.879 Td [(^ ^ m )]TJ/F15 10.9091 Tf 13.409 0.152 Td [(^ m eq i + T )]TJ/F33 7.9701 Tf 6.587 0 Td [(1 I )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(1 2 ^ ^ S t ; .8 wherethediagonalrelaxationtimematrix ^ isdenedas ^ =diag ; 0 ; 0 ;! 3 ;! 4 ;! 5 ;! 6 ;! 7 ;! 8 : .9 Thepreconditionedrawmomentsoftheequilibriumdistributionandsourcetermscanberepresentedas b eq 0 0 = ; b eq 0 x = u x ; b eq 0 y = u y ; b eq 0 xx = 1 3 + u 2 x ; b eq 0 yy = 1 3 + u 2 y ; b eq 0 xy = u x u y ; b eq 0 xxy = 1 3 u y + u 2 x u y 2 ; b eq 0 xyy = 1 3 u x + u x u 2 y 2 ; b eq 0 xxyy = 1 9 + 1 3 u 2 x + u 2 y + u 2 x u 2 y : .10 and b 0 0 =0 ; b 0 x = F x ; b 0 y = F y ; b 0 xx = 2 F x u x 2 ; b 0 yy = 2 F y u y 2 ; b 0 xy = F x u y + F y u x 2 ; b 0 xxy = F y u 2 x +2 F x u x u y ; b 0 xyy = F x u 2 y +2 F y u y u x ; b 0 xxyy =2 F x u x u 2 y + F y u y u 2 x : .11

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32 Thefollowingcommentsareinorderhere.Uptothesecondordermoments,theaboveexpressionscoincidewiththosepresentedinourpreviouswork[58].Inotherwords, u i u j termsin themomentequilibriaarepreconditionedby ,whiletherstandsecondordermomentterms, i.e. F i and F i u j arepreconditionedby and 2 ,respectively.Asarstnewelementtowardsa LBschemewithanimprovedGI,wepreconditionthethird-ordermomentequilibriaterms u i u 2 j termsby 2 seethetermsinsideboxesinEq..10.ThispartiallyrestoresGIwithoutcubic velocitydefectsforthepreconditionedLBmodelfortheo-diagonalcomponentsofthethirdordermoments.Infact,aswillbeshownlaterinthissection,inordertoremovethespurious cross-velocityderivativetermsappearingintheequivalentmacroscopicequationsofourpreconditionedLBschemee.g. u x u y @ x u y and u y u x @ y u x ,suchascalingofthecubicvelocitytermsinthe thirdordermomentequilibriaisessential.Then,applyingthestandardChapman-EnskogmultiscaleexpansiontoEq..8,i.e. b m = 1 X n =0 n b m n ; .12 @ t = 1 X n =0 n @ t n : .13 where is a smallbookkeepingperturbationparameter,andalsousingaTaylorexpansionto simplifythestreamingoperatorinEq..8,i.e. f x + e ;t + = n X n =0 n n ! @ t + e r f x ;t : .14 AfterconvertingalltheresultingtermsintothemomentspaceusingEq..6,wegetthefollowingmomentequationsatconsecutiveorderin : O 0 : b m = b m eq ; .15a O 1 : @ t 0 + b E i @ i b m = )]TJ/F41 10.9091 Tf 9.121 2.879 Td [(b b m + b S ; .15b O 2 : @ t 1 b m + @ t 0 + b E i @ i h I )]TJ/F33 7.9701 Tf 12.105 4.295 Td [(1 2 b i b m = )]TJ/F41 10.9091 Tf 9.121 2.879 Td [(b b m ; .15c where b E i = T e i I T )]TJ/F33 7.9701 Tf 6.587 0 Td [(1 ;i 2f x;y g .Therelevantcomponentsoftherst-order O equations Eq..15b,i.e.uptothesecondorderinmomentspaceneededforderivingthepreconditioned

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33 macroscopichydrodynamicsequationsaregivenas @ t 0 + @ x u x + @ y u y =0 ; .16a @ t 0 u x + @ x 1 3 + u 2 x + @ y u x u y = F x ; .16b @ t 0 u y + @ x u x u y + @ y 1 3 + u 2 y = F y ; .16c @ t 0 2 3 + u 2 x + u 2 y + @ x 4 3 u x + u x u 2 y 2 + @ y 4 3 u y + u 2 x u y 2 = )]TJ/F35 10.9091 Tf 8.485 0 Td [(! 3 b m 3 + 2 F x u x + F y u y 2 ; .16d @ t 0 u 2 x )]TJ/F36 7.9701 Tf 6.587 0 Td [(u 2 y + @ x 2 3 u x )]TJ/F36 7.9701 Tf 12.105 6.536 Td [(u x u 2 y 2 + @ y )]TJ/F33 7.9701 Tf 9.681 4.295 Td [(2 3 u y + u 2 x u y = )]TJ/F35 10.9091 Tf 8.485 0 Td [(! 4 b m 4 + 2 F x u x )]TJ/F36 7.9701 Tf 6.587 0 Td [(F y u y 2 ; .16e @ t 0 u x u y + @ x 1 3 u y + u 2 x u y 2 + @ y 1 3 u x + u x u 2 y 2 = )]TJ/F35 10.9091 Tf 8.484 0 Td [(! 5 b m 5 + F x u y + F y u x 2 : .16f Similarly,theleadingordermomentequationsat O 2 canbeobtainedfromEq..15cas @ t 1 =0 ; .17a @ t 1 u x + @ x h 1 2 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1 )]TJ/F33 7.9701 Tf 12.104 4.295 Td [(1 2 ! 3 b m 3 + 1 2 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1 )]TJ/F33 7.9701 Tf 12.105 4.295 Td [(1 2 ! 4 b m 4 i + @ y h )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1 )]TJ/F33 7.9701 Tf 12.105 4.295 Td [(1 2 ! 5 b m 5 i =0 ; .17b @ t 1 u y + @ x h )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1 )]TJ/F33 7.9701 Tf 12.104 4.296 Td [(1 2 ! 5 b m 5 i + @ y h 1 2 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1 )]TJ/F33 7.9701 Tf 12.105 4.296 Td [(1 2 ! 3 b m 3 )]TJ/F33 7.9701 Tf 12.105 4.296 Td [(1 2 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1 )]TJ/F33 7.9701 Tf 12.104 4.296 Td [(1 2 ! 4 b m 4 i =0 : .17c Intheaboveequations,thesecond-order,non-equilibriummoments b m 3 , b m 4 and b m 5 correspondingto, b 0 xx + b 0 yy , b 0 xx )]TJ/F41 10.9091 Tf 11.49 0 Td [(b 0 yy and b 0 xy ,respectivelyareunknowns.Ideally,theyshould onlyberelatedtothestrainratetensorcomponentstorecoverthecorrectphysicsrelatedtothe viscousstress.However,aswillbeshouldbelow,onthestandardD2Q9latticetherewillbenonGIcontributionsdependentonthepreconditioningparameter .Inwhatfollows, b m 3 , b m 4 and b m 5 willbeobtainedfromEq..16d,Eq..16eandEq.2.16f,respectively.

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34 Now,fromEq..16d,thenon-equilibriummoment b m 3 canbewrittenas b m 3 = 1 ! 3 h )]TJ/F35 10.9091 Tf 8.485 0 Td [(@ t 0 2 3 + u 2 x + u 2 y )]TJ/F35 10.9091 Tf 10.909 0 Td [(@ x 4 3 u x + u x u 2 y 2 )]TJ/F35 10.9091 Tf 10.909 0 Td [(@ y 4 3 u y + u 2 x u y 2 + 2 F x u x + F y u y 2 i : .18 InordertosimplifyEq..18further,oneneedstoobtainexpressions,inparticular,for @ t 0 u 2 x , @ t 0 u 2 y , @ x u x u 2 y 2 and @ y u 2 x u y 2 .ItfollowsfromEq..16bthat @ t 0 u x = )]TJ/F15 10.9091 Tf 9.68 7.381 Td [(1 3 @ x )]TJ/F35 10.9091 Tf 10.909 0 Td [(@ x u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(@ y u x u y + F x : .19 Rearranging @ t 0 u 2 x as @ t 0 u 2 x = 2 u x @ t 0 u x + u 2 x @ t 0 : UsingEq..19andEq..16atoreplacethetimederivativeintherstandsecondtermsrespectively,ontherighthandsideoftheaboveequation,weget. @ t 0 u 2 x = 2 u x h )]TJ/F33 7.9701 Tf 9.681 4.295 Td [(1 3 @ x )]TJ/F35 10.9091 Tf 10.909 0 Td [(@ x u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(@ y u x u y + F x i + u 2 x [ @ x u x + @ y u y ] : .20 Similarly,wemaywrite @ t 0 u 2 y ! = 2 u y h )]TJ/F33 7.9701 Tf 9.681 4.295 Td [(1 3 @ y )]TJ/F35 10.9091 Tf 10.909 0 Td [(@ y u 2 y )]TJ/F35 10.9091 Tf 10.909 0 Td [(@ x u x u y + F y i + u 2 y [ @ x u x + @ y u y ] : .21 Thus,thetimederivativecanbereplacedwiththespatialderivative.Also,itreadilyfollows that )]TJ/F35 10.9091 Tf 8.485 0 Td [(@ x u x u 2 y 2 = )]TJ/F36 7.9701 Tf 9.681 6.536 Td [(u 2 y 2 @ x u x )]TJ/F33 7.9701 Tf 12.105 5.54 Td [(2 u x u y 2 @ x u y ; .22a )]TJ/F35 10.9091 Tf 8.484 0 Td [(@ y u 2 x u y 2 = )]TJ/F36 7.9701 Tf 9.681 5.374 Td [(u 2 x 2 @ y u y )]TJ/F33 7.9701 Tf 12.104 5.54 Td [(2 u x u y 2 @ y u x : .22b RearrangingEq..20andsimplifyingitfurtherbyretainingallcubicvelocitytermsandneglectingallothershigherordertermsinvelocitye.g.fthorderandhigherweget )]TJ/F35 10.9091 Tf 8.485 0 Td [(@ t 0 u 2 x = 2 u x 3 @ x + 2 u x 2 @ x u 2 x + 2 u 2 x 2 @ y u y + 2 u x u y 2 @ y u x )]TJ/F33 7.9701 Tf 12.105 4.378 Td [(2 F x u x 2 )]TJ/F36 7.9701 Tf 9.681 5.374 Td [(u 2 x @ x u x )]TJ/F36 7.9701 Tf 12.105 5.374 Td [(u 2 x @ y u y : .23

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35 Similarly,itfollowsfromEq..21that )]TJ/F35 10.9091 Tf 8.485 0 Td [(@ t 0 u 2 y = 2 u y 3 @ y + 2 u y 2 @ y u 2 y + 2 u 2 y 2 @ x u x + 2 u x u y 2 @ x u y + 2 F y u y 2 )]TJ/F36 7.9701 Tf 9.681 6.537 Td [(u 2 y @ x u x )]TJ/F36 7.9701 Tf 12.105 6.537 Td [(u 2 y @ y u y : .24 Now,toobtainanexpressionfor b m 3 ,wegroupallthehigherordertermsgiveninEqs..22a,.22b,.23 and.24.Itfollowsthatowingtothechoiceoftheo-diagonalthird-orderequilibriummomentswiththecubicvelocitytermsscaledby 2 i.e. b eq 0 xxy = 1 3 u y + u 2 x u y 2 ; b eq 0 xyy = 1 3 u x + u x u 2 y 2 attheoutsetfollowingEq..9earlier,allthecross-derivativespuriousterms,i.e. )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 u x u y @ x u y and )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 u x u y @ y u x cancel.Then,simplifyingthegroupingofalltheremaininghigherorderterms inEq..22a,Eq..22b,Eq.2.23andEq..24andretainingallcubicvelocitytermsand neglectingtermsofnegligiblehigherordersandafterconsiderablerearrangement,weget )]TJ/F35 10.9091 Tf 8.485 0 Td [(@ t 0 u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(@ t 0 u 2 y )]TJ/F35 10.9091 Tf 10.909 0 Td [(@ x u x u 2 y 2 )]TJ/F35 10.9091 Tf 10.909 0 Td [(@ y u 2 x u y 2 2 3 u x @ x + u y @ y )]TJ/F33 7.9701 Tf 14.479 4.296 Td [(2 2 F x u x + F y u y + h 4 2 )]TJ/F33 7.9701 Tf 12.404 4.296 Td [(1 u 2 x + 1 2 )]TJ/F33 7.9701 Tf 12.404 4.296 Td [(1 u 2 y i @ x u x + h 4 2 )]TJ/F33 7.9701 Tf 12.403 4.295 Td [(1 u 2 y + 1 2 )]TJ/F33 7.9701 Tf 12.403 4.295 Td [(1 u 2 x i @ y u y : .25 BysubstitutingtheaboveequationEq..25inEq..18andusing @ t 0 = )]TJ/F35 10.9091 Tf 8.485 0 Td [(@ x u x )]TJ/F35 10.9091 Tf -435.184 -23.995 Td [(@ y u y fromEq..16atofurthersimplifytheresultingexpressions,wenallygettheformof thenon-equilibriummoment b m 3 as b m 3 = )]TJ/F33 7.9701 Tf 12.22 4.931 Td [(2 3 ! 3 @ x u x + @ y u y + 2 3 ! 3 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 u x @ x + u y @ y + ! 3 h 4 2 )]TJ/F33 7.9701 Tf 12.403 4.296 Td [(1 u 2 x + 1 2 )]TJ/F33 7.9701 Tf 12.404 4.296 Td [(1 u 2 y i @ x u x + ! 3 h 4 2 )]TJ/F33 7.9701 Tf 12.404 4.295 Td [(1 u 2 y + 1 2 )]TJ/F33 7.9701 Tf 12.403 4.295 Td [(1 u 2 x i @ y u y : .26 Similarly,usingEq..16eandfollowinganalogousprocedureasabovefor b m 4 andusingEq..16f for b m 5 afterconsiderablealgebraicmanipulationsandsimplicationswegettheexpressionsfor

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36 theremainingnon-equilibriumsecond-ordermomentsas b m 4 = )]TJ/F33 7.9701 Tf 12.22 4.932 Td [(2 3 ! 4 @ x u x )]TJ/F35 10.9091 Tf 10.909 0 Td [(@ y u y + 2 3 ! 4 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 u x @ x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y @ y + ! 4 h 4 2 )]TJ/F33 7.9701 Tf 12.403 4.295 Td [(1 u 2 x )]TJ/F41 10.9091 Tf 10.909 12.109 Td [( 1 2 )]TJ/F33 7.9701 Tf 12.404 4.295 Td [(1 u 2 y i @ x u x + ! 4 h )]TJ/F41 10.9091 Tf 10.303 12.109 Td [( 4 2 )]TJ/F33 7.9701 Tf 12.403 4.295 Td [(1 u 2 y + 1 2 )]TJ/F33 7.9701 Tf 12.404 4.295 Td [(1 u 2 x i @ y u y ; .27 and b m 5 = )]TJ/F36 7.9701 Tf 14.337 4.931 Td [( 3 ! 5 @ x u y + @ y u x + 1 3 ! 5 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 u x @ y + u y @ x + 1 ! 5 1 2 )]TJ/F33 7.9701 Tf 12.404 4.296 Td [(1 u x u y @ x u x + @ y u y : .28 Therstterms,whichareunderlined,intherighthandsidesofEq..26,Eq..27andEq..28 areassociatedwiththerequiredowphysicsrelatedtothecomponentsoftheviscousstresstensor.Alltheremainingtermsintheseequationsarenon-GalileaninvarianttermsforthepreconditionedLBscheme.Thesespurioustermsarisebecausethediagonalthird-ordermoments b eq 0 xxx and b eq 0 yyy arenotsupportedbythestandardD2Q9lattice.However,suchdiscreteeectsarenot observedintheC-EanalysisofthecontinuousBoltzmannequation.Inordertoeliminatethe non-GIerrortermsbyothermeansinthenextsectiononthestandardlattice,weexplicitlyidentifythevariousnon-GItermsinthecomponentsofthesecond-ordernon-equilibriummoments as E 3 g = 2 3 ! 3 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 u x @ x + u y @ y ; .29a E 3 gu = ! 3 4 2 )]TJ/F15 10.9091 Tf 12.505 7.38 Td [(1 u 2 x + 1 2 )]TJ/F15 10.9091 Tf 12.505 7.38 Td [(1 u 2 y @ x u x + ! 3 4 2 )]TJ/F15 10.9091 Tf 12.505 7.38 Td [(1 u 2 y + 1 2 )]TJ/F15 10.9091 Tf 12.504 7.38 Td [(1 u 2 x @ y u y : .29b E 4 g = 2 3 ! 4 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 u x @ x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y @ y ; .30a E 4 gu = ! 4 4 2 )]TJ/F15 10.9091 Tf 12.504 7.38 Td [(1 u 2 x )]TJ/F41 10.9091 Tf 10.909 15.382 Td [( 1 2 )]TJ/F15 10.9091 Tf 12.504 7.38 Td [(1 u 2 y @ x u x + ! 4 )]TJ/F41 10.9091 Tf 10.303 15.382 Td [( 4 2 )]TJ/F15 10.9091 Tf 12.504 7.38 Td [(1 u 2 y + 1 2 )]TJ/F15 10.9091 Tf 12.505 7.38 Td [(1 u 2 x @ y u y ; .30b

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37 and E 5 g = 1 3 ! 5 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 u x @ y + u y @ x ; .31a E 5 gu = ! 5 1 2 )]TJ/F15 10.9091 Tf 12.505 7.38 Td [(1 u x u y @ x u x + ! 5 1 2 )]TJ/F15 10.9091 Tf 12.505 7.38 Td [(1 u x u y @ y u y : .31b Then,wecanrewritethenon-equilibriumsecond-ordermoments b m 3 = b 0 xx + b 0 yy = )]TJ/F15 10.9091 Tf 12.622 7.38 Td [(2 3 ! 3 @ x u x + @ y u y + E 3 g + E 3 gu ; .32 b m 4 = b 0 xx )]TJ/F41 10.9091 Tf 11.022 0 Td [(b 0 yy = )]TJ/F15 10.9091 Tf 12.622 7.38 Td [(2 3 ! 4 @ x u x )]TJ/F35 10.9091 Tf 10.909 0 Td [(@ y u y + E 4 g + E 4 gu ; .33 b m 5 = b 0 xy = )]TJ/F35 10.9091 Tf 15.349 7.38 Td [( 3 ! 5 @ x u y + @ y u x + E 5 g + E 5 gu : .34 Someinterestingobservationscanbemadefromtheaboveanalysis: i whentheLBschemeis preconditioned,i.e. 6 =1,non-GItermspersistintermsofvelocityanddensitygradientsfor allthesecond-ordernon-equilibriummoments,includingtheo-diagonalmoment b m 5 = b 0 xy , unlikethatforthesimulationofthestandardNSequationsi.e.with =1.However,thenonGIcubicvelocitycontributionsin b m 5 vanishforincompressibleow r u =0,i.e. E 5 gu = 0. ii .Ingeneraltheprefactorsappearinginthenon-GItermsforthediagonalcomponents, i.e.in b m 3 and b m 4 exhibitdramaticallydierentbehaviourfortheasymptoticlimitcases:No preconditioningcase ! 1: 4 2 )]TJ/F33 7.9701 Tf 12.403 4.295 Td [(1 3, 1 2 )]TJ/F33 7.9701 Tf 12.404 4.295 Td [(1 0;strongpreconditioningcase ! 0: 4 2 )]TJ/F33 7.9701 Tf 12.404 4.295 Td [(1 4 2 , 1 2 )]TJ/F33 7.9701 Tf 12.403 4.295 Td [(1 1 2 .Thus,duetothecomplicatedstructureofthetruncation errorsandtheirdependenceon ,thenon-GItermsinthediagonalmomentcomponentsmodify signicantlyas variesduetopreconditioning. iii when =1,i.e.whenourpreconditioned LBschemerevertstothesolutionofthestandardNSequations, E 3 g = E 4 g = E 5 g =0, E 3 gu = 3 ! 3 )]TJ/F35 10.9091 Tf 5 -8.837 Td [(u 2 x @ x u x + u 2 y @ y u y , E 4 gu = 3 ! 4 )]TJ/F35 10.9091 Tf 5 -8.837 Td [(u 2 x @ x u x )]TJ/F35 10.9091 Tf 10.91 0 Td [(u 2 y @ y u y ,and E 5 gu =0.Thatis,thenon-GIterms becomeidenticaltotheresultsreportedby[42]and[43].

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38 2.4DerivationofCorrectionsviaExtendedMomentEquilibriaforElimination ofCubicVelocityerrorsinPreconditionedMacroscopicEquations Inordertoeectivelyeliminatethenon-GIerrortermsgiveninEq..29a-.31bthatappear inthenon-equilibriummoments b m 3 , b m 4 and b m 5 intheprevioussectionseeEqs..32.34arisingduetothethird-orderdiagonalequilibriummoments b eq 0 xxx and b eq 0 yyy notbeing independentlysupportedbytheD2Q9lattice,weconsideranapproachbasedontheextended momentequilibria.Inotherwords,weextendedthesecond-ordermomentequilibriabyincluding extragradienttermswithunknowncoecientsasfollows: b f eq = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 b m eq 0 b m eq 1 b m eq 2 b m eq 3 b m eq 4 b f eq 5 b m eq 6 b m eq 7 b m eq 8 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 + t 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 0 0 0 b m eq 3 b m eq 4 b m eq 5 0 0 0 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 b eq 0 0 b eq 0 x b eq 0 y b eq 0 xx + b eq 0 yy b eq 0 xx )]TJ/F41 10.9091 Tf 11.021 0 Td [(b eq 0 yy b eq 0 xy b eq 0 xxy b eq 0 xyy b eq 0 xxyy 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 + t 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 0 0 0 3 x @ x u x + 3 y @ y u y + 3 x @ x + 3 y @ y 4 x @ x u x )]TJ/F35 10.9091 Tf 10.909 0 Td [( 4 y @ y u y + 4 x @ x )]TJ/F35 10.9091 Tf 10.909 0 Td [( 4 y @ y 5 x @ x u x + 5 y @ y u y + 5 x @ x + 5 y @ y 0 0 0 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 : .35 Inotherwords,thecorrectionstothesecond-ordermomentsaregivenby b m eq 3 = 3 x @ x u x + 3 y @ y u y + 3 x @ x + 3 y @ y ; .36a b m eq 4 = 4 x @ x u x )]TJ/F35 10.9091 Tf 10.909 0 Td [( 4 y @ y u y + 4 x @ x )]TJ/F35 10.9091 Tf 10.909 0 Td [( 4 y @ y ; .36b b m eq 5 = 5 x @ x u x + 5 y @ y u y + 5 x @ x + 5 y @ y ; .36c wherethecoecients j x , j y , j x and j y ,where j =3 ; 4 ; 5aretobedeterminedfromamodied Chapman-Enskoganalysissothatthenon-GIcubicvelocitytermsareeectivelyremovedfrom theemergentpreconditionedmacroscopicmomentequations.

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39 WenowapplyaChapman-EnskogC-Eexpansionbytakingintoaccountthemodiedequilibriawhichisnowgivenas b m eq = b m eq + t b m eq ,where b m eq isthemomentequilibriapresentedintheprevioussectionand b m eq isthecorrectiontothisequilibria.Asaresult,theC-E expansiongivenasEq..12andEq..13arenowreplacedwith b m = b m eq + b m eq + b m + 2 b m + ;@ t = @ t 0 + @ t 1 + 2 @ t 2 + : .37 Then,byusingaTaylorexpansiongiveninEq..14forthestreamingoperatorinEq.b alongwithabovemodiedC-EexpansionEq..37,wegetthefollowinghierarchyofmoment equationsatdierentordersin : O 0 : b m = b m eq ; .38a O 1 : @ t 0 + b E i @ i b m = )]TJ/F41 10.9091 Tf 9.122 2.879 Td [(b b m )]TJ/F41 10.9091 Tf 13.106 0.152 Td [(b m eq + b S ; .38b O 2 : @ t 1 b m + @ t 0 + b E i @ i h I )]TJ/F33 7.9701 Tf 12.105 4.295 Td [(1 2 b i b m + @ t 0 + b E i @ i h 1 2 b b m eq i = )]TJ/F41 10.9091 Tf 9.121 2.879 Td [(b b m ; .38c where b E i = T e i I T )]TJ/F33 7.9701 Tf 6.587 0 Td [(1 and i 2f x;y g .Therelevant O equationsfortherstordermoments aregiveninEqs..16a-.16c.However,theequationsofthesecondordermomentsarenow modiedduetothepresenceoftheextendedmomentequilibria b m eq inEq..38bwhichare nowgivenbyinsteadofEqs..16d-.16f @ t 0 2 3 + u 2 x + u 2 y + @ x 4 3 u x + u x u 2 y 2 + @ y 4 3 u y + u 2 x u y 2 = )]TJ/F35 10.9091 Tf 8.485 0 Td [(! 3 b m 3 + ! 3 b m eq 3 + 2 F x u x + F y u y 2 ; .39a @ t 0 u 2 x )]TJ/F36 7.9701 Tf 6.587 0 Td [(u 2 y + @ x 2 3 u x )]TJ/F36 7.9701 Tf 12.105 6.537 Td [(u x u 2 y 2 + @ y )]TJ/F33 7.9701 Tf 9.681 4.295 Td [(2 3 u y + u 2 x u y = )]TJ/F35 10.9091 Tf 8.485 0 Td [(! 4 b m 4 + ! 4 b m eq 4 + 2 F x u x )]TJ/F36 7.9701 Tf 6.587 0 Td [(F y u y 2 ; .39b @ t 0 u x u y + @ x 1 3 u y + u 2 x u y 2 + @ y 1 3 u x + u x u 2 y 2 = )]TJ/F35 10.9091 Tf 8.485 0 Td [(! 5 b m 5 + ! 5 b m eq 5 + F x u y + F y u x 2 : .39c Similarly,theleadingordermomentequationsof O 2 whicharemodiedby b m eq asshownin

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40 Eq..38careobtainedasinsteadofEqs..17a-.17c @ t 1 =0 ; .40a @ t 1 u x + @ x h 1 2 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1 )]TJ/F33 7.9701 Tf 12.105 4.295 Td [(1 2 ! 3 b m 3 + 1 2 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1 )]TJ/F33 7.9701 Tf 12.104 4.295 Td [(1 2 ! 4 b m 4 i + @ y h )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1 )]TJ/F33 7.9701 Tf 12.104 4.295 Td [(1 2 ! 5 b m 5 i + @ x h 1 4 ! 3 b m eq 3 + 1 4 ! 4 b m eq 4 i + @ y h 1 2 ! 5 b m eq 5 i =0 ; .40b @ t 1 u y + @ x h )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1 )]TJ/F33 7.9701 Tf 12.105 4.296 Td [(1 2 ! 5 b m 5 i + @ y h 1 2 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1 )]TJ/F33 7.9701 Tf 12.105 4.296 Td [(1 2 ! 3 b m 3 )]TJ/F33 7.9701 Tf 12.105 4.295 Td [(1 2 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1 )]TJ/F33 7.9701 Tf 12.105 4.296 Td [(1 2 ! 4 b m 4 i + @ x h 1 2 ! 5 b m eq 5 i + @ y h 1 4 ! 3 b m eq 3 )]TJ/F33 7.9701 Tf 12.105 4.295 Td [(1 4 ! 4 b m eq 4 i =0 : .40c Thenon-equilibriummoment b m 3 isnowobtainedfromEq..39aas b m 3 = 1 ! 3 h )]TJ/F35 10.9091 Tf 8.485 0 Td [(@ t 0 2 3 + u 2 x + u 2 y )]TJ/F35 10.9091 Tf 10.909 0 Td [(@ x 4 3 u x + u x u 2 y 2 )]TJ/F35 10.9091 Tf 10.909 0 Td [(@ y 4 3 u y + u 2 x u y 2 + 2 F x u x + F y u y 2 i + b m eq 3 : .41 AllthetermswithinthesquarebracketsintheaboveequationexactlycorrespondstoEq..32. Hence,Eq..41reducesto b m 3 = )]TJ/F15 10.9091 Tf 12.622 7.38 Td [(2 3 ! 3 @ x u x + @ y u y + E 3 g + E 3 gu + b m eq 3 ; .42 wherethenon-GIerrorterms E 3 g and E 3 gu aregiveninEqs..29aand.29b,respectively, andtheextendedmomentequilibrium b m eq 3 inEq..36a.Similarly,thenon-equilibriummoment b m 4 isobtainedfromEq..39bandusingEq..33forsimplication,andfor b m 5 using Eqs..39cand.34,wenallyget b m 4 = )]TJ/F33 7.9701 Tf 12.221 4.932 Td [(2 3 ! 4 @ x u x )]TJ/F35 10.9091 Tf 10.909 0 Td [(@ y u y + E 4 g + E 4 gu + b m eq 4 ; .43 b m 5 = )]TJ/F36 7.9701 Tf 14.338 4.931 Td [( 3 ! 5 @ x u y + @ y u x + E 5 g + E 5 gu + b m eq 5 : .44 Here,thenon-GIerrorterms E 4 g and E 4 gu aregiveninEq..30aandEq..30b,respectively, andthecorrectionequilibriummoment b m eq 4 inEq..36b.Likewise, E 5 g and E 5 gu areobtained fromEqs..31aand.31brespectivelyand b m eq 5 ispresentedinEq..36c. Now,inordertoobtainthepreconditionedmomentsystemfortheconservedmoments,wecombine O equationsEqs..16a-.16cwith Eq..40a-.40cforthecorrespondingequa-

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41 tionsat O 2 ,andusing @ t = @ t 0 + @ t 1 ,weget @ t + @ x u x + @ y u y =0 ; .45a @ t u x + @ x 1 3 + u 2 x + @ y u x u y = F x )]TJ/F35 10.9091 Tf 10.909 0 Td [(@ x h 1 2 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1 )]TJ/F36 7.9701 Tf 12.105 4.488 Td [(! 3 2 b m 3 + 1 2 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1 )]TJ/F36 7.9701 Tf 12.104 4.488 Td [(! 4 2 b m 4 i )]TJ/F35 10.9091 Tf 10.91 0 Td [(@ y h 1 2 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1 )]TJ/F36 7.9701 Tf 12.104 4.488 Td [(! 5 2 b m 5 i )]TJ/F35 10.9091 Tf 8.485 0 Td [(@ x h ! 3 4 b m eq 3 + ! 4 4 b m eq 4 i )]TJ/F35 10.9091 Tf 10.909 0 Td [(@ y h ! 5 2 b m eq 5 i ; .45b @ t u y + @ x u x u y + @ y 1 3 + u 2 y = F y )]TJ/F35 10.9091 Tf 10.909 0 Td [(@ x h )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1 )]TJ/F36 7.9701 Tf 12.105 4.489 Td [(! 5 2 b m eq 5 i )]TJ/F35 10.9091 Tf 8.485 0 Td [(@ y h 1 2 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1 )]TJ/F36 7.9701 Tf 12.104 4.488 Td [(! 3 2 b m 3 )]TJ/F33 7.9701 Tf 12.104 4.295 Td [(1 2 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1 )]TJ/F36 7.9701 Tf 12.105 4.488 Td [(! 4 2 b m 4 i )]TJ/F35 10.9091 Tf 8.485 0 Td [(@ x h ! 5 2 b m eq 5 i )]TJ/F35 10.9091 Tf 10.909 0 Td [(@ y h ! 3 4 b m eq 3 )]TJ/F36 7.9701 Tf 12.105 4.489 Td [(! 3 4 b m eq 4 i : .45c OurgoalistoshowthattheaboveequationsEq..45a-.45cisconsistentwiththepreconditionedNSequationsEq.presentedinSec.2.1withouttheidentiedtruncationerrors,i.e. withoutinvolvingthenon-GIcubicvelocitydefects.Now,inordertorelatethemomentcorrections b m eq 3 , b m eq 4 and b m eq 5 appearingintheequilibriawiththenon-GIerrorterms,witha viewtoeliminatethem,considertherighthandsideofEq..45bi.e.the x -momentumequationandsubstitutefor b m 3 , b m 4 and b m 5 fromEq..42,Eq..43andEq..44,respectively,whichbecomes = F x + @ x h + 1 3 1 ! 3 )]TJ/F33 7.9701 Tf 12.105 4.295 Td [(1 2 @ x u x + @ y u y + 1 3 1 ! 4 )]TJ/F33 7.9701 Tf 12.105 4.295 Td [(1 2 @ x u x )]TJ/F35 10.9091 Tf 10.909 0 Td [(@ y u y i + @ y h 1 3 1 ! 5 )]TJ/F33 7.9701 Tf 12.104 4.296 Td [(1 2 @ x u y + @ y u x i )]TJ/F35 10.9091 Tf 8.485 0 Td [(@ x 1 2 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1 )]TJ/F36 7.9701 Tf 12.105 4.489 Td [(! 3 2 E 3 g + E 3 gu + 1 2 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1 )]TJ/F36 7.9701 Tf 12.104 4.489 Td [(! 4 2 E 4 g + E 4 gu )]TJ/F35 10.9091 Tf 10.909 0 Td [(@ y )]TJ/F15 10.9091 Tf 9.546 -8.837 Td [(1 )]TJ/F36 7.9701 Tf 12.105 4.489 Td [(! 5 2 E 5 g + E 5 gu )]TJ/F35 10.9091 Tf 8.485 0 Td [(@ x h 1 2 b m eq 3 + 1 2 b m eq 4 i )]TJ/F35 10.9091 Tf 10.909 0 Td [(@ y h b m eq 5 i : .46 Thersttwolinesintheaboveequationscorrespondtothephysics,whilethethirdlinecorrespondstothespuriousnon-GItermsarisingfromdiscretelatticeeectsandthefourthlineare relatedtoequilibriumcorrections. Inordertoeliminatethecubicvelocitytruncationerrors,itfollowsthatthethirdandfourth

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42 linesintheaboveequationEq..46sumtozero.Thisyields 1 )]TJ/F35 10.9091 Tf 12.105 7.38 Td [(! 3 2 E 3 g + E 3 gu + b m eq 3 =0 ; .47a 1 )]TJ/F35 10.9091 Tf 12.105 7.38 Td [(! 4 2 E 4 g + E 4 gu + b m eq 4 =0 ; .47b 1 )]TJ/F35 10.9091 Tf 12.105 7.38 Td [(! 5 2 E 5 g + E 5 gu + b m eq 5 =0 : .47c TheaboveequationsEqs..47a-.47c,representthekeyconstraintrelationsbetweenthenonGIerrortermsandthemomentequilibriacorrectiontermstoobtainapreconditionedcascaded centralmomentLBmodelwithoutcubicvelocitydefects. Furtheranalysisshowsthattheseconstraintsholdidenticallyforthe y -momentumaswellEq.2.45c. NowconsideringEq..47aandusingEq..29aand.29bfor E 3 g and E 3 gu ,respectively,the extendmomentequilibrium b m eq 3 isgivenas b m eq 3 = 3 x @ x u x + 3 y @ y u y + 3 x @ x + 3 y @ y ; wherethecoecientsobtainedaftermatchingaregivenby 3 x = )]TJ/F41 10.9091 Tf 10.303 12.109 Td [( 1 ! 3 )]TJ/F33 7.9701 Tf 12.104 4.295 Td [(1 2 h 4 2 )]TJ/F33 7.9701 Tf 12.403 4.295 Td [(1 u 2 x + 1 2 )]TJ/F33 7.9701 Tf 12.404 4.295 Td [(1 u 2 y i ; .48a 3 y = )]TJ/F41 10.9091 Tf 10.303 12.109 Td [( 1 ! 3 )]TJ/F33 7.9701 Tf 12.105 4.295 Td [(1 2 h 4 2 )]TJ/F33 7.9701 Tf 12.404 4.295 Td [(1 u 2 y + 1 2 )]TJ/F33 7.9701 Tf 12.404 4.295 Td [(1 u 2 x i ; .48b 3 x = )]TJ/F33 7.9701 Tf 9.68 4.296 Td [(2 3 1 ! 3 )]TJ/F33 7.9701 Tf 12.104 4.296 Td [(1 2 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 u x ; .48c 3 y = )]TJ/F33 7.9701 Tf 9.68 4.295 Td [(2 3 1 ! 3 )]TJ/F33 7.9701 Tf 12.105 4.295 Td [(1 2 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 u y : .48d Similarly,fromEq..30a,.30b,.36band.47b,wecanobtainthecoecientof b m eq 4 , andfromEq..31a,Eq..31b,Eq..36candEq..47c,thosefor b m eq 5 canbedetermined.Theresultsreadasfollows: b m eq 4 = 4 x @ x u x )]TJ/F35 10.9091 Tf 10.909 0 Td [( 4 y @ y u y + 4 x @ x )]TJ/F35 10.9091 Tf 10.909 0 Td [( 4 y @ y ;

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43 where 4 x = )]TJ/F41 10.9091 Tf 10.303 12.109 Td [( 1 ! 4 )]TJ/F33 7.9701 Tf 12.104 4.295 Td [(1 2 h 4 2 )]TJ/F33 7.9701 Tf 12.403 4.295 Td [(1 u 2 x )]TJ/F41 10.9091 Tf 10.909 12.109 Td [( 1 2 )]TJ/F33 7.9701 Tf 12.404 4.295 Td [(1 u 2 y i ; .49a 4 y = 1 ! 4 )]TJ/F33 7.9701 Tf 12.105 4.295 Td [(1 2 h )]TJ/F41 10.9091 Tf 10.303 12.109 Td [( 4 2 )]TJ/F33 7.9701 Tf 12.404 4.295 Td [(1 u 2 y + 1 2 )]TJ/F33 7.9701 Tf 12.404 4.295 Td [(1 u 2 x i ; .49b 4 x = )]TJ/F33 7.9701 Tf 9.68 4.296 Td [(2 3 1 ! 4 )]TJ/F33 7.9701 Tf 12.104 4.296 Td [(1 2 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 u x ; .49c 4 y = )]TJ/F33 7.9701 Tf 9.68 4.295 Td [(2 3 1 ! 4 )]TJ/F33 7.9701 Tf 12.105 4.295 Td [(1 2 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 u y ; .49d and b m eq 5 = 5 x @ x u x + 5 y @ y u y + 5 x @ x + 5 y @ y ; where 5 x = )]TJ/F41 10.9091 Tf 10.303 12.109 Td [( 1 ! 5 )]TJ/F33 7.9701 Tf 12.104 4.295 Td [(1 2 1 2 )]TJ/F33 7.9701 Tf 12.404 4.295 Td [(1 u x u y ; .50a 5 y = )]TJ/F41 10.9091 Tf 10.303 12.109 Td [( 1 ! 5 )]TJ/F33 7.9701 Tf 12.104 4.295 Td [(1 2 1 2 )]TJ/F33 7.9701 Tf 12.404 4.295 Td [(1 u x u y ; .50b 5 x = )]TJ/F33 7.9701 Tf 9.681 4.296 Td [(1 3 1 ! 5 )]TJ/F33 7.9701 Tf 12.105 4.296 Td [(1 2 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 u y ; .50c 5 y = )]TJ/F33 7.9701 Tf 9.68 4.295 Td [(1 3 1 ! 5 )]TJ/F33 7.9701 Tf 12.105 4.295 Td [(1 2 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 u x : .50d Notethat,asaspecialcase,when =1,i.e.theLBmodelisusedtosolvethestandardNS equationswithoutpreconditioning,then 3 x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 ! 3 )]TJ/F33 7.9701 Tf 11.353 4.295 Td [(1 2 u 2 x , 3 y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 ! 3 )]TJ/F33 7.9701 Tf 11.352 4.295 Td [(1 2 u 2 y , 4 x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 ! 4 )]TJ/F33 7.9701 Tf -458.32 -19.701 Td [(1 2 u 2 x , 4 x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 ! 4 )]TJ/F33 7.9701 Tf 13.159 4.295 Td [(1 2 u 2 y ,andalltheremainingcoecientgotozero.Insuchacase,these momentcorrectionstotheequilibriabecomeidenticaltotheGIcorrectionspresentedby[43]and equivalenttothealternativeGIformulationwithoutcubicvelocityerrorsintroducedby[42]. Finally,usingtheaboveextendedmomentequilibria b m eq 3 , b m eq 4 and b m eq 5 andtheexpressionforthenon-equilibriummoments b m 3 , b m 4 and b m 5 fromEq..42-Eq..44along withtheconstraintrelations,i.e.Eqs..47a-.47cinEqs..45a-.45c,weget @ t + r j =0 ;; .51 @ t j x + r j u x = )]TJ/F35 10.9091 Tf 8.485 0 Td [(@ x p + @ x # 4 @ x j x )]TJ/F55 10.9091 Tf 10.909 0 Td [(r j + # 3 r j + @ y # 5 @ x j y + @ y j x + F x ; .52

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44 @ t j y + r j u y = )]TJ/F35 10.9091 Tf 8.485 0 Td [(@ y p + @ x # 5 @ x j y + @ y j x + @ y # 4 @ y j y )]TJ/F55 10.9091 Tf 10.909 0 Td [(r j + # 3 r j + F y ; .53 where p = 3 isthepressure, j = u ,andthebulkandshearviscositiesare,respectivelygiven by # 3 = 3 1 ! 3 )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(1 2 ;# 4 = 3 1 ! 4 )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 2 ;# 5 = 3 1 ! 5 )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 2 : .54 Thus,Eqs..51-.53areconsistentwiththepreconditionedNSequationsgiveninEqs..1a.1bwithoutcubicvelocitydefectsinGIduetotheuseoftheextendedmomentequilibriapresentedearlier. 2.5GalileanInvariantPreconditionedCascadedCentralMomentLBMwithout CubicVelocityErrorsonaStandardLattice ThecascadedcentralmomentLBMwithforcingtermpresentedinEqs..2a,.2b,.3and modifytoenforceGIwithoutcubicvelocityerrorsasfollows.EquationsEq..2a,Eq..2b andEq..3remainsthesameasbeforeandthecollisionkernelgiveninEq.ismodiedto accountfortheextendedmomentequilibriainthesecondordermomentsaswellascorrections tothethird-orderequilibriummoments.Thechangeofmoments b g 3 , b g 4 and b g 5 forthesecondordercomponentsfollowbyaugmentingthecorrespondingmomentequilibriawiththeextended momentequilibriaincorporatingtheGIcorrectionsidentiedintheprevioussection.Onthe otherhand,owingtothecascadedstructureofthecollisionkernel,theGIcorrectionstothe thirdordermomentchanges b g 6 and b g 7 ,whichdependonthelowerordermomentchanges,for thepreconditionedcentralmomentLBschemeneedtobeconstructedcarefully.Theyareobtainedbyprescribingtherelaxationofthethirdordercentralmomentcomponentstotheircorrespondingcentralmomentequilibria.Followingthederivationgivenin[19],theycanthenbe representedas )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 u y b g 3 )]TJ/F15 10.9091 Tf 9.833 0 Td [(2 u y b g 4 )]TJ/F15 10.9091 Tf 9.834 0 Td [(8 u x b g 5 )]TJ/F15 10.9091 Tf 9.833 0 Td [(4 b g 6 = ! 6 [ b eq xxy )]TJ/F41 10.9091 Tf 9.946 0 Td [(b xxy ]and )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 u x b g 3 +2 u x b g 4 )]TJ/F15 10.9091 Tf 9.833 0 Td [(8 u y b g 5 )]TJ/F15 10.9091 Tf 9.833 0 Td [(4 b g 7 = ! 7 [ b eq xyy )]TJ/F41 10.9091 Tf 11.619 0 Td [(b xyy ],where b xxy and b xyy arethethirdordercentralmomentcomponents,and b eq xxy

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45 and b eq xyy ,respectively,aretheirequilibria.Rewritingthesecentralmomentrelaxationsinterms oftherelaxationsoftherawmomentcomponentsofthethirdandlowerordersviathebinomial theorem,itfollowsthat b g 6 = ! 6 4 h b 0 xxy )]TJ/F41 10.9091 Tf 11.022 0 Td [(b eq 0 xxy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x b 0 xy )]TJ/F41 10.9091 Tf 11.022 0 Td [(b eq 0 xy )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y b 0 xx )]TJ/F41 10.9091 Tf 11.021 0 Td [(b eq 0 xx i )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y 3 2 b g 3 + 1 2 b g 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x b g 5 ; b g 7 = ! 7 4 h b 0 xyy )]TJ/F41 10.9091 Tf 11.022 0 Td [(b eq 0 xyy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y b 0 xy )]TJ/F41 10.9091 Tf 11.021 0 Td [(b eq 0 xy )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x b 0 yy )]TJ/F41 10.9091 Tf 11.022 0 Td [(b eq 0 yy i )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x 3 2 b g 3 )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(1 2 b g 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y b g 5 : Now,usingthecomponentsofthepreconditionedrawmomentequilibria,includingthoseforthe thirdorderequilibriummomentswiththeGIcorrectionsfromEq..10,thenalexpressions forthechangeinmomentsforthecollisionkernel b g 6 and b g 7 canbederived.Thus,themodied preconditionedcollisionkernelwiththeGIcorrectionsreads b g 0 =0 ; b g 1 =0 ; b g 2 =0 ; b g 3 = ! 3 12 n 2 3 + u 2 x + u 2 y = )]TJ/F15 10.9091 Tf 10.909 0 Td [( b 0 xx + b 0 yy )]TJ/F33 7.9701 Tf 12.105 4.295 Td [(1 2 b 0 xx + b 0 yy + 3 x @ x u x + 3 y @ y u y t + 3 x @ x + 3 y @ y t o ; b g 4 = ! 4 4 n u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y = )]TJ/F15 10.9091 Tf 10.909 0 Td [( b 0 xx )]TJ/F41 10.9091 Tf 11.021 2.182 Td [(b 0 yy )]TJ/F33 7.9701 Tf 12.104 4.295 Td [(1 2 b 0 xx )]TJ/F41 10.9091 Tf 11.191 0 Td [(b 0 yy + 4 x @ x u x )]TJ/F35 10.9091 Tf 10.909 0 Td [( 4 y @ y u y t + 4 x @ x )]TJ/F35 10.9091 Tf 10.909 0 Td [( 4 y @ y t o ; b g 5 = ! 5 4 n u x u y = )]TJ/F41 10.9091 Tf 11.021 2.182 Td [(b 0 xy )]TJ/F33 7.9701 Tf 12.105 4.296 Td [(1 2 b 0 xy + 5 x @ x u x + 5 y @ y u y t + 5 x @ x + 5 y @ y t o ; b g 6 = ! 6 4 3 )]TJ/F33 7.9701 Tf 14.48 4.295 Td [(1 2 u 2 x u y + b 0 xxy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x b 0 xy )]TJ/F35 10.9091 Tf 10.91 0 Td [(u y b 0 xx )]TJ/F33 7.9701 Tf 12.104 4.295 Td [(1 2 u y b g 3 + b g 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x b g 5 ; b g 7 = ! 7 4 3 )]TJ/F33 7.9701 Tf 14.479 4.295 Td [(1 2 u x u 2 y + b 0 xyy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y b 0 xy )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x b 0 yy )]TJ/F33 7.9701 Tf 12.104 4.295 Td [(1 2 u x b g 3 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y b g 5 ; b g 8 = ! 8 4 n 1 9 +3 u 2 x u 2 y )]TJ/F41 10.9091 Tf 10.909 12.11 Td [(h b 0 xxyy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x b 0 xyy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y b 0 xxy + u 2 x b 0 yy + u 2 y b 0 xx +4 u x u y b 0 xy io )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 b g 3 )]TJ/F33 7.9701 Tf 12.104 4.295 Td [(1 2 u 2 y b g 3 + b g 4 )]TJ/F33 7.9701 Tf 12.105 4.295 Td [(1 2 u 2 x b g 3 )]TJ/F41 10.9091 Tf 10.98 0 Td [(b g 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 u x u y b g 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y b g 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x b g 7 : wherethevariouscoecients j x , j y , j x and j y where j =3 ; 4and5aregiveninEqs..48b.48d,and.49b-.49dand.50a-.50d.TheGIcorrectionsareidentiedbymeansof theunderlinedtermsinthecascadedcollisionkerneltermsintheaboveequation. ItmaybenotedthatotherGIpreconditionedLBschemeswithoutcubicvelocityerrorscanbe

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46 constructedfromourresultsintheprevioussection.Forexample,anon-orthogonalmoment basedmultiplerelaxationtimeLBmethodreadilyfollowsfromtheanalysispresentedbefore. Thespatialgradientsforthevelocitycomponentsandthedensityappearingintheextended momentequilibriacanbecalculatedusingisotropicnitedierenceschemes.Alternatively,the diagonalstrainratecomponents @ x u x and @ y u y canbelocallyobtainedfromnon-equilibrium momentsasfollows,whichisusedinoursimulationstudiespresentedinthenextsection.From Eqs..42and.47aandrearranging,onemaywritetheresultingexpressionasfollows: )]TJ/F35 10.9091 Tf 8.485 0 Td [(c 1 @ x u x )]TJ/F35 10.9091 Tf 10.909 0 Td [(c 2 @ y u y = b m 3 )]TJ/F35 10.9091 Tf 10.909 0 Td [(e : .55 Similarly,fromEq..43andEq..47b,itfollowsthat )]TJ/F41 10.9091 Tf 8.421 0 Td [(e c 1 @ x u x + e c 2 @ y u y = b m 4 )]TJ/F41 10.9091 Tf 12.835 0 Td [(e e ; .56 wherethecoecients c 1 , c 2 , e c 1 and e c 1 andtheparameters e and e e aredenedas c 1 = h 2 3 ! 3 + P i ; e c 1 = h 2 3 ! 4 + e P i ; .57 c 2 = h 2 3 ! 3 + Q i ; e c 2 = h 2 3 ! 4 + e Q i : .58 Here, P = )]TJ/F33 7.9701 Tf 9.681 4.295 Td [(1 2 )]TJ/F35 10.9091 Tf 5 -8.836 Td [(A u 2 x + B u 2 y ;Q = )]TJ/F33 7.9701 Tf 9.681 4.295 Td [(1 2 )]TJ/F35 10.9091 Tf 5 -8.836 Td [(A u 2 y + B u 2 x ; e P = )]TJ/F33 7.9701 Tf 9.68 4.296 Td [(1 2 )]TJ/F35 10.9091 Tf 5 -8.837 Td [(A u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(B u 2 y ; e Q = )]TJ/F33 7.9701 Tf 9.68 4.296 Td [(1 2 )]TJ/F35 10.9091 Tf 5 -8.837 Td [(A u 2 y )]TJ/F35 10.9091 Tf 10.909 0 Td [(B u 2 x where A = 4 2 )]TJ/F33 7.9701 Tf 12.404 4.295 Td [(1 , B = 1 2 )]TJ/F33 7.9701 Tf 12.404 4.295 Td [(1 , C = 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 and e = 1 3 C u x @ x + u y @ y ; e e = 1 3 C u x @ x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y @ y : .59 SolvingEqs..55and.56for @ x u x and @ y u y ,weget @ x u x = h e c 2 b m 3 )]TJ/F35 10.9091 Tf 10.909 0 Td [(e + c 2 b m 4 )]TJ/F41 10.9091 Tf 12.836 0 Td [(e e i = [ )]TJ/F35 10.9091 Tf 8.485 0 Td [(c 1 e c 2 )]TJ/F41 10.9091 Tf 10.845 0 Td [(e c 1 c 2 ] ; .60a @ y u y = )]TJ/F41 10.9091 Tf 10.303 12.109 Td [(h c 1 b m 4 )]TJ/F41 10.9091 Tf 11.024 0 Td [(e e + e c 1 b m 3 )]TJ/F35 10.9091 Tf 10.909 0 Td [(e i = [ )]TJ/F35 10.9091 Tf 8.485 0 Td [(c 1 e c 2 )]TJ/F41 10.9091 Tf 10.845 0 Td [(e c 1 c 2 ] : .60b

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47 Here,thedensitygradientsappearingin e and e e Eqs..59maybecomputedusingaisotropic nitedierencescheme.InEqs..60aand.60b,allthecoecientsinvolving needtobe computedonlyoncebeforethestartofcomputationsforecientimplementation;quantitiessuch as u 2 x and u 2 y appearinginthefactors P , Q , e P and e Q aboveneedtobereusedratherthanperformtheproductcalculationsforeveryoccurrence.Acomparisonofthecomputationalcosts fortheuncorrectedpreconditionedLBschemeandtheGIcorrectedpreconditionedformulation ispresentedforabenchmarkcasestudyonthefour-rollsmillowproblemattheendofthe numericalresultssectionseeSec.2.6.5,whichalsodemonstratesaquantitativeimprovement inaccuracyachievedwithcorrection.Thenon-equilibriummoments b m 3 and b m 4 requiredin Eqs..60aandEqs..60bareobtainedas b m 3 = X e 2 x + e 2 y f )]TJ/F41 10.9091 Tf 10.909 18.655 Td [(" 2 3 + u 2 x + u 2 y # ; .61a b m 4 = X e 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(e 2 y f )]TJ/F35 10.9091 Tf 12.105 8.501 Td [( u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y : .61b 2.6NumericalResults WewillnowpresentthevalidationofournewGalileaninvariantpreconditionedcascadedcentralmomentLBMbymakingcomparisonsagainstpriornumericalsolutionsforvariouscomplex owbenchmarkproblems.Theseincludethelid-drivencavityow,owoverasquarecylinder, backward-facingstepow,theHartmannowandthefour-rollmillsowproblem.Inaddition, wewillalsodemonstratetheconvergenceaccelerationachievedusingourpreconditioningLB modelforsomeofthebenchmarkowproblems. 2.6.1Lid-drivenCavityFlow Asthersttestproblem,theGIpreconditionedcentralmomentLBmodelisappliedforthe simulationofsteady,two-dimensionalowwithinasquarecavitydrivenbythemotionofthe

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48 toplid.Thisisoneoftheclassicalinternalowbenchmarkproblemswithcomplexowstructures.ThenumericalsimulationsarecomputedattwodierentReynoldsnumbersof3200and 5000,whichareresolvedbycomputationalmesheswitharesolutionof400 400.Toimplement themovingtopwallatavelocity U p ,thestandardmomentumaugmentedhalf-waybounceback schemeisconsidered.InordertovalidatethenumericalsimulationresultsobtainedwithourGI preconditionedLBscheme,thecomputeddimensionlesshorizontalandverticalvelocityproles alongtheverticalandhorizontalcenterlines,respectively,forReynoldsnumber Re =3200and 5000andpreconditioningparameter =0 : 1,arepresentedwithbenchmarksolutionsof[2]in Fig.2.1.TheMachnumberMaconsideredinthesimulationsis0 : 05.Itisclearthatthevelocity prolesforallthecasesagreeverywellwiththepriornumericaldata.Next,weinvestigatehow thesteadystateconvergencehistoriesareinuencedbytheuseofournewpreconditionedformulationforthisbenchmarkproblem.Figure2.2presentstheconvergencehistoriesfor Re =3200 obtainedbyvaryingthepreconditioningparameter .Here =1correspondstoresultswithout preconditioning.Obviously,theuseofpreconditioningacceleratesthesteadystateconvergence byatleastoneorderofmagnitude.Forexample,itcanbeseenthatwhencomparedtothecase withoutpreconditioning =1,thepreconditionedGIcascadedLBMwith =0 : 05,isatleast 15timesfaster. 2.6.2LaminarFlowoveraSquareCylinder Next,inordertovalidateourpreconditionedLBformulationforanexternalcomplexowexample,atwodimensionallaminarowoverasquarecylinderinachannelisstudied.Thegeometry detailsandthesetupoftheowproblemisprovidedinFig.2.3.Afullydevelopedvelocityproleisconsideredattheinlet,andattheoutlet,aconvectiveboundaryconditionisusedwhichis givenby @ t u i + U max @ x u i =0.62

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49 aRe=3200 bRe=3200 cRe=5000 dRe=5000 FIGURE2.1:Comparisonofthecomputedhorizontalvelocity u=U p andverticalvelocity v=U p prolesalongthegeometriccenterlinesofthecavityusingtheGalileaninvariantpreconditioned cascadedcentralmomentLBMwiththebenchmarkresultsof[2]symbolsforRe=3200and 5000and =0 : 1. FIGURE2.2:ConvergencehistoriesoftheGIpreconditionedcascadedcentralmomentLBMand thestandardcascadedLBM =1forlid-drivencavityowforRe=3200.

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50 where U max isthemaximumvelocityoftheinowprole.Computationswereperformedusing L =50 D , H =8 D and L 1 =12 : 5 D ,whereDissideofsquarethecylinder,LandHarethetotal lengthandwidthofcomputationdomain,respectivelyandthelocationofsquarecylinderfrom entranceisdenedby L 1 .Inordertovisualizethegeneralcomplexfeaturesandpatternsofthe FIGURE2.3:Schematicrepresentationoftheowoverasquarecylinderina2Dchannel. ow,thestreamlinesplotsatfourdierentReynoldsnumbers Re =1, Re =15, Re =30and Re =200arepresentedinFig.2.4.InFig.2.4a,asitmaybeexpected,atalowReynoldsnumber, Re =1,wheretheuidvelocityisrelativelyveryslowandontheotherhand,theviscosity islarge,theuidowiscreepingandsymmetricwithoutseparation.However,withincreasing Reynoldsnumberanadversepressuregradientisestablishedwhichleadstotheowseparation fromthesurfaceandavortexpairregimeisformedFig.2.4b.AstheReynoldsnumberisfurtherincreasedfurtherto Re =30,thesizeoftherecirculationzoneincreases;besidestheowis stillsteadyandsymmetricaboutthehorizontalcenterlineFig.2.4c.Thesegeneralfeatures andowpatternsareconsistentwiththepriorbenchmarkresultse.g.[63],[3]. Then,wepresentthevelocityprolesalongthecenterlineatdierentsectionsat Re =100with ameshresolutionof1000 320.Figure.2.5illustratesthehorizontalandverticalcomponents ofthevelocityprolesof u and v ,respectively.Bycomparingthepresentresultsagainstthe benchmarknumericalresultsobtainedusingtheGasKineticschemeGKS[3],agoodagreementbetweenthecomputationalresultsisobserved.Animportantglobalfeatureoftheowover

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51 aRe=1 bRe=15 cRe=30 FIGURE2.4:StreamfunctioncontoursforowoverasquarecylinderforfourdierentReynolds numbers;Re=1,Re=15andRe=30usingtheGIpreconditionedcascadedcentralmomentLBM with =0 : 5.

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52 acylinderisthelengthoftherecirculatingowpatternformedbehindthecylinder.Quantitative characterizationofthiswakelength L r anditsdependenceontheReynoldsnumberReisakey elementinthevalidationofnumericalscheme.Awidelyusedempiricalcorrelationforthewake length L r asalinearfunctionoftheReynoldsnumberisgivenby[63] L r D )]TJ/F15 10.9091 Tf 20 0 Td [(0 : 065+0 : 0554 Re;for 5
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53 a b c d FIGURE2.5:Comparisonofthecomputedvelocityprolesalongandacrossthesquarecylinder alongitscenterlineforboththehorizontal u andvertical v velocitycomponentsobtainedusing theGIpreconditionedcascadedcentralmomentLBMwith =0 : 5for Re =100withbenchmark resultsobtainedusingtheGasKineticSchemeGKS[3].

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54 a b FIGURE2.6:ComparisonofthecomputedReynoldsnumberdependenceoftherecirculating wakelength L r ontheleftaandtheReynoldsnumberdependenceofthedragcoecient C D ontherightwithbbenchmarkcorrelationEq..63andGKS-basednumericalresults[3] respectively. FIGURE2.7:ConvergencehistoriesoftheGIpreconditionedcascadedcentralmomentLBMand thestandardcascadedLBM =1forowoverthesquarecylinderforRe=30.

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55 2.6.3Backward-FacingStepFlow Asthethirdowbenchmarkowprobleminvolvingcomplexseparationandreattachmenteffects,weconsideratwo-dimensionallaminarowoverabackwardfacingstep,whichiscomputed usingtheGIpreconditionedcentralmomentLBM.Thegeometryandboundaryconditionsfor thesimulationareshowninFig.2.8.Forastepofheight h ,theowentryisplacedat L 1 =10 h behindthestepandtheexitislocated L 2 =30 h downstreamofthestep,andthechannelheight isdenedas H =2 h .Inthissimulation,thenumberofnodesinresolvingthestepowisdenedbyconsidering h =94.Attheentrance,aparabolicprole,and,attheoutlet,aconvectiveboundaryconditionareimposed,and,nally,thehalf-waybounce-backschemeisutilized fortheno-slipboundaryconditionatthewalls.Thecomputationalresultsarethenpresentedfor Reynoldsnumbersupto800,wheretheReynoldsnumberisdenedas Re = 2 hU max 3 .Here, U max isthemaximumspeedattheinletchannel.Forthepurposeofinvestigatingtheowbehavior FIGURE2.8:Schematicrepresentationoftheowoverabackward-facingstepina2Dchannel. inthevicinityofthestep,thedistributionsofstreamlinesareplottedatfourdierentReynolds numbersinFig.2.9.Initially,aprimaryrecirculationzoneiscreateddownstreamofthestepat Re =100Fig.2.9a.However,itcanbeseenfromFig.2.9atoFig.2.9dthattheReynolds numberhasaremarkableeectonthestructurerecirculationregimesandthelengthofthiszone isseentoincreasebyincreasingtheReynoldsnumber.Furthermore,asecondrecirculationzone occursalongthetopwallatthehigherReynoldsnumberof Re =500whichbecomesmorevis-

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56 ibleat Re =800.Alltheseobservedowpatternareconsistentwithpriorbenchmarkresults. InordertomorepreciselydeterminethequantitativeeectoftheReynoldsnumberonthereattachmentlengthintheprimaryrecirculationzone,ourcomputedresultsbasedontheGIpreconditionedcascadedcentralmomentLBMfordierentReynoldsnumbersarecomputedwiththe numericalresultsof[4],whicharepresentedinFig.2.10.Itcanbeobservedthattheagreement betweenthepredictionsbasedonourGIpreconditionedLBschemeandthebenchmarkresultsis excellent.Moreover,itcanbeclearlyseenthatbyincreasingtheReynoldsnumber,thereattachmentlengthincreased,consistentwithpriorobservations. 2.6.4HartmannFlow Inthissection,inordertovalidateourpreconditionedschemeforaprobleminvolvingabody force,theHartmannowofanincompressibleuidboundedbytwoparallelplatesisstudied.An externaluniformmagneticeld B z = B 0 isappliedperpendiculartotheplates.Sincethebody forcevariesspatiallyarisingduetotheinteractionoftheowvelocityandtheinducedmagnetic eld,i.e.theLorentzforce,itrepresentsappropriatetestproblemforthepresentstudy.Inour preconditionedLBmodel,themomentsofthesourcetermsatdierentordersarepreconditioned dierentlytocorrectlyrecoverthemacroscopicwithvariablebodyforces.Therelationshipbetweentheexternalmagneticeld B 0 andaninducedmagneticeld B x z acrossthechannelis givenby B x z = F b L B 0 sinh Ha z L sinh Ha )]TJ/F36 7.9701 Tf 12.846 4.295 Td [(z L ,where F b and L aredrivingforceduetoimposedpressuregradientandthehalfchannelwidth,respectively,andHaistheHartmannnumber,which measurestheratiooftheLorentzforcetoviscousforce. TheLorentzforcecomponentisthendenedas F mx = B 0 dB x dz .Inconsequence,theeectivevariablebodyforcecomponentisdenedas F x = F b + F mx .TheanalyticalsolutionfortheHartmann owhasthefollowingvelocityprole u x z = F b L B 0 q cothHa 1 )]TJ/F15 10.9091 Tf 12.105 6.843 Td [(cosh Ha z L cosh Ha ,where isthe magneticresistivitygivenby = B 0 2 L 2 = Ha 2 .Figure2.11presentscomparisonsofthecom-

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57 aRe=100 bRe=300 cRe=500 dRe=800 FIGURE2.9:Streamlinecontoursforowoverabackward-facingstepataRe=100,bRe =300,cRe=500,dRe=800computedusingtheGIpreconditionedcascadedcentralmomentLBMwith =0 : 3. FIGURE2.10:ComparisonofthereattachmentlengthasafunctionoftheReynoldsnumber Re computedusingtheGIpreconditionedcascadedcentralmomentLBMwith =0 : 3symbols againstthebenchmarkresultsof[4].

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58 putedvelocityprolesusingtheGIpreconditionedcascadedLBMwith =0 : 1andMachnumberMa=0 : 02againsttheexactsolutionforvariousvaluesofHa.Itcanbeobservedthatthe GIpreconditionedcascadedcentralmomentLBMisabletoreproducethebenchmarksolution verywell.Inparticular,asHaisincreased,theresultinghighermagnitudesoftheLorentzforce causessignicantatteringofthevelocityprolesandthiseectofHaonthevelocityprolesis representedbyourpreconditionedismodelwithverygoodaccuracy. FIGURE2.11:ComparisonofthecomputedvelocityproleusingthepreconditionedGIcascadedcentralmomentLBM =0.1withtheanalyticalsolutionforHartmannowforvariousHaatMa=0 : 02.Thelinesindicateanalyticalresults,andthesymbolsarethesolutions obtainedbytheGIpreconditionedcascadedLBM. 2.6.5Four-rollsMillFlowProblem:ComparisonbetweenGICorrectedandUncorrectedPreconditionedCascadedLBM AsseeninSec.2.3,theGIerrorsfortheLBMonthestandard,tensorproductlattices,such astheD2Q9lattice,aregenerallyrelatedtothestrainratesintheprincipaldirections @ x u x and @ y u y .Hence,inordertocomparetheGIcorrectedformulationSec.2.5,whichisconstructedtoeliminatesucherrors,withtheuncorrectedformulationSec.2.2,weconsiderthe four-rollsmillowproblem,whichischaracterizedbylocalextensional/compressionstrainrates i.e. @ x u x 6 =0 ;@ y u y 6 =0,andforwhichawell-denedanalyticalsolutionisavailable.Itisa

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59 modiedformoftheclassicalTaylor-Greenvortexowdrivenbyalocalbodyforce,whosecomponentsaregivenby F x x;y =2 u 0 sin x sin y;F y x;y =2 u 0 cos x cos y inaperiodicsquaredomainofsidelength2 x;y 2 ,resultinginasteadyvorticalmotionintheformofanarrayofcounterrotatingvortices.Here, and u 0 arethekinematicviscosityandthevelocityscale,respectively,andaunitreferencedensityisconsidered.Theanalytical solutionofthevelocityeld,whichfollowsfromasimplicationoftheNavier-Stokesequations impressedbytheabovebodyforce,reads u x x;y = u 0 sin x sin y;F y x;y = u 0 cos x cos y: Clearly,thelocaloweldissubjectedtolocaldiagonalstrainrates,i.e. @ x u x = )]TJ/F35 10.9091 Tf 8.485 0 Td [(@ y u y = u 0 cos x sin y ,and,asaresult,theuncorrectedLBschemeinducesadditionalGIerrors,which shouldbeannihilatedbythecorrectedLBmethod;andthus,thedierenceintheglobalow eldsagainsttheanalyticalsolutionunderasuitablenormineachcasecanbequantitatively studiedandcompared. Weperformedcomputationsonasquaredomainresolvedby251 251gridnodeswithavelocity scale u 0 =0 : 045foraReynoldsnumberRe= u 0 L= ,where L =2 ,of20.Figure2.12shows thestreamlinepatternsatthesteadystatecomputedusingtheGIcorrectedpreconditionedLB scheme =0 : 3,whichmanifestasasetofcounterrotatingvortices.Thecomputedvelocity prole u y x;y = obtainedusingtheGIcorrectedLBschemealongthehorizontalcenterline ofthedomainpresentedinFig.2.13arecomparedagainsttheanalyticalsolutiongivenabove, whichshowgoodagreement. Furthermore,Fig.2.14presentsasurfaceplotofthediagonalstrainratecomponent @ x u x ,which isseentohaveasignicantlocalvariation,duetowhichquantitativedierencesinthesolutions betweentheGIcorrectedanduncorrectedpreconditionedLBschemescanbeexpected,which willnowbedemonstratedinthefollowing.

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60 FIGURE2.12:Steadystatestreamlinepatternsforthefour-rollsmillowproblemat u 0 =0 : 045 andRe=20computedusingtheGIcorrectedpreconditionedcascadedLBschemewith251 251 gridnodesand =0 : 3. FIGURE2.13:Comparisonofthecomputedandanalyticalverticalvelocityproles u y x at y = forthefour-rollsmillowproblematRe=20obtainedusingtheGIcorrectedpreconditioned cascadedLBschemewith251 251gridnodes, u 0 =0 : 45and =0 : 3.

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61 FIGURE2.14:Distributionofthediagonalstrainratecomponent @ x u x = )]TJ/F35 10.9091 Tf 8.484 0 Td [(@ y u y forthefourrollsmillowproblemwith u 0 =0 : 045.

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62 InordertomakeaquantitativecomparisonbetweenthesolutionsobtainedusingthetwodierentLBmethods,werstdenetheglobalrelativeerrorsforthevelocityeld jj GRE GI u jj 2 and jj GRE GI v jj 2 betweenthecomponentsofthesolutionobtainedusingtheGIcorrectedpreconditionedLBschemei.e. u c ;v c andtheanalyticalsolutioni.e. u a ;v a underadiscrete ` 2 norm; andsimilarly jj GRE u jj 2 and jj GRE v jj 2 betweentheuncorrectedpreconditionedLBschemei.e. u uc ;v uc andtheanalyticalsolution.Thesearewrittenasfollows: jj GRE GI u jj 2 = s P u c )]TJ/F35 10.9091 Tf 10.909 0 Td [(u a 2 P u 2 a ; jj GRE GI v jj 2 = s P v c )]TJ/F35 10.9091 Tf 10.909 0 Td [(v a 2 P v 2 a ; jj GRE u jj 2 = s P u uc )]TJ/F35 10.9091 Tf 10.909 0 Td [(u a 2 P u 2 a ; jj GRE v jj 2 = s P v uc )]TJ/F35 10.9091 Tf 10.909 0 Td [(v a 2 P v 2 a ; wherethesummationsintheabovearecarriedoutforthewholecomputationaldomain.Table2.1presentstheaboveglobalrelativeerrorsforthevelocityeldcomponentsforboththe preconditionedcascadedLBformulationsfordierentvaluesofthepreconditioningparameter =0 : 2 ; 0 : 3 ; 0 : 4and0 : 5.Itcanbeseenthatsignicantimprovementsinaccuracyisachieved bytheGIcorrectedpreconditionedLBscheme.Inparticular,theerrorsrelativetotheanalytical solutionarereducedbyaboutafactoroftwowiththecorrectedpreconditionedLBschemefor theconditionsconsideredforthecomputationofthisproblem.Suchimprovementsareconsistent withthefactthatthecorrectedLBschemeeliminatestheadditionalGIerrorsarisinginthisow subjectedtothelocalvariationsofthediagonalcompression/extensionstrainrates,whichare presentintheuncorrectedLBscheme. TABLE2.1:ComparisonbetweentheglobalrelativeerrorsinthecomputedsolutionsforthevelocityeldusingtheGIcorrectedpreconditionedcascadedLBschemeandtheuncorrectedpreconditionedcascadedLBschemeforthefour-rollsmillowproblematRe=20, u 0 =0 : 045anda gridresolutionof251 251. GIcorrected u error Uncorrected u error GIcorrected v error Uncorrected v error jj GRE GI u jj 2 jj GRE u jj 2 jj GRE GI v jj 2 jj GRE v jj 2 0.2 3 : 386 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(3 6 : 662 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(3 3 : 377 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(3 6 : 665 10 )]TJ/F33 7.9701 Tf 6.586 0 Td [(3 0.3 1 : 850 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(3 4 : 104 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(3 1 : 854 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(3 4 : 126 10 )]TJ/F33 7.9701 Tf 6.586 0 Td [(3 0.4 1 : 384 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(3 2 : 851 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(3 1 : 389 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(3 2 : 865 10 )]TJ/F33 7.9701 Tf 6.586 0 Td [(3 0.5 1 : 135 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(3 2 : 113 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(3 1 : 140 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(3 2 : 123 10 )]TJ/F33 7.9701 Tf 6.586 0 Td [(3

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64 tionscontainingafreeparameter,herethepreconditioningparameter .Inthischapter,wehave presentedanewpreconditionedcentralmomentbasedcascadedLBschemethateliminatessuch non-GIcubic-velocityandparameterdependenterrorsforthesimulationofsteadystateows. AdetailedanalysisbasedontheChapman-Enskogexpansionrevealsthestructureofthenon-GI truncationerrorsthatappearinthesecond-ordernon-equilibriummomentcomponents,which arerelatedtotheviscousstress.Subsequently,weprescribeanextendedsecond-ordermoment equilibriathatrestoresGIfreeofcubic-velocityerrorsforthepreconditionedLBmodelonthe standardD2Q9lattice.Thefollowingareamongthemainndingsarisingfromouranalysis: Ingeneral,theuseofcentralmomentsinaLBschemeprovidesanaturalsettingtopartiallyrestoreGIforthethird-ordero-diagonalmoments.Inparticular,bysettingthe third-order central momentequilibriaoftheo-diagonalcomponentstozeroe.g. b eq xxy = 0,onenaturallyarrivesatthepreciseformsofthecorresponding raw momentequilibria e.g. b eq 0 xxy = c 2 s u y + u 2 x u y thatrestoresGIofsuchcomponentsintherepresentation ofthestandardNSequations.Ontheotherhand,inthepreconditionedLBscheme,the cubic-velocitytermsappearinginthethird-order,o-diagonalmomentequilibrianeeds tobescaledby 2 e.g. b eq 0 xxy = c 2 s u y + u 2 x u y = 2 tofullyeliminatethespuriouscubicvelocitycross-derivativetermse.g. u x u y @ y u x ;u y u x @ x u y appearinginthederivationof thepreconditionedmacroscopicequations. Inordertoeectivelyeliminatethenon-GI,diagonalvelocitygradienttermse.g. u 2 x @ x u x , thesecond-order,diagonalmomentequilibrianeedsadditionalcorrectionsinboththevelocityanddensitygradientswhen 6 =1,whichareprescribedviaextendedmoment equilibria.Thevelocitygradientscanbelocallyandecientlyobtainedusingthenonequilibriumsecondordermomentcomponents;ontheotherhand,thedensitygradients canbecomputedusinganite-dierenceapproximation. UnlikethatforthestandardNSequations,therepresentationofthepreconditionedNS

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65 equationsusingaLBschemeresultsinadditional,non-GI,cross-couplingvelocityterms e.g. u 2 y @ x u x ,whicharealsoeliminatedbyourGI-correctedpreconditionedLBscheme. Forthesecond-order,o-diagonalmomentequilibria,additionalgradientvelocitycorrectiontermsareneededtorestoreGIforthesecomponentswhen 6 =1.However,forincompressibleows r u =0,theyvanishregardlessofthevalueof .Suchasituation isuniquetotherepresentationofthepreconditionedNSequationsusingLBschemes,as thenon-GIcorrectionsaregenerallyrestrictedonlytothediagonalcomponentsofthe second-orderequilibriafortherepresentationofthestandardNSequations. Ingeneral,theprefactorsinGIdefecttermsexhibitdramaticallydierentbehaviorsfor theasymptoticlimitcases:Forexample, ! 1Nopreconditioning: 4 2 )]TJ/F33 7.9701 Tf 12.404 4.295 Td [(1 3and ! 0Strongpreconditioning: 4 2 )]TJ/F33 7.9701 Tf 12.403 4.295 Td [(1 4 2 . When =1,i.e.whenthepresentLBmodelisusedtosimulateowsrepresentedby standardNSequationsasaspecialcase,allourresultsfortheGIdefecttermsandcorrectionsbecomeidenticalwiththosederivedby[42]and[43]. Finally,theresultsofourpresentanalysiscanbeextendedtothree-dimensionse.g. D3Q27latticeandothercollisionmodelsforthesimulationofthepreconditionedNS equations. Inaddition,wehavepresentednumericalvalidationofournewGIpreconditionedLBscheme basedoncentralmomentsagainstseveralcomplexowbenchmarkproblemsincludingtheliddrivencavityow,owoverasquarecylinder,thebackwardfacingstepow,theHartmannow andthefour-rollmillsorproblem.Comparisonagainstpriornumericalsolutionsshowgood agreementforthemodiedpreconditionedscheme.Inaddition,itisdemonstratedthatourGI correctedpreconditionedcascadedLBschemeresultsinsignicantconvergenceaccelerationof complexowsimulations,andaquantitativeimprovementinaccuracywhencomparedtothe

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66 uncorrectedpreconditionedLBscheme.Finally,itmaybenotedthatouranalysisofnon-GIaspectsforthepreconditionedLBschemehasimplicationsforLBschemesforothersituationssuch astheporousmediaows.Forexample,thereisaformalanalogybetweenthepreconditioned NSequationsandtheBrinkman-Forchheimer-Darcyequations,wheretheporosityservesasa freeparametere.g.[64,65].LBmodelsconstructedforsuchowse.g.[66]canbefurtherimprovedbytheapproachpresentedinthischapter.

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70 CDEwithasourceterm,suchasthosearisingintheconvectivethermalowswithinternal heatgeneration.Inthisregard,anovelcascadedLBformulationforthesolutionoftheCDE withsourcetermusingtheStrangsplittingwillbeconstructed.Finally,wewillpresentanumericalvalidationstudyofthesymmetrizedoperatorsplitforcing/sourceschemesforthecascaded LBmethodforuidowi.e.,theNSequationsandpassivescalartransporti.e.,theCDEand indierentdimensions. Thischapterisorganizedasfollows.InthenextsectionSec.2,webrieyreviewthevarious operatorsplittingapproachesincludingtheStrangsplitting.Section3presentsthegeneralideas behindthesymmetrizedoperatorsplittingbasedforcingimplementationintheLBmethod.Section4discussesthederivationandthealgorithmicprocedureofthesymmetrizedoperatorsplit forcingschemeforthe2DcascadedLBmethodforrepresentinguidowsubjectedtolocalimpressedforces.Acorresponding3DformulationisoutlinedintheAppendixA.Section5presents asymmetrizedoperatorsplitapproachsourceincorporationschemefora2DcascadedLBscheme forrepresentingtheconvection-diusionbasedtransportofapassivescalareldwithlocalsources. Numericalvalidationresultsofvarioussymmetrizedoperatorsplitforcing/sourceschemearepresentedinSec.6.Finally,Sec.7summarizesourapproachandpresentsthemainconclusionarisingfromthiswork. 3.2OperatorSplittingMethods Wewillnowbrieyreviewthevarioustypicaloperatorsplittingmethods,includingtheStrang splittingwhichwillthenbeexploitedtoconstructecientsecondorderaccurateforcingschemes inthecascadedLBmethod.Forthepurposeofillustration,wewillconsiderthenumericalsolutionsofthefollowingevolutionproblem: d y dt = P y + Q y ; y t = y 0 on[ t;t + t ] ; .1

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71 where,foreaseofpresentation, P and Q areconsideredaslinearoperators.Nonlinearoperators canbedealtwithusingLieoperatorformalism[84].Here, t isthetimestep.Forreference,the unsplitsolution y U ofthefullproblemcanberepresentedas y U = e t P + Q y 0 : .2 Now,arstordersplittingscheme,whichissometimesknownastheLie-TrotterLTsplitting orastheGodunovsplittingschemeintheCFDliterature,canberepresentedbymeansofthe followingsteps,whichcomputesolutiontoeachsubprobleminvolving P and Q separately: Step P :Solve d y dt 0 = P y ; y t 0 = t = y 0 on[ t;t + t ] ; .3a Step Q :Solve d y dt 0 = Q y ; y t 0 = t = y t + t on[ t;t + t ] ; .3b Solution : y LT t + t = y t + t : .3c ThissolutionoftheLie-Trottersplittingorthe P Q splittingschememaybemorecompactly representedbymeansoftheexponentialoperatorsas y LT t + t = e t Q e t P y 0 : .4 Thelocalerror E l incurredoverasmalltimestep t duetosplittingwhencomparedtothe unsplitsolutionEq..2canbeestimatedbymeansofaLie-Taylorseriesfactoredproduct expansionsas[81] E l; LT = y LT )]TJ/F52 10.9091 Tf 10.909 0 Td [(y U = 1 2 [ P ; Q ] y 0 t 2 + O t 3 ; .5 wherethesymbol[ ; ]representsthecommutator,i.e.,[ X ; Y ]= XY )]TJ/F58 10.9091 Tf 11.913 0 Td [(YX foranytwooperators X and Y .Then,theglobalerror E g overatimeduration T or T= t numberofstepsis E g; LT = T= t E l; LT O t ,whichmeansthattheLie-Trotterschemeisrstorderaccurate. Thismeansthatevenifahigherordermethodisusedtosolveeachsubproblem Step P and Step Q ,theabovesplittingschemeisstilloverallrstorderaccurateduetothedecomposition errorarisingfromthenon-commutingoperators,whichisoftenthecaseinpractice.

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72 Onepossibilitytoimprovetheorderofaccuracyistosymmetrizethecomputationviatakingthe averageofthetwosequencesofcalculations,i.e. Step P Step Q and Step Q Step P results. Suchanaveragedschememayberepresentedas[88] y A = 1 2 e t P e t Q + e t Q e t P y 0 : .6 ThisapproachintroducesalocalerrorrelativetotheunsplitsolutionEq..2,whichcanbe writtenas[89] E l; A = y A )]TJ/F52 10.9091 Tf 10.909 0 Td [(y U = R 0 t 3 + O t 4 ; where R 0 = )]TJ/F15 10.9091 Tf 12.408 7.381 Td [(1 12 [ P ; [ P ; Q ]]+[ Q ; [ Q ; P ]] y 0 : Hence,theglobalerrorbecomes E g; A = T= t E l; A O t 2 .Whilethisistheoretically interestingtogainanorderofaccuracy,itiscomputationallyexpensiveas,foreachtimestep, doubletheeortisrequiredwhencomparedtothepreviousscheme P )]TJ/F58 10.9091 Tf 10.909 0 Td [(Q splitting. AmoreecientstrategytoachieveaglobalsecondorderaccuracyistodevisetheStrangS splitting[83].Inthisscheme,oneoftheoperatorssay P isappliedtwiceforatimestepof length t= 2,beforeandafterthesolutionoftheothersubproblemsay,involving Step Q ,which issolvedforfullsteplengthof t .Thismayberepresentedas Step P 1 = 2 :Solve d y dt 0 = P y ; y t 0 = t = y 0 on[ t;t + t= 2] ; .7a Step Q :Solve d y dt 0 = Q y ; y t 0 = t = y t + = 2on[ t;t + t ] ; .7b Step P 1 = 2 :Solve d y dt 0 = P y ; y t 0 = t = y t + t on[ t;t + t= 2] ; .7c Solution : y S t + t = y t + t= 2 : .7d Thissymmetricapplicationoftheoperatorsinthe P 1 = 2 )]TJ/F58 10.9091 Tf 11.011 0 Td [(Q )]TJ/F58 10.9091 Tf 11.01 0 Td [(P 1 = 2 schemeachievessecondorder accuracy,whichmaybededucedbyrstnotingthattheStrangsplittingsolutionmaybemore compactlywrittenintheexponentialformas y S t + t = e t= 2 P e t Q e t= 2 P y 0 : .8

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73 ItslocalerrorwhencomparedtotheunsplitsolutionEq..2thenfollowsviaaLie-Taylor seriesas[82] E l; S = y S )]TJ/F52 10.9091 Tf 10.909 0 Td [(y U = R t 3 + O t 4 ; .9 where R = 1 24 [[ P ; Q ] ; P ]+2[[ P ; Q ] ; Q ] y 0 : .10 Then,theglobalerror E g overatimeperiod T followsas E g; S = T= t E l; S O t 2 and hencethisschemeissecondorderaccurate.Anequallyvalidpossibilitytoachieveasimilarsecondorderaccuracyistoconsiderthe Q 1 = 2 )]TJ/F58 10.9091 Tf 11.485 0 Td [(P )]TJ/F58 10.9091 Tf 11.485 0 Td [(Q 1 = 2 splitting,whichisusefulwhen Step P is moreexpensivetocomputethan Step Q .Itmaybenotedthatasimilarschemewasindependentlydevisedby[90],whofurtheranalyzedandelaboratedonitsvariantsseealso[91],and henceitissometimesreferredtoastheStrang-Marchuksplittingscheme. 3.3StrangSplittingofLatticeBoltzmannMethodIncludingBodyForces LatticeBoltzmannLBschemesaregenerallyconstructedtorepresenttheevolutionofthedynamicsoftheuidmotionrepresentedby @ t + r u =0 ; .11a @ t u + r uu = )]TJ/F55 10.9091 Tf 8.485 0 Td [(r P + r V + F ; .11b where and u aretheuiddensityandvelocity,respectively, P isthepressureand V isthe viscousstresstensor.Here, F representstheeectofthelocalimpressedbodyforces,which canvaryspatiallyandmaybetimedependent,i.e.fore.g.in2D, F = F x ;F y where F x = F x x ;t and F y = F y x ;t .Anecientapproachtosolvetheaboveuidowequationinthe LBframeworkistosolvetheEqs..11aand.11b,butwithoutthebodyforce F usingthe usualstreamandcollideproceduresubproblemAandthenseparatelysolve @ t u = F asa forcingstepsubproblemBandsubsequentlycombinedappropriatelyinacertainsequenceto yieldasecondorderaccuratescheme.Thiscanbeachievedviasymmetrizationoftheoperator

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74 splittingoftheoneofthesubproblemsovertwohalftimesteps.Dellar[85]performedaderivationandanalysisoftheLBmethodviaStrangsplitting,whichwillbeusedasformalstarting pointtoconstructecientoperatorsplitforcingschemesforthecascadedLBmethodinthesubsequentsections. Inthefollowing, S , C and F areusedtodenotetheoperatorsusedtoperformthestreamingstep, collisionstepandtheforcingstep,respectively.Foralatticecontaining =0 ; 1 ; 2 ;:::b directions,thecollisionandstreamingstepscanberepresentedas Step C : f x ;t + t = C f x ;t = f x ;t + K b g ; .12a Step S : f x ;t + t = S f x ;t f x )]TJ/F25 10.9091 Tf 10.909 0 Td [(e t;t : .12b Here, f = f 0 ;f 1 ;f 2 :::f b y isavectorofsize b +1representingthedistributionfunctions,where y isthetransposeoperator, b g = b g 0 ; b g 1 ; b g 2 ::: b g b y isthevectorrepresentingthechangeofdierent momentsundercollision,and K isthetransformationmatrixofthecascadedLBmethodthat mapschangesinmomentsbacktochangesinthedistributionfunctions,whicharespeciedlater. Itmaybenotedthat C and S operatorsrepresentthesplitsolutionoperatorsofthediscreteanalogof @ t f = and @ t f + e r f =0,respectively,ofthediscretevelocityBoltzmannequation @ t f + e r f = ,whoseemergentbehaviorrepresentstheNSequationsgiveninEq..11a andEq..11b,butwithout F .Then,theforcingstepseparatelysolvesthefollowing: Step F : @ @t u = F : .13 Onepossibilitytocombinetheabovesplitstepstoeectivelyachievesecondorderaccuracyisto performasymmetricapplicationoftheforcingstepsovertwohalftimesteps,beforeandafter thecollisionstep,whichisakintothespinstepsfortheforcepresentedbySalmon[86]: f x ;t + t = SF 1 = 2 CF 1 = 2 f x ;t ; .14

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75 where F 1 = 2 representsperformingthesolutionofEq..13overtimestepoflength t= 2.Ref.[85] showedthatthisachievessecondorderaccuracysimilartotheStrangsplittingextendedtothree operators: f 0 x ;t + t = C 1 = 2 F 1 = 2 SF 1 = 2 C 1 = 2 f 0 x ;t ,wherethetwoarerelatedby f 0 = C 1 = 2 F 1 = 2 f .Sincethemomentumisconservedduringcollisions,asecondorderschemewith Eq..14canbeobtainedby u = P f 0 e = F 1 = 2 P f e .Wewilladopttheabove strategyinourderivationofthesymmetrizedoperatorsplitforcingschemeforthecascadedLB methodinthesubsequentsections.SimilarapproachwasrecentlyadoptedfortheMRTLBmodelse.g.,[87].Inaddition,Schiller[92]proposedavariantoftheStrangsplittingofforcingsteps aroundstreamingandcollisions,wherethehalfcollisionstepisvalidfortheregimeinvolvingthe relaxationtimebeingmuchgreaterthanthetimestep.Also,Dellar[42]showedthattheCrankNicolsonsolutionofthemomentequationsforcombinedcollisionsandtime-independentforcing obtainedbyStrangsplittingisequivalenttoKupershtokh'sexactdierencemethod[73]. 3.4BodyForceSchemefor2DCascadedLBMethodforFluidFlowviaStrang Splitting Wewillconsidera2DcascadedLBformulationforatwo-dimensional,ninevelocityD2Q9lattice.ThecomponentsoftheparticlevelocitiesarethenrepresentedbythefollowingvectorsusingthestandardDirac'sbra-ketnotation: j e x i = ; 1 ; 0 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 y ; .15a j e y i = ; 0 ; 1 ; 0 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 y : .15b Theircomponentsforanyparticlevelocitydirection where =0 ;:::; 8arereferredtoas e x and e y ,respectively.Furthermore,weneedthefollowing9-dimensionalvector: j 1 i = ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 y : .16 ThezerothmomentistheEuclideaninnerproductofthisvectorwiththedistributionfunction. Wethenconsiderthefollowingspecicsetoforthogonalbasisvectorsusedinthecollisionterm

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76 ofthecascadedLBmethode.g.,[19]: j K 0 i = j 1 i ; j K 1 i = j e x i ; j K 2 i = j e y i ; j K 3 i =3 j e 2 x + e 2 y i)]TJ/F15 10.9091 Tf 17.576 0 Td [(4 j 1 i ; j K 4 i = j e 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(e 2 y i ; j K 5 i = j e x e y i ; j K 6 i = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 j e 2 x e y i +2 j e y i ; j K 7 i = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 j e x e 2 y i +2 j e x i ; j K 8 i =9 j e 2 x e 2 y i)]TJ/F15 10.9091 Tf 17.575 0 Td [(6 j e 2 x + e 2 y i +4 j 1 i : .17 Intheabove,symbolsuchas j e 2 x e y i = j e x e x e y i representsavectorresultingfromtheelementwisevectormultiplicationHadamardproductofthesequenceofvectors j e x i , j e x i and j e y i .By combiningtheabove9vectors,wethenobtainthefollowingorthogonalmatrix K =[ j K 0 i ; j K 1 i ; j K 2 i ; j K 3 i ; j K 4 i ; j K 5 i ; j K 6 i ; j K 7 i ; j K 8 i ] : .18 Here, K mapschangesofmomentsundercollisionsbacktochangesinthedistributionfunctions. Inordertodeterminethestructureofthecascadedcollisionoperator,werstdenethefollowingsetofcentralmomentsofthedistributionfunctionsanditsequilibriaoforder m + n ,respectively,as 0 B @ ^ x m y n ^ eq x m y n 1 C A = X 0 B @ f f eq 1 C A e x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x m e y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y n : .19 ByequatingthediscretecentralmomentsoftheequilibriumdistributionfunctionwiththecorrespondingcontinuouscentralmomentsbasedonthelocalMaxwellian[15],[19],[20],weget b eq 0 = ; b eq x =0 ; b eq y =0 ; b eq xx = c 2 s ; b eq yy = c 2 s ; b eq xy =0 ; b eq xxy =0 ; b eq xyy =0 ; b eq xxyy = c 4 s : .20 where c 2 s =1 = 3with c s beingthesoundspeed.Thisissetbyapplyingtheusuallatticeunits,i.e. x = t =1ortheparticlespeed c = x= t =1,andbecause c 2 s = c 2 = 3fortheathermal LBschemeusedinthisworkseee.g.[18].Ontheotherhand,theactualcomputationsinthe cascadedformulationsarecarriedoutintermsofrawmoments,whicharedenedasdesignated herewiththe 0 symbol 0 B @ ^ 0 x m y n ^ eq 0 x m y n 1 C A = X 0 B @ f f eq 1 C A e m x e n y : .21

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77 Thecollideandstreamsteps C and S ofthe2DcascadedLBmethodcanthenbe,respectively, writtenas[15,19] Step C : f p = f + K b g .22a Step S : f x ;t = f p x )]TJ/F52 10.9091 Tf 10.909 0 Td [(e t;t ; .22b where f p representsthepost-collisiondistributionfunctionand b g = b g 0 ; b g 1 ; b g 2 ::: b g 8 y isthe changeofdierentmomentsundercollisions,whichisdeterminedbasedontherelaxationofvariouscentralmomentstotheircorrespondingequilibriainacascadedfashion[15].Sincethemass andmomentumarecollisioninvariants, b g 0 = b g 1 = b g 2 =0.Asaresult,thecascadedstructure startsfromthenon-conservedsecondordermoments,andthecorrespondingcomponentsofthe changeofdierentmomentsundercollisionsaregivenby b g 3 = ! 3 12 2 3 + u 2 x + u 2 y )]TJ/F15 10.9091 Tf 10.909 0 Td [( b 0 xx + b 0 yy ; b g 4 = ! 4 4 n u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y )]TJ/F15 10.9091 Tf 10.909 0 Td [( b 0 xx )]TJ/F41 10.9091 Tf 11.021 0 Td [(b 0 yy o ; b g 5 = ! 5 4 n u x u y )]TJ/F41 10.9091 Tf 11.022 0 Td [(b 0 xy o ; b g 6 = ! 6 4 n 2 u 2 x u y + b 0 xxy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x b 0 xy )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y b 0 xx o )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(1 2 u y b g 3 + b g 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x b g 5 ; b g 7 = ! 7 4 n 2 u x u 2 y + b 0 xyy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y b 0 xy )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x b 0 yy o )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 2 u x b g 3 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 4 )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 u y b g 5 ; b g 8 = ! 8 4 1 9 +3 u 2 x u 2 y )]TJ/F41 10.9091 Tf 10.909 12.109 Td [(h b 0 xxyy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x b 0 xyy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y b 0 xxy + u 2 x b 0 yy + u 2 y b 0 xx +4 u x u y b 0 xy io )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 3 )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 2 u 2 y b g 3 + b g 4 )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(1 2 u 2 x b g 3 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 4 )]TJ/F15 10.9091 Tf 10.91 0 Td [(4 u x u y b g 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y b g 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x b g 7 : .23 where ! 3 ;! 4 ;:::;! 8 aretherelaxationparameters.Theserelaxationstepsleadtothefollowing expressionsforthebulkandshearviscosities,respectively,as = 1 3 1 ! 3 )]TJ/F33 7.9701 Tf 12.545 4.295 Td [(1 2 t and = 1 3 1 ! j )]TJ/F33 7.9701 Tf -450.398 -19.701 Td [(1 2 t where j =4 ; 5,andthepressureeld P isobtainedviaanequationofstateas P = 1 3 . Afterthestreamingstep,i.e.,Eq..22b,weobtaintheoutputvelocityeldcomponentsdesig-

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78 natedwithasuperscript" o "astherstmomentof f : u o x = P 8 =0 f e x ;u o y = P 8 =0 f e y : .24 Wethenintroducetheeectofthebodyforce F = F x ;F y asasolutionofthesubproblemin Eq..13.Thisisaccomplishedbyperformingtwosymmetricstepsofhalftimestepsoflength t= 2,onebeforeandtheotherafterthecollisionstep.Boththesestepsincorporatetheeect offorcesdirectlyintothemomentspace.SolvingEq..13fortherstpartofthesymmetric sequenceofstepyields u x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u o x = F x t 2 and u y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u o y = F y t 2 .Thus, Pre-collisionForcingStep F 1 = 2 : u x = 1 u o x + F x 2 t ;u y = 1 u o y + F y 2 t : .25 Then,weusethisupdatedvelocityeld u x ;u y inEq..23toperformthecascadedrelaxationcollisionsteptodeterminethechangeofdierentmomentsundercollisions,i.e. b g , = 3 ; 4 ;:::; 8.Asaresultofcorrectlyprojectingtheeectoftheforcesinthevarioushigherorder moments,itnaturallyeliminatesthediscreteeectsidentiedearlier[71]seethediscussionat theendofthissection.Then,toimplementtheotherpartofthesymmetrizedforcestepwith halfsteptosolveEq..13postcollision,weset u p x )]TJ/F35 10.9091 Tf 11.069 0 Td [(u x = F x t 2 and u p y )]TJ/F35 10.9091 Tf 11.07 0 Td [(u y = F y t 2 ,where u p x ;u p y istheresultofthetargetvelocityeldduetotheforcingstepaftercollision.Thus, Post-collisionForcingStep F 1 = 2 : u p x = u x + F x 2 t;u p y = u y + F y 2 t: .26 Notethatthiscanalsoberewrittenintermsoftheoutputvelocityeld u o = u o x ;u o y givenin Eq..24byusingEq..25as u p x = u o x + F x t;u p y = u o y + F y t: .27 Amainissuehereishowtoeectivelydesignthepost-collisiondistributionfunction f p inthe cascadedLBmethodsothatEq..27ispreciselysatised.Now,using f p = f + K b g and takingitsrstmoments,weget u p x = f p e x = f e x + h K j e x i b g ; .28a u p y = f p e y = f e y + h K j e y i b g : .28b

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79 Basedontheorthogonalbasisvectors j K i giveninEq..17,itfollowsthat h K j e x i g =6 b g 1 ; h K j e y i g =6 b g 2 : .29 UsingEqs..24and.29inEqs..28aand.28bwe,getthedesiredvelocityeldas u p x = u o x +6 b g 1 ;u p y = u o y +6 b g 2 : .30 Comparingtheresultofthetargetvelocityeldfollowingthesecondhalfofthesymmetrized forcingstepsgiveninEq..27withthechangeofmomentsbasedexpressionsinEq..30,we obtain b g 1 = F x 6 t; b g 2 = F y 6 t: .31 Equation.31representsanalgorithmicresultthateectivelyimplementstheeectofthe post-collisionforcingstepoveradurationofhalftimestepfollowingcollision.Thisisaconsequenceofthemomentumneedingtochangeby F t overatimestep,andthenormalization isimpliedbyourchoiceofbasisforthemoments.Then,theaboverelationEq..31forthe post-collisionchangeofrstmomentsduetotheforceeld b g 1 and b g 2 alongwiththechangeof dierenthighermomentsundercollisions b g ,where =3 ; 4 ;:::; 8,giveninEq..23eectively providethedesiredpost-collisionstatesofthedistributionfunction f p .ExpandingEq..22a,

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80 wegettheexpressionsforthepost-collisiondistributionfunctionsas f p 0 = f 0 +[ b g 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 b g 3 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 8 ] ; f p 1 = f 1 +[ b g 0 + b g 1 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 3 + b g 4 +2 b g 7 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 8 ] ; f p 2 = f 2 +[ b g 0 + b g 2 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 3 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 4 +2 b g 6 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 8 ] ; f p 3 = f 3 +[ b g 0 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 1 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 3 + b g 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 7 + b g 8 ] ; f p 4 = f 4 +[ b g 0 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 2 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 3 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 6 + b g 8 ] ; f p 5 = f 5 +[ b g 0 + b g 1 + b g 2 +2 b g 3 + b g 5 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 6 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 7 + b g 8 ] ; f p 6 = f 6 +[ b g 0 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 1 + b g 2 +2 b g 3 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 5 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 6 + b g 7 + b g 8 ] ; f p 7 = f 7 +[ b g 0 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 1 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 2 +2 b g 3 + b g 5 + b g 6 + b g 7 + b g 8 ] ; f p 8 = f 8 +[ b g 0 + b g 1 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 2 +2 b g 3 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 5 + b g 6 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 7 + b g 8 ] : .32 Then,thealgorithmicprocedureofoursymmetrizedoperatorsplitforcingschemeforthe2Dcascadedmethodcanbesummarizedintermsofthefollowingsequenceofstepstoevolveforatime duration[ t;t + t ]: iObtaintheupdatedthevelocity u = u x ;u y basedonthepre-collisionforcingwithhalf stepusingEq..25. iiComputethechangeofmomentsundercollisions, b g , =3 ; 4 ;:::; 8usingEq..23 basedontheupdatedvelocity u x ;u y obtainedinStepi. iiiPerformpost-collisionforcingwithahalfstepeectivelyviathecalculationofchangeof rstordermoments,i.e. b g 1 and b g 2 usingEq..31. ivComputethepost-collisiondistributionfunctions f p , =0 ; 1 ;:::; 8usingEq..32. vPerformthestreamingstepusingEq..22btoobtaintheupdateddistributionfunctions f , =0 ; 1 ;:::; 8.

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81 viFinally,obtaintheoutputvelocityeld u o = u o x ;u o y viaEq..24andthedensity using = P 8 =0 f . Someofthemainadvantagesofthissymmetrizedoperatorsplitforcingschemeofthecascaded LBmethodare: aUsingsymmetrizationprinciplewithhalf-timestepapplicationofthebodyforcebefore andaftercollisionisconsistentwithStrangsplittingandtheschemeisformallysecond orderaccurateintime. bTheapproachcorrectlyprojectstheeectsofthebodyforceonthehigherordermomentsviastepiiaboveandhencenaturallyeliminatesthediscreteeectsidentiedin priorworks[71]seebelowfordetails. cTheprocedureissimpleandecientbyinvolvingthebodyforceimplementationdirectly onlyinthemomentspaceanddoesnotrequireadditionaltermsduetoforcinginthevelocityspace,whichisusuallyobtainedviacumbersometransformationfromthemoment spaceasinpriorforcingschemesforthecascadedLBmethod.Thisaspectisespecially advantageousin3D.AppendixAoutlinestheimplementationofthisapproachfora3D centralmomentbasedLBscheme. Wewillnowpresentananalysisonhowthespuriousterm F i u j + F j u i thatcanappearinthe viscousstressiseliminatedinourpresentcentralmoments-basedcascadedLBformulationusing asplitforceimplementation.Thiscanbeachievedbyacontinuoustimeequationforthesecond centralmomentwhoseevolutionisindependentofthebodyforce.Asaresult,itcanintroducea cancelingsecondmomentofthebodyforcetermattheleadingorderintheemergentPDEofthe secondrawmomentofthedistributionfunctionsrecoveringcorrectowphysics.Wewillstart withthislatteraspectrstandidentifythiscompensatingsecondrawmomentofthebodyforce byconsideringthediscretevelocityBoltzmannequation @ t f + e r f = + S ,where and

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82 S arethecollisionoperatorandthesourcetermduetothebodyforce,respectively.Takingits zerothandrstmomentsleadto @ t + r u =0 ;@ t u + r )]TJ/F15 10.9091 Tf 9.364 0 Td [(= F ; .33 andthentakingitssecondmoment,weobtainthefollowingevolutionequation @ t )]TJ/F15 10.9091 Tf 8.757 0 Td [(+ r = )]TJ/F15 10.9091 Tf 9.955 7.38 Td [(1 )]TJ/F33 7.9701 Tf 6.333 4.504 Td [( neq + ; .34 where )]TJ/F15 10.9091 Tf 9.97 0 Td [(and arethesecondandthirdmomentsofthedistributionfunctions,i.e., P f e i e j and P f e i e j e k ,respectively,and istherequiredcancelingsecondmomentofthebody forceterm,i.e., P S e i e j ,whichshouldariseviaaconditiononthesecondcentralmoment giveninthefollowing.InEq..34, )]TJ/F33 7.9701 Tf 6.334 4.421 Td [( neq isthenon-equilibriumpartofthesecondrawmoment and =1 =! j ,where j =4 ; 5,isthecorrespondingrelaxationtime,whicharerelatedtothe viscousstress. Inordertodeterminetheevolutionequationforhydrodynamicsattheleadingorder,wenow applytheChapman-EnskogC-EexpansionsofthedistributionfunctionsaboutitsequilibrialocalMaxwellianandthetimederivative,i.e., f = f + f + 2 f + and @ t = @ t 0 + @ t 1 + 2 @ t 2 + ,respectively,where isasmallperturbationparameter.Thisisequivalent tothefollowingexpansionsonthehigher,non-conserved,rawmoments )]TJ/F15 10.9091 Tf 9.364 0 Td [(= )]TJ/F33 7.9701 Tf 6.333 4.504 Td [( + )]TJ/F33 7.9701 Tf 6.334 4.504 Td [( + 2 )]TJ/F33 7.9701 Tf 6.333 4.504 Td [( + ; = + + 2 + ; .35 intheabovemomentsystem.Totheleadingorder,themassandmomentumequationsinEq..33 become @ t 0 + r u =0 ;@ t 0 u + r )]TJ/F33 7.9701 Tf 6.333 4.504 Td [( = F ; .36 where )]TJ/F33 7.9701 Tf 6.334 4.421 Td [( = c 2 s I + uu istheequilibriumpartofthesecondrawmoment.Ontheotherhand, theleadingordersecondrawmomentequation,viaEq..34,readsas @ t 0 )]TJ/F33 7.9701 Tf 6.333 4.504 Td [( + r = )]TJ/F15 10.9091 Tf 9.955 7.38 Td [(1 )]TJ/F33 7.9701 Tf 6.333 4.504 Td [( + : .37

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83 Inordertorecoverthephysicallycorrectviscousstress,thenon-equilibriumpartofthesecond moment )]TJ/F33 7.9701 Tf 6.334 4.421 Td [( intheaboveequation,Eq..37,shouldonlyberelatedto r ,whichdepends onthevelocitygradients.However,thepresenceofthetimederivativeterminEq..37,i.e., @ t 0 )]TJ/F33 7.9701 Tf 6.333 4.421 Td [( = c 2 s @ t 0 I + @ t 0 uu ,inwhichthetimederivativesofthevelocity @ t 0 uu viatheleading momentumequationEq..36giverisetoanadditionaltermoftheform Fu + uF .Thiscan beeliminatedonlyifthecorrespondingmomentofthebodyforce becomesequalto = Fu + uF : .38 Thisnecessaryconditionforthesecondrawmomentofthebodyforce P S e i e j = F i u j + F j u i ,whichisaclassicresultoftheaccelerationtermintheBoltzmannequation,wasgiven in[93].Thisimpliesavanishingsecondcentralmomentofthebodyforce,i.e., P S e x )]TJ/F35 10.9091 Tf -434.811 -23.995 Td [(u x m e y )]TJ/F35 10.9091 Tf 12.057 0 Td [(u y n =0for m + n =2,whichappearsexplicitlyin[94]andwasconsideredin thepreviousunsplitforcingapproachforthecascadedLBscheme[19]. Inviewoftheabove,inourpresentoperator-splitforcingbasedcascadedLBformulation,the PDEneededforthesolvingthesplitforcestepgiveninEq..13isacentralmomentrepresentationofthesplitkineticequation @ t f = S .Thatis,takingthecentralmomentsofthis equationoforder m + n ,wegetanevolutionequationasfollows: Step F : @ @t b x m y n = b x m y n ; .39 where b x m y n = P f e x )]TJ/F35 10.9091 Tf 12.056 0 Td [(u x m e y )]TJ/F35 10.9091 Tf 12.056 0 Td [(u y n and b x m y n = P S e x )]TJ/F35 10.9091 Tf 12.056 0 Td [(u x m e y )]TJ/F35 10.9091 Tf 12.056 0 Td [(u y n arethecentralmomentsofthedistributionfunctionsandthesourcetermduetothebodyforce, respectively.Itthusfollowsthat,inparticular,thecontinuoustimeequationsforthechangein thesecondcentralmomentcomponentsforthesplitbodyforcesteparegivenas Step F : @ @t b xx =0 ; @ @t b yy =0 ; @ @t b xy =0 ; .40 whichimpliesthenecessaryconditionforintroducingthecancelingsecondrawmomentcomponentsofthebodyforce,i.e.,2 F x u x ,2 F y u y and F x u y + F y u x toeliminatethespuriouseectsin theviscousstressandtherebycorrectlyrecovertheNavier-Stokesequationsasmentionedabove.

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84 3.5ExtensionoftheSymmetrizedOperatorSplitImplementationforCascaded LBMethodforPassiveScalarTransportIncludingSources Inmanyapplications,thetransportofapassivescalare.g.,temperatureorspeciesconcentrationoccurs,whichisgenerallyrepresentedbymeansofthefollowingconvection-diusionequationCDEwithasourceterm @ t + r u = r D r + S : .41 Here, isthepassivescalarvariable, D isthediusioncoecient,and S isthelocalsource terme.g.duetoviscousdissipation,internalheatgenerationorchemicalreaction.VariousLB schemeshavebeeninvestigatedformodelingtheCDEduringthelasttwodecadese.g.,[95,21, 96,97,98,99,100,101,102,87].Anovelnumericalapproachconsideredinthisstudyforthe solutionofEq..41isasfollows.Thevelocity u intheaboveequationcanbeobtainedfrom thecascadedLBschemefortheD2Q9latticepresentedintheprevioussection.Ourgoalisto solveforthepassivescalareld whoseevolutionisrepresentedbytheaboveCDE,butwithout thesourcetermusingaseparate2Dcascadedschemewithcollideandstreamstepsinvolvinganotherdistributionfunction;thenimplementtheeectofthesourceterm S viaadditionalsource stepsusinganoperatorsplitschemebasedonasymmetrizationprinciple.Tomeetthisobjective,weconsideranewcascadedLBschemeforcoupleduidowandscalartransportthatwe developedrecentlyindierentdimensions[33]andfurtheracceleratedbyusingmultigrid[60]. Here,atwo-dimensional,vevelocityD2Q5latticebasedcascadedLBmethodisintroducedto representtheevolutionofthepassivescalareldviatheCDE,whichisadoptedinthisworkfor furtherextensionusinganoperatorsplitsourceimplementation. TheD2Q5latticeisrepresentedbymeansofthefollowingcomponentsoftheparticlevelocity vectors j e x i and j e y i : j e x i = ; 1 ; 0 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0 y ; .42a j e y i = ; 0 ; 1 ; 0 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 y : .42b

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85 Inaddition,weintroducethefollowing j 1 i vector: j 1 i = ; 1 ; 1 ; 1 ; 1 y : .43 ThezerothmomentistheEuclideaninnerproductofthisvectorwiththedistributionfunctions. Thecorrespondingveorthogonalbasisvectorsaregivenby[60] j L 0 i = j 1 i ; j L 1 i = j e x i ; j L 2 i = j e y i ; j L 3 i =5 j e 2 x + e 2 y i)]TJ/F15 10.9091 Tf 17.576 0 Td [(4 j 1 i ; j L 4 i = j e 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(e 2 y i ; .44 whichcanbegroupedtogetherasthefollowingtransformationmatrix L formappingchangesin themomentspacetothoseinthevelocityspace L =[ j L 0 i ; j L 1 i ; j L 2 i ; j L 3 i ; j L 4 i ] : .45 Inordertorepresentthestructureofthecascadedcollisionoperatorforthepassivescalareld, wedenethefollowingcentralmomentsandrawmoments,respectively,ofthedistributionfunction g anditsequilibrium g eq as 0 B @ ^ x m y n ^ eq; x m y n 1 C A = X 0 B @ g g eq 1 C A e x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x m e y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y n ; .46 and 0 B @ ^ 0 x m y n ^ eq; 0 x m y n 1 C A = X 0 B @ g g eq 1 C A e m x e n y : .47 ByequatingthediscretecentralmomentsoftheequilibriumdistributionfunctionwiththecorrespondingcontinuouscentralmomentsbasedonthelocalMaxwellianwhereinthedensityis replacedby ,weget b eq; 0 = ; b eq; x =0 ; b eq; y =0 ; b eq; xx = c 2 s ; b eq; yy = c 2 s ; .48 whichwillbeusedintheconstructionofthecollisionoperatorlater.Inthiswork,wetset c 2 s = 1 = 3.Then,the2DcascadedLBschemeforthepassivescalartransportwithoutthesourceterm

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86 canberepresentedbymeansofthefollowingcollisionandstreamingsteps: g p = g + L b h ; .49a g x ;t = g p x )]TJ/F52 10.9091 Tf 10.909 0 Td [(e t;t : .49b Theproceduretoobtainthechangeofdierentmomentsundercascadedcollision,i.e. b h based onthecentralmomentequilibriaEq..48isanalogoustothatusedintheprevioussectionfor uidow,withthemaindierencebeingthatinthepresentcase,thereisonlyonecollisional invariant,i.e. ,andhence b h 0 =0.Then,itfollowsthat[60]seealso[33]thatelaboratessucha formulationfora3DcascadedLBMforCDE b h 1 = ! 1 2 h u x )]TJ/F41 10.9091 Tf 11.022 0 Td [(b 0 x i ; b h 2 = ! 2 2 h u y )]TJ/F41 10.9091 Tf 11.021 0 Td [(b 0 y i ; b h 3 = ! 3 4 h 2 c 2 s )]TJ/F15 10.9091 Tf 10.909 0 Td [( b 0 xx + b 0 yy +2 u x b 0 x + u y b 0 y + u 2 x + u 2 y i + u x b h 1 + u y b h 2 ; b h 4 = ! 4 4 h )]TJ/F15 10.9091 Tf 8.485 0 Td [( b 0 xx )]TJ/F41 10.9091 Tf 11.021 0 Td [(b 0 yy +2 u x b 0 x )]TJ/F35 10.9091 Tf 10.91 0 Td [(u y b 0 y + u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y i + u x b h 1 )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y b h 2 : .50 where ! 1 , ! 2 , ! 3 and ! 4 aretherelaxationparameters.Noticethatthecascadedstructureof theexpressionsforthechangeofmoments b h startsfromtherstordermomentsfortheCDE, unlikethosefortheNSEgiventheprevioussection.Therelaxationparametersfortherstorder momentsintheabovedeterminethemoleculardiusivity D : D = c 2 s 1 ! j )]TJ/F33 7.9701 Tf 12.223 4.295 Td [(1 2 t , j =1 ; 2.AfterthestreamingstepinEq..49b,theoutputpassivescalareld o isobtainedasthezeroth momentof g as o = 4 X =0 g : .51 Theeectofthesourceterm S canthenbeintroducedasthesolutionofthesourcesubproblemsplitfromEq..41: @ t = S .Asbefore,thiscanbeimplementedbymeansoftwosymmetrizedsequenceofstepsbeforeandaftercollision,eachusingatimestep t= 2andsucha sourceoperatorwillbedenotedby R 1 = 2 .Thus,theextensionoftheStrangsplittingapproach forthecascadedLBMtorepresentthesourcetermintheCDEcanbeformulatedas g x ;t + t = SR 1 = 2 CR 1 = 2 g x ;t : .52

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87 Solvingtheabovesubproblemrepresentingtheevolutionofthescalareld duetothesource term S yieldsthefollowingstepbeforecollision Pre-collisionSourceStep R 1 = 2 : = o + S 2 t: .53 Thisupdated isthenusedtoperformthecascadedcollisionrelaxationstepanddeterminethe changeofdierentmomentsundercollision b h ,where =1 ; 2 ; 3 ; 4 ; giveninEq..50.Analogously,theothersourcehalfstepfollowingcollisioncanberepresentedas Post-collisionSourceStep R 1 = 2 : p = + S 2 t = o + S t: .54 Inordertoeectivelyimplementthisinthecascadedformulation,wetakethezerothmomentof thepost-collisiondistribution g p givenby g p = g + L b h ,whichyields p = X g p = X g + X h K j 1 i b h : .55 BasedontheorthogonalbasisvectorsgiveninEq..44,itfollowsthat P h K j 1 i b h =5 b h 0 , whichwhensubstitutedinEq..55,andalongwithEq..51,weobtain p = o +5 b h 0 : .56 ComparingthetargetresultEq..54withtheaboveconstructedeldEq..56,wegetthe followingresultforthezerothordermomentchangeduetothesource S b h 0 = S 5 t: .57 Thiseectivelyimplementstheeectofthepost-collisionsourcestepoverasteplengthof t= 2. UsingthisresultEq..57alongwithEq..50forthechangeofmomentsundercollisionin Eq..49aandexpanding K b h ,weobtainthepost-collisiondistributionfunctions,whichread

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88 as g p 0 = g 0 + h b h 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 b h 3 i ; g p 1 = g 1 + h b h 0 + b h 1 + b h 3 + b h 4 i ; g p 2 = g 2 + h b h 0 + b h 2 + b h 3 )]TJ/F41 10.9091 Tf 10.719 2.878 Td [(b h 4 i ; g p 3 = g 3 + h b h 0 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 1 + b h 3 + b h 4 i ; g p 4 = g 4 + h b h 0 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 2 + b h 3 )]TJ/F41 10.9091 Tf 10.719 2.879 Td [(b h 4 i : .58 Theoverallsequenceofcomputationalstepsforthe2DcascadedLBschemeforpassivescalar transportwithasourceimplementationbasedontheStrangsplittingissimilartothatforthe uidowpresentedintheprevioussection.Moreover,suchasymmetrizedoperatorsplittingformulationcanalsobeusedtorepresentforces/sourcesinthe3DcentralmomentbasedLBMfor thermalconvectiveowsdevelopedrecently[33]. 3.6ResultsandDiscussion Wewillnowpresentanumericalvalidationstudyofthevarioussymmetrizedoperatorsplitschemes toincorporateforces/sourcesinthecascadedLBmethodpresentedearlierbycomparisonoftheir computedresultsagainstasetofbenchmarkproblemswithanalyticalsolutions.Inthefollowing,allthenumericalresultswillbegenerallyreportedinthelatticeunitstypicalforLBsimulations[18].Thatis,unlessotherwisespecied,weconsider x = t =1andhencetheparticle speed c = x= t istakentobeunity.Theuidvelocitywillbescaledbytheparticlespeed c , andthereferencescaleforthedensity 0 is1.0.ForthecascadedLBmethodforuidowpresentedinSec.3.4,theconsiderationsfortherelaxationparametersareasfollows: ! 4 and ! 5 determinetheshearkinematicviscosityvia ! 4 = ! 5 =1 = and = 1 3 )]TJ/F33 7.9701 Tf 13.17 4.295 Td [(1 2 t ,whichcan bespeciedfromtheproblemstatement.Theparameter ! 3 isrelatedtothebulkviscositysee e.g.[19],whiletheremainingparametersforthehigherordermoments ! 6 ;! 7 and ! 8 ,alongwith

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89 ! 3 canbetunedtoimprovenumericalstability.AdetailedstudyoftheinuenceofsuchparametersinthecascadedLBschemewasperformedin[17].Forturbulentowcomputations,care needstobeexercisedinchoosingtherelaxationparametersforthehigherordermomentsinordertoavoidbeingover-dissipative.Inthiswork,fortheincompressible,laminarowbenchmark owproblemsconsideredinthefollowing,weuse ! 3 = ! 6 = ! 7 = ! 8 =1 : 0.Ontheotherhand, forthecascadedLBmethodforthesolutionofthepassivescalartransportpresentedinSec.3.5, theparameters ! 1 and ! 2 ,whicharerelatedtothecoecientofdiusivityi.e. ! 1 = ! 2 =1 = and D = 1 3 )]TJ/F33 7.9701 Tf 12.917 4.295 Td [(1 2 t ,areassignedfromtheproblemstatementbasedonthecharacteristic dimensionlessgroup;relaxationparameters ! j ,where j =3 ; 4 ; 5,whichinuencethenumerical stability,aresettounityinthiswork. 3.6.1PoiseuilleFlow Inthesesections,wevalidateour2Doperatorsplitforcingapproachbyconsideringvarioustest problemsinvolvingdierenttypesofbodyforceelds.Fortherstproblem,atwo-dimensional Poiseuilleowinachanneldiscretizedwith3 100latticenodesisconsidered.Inourcomputations,atthetopandbottomwalls,ano-slipboundarycondition,andattheinletandoutlet,periodicboundaryconditionsareapplied.Theno-slipboundaryconditionisimplemented byusingtheclassicalhalf-waybouncebackschemeinthiswork[103,18].Theanalyticalsolutionofthevelocityproleowforthislaminarowproblemcanbewrittenasfollows: u y = U max [1 )]TJ/F15 10.9091 Tf 11.127 0 Td [( y L 2 ],where U max = F x L 2 = isthemaximumvelocityalongthecentralline.Here, L , and arethechannelhalf-width,uiddensityandkinematicviscosityrespectively. F x isa constantbodyforceactinginthe x -directionwhichdrivestheow.Comparisonofthesimulation resultsofthevelocityproleagainsttheanalyticalsolutionisshowninFig.3.1,wherethebody forcesfortwocaseswithmaximumvelocitiesof0 : 02and0 : 08aresettothevaluesof10 )]TJ/F33 7.9701 Tf 6.586 0 Td [(8 and 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(7 ,respectively.Fortheformercase,therelaxationtime ischosentobe0 : 5019,whichfor thelatteritis0 : 5047.ThecorrespondingMachnumbersMaare0 : 034and0 : 138,respectively.

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90 Itcanbeclearlyseenthatthereisanexcellentagrementbetweenthenumericalsimulationcarriedoutusingthe2DsymmetrizedoperatorsplitcascadedLBforcingschemeandtheanalytical solutionforthebothcases. FIGURE3.1:Comparisonofthecomputedvelocityprolesusingthe2Dsymmetrizedoperator splitcascadedLBforcingschemewiththeanalyticalsolutionforPoiseuilleowforbodyforce magnitudesof10 )]TJ/F33 7.9701 Tf 6.586 0 Td [(7 and10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(8 .Thelinesindicatetheanalyticalresults,andthesymbolsarethe solutionsobtainedbyourpresentnumericalscheme. GridConvergenceStudy Inordertodeterminetheorderofaccuracyofoursymmetrizedoperatorsplitforcingscheme,we performagridconvergencetestbyapplyingadiusivescaling.Accordingtothisscaling,Mach numberMa= U=c s reducesproportionallywiththeincreaseinthegridresolutionataxed viscosityorxedrelaxationtime =1 =! j , j =4 ; 5,where ! 4 and ! 5 representtherelaxation parametersforthesecondordermomentsinthe2DcascadedLBschemeseeSec.4,sothat theschemehasasymptoticconvergencetotheincompressibleowlimit.Foroursimulation,we consideraPoiseuilleowwiththesamesetupasconsideredearlier.Weconsiderasequenceof 3 15 ; 3 31 ;:::; 3 121latticenodestostudygridconvergenceunderdiusivescalingwhenthe

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91 relaxationtimeandReynoldsnumberaresetto =0 : 55and100,respectively.Next,toquantify thegridconvergence,weconsidertheglobalrelativeerror E g;u oftheoweldunderadiscrete ` 2 -normasfollows: k E g;u k 2 = s u c )]TJ/F35 10.9091 Tf 10.909 0 Td [(u a 2 u a 2 ; .59 where u c and u a isthecomputedandanalyticalsolutions,respectively,andthesummationis carriedoutfortheowdomain.TherelativeerrorbetweenthecomputedresultsandtheanalyticalsolutionagainstdierentgridresolutionsisillustratedinFig.3.2.Therelativeerrorshave aslopeof2 : 00whichindicatesthatournewapproachbasedonthesymmetrizedoperatorsplit forcingschemeforthecascadedLBmethodisspatiallysecond-orderaccurate. FIGURE3.2:Gridconvergencefor2DPoiseuilleowwithaconstantReynoldsnumber Re = 100andrelaxationtime =0 : 55computedusingthe2DsymmetrizedoperatorcascadedLB forcingscheme. 3.6.2HartmannFlow Asthenextbenchmarkcasestudy,anumericalcomparisonoftheresultswithour2Doperator splitforcingapproachismadeforaspecictypeofmagnetohydrodynamicMHDow,i.e.,the owbetweentwounboundedplatessubjectedtoatransversemagneticeldknownastheHartmannow.ThistypeofowarisesinavarietyofengineeringdevicesincludingMHDpumps,

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92 fusiondevices,generatorsandmicrouidicdevices.Furthermore,aninherentspatially-varying bodyforcemakesthisbenchmarkaparticularlysuitabletestproblemforthepresentstudy.The uidisdrivenbyaconstantbodyforce F b andretardedbyalocalvariableforcei.e.Lorentz forcearisingbyaninteractionbetweenauniformsteadymagneticeld B y = B 0 ,actingperpendiculartothechannelwallsandtheuidmotion.Bychoosingthe x axisfortheowdirection andthe y axistobeco-directionalwiththeexternalmagneticeld B y = B 0 ,theinducedmagneticeldresultingfromsuchaninteractioncanberepresentedas B x y = F b L B 0 sinh Ha y L sinh Ha )]TJ/F36 7.9701 Tf 12.748 4.932 Td [(y L . Here,theHartmannnumber,H a isthesquarerootoftheratiooftheelectromagneticforceto theviscousforceand L and F b arethechannelhalf-widthandtheuniformdrivingforce,respectively.Consequently,theeectivelyspatiallyvaryingbodyforcewhichactontheowis F x = F b + F mx .ThisisacombinationoftheLorentzforce F mx = B 0 dB x dy ,andtheuniformdriving force F b .Theanalyticalsolutionforsuchaproblemis u x y = F b L B 0 q cothHa 1 )]TJ/F15 10.9091 Tf 12.104 6.843 Td [(cosh Ha y L cosh Ha . Here, isthekinematicviscosityand isthemagneticresistivity,whichcanberepresentedby = B 0 2 L 2 = H a 2 .WeconsiderthesamesetupasconsideredforthePoiseuilleowsimulation fortheboundaryconditionsbutnowwithspatiallyvaryingbodyforces.Fortwodierentvalues ofH a ,3and10,correspondingtoMachnumbersof0.013and0.004,respectively,thecomputed velocityprolesagainsttheanalyticalsolutionareillustratedinFig.3.3.Itcanbeobservedthat thepresentsimulationisabletoreproducetheanalyticalsolutionverywell.Inparticular,the signicantatteningofthevelocityproleathigherH a iswellreproducedbyourforcingscheme. 3.6.3WomersleyFlow WenowturntostudytheWomersleyow,whichisaowbetweentwoinniteparallelplates drivenbyatemporallyoscillatoryexternalforce.Thisbenchmarkproblemisusedtoassessthe abilityofoursymmetrizedoperatorsplitforcingschemeforrepresentingtime-dependentbody forces.Theexternalforce F m cos !t oscillateswithanamplitude F m andwithanangularfre-

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94 quency ! =2 =T ,where T isthetimeperiod.Supposingthattheowislaminarandincompressible,theanalyticalsolutionforthevelocityeldisgivenas u y;t =Re i F m ! 1 )]TJ/F15 10.9091 Tf 12.105 5.374 Td [(cos y=L cos ] e i!t g ; .60 where = p i Wo 2 ,Wo= L p != beingtheWomersleynumber,whichisanon-dimensional parameterrepresentingtheratioofthechannelhalfwidth L tothediusionlengthoveranoscillationperiodi.e.,theStokeslayerthickness.Re fg representstakingtherealpartoftheexpressionwithinthebrackets.Thesimulationparametersaresetasfollows.Thecomputational domainisresolvedbya3 100mesh,thetimeperiod T =10000andthemaximumforceamplitudeissetto F m =1 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(5 .Theboundaryconditionattheinletandtheoutletisperiodic andthehalf-waybounce-backschemetorepresenttheno-slipconditionisusedatthewalls.The bodyforceforthiscaseisimplementedasasolutionofEq..13toupdatethevelocityeld. Sincetheexplicitformofthetime-dependentforceisknownhere,itcanbeeitheranalytically integratedtoperformthevelocityupdateintheforcesteporsolvednumericallybyrepresenting thebodyforce F x viathetrapezoidalruleas 1 2 F m cos !t +cos !t + t= 2.Thelatterapproach isusedinthepresentstudy.Ingeneralcases,ifthebodyforce F dependson u ,thenEq..13 needstobenumericallyintegratedandusedasanimplicitequationtosolvefor u .Simulations arecarriedouttoobtainthevelocityprolesacrossthechannelatdierenttimeinstantswith thetimeperiod T .Figure6.5showsacomparisonforthevelocityprolesfortwovaluesofthe Womersleynumber,i.e.4and10 : 7atdierenttimeinstants.ItcanbeclearlyseenthatthenumericalresultsagreewellwiththeanalyticalsolutionrepresentedbyEq..60.Thus,thesymmetrizedoperatorsplitforcingschemeisabletorepresentowprolesdrivenbytimevarying bodyforceswithexcellentaccuracy.

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96 3.6.4FlowthroughaSquareDuct Inordertovalidateour3Dsymmetrizedoperatorsplitforcingschemeforamultidimensional owsubjectedtoabodyforce,weconsiderowthroughasquareductdrivenbyaconstantbody force F x .Inourcomputations,weapplyperiodicboundaryconditionsattheinletandoutletand ano-slipboundaryconditionatthefourwallsurfaces.Forachannelwithwidth2 a ,thistest problemhasananalyticalsolutionbasedonaFourierseriesforthevelocityeld,whichreads as u y;z = 16 a 2 F x 3 1 X n =1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 n )]TJ/F33 7.9701 Tf 6.586 0 Td [(1 2 4 1 )]TJ/F15 10.9091 Tf 12.104 11.345 Td [(cosh n )]TJ/F33 7.9701 Tf 6.587 0 Td [(1 z 2 a cosh n )]TJ/F33 7.9701 Tf 6.587 0 Td [(1 2 3 5 cos n )]TJ/F33 7.9701 Tf 6.586 0 Td [(1 y 2 a n )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 3 ; .61 where and aretheuiddensityandkinematicviscosity,respectivelyand x isthedirectionof theow,and )]TJ/F35 10.9091 Tf 8.484 0 Td [(a
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97 FIGURE3.5:Comparisonofthecomputedvelocityprolesusingthe3DsymmetrizedoperatorsplitcascadedLBforcingschemeandtheanalyticalsolution,forowthroughasquareduct inpresenceofabodyforcemagnitudeof F x =10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(7 fordierentvaluesof y .Here,linesrepresenttheanalyticalsolutionandsymbolsrefertotheresultsobtainedusingthepresentnumerical scheme.

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98 uidmotionisestablishedbyimposingthefollowinglocalbodyforcecomponents: F x x;y =2 0 u 0 sin x sin y;F y x;y =2 0 u 0 cos x cos y; where0 x;y 2 , 0 isthereferencedensity, isthekinematicviscosity,and u 0 isthevelocityscale.AsimplicationoftheNavier-Stokesequationswiththeabovelocalbodyforceleadsto thefollowinganalyticalsolutionforthevelocityeld: u x x;y = u 0 sin x sin y;u y x;y = u 0 cos x cos y: First,inordertovalidatetheStrangsplitting-basedforcingschemeforthecascadedLBM,we consider u 0 =0 : 01, 0 =1 : 0and =0 : 0011,andthesquaredomainofside2 isresolvedby N N meshgrids,where N =24 ; 48 ; 96 ; 192.Themeshspacing x thenisgivenby x = 2 =N .Consideringtheconvectivescaling x= t = c =1,thekinematicviscositymaybewritten as = 1 3 )]TJ/F33 7.9701 Tf 13.216 4.295 Td [(1 2 x ,where =1 =! 4 =1 =! 5 .Figure3.6showsthevelocityeld u y x;y = computedusing N =96alongthehorizontalcenterlineofthedomainandcomparedagains theanalyticalsolutiongivenabove.Excellentagreementisseen.Furthermore,Fig.3.7presents the2Dcomputedandanalyticalresultsforthestreamlines,whichareinverygoodagreement witheachother.Evidently,counter-rotatingpairsofvorticesarewellreproducedbythepresent forcingschemeforthecascadedLBMbasedonStrangsplitting. GridConvergenceStudy InordertoverifythehigherorderaccuracyprovidedbytheStrangsplitting,i.e., O x 2 O t 2 ,weusethe convective or acoustic scalingtostudytheconvergencerateofthepresent operator-splitforcingformulationfordierentgridresolutions,ratherthanthe diusive scalingconsideredearlier.Thus,weagainuse u 0 =0 : 01, =0 : 0011and N =24 ; 48 ; 96and192. Bymaintaining x= t = c =1,foranypairofgridresolutions, N i N i and N j N j ,the correspondingrelaxationparameters i and j ,respectively,undertheconvectivescalingarerelatedby j = 1 2 + i )]TJ/F33 7.9701 Tf 12.753 4.295 Td [(1 2 N j N i .Figure3.8illustratesrateofconvergenceusingtherelativeerror

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99 FIGURE3.6:Comparisonofthecomputedandanalyticalverticalvelocityproles u y x at y = forthefour-rollsmillowproblemat u 0 =0 : 01, =0 : 0011and N =96.Here,line representstheanalyticalsolutionandthesymbolreferstothenumericalresultsobtainedusing the2DsymmetrizedoperatorsplitcascadedLBforcingscheme. aPresentwork bAnalytical FIGURE3.7:Streamlinesacomputedusingthe2DsymmetrizedoperatorsplitcascadedLB forcingschemeandbobtainedusingtheanalyticalsolutionforthefour-rollsmillowproblem at u 0 =0 : 01, =0 : 0011and N =96.

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100 betweenthecomputedandanalyticalsolutionforthe x -componentofthevelocityeldsummed fortheentiredomainunderthediscrete ` 2 normseeEq..59fortheabovefourdierentgrid resolutions.Itcanbeseenthattherelativeerrorvarieswiththegridresolutioninthelog-log scalewithaslopeof )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 : 0.Hence,thepresentforcingschemebasedontheStrangsplittingfor thecascadedLBMissecondorderaccurateundertheconvectivescaling.Inotherwords,this testdemonstratessecondorderaccuracyintime,whiletheearliertestforPoiseuilleowunder diusivescalinginFig.3.2isnot. FIGURE3.8:Gridconvergenceforthefour-rollsmillowproblemat u 0 =0 : 01, =0 : 0011computedusingthe2DsymmetrizedoperatorsplitcascadedLBforcingschemeundertheconvective scaling. 3.6.6ThermalCouetteFlowwithViscousHeatDissipation ForthepurposeofvalidatingthesymmetrizedoperatorsplitcascadedsourceschemeforthesolutionofascalarpassiveeldrepresentedbytheCDEwithasourceterminSec.5,weperform thesimulationofathermalCouetteowwithviscousheatdissipation.Here,thepassivescalar eld isthetemperature T ,whichisevolvedunderathermaldiusivityD,andmodiedbya

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101 sourceterm S r duetoviscousdissipationarisingfromtheshearow.Forsuchaone-dimensional Couetteow,thetopwallmoveswithaconstantvelocity U 0 inahorizontaldirection,whichis maintainedatahighertemperature T H andthebottomwallisatalowertemperature T L and remainsstationary.Thescalarsourceterm S r resultingfromtheviscousheatdissipationisgiven by S r = 2 C v S : S ; .62 where S =[ r u + r u T ] = 2isthestrainratetensorand C v isspecicheatatconstantvolume. Thesourcetermduetotheviscousheating S r inEq..62isobtainedinsimulationsfromthe cascadedLBsolutionfortheoweldpresentedinSec.3.3.Inparticular,thestrainratetensor S inthecascadedLBformulationcanbereadilyrelatedtothesecond-ordernon-equilibriummomentcomponentsseee.g.,[19,17].Forexample, S xy = 1 2 @ x u y + @ y u x = )]TJ/F33 7.9701 Tf 9.68 4.489 Td [(3 ! 5 2 0 P f e x e y )]TJ/F35 10.9091 Tf -454.063 -23.995 Td [(u x u y .Thisproblemhasthefollowinganalyticalsolutionforthetemperatureprole[105] T )]TJ/F35 10.9091 Tf 10.909 0 Td [(T L T H )]TJ/F35 10.9091 Tf 10.909 0 Td [(T L = y H + P r E c 2 y H 1 )]TJ/F35 10.9091 Tf 14.212 7.38 Td [(y H ; .63 whereP r = =D isthePrandtlnumberandEc= U 2 0 = [ C v T H )]TJ/F35 10.9091 Tf 11.536 0 Td [(T L ]istheEckertnumber.In Fig.3.9,thePrisxedat0 : 71whiletheEcvariesfrom10to100andthedomainisdiscretized with3 64latticenodes.Thevelocityofthetopwall U 0 istakenas0 : 05,theboundarytemperature T L and T H arespeciedas0 : 0and1 : 0,respectively,andtherelaxationparameters and arechosenas0 : 70and0 : 782,respectively.ComputedresultsobtainedusingthesymmetrizedoperatorsplitcascadedsourceschemearecomparedwiththeanalyticalsolutiongiveninEq..63. Itisfoundthatthenumericalresultsareinexcellentagreementwiththeanalyticalsolutionfor variousvaluesofEc,representingthesourcestrengthforthisproblem.Inaddition,therelative errorbetweenthecomputedresultsobtainedusingtheStrangsplitting-basedsourceschemeand theanalyticalsolutionmeasuredunderthediscrete ` 2 -normEq..59forthesimulationofthe thermalCouetteowarereportedinTable3.1.

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102 TABLE3.1:Relativeerrorbetweenthenumericalresultsobtainedusingthe2DsymmetrizedoperatorsplitcascadedLBsourceschemeforapassivescalartransportandtheanalyticalsolution forthesimulationofthethermalCouetteowatvariousEckertnumbersEc. EckertnumberEc Relativeerror 10 2 : 840 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(5 20 3 : 695 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(5 40 4 : 317 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(5 60 4 : 561 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(5 80 4 : 691 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(5 100 4 : 778 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(5 FIGURE3.9:Comparisonbetweennumericalresultsofthetemperatureprolecomputedusing the2DsymmetrizedoperatorsplitcascadedLBsourceschemeforapassivescalartransportand theanalyticalsolutionforthethermalCouetteowforvariousvaluesoftheEckertnumberEc. Here,linesrepresenttheanalyticalsolutionandsymbolsrefertotheresultsobtainedusingthe presentnumericalscheme.

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103 3.7SummaryandConclusions Symmetrizedoperatorsplitforcingschemesforowsimulationsin2Dand3Dandamethodfor incorporatingsourcesinaconvection-diusiontransportofascalareldusingthecascadedlatticeBoltzmannformulationsaredevelopedinthischapter.Theyinvolveforce/sourceimplementationstepsbeforeandfollowingthecollisionstepeachtakingahalftimestep,andareconsistentwiththeStrangsplitting,whichhassecondorderrateofconvergencebyconstruction.The post-collisionhalfsource/forcingstepiseectivelyimplementedintermsofthechangeofmomentsatthezeroth/rstorderthatisafunctionofthesource/bodyforceandthetimestep, andanormalizationfactorarisingfromthechoiceofthebasisformomentsforthelatticeset considered.Theimplementationofthepre-collisionhalfsource/forcingstepproperlyprojects theeectsoftheforce/sourcetothehigherordermomentsthatundergorelaxationbycollision andnaturallyeliminatesthediscreteeects.IncontrasttothepriorforcingschemesforthecascadedLBmethodthatrequiredusingextratermsatdierentordersinthemomentspaceand cumbersomelattice-dependenttransformationstomapthemtothevelocityspace,thepresent symmetrizedoperatorsplitforcing/sourceschemesresultinasimplerformulation,withallthe force/sourcerelatedcomputationsperformedonlyinthemomentspace,whichfacilitatesimplementation.However,itmaybenotedthatforecientimplementationsoftheLBalgorithms, theirperformanceoncurrenthardwareislimitedentirelybymemorybandwidthratherthan byoatingpointoperations,andthecomplexityoftheaggregatecollisionoperatorincluding forcingdoesnotaectperformance.Comparisonsofthenumericalsolutionsobtainedusingthe Strangsplittingbasedforcing/sourceimplementationmethodsforcascadedLBschemesagainst variousbenchmarksolutionsvalidatethemforowcomputationsinboth2Dand3Daswellas forthepassivescalartransportwithalocalsource.Furthermore,thenumericalresultsdemonstratethesecondorderaccuracyfortheconvergencerateintimeundertheacousticscalingof thesymmetrizedoperatorsplitforcingscheme.

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105 siderableattentionmorerecently.Generally,thefollowingtypesofapproacheshavebeenconsideredintheLBframeworkforsimulationofthermalconvectiveows:aMultiphaseMSLB schemes[106,107,108,109],bhybridapproache.g.[110],andcdoubledistributionfunctionDDFbasedLBM[21,111,112].MS-ThermalLBmodelsareobtainedbyincludingadditionaldiscretevelocitiestothedistributionfunctionandusingahigherordervelocityexpansionoftheMaxwellianformodelingtheequilibrium;hereasingledistributionfunctionisused torepresenttheevolutionofbothvelocityandtemperatureelds.Suchapproacheshavesevere restrictionsinnumericalstabilityandhenceresultsinanarrowrangeoftemperaturevariation. ThehybridapproachconsidersusingaLBmodelfortheoweldandsolversthethermalenergyequationbymeansofanothernumericalschemesuchasthenitedierencemethod.The DDF-LBschemesconsiderstheevolutionoftwodierentdistributionfunctions,whichhaveovercomemanyofthelimitationofotherformulations,andarenowmorewidelyused. MostofthepriorstudiesrelatedtothedevelopmentofapplicationsofDDF-basedLBmodels considerSRTmodelsandgenerallylimitedtotwo-dimensionsD[113,114,102].ThecorrespondingMRTbasedDDF-LBformulationswereinvestigatedby[115,116,117,115,112].For practicalapplications,itisimportanttoexpandthecapabilitiesoftheLBMforthermalconvectiveowsin3D.However,onlylimitedstudieshavesoforbeenconductedinliteratureinthis regard.Oneoftheearliest3DLBmodelsforheattransferbasedonapassivescalarapproach waspresentedby[118],whoperformedsimulationsofRayleigh-BenardconvectionusingaSRT model.Subsequently,[119]developeda3DSRTLBMbasedonDDFapproachandstudiednaturalconvectioninacubiccavity.Morerecently,[116]presentedaMRT-LBEin3DforCDE. Furthermore,[120,121]and[122]employedDDF-basedLBMin3DusingMRTformulationfor heattransferproblems. Inthischapter,wepresentnew3Dformulations,basedoncascadedapproachusingcentralmomentswithinaDDFapproachtorepresentowswiththermalconvection.Suchacollisionmodel isconstructedusingamovingframeofreferenceandinvolvingcentralmomentrelaxationbased

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108 orderaccuracy,andthenavariabletransformation f = f )]TJ/F33 7.9701 Tf 12.762 4.295 Td [(1 2 S isintroducedtoremoveimplicitness.Briey,the3DcascadedLBMforuidowwithabodyforcemaythenbewritten as[125] ~ f x ;t = f x ;t + K : ^ g + S x ;t ; .2a f x + e ;t +1= ~ f x ;t : .2b Here,Eqs..2aand.2brepresentthecollisionandstreamingsteps,respectively. ~ f representsthepost-collisiondistributionfunction, K istheorthogonalcollisionmatrix,and ^ g isthe collisionkernel,whichisobtainedfromcentralmomentrelaxationatdierentorderstotheircorrespondinglocalequilibria.WhiletheforcehereisonthederivationofanewcascadedLBEfor 3DCDEasdiscussedinwhatfollows,for K , ^ g , S ,and ~ f forthesolutionoftheNSEinAppendixB.Oncethedistributionfunctionisupdated,thehydrodynamicvariablesareobtained fromthevariouskineticmomentsas = f ; u = f e + 1 2 F : .3 4.2.23DCascadedLBModelforTransportofTemperatureField Wenowpresentaderivationofa3DcascadedLBMonaD3Q15latticeforthetransportofany genericscalareld suchastemperature,where = T ,thatsatisesthefollowing CDE : @ t + r : u = r D r + R: .4 where = x ;t , r = @ x ;@ y ;@ z , D isthecoecientofdiusivity, R = R x ;t isthelocal sourceterm,andthevelocityeld u canbeobtainedfromtheLBmodeldiscussedearlier.The approachthatcanbetakeninthisregardconsistsofthefollowingoverallsteps:Constructan orthogonalmomentbasisstartingfromaninitialsetoflinearlyindependentnonorthogonalbasisvectorsfortheD3Q15lattice.Prescribeexpressionsforthecontinuouscentralmoments

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109 ofequilibriaandthesourcetermatdierentordersandsetthemequaltotheirdiscretecentral momentsusedinthecascadedLBformulation;obtaincorrespondingrawmomentsatdierent orders.Determinethestructureofthecascadedcollisionkernelviaconsideringacentralmomentrelaxationatdierentorders,andobtainthesourcetermsinthevelocityspace.ThecomponentsoftheparticlevelocityfortheD3Q15latticecanbewrittenas j e x i = ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0 ; 0 ; 0 ; 0 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 y ; j e y i = ; 0 ; 0 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0 ; 0 ; 1 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; 1 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 y ; j e z i = ; 0 ; 0 ; 0 ; 0 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 ; 1 ; 1 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 y ; .5 andacorrespondingunitvectormayberepresentedby j i = ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 y : .6 Here,wehaveusedthenotations hj and ji torepresenttherowandthecolumnvectorsrespectively,and y isthetransposeoperator. h a j b i representthedotproductofanytwovectors a and b .Usingsuccessivelyhigherorderordersofthemonomials e m x e n y e p z ,wecanwritethefollowing nonorthogonalbasisvectors j T 0 i = j i ; j T 1 i = j e x i ; j T 2 i = j e y i ; j T 3 i = j e z i ; j T 4 i = j e x e y i ; j T 5 i = j e x e z i ; j T 6 i = j e y e z i ; j T 7 i = j e 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(e 2 y i ; j T 8 i = j e 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(e 2 z i ; j T 9 i = j e 2 x + e 2 y + e 2 z i ; j T 10 i = j e x e 2 x + e 2 y + e 2 z i ; j T 11 i = j e y e 2 x + e 2 y + e 2 z i ; .7 j T 12 i = j e z e 2 x + e 2 y + e 2 z i ; j T 13 i = j e x e y e z i ; j T 14 i = j e 2 x e 2 y + e 2 x e 2 z + e 2 y e 2 z i : ByapplyingtheGram-Schmidtorthogonalizationmethodontheaboveset,wecanobtainthe

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110 correspondingsetoforthogonalbasisvectors,whicharegroupedtogetherintothefollowingcollisionmatrix K as K =[ K 0 ; K 1 ; K 2 ; K 3 ; K 4 ; K 5 ; K 6 ; K 7 ; K 8 ; K 9 ; K 10 ; K 11 ; K 12 ; K 13 ; K 14 ] where K 0 = j i ; K 1 = j e x i ; K 2 = j e y i ; K 3 = j e z i ; K 4 = j e x e y i ; K 5 = j e x e z i ; K 6 = j e y e z i ; K 7 = j e 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(e 2 y i ; K 8 = j e 2 x + e 2 y + e 2 z i)]TJ/F15 10.9091 Tf 17.576 0 Td [(3 j e 2 z i ; K 9 = j e 2 x + e 2 y + e 2 z i)]TJ/F15 10.9091 Tf 17.576 0 Td [(2 j i ; K 10 =5 j e x e 2 x + e 2 y + e 2 z i)]TJ/F15 10.9091 Tf 17.576 0 Td [(13 j e x i ; K 11 =5 j e y e 2 x + e 2 y + e 2 z i)]TJ/F15 10.9091 Tf 17.576 0 Td [(13 j e y i ; K 12 =5 j e z e 2 x + e 2 y + e 2 z i)]TJ/F15 10.9091 Tf 17.575 0 Td [(13 j e z i ; K 13 = j e x e y e z i ; K 14 =30 j e 2 x e 2 y + e 2 x e 2 z + e 2 y e 2 z i)]TJ/F15 10.9091 Tf 17.576 0 Td [(40 j e 2 x + e 2 y + e 2 z i +32 j i : .8 Then,wedenethecontinuouscentralmomentsofequilibrianeededintheconstructionofthe cascadedcollisionkernelforthe3DCDEasfollows: b eq; x m y n z p = Z 1 Z 1 Z 1 g eq x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x m y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y n z )]TJ/F35 10.9091 Tf 10.909 0 Td [(u z p d x d y d z ; .9 where g eq istheequilibriumdistributionfunctioninthecontinuousvelocityspace x ; y ; z for thescalareld ,whichisgivenby g eq g eq ; u ; = 2 c 2 s exp h )]TJ/F33 7.9701 Tf 9.68 5.374 Td [( )]TJ/F53 7.9701 Tf 6.586 0 Td [(u 2 2 c 2 s i .Here c s isafree parameter,whichwillberelatedtothedesiredcoecientofdiusivity D later.Typically,we set c 2 s = 1 3 ,throughitcanbechosentobeatothervaluesdierentfromthatforthecascaded LBmodelfortheoweld.Moreover, u intheaboveistheuidvelocityasdenedintheprevioussection.ItmaybenotedthattheaboveequilibriumisobtainedfromthelocalMaxwellianby replacingthedensitywiththescalarled usedinourDDFscheme.Then,rewritingthecom-

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111 ponentofEq..9intheincreasingorderofmomentsas b eq; 0 = ; b eq; x = b eq; y = b eq; z =0 ; b eq; xx = b eq; yy = b eq; zz = c 2 s ; b eq; xy = b eq; xz = b eq; yz = b eq; xyy = b eq; xxy = b eq; xxz = b eq; xyz =0 ; b eq; xxyy = c 4 s : Here,andhenceforth,theuseofhatoverasymbolrepresentsanyquantityinthemomentspace. Similarly,forthecontinuouscentralmomentsduetosourceterm R maybedenedas b )]TJ/F36 7.9701 Tf 6.818 4.504 Td [(R x m y n z p = Z 1 Z 1 Z 1 g R x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x m y )]TJ/F35 10.9091 Tf 10.91 0 Td [(u y n z )]TJ/F35 10.9091 Tf 10.909 0 Td [(u z p d x d y d z : .10 where g R isthechangeinthedistributionforthescalareldduetothesourceterm.Asthe sourceterm R canonlyeectthelowest,i.e.zerothmoment,thecomponentofEq..10maybe writtenas b )]TJ/F36 7.9701 Tf 6.818 3.959 Td [(R 0 = R; b )]TJ/F36 7.9701 Tf 6.818 3.959 Td [(R x = b )]TJ/F36 7.9701 Tf 6.818 3.959 Td [(R y = b )]TJ/F36 7.9701 Tf 6.818 3.959 Td [(R z = b )]TJ/F36 7.9701 Tf 6.818 3.959 Td [(R xx = b )]TJ/F36 7.9701 Tf 6.818 3.959 Td [(R yy = b )]TJ/F36 7.9701 Tf 6.818 3.959 Td [(R zz = b )]TJ/F36 7.9701 Tf 6.818 3.959 Td [(R xy = b )]TJ/F36 7.9701 Tf 6.818 3.959 Td [(R xz = b )]TJ/F36 7.9701 Tf 6.819 3.959 Td [(R yz =0 ; b )]TJ/F36 7.9701 Tf 6.818 3.959 Td [(R xyy = b )]TJ/F36 7.9701 Tf 6.818 3.959 Td [(R xxy = b )]TJ/F36 7.9701 Tf 6.818 3.959 Td [(R xxz = b )]TJ/F36 7.9701 Tf 6.818 3.959 Td [(R xyz = b )]TJ/F36 7.9701 Tf 6.818 3.959 Td [(R xxyy =0 : .11 ThecascadedLBErepresentingthetransportofthe3DCDEcanbeobtainedbyapplyinga trapezoidalruleforthetreatmentofthesourceterminthecharacteristicintegrationtomaintain secondorderaccuracy.Thus,wehavae g x + e ;t +1= g x ;t + g x ;t + 1 2 h S x ;t + S x + e ;t +1 i ; .12 where S isthesourceterminthevelocityspacethateectivelyaccountsfortheterm R x ;t in themacroscopicCDE.Intheaboveequation,thecollisionterm g x ;t canbemodeledby g g g ; b h = K b h ; .13

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112 where g = g 0 ;g 1 ; ;g 14 y isthevectorofthedistributionfunction,and b h = b h 0 ; b h 1 ; ; b h 14 y is thevectoroftheunknowncollisionkernelwhichwillbedeterminedlater.Forremovingtheimplicitness,whilemaintainingasecond-orderaccuracy,byapplyingavariabletransformation[126], g = g )]TJ/F33 7.9701 Tf 12.105 4.296 Td [(1 2 S inEq..12,weobtain g x + e ;t +1= g x ;t + g x ;t + S x ;t : .14 This3DcentralmomentLBEmayberewrittenintermsofthefollowingcollisionandstreaming stepsforthepurposeofimplementationas ~ g x ;t = g x ;t + K : ^ h + S x ;t ; .15a g x + e ;t +1= ~ g x ;t ; .15b wherethesymbol intheaboverepresentsthepostcollisiondistributionfunction.Inorderto buildthestructureofthecascadedcollisionandthesourcetermsforrepresentingthe3DCDE, werstdenethefollowingsetofdiscretecentralmomentsas 0 B B B B B B B B @ ^ x m y n z p ^ eq; x m y n z p ^ x m y n z p ^ x m y n z p 1 C C C C C C C C A = X 0 B B B B B B B B @ g g eq S g 1 C C C C C C C C A e x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x m e y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y n e z )]TJ/F35 10.9091 Tf 10.909 0 Td [(u z p ; .16 where ^ x m y n z p =^ x m y n z p )]TJ/F33 7.9701 Tf 12.829 4.296 Td [(1 2 ^ x m y n z p .Then,byequatingthediscretecentralmomentsofthe equilibriumdistributionfunctionandsourcetermwiththeircorrespondingcontinuouscentral

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113 momentsatdierentorders,i.e.^ eq; x m y n z p = b eq; x m y n z p and^ x m y n z p = b )]TJ/F36 7.9701 Tf 6.818 3.959 Td [(R x m y n z p ,respectively,weget ^ eq; 0 = ; ^ eq; x =^ eq; x =^ eq; x =0 ; ^ eq; xx =^ eq; yy =^ eq; zz = c 2 s ; ^ eq; xy =^ eq; xz =^ eq; yz =^ eq; xyy =^ eq; xxy =^ eq; xxz =^ eq; xyz =0 ; ^ eq; xxyy = c 4 s ; .17 and ^ 0 = R; ^ x =^ y =^ z =0 ; ^ xx =^ yy =^ zz =^ xy =^ xz =^ yz =0 ; ^ xyy =^ xxy =^ xxz =^ xyz =0 ; ^ xxyy =0 : .18 Sincetheactualcalculationsarecarriedoutintermofvariousrawmoments,wedenethefollowingsetoftherawmomentatdierentordersas 0 B B B B B B B B @ ^ 0 x m y n z p ^ eq; 0 x m y n z p ^ 0 x m y n z p ^ 0 x m y n z p 1 C C C C C C C C A = X 0 B B B B B B B B @ g g eq S 0 g 1 C C C C C C C C A e m x e n y e p z ; .19 where ^ 0 x m y n z p =^ 0 x m y n z p )]TJ/F33 7.9701 Tf 10.053 4.296 Td [(1 2 ^ 0 x m y n z p ,andtheuseofprimesoverthesymbolhereandhenceforth refertorawmoments.Fromtheabove,werstdeterminetheexpressionsforthesourcetermsin thevelocityspaceinEq..14.Inthisregard,asanintermediatestep,byapplyingthebinomialtheoremonEq..18,weobtainthediscreterawmomentsofthesourcetermsatdierent

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114 ordersas ^ 0 0 = R; ^ 0 x = u x R; ^ 0 y = u y R;; ^ 0 z = u z R; ^ 0 xx = u 2 x R; ^ 0 yy = u 2 y R; ^ 0 zz = u 2 z R; ^ 0 xy = u x u y R; ^ 0 xz = u x u z R; ^ 0 yz = u y u z R; ^ 0 xyy = u x u 2 y R; ^ 0 xxy = u 2 x u y R; ^ 0 xxz = u 2 x u z R; ^ 0 xyz = u x u y u z R; ^ 0 xxyy = u 2 x u 2 y R: Next,fromthis,weobtainthesourcetermsprojectedtotheorthogonalbasisvector K ,i.e. b m s; = K S ,where S = S 0 ;S 1 ;S 2 ;:::;S 14 .Thatis, b m s; 0 = R; b m s; 1 = h K 1 j S i = u x R; b m s; 2 = h K 2 j S i = u y R; b m s; 3 = h K 3 j S i = u z R; b m s; 4 = h K 4 j S i = u x u y R; b m s; 5 = h K 5 j S i = u x u z R; b m s; 6 = h K 6 j S i = u y u z R; b m s; 7 = h K 7 j S i = u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y R; b m s; 8 = h K 8 j S i = u 2 x + u 2 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u 2 z )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 R; b m s; 9 = h K 9 j S i = u 2 x + u 2 y + u 2 z R; b m s; 10 = h K 10 j S i =5 u 3 x + u x u 2 y + u x u 2 z R )]TJ/F15 10.9091 Tf 10.909 0 Td [(13 u x R; b m s; 11 = h K 11 j S i =5 u 2 x u y + u 3 y + u y u 2 z R )]TJ/F15 10.9091 Tf 10.909 0 Td [(13 u y R; b m s; 12 = h K 12 j S i =5 u 2 x u z + u 2 y u z + u 3 z R )]TJ/F15 10.9091 Tf 10.909 0 Td [(13 u z R; b m s; 13 = h K 13 j S i = u x u y u z R; b m s; 14 = h K 14 j S i =30 u 2 x u 2 y + u 2 x u 2 z + u 2 y u 2 z R )]TJ/F15 10.9091 Tf 10.909 0 Td [(40 u 2 x + u 2 y + u 2 z R +32 R: .20 Finally,byinvertingtheabove,i.e. S = K )]TJ/F33 7.9701 Tf 6.586 0 Td [(1 b m s; ,andexploitingtheorthogonalityof K ,wecan determinetheexplicitexpressionsforthesourcetermsinthevelocityspace S ,whicharelisted

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115 inAppendixC.Inordertoconstructthecollisionkernel b h forthe3Dcascadedcollisionoperatorforthescalareld ,weneedtherawmomentsofthecollisionkernelofdierentorders,i.e. P K : b h x m y n z p .Usingtheorthogonalitypropertyof K ,andconsideringthattheonlyconserved invariantofthis3DcascadedLBEisthescalareld correspondingtothezerothmomenti.e. b h 0 =0,weget P K b h =0 ; P K b h e x =10 b h 1 ; P K b h e y =10 b h 2 ; P K b h e z =10 b h 3 ; P K b h e x e y =8 b h 4 ; P K b h e x e z =8 b h 5 ; P K b h e y e z =8 b h 6 ; P K b h e 2 x =2 b h 7 +2 b h 8 +6 b h 9 ; P K b h e 2 y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 b h 7 +2 b h 8 +6 b h 9 ; P K b h e 2 z = )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 b h 8 +6 b h 9 ; P K b h e x e 2 y =16 b h 10 +8 b h 1 ; P K b h e x e 2 z =16 b h 10 +8 b h 1 ; P K b h e 2 x e y =16 b h 11 +8 b h 2 ; P K b h e y e 2 z =16 b h 11 +8 b h 2 ; P K b h e x e 2 z =16 b h 12 +8 b h 3 ; P K b h e y e 2 z =16 b h 12 +8 b h 3 ; P K b h e x e y e z =8 b h 13 ; P K b h e 2 x e 2 y =8 b h 9 +16 b h 14 ; P K b h e 2 x e 2 z =8 b h 9 +16 b h 14 ; P K b h e 2 y e 2 z =8 b h 9 +16 b h 14 : Atthispoint,itisimportanttohighlightthesignicantdierenceinthederivationofthecascadedLBEfortheuidvelocity u giveninAppendixBandthatforthescalareld consideredhere.Inthecaseoftheuidow,themassandmomentumaretheconservedinvariantsfor collision,and,henceitscorrespondingcollisioninvariantskernelcomponentswillbezeroth,i.e. b g 0 = b g 1 = b g 2 = b g 3 .However,inthepresentcase,onlythezerothmoment,i.e.thepassivescalar eldistheconservedmomentduringcollision.Hence, b h 0 ,but b h 1 6 = b h 2 6 = b h 3 .Duetothesedifference,itwillbeevidentinthefollowingthattheexpressionsforthecascadedcollisionoperator forthescalareld isquitedierentfromthatfortheuidvelocity u giveninAppendixB. Finallybasedontheabove,wedeterminethestructureofthe3Dcascadedcollisionoperatorfor thescalareld satisfyingtheCDEasfollows:Beginningrstatthelowestordernon-conserved post-collisioncentralmoments,i.e.thosefortherstordermomentcomponenthere,wesetthem

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116 equaltotheircorrespondingequilibriumstatesasanintermediatestep.Whentheexpressionfor aparticularcollisionkernelcomponent b h > 1isobtainedinthismanner,wediscardthe equilibriumassumptionandmultiplyitbyacorrespondingrelaxationparameter ! .Thisstep allowsforarelaxationprocessintermsofthecentralmomentstorepresenttheeectofcollision[15,125].Afterconsiderablealgebraicmanipulationsandsimplications,andusingthenotation b 0 x m y n z p = b 0 x m y n z p + b 0 x m y n z p .21 For,wesummarizethenalexpressionsforthecollisionkernelcomponents b h as b h 1 = ! 1 10 u x )]TJ/F41 10.9091 Tf 11.022 2.181 Td [(b 0 x )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(1 2 u x R ; .22 b h 2 = ! 2 10 u y )]TJ/F41 10.9091 Tf 11.022 2.181 Td [(b 0 y )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(1 2 u y R ; .23 b h 3 = ! 3 10 u z )]TJ/F41 10.9091 Tf 11.021 2.181 Td [(b 0 z )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(1 2 u z R ; .24 b h 4 = ! 4 8 )]TJ/F41 10.9091 Tf 8.359 2.181 Td [(b 0 xy + u y b 0 x + u x b 0 y )]TJ/F15 10.9091 Tf 10.909 0 Td [( + 1 2 R u x u y + 5 4 u y b h 1 + 5 4 u x b h 2 ; .25 b h 5 = ! 5 8 )]TJ/F41 10.9091 Tf 8.359 2.181 Td [(b 0 xz + u z b 0 x + u x b 0 z )]TJ/F15 10.9091 Tf 10.909 0 Td [( + 1 2 R u x u z + 5 4 u z b h 1 + 5 4 u x b h 3 ; .26 b h 6 = ! 6 8 )]TJ/F41 10.9091 Tf 8.359 2.181 Td [(b 0 yz + u z b 0 y + u y b 0 z )]TJ/F15 10.9091 Tf 10.909 0 Td [( + 1 2 R u y u z + 5 4 u z b h 2 + 5 4 u y b h 3 ; .27

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117 b h 7 = ! 7 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [( b 0 xx )]TJ/F41 10.9091 Tf 10.782 2.182 Td [(b 0 yy +2 u x b 0 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y b 0 y )]TJ/F15 10.9091 Tf 10.909 0 Td [( + 1 2 R u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y + 5 u x b h 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 u y b h 2 ; .28 b h 8 = ! 8 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [( b 0 xx + b 0 yy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b 0 zz +2 u x b 0 x + u y b 0 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u z b 0 z )]TJ/F15 10.9091 Tf 10.909 0 Td [( + 1 2 R u 2 x + u 2 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u 2 z + 5 3 u x b h 1 + 5 3 u y b h 2 )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(10 3 u z b h 3 ; .29 b h 9 = ! 9 18 )]TJ/F15 10.9091 Tf 8.485 0 Td [( b 0 xx + b 0 yy + b 0 zz +2 u x b 0 x + u y b 0 y + u z b 0 z )]TJ/F15 10.9091 Tf 10.91 0 Td [( + 1 2 R u 2 x + u 2 y + u 2 z + 10 9 u x b h 1 + u y b h 2 + u z b h 3 ; .30 b h 10 = ! 10 16 )]TJ/F41 10.9091 Tf 8.359 2.182 Td [(b 0 xyy +2 u y b 0 xy )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y b 0 x + u x b 0 yy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x u y b 0 y + + 1 2 R u x u 2 y )]TJ/F15 10.9091 Tf 8.485 0 Td [( 5 8 u 2 y + 1 2 b h 1 )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(5 8 u x u y b h 2 + u y b h 4 + 1 8 u x )]TJ/F41 10.9091 Tf 8.295 2.879 Td [(b h 7 + b h 8 +3 b h 9 ; .31 b h 11 = ! 11 16 )]TJ/F41 10.9091 Tf 8.359 2.181 Td [(b 0 xxy +2 u x b 0 xy + u y b 0 xx )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 x b 0 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x u y b 0 x + + 1 2 R u 2 x u y )]TJ/F15 10.9091 Tf 9.681 7.38 Td [(5 4 u x u y b h 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [( 5 8 u 2 x + 1 2 b h 2 + u x b h 4 + 1 8 u y b h 7 + b h 8 +3 b h 9 ; .32 b h 12 = ! 12 16 )]TJ/F41 10.9091 Tf 8.359 2.182 Td [(b 0 xxz +2 u x b 0 xz + u z b 0 xx )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x u z b 0 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 x b 0 z + + 1 2 R u 2 x u z )]TJ/F15 10.9091 Tf 9.681 7.38 Td [(5 4 u x u z b h 1 )]TJ/F41 10.9091 Tf 11.913 10.258 Td [(b h 3 2 ; .33 b h 13 = ! 13 8 )]TJ/F41 10.9091 Tf 8.359 2.182 Td [(b 0 xyz + u x b 0 yz + u y b 0 xz u z b 0 xy )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y u z b 0 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x u z b 0 y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x u y b 0 z + + 1 2 R u x u y u z )]TJ/F15 10.9091 Tf 9.681 7.38 Td [(5 4 u y u z b h 1 )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(5 4 u x u z b h 2 )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(5 4 u x u y b h 3 + u z b h 4 + uy b h 5 + u x b h 6 ; .34 b h 14 = ! 14 16 h )]TJ/F41 10.9091 Tf 8.359 2.181 Td [(b 0 xxyy +2 u y b 0 xxy +2 u x b 0 xyy )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y b 0 xx )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 u x u y b 0 xy +2 u x u y b 0 x +2 u 2 x u y b 0 y + c 4 s )]TJ/F15 10.9091 Tf 8.485 0 Td [( + 1 2 R u 2 x u 2 y 5 4 u x u 2 y b h 1 + 5 4 u 2 x u y b h 2 )-222()]TJ/F15 10.9091 Tf 19.394 0 Td [(2 u x u y b h 4 + 1 8 u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y b h 7 )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 8 u 2 x + u 2 y b h 8 )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(3 8 u 2 x + u 2 y )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(1 2 b h 9 +2 u x b h 10 +2 u y b h 11 + u y b h 2 + u x b h 1 : .35 Here,therelaxationparameters ! ,where =1 ; 2 ; 14satisfythebounds0
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118 Theremainingrelaxationparametersathigherordersinuencenumericalstabilityandcanbe tunedindependently.Inthischapter,wesetthemtounity.Noticethatthestructureofthecollisionkernelofthe3DcascadedLBEforthescalareld ismarkedlydierentfromthatforthe uidowseeAppendixB.Inparticular,the"cascaded"structurestartsforthescalareld fromthesecondordermomentcomponentson,whichthatfortheuidowbeginsfromthe thirdordermomentsowingtothedierencesinthenumberofcollisioninvariantsasmentioned earlier.Finally,byexpandingtheelementsoftheproduct K : b h inEq..13andusingitinEq..15a, thepost-collisiondistributionfunction e g isgivenby e g 0 = g 0 + h b h 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b h 9 +32 b h 14 i + S 0 ; e g 1 = g 1 + h b h 0 + b h 1 + b h 7 + b h 8 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 9 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 b h 10 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 b h 14 i + S 1 ; e g 2 = g 2 + h b h 0 )]TJ/F41 10.9091 Tf 10.718 2.878 Td [(b h 1 + b h 7 + b h 8 )]TJ/F41 10.9091 Tf 10.718 2.878 Td [(b h 9 +8 b h 10 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 b h 14 i + S 2 ; e g 3 = g 3 + h b h 0 + b h 2 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 7 + b h 8 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 9 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 b h 11 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 b h 14 i + S 3 ; e g 4 = g 4 + h b h 0 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 2 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 7 + b h 8 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 9 +8 b h 11 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 b h 14 i + S 4 ; e g 5 = g 5 + h b h 0 + b h 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b h 8 )]TJ/F41 10.9091 Tf 10.719 2.878 Td [(b h 9 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 b h 12 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 b h 14 i + S 5 ; e g 6 = g 6 + h b h 0 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b h 8 )]TJ/F41 10.9091 Tf 10.719 2.879 Td [(b h 9 +8 b h 12 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 b h 14 i + S 6 ; e g 7 = g 7 + h b h 0 + b h 1 + b h 2 + b h 3 + b h 4 + b h 5 + b h 6 + b h 9 +2 b h 10 +2 b h 11 +2 b h 12 + b h 13 +2 b h 14 i + S 7 ; e g 8 = g 8 + h b h 0 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 1 + b h 2 + b h 3 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 4 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 5 + b h 6 + b h 9 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b h 10 +2 b h 11 +2 b h 12 )]TJ/F41 10.9091 Tf 8.294 2.879 Td [(b h 13 +2 b h 14 i + S 8 ; e g 9 = g 9 + h b h 0 + b h 1 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 2 + b h 3 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 4 + b h 5 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 6 + b h 9 +2 b h 10 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b h 11 +2 b h 12 )]TJ/F41 10.9091 Tf 8.294 2.879 Td [(b h 13 +2 b h 14 i + S 9 ;

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119 e g 10 = g 10 + h b h 0 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 1 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 2 + b h 3 + b h 4 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 5 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 6 + b h 9 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b h 10 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b h 11 +2 b h 12 + b h 13 +2 b h 14 i + S 10 ; e g 11 = g 11 + h b h 0 + b h 1 + b h 2 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 3 + b h 4 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 5 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 6 + b h 9 +2 b h 10 +2 b h 11 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b h 12 )]TJ/F41 10.9091 Tf 8.294 2.879 Td [(b h 13 +2 b h 14 i + S 11 ; e g 12 = g 12 + h b h 0 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 1 + b h 2 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 3 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 4 + b h 5 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 6 + b h 9 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b h 10 +2 b h 11 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b h 12 )]TJ/F41 10.9091 Tf 8.294 2.879 Td [(b h 13 +2 b h 14 i + S 12 ; e g 13 = g 13 + h b h 0 + b h 1 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 2 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 3 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 4 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 5 + b h 6 + b h 9 +2 b h 10 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b h 11 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b h 12 + b h 13 +2 b h 14 i + S 13 ; e g 14 = g 14 + h b h 0 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 1 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 2 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 3 + b h 4 + b h 5 + b h 6 + b h 9 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b h 10 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b h 11 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b h 12 )]TJ/F41 10.9091 Tf 8.294 2.878 Td [(b h 13 +2 b h 14 i + S 14 : .37 UponperformingthestreamingstepasgiveninEq..15b,usingtheupdateddistributionfunction,thescalareld canbenallycomputedas = 14 X =0 g + 1 2 R .38 Asimplied3DcascadedLBformulationfortheD3Q7latticeispresentedinAppendixDfor completeness. 4.3ResultsandDiscussion Amainobjectiveofthissectionistovalidatethenew3DcascadedLBmethoddiscussedearlierforsimulationofconvectivethermalows.Inthisregard,weconsidersimulationofnatural convectioninacubiccavityandacomparisonofthecomputedowandthermalcharacteristic against3Dbenchmarknumericalsolutions.Naturalconvectionofuidsindierentiallyheated enclosureshasnumerousengineeringapplicationsandariseinvariousnaturalsettings.Theseincludesolarenergycollectors,thermalenergystoragesystems,coolingofelectronicdevices,venti-

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120 lationofbuildingsandcrystalgrowthprocesses.ItischieycharacterizedbytheRayleighnumberrepresentingthestrengthofthebuoyancyeectsrelativetothecounteractingthermaland momentumdiusioneects,andthePrandtlnumber.Someoftheclassic2Dbenchmarksolutionsforthisproblemincludetheresultsreportedby[127].Giventhatthenaturalconvective uidmotioninvariouscasesofpracticalinterestarethreedimensionalinnature,therehavebeen considerableprogressinobtainingbenchmarknumericalresultsforthe3Dnaturalconvectionin acubicenclosure[1]andourpresentstudyusessuchdataaspartofthevalidationinthefollowing. Aschematicofthegeometriccongurationforthephysicalmodelofthe3DcubiccavityconsideredandthecoordinatesystemisshowninFig.1.Itconsistsofacubicenclosureofsidelength L andtheleftwallandtherightwallsurfacearemaintainedattemperaturesof T L and T H , where T H >T C ;alltheotherfourwallsurfacearemaintainedtobeadiabatic.Theconvective uidmotionthenarisesnaturallyfromthebuoyancyforceduetoalocaltemperaturedierence withrespecttoareferencetemperatureinthepresenceofagravityeld.Thisthermallydriven owmayberepresentedbymeansofthefollowingbodyforce F inthe NSE inEq..1bunder Boussinesqapproximationas F = g T )]TJ/F35 10.9091 Tf 10.91 0 Td [(T 0 b k .39 where isthecoecientofthermalexpansion, T = T x;y;z;t isthelocaltemperatureeld, T 0 = T L + T H = 2isthereferenceetemperature, g istheaccelerationduetogravity,and b k isthe unitvectorinthepositive z -directioninFig.1.Thisbodyforceisusedinthe3DcascadedLBE foruidowisdiscussesinSec.2.1,whichthelocaltemperatureeldneededinEq..38isobtainedfromtheother3DcascadedLBEforthethermalenergyequationpresentedinSec.2.2. Thevelocityandthetemperatureboundaryconditionsmaybesummarizedas u x = u y = u z =0forallwalls.40 T x;y =0 ;z = T L ;T x;y = L;z = T H ; .41 @T @ b n =0forallotherwalls.42

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121 FIGURE4.1:Geometriccongurationforthephysicalmodelofthe3Dcubiccavityandthecoordinatesystem.modelof3Dcavitywithcoordinatesystem where b n isthewallnormaldirection.Halfbouncebackschemeisemployedtoimplementthe velocityboundarycondition,andananti-bouncebackschemeisusedtorepresenttheDirichlet boundaryconditionsforthescalartemperatureeld[116]andtheNeumannboundarycondition isimplementedusingtheschemegivenin[128].ThecharacteristicdimensionlessRayleighnumber Ra andthePrandtlnumber Pr forthisproblemaregivenby Ra = g TL 3 = Pr = = .43 where T = T H )]TJ/F35 10.9091 Tf 12.012 0 Td [(T L isthetemperaturedierencebetweenthehotandcoldsurface, and arethethermaldiusivityandkinematicviscosityoftheuid,respectively.Inthefollowing, wewillnon-dimensionalizthecoordinatelengthsbythescale L ,componentsofthevelocityby [ gL T H )]TJ/F35 10.9091 Tf 11.258 0 Td [(T L ] 1 = 2 andthetemperatureby T 0 .Thecorrespondingdimensionlesscoordinatesare

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122 thendenotedby x;y;z ,thevelocityeldby u x ;u y ;u z andthetemperatureeldby T .Akey parametercharacterizingthethermaltransportduringnaturalconvectionistheNusseltnumber. ThemeanNusseltnumberateitherthehotorcoldwallmayberepresentedas Nu mean z = Z 1 x =0 @T x;y @y y =0 ory =1 dx: .44 Inthefollowing,wewillconsidersimulationsofnaturalconvectionofair Pr =0 : 71atdierentvaluesoftheRayleighnumbers Ra .Wewillusethe3DcascadedLBMbasedontheD3Q15 latticeinthisregard,andusingagridresolutionof91 91 91.Figure4.2presentsthetemperatureandvelocityprolesbetweentheadiabaticbottomandtopwallsinthe z -directiona andc,respectivelyandbetweenthecoldandhotsurfacesinthe y )]TJ/F15 10.9091 Tf 8.485 0 Td [(directionbandd, respectively,alongthesymmetryplanex=0.5at Ra =10 5 computedusingourpresent3D cascadedLBM.Also,plottedintheseconguresinsymbolsarethepriorreferencebenchmark solutionbasedontheNSE[1].Itcanbeseenthatthecomputedstructureofboththetemperatureandvelocityeldsalongdierentdirectionsareinverygoodagrementwiththebenchmark numericalresults.Theslopesofthetemperatureeldsnearboththeadiabaticandisothermal wallsarefoundtobewallcapturedbyour3DcascadedLBMbasedonDDFformulation.Furthermore,fromthevelocityprolesshowninFigs.2cand2d,itisevident,inparticular,that boththepeakmagnitudesandtheirlocationsoftheuidconvectionvelocityarewellreproduced byour3DcascadedLBmodel. Figure3presentsthedistributionofstreamlinesarisingduetonaturalconvectiveuidcurrents fromdierentiallyheatedenclosuresattwodierentRayleighnumbersof Ra =10 4 and Ra =10 5 alongcenterplanesindierentcoordinatedirections.Inthevertical y )]TJ/F35 10.9091 Tf 11.305 0 Td [(z midplane x =0 : 5,it canbeseenthatatlower Ra of10 4 ,acentralvortexappearasadominantcharacteristicofthe uidmotion.However,withincreasingtheRayleighnumberto10 5 ,whenthenaturalconvectioneectsbecomemorepronounced,thecentralvortexbreakupintoasetoftwovortices.In addition,itisevidentthatthereisaclusteringofstreamlinesnearthewallsurfaces.Scaleanal-

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123 FIGURE4.2:ComparisonofthetemperaturetopandvelocityprolesbottomforRayleigh number Ra =10 5 onthesymmetrycenterplane x )]TJ/F35 10.9091 Tf 11.067 0 Td [(z ;symbols" "denotethereferencebenchmarksolutions[1],andline" )]TJ/F15 10.9091 Tf 10.909 0 Td [("bypresentwork. ysispredictstheboundarylayerthickness nearanisothermalwallsetupbynaturalconvection scalesas Ra )]TJ/F33 7.9701 Tf 6.587 0 Td [(1 = 4 .Hence,thereisathinnerlayerofuidnearthatdrivesamorevigorousnatural convectionathigher Ra .Onthe x )]TJ/F35 10.9091 Tf 11.394 0 Td [(z midplaney=0.5,inwhichsidewallsareadiabatic,itis seenthattheheateduidrisesup,withthecolderuidmovingdownandbeingreplacedbythe heatedone.Theowpatternisfoundtobethree-dimensionalinnature.Onthe x )]TJ/F35 10.9091 Tf 11.2 0 Td [(y midplane z=0.5,nearthehotwalltheuid,whichrisesfromthebottomofthecavity,movestowardsthe coldwallandaftersomedistancechangesthedirection.Forboththemidplaneasweincrease theRayleighnumber,thethermalconvectiveeectsarefoundtobemoredominate.Theseow patternsareconsistentwithpriornumericalsolutions[1,124]. Thetemperaturedistributions,representedbyisotherms,inmidplanesalongdierentdirections

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124 FIGURE4.3:Projectionsofstreamlinesinnaturalconvectionina3Dcavitycomputedusing3D cascadedLBMondierentcenterplanesatRayleighnumber Ra =10 4 leftand Ra =10 5 right.Toprow: y )]TJ/F35 10.9091 Tf 10.909 0 Td [(z plane,Middlerow x )]TJ/F35 10.9091 Tf 10.909 0 Td [(z planeBottomraw: y )]TJ/F35 10.9091 Tf 10.909 0 Td [(x plane.

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125 FIGURE4.4:Temperaturedistributioninnaturalconvectionina3Dcavitycomputedusing3D cascadedLBMondierentcenterplanesatRayleighnumbers Ra =10 4 leftand Ra =10 5 right.Toprow: y )]TJ/F35 10.9091 Tf 10.909 0 Td [(z plane,Middlerow x )]TJ/F35 10.9091 Tf 10.909 0 Td [(z planeBottomrow: y )]TJ/F35 10.9091 Tf 10.909 0 Td [(x plane.

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126 for Ra of10 4 and10 5 areshowninFig.4.Itcanbeseenthatasthenaturalconvectioneectbecomemoresignicant,athigher Ra =10 5 ,theisothermsbecomemorehorizontalintheregion aroundthecentraofthecavity,andbecomesnearlyverticalinthethinboundarylayerscloserto thehotandcoldwalls.Ingeneral,asexpected,signicanttemperaturevariationsappearinthe thinregionsinthevicinityoftheisothermalwallsurfacesandmoreuniformdistributionsnear theadiabaticwallsurfaces. Inaddition,inordertoprovideaquantitativelystudyofthenumericalresultswecomparethe followingmainowandthermalcharacteristicsofnaturalconvectioninacubiccavityinthe symmetryplane x =0 : 5at Ra =10 3 ,10 4 and10 5 computedusingour3DcascadedLBMwith thereferencebenchmarknumericalsolution[1].Themaximumhorizontalvelocity u max andits coordinatelocation y;x ,themaximumverticalvelocity w max anditscoordinatelocationy,x; themaximumandminimumNusseltnumbers Nu mean and Nu min andtheirlocation,and,nally,theaverageNusseltnumber Nu mean .Thecomputedresult,benchmarksolutionsofthese quantitiesarepresentedinTable1.ItcanbeseenthattheagrementbetweentheDDF-based3D cascadedLBMresultsandthebenchmarksolutionsareinverygoodagreement. TABLE4.1:Qualitativecomparisonofkeyowandthermalcharacteristicsinnaturalconvection inacubiccavityinthesymmetryplanez=0.5betweenthe3DcascadedLBMandthereference benchmarkresultsobtainedusingaNSEsolver[1] Ra 10 3 10 4 10 5 Method 3DcascadedLBMReferenceSolution[1] 3DcascadedLBMReferenceSolution[1] 3DcascadedLBMReferenceSolution[1] Gridesize 91 91 9131 31 31 91 91 9162 62 62 91 91 9162 62 62 u max 0.13080.1314 0.19650.2013 0.14410.1468 Position y;x .5,0.1910.5,0.2000 .5,0.1910.5,0.1833 .5,0.1460.5,0.1453 v max 0.13080.1320 0.22320.2252 0.24470.2471 Position y;x .8426,0.5.8333,0.5 .8876,0.5.8833,0.5 .9325,0.5.9353,0.5 Nu max 1.41701.420 3.58153.652 7.7457.795 Position y;x ,0.0786,0.08333 0,0.1685,0.1623 ,0.0786,0.08256 Nu min 0.7300.7639 0.59250.6110 0.7720.7867 Position y;x ,1.0,1.0 ,1.0,1.0 ,1.0,1.0 Nu mean 1.09771.105 2.26472.302 4.45954.646

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128 CHAPTERV CASCADEDLATTICEBOLTZMANNMETHODBASEDONCENTRAL MOMENTSFORAXISYMMETRICTHERMALFLOWSINCLUDING SWIRLINGEFFECTS 5.1Introduction Fluidmotionincylindricalcoordinateswithaxialsymmetrythatisdrivenbyrotationaleects and/orthermalbuoyancyeectsarisewidelyinanumberofengineeringapplicationsandgeophysicalcontextse.g.,[129,130,131,132,133].Someexamplesoftechnologicalapplications encounteringheatandmasstransfereectsinaxisymmetricowsincludepipelinesystems,heat exchangers,solarenergyconversiondevices,crystalgrowthandmaterialprocessingsystems,electroniccoolingequipmentandturbomachinery.Computationalmethodsplayanimportantrole forbothfundamentalstudiesoftheuidmechanicsandheattransferaspectsandaspredictive toolsforengineeringdesignofsuchsystems.Ingeneral,uidmotionincylindricalcoordinates duetoswirlingeectsandbuoyancyforces,andaccompaniedbythermalandmasstransportis three-dimensionalDinnature.Computationaleortforsuchproblemscanbesignicantlyreducedifaxialsymmetry,whichariseinvariouscontexts,canbeexploited;insuchcasesthesystemofequationscanbereducedtosetofquasi-two-dimensionalDproblemsinthemeridian plane.Traditionally,numericalschemesbasedonnitedierence,nitevolumeorniteelements wereconstructedtosolvetheaxisymmetricNavier-StokesNSequationsfortheuidowalong withtheadvection-diusionequationfortheenergytransporte.g.,[134,135]. Ontheotherhand,latticeBoltzmannLBmethods,whichariseasminimalkineticmodelsof theBoltzmannequation,hasattractedmuchattentionandapplicationtoawiderangeofuid owsandheatandmasstransferproblems[126,6,11,8].

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133 150] @ t + @ r u r + @ z u z = )]TJ/F35 10.9091 Tf 8.485 0 Td [( u r r ; .1a @ t u r + @ r u 2 r + @ z u r u z = )]TJ/F35 10.9091 Tf 8.485 0 Td [(@ r p + @ r @ r u r + @ z [ @ z u r + @ r u z ].1b + u 2 r )]TJ/F35 10.9091 Tf 10.909 0 Td [( u 2 r r +2 @ r u r r )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u r r 2 + F b r ; @ t u z + @ r u r u z + @ z u 2 z = )]TJ/F35 10.9091 Tf 8.485 0 Td [(@ z p + @ r [ @ r u z + @ z u r ]+ @ z @ z u z .1c )]TJ/F35 10.9091 Tf 8.485 0 Td [( u r u z r + @ z u r + @ r u z r + F b z ; @ t u + @ r u r u + @ z u z u = @ 2 @r 2 u + @ 2 @z 2 u )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u r u r .1d + r @ r u )]TJ/F35 10.9091 Tf 10.909 0 Td [( u r 2 ; @ t + @ r u r + @ z u z = @ r D @ r + @ z D @ z )]TJ/F35 10.9091 Tf 12.104 7.38 Td [(u r r + D r @ r : .1e Here, r , z and representthecoordinatesintheradial,axialandazimuthaldirections,respectively;accordingly, u r , u z and u denotetheuidvelocitycomponentsintherespectivedirections,and F b r and F b z areradialandaxialcomponentsoftheexternalbodyforces,respectively. and p representthedensityandpressure,respectively,while and = correspondtothe kinematicanddynamicviscositiesoftheuid,respectively. isthepassivescalarvariable,which isthetemperatureeld T inthepresentstudyi.e. = T and D isthecoecientofdiusivity.Equations.1a-.1crepresenttheaxisymmetricNSequationsfortheaxialandradial componentsofthevelocityeldinthemeridianplane.Thestructureoftheevolutionequations fortheazimuthalmomentum u andthescalareld giveninEqs..1dand.1erespectively,issimilarinform,viz.,advection-diusionequationwithasource,andhencetheycan besolvedusingthesamenumericalprocedures. Inordertorepresenttheabovemacroscopicequationsincylindricalcoordinatesinasetofpseudo2DCartesianforms,weapplythefollowingcoordinate/variabletransformations: r;z 7)167(! y;x ; u r ;u z 7)167(! u y ;u x ;u 7)167(! : .2 Then,theresultingequationsinpseudo-Cartesianformsinvolveadditionaltermswhencompared

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134 tothestandardowandthermaltransportequationsin2D,whichcanberegardedasgeometric sourceterms.Thelatterwillbeintroducedviaasymmetricoperatorsplittingtechniqueinthe respectivecascadedLBformulationinthefollowing.Thus,themassandmomentumequations fortheuidmotioninthemeridianplaneEqs..1a-.1ccanbewritteninpseudo-2DCartesianformsas @ t + @ y u y + @ x u x = M A ; .3a @ t u x + @ r u 2 x + @ y u x u y = )]TJ/F35 10.9091 Tf 10.91 0 Td [(@ x p + @ x [2 @ x u x ]+ @ y [ @ y u x + @ x u y ] + F A x + F b x ; .3b @ t u y + @ x u x u y + @ y u 2 y = )]TJ/F35 10.9091 Tf 10.909 0 Td [(@ y p + @ x [ @ x u y + @ y u x ]+ @ y [2 @ y u y ] + F A y + F b y ; .3c wherethegeometricmasssource M A andthemomentumsourcevector F A = F A x ;F A y canbe representedas M A = )]TJ/F35 10.9091 Tf 8.485 0 Td [( u y y ; .4a F A x = )]TJ/F35 10.9091 Tf 8.485 0 Td [( u x u y y + @ x u y + @ y u x y ; .4b F A y = 2 y )]TJ/F35 10.9091 Tf 10.909 0 Td [( u 2 y y +2 @ y u y y )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y y 2 : .4c Then,thetotalforce F = F x ;F y inthisapproachbecomes F x = F A x + F b x ;F y = F A y + F b y : .5 Here,thebodyforce F b = F b x ;F b y couldbeavolumetricforcesuchasthebuoyancyforceorthe Lorentzforce.Similarly,theazimuthalmomentumequationfor = u canbewrittenas @ t + @ x u x + @ y u y = D @ 2 x + @ 2 y + S ; .6 wherethecorrespondinggeometricsourceterm S canbeexpressedas S = )]TJ/F15 10.9091 Tf 9.68 7.441 Td [(2 u y y + y @ y )]TJ/F35 10.9091 Tf 12.105 7.38 Td [( y 2 ; .7

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136 where y isthetransposeoperatorandtheircomponentsforanyparticledirection aredenoted by e x and e y ,where =0 ; 1 ; 8.Wealsoneedthefollowing9-dimensionalvector j 1 i = ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 y .11 whoseinnerproductwiththedistributionfunction f denesitszerothmoment.Here,andin thefollowing,wehaveusedthestandardDirac'sbra-ketnotationtorepresentthevectors.The correspondingnineorthogonalbasisvectorsmayberepresentedbye.g.[19]: K 0 = j 1 i ;K 1 = j e x i ;K 2 = j e y i ;K 3 =3 j e 2 x + e 2 y i)]TJ/F15 10.9091 Tf 17.575 0 Td [(4 j 1 i ; K 4 = j e 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(e 2 y i ;K 5 = j e x e y i ;K 6 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 j e 2 x e y i +2 j e y i ; K 7 = )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 j e x e 2 y i +2 j e x i ;K 8 =9 j e 2 x e 2 y i)]TJ/F15 10.9091 Tf 17.575 0 Td [(6 j e 2 x + e 2 y i +4 j 1 i : .12 Hereandhenceforth,symbolssuchas j e 2 x e y i = j e x e x e y i denoteavectorthatresultfromtheelementwisevectormultiplicationofvectors j e x i , j e x i and j e y i .Theabovesetofvectorscanbe organizedbythefollowingorthogonalmatrix K =[ K 0 ;K 1 ;K 2 ;K 3 ;K 4 ;K 5 ;K 6 ;K 7 ;K 8 ] ; .13 whichmapschangesofmomentsundercollisionsduetoacascadedcentralmomentrelaxation backtochangesinthedistributionfunctionseebelow.Asthecascadedcollisionoperatoris builtonthemomentspace,werstdenethecentralmomentsandrawmomentsoforder m + n ofthedistributionfunction f anditsequilibrium f eq as 0 B @ ^ x m y n ^ eq x m y n 1 C A = X 0 B @ f f eq 1 C A e x )]TJ/F35 10.9091 Tf 10.91 0 Td [(u x m e y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y n ; .14 and 0 B @ ^ 0 x m y n ^ eq 0 x m y n 1 C A = X 0 B @ f f eq 1 C A e m x e n y ; .15 respectively.Hereandinwhatfollows,theprime 0 symbolsdenotevariousrawmoments.The centralmomentsoftheequilibriumareconstructedtobeequaltothosefortheMaxwellian,which

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137 thenserveasattractorsduringthecascadedcollisionrepresentedasarelaxationprocess[15]. Inthefollowing,anoperatorsplittingbasedcascadedLBschemewillbeconstructedtosolve Eqs..3-.5.First,werepresentthesolutionofthemassandmomentumequationsinthe meridianplaneEq..3withouttherespectivesourcetermsi.e. M A ;F A x ;F b x ;F A y ;F b y bymeans oftheevolutionofthedistributionfunction f usingtheusualcollisionandstreamingsteps C and S ,respectivelyas Step C : f p = f + K b p ; .16a Step S : f x ;t = f p x )]TJ/F52 10.9091 Tf 10.909 0 Td [(e t;t ; .16b where e = e x ; e y , t isthetimestep, f p isthepost-collisiondistributionfunctionatalocation x andtime t . b p = b p 0 ; b p 1 ; b p 2 ::: b p 8 denotesthechangesofdierentmomentsundercollision basedontherelaxationofcentralmomentstotheirequilibriainacascadedfashion[15].With themassandmomentumbeingconservedduringcollision b p 0 = b p 1 = b p 2 =0,andthechangesin thehigherordernon-conservedmomentsaregivenby[15,157,19] b p 3 = ! 3 12 n 2 c 2 s + u 2 x + u 2 y )]TJ/F15 10.9091 Tf 10.909 0 Td [( b 0 xx + b 0 yy o ; b p 4 = ! 4 4 n u 2 x )]TJ/F35 10.9091 Tf 10.91 0 Td [(u 2 y )]TJ/F15 10.9091 Tf 10.909 0 Td [( b 0 xx )]TJ/F41 10.9091 Tf 11.021 0 Td [(b 0 yy o ; b p 5 = ! 5 4 n u x u y )]TJ/F41 10.9091 Tf 11.021 0 Td [(b 0 xy o ; b p 6 = ! 6 4 n 2 u 2 x u y + b 0 xxy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x b 0 xy )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y b 0 xx o )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 2 u y b p 3 + b p 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x b p 5 ; b p 7 = ! 7 4 n 2 u x u 2 y + b 0 xyy )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 u y b 0 xy )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x b 0 yy o )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(1 2 u x b p 3 )]TJ/F41 10.9091 Tf 11.532 0 Td [(b p 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y b p 5 ; b p 8 = ! 8 4 n c 4 s +3 u 2 x u 2 y )]TJ/F41 10.9091 Tf 10.909 12.11 Td [(h b 0 xxyy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x b 0 xyy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y b 0 xxy + u 2 x b 0 yy + u 2 y b 0 xx +4 u x u y b 0 xy io )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b p 3 )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(1 2 u 2 y b p 3 + b p 4 )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 2 u 2 x b p 3 )]TJ/F41 10.9091 Tf 11.533 0 Td [(b p 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 u x u y b p 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y b p 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x b p 7 : .17 Here, ! 3 ;! 4 : ! 8 arerelaxationparameters,where ! 3 ;! 4 and ! 5 arerelatedtothebulkand shearviscositiesandtheother ! i inuencethenumericalstabilityofthemethod.Inparticular, thebulkviscosityisgivenby = c 2 s 1 ! 3 )]TJ/F33 7.9701 Tf 12.742 4.295 Td [(1 2 t andtheshearviscosityby = c 2 s 1 ! j )]TJ/F33 7.9701 Tf 12.743 4.295 Td [(1 2 = t ,

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138 where j =4 ; 5,and c 2 s = c 2 = 3 ; where c = x= t .Inthiswork,weconsiderthelatticeunits, where x = t =1andhencethespeedofsound c s =1 = p 3,andthehigherorderrelaxationparameters ! 6 , ! 7 and ! 8 aresettounityforsimplicity.AfterthestreamingstepseeEq..16b, theoutputdensityeldandthevelocityeldcomponentsdesignatedwithasuperscript" o "as thezerothandrstmomentsof f ,respectively: o = P 8 =0 f ; o u o x = P 8 =0 f e x ; o u o y = P 8 =0 f e y .18 Wethenintroducetheinuenceofthemasssource M A inEq..3aandthemomentumsources F A x = F A x + F b x and F y = F A y + F b y inEqs..3band.3c,respectively,asthesolutionofthe followingtwosubproblems,referredtoasthemasssourcestep M andmomentumsourcestep F , respectively: Step M : @ t = M A ; .19a Step F : @ t u = F = F A + F b ; .19b where u = u x ;u y and F A = F A x ;F b y etc.Inourpreviouswork[163],weconstructedasymmetricoperatorsplittingbasedapproachtoincorporateasinglemomentumsourceinacascadedLB method.Inthepresentwork,wefurtherextendthisapproachtosymmetricsplittingofmultipleoperatorsrelatedtomassandmomentumsources.Inotherwords,weperformtwosymmetric stepsofhalftimestepsoflength t= 2of M and F ,onebeforeandtheotherafterthecollision step.TheoverallsymmetrizedoperatorsplittingbasedcascadedLBalgorithmimplementingall thefouroperators C ; S ; M and F duringthetimeinterval[ t;t + t ]maybewrittenas f x ;t + t = M 1 = 2 F 1 = 2 CF 1 = 2 M 1 = 2 S f x ;t ; .20 where M 1 = 2 and F 1 = 2 representsolvingEqs..19aand.19b,respectively,overtimestep t= 2. Bothofthesestepsintroducetheeectofgeometricmassandmomentumsourceandthebody forcesdirectlyinthemomentumspace.

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139 SolvingEqs..19aand.19bfortherstpartofsymmetricsequenceneededinEq..20 yields )]TJ/F35 10.9091 Tf 10.909 0 Td [( o = M A t 2 , u x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u o x = F x t 2 and u y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u o y = F y t 2 .Thus,wehave Pre-collisionMassSourceStep M 1 = 2 : = o + M A t 2 .21a Pre-collisionMomentumSourceStep F 1 = 2 : u x = u o x + F x t 2 ; .21b u y = u o y + F y t 2 ; .21c where M A ;F x and F y aregiveninEqs..4a-.4cand.5.BasedonEq..20,thenext stepisthecollisionstep,whichisperformedusingtheupdateddensityandvelocityelds ;u x ;u y giveninEqs..21a-.21candthendeterminingthechangeofmomentsundercollision b p = 3 ; 4 ::: 8usingEq..17.Then,implementingtheotherpartofthesymmetrizedmassandmomentumstepswithusingahalftimesteptosolveEqs..19aand.19b,weobtainthetarget densityandvelocityeldaftercollisionrepresentedas p ;u p x ;u p y via p )]TJ/F35 10.9091 Tf 8.725 0 Td [( = M A t 2 ;u p x )]TJ/F35 10.9091 Tf 8.725 0 Td [(u x = F x t 2 and u p y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y = F y t 2 .Thus,wehave Post-collisionMomentumSourceStep F 1 = 2 : u p x = u x + F x t 2 u p y = u y + F y t 2 ; .22a Post-collisionMassSourceStep M 1 = 2 : p = + M A t 2 ; .22b Byrewritingtheaboveresultsforthepost-collisionsourcestepsintermsoftheoutputdensity o andvelocityeld u o = u o x ;u o y viaEqs..21a-.21c,weget p = o + M A t;u p x = u o x + F x t;u p y = u o y + F y t: .23 Toeectivelydesignthepost-collisiondistributionfunction f p inthecascadedLBschemeso thatEq..23ispreciselysatised,weconsider f p = f + K b p andtakingitszerothand rstmoments,weobtain p = f p = f + h K j 1 i b p ; .24a u p x = f p e x = f e x + h K j e x i b p ; .24b u p y = f p e y = f e y + h K j e y i b p : .24c

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140 Sincetheorthogonalbasisvectors j K i giveninEq..12satisfy h K j 1 i =9 b p 0 ; h K j e x i = 6 b p 1 ; h K j e y i =6 b p 2 ; Eqs..24a-.24cbecome p = o +9 b p 0 ;u p x = u o x +6 b p 1 ;u p y = u o y +6 b p 2 : .25 ComparingEqs..23and.25,itfollowsthatthechangeofthezerothmoment b p 0 andthe rstmoments b p 1 and b p 2 duetomassandmomentumsourcecanbewrittenas b p 0 = M A 9 t; b p 1 = F x 6 t; b p 2 = F y 6 t: .26 where M A followsfromEq..4a, F x and F y aregiveninEq..5and.4b-.4c.These expressionseectivelyprovidethedesiredpost-collisionstatesofthedistributionfunction,i.e. f p duetomassandmomentumsources.Theoverallschemepresentedabovebasedonoperatorsplittingprovidesaconsistentapproachtorepresentmassandmomentumsourcesinthe uidmotion.Inparticular,thepre-collisionstepseachoverahalftimesteplengthshownin Eqs..21a-.21cintroducetheeectofmasssourceandtheforcesintothemomentequilibriaofallordersbeforetheyundergocentralmomentrelaxationEq..17.Asaresult,inparticular,theyeliminatethespurioustermssuchas F i u j + F j u i arisinginthesecondordernonequilibriummomentsrelatedtotheviscousstressintheChapman-Enskoganalysisandcorrectly recovertheNavier-Stokesequations[85,163].Inaddition,theuseoftwohalfmasssource/force stepsaroundthecollisionstepisconsistentwiththeclassicalStrangsplittingofmultipleoperatorsandissecondorderaccuratesee[85,163]fordetailsandalso[87]foritsapplicationtoa multiplerelaxationformulation.Analternativeapproachtointroduceforcingtermsbasedonan unsplitformulationforthecascadedLBmethodhasbeenpresentedrecentlyin[80,161].Thus, nallyexpanding K b p inEq..16a,thecomponentsofthepost-collisiondistributionfunc-

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141 tionsreadas f p 0 = f 0 +[ b p 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 b p 3 )]TJ/F41 10.9091 Tf 11.532 0 Td [(b p 8 ] ; f p 1 = f 1 +[ b p 0 + b p 1 )]TJ/F41 10.9091 Tf 11.532 0 Td [(b p 3 + b p 4 +2 b p 7 )]TJ/F41 10.9091 Tf 11.532 0 Td [(b p 8 ] ; f p 2 = f 2 +[ b p 0 + b p 2 )]TJ/F41 10.9091 Tf 11.532 0 Td [(b p 3 )]TJ/F41 10.9091 Tf 11.532 0 Td [(b p 4 +2 b p 6 )]TJ/F41 10.9091 Tf 11.532 0 Td [(b p 8 ] ; f p 3 = f 3 +[ b p 0 )]TJ/F41 10.9091 Tf 11.532 0 Td [(b p 1 )]TJ/F41 10.9091 Tf 11.532 0 Td [(b p 3 + b p 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b p 7 + b p 8 ] ; f p 4 = f 4 +[ b p 0 )]TJ/F41 10.9091 Tf 11.532 0 Td [(b p 2 )]TJ/F41 10.9091 Tf 11.532 0 Td [(b p 3 )]TJ/F41 10.9091 Tf 11.532 0 Td [(b p 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b p 6 + b p 8 ] ; f p 5 = f 5 +[ b p 0 + b p 1 + b p 2 +2 b p 3 + b p 5 )]TJ/F41 10.9091 Tf 11.532 0 Td [(b p 6 )]TJ/F41 10.9091 Tf 11.532 0 Td [(b p 7 + b p 8 ] ; f p 6 = f 6 +[ b p 0 )]TJ/F41 10.9091 Tf 11.532 0 Td [(b p 1 + b p 2 +2 b p 3 )]TJ/F41 10.9091 Tf 11.532 0 Td [(b p 5 )]TJ/F41 10.9091 Tf 11.532 0 Td [(b p 6 + b p 7 + b p 8 ] ; f p 7 = f 7 +[ b p 0 )]TJ/F41 10.9091 Tf 11.533 0 Td [(b p 1 )]TJ/F41 10.9091 Tf 11.532 0 Td [(b p 2 +2 b p 3 + b p 5 + b p 6 + b p 7 + b p 8 ] ; f p 8 = f 8 +[ b p 0 + b p 1 )]TJ/F41 10.9091 Tf 11.532 0 Td [(b p 2 +2 b p 3 )]TJ/F41 10.9091 Tf 11.532 0 Td [(b p 5 + b p 6 )]TJ/F41 10.9091 Tf 11.532 0 Td [(b p 7 + b p 8 ] : .27 where b p 0 , b p 1 and b p 2 areobtainedfromEq.and b p 3 ; b p 4 ; ; b p 8 fromEq..17* 5.2.3CascadedLBschemeforazimuthalvelocityeld:operatorsplittingforsourceterm *WenowconstructanovelcascadedLBschemeforthesolutionoftheequationoftheazimuthal momentumcomponent = u giveninEqs..6and.7usingaD2Q5lattice[163].First, deningthevectorscorrespondingtoparticlevelocitycomponentsanda5-dimensionalvector j 1 i as j e x i = ; 1 ; 0 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0 y ; .28a j e y i = ; 0 ; 1 ; 0 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 y ; .28b j 1 i = ; 1 ; 1 ; 1 ; 1 y ; .28c

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142 wheretakingtheinnerproductofthedistributionfunction g with j 1 i denesitszerothmoment.Usingthese,theveorthogonalbasisvectorscanbewrittenas L 0 = j 1 i ;L 1 = j e x i ;L 2 = j e y i ; L 3 =5 j e 2 x + e 2 y i)]TJ/F15 10.9091 Tf 17.576 0 Td [(4 j 1 i ;L 4 = j e 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(e 2 y i ; .29 whichcanbegroupedtogetherasthefollowingtransformationmatrixthatconvertsthechanges inmomentstothoseinthedistributionfunctions: L =[ L 0 ;L 1 ;L 2 ;L 3 ;L 4 ] : .30 Inordertodesignacascadedcollisionoperatortosolvefortheazimuthalmomentum,whichacts asapassivescalareld = u describedbyonadvection-diusionequationundertheaction ofalocalsourcetermEqs..6and.7,wedenethefollowingcentralmomentsandraw momentsofthedistributionfunction g anditsequilibrium g eq as 0 B @ ^ x m y n ^ eq; x m y n 1 C A = X 0 B @ g g eq 1 C A e x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x m e y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y n ; .31 and 0 B @ ^ 0 x m y n ^ eq; 0 x m y n 1 C A = X 0 B @ g g eq 1 C A e m x e n y : .32 respectively.Thecentralmomentsoftheequilibrium^ eq; x m y n aredevisedbeequaltothoseforthe Maxwellianafterreplacingthedensitywiththescalareldinitsexpression.Thenthecascaded collisionstepiswrittenintermsofrelaxationofdierentcentralmomentstotheirequilibria. Similartotheprevioussection,asymmetrizedoperatorsplitschemewillnowbedevelopedto solveEqs..6and.7inthecascadedLBformulation.First,werepresentthesolutionof Eq..6withoutthesourcetermEq..7throughthecollisionandstreamingstepsofthe distributionfunction g as Step C : g p = g + L b q ; .33a Step S : g x ;t = g p x )]TJ/F52 10.9091 Tf 10.909 0 Td [(e t;t : .33b

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143 where g p isthepost-collisiondistributionfunctionand b q = b q o ; b q 1 ; b q 4 representsthechanges ofdierentmomentsunderacascadedcollisionprescribedasarelaxationprocessintermsof centralmoments,whichreadsas[163] b q 1 = ! 1 2 h u x )]TJ/F41 10.9091 Tf 11.021 0 Td [(b 0 x i ; b q 2 = ! 2 2 h u y )]TJ/F41 10.9091 Tf 11.022 0 Td [(b 0 y i ; b q 3 = ! 3 4 h 2 c 2 s )]TJ/F15 10.9091 Tf 10.909 0 Td [( b 0 xx + b 0 yy +2 u x b 0 x + u y b 0 y + u 2 x + u 2 y i + u x b q 1 + u y b q 2 ; b q 4 = ! 4 4 h )]TJ/F15 10.9091 Tf 8.485 0 Td [( b 0 xx )]TJ/F41 10.9091 Tf 11.021 0 Td [(b 0 yy +2 u x b 0 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y b 0 y + u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y i + u x b q 1 )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y b q 2 ; .34 where ! 1 , ! 2 , ! 3 and ! 4 aretherelaxationparameters.Since isconservedduringcollision, b q o =0.Therelaxationparametersfortherstordermoments ! 1 and ! 2 arerelatedtodiusivity D = = c 2 s 1 ! j )]TJ/F33 7.9701 Tf 12.227 4.295 Td [(1 2 t ,j=1,2where c 2 s isafreeparameter,whichissetto1 = 3.Therelaxationparametersforthehigherordermoments,whichinuencenumericalstability,aretaken tobeunityinthisstudy.AfterthestreamingstepinEq..33b,theoutputpassiveazimuthal momentumeld o iscomputedasthezerothmomentof g as o = 4 X g : .35 Thesourceterm S ,whichwaseliminatedintheabove,willnowbeintroducedbyappropriately combiningitseectaftersolutionofthefollowingsuchproblem: Step R : @ t = S .36 Itssolutionwillnowbecombinedwiththesplitsolutionobtainedintheabsenceofthesource terminEqs..33aand.33bviaasymmetricoperatorsplittingtechniqueoveratimeinterval [ t;t + t ],analogoustothatconsideredintheprevioussubsection.Thiscanberepresentedas g x ;t + t = SR 1 = 2 CR 1 = 2 g x ;t ; .37 Thepre-collisionsourcestep R 1 = 2 isexecutedviaasolutionofEq..36overaduration t= 2, whichyields )]TJ/F35 10.9091 Tf 10.91 0 Td [( o = S t 2 ,andhence Pre-collisionSourceStep R 1 = 2 : = o + S t 2 .38

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144 Basedonthisupdatedscalareld,thechangesofdierentmomentsundercollision b q ; =1 ; 2 ; 3 ; 4, giveninEq..34canbecomputed.Similarly,theotherpartofthesourcestep R 1 = 2 withhalf timestepfollowingcollisioncanbeperformedbysolvingEq..36,whichcanbeexpressedas Post-collisionSourceStep R 1 = 2 : p = + S t 2 .39 where p isthetargetscalareldaftercollision.Byrewritingitintermsoftheoutputscalar eld o usingEq..38,wehave p = o + S t: .40 Inorderforthepost-collisiondistributionfunction g p = g + L b q tosatisfyEq..40,we writeitszerothmomentas p = g p = g + h L j 1 i b q : .41 Since h L j 1 i q =5 b q o viaorthogonalofbasisvectorsseeEq..29,itfollowsfromEqs..35 and.41that p = o +5 b q 0 .ComparingthiswithEq..40,wegetthechangeofthezeroth moment b q o duetothepresenceofthesourceterm S as b q 0 = S 5 t: .42 Themodelingofthesourcetermforthetransportofascalareldviatheaboveoperatorsplittingapproachprovidesaconsistentrepresentationoftheireectintheconvection-diusionequation.Thisisduetothefactthattheuseofthepre-collisionsourcestepoverahalftimestep Eq..38introducestheeectofthescalarsourcetermintothemomentequilibriaofallordersbeforetheyparticipateinthecollisionstepviatherelaxationofvariouscentralmoments Eq..34.Moreover,thesymmetricapplicationofpre-andpost-collisionsourcestepsovera halftimesteplengthineachcasemakesitconsistentwiththeStrangsplitting[163].Similarconsiderationsholdforthestrategyforincludingthelocalheatsourcespresentedinthenextsection. Anotherequivalentapproachbasedonanunsplitformulationtoincorporatethesourcetermfor thescalareldisdiscussedin[162].Finally,thecomponentsofthepost-collisiondistribution

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145 functioninEq..33acanbeexpressedafterexpanding L b q as g p 0 = g 0 +[ b q 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 b q 3 ] ; g p 1 = g 1 +[ b q 0 + b q 1 + b q 3 + b q 4 ] ; g p 2 = g 2 +[ b q 0 + b q 2 + b q 3 )]TJ/F41 10.9091 Tf 11.418 0 Td [(b q 4 ] ; g p 3 = g 3 +[ b q 0 )]TJ/F41 10.9091 Tf 11.418 0 Td [(b q 1 + b q 3 + b q 4 ] ; g p 4 = g 4 +[ b q 0 )]TJ/F41 10.9091 Tf 11.418 0 Td [(b q 2 + b q 3 )]TJ/F41 10.9091 Tf 11.418 0 Td [(b q 4 ] ; .43 where b q o i.e.,thechangeofthezerothmomentduetosourceisgiveninEq..42and b q ; = 1 ; 2 ; 3 ; 4i.e.,thechangesofthehigher,non-conserved,momentsundercollisionisobtainedfrom Eq..34. 5.2.4CascadedLBschemefortemperatureeld:operatorsplittingforsourceterm Asintheprevioussection,weconsideraD2Q5lattice,andusetheorthogonalbasisvectors L andthetransformationmatrix L giveninEqs..29and.30,respectively,todesignacascadedLBschemeforthesolutionofthetemperatureeld = T .Itsevolutionispresentedby theadvection-diusionequationwithasourcetermgiveninEqs..8and.9.Thevarious centralmomentsandrawmomentsofthecorrespondingdistributionfunction h anditsequilibrium h eq aredenedas 0 B @ ^ x m y n ^ eq; x m y n 1 C A = X 0 B @ h h eq 1 C A e x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x m e y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y n ; .44 and 0 B @ ^ 0 x m y n ^ eq; 0 x m y n 1 C A = X 0 B @ h h eq 1 C A e m x e n y : .45 Asbefore,weusethesymmetrizedoperatorsplittingtoincludethesourceterm S inthecascadedLBscheme,whichcanbepresentedas: h x ;t + t = SR 1 = 2 CR 1 = 2 h x ;t ; .46

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146 where C and S denotethecollisionandstreamingsteps,respectively,of g usedtosolveEq..8 without S StepC: h p = h + L b r ; .47a StepS: h x ;t = h p x )]TJ/F52 10.9091 Tf 10.909 0 Td [(e t;t : .47b Here, h p isthepost-collisiondistributionfunctionand b r = b r o ; b r 1 ; b r 2 ; b r 3 ; b r 4 isthechangeofdifferentmomentsundercollision,with b r o =0dueto beingacollisioninvariant.Intheabove, afterthestreamingstep,thesolutionoftheoutputscalareld o iscomputedviathezerothmomentof h as o = 4 X =0 h : .48 Theoperator R 1 = 2 appliedtwiceinEq..46representsthesplitsolutionofthescalarelddue tothesourcetermoftheevolutionequation @ t = S beforeandaftercollisionoverahalftime step t= 2.Thus,thepre-collisionsourcestepcanbeexpressedas Pre-collisionSourceStep R 1 = 2 : = o + S 2 t: .49 Thisupdatedscalareld isthenusedtocomputethechangesofdierentmomentsundercollision b r ; =1 ; 2 ; 3 ; 4,whichcanbewrittenas b r 1 = ! 1 2 h u x )]TJ/F41 10.9091 Tf 11.021 0 Td [(b 0 x i ; b r 2 = ! 2 2 h u y )]TJ/F41 10.9091 Tf 11.021 0 Td [(b 0 y i ; b r 3 = ! 3 4 h 2 c 2 s )]TJ/F15 10.9091 Tf 10.909 0 Td [( b 0 xx + b 0 yy +2 u x b 0 x + u y b 0 y + u 2 x + u 2 y i + u x b r 1 + u y b r 2 ; b r 4 = ! 4 4 h )]TJ/F15 10.9091 Tf 8.484 0 Td [( b 0 xx )]TJ/F41 10.9091 Tf 11.022 0 Td [(b 0 yy +2 u x b 0 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y b 0 y + u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y i + u x b r 1 )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y b r 2 ; .50 wheretherelaxationparameters ! 1 and ! 2 arerelatedtothethermaldiusivity D via D = c 2 s 1 ! j )]TJ/F33 7.9701 Tf 13.201 4.295 Td [(1 2 t;j =1 ; 2,where c 2 s = 1 3 and ! 3 = ! 4 =1inthiswork.Followingthis,the post-collisionsourcestep R 1 = 2 canberepresentedas Post-collisionSourceStep R 1 = 2 : p = + S 2 t .51

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147 where p isthetargetscalareldfollowingcollision,whichviaEq..49readsas p = o + S t . Thepost-collisiondistributionfunction h p = h + L b r canbemadetosatisfythiscondition usingEq..48andusing h L j 1 i b r =5 b r 0 aftertakingitszerothmoment,i.e. p = P h p . Thisprovidesthefollowingzerothmomentchangedueto S aftercollision b r 0 = S 5 t: .52 Finally,thepost-collisiondistributionfunction h p canbeexplicitlywrittenafterexpanding L b r inEq..47aasfollows: h p 0 = h 0 +[ b r 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 b r 3 ] ; h p 1 = h 1 +[ b r 0 + b r 1 + b r 3 + b r 4 ] ; h p 2 = h 2 +[ b r 0 + b r 2 + b r 3 )]TJ/F41 10.9091 Tf 11.097 0 Td [(b r 4 ] ; h p 3 = h 3 +[ b r 0 )]TJ/F41 10.9091 Tf 11.098 0 Td [(b r 1 + b r 3 + b r 4 ] ; h p 4 = h 4 +[ b r 0 )]TJ/F41 10.9091 Tf 11.098 0 Td [(b r 2 + b r 3 )]TJ/F41 10.9091 Tf 11.097 0 Td [(b r 4 ] ; .53 where b r o isobtainedformEq..52and b r , =1 ; 2 ; 3and4,followsfromEq..50dueto variousnon-conservedmomentchangesundercollision. 5.3ResultsandDiscussion Inthissection,thecascadedLBschemesdescribedabovewillbeappliedtoandstudiedfordifferentcomplexowbenchmarkproblemstovalidatethemforsimulationsofaxisymmetricows withheattransferandincludingrotational/swirlingeects.Theseincludethefollowing:aTaylorCouetteowbetweentworotatingcircularcylinders,bnaturalconvectioninanannulusbetweentwostationarycoaxialverticalcylinders,cRayleigh-Benardconvectioninsidevertical cylinderheatedatthebottomandcooledatthetop,dcylindricalcavityowdrivenbythe motionofthetoplid,emixedconvectioninaslenderverticalannulussubjectedtotheinner cylinderrotation,andfmeltowinacylinderduringCzochralskicrystalgrowthprocess.

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148 5.3.1Taylor-Couetteow Asthersttestproblem,theclassicalshear-drivencircularCouetteowbetweentwocircular cylindersisconsidered[164].ThisproblemisusedtoassessthecascadedLBschemefortheazimuthalvelocitycomponent u giveninSec.2.3,whoseevolutionisrepresentedbyEqs..6and .Theradiioftheinnerandoutercylindersaredenedas R i and R o ,respectively.Letthe angularvelocitiesoftheinnerandoutercylindersbe i and o ,respectively,whichinduceanazimuthalowwithintheirannulusgap.TheanalyticalsolutionforsuchacylindricalCouetteow isgivenintermsoftheradialvariationoftheazimuthalvelocityasfollows: u r = Ar + B r ; where A = o R 2 o )]TJ/F33 7.9701 Tf 6.587 0 Td [( i R 2 i R 2 o )]TJ/F36 7.9701 Tf 6.587 0 Td [(R 2 i , B = i )]TJ/F33 7.9701 Tf 6.587 0 Td [( o R 2 i R 2 o R 2 o )]TJ/F36 7.9701 Tf 6.586 0 Td [(R 2 i .Here, r istheradialdistancefromthecylindricalaxis. Foreaseofrepresentation,thiscanbewritteninanon-dimensionalformas u r u o = 1 1 )]TJ/F35 10.9091 Tf 10.909 0 Td [( 2 [ )]TJ/F35 10.9091 Tf 10.909 0 Td [( 2 r R i + R i r )]TJ/F35 10.9091 Tf 10.909 0 Td [( ] ; where u o = i R i , istheradiusratiogivenby = R i =R o and denotestheangularvelocity ratio,i.e., = o = i : Inoursimulation,periodicboundaryconditionsareappliedintheaxialdirectionandthevalues oftheazimuthalvelocitiesattheinnerandoutercylinderareprescribedas u r = R i = i R i = u o and u r = R o = o R o = u o ,respectivelyusingtheDirichletboundaryconditionimplementationschemeassociatedwiththeadvection-diusionequationrepresentingthedynamicsof u [100].Theoutercylinderradiusisresolvedby200latticenodesandthelatticelocationfor theinnercylinderxedusing R i = R o fordierentchoicesof .Theperiodicaxialdirectionis discretizedusing3latticenodes.TherelaxationtimesinthecascadedLBschemerepresenting thekinematicshearviscosityaresetas ! j =1 =;j =4 ; 5,where =0 : 6,and u o ischosensuch thattherotationalReynoldsnumber Re = u o R i = becomes5.Figure1presentsacomparison ofthevelocityprolescomputedusingthecascadedLBschemeagainsttheanalyticalsolutionat

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149 theangularvelocityratio =0 : 1forvariousvaluesoftheradiusratio =0 : 103 ; 0 : 203 ; 0 : 303 and0 : 503.Itisclearthattheagreementbetweenthenumericalandanalyticalsolutionisvery good. FIGURE5.1:Comparisonbetweentheanalyticalvelocityprolesolidlinesandthecascaded LBsolutionsymbolsfortheTaylor-Couetteowbetweentwocircularcylindersatanangular velocityratio =0 : 1andforvariousvaluesoftheradiusratio . Orderofaccuracy WewillnowexamineournewaxisymmetriccascadedLBschemeforthisbenchmarkproblemto establishitsorderofaccuracy.Inthisregard,weconsiderthediusivescaling,i.e.,anincrease inthegridresolutionisaccompaniedbyaproportionaldecreaseintheMachnumberataxed viscosityoraxedrelaxationtime,whichcorrespondstoanasymptoticconvergencetotheincompressibleowlimit.Forthispurpose,theglobalrelativeerror E g;u isdenedasfollows: E g;u = s u c )]TJ/F35 10.9091 Tf 10.909 0 Td [(u a 2 u a 2 ; .54

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150 where u c isthenumericalvelocityeldcomputedusingtheaxisymmetriccascadedLBscheme and u a istheanalyticalsolutiongivenabove,andthesummationiscarriedoutforthewholedomain.WeconsiderxedvaluesoftheReynoldsnumberRe=5,radiusratio =1 = 3,relaxation timethatdeterminesthemomentumdiusivity,i.e.,theviscosityas =0 : 6.Fourdierentgrid resolutionsof24 3,48 3,96 3,and192 3areconsideredandthecorrespondingrelativeerrorsarecomputed.AsdisplayedinFig.5.2,theglobalrelativeerrorshaveaslopeof )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 : 0inthe log-logscale,andthusevidentlyouraxisymmetriccascadedLBschemeissecondorderaccurate. FIGURE5.2:EvaluationoforderofaccuracyforTaylor-CouetteowwithaconstantReynolds numberRe=5,radiusratio =1 = 3andrelaxationtime =0 : 6atdierentgridresolutions computedusingtheaxisymmetriccascadedLBscheme. 5.3.2Naturalconvectioninanannulusbetweentwocoaxialverticalcylinders InordertovalidateourcascadedLBschemesforaxisymmetricowswithheattransfer,wesimulateabuoyancy-drivenowbetweentwocoaxialstationarycylinders,whichisaprototypeproblemofbothfundamentalandpracticalinterest.Sincetheoweldiscoupledtothetemperature

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151 eldviathebuoyancyforceinviewofEqs..3a-.3c,.4a-.4c,.5,.8and.9,this problemfacilitatesathoroughexaminationoftheecacyofthecouplingbetweenthecascaded LBschemespresentedinSec.2.2and2.4.TheschematicofthisproblemisdepictedinFig.5.3, where R i , R o , H and g aretheradiioftheinnercylinderandoutercylinder,theheightofthe cylinderandthegravitationacceleration,respectively. FIGURE5.3:Schematicillustrationofthegeometryandboundaryconditionsfornaturalconvectioninaverticalannulus. Forthevelocityeld,no-slipboundaryconditionsareconsideredonallfourwallsinvolvingthe innerandoutercylindricalsurfaces,andtopandbottomwalls.Theinnerandouterwallsofthe lateralcylindricalsidewallsaremaintainedattemperaturesof T H and T L ,respectively,where T H >T L ,whilethetopandbottomwallsareconsideredtobethermallyinsulatedadiabatic. Asaresult,thisgeneratesabodyforceduetobuoyancyintheaxialdirection,whichunderthe Boussinesqapproximation,canbewrittenas g T )]TJ/F35 10.9091 Tf 10.619 0 Td [(T o ,where isthethermalexpansioncoecient,and T o = T H + T L = 2.Thisbodyforcecomponentisaddedtothegeometricsourceterms inEq..5for F b x ,whichthensetsupnaturalconvectionwithintheannulusoftheaxisymmetricgeometry.Thisthermallydrivenowproblemischaracterizedbytwodimensionlessnumbers, viz.,theRayleighnumberRaandPrandtlnumber Pr denedas Ra = g T H )]TJ/F35 10.9091 Tf 10.909 0 Td [(T L L 3 ;Pr = ; where L = R o )]TJ/F35 10.9091 Tf 11.629 0 Td [(R i istheannualgapservingasthecharacteristiclength,and and arethe

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152 kinematicviscosityandthermaldiusivity,respectively.Inaddition,thegeometricparameters inuencingthisproblemaretheaspectratio H=L andtheradiusratio R o =R i ,bothofwhichare setto2inthepresentstudy.Theno-slipconditionsforthevelocityeldareimplementedusingthestandardhalf-waybouncebackschemeinthecascadedLBmethod,whiletheimposed temperatureandnoheatuxconditionsontheboundariesarerepresentedusingtheapproach presentedin[100].AllthespatialderivativesneededinthesourcetermsinEqs..4b,.4cand .9arecomputedusingacentraldierencescheme.Thecharacteristicvelocityduetonatural convection p g T H )]TJ/F35 10.9091 Tf 10.91 0 Td [(T L R i iskeptsmallsothattheowcanberegardedasincompressible.We performedsimulationsat Pr =0 : 7and Ra =10 3 ,10 4 and10 5 correspondingtotheparameter spacesconsideredinpriorstudies[165,166,152].Inaddition,wehavealsoperformedadditional simulationsathigherRayleighnumbersof Ra =10 6 and Ra =10 7 thatcouldserveaspossiblereferenceresultsforfutureresearchworkinthisarea.Thecomputationaldomainisresolved usingagridresolutionof200 200intheaxialandradialdirections,respectively,forthelower Rayleighnumbercasesi.e., Ra =10 3 ,10 4 and10 5 andusing300 300forhigherRayleigh numberconsideredi.e., Ra =10 6 and Ra =10 7 . Figure5.4presentsthecomputedstreamlinesandisothermsforthreedierent Ra =10 3 ,10 4 and 10 5 .Itisclearthatas Ra increases,thevorticalpatternsturntobeprogressivelymorecomplex, withthe Ra =10 5 casegeneratingadditionalpairsofvorticesaroundthemiddleoftheannulus. Furthermore,as Ra increase,theisothermsaregreatlydistorted,andthevelocityandthermal boundarylayersbecomethinnernearthehotandcoldlateralwallssignifyingthestrengthened convectionmodeofheattransfer.Itmaybenotedthatalltheseobservationsareconsistentwith priorstudiesbasedonothernumericalmethodse.g.,[165,166,152].Furthermore,theresults forthecomputedstreamlinesandtheisothermsatthehigher Ra =10 6 and Ra =10 7 arepresentedinFig.5.5.Itcanbeseenthatthestreamlinepatternsbecomemorecomplex,withtheresultingbuoyancy-drivenconvectionbecomingmoreintense.Also,ascanbeexpectedfromscaling arguments,thethermalboundarylayersnearthecylinderwallsbecomethinnertherebymono-

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153 aRe=10 3 bRe=10 4 cRe=10 5 dRe=10 3 eRe=10 4 fRe=10 5 FIGURE5.4:Streamlinesandisothermsforthenaturalconvectionbetweentwoco-axialvertical cylindersat Pr =0 : 7anda,d Ra =10 3 ; b,e Ra =10 4 andc,f Ra =10 5 computedusing cascadedLBschemes.Toprowpresentsstreamlinesandthebottomrowtheisotherms.

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154 tonicallyincreasingtheheattransferrateforthesehigher Ra cases. Theninordertoquantifytheratesofheattransferonthelateralwalls,theoverallNusseltnumbers Nu i and Nu o ontheinnerandoutercylinderscanbedenedas Nu i = )]TJ/F35 10.9091 Tf 8.485 0 Td [(R i H T H )]TJ/F35 10.9091 Tf 10.909 0 Td [(T L Z H o @ y T i dx;Nu o = )]TJ/F35 10.9091 Tf 8.485 0 Td [(R o H T H )]TJ/F35 10.9091 Tf 10.909 0 Td [(T L Z H o @ y T o dx; andhencetheaverageNusseltnumber Nu = Nu i + Nu o = 2. First,wehaveconductedgridsensitivityanalysistoidentifytheminimumgridresolutionnecessarytoprovideconvergedheattransferrateresultsforeachRayleighnumber.Forexample, Table5.1reportstheresultsofagridconvergencetestintermsoftheaverageNusseltnumber Nu foratypical Ra =10 4 byrepeatingthesimulationsforthreedierentmeshresolutionsof 100 100 ; 200 200 ; 250 250.Fromthistable,itcanbeseenthatforthisbenchmarkproblem,whilethe Nu resultsbetweenthegridresolutioncases100 100and200 200casesvary appreciably,thatbetween200 200and250 250showrelativelynegligiblevariations.Hence, theresultswithusing200 200canbedeemedtohaveshowngridconvergencefor Ra =10 4 .In asimilarmanner,wehaveperformedasensitivitystudyandestablishedthegridconvergencefor various Ra fordierentbenchmarkproblems. TABLE5.1:GridconvergencestudygivenintermsoftheaverageNusseltnumber Nu for Ra = 10 4 fornaturalconvectioninacylindricalannuluscomputedusingaxisymmetriccascadedLB schemes. GridResolution Nu 100 100 3.172 200 200 3.199 250 250 3.202 Table5.2showsacomparisonoftheaverageNusseltnumbercomputedusingthecascadedLB schemefor Ra =10 3 ; 10 4 ,and10 5 againstpriornumericalbenchmarkresults[165,166,152].It canbeseenthatourpredictionsfortheaverageNusseltnumbersagreewellwiththoseobtained byothermethods.Inaddition,thistablealsoincludesnewresultsforthehigher Ra of10 6 and

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155 a Ra =10 6 b Ra =10 7 c Ra =10 6 d Ra =10 7 FIGURE5.5:Streamlinesandisothermsforthenaturalconvectionbetweentwoco-axialvertical cylindersat Pr =0 : 7anda,c Ra =10 6 ,b,d Ra =10 7 computedusingcascadedLBschemes. Toprowpresentsstreamlinesandthebottomrowtheisotherms.Gridresolutionusedis300 300.

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156 10 7 .ItisevidentthatincreasingtheRayleighnumberincreasestheaverageNusseltnumber.For example,theheattransferrateincreasesbyaboutsixtimeswhen Ra increasesfrom10 4 to10 7 . Thenewquantitativedatafor Nu for Ra =10 6 and10 7 ,inparticular,couldserveaspossible benchmarkresultsforfutureresearchwork. TABLE5.2:ComparisonoftheaverageNusseltnumber Nu fordierent Ra fornaturalconvectioninacylindricalannuluscomputedusingaxisymmetriccascadedLBschemeswithotherreferencenumericalsolutionsandnewresultsfor Ra =10 6 and10 7 . Ra CascadedLBschemesRef.[165]Ref.[166]Ref.[152] 10 3 1.688--1.692 10 4 3.1993.0373.1633.215 10 5 5.7815.7605.8825.798 10 6 10.421--10 7 18.411--5.3.3Swirlingowinalid-drivencylindricalcontainer Inthissection,weinvestigatetheabilityoftheaxisymmetriccascadedLBschemestoaccurately simulatethedominantroleplayedbytheswirlingmotionanditscouplingwiththecomplex radialandaxialowinducedinthemeridianplane.Inthisregard,weconsiderthesymmetry breakingowinacylindricalcontainerofradius R andheight H drivenbyarotatingtopend wallatangularvelocityseeFig.6. ThedynamicsofthisowispresentedbyEqs..3a-.3c,.4a-.4c,.6and.7,whose solutionschemeviaourcascadedLBformulationispresentedinSec.2.2and2.3.Briey,asthe uidinthevicinityofthetoplidgainsazimuthalmotion,itisejectedradiallyoutward,andthen downwardduetotheconstrainingeectofthesidewall.Subsequentlyastheuidreachesthe bottomitispushedradiallyinward,andwhenitisclosertotheaxis,ittravelsupward,thereby completingowcirculationinthemeridianplane.Thedetailsofthephysicsandtheowpattern dependontheaspectratio R A = H=R andtherotationalReynoldsnumber Re = R 2 = .Variousexperimentse.g.,[167,168]andnumericalsimulationse.g.,[169,170,171]haverevealed

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157 FIGURE5.6:Schematicofswirlingowinaconnedcylinderdrivenbyarotatingtoplid.

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158 thatforcertaincombinationsofthecharacteristicparameters R A and Re ,distinctrecirculation regainaroundthecylinderaxis,designatedasthevortexbreakdownbubble,mayoccur.Forexample,Refs.[168,5]showthatforcases R A ;Re equalto.5,990and.5,1010,novortex breakdownbubblesoccurwhereasfor.5,1290,theydooccur. InordertoassesandvalidateourcascadedLBschemespresentedearliertosimulatesuchcomplexswirlingow,weconsiderthefollowingfourtestcases: Re =990and Re =1290with R A =1 : 5and Re =1010and Re =2020with R A =2 : 5.Thecomputationaldomainisresolved usingameshresolutionof100 150for R A =1 : 5and100 250for R A =2 : 5.No-slipboundary conditionsareusedatbottom,lateralandtopwalls: u = u r = u z =0at z =0and r = R , and u = r , u r = u z =0at z = H .ThestreamlinescomputedusingthecascadedLBschemes fortheabovefourcasesareinFig.7.Itcanbeseenthatnovortexbreak-downbubblesappear for R A ;Re equalto.5,990and.5,1010.Ontheotherhand,onevortexbreakdownbubbleisseenat.5,1290andtwobreakdownbubblesoccurinthevicinityofthecylinderaxis. Thesedistinctregimesinswirlingowsandthecomplexowstructurefordierent R A ;Re casesarestrikinglyconsistentwithpriornumericalsolutione.g.,[5,143,145,172].QuantitativecomparisonofthecomputedstructureoftheaxialvelocitiesalongtheaxisofsymmetryobtainedusingtheaxisymmetriccascadedLBschemesfortheabovefoursetsoftheaspectratios R A andReynoldsnumber Re againsttheresultsfromaNS-basedsolvergivenin[5]areshown inFig.8.Here,theaxialvelocityisscaledbythemaximumimposedazimuthalvelocity u o = R ontherotatinglidandtheaxialdistance z bythecylinderheight H .Thenumericalresultsof ourcentralmomentsbasedcascadedLBmethodfortheaxialvelocityprolesareinverygood agrementwiththeNS-basedsolutionapproach[5].Also,inparticular,noticelocalnegativevaluesfortheaxialvelocitiesforthecases Re =1290and R A =1 : 5and Re =2200and R A =2 : 5, whichisanindicationofthepresenceofoneormorevortexbreakdownbubbles.Assuch,both themagnitudesandtheshapesoftheaxialvelocitydistributionsarewellreproducedbyourcascadedLBapproachusingoperatorsplittingtorepresentcomplexowsincylindricalcoordinates.

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159 a R A =1 : 5,Re=990 b R A =1 : 5,Re=1290 c R A =2 : 5,Re=1010 d R A =2 : 5,Re=2200 FIGURE5.7:ComputedstreamlinepatternsinthemeridianplaneduetoswirlingowinaconnedcylinderdrivenbyarotatinglidatvariousaspectratiosandReynoldsnumbersusingthe axisymmetriccascadedLBsachems:a R A =1 : 5and Re =990,b R A =1 : 5and Re =1290 c R A =2 : 5and Re =1010andd R A =2 : 5and Re =2200.

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160 aRe=990, R A =1 : 5 bRe=1290, R A =1 : 5 cRe=1010, R A =2 : 5 dRe=2200, R A =2 : 5 FIGURE5.8:Dimensionlessaxialvelocityprole u z =u o asafunctionofthedimensionalaxial distance z=H fora R A =1 : 5and Re =990,b R A =1 : 5and Re =1290c R A =2 : 5and Re =1010andd R A =2 : 5and Re =2200:ComparisonbetweenaxisymmetriccascadedLB schemepredictionsandNS-basedsolverresults[5]

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161 5.3.4Mixedconvectioninaslenderverticalannulusbetweentwocoaxialcylinders WewillnowassessournewaxisymmetricLBcomputationalapproachbasedoncentralmoments tosimulatethecombinedeectsofrotationandbuoyancyforcesontheowandheattransfer inconnedcylindricalspaces.Inthisregard,weinvestigatemixedconvectioninaslenderverticalannulusbetweentwocoaxialcylindersarisingduetoinnersidewallrotation,whichhasnumerousapplicationsrelatedtorotatingmachineryandvariousotherheattransfersystems.This probleminvolvingbothnaturalconvectionandforcedconvectionduetorotationcantestallthe threeaxisymmetriccascadedLBformulationsSecs.2.2-2.4inauniedmanner. AschematicarrangementofthisaxisymmetricthermalowproblemisshowninFig.9.ItconFIGURE5.9:Schematicofthearrangementformixedconvectioninaslendercylindricalannulus withinnerlateralwallrotation. sistoftwocoaxialcylindersofheight H ,withanannulargap D = R o )]TJ/F35 10.9091 Tf 11.027 0 Td [(R i ,where R i and R o are theradiioftheinnerandoutercylinders,respectively.Thelateralwallsoftheinnerandouter cylindersaremaintainedattemperatures T H and T L ,respectively,where T H >T L ,andtheir bottomandtopendsarethermallyinsulated.Theinnercylinderissubjectedtorotationatan angularvelocity i ,whiletheoutercylinderandtheendwallsareconsideredtoberigidlyxed.

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162 Asnotedinarecentstudy[154],thisproblemisgovernedbythefollowingcharacteristicdimensionlessparameters:Prandtlnumber Pr = = ,radiusratio R io = R o =R i slendernessratio = H= R o )]TJ/F35 10.9091 Tf 11.198 0 Td [(R i ,Reynoldsnumber Re = i R i D= ,Grashofnumber Gr = g T H )]TJ/F35 10.9091 Tf 11.198 0 Td [(T L D 3 = 2 , and = Gr=Re 2 ,wheretheparameter isusedtomeasurethestrengthofthebuoyancyforce relativetothecentrifugalforce.Hence, characterizesthedegreeofmixedconvection. Inthepresentstudy,weset Pr =0 : 7, R io =2, =10, Re =100,andthreecasesof areconsidered: =0 ; 0 : 01and0 : 05.Thegridresolutionusedforallthethreecasesis40 400,inwhich thelocationoftheinnercylinderfromtheaxis R i isat40.Figure10showsthecomputedcontoursoftheazimuthalvelocity,temperatureeld,vorticityandstreamlinesfortheabovethree valuesof .When =0,thereisnobuoyancyforceandtheowandthetemperatureelds areinuencedbythecentrifugalforceandtheforcedconvectioneects,whichmanifestinthe formofvepairsofcounter-rotatingcells,viz.,theclassicalTaylorvortexcellsarisingfromcentrifugalowinstabilitybetweencurvedwalls[130].As isincreased,thepresenceofbuoyancy forcesandtheassociatednaturalconvectiveuidcurrentsaltertheoverallowstructureand thetemperatureeldbytheircomplicatedinteractionswithprimaryvortexcellsinducedbythe swirlingeectsfrominnerwallrotation.Forexample,when =0 : 05,afour-pairsbasedTaylor vortexstructure,ratherthanve-pairofvortexcellsobservedfor =0,arisesfromtherelative weakeningeectsofthecentrifugalforcesinthepresenceofheating.ThestrengthoftheTaylor vortexinthepositiveazimuthaldirection isseentobeenhanced,whilethatnegative directionappeartobediminishedandtheseobservationsareconsistentwiththebenchmarksresults [173,174]andrecentnumericalsimulations[153].Inordertoquantifytheheattransferratein thepresenceofmixedconvection,ameanequivalentthermalconductivityattheinnercylinder canbedenedas k eq j i = lnR io Z H o )]TJ/F35 10.9091 Tf 8.485 0 Td [(r @T @r j r = R i dr: Table3presentsacomparisonoftheequivalentthermalconductivelycomputedusingtheaxisymmetriccascadedLBformulationsagainstthebenchmarkresults[173,174]fordierentvalues

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164 of .VerygoodquantitativeagrementisseenandthisvalidatestheabilityofthecascadedLB schemesinthecylindricalcoordinatesystemtorepresentcomplexowswithheattransfer. TABLE5.3:Comparisonofthemeanequivalentthermalconductivityattheinnercylinderina slenderverticalcylindricalannulusduringmixedconvectionfor Re =100 ;Pr =0 : 7 ;R io =2 ; = 10atdierentvaluesof . Ref.[174]Ref.[173]PresentWork 01.4731.3931.395 0.011.3701.3831.378 0.051.3241.3231.321 5.3.5MeltowandconvectionduringCzochralskicrystalgrowthinarotatingcylindricalcrucible Asthelasttestproblem,wesimulatemeltowandconvectionduringCzochalskicrystalgrowth, basedonacongurationreportedbyWheeler[175],usingouraxisymmetriccascadedLBschemes. ThisWheeler'sbenchmarkprobleminvolvedbothforcedconvectionduetotherotationofthe crucibleandthecrystalandnaturalconvectionarisingfromheatingeectsinthepresenceof gravity.Ithasbeenstudiedbyavarietyofnumericalschemese.g.,[176,177,146,154].The geometricarrangementofthisproblemisshowninFig.11. Liquidmeltinacylindricalrotatingcrucibleofradius R c andheight H atanangularrotation rateof c undergoesstirredvorticalmotioninthemeridianplane,whichisaidedbytheangularrotationofthesolidcrystalofradius R x atrate x .Inaddition,naturalconvectionissetup duetothebuoyancyforcegeneratedfromadierentialheating,wherethebottomisinsulated anditscruciblesideismaintainedatatemperature T H ,whilethecrystalisatalowertemperature T L i:e:;T L
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165 FIGURE5.11:GeometricarrangementofmeltowandconvectionduringCzochralskicrustal growthinarotatingcrucible )]TJ/F15 10.9091 Tf 8.485 0 Td [(Wheeler'sbenchmarkproblem.

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166 wherethe x;z coordinatesarescaledby R c : u r = u = @u z @r = @T @r =0for r =00 z u r = u z =0 ;u = c R c ;T = T H for r =10 z u r = u z =0 ;u = r c ; @T @z =0for z =00 r 1 u r = u z =0 ;u = r x ;T = T L for z = 0 r @u r @r = @u @z =0 ;u z =0 ;T = T L + r )]TJ/F35 10.9091 Tf 10.909 0 Td [( 1 )]TJ/F35 10.9091 Tf 10.909 0 Td [( T H )]TJ/F35 10.9091 Tf 10.909 0 Td [(T L for z = r 1 where = H=R c ; = R x =R c .Thisowproblemischaracterizedbythefollowingdimensionlessparameters:Reynoldsnumbersduetocrucibleandcrystalrotations Re c = R 2 c c = and Re x = R 2 x x = ,andPrandtlnumber Pr = = .WeinvestigatetheabilityoftheaxisymmetriccascadedLBschemesforthesimulationofmixedconvectionassociatedwiththeWheeler's benchmarkproblemforthefollowingtwocases:a Re x =100, Re c = )]TJ/F15 10.9091 Tf 8.485 0 Td [(25andb Re x =1000, Re c = )]TJ/F15 10.9091 Tf 8.485 0 Td [(250,wherethenegativesigndenotesthatthesenseofrotationofthecrystalisappositetothatofthecrucible.Wetake Pr =0 : 05 ; =1,and =1anduseagridresolutionof 100 200forthesimulationofboththecases. Figure12showsthestreamlinesandisothermcontoursinthemeridianplaneoftheliquidmelt motionforthetwocases.Itcanbeseenthatarecirculatingvortexappearsaroundtheupper leftregionbelowthecrystalinbothcasesinadditiontotheprimaryvortex.Thecenterofthis secondaryvortexisfoundtomovetotherightathigherReynoldsnumbersasaresultofhigher associatedcentrifugalforces.Ontheotherhand,theforcedconvectionhasmodesteectonthe temperaturedistribution,astheyarelargelyalikeforboththecasesduetotherelativelylow Reynoldsnumbersconsidered. Table4showsthecomputedabsolutemaximumvaluesofthestreamfunction max fortheabove twocasesandcomparedwithpriornumericalresultspresentedin[146,176].Inthepseudo-2D

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167 a Re x =100 ;Re c = )]TJ/F61 8.9664 Tf 7.167 0 Td [(25 b Re x =1000 ;Re c = )]TJ/F61 8.9664 Tf 7.168 0 Td [(250 c Re x =100 ;Re c = )]TJ/F61 8.9664 Tf 7.167 0 Td [(25 d Re x =1000 ;Re c = )]TJ/F61 8.9664 Tf 7.168 0 Td [(250 FIGURE5.12:Streamlinesupperrowandisothermsbottomrowcorrespondingtotwocases oftheWheeler'sbenchmarkproblemofmeltowandconvectionduringCzochralshicrystal growth: Re x =100, Re c = )]TJ/F15 10.9091 Tf 8.485 0 Td [(25leftand Re x =1000 ;Re c = )]TJ/F15 10.9091 Tf 8.485 0 Td [(250right.

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168 Cartesiancoordinates,thisisobtainedbysolvingfor using @=@y = )]TJ/F35 10.9091 Tf 8.485 0 Td [(yu x and @=@x = yu y . ThegoodagreementconrmsthatthenewaxisymmetriccascadedLBschemaspresentedinthis studycaneectivelysimulatecomplexowandheattransferproblemsincylindricalgeometries. TABLE5.4:Comparisonofthemaximumvalueofthestreamfunction max computedusingthe axisymmetriccascadedLBschemeswithreferencenumericalsolutionsfortheWheeler'sbenchmarkproblem. Reference Re x =10 2 , Re c =-25 Re x =10 3 , Re c =-250 PresentWork0.11831.123 Ref.[119]0.11401.114 Ref.[176]0.11771.148 5.3.6ComparisonofsinglerelaxationtimeandcascadedLBmodelsforaxisymmetricowsimulations Wewillnowmakeadirectcomparisonbetweentheperformanceofthesinglerelaxationtime SRTLBschemeandtheproposedcascadedLBformulationbasedoncentralmomentsforaxisymmetricowsimulations.Inparticular,astudyofthenumericalstabilitycharacteristicsof theseschemesforowsimulationsinthecylindricalcoordinatesrepresentsakeyaspectwithimportantpracticalimplications.Inthisregard,weconsiderthecomputationoflid-drivenswirling owinacylindricalcontainerdiscussedinSec.5.3.3,astheowissetupbyashearinthepresenceofageometricsingularitybetweenthestationarycylindricalwallsandtherotatinglid. Foraxedangularrotationrateofthelidofthecylindricalcontainerofradius R andheight H seeSec.5.3.3forthenomenclatureatdierentaspectratios R A = H=A ,wewillnowmake acomparisonbetweenthemaximumattainableReynoldsnumber Re = R 2 = withtheaxisymmetricSRT[145]andcascadedLBschemes.Inthisregard,wegraduallyreducetheviscosity,or equivalently,therelaxationtime foraspecicgridresolutionuntilthecomputationbecomes numericallyunstable.InthecaseofthecascadedLBformulation,thisisequivalentvaryingthe relaxationtimesforthesecondordermoments ! j =1 = ,where j =4 ; 5,andtheremainingrelax-

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173 CHAPTERVI LOCALVORTICITYCOMPUTATIONINDOUBLEDISTRIBUTION FUNCTIONSBASEDLATTICEBOLTZMANNMODELSFORFLOWAND SCALARTRANSPORT 6.1Introduction Qualitativedistributionandquantitativemeasuresofvorticityareoffundamentalinterestin uidmechanics.Indeed,uidmotionsareoftenassociatedwithvorticalstructures,whichcan becharacterizedbyvorticity,and,moregenerally,bycertaininvariantsofthevelocitygradient tensor[178,179].Thesignicanceoftherigid-bodylikerotationalcomponentoftheuidelementwasrstidentiedinapioneeringworkbyHelmboltz[180]andthesubjecthasalongand richhistory[181,182].Thislocalrotationalpropertyoftheow,givenbythecurlofthevelocityeld,wastermedvorticitybyLamb[183].Whilethereisnoconsensusonarigorousdenitionofavortex,variousquantitativemeasureshavebeendevisedtoidentifyregionsassociated withmorerigid-bodyrotationsthanstretchingorshearingmotionsthataidinowclassication[184,185,186,187,188,189,190,191,192,193,194,195,196].Suchapproachesarebased onacompleteknowledgeofthevelocitygradienttensor,andthelocal,EulerianbasedmethodsforcoherentstructureidenticationarepopularInmoredetail,thevelocitygradienttensor A ij @ j u i ofthevelocityeld u i canbedecomposedintosymmetric S ij andanti-orskewsymmetricparts ij as @ j u i = 1 2 @ j u i + @ i u j + 1 2 @ j u i )]TJ/F35 10.9091 Tf 10.909 0 Td [(@ i u j = S ij + ij ; .1 where S ij isthestrainratetensorand ij istheintrinsicrotationratespintensor,with ij = )]TJ/F33 7.9701 Tf 9.68 4.295 Td [(1 2 ijk ! k .Here, ! k istheCartesiancomponentofthevorticityand ijk istheLevi-Civitapermutationtensor,andthevorticitycanbedenedas ! i = ijk @ j u k or ! = r u .Both ! i and

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174 S ij ,or,ingeneral, @ j u i playanimportantroleineductiontechniquesforvortexstructureidentication.Inparticular,manyofthesemethods[197]arebasedonthesecondandthirdinvariants ofthevelocitygradienttensor @ j u i ,i.e., Q = )]TJ/F33 7.9701 Tf 9.681 4.295 Td [(1 2 S ij S ij + 1 4 ! k ! k and R = 1 3 S ij S jk S ki + 3 4 ! i ! j S ij . Similarly,sometimestheLambvector L i = ijk ! j u k playsaprominentroleintheanalysisof vortexdynamics[198].Thus,acompleteknowledgeofthelocalvelocitygradienttensor @ j u i ,or equivalently, S ij and ij or ! k isofbasicinterestinstructureidenticationandclassicationof ows.Thisalsoallowsalocaldeterminationofthecomponentsoftheconvectiveaccelerationof theuidelements.Inaddition,thedistributionofvorticityisrelatedtothesoundgeneration andpropagationinowgeneratedacoustics[199].Furthermore,manymodelsfortherepresentationofturbulence,rheologicaluidowssuchasthoseinvolvingviscoelasticity,andcomplex uidsystemssuchasliquidcrystalsandpolaruidsdependonthelocalmeasuresofthecompletevelocitygradienttensor @ j u i [200].Itisthushighlydesirableforcomputationalmethodsfor uiddynamicsthatallowespeciallylocaldeterminationofallcomponentsofthevelocitygradienttensor,includingtheskewsymmetricparti.e.,thevorticity. ThelatticeBoltzmannLBmethodisakineticcomputationalapproachforavarietyofuid mechanicsandtransportproblems[201,14,24,11,12,7,43,18].Generally,thestandardversionsoftheLBmodelscanonlyrepresentthesymmetricpartof @ j u i ,i.e.,thestrainratetensor S ij viathesecondordernon-equilibriummomentsofthedistributionfunction,whichare,in turn,relatedtothespatialderivativesoftherstandthirdordermomentequilibria.Thelatterareconstructedbasedonsymmetryandisotropyconsiderationsthatrespecttheunderlying isotropyoftheviscousstresstensoroftheuidmotionrepresentedbytheNavier-Stokesequations.ItisknownthatsuchLBapproachescanrecoverthestrainratetensorcomponentslocallywithsecondorderaccuracyseee.g.,Refs.[202,203,17].However,mostoftheexisting LBmodelsarenotconstructedtorecovertheantisymmetricvelocitygradienttensor ij locally withoutrelyingontheuseofnitedierenceapproximationforthespatialderivativesofthevelocityeldcomponents.Onenotableexceptionistherecentandinterestingwork[204],which

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175 introducedanapproachbasedonmodifyingthefthordermomentequilibriaoftheLBsolverfor uidowthatenablesvorticitycomputation.Thisapproachisrestrictedtoonlylatticesthatcan supportfthorderindependentmomentsandthusisapplicableonlytothethree-dimensional, twentysevenvelocityD3Q27lattice,andnotforotherstandardlatticesets,includingthecommontwo-dimensional,ninevelocityD2Q9lattice.Furthermore,sinceitisbasedonacertain prescribedformofthehigherordermomentequilibria,itmaybechallengingtoextenditfor thermalowsaswellasthosewithsignicantcompressibilityeectsthatinvolveconstraintson thehighermomentsofthesingledistributionfunction,andmayalsoimpactitsGalileaninvarianceofsolvingtheuidmotion. Ourapproachisbasedondierentconsiderationsthentheabovementionedworkforvorticity computation.Thegoalistosimulatetheuidmotionalongwithanadvection-diusiontransportofascalareld,representedbythefollowingNavier-StokesequationsNSEandtheconvectiondiusionequationCDE,respectively: @ t + r u =0 ; .2 @ t u + r uu = )]TJ/F55 10.9091 Tf 8.485 0 Td [(r p + r T + F ; .3 @ t + r u = r D r ; .4 where , u and p aretheuiddensity,velocity,andpressure,respectively, T ij =2 S ij )]TJ/F33 7.9701 Tf 9.742 4.295 Td [(2 d @ k u k ij + @ k u k ij isthedeviatoricstresstensorwith and beingthekinematicshearandbulkviscosities,respectively,and d beingthenumberofspatialdimensions, F isthelocalbodyforce, and isthescalareldwith D beingitsdiusivity.Theseequationscanbesolvedbymeans ofadoubledistributionfunctionsDDFbasedapproachusingtwolatticeBoltzmannequations LBEs{onefortheoweldandtheotherforthescalareld.Suchsituationsrelatedtosolvingtheadditionalpassivescalarelddynamicsarisewidely,includingthoserelatedtothetransportofenergyortemperatureeldinthermalconvection,andoftheconcentrationeldofa chemicalspeciesinreactingsystems,andintheinterfacecapturingusingphaseeldmodelsin

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177 Forthepurposeofillustrationwithoutlosinggenerality,inthiswork,wewillspecializeourDDF approachbyformulatingtwoLBEsusingnaturalnon-orthogonalmomentbasisandmultiple relaxationtimesMRTforthesolutionofowandscalartransportusingthestandardD2Q9 latticetolocallycomputethecompleteinformationabouttheowkinematics,i.e.,allthe1291 componentsofthevelocitygradienttensor,includingtheskew-symmetriccomponents.However, ourmethodcanbereadilyextendedtoLBEbasedonothercollisionmodels,suchasthesingle relaxationtimeLBE,MRT-LBEwithothermomentbasis,cascadedLBEandcumulantLBE, andvariousotherlatticesetsindierentdimensions.Thischapterisorganizedasfollows.The nextsectionSec.6.2willpresentaMRT-LBEforcomputingtheuidmotion,anditsChapmanEnskogC-Eanalysistodeterminethesymmetriccomponentsofthevelocitygradienttensor. Section6.3.1willthendiscussanotherMRT-LBEforrepresentingtheadvection-diusiontransportofascalareld,anditsC-Eanalysistoobtainthenecessaryrelationsfortheskew-symmetric componentsofthevelocitygradienttensor.TheexpressionforthelocalcomputationofthevorticityeldisderivedinSec.6.4.Then,resultsanddiscussionofthecomparisonsofthecomputed vorticityeldsagainsttheanalyticalsolutionsforvariousrepresentativeuidowproblemsare giveninSec.6.5.Finally,Sec.7.8presentsasummaryandconclusionsofthischapter. 6.2MRT-LBEforFluidMotion Inordertosolvetheuidmotionintwo-dimensionsDrepresentedbythemassandmomentumconservationequationsgiveninEqs..2and.3,respectively,wewillnowpresenta MRT-LBEusinganatural,non-orthogonalmomentbasis[19].Inthisregard,aD2Q9latticeis used,andwhoseparticlevelocitiesaregivenbythefollowing: j e x i = ; 1 ; 0 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; 0 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 y ; .5a j e y i = ; 0 ; 1 ; 0 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 y ; .5b

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178 where y isthetransposeoperatorandthestandardDirac'sbra-ketnotationisusedtorepresent thevectors.TheCartesiancomponentsforanyparticledirection arerepresentedby e x and e y ,where =0 ; 1 ;:::; 8.Inaddition,weneedthefollowing9-dimensionalvectorwhoseinner productwiththeparticledistributionfunction f yieldsitszerothmoment: j 1 i = ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 y : .6 Thenon-orthogonalbasisvectorscanthenbewrittenas T 0 = j 1 i ;T 1 = j e x i ;T 2 = j e y i ;T 3 = j e 2 x + e 2 y i ;T 4 = j e 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(e 2 y i ; T 5 = j e x e y i ;T 6 = j e 2 x e y i ;T 7 = j e x e 2 y i ;T 8 = j e 2 x e 2 y i : .7 Intheabove,symbolssuchas j e 2 x e y i = j e x e x e y i denoteavectorthatarisefromtheelementwise vectormultiplicationofvectors j e x i , j e x i and j e y i .Inordertomapchangesofmomentsbackto changesinthedistributionfunction,wegrouptheabovesetofvectorsasatransformationmatrix T ,whichreadsas T =[ T 0 ;T 1 ;T 2 ;T 3 ;T 4 ;T 5 ;T 6 ;T 7 ;T 8 ] : .8 Wethendenetherawmomentsoforder m + n ofthedistributionfunction f ,itsequilibrium f eq ,andthesourceterms S torepresentthebodyforce,respectively,as 0 B B B B B @ ^ 0 x m y n ^ eq 0 x m y n ^ eq 0 x m y n 1 C C C C C A = 8 X =0 0 B B B B B @ f f eq S 1 C C C C C A e m x e n y ; .9 Here,andinwhatfollows,theprime 0 symbolsdenotevariousrawmoments.Intermsofthe nominal,nonorthogonaltransformationmatrix T therelationbetweenthevariousmomentsand theircorrespondingstatesinthevelocityspacecanbewrittenas b m = T f ; b m eq = T f eq ; b S = T S ; .10

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179 where f = f 0 ;f 1 ;f 2 ;:::;f 8 y ; f eq = f eq 0 ;f eq 1 ;f eq 2 ;:::;f eq 8 y ; S = S 0 ;S 1 ;S 2 ;:::;S 8 y arethevariousquantitiesinthevelocityspace,and b m = b m 0 ; b m 1 ; b m 2 ;:::; b m 8 y = b 0 0 ; b 0 x ; b 0 y ; b 0 xx + b 0 yy ; b 0 xx )]TJ/F41 10.9091 Tf 11.021 0 Td [(b 0 yy ; b 0 xy ; b 0 xxy ; b 0 xyy ; b 0 xxyy y ; .11a b m eq = b m eq 0 ; b m eq 1 ; b m eq 2 ;:::; b m eq 8 y = b eq 0 0 ; b eq 0 x ; b eq 0 y ; b eq 0 xx + b eq 0 yy ; b eq 0 xx )]TJ/F41 10.9091 Tf 11.021 0 Td [(b eq 0 yy ; b eq 0 xy ; b eq 0 xxy ; b eq 0 xyy ; b eq 0 xxyy y ; .11b b S = b S 0 ; b S 1 ; b S 2 ;:::; b S 8 y = b 0 0 ; b 0 x ; b 0 y ; b 0 xx + b 0 yy ; b 0 xx )]TJ/F41 10.9091 Tf 11.191 0 Td [(b 0 yy ; b 0 xy ; b 0 xxy ; b 0 xyy b 0 xxyy y .11c arethecorrespondingstatesinthemomentspace. TheMRT-LBEwithtrapezoidalruletorepresentthesourcetermwithsecondorderaccuracy canbewrittenas f x + e t ;t + t )]TJ/F25 10.9091 Tf 10.909 0 Td [(f x ;t = T )]TJ/F33 7.9701 Tf 6.587 0 Td [(1 h )]TJ/F15 10.9091 Tf 9.424 2.879 Td [(^ b m )]TJ/F41 10.9091 Tf 13.106 0.152 Td [(b m eq i + 1 2 T )]TJ/F33 7.9701 Tf 6.587 0 Td [(1 h b S x + e t ;t + t + b S x ;t i t ; .12 wherethediagonalrelaxationtimematrix ^ canberepresentedas ^ =diag ; 0 ; 0 ;! 3 ;! 4 ;! 5 ;! 6 ;! 7 ;! 8 : .13 Inordertoobtainaneectivelyexplicitscheme,weapplythetransformation[69,23] f = f )]TJ/F33 7.9701 Tf -452.146 -19.701 Td [(1 2 S t ,orequivalently b m = b m )]TJ/F33 7.9701 Tf 12.249 4.295 Td [(1 2 b S t and b 0 x m y n =^ 0 x m y n )]TJ/F33 7.9701 Tf 12.249 4.295 Td [(1 2 ^ 0 x m y n t ,andtheMRT-LBEcanbe writtenas f x + e t ;t + t )]TJ/F15 10.9091 Tf 10.693 2.878 Td [( f x ;t = T )]TJ/F33 7.9701 Tf 6.587 0 Td [(1 h )]TJ/F15 10.9091 Tf 9.424 2.879 Td [(^ )]TJ 7.197 -7.188 Td [(b m )]TJ/F41 10.9091 Tf 13.106 0.151 Td [(b m eq i + T )]TJ/F33 7.9701 Tf 6.586 0 Td [(1 I )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 2 ^ b S t ; .14

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180 Themomentequilibria^ eq 0 x m y n atdierentorderscanbewrittenas[19] b eq 0 0 = ; b eq 0 x = u x ; b eq 0 y = u y ; b eq 0 xx = c 2 s + u 2 x ; b eq 0 yy = c 2 s + u 2 y ; b eq 0 xy = u x u y ; b eq 0 xxy = c 2 s u y + u 2 x u y ; b eq 0 xyy = c 2 s u x + u x u 2 y ; b eq 0 xxyy = c 4 s + c 2 s u 2 x + u 2 y + u 2 x u 2 y ; .15 whichareobtainedfromthediscreterepresentationofthelocalMaxwellianbytransformingback theircentralmomentsatagivenordertotheircorrespondingrawmoments.Here, c s isthespeed ofsound,andinthepresentwork,wetypicallyset c 2 s =1 = 3.Also,momentsofthesourceterms ^ eq 0 x m y n followsas[19] b 0 0 =0 ; b 0 x = F x ; b 0 y = F y ; b 0 xx =2 F x u x ; b 0 yy =2 F y u y ; b 0 xy = F x u y + F y u x ; b 0 xxy = F y u 2 x +2 F x u x u y ; b 0 xyy = F x u 2 y +2 F y u y u x ; b 0 xxyy =2 F x u x u 2 y + F y u y u 2 x ; .16 where F = F x ;F y .Thehydrodynamiceldsaregivenby = 8 X =0 f ; u = 8 X =0 f e + 1 2 F t ;p = c 2 s : .17 TheaboverepresentsthesolutionoftheNSEEqs..2and.3,withthekinematicbulkand shearviscositiesrelatedtotherelaxationtimesvia = c 2 s 1 ! 3 )]TJ/F33 7.9701 Tf 12.104 4.295 Td [(1 2 t and = c 2 s 1 ! j )]TJ/F33 7.9701 Tf 12.105 4.295 Td [(1 2 t , where j =4 ; 5respectively.Theremainingrelaxationtimesforthehigherordermoments,which inuencethenumericalstability,aresettounityinthiswork. 6.2.1Momentrelationshipsforthesymmetricvelocitygradienttensor:Chapman-EnskogAnalysis WewillnowperformaChapman-Enskoganalysis[207]todeterminetheexpressionsthatrelate thesymmetricvelocitygradienttensortocertaincomponentsofthelocalnon-equilibriummo-

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181 ments.Expandingthemomentsaboutitsequilibriaaswellasapplyingthestandardmultiscale expansionofthetimederivativesintheMRT-LBEgivenintheprevioussection b m = 1 X j =0 j b m j ;@ t = 1 X j =0 j @ t j ; .18 where is a smallbookkeepingperturbationparameter,andalsoperformingaTaylorseriesexpansionofthestreamingoperatorinEq..14,i.e., f x + e ;t + = 1 X j =0 j j ! @ t + e r j f x ;t : .19 andconvertingallquantitiesinthevelocityspacetothemomentspaceviaEq..10andusing b m = b m )]TJ/F33 7.9701 Tf 12.478 4.295 Td [(1 2 b S t ,weobtainthefollowingsystemofmomentequationsatconsecutiveorderin : O 0 : b m = b m eq ; .20a O 1 : @ t 0 + b E i @ i b m = )]TJ/F41 10.9091 Tf 9.121 2.879 Td [(b b m + b S ; .20b O 2 : @ t 1 b m + @ t 0 + b E i @ i h I )]TJ/F33 7.9701 Tf 12.105 4.296 Td [(1 2 b i b m = )]TJ/F41 10.9091 Tf 9.121 2.879 Td [(b b m ; .20c where b E i = T e i I T )]TJ/F33 7.9701 Tf 6.586 0 Td [(1 ;i 2f x;y g .Inordertoobtainthehydrodynamicmacroscopicequations,in the O systemseeEq..20b,theequationsrepresentingtheevolutionofthemomentcomponentsuptothesecondorderarenecessary,whichreadasseeAppendixEfordetails @ t 0 + @ x u x + @ y u y =0 ; .21a @ t 0 u x + @ x )]TJ/F35 10.9091 Tf 5 -8.836 Td [(c 2 s + u 2 x + @ y u x u y = F x ; .21b @ t 0 u y + @ x u x u y + @ y )]TJ/F35 10.9091 Tf 5 -8.837 Td [(c 2 s + u 2 y = F y ; .21c @ t 0 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(2 c 2 s + u 2 x + u 2 y + @ x + c 2 s u x + u x u 2 y + @ y + c 2 s u y + u 2 x u y = )]TJ/F35 10.9091 Tf 8.484 0 Td [(! 3 b m 3 +2 F x u x + F y u y ; .21d @ t 0 )]TJ/F35 10.9091 Tf 5 -8.837 Td [( u 2 x )]TJ/F35 10.9091 Tf 10.91 0 Td [(u 2 y + @ x )]TJ/F35 10.9091 Tf 10.909 0 Td [(c 2 s u x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x u 2 y + @ y )]TJ/F15 10.9091 Tf 8.485 0 Td [(1+ c 2 s u y + u 2 x u y = )]TJ/F35 10.9091 Tf 8.484 0 Td [(! 4 b m 4 +2 F x u x )]TJ/F35 10.9091 Tf 10.909 0 Td [(F y u y ; .21e @ t 0 u x u y + @ x )]TJ/F35 10.9091 Tf 5 -8.836 Td [(c 2 s u y + u 2 x u y + @ y )]TJ/F35 10.9091 Tf 5 -8.836 Td [(c 2 s u x + u x u 2 y = )]TJ/F35 10.9091 Tf 8.485 0 Td [(! 5 b m 5 + F x u y + F y u x : .21f

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182 Analogously,atthe O 2 levelseeEq..20c,therelevantmomentequationstorecoverthe equationsoftheuidmotionwrittenuptotherstorderas @ t 1 =0 ; .22a @ t 1 u x + @ x h 1 2 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1 )]TJ/F33 7.9701 Tf 12.104 4.296 Td [(1 2 ! 3 b m 3 + 1 2 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1 )]TJ/F33 7.9701 Tf 12.105 4.296 Td [(1 2 ! 4 b m 4 i + @ y h )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1 )]TJ/F33 7.9701 Tf 12.105 4.296 Td [(1 2 ! 5 b m 5 i =0 ; .22b @ t 1 u y + @ x h )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1 )]TJ/F33 7.9701 Tf 12.104 4.295 Td [(1 2 ! 5 b m 5 i + @ y h 1 2 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1 )]TJ/F33 7.9701 Tf 12.105 4.295 Td [(1 2 ! 3 b m 3 )]TJ/F33 7.9701 Tf 12.105 4.295 Td [(1 2 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1 )]TJ/F33 7.9701 Tf 12.104 4.295 Td [(1 2 ! 4 b m 4 i =0 : .22c Here,thecomponentsofthesecond-ordernon-equilibriummoments b m 3 , b m 4 and b m 5 which represent b 0 xx + b 0 yy , b 0 xx )]TJ/F41 10.9091 Tf 11.71 0 Td [(b 0 yy and b 0 xy ,respectivelyareunknowns.Theycanbeobtained fromEqs..21d,.21eand.21f,respectively,wherethetimederivatives @ t 0 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(2 c 2 s + u 2 x + u 2 y , @ t 0 )]TJ/F35 10.9091 Tf 5 -8.837 Td [( u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y and @ t 0 u x u y areeliminatedinfavorthespatialderivativesusingtheleading ordermassandmomentumequationsi.e.,Eqs..21a{.21c,respectively.Fordetails,see e.g.,Refs.[19,68].Neglectingalltermsof O u 3 andhigher,wecanobtaintheexpressionsfor thevariouscomponentsofthenon-equilibriumsecondordermomentsrelatedtothesymmetric partofthevelocitygradienttensor S ij = 1 2 @ j u i + @ i u j i.e., @ x u x , @ y u y and @ y u x + @ y u x ,which readas[19,68] b m 3 = b 0 xx + b 0 yy = )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(2 c 2 s ! 3 @ x u x + @ y u y ; .23a b m 4 = b 0 xx )]TJ/F41 10.9091 Tf 11.021 0 Td [(b 0 yy = )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(2 c 2 s ! 4 @ x u x )]TJ/F35 10.9091 Tf 10.909 0 Td [(@ y u y ; .23b b m 5 = b 0 xy = )]TJ/F35 10.9091 Tf 9.68 7.381 Td [(c 2 s ! 5 @ x u y + @ y u x : .23c WhentheseexpressionsaresubstitutedinEqs..22band.22c,andthencombiningthe O and O 2 momentequationsuptotherstorder,theNSEgivenEqs..2and.3follows. Thenon-equilibriummomentrelationsgiveninEqs..23a{.23cwillbecombinedfurther withthedevelopmentsgiveninthenextsectiontodevelopalocalcomputingapproachforthe vorticityeldlaterinSec.6.4.

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183 6.3MRT-LBEforTransportofaPassiveScalar Thesolutionoftheadvection-diusionofthepassivescalareld givenbytheCDEinEq..4 willnowberepresentedbyusinganotherMRT-LBE.ConsideringtheD2Q9latticeagain,which, asrequired,supportstheo-diagonalthirdordermomentequilibriaindependentlyasnotedin theIntroduction,weusethesamenaturalmomentbasisgiveninEq..7aswellastheresultingtransformationmatrix T seeEq..8.First,wedenetherelationbetweenthevariousraw momentsandthecorrespondingdistributionfunction g andtheirequilibria g eq forthisMRTLBEas ^ n = T g ; ^ n eq = T g eq ; .24 where g = g 0 ;g 1 ;g 2 :::g 8 y ; g eq = g eq 0 ;g eq 1 ;g eq 2 :::g eq 8 y .25 aregiveninthevelocityspace,and ^ n =^ n 0 ; ^ n 1 ; ^ n 2 ::: ^ n 8 y =^ 0 0 ; ^ 0 x ; ^ 0 y ; ^ 0 xx +^ 0 yy ; ^ 0 xx )]TJ/F15 10.9091 Tf 11.692 0 Td [(^ 0 yy ; ^ 0 xy ; ^ 0 xxy ; ^ 0 xyy ; ^ 0 xxyy y .26 ^ n eq =^ n eq 0 ; ^ n eq 1 ; ^ n eq 2 ::: ^ n eq 8 y =^ eq 0 0 ; ^ eq 0 x ; ^ eq 0 y ; ^ eq 0 xx +^ eq 0 yy ; ^ eq 0 xx )]TJ/F15 10.9091 Tf 11.692 0 Td [(^ eq 0 yy ; ^ eq 0 xy ; ^ eq 0 xxy ; ^ eq 0 xyy ; ^ eq 0 xxyy y .27 representtheequivalentstatesinthemomentspace.Here,thevarioussetsofrawmomentsare denedasfollows: 0 B @ ^ 0 x m y n ^ eq 0 x m y n 1 C A = 8 X =0 0 B @ g g eq 1 C A e m x e n y ; .28 ThentheMRT-LBEusinganon-orthogonalmomentbasisforthesolutionoftheCDEcanbe writtenas g x + e t ;t + t )]TJ/F25 10.9091 Tf 10.909 0 Td [(g x ;t = )]TJ/F58 10.9091 Tf 8.485 0 Td [(T )]TJ/F33 7.9701 Tf 6.586 0 Td [(1 [ ^ ^ n )]TJ/F15 10.9091 Tf 11.666 0.152 Td [(^ n eq ] ; .29

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184 where ^ isthediagonalrelaxationtimematrixgivenby ^ =diag ;! 1 ;! 2 ;! 3 ;! 4 ;! 5 ;! 6 ;! 7 ;! 8 ; .30 Akeyelementinthisworkistheprescriptionofthemomentequilibria ^ n eq Eq..27usedin Eq..29toenablealocalcomputationoftheantisymmetricvelocitygradienttensororthevorticityeld.Thepassivescalar isadvectedbythelocalvelocityeld u ,and henceitssolution procedure,inprinciple,hasacompleteinformationonthekinematicsoftheuidelementsundergoingavarietyofmotionwhenitiscarefullydesigned.Assuch,mostofthecomponentsof themomentequilibria ^ n eq canbeconstructedinanalogywith ^ m eq giveninEq..15,wherethe density isreplacedbythescalareld .Ontheotherhand,inviewoftheaboveconsideration, inordertoextractthelocalintrinsicrotationrateoftheuidelementrelatedtotheantisymmetricvelocitygradienttensor,weprescribeanisotropyinthescalarux u componentsusedin thethirdordermomentequilibria,which,asweshallseeinthefollowing,doesnotaectrecoveringofthemacroscopicCDE.Thus,weset b eq 0 0 = ; b eq 0 x = u x ; b eq 0 y = u y ; b eq 0 xx = c 2 s + u 2 x ; b eq 0 yy = c 2 s + u 2 y ; b eq 0 xy = u x u y ; b eq 0 xxy = 1 c 2 s u y + u 2 x u y ; b eq 0 xyy = 2 c 2 s u x + u x u 2 y ; b eq 0 xxyy = c 4 s + c 2 s u 2 x + u 2 y + u 2 x u 2 y ; .31 where c s isanindependentparameterrelatedtothediusivity D seebelow,andwetypically set c 2 s =1 = 3inthiswork.Here, 1 and 2 arefreeparametersthatprescribeanisotropyonthe scalaruxappearinginthethirdordermomentequilibria.Typically, 1 1and 2 1,but 1 )]TJ/F35 10.9091 Tf 11.569 0 Td [( 2 6 =0,i.e.,asmallintentionalanisotropyisintroducedtolocallyrecoverthemagnitude oftheintrinsicrotationrateoftheuidmotionseethefollowingsection.Thescalareld is thenobtainedasthezerothmomentofthedistributionfunction g ,whichevolvesaccordingto

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185 Eq..29intheformofthestandardcollide-and-steamsteps: = 8 X =0 g : .32 Then,theaboverepresentsthesolutionoftheCDEEq..4,withthediusivityrelatedto therelaxationtimesvia D = c 2 s 1 ! j )]TJ/F33 7.9701 Tf 12.104 4.296 Td [(1 2 t where j =1 ; 2. 6.3.1Momentrelationshipsforthescalargradientvectorandskew-symmetricvelocitygradienttensor:Chapman-EnskogAnalysis WewillnowperformaC-EanalysisoftheMRT-LBEforthepassivescalareld.Applyingthe momentexpansionaboutitsequilibriaandamultiscaleexpansionofthetimederivativetoEq..29 ^ n = 1 X j =0 j ^ n j ;@ t = 1 X j =0 j @ tj ; .33 where = t andalsousingaTaylorexpansionofthestreamingoperator g x + e ;t + = P 1 j =0 j j ! @ t + e r j g x ;t ,thefollowingmomentequationsatconsecutiveorderin canbe obtained: O 0 : b n = b n eq ; .34a O 1 : @ t 0 + b E i @ i b n = )]TJ/F41 10.9091 Tf 9.121 2.879 Td [(b b n ; .34b O 2 : @ t 1 b n + @ t 0 + b E i @ i I )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(1 2 b b n = )]TJ/F41 10.9091 Tf 9.121 2.879 Td [(b b n ; .34c

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186 where b E i isthesameasthatgivenearlier.Someoftherelevantcomponentsattheleadingorder i.e., O ofthemomentsystemseeEq..34baregivenas @ t 0 + @ x u x + @ y u y =0 ; .35a @ t 0 u x + @ x c 2 s + u 2 x + @ y u x u y = )]TJ/F35 10.9091 Tf 8.485 0 Td [(! 1 ^ n 1 ; .35b @ t 0 u y + @ x u x u y + @ y c 2 s + u 2 y = )]TJ/F35 10.9091 Tf 8.485 0 Td [(! 2 ^ n 2 ; .35c @ t 0 c 2 s + u 2 x + u 2 y + @ x h + 2 c 2 s u x + u x u 2 y i + @ y h + 1 c 2 s u y + u 2 x u y i = )]TJ/F35 10.9091 Tf 8.485 0 Td [(! 3 ^ n 3 ; .35d @ t 0 u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y + @ x h )]TJ/F35 10.9091 Tf 10.909 0 Td [( 2 c 2 s u x + u x u 2 y i + @ y h )]TJ/F15 10.9091 Tf 8.485 0 Td [(1+ 1 c 2 s u y + u 2 x u y i = )]TJ/F35 10.9091 Tf 8.485 0 Td [(! 4 ^ n 4 ; .35e @ t 0 u x u y + @ x h 1 c 2 s u y + u 2 x u y i + @ y h 2 c 2 s u x + u x u 2 y i = )]TJ/F35 10.9091 Tf 8.485 0 Td [(! 5 ^ n 5 ; .35f wheretheabovecanbeobtainedbyreplacing b eq 0 x m y n inthecorrespondingC-Eanalysisforthe uidmotionwith^ eq 0 x m y n seetheprevioussectionandfordetailsandallowingfortherelaxation oftherstordermoments,sinceonlythescalareld isconservedinthepresentcase.Similarly,theleadingcomponenti.e.,thezerothorderofthemomentsystematthe O 2 levelto recovertheCDEisobtainedfromEqs..34ccanbewrittenas @ t 1 + @ x " 1 )]TJ/F35 10.9091 Tf 12.105 7.38 Td [(! 1 2 ! ^ n 1 # + @ y " 1 )]TJ/F35 10.9091 Tf 12.105 7.38 Td [(! 2 2 ! ^ n 2 # =0 : .36 Now,inordertoderivetheCDE,weneedtocombineEq..35aand timesEq..36byusing @ t = @ t 0 + @ t 1 ,whichrequires^ n 1 and^ n 2 .Theserstordernon-equilibriummoments^ n 1 and^ n 2 canbeobtainedfromEqs..35band.35c,respectively,wherethetimederivatives areeliminatedinfavorofthespatialderivativesbyusingtheleadingordermass,momentumand scalarconservationequationsi.e.,Eqs..21a,.21b,.21cand.35a.Henceaftersome simplication,andneglectingtermsof O u 2 andhigher,wegetthecomponentsoftherstor-

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187 dernon-equilibriummomentsintermsofthecomponentsofthescalargradientvector @ i as ^ n 1 =^ 0 x = )]TJ/F15 10.9091 Tf 13.31 7.38 Td [(1 ! 1 c 2 s @ x .37a ^ n 2 =^ 0 y = )]TJ/F15 10.9091 Tf 13.31 7.38 Td [(1 ! 2 c 2 s @ y ; .37b Itmaybenotedthatinthederivationofthesenon-equilibriummomentcomponents,onlythe spatialderivativesofthesecondordermomentequilibriai.e.,^ eq 0 xx ,^ eq 0 yy and^ eq 0 xy areinvolved anddonotinvolvetheintroducedanisotropy,whichappearsatahigherorderlevel,i.e.,forthe thirdordermomentsoftheequilibriumdistributionviathefactors 1 and 2 andhencetheadvectiondiusionofthepassivescalartransportiscorrectlyrecovered. Asshownintheprevioussection,thesymmetriccomponentsofthevelocitygradienttensor @ x u x ;@ y u y and @ x u y + @ y u x canbeobtainedfromtheMRT-LBEforuidow.Inordertoobtaintheskewsymmetriccomponent,i.e., @ x u y )]TJ/F35 10.9091 Tf 11.029 0 Td [(@ y u x ,whichwouldthenprovideacompleteinformationabout thevelocitygradienttensor @ j u i andhencethevorticityeld,wenowexploittheadditionaldegreeoffreedomavailableintheo-diagonal,second-ordernon-equilibriummomentequation resultingfromtheMRT-LBEforCDE,i.e.,Eq..35f.Simplifyingthisequationbyeliminatingthetimederivativeinfavorofspatialderivativesandeliminatinghigherordertermsi.e., O u 2 andabove,weget 1 c 2 s @ x u y + 2 c 2 s @ y u x = )]TJ/F35 10.9091 Tf 8.484 0 Td [(! 5 ^ n 5 ; .38 whichcanberewrittenas ^ n 5 = )]TJ/F35 10.9091 Tf 9.68 9.126 Td [(c 2 s ! 5 [ 1 @ x u y + 2 @ y u x + 1 u y @ x + 2 u x @ y ] : .39 Clearly,theanisotropyintroducedintothescalaruxcomponentsinthethirdordermoment equilibriaresultsinanadditionalexibilityviaanindependentequationgivenaboveEq..39. Inthisequation,thegradientsofthescalareldintheCartesiancoordinatedirections @ x and @ y canbeobtainedlocallyfromEq..37aandEq..37b;andwiththeknowledgeoftheodiagonalsecond-ordernon-equilibriummomentcomponent^ n 5 ,thenEq..39representsanad-

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188 ditionalindependentequationtocomputetheantisymmetricvelocitygradienttensorcomponent, whichwillbeexploitedfurtherinthenextsection. 6.4Derivationoflocalexpressionsforthecompletevelocitygradienttensorand vorticityeld Inordertoseparatelydeterminethecross-derivativecomponentsofthevelocitygradienttensor, i.e., @ y u x and @ x u y ,wecombinetheanalysispresentedinthetwoearliersections.Inparticular, theEq..23cresultingfromthesolutionoftheMRT-LBEforuidowandEq..39from theMRT-LBEforCDE,canberewrittenas @ x u y + @ y u x = N xy ; .40a 1 @ x u y + 2 @ y u x = N xy ; .40b where,when 6 =0, N xy = )]TJ/F35 10.9091 Tf 13.823 7.38 Td [(! 5 c 2 s ^ m 5 ; .41a N xy = )]TJ/F35 10.9091 Tf 13.657 7.38 Td [(! 5 c 2 s ^ n 5 )]TJ/F15 10.9091 Tf 12.628 7.38 Td [(1 1 u y @ x + 2 u x @ y : .41b SolvingEqs..40aand.40b,wegetfollowingindependentandlocalexpressionsfortheodiagonalcomponentsorthecrossderivativesofthevelocityeld,whichisoneofthemainresults ofthiswork: @ x u y = N xy )]TJ/F35 10.9091 Tf 10.909 0 Td [( 2 N xy 1 )]TJ/F35 10.9091 Tf 10.909 0 Td [( 2 ; .42a @ y u x = 1 N xy )]TJ/F35 10.9091 Tf 10.909 0 Td [(N xy 1 )]TJ/F35 10.9091 Tf 10.909 0 Td [( 2 : .42b Thediagonalcomponentsofthevelocitygradienttensor,i.e., @ x u x and @ y u y followfromsolving theEqs..23aand.23bresultingfromtheMRT-LBEfortheuidmotion,whichreadsas @ x u x = )]TJ/F15 10.9091 Tf 17.227 7.38 Td [(1 4 c 2 s h ! 3 ^ m 3 + ! 4 ^ m 4 i ; .43a @ y u y = )]TJ/F15 10.9091 Tf 17.227 7.38 Td [(1 4 c 2 s h ! 3 ^ m 3 )]TJ/F35 10.9091 Tf 10.909 0 Td [(! 4 ^ m 4 i ; .43b

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189 andthiscompletesthedeterminationofallthecomponentsofthevelocitygradienttensor.Finally,alocalexpressionforthepseudo-vector,viz.,thevorticityeld ! = r u = ; 0 ;! z can beobtainedbycombiningEqs..42aand.42bas ! z = @ x u y )]TJ/F35 10.9091 Tf 10.909 0 Td [(@ y u x = 2 N xy )]TJ/F15 10.9091 Tf 10.909 0 Td [( 1 + 2 N xy 1 )]TJ/F35 10.9091 Tf 10.909 0 Td [( 2 ; .44 whichisanotherkeyresultarisingfromouranalysis. Theterms N xy and N xy giveninEqs..41aand.41b,respectively,whichareneededinEqs..42a, .42band.44canbeevaluatedlocallyusing ^ m 5 = b 0 xy )]TJ/F41 10.9091 Tf 11.021 0 Td [(b eq 0 xy = b 0 xy )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x u y ; .45a ^ n 5 = b 0 xy )]TJ/F41 10.9091 Tf 11.389 0 Td [(b eq 0 xy = b 0 xy )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x u y ; .45b andalsosince^ n 1 = b 0 x )]TJ/F41 10.9091 Tf 11.456 0 Td [(b eq 0 x = b 0 x )]TJ/F35 10.9091 Tf 10.976 0 Td [(u x and^ n 2 = b 0 y )]TJ/F41 10.9091 Tf 11.456 0 Td [(b eq 0 y = b 0 y )]TJ/F35 10.9091 Tf 10.976 0 Td [(u y ,andfromEqs..37a and.37b,wehavetherequiredlocalexpressionsforthederivativesofthescalareld,which readas @ x = )]TJ/F35 10.9091 Tf 10.408 7.38 Td [(! 1 c 2 s [ b 0 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x ] ;@ y ; = )]TJ/F35 10.9091 Tf 10.408 7.38 Td [(! 2 c 2 s [ b 0 y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y ].46 Notethat 1 1and 2 1,but 1 6 = 2 andareotherwisefreeparameters.Wetypically set 1 =1 ; 2 =0 : 9inthiswork.Inaddition,theexpressionsfor^ m 3 and^ m 4 neededinthe diagonalcomponentsofthevelocitygradienttensor,i.e.,Eqs..43aand.43bcanbewritten as ^ m 3 = b 0 xx + b 0 yy )]TJ/F15 10.9091 Tf 10.909 0 Td [( c 2 s + u 2 x + u 2 y ; .47a ^ m 4 = b 0 xx )]TJ/F41 10.9091 Tf 11.021 0 Td [(b 0 yy )]TJ/F35 10.9091 Tf 10.909 0 Td [( u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y : .47b Intheabove, b 0 xx , b 0 yy , b 0 xy , b 0 x , b 0 y and b 0 xy aretherawmomentcomponentsofdierentordersof therespectivedistributionfunctions.Theformulationpresentedabovethusallowslocalcomputationofthecompletevelocitygradienttensorandthevorticityeld.

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190 6.5Resultsanddiscussion Inthissection,wewillperformanumericalvalidationstudyofthenewDDFMRT-LBscheme forvorticitycomputation.Inthisregard,wewillconsiderasetofwell-denedbenchmarkow problemsforwhichtheanalyticalsolutionsforthevorticityeldareavailableorcanbederived. Inthesimulationsresultspresentedinthefollowing,therelaxationtimesforthesecondorder momentsoftheMRT-LBEfortheoweld ! 4 = ! 5 =1 = arechosentospecifythedesireduidviscosity,whilethosefortherstordermomentsoftheMRT-LBEforthescalareld ! 1 = ! 2 =1 = areprescribedtoselectthediusivity.Therelaxationtimesofallthehigher ordermomentsforboththeLBEsaresettounityforsimplicity.Unlessotherwisespecied,we considertheuseoflatticeunits,i.e., x = t =1 : 0typicalforLBsimulationsandareference densityofunityisconsideredinthiswork.Forallthebenchmarkproblemsreportedinwhatfollows,wesetthecoecientsforthescalaruxtermsinthethirdordermomentequilibriaofthe MRT-LBEforthescalareldto 1 =1 : 0and 2 =0 : 9. 6.5.1Poiseuilleow Astherstbenchmarkproblem,asteadyowbetweentwoparallelplateswithawidth2 L driven byaconstantbodyforce F x ,i.e.,thePoiseuilleow,issimulated.Thisowproblemhasananalyticalsolutionforthevorticityeldasthelinearprole ! z y =2 U max y=L 2 ,whichcanbe derivedfromtheparabolicvelocityprole u x y = U max [1 )]TJ/F36 7.9701 Tf 13.824 4.932 Td [(y 2 L 2 ],where U max = F x L 2 2 isthe maximumcenterlinevelocity, and areuidkinematicviscosityanddensity,respectively.Periodicboundaryconditionsareemployedinthestreamwisedirectionandno-slipconditionforthe velocityeldareimposedusingthehalf-waybouncebackscheme.Thecomputationaldomainis resolvedusing3 151latticenodes.Forthescalareld,weconsiderxedvaluesatthebottom andtopwallsas L =1 : 0and H =2 : 0,respectively,anditsdiusivityisspeciedbychoosing =0 : 57.Ataxedbodyforce F x =3 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(6 ,computationsarecarriedbyadjustingtheuid

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191 kinematicviscositysuchthatthefollowingvesetsofmaximumcenterlinevelocitiesareconsidered: U max =0 : 01 ; 0 : 03 ; 0 : 05,and0 : 08.Figure6.1showsacomparisonbetweennumericalresults forthevorticityprolesobtainedusingtheDDFMRT-LBschemeandtheanalyticalsolutions fortheabovesetofvaluesfor U max .Excellentagreementisseen. FIGURE6.1:ComparisonofthecomputedprolesofthevorticityeldandtheanalyticalsolutioninaPoiseuilleowfordierentvaluesofthecenterlinevelocity U max =0 : 01 ; 0 : 03 ; 0 : 05, and0 : 08obtainedbyvaryingtheuidviscosityataxedbodyforce F x =3 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(6 .Here,the linesrepresenttheanalyticalsolutionandsymbolsrefertothenumericalresultsobtainedbythe presentDDFMRT-LBscheme. 6.5.2Four-rollsmillow Inordertoexaminethevalidityofourapproachforaowproblemwithfullytwo-dimensional Dspatiallyvaryingdistributionofthevorticityeld,weconsidernextthefour-rollsmillow. Itisasteady,rotationalowconsistingofanarrayofcounter-rotatingvorticesgeneratedbythe stirringactionofasuitablyspeciedlocalbodyforce F x = F x x;y and F y = F y x;y ina periodicsquaredomainofsize2 2 .ItisamodiedformoftheTaylor-Greenvortexow. Thespatiallyvaryingdrivingbodyforcecanbewrittenas F x x;y =2 0 u 0 sin x sin y and

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192 F y x;y =2 0 u 0 cos x cos y ,where 0 isthereferencedensity, iskinematicviscosityand u 0 isthevelocityscaleand0 x;y 2 .AsolutionofthesimpliedformoftheNavier-Stokes equationswiththeabovedescribedbodyforceyieldstheexplicitformofthelocalvelocityeld, whichreadsas u x x;y = u 0 sin x sin y and u y x;y = u 0 cos x cos y .Then,theanalyticalsolution forthelocalvorticityeld ! z x;y canbederivedbytakingthecurloftheabovevelocityeld, whichcanbewrittenas ! z x;y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 u 0 sin x cos y: .48 Forthepurposeofsettingupsimulations,theReynoldsnumberforthisowproblemcanbedenedasRe= u 0 2 = andtheviscositycanbewrittenas = 1 3 )]TJ/F33 7.9701 Tf 12.695 4.295 Td [(1 2 x ,where x = t = 2 =N ,where N isthenumberofgridnodesineachdirection.Weconsideragridresolutionof 84 84andavelocityscale u 0 =0 : 035tosimulatefour-rollsmillowatRe=40.Thescalareld isinitializedtoauniformvalueof2 : 0inthisperiodicdomainwiththerelaxationtime =0 : 57. Figure6.2presentsacomparisonbetweenthespatialdistributionofthecomputedvorticityeld obtainedusingtheDDFMRT-LBschemeandtheanalyticalsolution.Duetothepresenceofa systemofcounter-rotatingvortices,thevorticityeld,representedbyharmonicfunctionsanalytically,dramaticallyvariesbothinitsmagnitudeandsign.Goodagreementbetweenthetworesultsareevident.Furthermore,inordertomakeamorehead-oncomparison,Fig.6.3showsthe computedvorticityproles ! z x;y computedusingourLBschemealongvarioushorizontalsectionsat y =0 ;= 4 ;= 2 ;; 5 = 4alongwithresultsbasedontheanalyticalsolution.Itisevident thatthereisaverygoodagreementbetweenournumericalresultsandtheanalyticalsolution. Gridconvergencestudy Wewillnowassesstheorderofaccuracyoftheconvergenceofthevorticitycomputationviaour DDFMRT-LBscheme.Inthisregard,ataxedviscosityof =0 : 00218withavelocityscale u 0 =0 : 045,weconsiderthefollowingsequenceoffourdierentresolutions:24 24,48 48,

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193 FIGURE6.2:Comparisonofthespatialdistributionofthecomputedvorticityeldwiththeanalyticalsolutioninafour-rollsmillowwithinasquaredomainofsize2 2 forRe=40.The surfaceplotontheleftcorrespondstothenumericalresultsobtainedbythepresentDDFMRTLBschemeandthatontherightisbasedontheanalyticalsolution. FIGURE6.3:Comparisonofcomputedprolesofthevorticityeldandtheanalyticalsolution inafour-rollsmillowalongvarioushorizontalsectionsat y =0 ;= 4 ;= 2 ;; 5 = 4.Here,the linesrepresenttheanalyticalsolutionandsymbolsrefertothenumericalresultsobtainedbythe presentDDFMRT-LBscheme.

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194 96 96and192 192.Foreachcase,wemeasurethefollowingglobalrelativeerror E g;! between thevorticityeldcomputedusingtheDDFMRT-LBschemegivenby ! c andthecorresponding analyticalsolutiondenotedby ! a : E g;! = s ! c )]TJ/F35 10.9091 Tf 10.909 0 Td [(! a 2 ! a 2 ; .49 wherethesummationsintheaboveareforthewholecomputationaldomain.Therateofconvergenceoftheglobalrelativeerrorisdepictedusingalog-logscaleinFig.6.4.Fromthisgure,it canbeseenthattherelativeerrorexhibitsaslopeof-2.0,whichdemonstratesthatthevorticity computationusingourapproachissecondorderaccurate. 6.5.3Womersleyow Inordertovalidateourapproachforthecalculationofthevorticityeldinunsteadyows,a2D pulsatileowbetweentwoparallelplatesseparatedbyawidth2 L drivenbyasinusoidallytimedependentbodyforce F x t isconsidered.ThisclassicalWomersleyowproblemissubjectedto aperiodicbodyforcegivenby F x = F m cos t ,where F m isthemaximumamplitudeofthe forceand=2 =T istheangularfrequencyand T beingthetimeperiod.Consideringthat thispulsatileowislaminarandincompressible,theanalyticalsolutionforvelocityeldisgiven as[208] u y;t = R i F m 1 )]TJ/F15 10.9091 Tf 12.104 5.374 Td [(cos y=L cos ] e i t g ; .50 where = p i Wo 2 andWo= L p = istheWomersleynumber.Here,andinthefollowing Rfg referstotherealpartoftheexpression.Then,theanalyticalsolutionforthelocaltimedependentvorticityeld ! z y;t canbereadilyobtainedbytakingthecurlofthevelocityeldas ! z y;t = R n i F m L h sin y=L cos ] e i t g : .51 Weconsideragridresolutionof3 101,maximumforceamplitude F m =1 : 0 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(5 witha timeperiod T =10 ; 000andtwodierentvaluesoftheWomersleynumber,i.e.,Wo=4 : 0and

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195 FIGURE6.4:EvaluationoftheorderofaccuracyofthepresentDDFMRT-LBschemefor vorticitycomputationinthefour-rollsmillowproblemwithaconstantkinematicviscosity =0 : 00218atdierentgridresolutions.

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196 Wo=7 : 0,whicharespeciedbysettingtherelaxationtimesfortheMRT-LBEfortheow eldtobe =0 : 781and =0 : 596,respectively.Periodicboundaryconditionsandthenoslipboundaryconditionsareconsideredfortheinlet/outletinthestreamwisedirectionandalong thetwoparallelwalls,respectively.Theparametersandtheboundaryconditionsforthescalar eldarethesameasthoseconsideredforthePoiseuilleowsimulationsdiscussedearlier.Figure6.5presentsacomparisonbetweenthecomputedvorticityprolesobtainedusingtheDDF MRT-LBschemeandthecorrespondinganalyticalsolutionatdierenttimeinstantswithina timeperiod T .Itisevidentthatthevorticityeldissubjectedtostrongtemporalandspatial variations,whichareverywellreproducedquantitativelybyourlocalcomputationalapproach. FIGURE6.5:Comparisonofcomputedprolesofthevorticityeldandtheanalyticalsolution inapulsatileowinachanneli.e.,Womersleyowatdierentinstantswithinatimeperiod fortwodierentWomersleynumbersofWo=4 : 0andWo=7 : 0.Here,linesrepresenttheanalyticalsolutionandthesymbolsrefertothenumericalresultsobtainedusingthepresentDDF MRT-LBscheme. 6.6SummaryandConclusions Aquantitativeknowledgeofthelocalskew-symmetricvelocitygradienttensor,orequivalently thevorticityeld,inconjunctionwiththesymmetricvelocitygradienttensor,inowsimulationmethodsiscrucialforvariousapplications,includingthoserelatedtotechniquesforthe identicationofowstructuresandinthemodelingofcomplexuids.Inmanysituations,itis requiredtocomputetheuidmotioncoupledtothetransportbyadvectionanddiusionofa

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200 basedformulation[216]andtheirthermodynamicconsistencywereanalyzedin[70,217,94].A signicantlyimprovedLBmethodusingakinetictheorybasedmeaneldmodelwaspresented in[22],whichallowedaccuratesimulationofmultiphaseowsatmoderatedensityratios.This approachusedoneLBschemefortheuidmotionandcapturedtheinterfacialmotionviaan indexfunction,whoseevolutionwasrepresentedbyanotherLBschemewherethephasesegregationwasachievedusingaCarnahan-Starlingnonidealequationofstate.Thiswasfurtherimprovedforsimulationoftwo-phaseowsathighdensityratiosbymeansofastablediscretization scheme[218].ThelatterworkmotivateddevelopmentsofconsistentLBtechniquesforinterfacial capturingtechniquesbasedonphaseeldmodels. Phaseeldmodelsrepresentinterfacestobediuse,whichcomprisethintransitionalregionsof nonzerothicknessacrosswhichvariousuidpropertiesvarycontinuouslyfromonephasetothe other[219,220,221,222].Suchdiuseinterfacemethodscaptureinterfacialmotionimplicitlyvia theevolutionofanorderparameter,whichservesasaphaseeldtodistinguishbetweendierentuidphases.Thedynamicsoftheorderparameterisoftenbasedonathermodynamicfree energyfunctionalformulation,ofwhichtheCahn-HilliardequationCHE[223]isacommon choice.ALBschemetorepresenttheconvectiveCHEwaspresentedin[224],whichwasshown tobeapplicableonlyfordensity-matchedtwo-uidsystemsin[225],whothenproposedamodicationtohandlemultiphaseowsatmoderatedensityratios.Thelatterworkwasfurtherimprovedintheinvestigationspresentedin[226,227]torepresentincompressiblemultiphaseows basedonmodiedCHEforcapturingofinterfaces. ThechallengesassociatedwiththeuseofCHE,suchastheneedtocalculatefourthorderderivatives,motivatedotherphaseeldtypeapproaches.TheAllen-CahnequationACEisanother typeofdiuseinterfacemodelusedthatwasoriginallydevelopedformaterialscienceapplications[228].Morerecently,theACEwasreformulatedbasedonacountertermapproach[229]to eliminatecurvaturedriveninterfacialmotioninordertomakeitapplicablefortwo-phaseows[230], inwhichthegeometricinformationsuchastheinterfacenormalandcurvaturearecomputed

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204 writtenas @ @t + r u = r [ M r )]TJ/F35 10.9091 Tf 10.909 0 Td [( n ] ; .1 where u istheuidvelocity, n istheunitnormalvector,whichcanbecomputedviatheorder parameter as n = r j r j ,and M isthemobility.Intheabove,thevariable canbeexpressed as = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F35 10.9091 Tf 10.909 0 Td [( A )]TJ/F35 10.9091 Tf 10.909 0 Td [( B W A )]TJ/F35 10.9091 Tf 10.909 0 Td [( B ; .2 wheretheparameter W isrelatedtothewidthoftheinterface.TherighthandsideofEq..1 isobtainedbyremovingthecurvature-driveninterfacemotion u n = )]TJ/F35 10.9091 Tf 8.485 0 Td [(M m n bycancelingit outbyaddingacounteractingtermbasedoncomputingthecurvature m ,where m = r n with n = r j r j ,directlyintermsofakernelfunctiongivenbythefollowinghyperbolictangent proleoftheorderparameter = 1 2 A + B + 1 2 A )]TJ/F35 10.9091 Tf 10.91 0 Td [( B tanh 2 W ; .3 whichrepresentstheequilibriumproleofthephaseeldvariable,where isaspatialcoordinate alongthenormalwiththeoriginattheinterface.Thus,Eq..1eectivelyrepresentstherelaxationofanyarbitraryinitialdistributionoftheorderparametertoahyperbolictangentprole acrosstheinterface,whichisthensustainedduringinterfacialadvection.Equivalently,thisequationcanbeinterpretedastheinterfacepropagatingviaadvectiongivenbyitsLHSunderthe competingeectsofadiusiontermandaninterfacesharpeningtermoraseparationuxterm givenbytherstandsecondtermsontheRHS,respectively.Intheabove, W and M arenumericalparameters,with W representingtheinterfacethickness,while M controllingtherelaxationrateofanyinitial toitsequilibriumproleacrosstheinterfacesEq..3aswellasthe dissipationofanyinterfacesingularitiesviadiusion. Ontheotherhand,thetwo-phaseuidowisrepresentedbythefollowingincompressibleNavier-

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205 StokesequationsNSE: r u =0 ; .4 @ u @t + r uu = )]TJ/F55 10.9091 Tf 8.485 0 Td [(r p + r h r u + r u y i + F s + F ext ; .5 where p isthehydrodynamicpressure, istheuiddensity, isitsviscosity, F s isthesmoothed formulationofthesurfacetensionforceand F ext isanexternalbodyforcee.g.,gravity. Intheabove,thereareseveralwaystoexpressthesurfacetensionforce F s asasmoothedrepresentationbasedontheorderparameter.OneapproachisbasedonathermodynamicGibbsDuhemformulationinwhichthesurfacetensionforceiscalculatedfromthenegativeproductof thegradientofthechemicalpotential~ andthephaseeldvariable asfollowsseee.g.,[220]: F s = )]TJ/F35 10.9091 Tf 8.485 0 Td [( r ~ ; ~ =4 )]TJ/F35 10.9091 Tf 10.909 0 Td [( A )]TJ/F35 10.9091 Tf 10.909 0 Td [( B )]TJ/F15 10.9091 Tf 10.91 0 Td [( A + B = 2 )]TJ/F35 10.9091 Tf 10.909 0 Td [( r 2 : .6 Here,theparameters and areusedtocontrolthesurfacetension andtheinterfacethickness W viathefollowingrelations = 3 2 W; = 12 W : .7 Alternatively,geometricapproachessuchasthecontinuoussurfaceforceformulationcanbeconsidered[235].Inparticular,ageometricapproachforthesurfacetensionforcedevelopedoriginallyforlevelsetmethodsandadaptedforphaseeldmethods[236]canbewrittenas F s = )]TJ/F15 10.9091 Tf 8.901 0 Td [(~ j r j 2 r n n : .8 Here,theparameter~ isrelatedtothesurfacetension via~ = W ,wherethecoecient satises R 1 d=d 2 d =1,whicharisesfrominterpretingthesurfacetensionintermsofinterfacialenergyperunitsurfaceareabyconsideringtheequilibriumphaseeldvariableprole giveninEq..3andmatchingitwiththesharpinterfacelimitforaatinterface[236].Inthis work,thislattergeometricapproachisadoptedforrepresentingthesurfacetensionforce F s forperformingtwo-phaseowsimulationsusingcascadedLBformulationsdiscussedinwhatfollows.Finally,thejumpsinuidpropertiessuchasthedensityandviscosityacrosstheinterface

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206 aresmoothedaswellandcanbewrittenasacontinuousfunctionofthephaseeldvariable andthenusedinEq..5indierentways.Inthisstudy,weemployalinearinterpolationfor representingtheinterfacialvariationsoftheuidpropertiesseee.g.,[221].Thus, = B + )]TJ/F35 10.9091 Tf 10.909 0 Td [( A A )]TJ/F35 10.9091 Tf 10.909 0 Td [( B A )]TJ/F35 10.9091 Tf 10.909 0 Td [( B ; = B + )]TJ/F35 10.9091 Tf 10.909 0 Td [( A A )]TJ/F35 10.9091 Tf 10.909 0 Td [( B A )]TJ/F35 10.9091 Tf 10.909 0 Td [( B ; .9 where A and B arethedensitiesand A and B arethedynamicviscositiesintheuidphases denotedby A and B ,respectively.Inthiswork,weconsider B =0and A =1. 7.3ModiedContinuousBoltzmannEquationforTwo-PhaseFlowsandCentral MomentsofEquilibriaandSources TosolvetheincompressibleNavier-StokesequationsNSEfortwo-phaseowsEqs..4and .5inakineticformulation,thestartingpointisthetwo-dimensionalDcontinuousBoltzmannequationgivenby[22] Df Dt @f @t + r f = )]TJ/F15 10.9091 Tf 9.955 7.38 Td [(1 )]TJ/F35 10.9091 Tf 5 -8.837 Td [(f )]TJ/F35 10.9091 Tf 10.909 0 Td [(f M + )]TJ/F52 10.9091 Tf 10.909 0 Td [(u c 2 s F t )]TJ/F55 10.9091 Tf 10.909 0 Td [(r f M ; .10 where f = f x ;t ; isthedensitydistributionfunctionatalocation x andattime t ,correspondingtotheparticlevelocity = x ; y .Here, f M isthelocalMaxwelldistributionfunctiondenedas f M f M ; u = 2 c 2 s exp )]TJ/F15 10.9091 Tf 9.68 7.38 Td [( )]TJ/F52 10.9091 Tf 10.909 0 Td [(u 2 2 c 2 s ; .11 where c s isthespeedofsoundanduidvelocity u = u x ;u y .Theeectofcollisionsistypicallyrepresentedasarelaxationof f toitsequilibrium,i.e., f M withacharacteristictimescale .Thecontinuousformulationoftheinterfacialtensionforce F s ,whichisdiscussedintheprevioussection,alongwithanylocalbodyforce F ext aregroupedasthetotalforce F t = F s + F ext . Thistotalforcealongwiththegradientcontributionoftheneteectofthehydrodynamicpressure p relativetothatfromtheidealequationofstate c 2 s ,i.e., = p )]TJ/F35 10.9091 Tf 11.859 0 Td [(c 2 s areaccounted forviaasourceterminEq..10.Ingeneral,multiphaseowscanbeassociatedwithrelatively

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207 largejumpsinuidpropertiesacrosstheinterfaces.Inparticular,asthedensitygradients r or r becomerelativelylarge,Eq..10becomesnumericallysti. Toalleviatesuchnumericalstiness,thefollowingkinetictransformationtothedistributioncan beintroduced[22] g = fc 2 s + p )]TJ/F35 10.9091 Tf 10.909 0 Td [(c 2 s f M ; 0 ; .12 where g canberegardedasthepressuredistributionfunction.Here, f M ; 0 isthelocalMaxwellian withnullmacroscopicuidvelocity,whichfollowsfromEq..11as f M ; 0 = 2 c 2 s exp )]TJ/F52 10.9091 Tf 11.831 7.381 Td [( 2 2 c 2 s : .13 Then,byapplyingtheabovetransformationEq..12tothecontinuousBoltzmannequation Eq..10andassumingtwo-phaseowsintheincompressiblelimit,i.e., j u j 1,thefollowing kineticequationforthedistributionfunction g canbeobtained[22] Dg Dt = )]TJ/F15 10.9091 Tf 9.955 7.38 Td [(1 g )]TJ/F35 10.9091 Tf 10.91 0 Td [(g eq + )]TJ/F52 10.9091 Tf 10.909 0 Td [(u F t f M ; u + )]TJ/F52 10.9091 Tf 10.909 0 Td [(u F p f M ; u )]TJ/F35 10.9091 Tf 12.105 7.38 Td [(f M ; 0 | {z } O u ; .14 whichisreferredtoasthemodiedcontinuousBoltzmannequationMCBEinthiswork.In Eq..14, g eq isthetransformedlocalMaxwellianorthemodiedequilibriumdistributionfunction,whichreadsas g eq = c 2 s f M ; u + p )]TJ/F35 10.9091 Tf 10.909 0 Td [(c 2 s f M ; 0 .15 and F p istheneteectofthehydrodynamicpressure p relativetothecontributionfromthe idealequationofstatedependentondensity,whichisreferredasthenetgradientpressureforce, andcanbeexpressedas F p = )]TJ/F55 10.9091 Tf 8.485 0 Td [(r p )]TJ/F35 10.9091 Tf 10.909 0 Td [(c 2 s )]TJ/F55 10.9091 Tf 20 0 Td [(r : .16 IntheMCBEEq..14,eventhough F p canbelargeathighdensityratios,sinceitismultipliedby n f M ; u )]TJ/F36 7.9701 Tf 12.105 5.374 Td [(f M ; 0 o ,whichis O u andsmall,theassociatednumericalstinessissues

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208 ontheevolutionofthedistributionfunction g isreducedsignicantly.Hence,theMCBEserves asthestartingpointintheconstructionofadiscretekineticschemeforthesolutionoftheincompressibletwo-phaseowswithhighphasedensitycontrasts.Thehydrodynamicpressureand velocityeldsarethenobtainedasthezerothandrstkineticmomentsofthedistributionfunction g ,respectively.Thatis, p = Z 1 Z 1 gd x d y ;c 2 s u = Z 1 Z 1 g d x d y : .17 7.3.1ContinuousCentralMomentsofEquilibriaandSourcesofMCBE AsapreludetoconstructingacascadedLBschemefromthediscretizationoftheMCBE,which isdiscussedinthenextsection,wewillrstneedthecontinuouscentralmomentsofitsequilibriaandvarioussources.Theyarebasedonthecontributionsfromthecorrespondingcontinuous Maxwelldistributionfunctionevaluatedwithandwithoutthemacroscopicuidvelocityinview ofthekinetictransformationintroducedabove. First,deningthecontinuouscentralmomentsofthelocalMaxwellianforamovinguid,i.e., withthemacroscopicuidvelocity,oforder m + n as ^ M mn = Z 1 Z 1 f M ; u x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x m y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y n d x d y ; .18 andthendeningthecontinuouscentralmomentsofthelocalMaxwellianwiththenullmacroscopicuidvelocityoforder m + n as ^ M 0 mn = Z 1 Z 1 f M ; 0 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x m y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y n d x d y : .19 ThedeniteintegralsgiveninEqs..18and.19canbeevaluatedexactlyviastandardquadraturerules.TheD2Q9latticeusedintheconstructionofthecascadedLBschemebasedona matchingprincipleinthenextsectionsupportsnineindependentmomentcomponents.Inthis regard,wewillneedthecorrespondingcomponentsof ^ M mn and ^ M 0 mn asintermediateresults.

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209 Thus,calculatingthecomponentsofthecontinuouscentralmomentsoftheMaxwellian ^ M mn Eq..18atvariousorders,whichreadsas ^ M 00 = ; ^ M 10 =0 ; ^ M 01 =0 ; ^ M 20 = c 2 s ; ^ M 02 = c 2 s ; ^ M 11 =0 ; ^ M 21 =0 ; ^ M 12 =0 ; ^ M 22 = c 4 s : .20 andthoseof ^ M 0 mn Eq..19maybewrittenas ^ M 0 00 =1 ; ^ M 0 10 = )]TJ/F35 10.9091 Tf 8.485 0 Td [(u x ; ^ M 0 01 = )]TJ/F35 10.9091 Tf 8.485 0 Td [(u y ; ^ M 0 20 = u 2 x + c 2 s ; ^ M 0 02 = u 2 y + c 2 s ; ^ M 0 11 = u x u y ; ^ M 0 21 = )]TJ/F15 10.9091 Tf 8.485 0 Td [( u 2 x + c 2 s u y ; ^ M 0 12 = )]TJ/F15 10.9091 Tf 8.485 0 Td [( u 2 y + c 2 s u x ; ^ M 0 22 = u 2 x + c 2 s u 2 y + c 2 s : Then,inordertodiscretizeEq..14inacascadedLBformulation,weneedthecontinuouscentralmomentsoftheequilibriumpressuredistributionfunctionorthetransformedMaxwellian g eq Eq..15oforder m + n .Bydeningitas ^ eq;g mn = Z 1 Z 1 g eq x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x m y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y n d x d y ; .21 itreadilyfollowsthatEq..21satisesthefollowingrelation ^ eq;g mn = c 2 s ^ M mn + ^ M 0 mn : Evaluatingitsninecomponents,weget ^ eq;g 00 = p; ^ eq;g 10 = )]TJ/F35 10.9091 Tf 8.485 0 Td [( u x ; ^ eq;g 01 = )]TJ/F35 10.9091 Tf 8.485 0 Td [( u y ; ^ eq;g 20 = pc 2 s + u 2 x ; ^ eq;g 02 = pc 2 s + u 2 y ; ^ eq;g 11 = u x u y ; ^ eq;g 21 = )]TJ/F35 10.9091 Tf 8.485 0 Td [( c 2 s + u 2 x u y ; ^ eq;g 12 = )]TJ/F35 10.9091 Tf 8.485 0 Td [( c 2 s + u 2 y u x ; ^ eq;g 22 = c 6 s + u 2 x + c 2 s u 2 y + c 2 s : .22 Next,weneedthecontinuouscentralmomentsofthesourcetermduetothetotalinterfacial andlocalbodyforce F t = F tx ;F ty oforder m + n inMCBEEq..14,whichcanbedenedas ^ )]TJ/F36 7.9701 Tf 6.818 4.505 Td [(t mn = Z 1 Z 1 S t x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x m y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y n d x d y ; .23

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210 where S t = )]TJ/F52 10.9091 Tf 10.909 0 Td [(u F t f M ; u : .24 Itcanbeshownthatthiscontinuouscentralmomentsatisesthefollowingidentitythatdepends onthethoseoftheMaxwellian ^ )]TJ/F36 7.9701 Tf 6.818 4.504 Td [(t mn = F tx ^ M m +1 ;n + F ty ^ M m;n +1 : Byevaluatingitscomponentsanddealiasingtheresultingcentralmomentcomponentshigher thanthesecondorderbysettingthemtozero,astheydonotinuencetherecoveryofthehydrodynamicsviatheChapman-Enskogexpansion[19],theresultscanbesummarizedas ^ )]TJ/F36 7.9701 Tf 6.818 4.504 Td [(t 00 =0 ; ^ )]TJ/F36 7.9701 Tf 6.819 4.504 Td [(t 10 = c 2 s F tx ; ^ )]TJ/F36 7.9701 Tf 6.818 4.504 Td [(t 01 = c 2 s F ty ; ^ )]TJ/F36 7.9701 Tf 6.818 4.504 Td [(t 20 =0 ; ^ )]TJ/F36 7.9701 Tf 6.818 4.504 Td [(t 02 =0 ; ^ )]TJ/F36 7.9701 Tf 6.818 4.504 Td [(t 11 =0 ; ^ )]TJ/F36 7.9701 Tf 6.819 4.504 Td [(t 21 =0 ; ^ )]TJ/F36 7.9701 Tf 6.818 4.504 Td [(t 12 =0 ; ^ )]TJ/F36 7.9701 Tf 6.818 4.504 Td [(t 22 =0 : .25 Finally,wedenethecontinuouscentralmomentsofthesourcetermduetothenetgradient pressureforce F p = F px ;F py inEq..14oforder m + n as ^ )]TJ/F36 7.9701 Tf 6.818 4.504 Td [(p mn = Z 1 Z 1 S p x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x m y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y n d x d y ; .26 where S p = )]TJ/F52 10.9091 Tf 10.909 0 Td [(u F p f M ; u )]TJ/F35 10.9091 Tf 12.105 7.38 Td [(f M ; 0 : .27 Basedonitsdenition,thiscentralmoment ^ )]TJ/F36 7.9701 Tf 6.818 5.308 Td [(p mn canbedemonstratedtosatisfythefollowing identity ^ )]TJ/F36 7.9701 Tf 6.818 4.504 Td [(p mn = F px ^ M m +1 ;n )]TJ/F15 10.9091 Tf 13.468 11.686 Td [(^ M 0 m +1 ;n ! + F py ^ M m;n +1 )]TJ/F15 10.9091 Tf 13.469 11.686 Td [(^ M 0 m;n +1 ! : Byusingthisandderivingtheexpressionsfortheninecomponents,where,asbefore,weretain theresultsonlyuptothesecondordermomentsthatdeterminethetwo-phaseuidmotionand setthehigherorderonestozero,theycanbesummarizedas ^ )]TJ/F36 7.9701 Tf 6.818 5.307 Td [(p 00 = F px u x + F py u y ; ^ )]TJ/F36 7.9701 Tf 6.818 5.307 Td [(p 10 = )]TJ/F35 10.9091 Tf 8.485 0 Td [(u x ^ )]TJ/F36 7.9701 Tf 6.818 5.307 Td [(p 00 ; ^ )]TJ/F36 7.9701 Tf 6.819 5.307 Td [(p 01 = )]TJ/F35 10.9091 Tf 8.485 0 Td [(u y ^ )]TJ/F36 7.9701 Tf 6.818 5.307 Td [(p 00 ; ^ )]TJ/F36 7.9701 Tf 6.819 5.307 Td [(p 20 =2 c 2 s F px u x + u 2 x + c 2 s ^ )]TJ/F36 7.9701 Tf 6.818 5.307 Td [(p 00 ; ^ )]TJ/F36 7.9701 Tf 6.818 5.308 Td [(p 02 =2 c 2 s F py u y + u 2 y + c 2 s ^ )]TJ/F36 7.9701 Tf 6.818 5.308 Td [(p 00 ; ^ )]TJ/F36 7.9701 Tf 6.818 5.308 Td [(p 11 = c 2 s F px u y + F py u x + u x u y ^ )]TJ/F36 7.9701 Tf 6.818 5.308 Td [(p 00 ; ^ )]TJ/F36 7.9701 Tf 6.818 5.307 Td [(p 21 =0 ; ^ )]TJ/F36 7.9701 Tf 6.818 5.307 Td [(p 12 =0 ; ^ )]TJ/F36 7.9701 Tf 6.818 5.307 Td [(p 22 =0 : .28

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211 7.4CascadedLBMethodforSolutionofTwo-PhaseFluidMotion WewillnowpresentacascadedcentralmomentLBmethodbasedonthediscretizationofthe MCBEdiscussedintheprevioussectionforthesolutionofincompressibletwo-phaseow.Inthis regard,weconsidertheD2Q9lattice,whosecomponentsoftheparticlevelocitiesarerepresented bythefollowingvectorsusingthestandardDirac'sbra-ketnotation: j e x i = ; 1 ; 0 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; 0 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 y ; .29a j e y i = ; 0 ; 1 ; 0 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 y : .29b Inaddition,weneedtodenethefollowingnine-dimensionalvector j 1 i = ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 y ; .30 whoseinnerproductwithadiscretedistributionfunction g seebelow,where =0 ; 1 ; 2 ; ; 8 representstheparticlevelocitydirection,i.e.,itszerothmomentyieldsthepressureeld.Using theabove,thefollowingsetoforthogonalmomentbasisvectorscanbeusedtoconstructedthe cascadedLBformulation: j K 0 i = j 1 i ; j K 1 i = j e x i ; j K 2 i = j e y i ; j K 3 i =3 j e 2 x + e 2 y i)]TJ/F15 10.9091 Tf 17.575 0 Td [(4 j 1 i ; j K 4 i = j e 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(e 2 y i ; j K 5 i = j e x e y i ; j K 6 i = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 j e 2 x e y i +2 j e y i ; j K 7 i = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 j e x e 2 y i +2 j e x i ; j K 8 i =9 j e 2 x e 2 y i)]TJ/F15 10.9091 Tf 17.576 0 Td [(6 j e 2 x + e 2 y i +4 j 1 i : .31 Intheabove,asymbolsuchas j e 2 x e y i = j e x e x e y i representsavectorresultingfromtheelementwisevectormultiplicationofthesequenceofvectors j e x i , j e x i and j e y i .Bycombiningtheabove nineindependentvectors,wethenobtainthefollowingorthogonalmomentbasismatrix K =[ j K 0 i ; j K 1 i ; j K 2 i ; j K 3 i ; j K 4 i ; j K 5 i ; j K 6 i ; j K 7 i ; j K 8 i ] : .32 Then,weperformthestandardspatialandtemporaldiscretizationoftheMCBEEq..14 alongthecharacteristicdirectionsoftheparticlevelocitiesoveratimestep t typically t =1in

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212 latticeunits,whereweapplyatrapezoidalruleforthetreatmentofthesourcetermtomaintain asecondorderaccuracy[22],whichyields g x + e t ;t + t = g x ;t + K b h + 1 2 [ S x ;t + S x + e t ;t + t ] t : .33 Here, K b h isthecascadedcollisionoperator,where b h = j b h i = b h 0 ; b h 1 ; b h 2 ;:::; b h 8 y isavector representingthechangesinalltheninemomentssupportedbythelatticeundercollisionwhich willbedeterminedinwhatfollows. S isthetotalsourcetermrepresentingthecumulativeeect ofthediscreteversionofthesourceduetotheinterfacialandlocalbodyforce S t viaEq..24 andthatduetothenetgradientpressureforce S p viaEq..27: S = S t + S p : .34 InordertoremoveimplicitnessinEq..33,weapplyavariabletransformation g = g )]TJ/F33 7.9701 Tf 10.531 4.295 Td [(1 2 S t , whichthenresultsinthefollowingeectivelyexplicitcascadedLBscheme g x + e t ;t + t = g x ;t + K b h + g s ; .35 where g s isamodiedcumulativesourcetermunderthevariabletransformation,whichweprescribetobethefollowing: g s = K )]TJ/F33 7.9701 Tf 6.587 0 Td [(1 I )]TJ/F15 10.9091 Tf 12.104 7.381 Td [(1 2 ^ K S : .36 Here, S = S 0 ;S 1 ;S 2 ;:::;S 8 y representsavectorofalltheninecomponentsofthediscrete sourcetermand ^ =diag ! 0 ;! 1 ;! 2 ;:::;! 8 isarelaxationtimematrixusedinthedevelopment ofthecascadedcollisionoperatorunderrelaxationofdierentcentralmomentslater.Sincethe eectsofthetwosources S t and S p appearinginthecumulativesourceterm S onthechanges ofvariousmomentsaredierent,weconsideramodicationtotheearliercentralmomentsbased strategy[19]inthisregard.TheexpressiongiveninEq..36ismotivatedtoremoveanyspuriouseectsduetothesourceterminthesecondordernon-equilibriummoments,whichare relatedtotheviscousstresstensor,inordertoconsistentlyrecovertheincompressibleNSEfor two-phaseows.SimilarapproachhasbeenconsideredintheMRT-LBEwithforcingtermpre-

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213 viouslyseee.g.,[75],buttheformof g s inEq..36willbestilldeterminedbyacentralmomentsbasedstrategyinwhatfollows. Inorderofderivetheexpressionsfor b h and g s tocompletetheformulationofthecascadedLB schemefortwo-phaseuidmotion,werstdenethediscretecentralmomentsofthedistributionfunction,itsequilibriumandthesourcetermas 0 B B B B B B B B @ ^ mn ^ eq mn ^ mn ^ mn 1 C C C C C C C C A = X 0 B B B B B B B B @ g g eq S g 1 C C C C C C C C A e x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x m e y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y n ; .37 where ^ mn =^ mn )]TJ/F33 7.9701 Tf 12.104 4.296 Td [(1 2 ^ mn t andthecorrespondingrawmomentsas 0 B B B B B B B B @ ^ 0 mn ^ eq 0 mn ^ 0 mn ^ 0 mn 1 C C C C C C C C A = X 0 B B B B B B B B @ g g eq S g 1 C C C C C C C C A e m x e n y ; .38 where ^ 0 mn =^ 0 mn )]TJ/F33 7.9701 Tf 12.418 4.296 Td [(1 2 ^ 0 mn t .Then,weneedtodeterminetheexpressionsforthediscretecentral momentsoftheequilibriumdistributionfunctionandthesourceterm.Inthisregard,weapply amatchingprinciple[15,19],wheretheyarerespectivelysetequaltotheircontinuousvaluesfor allorderssupportedbythelattice.Thatis, ^ eq mn = ^ eq;g mn ; ^ mn = ^ )]TJ/F36 7.9701 Tf 6.818 -1.637 Td [(mn ^ )]TJ/F36 7.9701 Tf 6.818 4.504 Td [(t mn + ^ )]TJ/F36 7.9701 Tf 6.819 4.504 Td [(p mn ; .39 wherethecontinuouscentralmomentcomponentsoftheequilibrium ^ eq;g mn isgiveninEq..22, whilethoseforthesourceterms ^ )]TJ/F36 7.9701 Tf 6.819 3.959 Td [(t mn and ^ )]TJ/F36 7.9701 Tf 6.818 5.307 Td [(p mn canbefoundinEqs..25and.28,respectively.ThisstepeectivelypreservetheGalileaninvarianceofallthemomentsindependently supportedbythelattice. BasedonEq..39,therststepinderivingthemodiedcumulativesourceterminthevelocity spaceduetovarioussources/forces g s istoconvertthecentralmoments^ mn tothecorrespond-

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214 ingrawmoments^ 0 mn atvariousordersviathebinomialtransform.Performingthisandsetting allthecumulativesourcemomentsofsecondandhigherordertozeroastheydonotaectrecoveringthehydrodynamicsofthetwo-phaseuidsintheChapman-Enskoganalysis[19,205],we get ^ 0 00 = ^ )]TJ/F36 7.9701 Tf 6.818 5.307 Td [(p 00 F px u x + F py u y ; ^ 0 10 = c 2 s F tx ; ^ 0 01 = c 2 s F ty ; ^ 0 20 =2 c 2 s F tx u x + F px u x + c 2 s ^ )]TJ/F36 7.9701 Tf 6.818 5.308 Td [(p 00 ; ^ 0 02 =2 c 2 s F ty u y + F py u y + c 2 s ^ )]TJ/F36 7.9701 Tf 6.818 5.308 Td [(p 00 ; ^ 0 11 = c 2 s F tx u y + F ty u x + c 2 s F px u y + F py u x ; ^ 0 21 =0 ; ^ 0 12 =0 ; ^ 0 22 =0 : Usingthis,wethenevaluatethevarioussourcemomentsprojectedtotheorthogonalbasisvectorsandwithascalingbasedontherelaxationtimeforavoidinganyspuriouseectsinthesecondordernon-equilibriummomentsasmentionedearlier,i.e.,^ m s 0 j = )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1 )]TJ/F33 7.9701 Tf 12.105 4.296 Td [(1 2 ! j h K j j S i ,which yields ^ m s 0 0 = 1 )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 2 ! 0 ^ 0 00 ; ^ m s 0 1 = 1 )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(1 2 ! 1 ^ 0 10 ; ^ m s 0 2 = 1 )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(1 2 ! 2 ^ 0 01 ; ^ m s 0 3 = 1 )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 2 ! 3 h 3^ 0 20 +^ 0 02 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4^ 0 00 i ; ^ m s 0 4 = 1 )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(1 2 ! 4 h ^ 0 20 )]TJ/F15 10.9091 Tf 11.494 0 Td [(^ 0 02 i ; ^ m s 0 5 = 1 )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 2 ! 5 ^ 0 11 ; ^ m s 0 6 = 1 )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(1 2 ! 6 h )]TJ/F15 10.9091 Tf 8.485 0 Td [(3^ 0 21 +2^ 0 01 i ; ^ m s 0 7 = 1 )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 2 ! 7 h )]TJ/F15 10.9091 Tf 8.485 0 Td [(3^ 0 12 +2^ 0 10 i ; ^ m s 0 8 = 1 )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 2 ! 8 h 9^ 0 22 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6^ 0 20 +^ 0 02 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8^ 0 00 i : Finally,byexploitingtheorthogonalpropertyof K in g s = K )]TJ/F33 7.9701 Tf 6.586 0 Td [(1 ^ m s 0 ,where ^ m s 0 = I )]TJ/F33 7.9701 Tf 12.105 4.296 Td [(1 2 ^ K S , with ^ m s 0 =^ m s 0 0 ; ^ m s 0 1 ; ^ m s 0 2 ; ; ^ m s 0 8 y ,wegetthemodiedcumulativesourcetermduetovarious

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215 sources/forcesinthecascadedLBschemefortwo-phaseowas g s 0 = 1 9 h ^ m s 0 0 )]TJ/F15 10.9091 Tf 12.971 0 Td [(^ m s 0 3 +^ m s 0 8 i ; g s 1 = 1 36 h 4^ m s 0 0 +6^ m s 0 1 )]TJ/F15 10.9091 Tf 12.971 0 Td [(^ m s 0 3 +9^ m s 0 4 +6^ m s 0 7 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2^ m s 0 8 i ; g s 2 = 1 36 h 4^ m s 0 0 +6^ m s 0 2 )]TJ/F15 10.9091 Tf 12.971 0 Td [(^ m s 0 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9^ m s 0 4 +6^ m s 0 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2^ m s 0 8 i ; g s 3 = 1 36 h 4^ m s 0 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6^ m s 0 1 )]TJ/F15 10.9091 Tf 12.971 0 Td [(^ m s 0 3 +9^ m s 0 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6^ m s 0 7 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2^ m s 0 8 i ; g s 4 = 1 36 h 4^ m s 0 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6^ m s 0 2 )]TJ/F15 10.9091 Tf 12.971 0 Td [(^ m s 0 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9^ m s 0 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6^ m s 0 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2^ m s 0 8 i ; g s 5 = 1 36 h 4^ m s 0 0 +6^ m s 0 1 +6^ m s 0 2 +2^ m s 0 3 +9^ m s 0 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3^ m s 0 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3^ m s 0 7 +^ m s 0 8 i ; g s 6 = 1 36 h 4^ m s 0 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6^ m s 0 1 +6^ m s 0 2 +2^ m s 0 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9^ m s 0 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3^ m s 0 6 +3^ m s 0 7 +^ m s 0 8 i ; g s 7 = 1 36 h 4^ m s 0 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6^ m s 0 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6^ m s 0 2 +2^ m s 0 3 +9^ m s 0 5 +3^ m s 0 6 +3^ m s 0 7 +^ m s 0 8 i ; g s 8 = 1 36 h 4^ m s 0 0 +6^ m s 0 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6^ m s 0 2 +2^ m s 0 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9^ m s 0 5 +3^ m s 0 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3^ m s 0 7 +^ m s 0 8 i : .40 Next,thestructureofthecascadedcollisionoperator K b h basedonthediscreteequilibrium centralmoments^ eq mn giveninEq..39isdeterminedasfollows.Forallnon-conservedmoments,i.e.,for m + n 2,weprescribetherelaxationofthediscretecentralmoments ^ mn totheircorrespondingcentralmomentequilibria^ eq mn atarelaxationtime ! [15,19].Thatis, P K b h e x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x m e y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y n = ! ^ eq mn )]TJ/F15 10.9091 Tf 11.44 2.182 Td [(^ mn .Forthetransformeddistributionfunction g employedinthecascadedLBschemeEq..35,duringatimestep t ,itszerothmoment changeneedstobe^ 0 00 ,whileitsrstordermomentsarerequiredtochangeby^ 0 10 and^ 0 01 in ordertoconsistentlyupdatethepressureeldandtheuidmomentumviatheinterfacialand bodyforces.Ontheotherhand,therespectivemomentchangesduetothesourcesgivenearlier are^ m s 0 0 = )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1 )]TJ/F33 7.9701 Tf 12.105 4.295 Td [(1 2 ! 0 ^ 0 00 ,^ m s 0 1 = )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1 )]TJ/F33 7.9701 Tf 12.104 4.295 Td [(1 2 ! 1 ^ 0 10 ,and^ m s 0 2 = )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1 )]TJ/F33 7.9701 Tf 12.105 4.295 Td [(1 2 ! 2 ^ 0 01 .Hence,tomeettheabove physicalconstraints,weeectivelyneedtosatisfythefollowingconstraints: P K b h = ! 0 2 ^ 0 00 , P K b h e x = ! 1 2 ^ 0 10 and P K b h e y = ! 2 2 ^ 0 01 .Basedontheseconsiderationsforthe lowerordermomentchangesandthecentralmomentrelaxationforthehigherordermoments undercollisionmentionedabove,theexpressionsforthecomponentsofthemomentchangevec-

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216 tor b h = b h 0 ; b h 1 ; b h 2 ;:::; b h 8 y canbedetermined,whicharesummarizedasfollows: ^ h 0 = ! 0 2 ^ )]TJ/F36 7.9701 Tf 6.818 5.307 Td [(p 00 9 ; ^ h 1 = ! 1 2 c 2 s F tx 6 ; ^ h 2 = ! 2 2 c 2 s F ty 6 ; ^ h 3 = ! 3 12 h 2 pc 2 s + c 2 s u 2 x + u 2 y )]TJ/F15 10.9091 Tf 10.909 0 Td [( ^ 0 20 + ^ 0 02 i ; ^ h 4 = ! 4 4 h c 2 s u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y )]TJ/F15 10.9091 Tf 10.909 0 Td [( ^ 0 20 )]TJ/F15 10.9091 Tf 11.086 2.182 Td [(^ 0 02 i ; ^ h 5 = ! 5 4 h c 2 s u x u y )]TJ/F15 10.9091 Tf 11.086 2.182 Td [(^ 0 11 i ; ^ h 6 = ! 6 4 h c 2 s + u 2 x u y + ^ 0 21 )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y ^ 0 20 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x ^ 0 11 +3 c 2 s u 2 x u y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 x u y p i )]TJ/F35 10.9091 Tf 8.484 0 Td [(u y 3 2 h 3 + 1 2 ^ h 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x ^ h 5 ; ^ h 7 = ! 7 4 h c 2 s + u 2 y u x + ^ 0 12 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y ^ 0 11 )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x ^ 0 02 +3 c 2 s u x u 2 y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x u 2 y p i )]TJ/F35 10.9091 Tf 8.484 0 Td [(u x 3 2 h 3 )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(1 2 ^ h 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y ^ h 5 ; ^ h 8 = ! 8 4 h c 6 s + c 2 s + u 2 x c 2 s + u 2 y )]TJ/F15 10.9091 Tf 11.086 2.182 Td [(^ 0 22 +2 u y ^ 0 21 + u x ^ 0 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [( u 2 y ^ 0 20 + u 2 x ^ 0 02 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 u x u y ^ 0 11 +4 c 2 s u 2 x u 2 y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 x u 2 y p i )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 ^ h 3 )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 2 u 2 y ^ h 3 + ^ h 4 )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(1 2 u 2 x ^ h 3 )]TJ/F15 10.9091 Tf 11.021 2.879 Td [(^ h 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 u x u y ^ h 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y ^ h 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x ^ h 7 : .41 Finally,thepost-collisiondistributionfunctionrepresentedby e g canbeobtainedbyexpanding K b h inEq..35,whichreadas e g 0 = g 0 + h ^ h 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 ^ h 3 )]TJ/F15 10.9091 Tf 11.022 2.879 Td [(^ h 8 i + g s 0 ; e g 1 = g 1 + h ^ h 0 + ^ h 1 )]TJ/F15 10.9091 Tf 11.022 2.878 Td [(^ h 3 + ^ h 4 +2 ^ h 7 )]TJ/F15 10.9091 Tf 11.022 2.878 Td [(^ h 8 i + g s 1 ; e g 2 = g 2 + h ^ h 0 + ^ h 2 )]TJ/F15 10.9091 Tf 11.022 2.879 Td [(^ h 3 )]TJ/F15 10.9091 Tf 11.022 2.879 Td [(^ h 4 +2 ^ h 6 )]TJ/F15 10.9091 Tf 11.022 2.879 Td [(^ h 8 i + g s 2 ; e g 3 = g 3 + h ^ h 0 )]TJ/F15 10.9091 Tf 11.021 2.879 Td [(^ h 1 )]TJ/F15 10.9091 Tf 11.022 2.879 Td [(^ h 3 + ^ h 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 ^ h 7 + ^ h 8 i + g s 3 ; e g 4 = g 4 + h ^ h 0 )]TJ/F15 10.9091 Tf 11.021 2.879 Td [(^ h 2 )]TJ/F15 10.9091 Tf 11.022 2.879 Td [(^ h 3 )]TJ/F15 10.9091 Tf 11.022 2.879 Td [(^ h 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 ^ h 6 + ^ h 8 i + g s 4 ; e g 5 = g 5 + h ^ h 0 + ^ h 1 + ^ h 2 +2 ^ h 3 + ^ h 5 )]TJ/F15 10.9091 Tf 11.021 2.879 Td [(^ h 6 )]TJ/F15 10.9091 Tf 11.021 2.879 Td [(^ h 7 + ^ h 8 i + g s 5 ; e g 6 = g 6 + h ^ h 0 )]TJ/F15 10.9091 Tf 11.021 2.879 Td [(^ h 1 + ^ h 2 +2 ^ h 3 )]TJ/F15 10.9091 Tf 11.021 2.879 Td [(^ h 5 )]TJ/F15 10.9091 Tf 11.021 2.879 Td [(^ h 6 + ^ h 7 + ^ h 8 i + g s 6 ; e g 7 = g 7 + h ^ h 0 )]TJ/F15 10.9091 Tf 11.021 2.879 Td [(^ h 1 )]TJ/F15 10.9091 Tf 11.022 2.879 Td [(^ h 2 +2 ^ h 3 + ^ h 5 + ^ h 6 + ^ h 7 + ^ h 8 i + g s 7 ; e g 8 = g 8 + h ^ h 0 + ^ h 1 )]TJ/F15 10.9091 Tf 11.022 2.879 Td [(^ h 2 +2 ^ h 3 )]TJ/F15 10.9091 Tf 11.021 2.879 Td [(^ h 5 + ^ h 6 )]TJ/F15 10.9091 Tf 11.021 2.879 Td [(^ h 7 + ^ h 8 i + g s 8 : .42

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218 sideraD2Q9latticeusingthesameorthogonalmomentbasisvectorsandthematrixgivenin Eqs..31and.32,respectively. Then,thecollisionandstreamingstepsofsuchacascadedLBschemefortheevolutionofthe discretedistributionfunction f canberespectivelyrepresentedas e f x ;t = f x ;t + K b g ; .45a f x ;t + t = e f x )]TJ/F52 10.9091 Tf 10.909 0 Td [(e t ;t : .45b Inordertodesignacascadedcollisionoperatortosolveforthephaseeldvariable described byanconservativeAllen-CahnequationACE,werstdenethefollowingcentralmomentsand rawmomentsofthedistributionfunction f anditsequilibrium f eq ,respectively,as 0 B @ ^ mn ^ eq mn 1 C A = X 0 B @ f f eq 1 C A e x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x m e y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y n ; .46 0 B @ ^ 0 mn ^ eq 0 mn 1 C A = X 0 B @ f f eq 1 C A e m x e n y : .47 Then,weconsiderthecontinuouscentralmomentsoftheequilibria b eq; mn = Z 1 Z 1 f eq x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x m y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y n d x d y .48 bydeningtheequilibriumdistributionfunction f eq inanalogywiththelocalMaxwelldistributionfunctionbyreplacingthedensitywiththephaseeldvariable : f eq f eq ; u ; = 2 c s exp )]TJ/F33 7.9701 Tf 9.68 5.375 Td [( )]TJ/F53 7.9701 Tf 6.587 0 Td [(u 2 2 c 2 s .Here c s isafreeparameter,whichwillberelatedtothecoecientofdiffusivity M later.Typically,weset c 2 s = 1 3 .Therelaxationofthecentralmomentstothecorrespondingequilibriagivenaboveonlymodelsadiusionprocess.Inordertoaccountforthecounteractingphaseseparationuxcomponentsn x andn y appearingintheconservativeACE Eq..1,where n = n x ;n y istheinterfacenormal,wemodifytherstordercontinuouscentralmomentsfrombeingnullto b eq; 10 = M n x and b eq; 01 = M n y .Then,bymatchingofthe

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219 discreteandcontinuouscentralmomentsoftheequilibria,i.e., b eq mn = b eq; mn forallthenineindependentmomentssupportedbythelattice,weobtainthecomponentsof b eq mn as ^ eq 00 = ; ^ eq 10 = M n x ; ^ eq 01 = M n y ; ^ eq 20 = c 2 s ; ^ eq 02 = c 2 s ; ^ eq 11 =0 ; ^ eq 21 =0 ; ^ eq 12 =0 ; ^ eq 22 = c 4 s : Thecascadedcollisionoperatorcanthenbeconstructedbyprescribingtherelaxationofcentral momentsofdierentorderstotheirequilibria,i.e., P K b g e x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x m e y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y n = ! ^ eq mn )]TJ/F15 10.9091 Tf -459.1 -23.996 Td [(^ mn ,whereonlythezerothmomentbeingconserved^ 00 =^ eq 00 = ,and ! arethevarious relaxationtimes.Theresultingchangesinalltheninecomponentsofmomentsundercollision, i.e., ^g =^ g 0 ; ^ g 1 ; ^ g 2 ; ; ^ g 8 canbesummarizedasfollows: ^ g 0 =0 ; ^ g 1 = ! 1 6 h u x + M n x )]TJ/F15 10.9091 Tf 11.324 0 Td [(^ 0 10 i ; ^ g 2 = ! 2 6 h u y + M n y )]TJ/F15 10.9091 Tf 11.324 0 Td [(^ 0 01 i ; ^ g 3 = ! 3 12 h 2 c 2 s )]TJ/F15 10.9091 Tf 10.909 0 Td [( u 2 x + u 2 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(^ 0 20 +^ 0 02 +2 u x 0 10 + u y ^ 0 01 i + u x ^ g 1 + u y ^ g 2 ; ^ g 4 = ! 4 4 h )]TJ/F15 10.9091 Tf 8.485 0 Td [( u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(^ 0 20 )]TJ/F15 10.9091 Tf 11.325 0 Td [(^ 0 02 +2 u x 0 10 )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y ^ 0 01 i +3 u x ^ g 1 )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y ^ g 2 ; ^ g 5 = ! 5 4 h )]TJ/F35 10.9091 Tf 8.485 0 Td [(u x u y )]TJ/F15 10.9091 Tf 11.324 0 Td [(^ 0 11 + u x 0 01 + u y ^ 0 10 i + 3 2 u x ^ g 2 + u y ^ g 1 ; ^ g 6 = ! 6 4 h )]TJ/F35 10.9091 Tf 8.485 0 Td [(u 2 x u y +^ 0 21 )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y 0 20 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x 0 11 +2 u x u y 0 10 + u 2 x 0 01 i +3 u x u y ^ g 1 + 3 2 u 2 x +1 ^ g 2 )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(3 2 u y ^ g 3 )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 2 u y ^ g 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x ^ g 5 ; ^ g 7 = ! 7 4 h )]TJ/F35 10.9091 Tf 8.485 0 Td [(u x u 2 y +^ 0 12 )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x 0 02 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y 0 11 +2 u x u y 0 01 + u 2 y 0 10 i + 3 2 u 2 y +1 ^ g 1 +3 u x u y ^ g 2 )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(3 2 u y ^ g 3 + 1 2 u x ^ g 4 )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 u y ^ g 5 ; ^ g 8 = ! 8 4 h c 4 s )]TJ/F15 10.9091 Tf 11.325 0 Td [(^ 0 22 +2 u x ^ 0 12 + u y ^ 0 21 )]TJ/F15 10.9091 Tf 10.909 0 Td [( u 2 y ^ 0 20 + u 2 x ^ 0 02 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 u x u y ^ 0 11 +2 u x u 2 y ^ 0 10 + u 2 x u y ^ 0 01 )]TJ/F35 10.9091 Tf 8.485 0 Td [(u 2 x u 2 y + u x +3 u x u 2 y ^ g 1 + u y +3 u 2 x u y ^ g 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(+ 3 2 u 2 x + u 2 y ^ g 3 + 1 2 u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y ^ g 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 u x u y ^ g 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y ^ g 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x ^ g 7 ; .49 wheretherelaxationtimesoftherstordermoments ! 1 and ! 2 arerelatedtothemobilityparameter M inEq..1via M = c 2 s 1 ! j )]TJ/F33 7.9701 Tf 12.104 4.295 Td [(1 2 t ;j =1 ; 2,andtherestoftherelaxationtimes aresettounity.Finally,thepost-collisiondistributionfunction e f canbeexplicitlywrittenafter

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220 expanding K b g inEq..45aas e f 0 = f 0 +[^ g 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4^ g 3 )]TJ/F15 10.9091 Tf 11.283 0 Td [(^ g 8 ] ; e f 1 = f 1 +[^ g 0 +^ g 1 )]TJ/F15 10.9091 Tf 11.282 0 Td [(^ g 3 +^ g 4 +2^ g 7 )]TJ/F15 10.9091 Tf 11.282 0 Td [(^ g 8 ] ; e f 2 = f 2 +[^ g 0 +^ g 2 )]TJ/F15 10.9091 Tf 11.282 0 Td [(^ g 3 )]TJ/F15 10.9091 Tf 11.283 0 Td [(^ g 4 +2^ g 6 )]TJ/F15 10.9091 Tf 11.282 0 Td [(^ g 8 ] ; e f 3 = f 3 +[^ g 0 )]TJ/F15 10.9091 Tf 11.282 0 Td [(^ g 1 )]TJ/F15 10.9091 Tf 11.282 0 Td [(^ g 3 +^ g 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2^ g 7 +^ g 8 ] ; e f 4 = f 4 +[^ g 0 )]TJ/F15 10.9091 Tf 11.282 0 Td [(^ g 2 )]TJ/F15 10.9091 Tf 11.282 0 Td [(^ g 3 )]TJ/F15 10.9091 Tf 11.283 0 Td [(^ g 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2^ g 6 +^ g 8 ] ; e f 5 = f 5 +[^ g 0 +^ g 1 +^ g 2 +2^ g 3 +^ g 5 )]TJ/F15 10.9091 Tf 11.282 0 Td [(^ g 6 )]TJ/F15 10.9091 Tf 11.282 0 Td [(^ g 7 +^ g 8 ] ; e f 6 = f 6 +[^ g 0 )]TJ/F15 10.9091 Tf 11.282 0 Td [(^ g 1 +^ g 2 +2^ g 3 )]TJ/F15 10.9091 Tf 11.282 0 Td [(^ g 5 )]TJ/F15 10.9091 Tf 11.282 0 Td [(^ g 6 +^ g 7 +^ g 8 ] ; e f 7 = f 7 +[^ g 0 )]TJ/F15 10.9091 Tf 11.282 0 Td [(^ g 1 )]TJ/F15 10.9091 Tf 11.282 0 Td [(^ g 2 +2^ g 3 +^ g 5 +^ g 6 +^ g 7 +^ g 8 ] ; e f 8 = f 8 +[^ g 0 +^ g 1 )]TJ/F15 10.9091 Tf 11.282 0 Td [(^ g 2 +2^ g 3 )]TJ/F15 10.9091 Tf 11.282 0 Td [(^ g 5 +^ g 6 )]TJ/F15 10.9091 Tf 11.282 0 Td [(^ g 7 +^ g 8 ] : .50 ThisisfollowedbyperformingthestreamingstepshowninEq..45b,whichthenupdatesthe phaseeldvariable viatakingthezerothmomentof f as = X f : .51 7.6ResultsandDiscussion WewillnowpresentavalidationstudyofournewcascadedLBapproachdevelopedforincompressibletwo-phaseowsforavarietyofbenchmarkproblems.SincetheLBformulationforthe interfacecapturingbasedontheconservativeACEhasbeenanalyzedinRef.[233],wewilllimit thevalidationofourimplementationinthisregardforonebenchmarkproblembelowSec.7.6.1. Instead,mostofourfocusinwhatfollowswillbeoninvestigatingthecascadedLBmethodspresentedintheprevioustwosectionsforthecoupledsolutionofthetwo-phaseuidmotionwith interfacialdynamics,especiallyathighdensityratiosandunderdierentinterfacialowcongurations.

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221 7.6.1Evolutionofacircularinterfaceinimposedshearow WewillrstassesstheabilityofthecascadedLBschemeforconservativeACEseeSec.7.5to capturethekinematicaleectsoftheinterfacialmotionunderdeformationandrotationaleects withgooddelity.Inthisregard,weconsideracircularinterfacesubjectedtoanimposedshear owgivenbythefollowingvelocityeldinaperiodicsquaredomainofsize L 0 [237] u x x;y = )]TJ/F35 10.9091 Tf 8.485 0 Td [(U 0 cos[ x=L 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = 2]sin[ y=L 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = 2] u y x;y = U 0 sin[ x=L 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = 2]cos[ y=L 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = 2] ; where U 0 isthevelocityscale.Inoursimulations,wetaketheradiusofthecircularinterfaceto be R = L 0 = 5,whosecenterisinitiallylocatedat x c ;y c = L 0 = 2 ; 3 L 0 = 10inasquarecomputationaldomainresolvedwith L 0 =200.Moreover,thenumericalparametersoftheconservative ACE,i.e.,thewidth W andthemobility M aresetasfollows: W =3andlatterisobtained byconsideringaPecletnumber Pe = U 0 W=M =60.Toguideinterfaceundergoingdeformationandrotationtoreturntoitsoriginalpositionat T =2 T f ,where T f = L 0 =U 0 ,thevelocity eldgivenaboveisreversedat T = T f .Figure7.1presentssnapshotsoftheinterface,identied bythecontoursof A + B = 2attheinstants T =0 ; 0 : 5 T f ;T f ; 1 : 5 T f ; 2 T f .Itcanbeseenthat theinterfaceundergoesadvectionwithcomplexshapechangesundershear,andthecascadedLB methodfaithfullyrecoverstheoriginalcircularshapewithgoodaccuracyaftercompletingacycle. 7.6.2Laplace-Youngrelationofastaticdrop WewillnowmakeaquantitativevericationoftheabilityofthecoupledcascadedLBformulationsinthecomputationofthevariousforcesandtheirbalancesinastaticdropimmersedin auidmediumbyconsideringhighdensityratios.Inthisregard,accordingtheanalyticalpredictionsoftheLaplace-Young'srelation,fora2Ddropatrest,thepressuredierencebetween theitandtheambientuid P isrelatedtothesurfacetension anditsradiusofcurvature

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223 =R via P = =R ,whichwewilluseforcomparison.Inthesimulations,weconsideradrop ofdensity A surroundedbyanambientuidofdensity B andplacedinthecenterofaperiodicsquaredomainresolvedby200 200gridnodes.Werstperformedsimulationswithadrop ofradius R =30byconsideringasurfacetension =1 10 )]TJ/F33 7.9701 Tf 6.586 0 Td [(3 atvariousdensityratiosof A = B =10 ; 100 ; 1000tilltheyreachedequilibrium.Figure7.2showsthesurfacecontoursof thepressuredierencesbetweenthedropandtheambientuid.Itisevidentthatthepressure distributionwithinthedropissmoothanduniformandthejumpacrosstheinterfaceissharp andindependentofthedensityratioasexpected.ThecascadedLBmethodisseentoberobust evenatrelativelyhighdensitycontrasts.Then,Fig.7.3showsacomparisonbetweenthecoma FIGURE7.2:Surfacecontoursofthepressuredistributionofasinglestaticdropofradius R =30atdierentdensityratios A = B withsurfacetension =1 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(3 inaperiodicsquare domain. putedpressuredierencesbetweenthedropandtheambientuidasafunctionofitscurvature forthreedierentvaluesofthesurfacetension =1 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(3 ; 5 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(3 ; 1 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(4 atadensityratioof 1000againstthepredictionsgivenbytheLaplace-Youngrelation.Itveriestheexpectedlinear dependencebetween P and1 =R andthecomputedresultsarefoundtobeingoodquantitative agreementwiththeanalyticalsolution.

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224 a FIGURE7.3:ComparisonofthecomputedpressuredierencessymbolsobtainedusingthecascadedLBmethodagainsttheanalyticalpredictionsusingtheLaplace-Youngrelationforvarious valuesofthedropcurvature1 =R withsurfacetension =5 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(3 ; 1 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(3 ; 1 10 )]TJ/F33 7.9701 Tf 6.586 0 Td [(4 .

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225 7.6.3Rayleigh-Taylorinstability Next,wewillinvestigatethecascadedLBmethodsforsimulationoftheclassicalRayleigh-Taylor R-Tinstability.Suchagravitationalacceleration-driveninstabilityariseswhenaheavieruid ofdensity A isplacedontopofalighteruidofdensity B inthepresenceofgravity,andthe interfacebetweenthetwouidsundergoescomplexunsteadymotion.Ameshsizeof L 4 L , where L =201,isemployed,andperiodicboundaryconditionsalongthelateralverticalsidesand no-slipboundaryconditionsatthetopandbottomboundariesareimposed.Theinitialperturbationattheinterfacebetweenthetwouidstoinitiateinstabilityisdescribedbyacosinusoidal functiongivenby y 0 =2 L +0 : 1 L cos x=L ,wheretheoriginofthecoordinatesystemisxed attheleftbottomcornerofthecomputationaldomain.TheinterfacialinstabilityischaracterizedbytheReynoldsnumberRe= A p gLL= basedonavelocityscale U c = p gL ,andthe AtwoodnumberAt= A )]TJ/F35 10.9091 Tf 11.736 0 Td [( B = A + B .Here, isthedynamicviscosityand g istheaccelerationduetogravity.Thedimensionlesstimescale T isthendenedbasedon U c and L as T = U c = L p At.Inaddition,forinterfacecapturing,weconsider W =5,andthePecletnumber Pe= U c L=M =3000. ByxingAt=0 : 5,weperformedsimulationsfortwocasesoftheReynoldsnumber,i.e.,Re= 256and3000.Figure7.4presentstheevolutionoftheinterfaceunderowinstabilityatthese twoReynoldsnumbers.Ingeneral,thespikeformationbytheheavieruidmovingdownward isaccompaniedbyabubbleofthelighteruidrisingupwards.Theinterfacebetweentheuids undergoescomplexshapechangesleadingtoaroll-upofitstailsunderthedynamicaleectsof thetwomovinguids.Moreover,athigherRe,whentheinertialeectspredominateoverthe viscouseects,smallscaleowstructuresemerge.Thesnapshotsofthesimulatedresultsofthe R-Tinstabilityatvarioustimeinstantsareinoverallagreementwiththepriornumericalresults atRe=256e.g.,[22,238]andRe=3000e.g.,[221,239].Moreover,Fig.7.5showsquantitativecomparisonsofthecomputedvaluesofthenon-dimensionallocationsofthespikeand

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226 a b FIGURE7.4:SnapshotsofsimulationofRayleigh-TaylorinstabilityatAt=0 : 5andaRe= 256andbRe=3000.

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227 bubblefrontsscaledby L atbothReagainstpriornumericalreferencedata.Itcanbethatthe numericalresultsobtainedusingthecascadedLBformulationsfortimeevolutionoftheinterfacelocationsevaluatedatthecenterspikeandattheedgesbubbleareingoodquantitative agreementwiththerespectivereferenceresultsatbothRe=256andRe=3000. a b FIGURE7.5:TimeevolutionofthepositionsofthebubblefrontandthespiketipforRayleighTaylorinstabilityatAt=0 : 5andaRe=256andbRe=3000. 7.6.4Fallingdropundergravity Wewillnowconsideranotherunsteadytwo-phaseowprobleminvolvingadropfallingundera gravitationaleld.Insuchacase,duringthedescentofthedrop,itundergoessignicantshape changesduetodeformation,whicharisesfromacomplexinterplaybetweenthegravityforforce, surfacetensionforceandtheviscousforce.Adropofdiameter D =30withadensity A is placedinitiallyatalocationof ; 300inarectangulardomainthatisdividedinto151 451 latticenodeswiththeoriginofthecoordinatesystembeinglocatedattheleftbottomcorner, andlledwithalighterambientuidofdensity B .Free-slipboundaryconditionsareimposed onthetopandbottomboundariesandlateralverticalsidesaretakentobeperiodic.Forthis

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228 computationalsetup,thegravitationalforceisappliedeverywherebysetting F ext = )]TJ/F15 10.9091 Tf 8.485 0 Td [( )]TJ/F35 10.9091 Tf 8.981 0 Td [( B g j . Thedropdynamicsischaracterizedbythefollowingnon-dimensionalnumbers:EotvosnumberEo= g A )]TJ/F35 10.9091 Tf 12.195 0 Td [( B D 2 = representingthegravityforcerelativetothesurfacetensionand theOhnesorgenumberOh= A = p A D representingtheviscouseects.FollowingRef.[240], wex A = B =5,Eo=43andstudytheinuenceofOhbyconsideringOh=0 : 3 ; 0 : 7and 1 : 0,with A = B = .ThesethreevaluesofOhareobtainedbysetting =0 : 1 ; 0 : 2333and 0 : 3333,respectively.Forreportingresults,theinstantaneoustime t isnon-dimensionalizedas T = t= p D=g . Figure7.6presentsthesnapshotsoftheevolutionoftheinterfaceofthefallingdropfortheabove threecasesofOh.Ingeneral,itcanbeseenthatas Oh increases,theviscousforceincreasesrelativetothesurfacetensionforceandhencethedropdeformationisreduced.Thus,atalarge valueofOh=1 : 0,thedropundergoesrelativesmalldeformationattainingasteadystate,while atOh=0 : 7,itisstretchedmorealongthehorizontaldirectionbythesurfacetensionforceafterinitiallytakinganellipsoidalshape.Ontheotherhand,atastilllowerOh=0 : 3,thedrop becomesconsiderablyslenderalongthesides,whileexhibitingbag-likeshapeduetoshearunder gravityinthepresenceoftheprevailingsurfacetensionforcewithsmallerviscousforceeects atlaterstages.ThesecomputeddropshapevariationsatdierenttimeswithOhareconsistent withthendingsreportedinRef.[240]. 7.6.5Buoyancy-drivenrisingbubble Next,weexaminetheabilityofourcascadedLBformulationstosimulateawell-denedtwophaseowprobleminvolvingamovingdispersedphaseinacontinuousphasewithhighdensity contraststhanthoseconsideredintheprevioustwocases.Inthisregard,weconsiderabubbleof diameter D anddensity B risinginanambientuidofdensity A ,with A = B being1000,by buoyancyforcesundervariousparametricconditions.Thisrepresentsthebuoyantmotionofan

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229 a FIGURE7.6:EvolutionofadeformingdropfallingundergravityforvariousvaluesoftheOhnesorgenumberOhof0.3,0.7and1.0ataxedEotvosnumberEo=43shownattimeinstants T =0 ; 2 : 04 ; 3 : 05 ; 4 : 07 ; 5 : 09 ; 6 : 11 ; 7 : 13 ; 8 : 14,and9 : 16fromtoptobottom.

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230 airbubbleinwaterandisofpracticalinterest.Ourgoalistotesttherobustnessofthecascaded LBapproachtocapturethevariousshapechangesthebubbleundergoesduetothebalancebetweenthedierentcompetingforcesaswellassimulatethetimehistoryofthebubblepathwith quantitativeaccuracy. Thecomputationalcongurationconsistsofarectangulardomainwithagridresolutionof161 481inwhichabubbleofdiameterresolvedwith64gridnodesisinitiallycenteredatalocation ; 120withthecoordinatesystem'soriginbeingsituatedatthebottomleftcornerof thedomain.Freeslipboundaryconditionsareimposedonthetwoverticalsidesandthenoslipconditionsareconsideredonthetopandbottomboundaries.Thissetupcorrespondsto thatdiscussedinRefs.[241,242].Thebubbleissetinmotionbyapplyingabodyforcegivenby F ext = )]TJ/F15 10.9091 Tf 8.485 0 Td [( )]TJ/F35 10.9091 Tf 11.919 0 Td [( A g j .Thecharacteristicscalesofthistwo-phaseowproblemare:thelength scale L = D ,thevelocityscale U g = p gD ,whichrepresentsthegravitationalvelocity,and thetimescale T = L=U g .Basedontheseandthevariouscompetingforcesi.e.,buoyancy,viscousandsurfacetension,thenon-dimensionalparametersofthistwo-phaseowproblemarethe ReynoldsnumberRe= A U g D= A andtheEotvosnumberEo= A U 2 g D= ,alongwiththe ratiosoftheuidproperties A = B and A = B .Thenon-dimensionaltimeforreportingtime historiesisrepresentedby t = t=T .Dependingonthemagnitudesofthesedimensionlessgroups, thebubbleundergoescomplexinterfacialshapechanges,attainingeitherspherical-cap,dimpled ellipsoidal-caporskirtedcongurations,amongvariouspossibilities[243]. Bysetting A = B =1000and A = B =100ataxedReynoldsnumberRe=35,weperformed buoyancy-drivenbubblerisesimulationsatvariousvaluesoftheEotvosnumbersEo=10 ; 50 and125asinRefs.[242,238]usingthecascadedLBmethods.Figure7.7presentsthecomputedevolutionoftheinterfaceoftherisingbubbleatthesethreevaluesofEo.Whentheroleof thesurfacetensionforceisrelativelysignicantincomparisonwiththeotherforces,aswhenthe EotvosnumberislowEo=10,thebubbleundergoessmallerdeformationthatisinitiatedatits rearend,whichthenresultsinaatteningofthatsideasthebubblerisesFortheintermediate

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231 caseEo=35,thedrivingbuoyancyforcepredominatesthesurfacetensionundertheprevailing viscousforce,resultinginamuchlargerdeformationbystretchingthatleadstotheformationof tailsthatelongatesatlatertimes.AtevenhigherEo=125,thisprocessismorepronouncedand theskirtedshapeaccompaniedbythepairoftailsisfurtherelongatedandstraightened.These computedshapevariationswithdierentEoatvarioustimeareverysimilarwiththeresults basedonothermethods[242,238].Furthermore,inordertomakeaquantitativecomparison, wethencomputetheverticalcoordinateofthecenterofmassoftherisingbubbleasitundergoesshapechangesusing y c = R b ydx= R b 1 dx ,where b representstheregionoccupiedbythe bubble,forthecaseRe=35andEo=125.Figure7.8showsthenon-dimensionalcenterofmass asafunctionofthenon-dimensionaltimecomputedusingthecascadedLBschemesagainstthe referencenumericalresultsfromRef.[238].Itisevidentthatourapproachisingoodquantitativeagreementwiththeavailablenumericaldataforthetemporalevolutionofthebubblepaths, therebyverifyingitsaccuracyandrobustnessforthishighdensityratiotwo-phaseowproblem. 7.6.6Impactofadroponathinliquidlayer Asanalbenchmarkproblem,weconsideraninertia-driventwo-phaseowproblematahigh densityratio,i.e.,theimpactofacirculardroponathinlayerofuidandthestudyofitssubsequentoutcomes.Suchimpactdynamicsofdropsleadstoarichvarietyofoutcomesdepending onthecharacteristicparametersrepresentingtheratiosofvariousattendantforces[244].The computationalsetupconsideredforthisexampleisdescribedinRef.[245].Boththedropand thethinlayerareconsideredtobetheofthesameliquidofdensity A andtheambientuidis ofdensity B .Weconsiderahighdensityratio A = B =1000torepresenttheimpactofawaterdropsurroundedbyair.Thecomputationaldomainisresolvedwith501 1501gridnodes, inwhichtheliquidlayerisdiscretizedby150gridnodes,whilethedropradius R isrepresented by100meshnodes.Theinterfacethickness W issettobe5.Weimposeperiodicconditionson thetwoverticalboundaries,no-slipboundaryconditiononthebottomwall,andfree-slipcon-

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232 aEo=10 bEo=50 cEo=125 FIGURE7.7:Evolutionoftheinterfaceofabuoyancy-drivenrisingbubbleatRe=35anda Eo=10,bEo=50,cEo=125.

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233 FIGURE7.8:Timehistoryofthenon-dimensionalcenterofmassofabuoyancy-drivenrising bubbleatRe=35andEo=125.

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235 a b FIGURE7.9:EvolutionofthesplashingofadroponathinlmatWe=8000and A = B = 1000foraRe=20bRe=100.

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236 liquidcylinderinanotherambientlighteruid.TheSRTformulationfortwo-phaseowsused forcomparisonisbasedononeSRTLBsolverobtainedasadiscretizationoftheMCBEfortwophaseuidmotionandanotherSRTLBschemeforcapturinginterfacialdynamicsrepresented bytheconservativeACE.Weconsideraperiodicdomainofresolution200 200inwhichaliquid cylinderofdensity A isplacedinanotherlighterambientuidofdensity B ,where A = B for simplicity,undergoesfreeoscillations.Theoscillationsareinitiatedfromaninitiallyellipticcongurationofthecylindersemi-majoraxis a =25andsemi-minoraxis b =15viathecapillary eectsonitsinterface.Figure7.10showsatypicalexampleoftheevolutionoftheinterfaceof theliquidcylinderundergoingfreeoscillations.Now,employingeachofthetwocollisionmodels, a FIGURE7.10:Evolutionoftheinterfaceofanoscillatingliquidcylinderstartingfromaninitial ellipticshapecongurationwithsemi-majoraxis a =25andsemi-minoraxis b =15;surface tensionparameter~ =0 : 1,kinematicviscosity A = B =0 : 01anddensityratio A = B =100. fortheaboveinitialgeometriccongurationoftheliquidcylinderwithsurfacetensionparameter~ =0 : 01,andforfoursetsofvaluesofthedensityratios A = B =500 ; 600 ; 800and900,the kinematicviscosityoftheuids A = B aregraduallyreducedtillthesimulationsbecomesunstable.Figure7.11reportstheratiosoftheminimumachievableviscositiesforSRTandcascaded

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239 acentralmomentsbasedformulationisdemonstrated.

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265 APPENDIXA 1.1StrangSplittingImplementationofBodyForcesin3DCentralMomentLB Method Fortheproposeofillustration,wewillconsiderthe3DcentralmomentLBmethodusingthe three-dimensional,fteenvelocityD3Q15[125]lattice,butcanbereadilyextendedforother latticessuchastheD3Q27lattice.Thecomponentsoftheparticlevelocityvectorsalongwith the j 1 i vectorwhichisusedtorepresentthezerothmomentwiththedistributionfunctionfor thislatticeare j e x i = ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0 ; 0 ; 0 ; 0 ; 1 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 y ; j e y i = ; 0 ; 0 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0 ; 0 ; 1 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 y ; j e z i = ; 0 ; 0 ; 0 ; 0 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 1 ; 1 ; 1 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 y ; j 1 i = ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 y : .1 Thecorrespondinglinearlyindependentorthogonalbasisvectorsaregivenby[125] j K 0 i = j 1 i ; j K 1 i = j e x i ; j K 2 i = j e y i ; j K 3 i = j e z i ; j K 4 i = j e x e y i ; j K 5 i = j e x e z i ; j K 6 i = j e y e z i ; j K 7 i = j e 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(e 2 y i ; j K 8 i = j e 2 x + e 2 y + e 2 z i)]TJ/F15 10.9091 Tf 17.575 0 Td [(3 j e 2 z i ; j K 9 i = j e 2 x + e 2 y + e 2 z i)]TJ/F15 10.9091 Tf 17.575 0 Td [(2 j 1 i ; j K 10 i =5 j e x e 2 x + e 2 y + e 2 z i)]TJ/F15 10.9091 Tf 17.576 0 Td [(13 j e x i ; j K 11 i =5 j e y e 2 x + e 2 y + e 2 z i)]TJ/F15 10.9091 Tf 17.575 0 Td [(13 j e y i ; j K 12 i =5 j e z e 2 x + e 2 y + e 2 z i)]TJ/F15 10.9091 Tf 17.576 0 Td [(13 j e z i ; j K 13 i = j e x e y e z i ; j K 14 i =30 j e 2 x e 2 y + e 2 x e 2 z + e 2 y e 2 z i)]TJ/F15 10.9091 Tf 17.575 0 Td [(40 j e 2 x + e 2 y + e 2 z i +32 j 1 i : .2

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266 Then,theorthogonalmatrix K followsas K =[ j K 0 i ; j K 1 i ; j K 2 i ; j K 3 i ; j K 4 i ; j K 5 i ; j K 6 i ; j K 7 i ; j K 8 i j K 9 i ; j K 10 i ; j K 11 i ; j K 12 i ; j K 13 i ; j K 14 i ] ; .3 whichmapsthechangeofmomentsundercollisionsbacktothechangesinthedistributionfunctions.Thecentralmomentsandrawmomentsofthedistributionfunctionanditsequilibriumof order m + n + p aredened,respectively,as 0 B @ ^ x m y n z p ^ eq x m y n z p 1 C A = X 0 B @ f f eq 1 C A e x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x m e y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y n e z )]TJ/F35 10.9091 Tf 10.909 0 Td [(u z p ; .4 and 0 B @ ^ 0 x m y n z p ^ eq 0 x m y n z p 1 C A = X 0 B @ f f eq 1 C A e m x e n y e p z : .5 Thecentralmomentequilibriausedfortheconstructionofthe3Dcascadedcollisionoperatorfor theD3Q15latticeispresentedin[125].Thecollideandstreamstepsofthe3Dcascadedmethod areformallyrepresentedinEqs..22aand.22b,respectively.Owingtothemassandmomentumbeingcollisioninvariants,itfollowsthat b g 0 = b g 1 = b g 2 = b g 3 =0.Forthenon-conserved

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267 moments,thechangeofmomentsundercascadedcollisionaregivenby b g 4 = ! 4 8 h )]TJ/F41 10.9091 Tf 8.597 0 Td [(b 0 xy + u x u y i ; b g 5 = ! 5 8 h )]TJ/F41 10.9091 Tf 8.597 0 Td [(b 0 xz + u x u z i ; b g 6 = ! 6 8 h )]TJ/F41 10.9091 Tf 8.597 0 Td [(b 0 yz + u y u z i ; b g 7 = ! 7 4 h )]TJ/F15 10.9091 Tf 8.485 0 Td [( b 0 xx )]TJ/F41 10.9091 Tf 11.021 0 Td [(b 0 yy + u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y i ; b g 8 = ! 8 12 h )]TJ/F15 10.9091 Tf 8.485 0 Td [( b 0 xx + b 0 yy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b 0 zz + u 2 x + u 2 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u 2 z b g 9 = ! 9 18 h )]TJ/F15 10.9091 Tf 8.485 0 Td [( b 0 xx + b 0 yy + b 0 zz + u 2 x + u 2 y + u 2 z i : b g 10 = ! 10 16 h )]TJ/F41 10.9091 Tf 8.597 0 Td [(b 0 xyy +2 u y b 0 xy + u x b 0 yy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x u 2 y i + u y b g 4 + 1 8 u x )]TJ/F41 10.9091 Tf 8.555 0 Td [(b g 7 + b g 8 +3 b g 9 ; b g 11 = ! 11 16 h )]TJ/F41 10.9091 Tf 8.597 0 Td [(b 0 xxy +2 u x b 0 xy + u y b 0 xx )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u 2 x u y i + u x b g 4 + 1 8 u y b g 7 + b g 8 +3 b g 9 ; b g 12 = ! 12 16 h )]TJ/F41 10.9091 Tf 8.597 0 Td [(b 0 xxz +2 u x b 0 xz + u z b 0 xx )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 u 2 x u z i + u x b g 5 + 1 8 u z b g 7 + b g 8 +3 b g 9 ; b g 13 = ! 13 8 h )]TJ/F41 10.9091 Tf 8.598 0 Td [(b 0 xyz + u x b 0 yz + u y b 0 xz + u z b 0 xy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x u y u z i + u z b g 4 + u y b g 5 + u x b g 6 ; b g 14 = ! 14 16 h )]TJ/F41 10.9091 Tf 8.597 0 Td [(b 0 xxyy +2 u x b 0 xyy +2 u y b 0 xxy )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 x b 0 yy )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y b 0 xx )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 u x u y b 0 xy + e b xx e b yy +3 u 2 x u 2 y i )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x u y b g 4 + 1 8 u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y b g 7 + 1 8 )]TJ/F35 10.9091 Tf 8.484 0 Td [(u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y b g 8 + 3 8 )]TJ/F35 10.9091 Tf 8.485 0 Td [(u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 2 b g 9 +2 u x b g 10 +2 u y b g 11 ; .6 Theoutputvelocityeld u 0 = u o x ;u o y ;u o z isobtainedfollowingthestreamingstepas u o x = P 14 =0 f e x ;u o y = P 14 =0 f e y u o z = P 14 =0 f e z : .7 Asinthe2Dcase,thepre-collisionforcingstep F 1 = 2 involvesthefollowingupdatetothevelocity eld: u x = 1 u o x + F x 2 t ;u y = 1 u o y + F y 2 t ;u z = 1 u o z + F z 2 t ; .8 whichwillbeusedinthedeterminationofthecascadedcollisionbasedchangeofdierentmoments,i.e. b g ,where =4 ; 5 ; ; 14asgiveninEq..6.Analogously,theotherpost-collision step F 1 = 2 inthesymmetrizedoperatorsplittingcanbewrittenas u p x = u x + F x 2 t;u p y = u y + F y 2 t;u p z = u z + F z 2 t; .9

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268 which,viaEq..8,readsas u p x = u o x + F x t;u p y = u o y + F y t;u p z = u o z + F z t: .10 Inordertoeectivelyintroducethiseectintothe3Dcascadedformulation,wetaketherst ordermomentsofthepost-collisiondistributionfunction f p = f + K b g ,whichyields u p x = f p e x = f e x + h K j e x i b g ; .11a u p y = f p e y = f e y + h K j e y i b g ; .11b u p z = f p e z = f e z + h K j e z i b g : .11c Basedontheorthogonalbasisvectors K giveninEq..2,itfollowsthat h K j e x i g =10 b g 1 ; h K j e y i g =10 b g 2 ; h K j e z i g =10 b g 3 : .12 UsingEqs..11a-.11calongwithEqs..7and.12andcomparingwith.10,weobtainthefollowingresultforthechangeofrstordermomentsduetotheforceeld: b g 1 = F x 10 t; b g 2 = F y 10 t; b g 3 = F z 10 t: .13 Finally,usingEq..13andEq..6forthechangeofmomentsundercascadedcollisionin K b g andexpandingit,wegettheexpressionsforthepostcollision-distributionfunction,which

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269 readas f p 0 = f 0 +[ b g 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 9 +32 b g 14 ] ; f p 1 = f 1 +[ b g 0 + b g 1 + b g 7 + b g 8 )]TJ/F41 10.9091 Tf 10.98 0 Td [(b g 9 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 b g 10 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 b g 14 ] ; f p 2 = f 2 +[ b g 0 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 1 + b g 7 + b g 8 )]TJ/F41 10.9091 Tf 10.98 0 Td [(b g 9 +8 b g 10 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 b g 14 ] ; f 3 = f 3 +[ b g 0 + b g 2 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 7 + b g 8 )]TJ/F41 10.9091 Tf 10.98 0 Td [(b g 9 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 b g 11 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 b g 14 ] ; f p 4 = f 4 +[ b g 0 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 2 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 7 + b g 8 )]TJ/F41 10.9091 Tf 10.98 0 Td [(b g 9 +8 b g 11 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 b g 14 ] ; f p 5 = f 5 +[ b g 0 + b g 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 8 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 9 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 b g 12 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 b g 14 ] ; f p 6 = f 6 +[ b g 0 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 8 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 9 +8 b g 12 )]TJ/F15 10.9091 Tf 10.91 0 Td [(8 b g 14 ] ; f p 7 = f 7 +[ b g 0 + b g 1 + b g 2 + b g 3 + b g 4 + b g 5 + b g 6 + b g 9 +2 b g 10 +2 b g 11 +2 b g 12 + b g 13 +2 b g 14 ] ; f p 8 = f 8 +[ b g 0 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 1 + b g 2 + b g 3 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 4 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 5 + b g 6 + b g 9 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 10 +2 b g 11 +2 b g 12 )]TJ/F41 10.9091 Tf 8.555 0 Td [(b g 13 +2 b g 14 ] ; f p 9 = f 9 +[ b g 0 + b g 1 )]TJ/F41 10.9091 Tf 10.98 0 Td [(b g 2 + b g 3 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 4 + b g 5 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 6 + b g 9 +2 b g 10 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 11 +2 b g 12 )]TJ/F41 10.9091 Tf 8.555 0 Td [(b g 13 +2 b g 14 ] ; f p 10 = f 10 +[ b g 0 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 1 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 2 + b g 3 + b g 4 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 5 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 6 + b g 9 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 10 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 11 +2 b g 12 + b g 13 +2 b g 14 ] ; f p 11 = f 11 +[ b g 0 + b g 1 + b g 2 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 3 + b g 4 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 5 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 6 + b g 9 +2 b g 10 +2 b g 11 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 12 )]TJ/F41 10.9091 Tf 8.555 0 Td [(b g 13 +2 b g 14 ] ; f p 12 = f 12 +[ b g 0 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 1 + b g 2 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 3 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 4 + b g 5 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 6 + b g 9 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 10 +2 b g 11 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 12 )]TJ/F41 10.9091 Tf 8.555 0 Td [(b g 13 +2 b g 14 ] ; f p 13 = f 13 +[ b g 0 + b g 1 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 2 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 3 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 4 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 5 + b g 6 + b g 9 +2 b g 10 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 11 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 12 + b g 13 +2 b g 14 ] ; f p 14 = f 14 +[ b g 0 )]TJ/F41 10.9091 Tf 10.98 0 Td [(b g 1 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 2 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 3 + b g 4 + b g 5 + b g 6 + b g 9 )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 b g 10 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 11 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 12 )]TJ/F41 10.9091 Tf 8.555 0 Td [(b g 13 +2 b g 14 ] : .14

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271 APPENDIXB 2.1Structureofthe3DCentralMoment-basedCollisionKernelforFluidFlowusingaD3Q15Lattice Thedetailsofthederivationofthe3DcascadedLBMforuidmotionwithforcingtermsusing theD3Q15latticeispresentedin[125].Here,wesummarizethemainresultsforcompleteness andforcomparisonwiththecorresponding3DcascadedLBmodelforthesolutionofthetransportofascalareldrepresentedbytheCDE.Thecollisionkernel b g = b g 0 ; b g 1 ; ; b g 14 inthe3D cascadedLBEforeldowgiveninEq..2adependsonthefollowingsetofmoments: 0 B B B B B B B B @ ^ 0 x m y n z p ^ eq 0 x m y n z p ^ 0 x m y n z p ^ 0 x m y n z p 1 C C C C C C C C A = X 0 B B B B B B B B @ f f eq S f 1 C C C C C C C C A e m x e n y e p z : .1 where ^ 0 x m y n z p =^ 0 x m y n z p )]TJ/F33 7.9701 Tf 12.837 4.295 Td [(1 2 ^ 0 x m y n z p .Thespecicexpressionsforrawmomentsofthesource term^ 0 x m y n z p aswellasthecorrespondingsourcetermsinthevelocityspace S representingthe eectofabodyforceispresentedin[125].Usingthenotation ^ 0 x m y n z p =^ 0 x m y n z p +^ 0 x m y n z p and byprescribingacentralmomentrelaxationatdierentordersfortheD3Q15lattice,thestructureofthecollisionkernelcomponentsfor b g canbeexpressedassee[125]fordetails

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272 b g 0 = b g 1 = b g 2 = b g 3 =0 ; .2 b g 4 = ! 4 8 )]TJ/F41 10.9091 Tf 8.359 2.182 Td [(b 0 xy + u x u y + 1 2 b 0 x u y + b 0 y u x ; .3 b g 5 = ! 5 8 )]TJ/F41 10.9091 Tf 8.359 2.182 Td [(b 0 xz + u x u z + 1 2 b 0 x u z + b 0 z u x ; .4 b g 6 = ! 6 8 )]TJ/F41 10.9091 Tf 8.359 2.182 Td [(b 0 yz + u y u z + 1 2 b 0 y u z + b 0 z u y ; .5 b g 7 = ! 7 4 h )]TJ/F15 10.9091 Tf 8.485 0 Td [( b 0 xx )]TJ/F41 10.9091 Tf 10.783 2.182 Td [(b 0 yy + u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y + b 0 x u x )]TJ/F41 10.9091 Tf 11.191 0 Td [(b 0 y u y i ; .6 b g 8 = ! 8 12 h )]TJ/F15 10.9091 Tf 8.485 0 Td [( b 0 xx + b 0 yy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b 0 zz + u 2 x + u 2 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u 2 z + b 0 x u x + b 0 y u y )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b 0 z u z i ; .7 b g 9 = ! 9 18 h )]TJ/F15 10.9091 Tf 8.485 0 Td [( b 0 xx + b 0 yy + b 0 zz + u 2 x + u 2 y + u 2 z + b 0 x u x + b 0 y u y + b 0 z u z + i ; .8 b g 10 = ! 10 16 )]TJ/F41 10.9091 Tf 8.359 2.181 Td [(b 0 xyy +2 u y b 0 xy + u x b 0 yy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x u 2 y )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(1 2 b 0 x u 2 y )]TJ/F41 10.9091 Tf 11.192 0 Td [(b 0 y u y u x + u y b g 4 + 1 8 u x )]TJ/F41 10.9091 Tf 8.555 0 Td [(b g 7 + b g 8 +3 b g 9 ; .9 b g 11 = ! 11 16 )]TJ/F41 10.9091 Tf 8.359 2.182 Td [(b 0 xxy +2 u x b 0 xy + u y b 0 xx )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u 2 x u y )]TJ/F15 10.9091 Tf 12.104 7.38 Td [(1 2 b 0 y u 2 x )]TJ/F41 10.9091 Tf 11.191 0 Td [(b 0 x u x u y + u x b g 4 + 1 8 u y b g 7 + b g 8 +3 b g 9 ; .10 b g 12 = ! 12 16 )]TJ/F41 10.9091 Tf 8.359 2.182 Td [(b 0 xxz +2 u x b 0 xz + u z b 0 xx )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u 2 x u z )]TJ/F15 10.9091 Tf 12.104 7.381 Td [(1 2 b 0 z u 2 x )]TJ/F41 10.9091 Tf 11.191 0 Td [(b 0 x u x u z + u x b g 5 + 1 8 u z b g 7 + b g 8 +3 b g 9 ; .11 b g 13 = ! 13 8 )]TJ/F41 10.9091 Tf 8.359 2.182 Td [(b 0 xyz + u x b 0 yz + u y b 0 xz + u z b 0 xy )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 u x u y u z )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(1 2 b 0 x u y u z + b 0 y u x u z + b 0 z u x u y i + u z b g 4 + u y b g 5 + u x b g 6 ; .12 b g 14 = ! 14 16 h )]TJ/F41 10.9091 Tf 8.359 2.182 Td [(b 0 xxyy +2 u x b 0 xyy +2 u y b 0 xxy )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 x b 0 yy )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y b 0 xx )]TJ/F15 10.9091 Tf 10.91 0 Td [(4 u x u y b 0 xy + e b xx e b yy +3 u 2 x u 2 y + b 0 x u x u 2 y + b 0 y u y u 2 x i )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x u y b g 4 + 1 8 u 2 x )]TJ/F35 10.9091 Tf 10.91 0 Td [(u 2 y b g 7 + 1 8 )]TJ/F35 10.9091 Tf 8.485 0 Td [(u 2 x )]TJ/F35 10.9091 Tf 10.91 0 Td [(u 2 y b g 8 + 3 8 )]TJ/F35 10.9091 Tf 8.485 0 Td [(u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y )]TJ/F15 10.9091 Tf 12.104 7.381 Td [(1 2 b g 9 +2 u x b g 10 +2 u y b g 11 ; .13 where ! 4 ;! 5 ;:::;! 14 aretherelaxationparameters
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273 simulatedthrough = c 2 s 1 ! j )]TJ/F33 7.9701 Tf 12.852 4.295 Td [(1 2 ,where j =5 ; ; 9.Therestoftherelaxationparameters, whichinuencenumericalstability,aresettounityinthepresentwork.Finally,byexpanding theproduct K : b g inEq..2a,thepost-collisionvaluesofthedistributionfunctionaregivenby e f 0 = f 0 +[ b g 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 9 +32 b g 14 ]+ S 0 ; e f 1 = f 1 +[ b g 0 + b g 1 + b g 7 + b g 8 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 9 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 b g 10 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 b g 14 ]+ S 1 ; e f 2 = f 2 +[ b g 0 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 1 + b g 7 + b g 8 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 9 +8 b g 10 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 b g 14 ]+ S 2 ; e f 3 = f 3 +[ b g 0 + b g 2 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 7 + b g 8 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 9 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 b g 11 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 b g 14 ]+ S 3 ; e f 4 = f 4 +[ b g 0 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 2 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 7 + b g 8 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 9 +8 b g 11 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 b g 14 ]+ S 4 ; e f 5 = f 5 +[ b g 0 + b g 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 8 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 9 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 b g 12 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 b g 14 ]+ S 5 ; e f 6 = f 6 +[ b g 0 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 8 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 9 +8 b g 12 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 b g 14 ]+ S 6 ; e f 7 = f 7 +[ b g 0 + b g 1 + b g 2 + b g 3 + b g 4 + b g 5 + b g 6 + b g 9 +2 b g 10 +2 b g 11 +2 b g 12 + b g 13 +2 b g 14 ]+ S 7 ; e f 8 = f 8 +[ b g 0 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 1 + b g 2 + b g 3 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 4 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 5 + b g 6 + b g 9 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 10 +2 b g 11 +2 b g 12 )]TJ/F41 10.9091 Tf 8.555 0 Td [(b g 13 +2 b g 14 ]+ S 8 ; e f 9 = f 9 +[ b g 0 + b g 1 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 2 + b g 3 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 4 + b g 5 )]TJ/F41 10.9091 Tf 10.98 0 Td [(b g 6 + b g 9 +2 b g 10 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 11 +2 b g 12 )]TJ/F41 10.9091 Tf 8.555 0 Td [(b g 13 +2 b g 14 ]+ S 9 ; e f 10 = f 10 +[ b g 0 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 1 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 2 + b g 3 + b g 4 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 5 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 6 + b g 9 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 10 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 11 +2 b g 12 + b g 13 +2 b g 14 ]+ S 10 ; e f 11 = f 11 +[ b g 0 + b g 1 + b g 2 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 3 + b g 4 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 5 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 6 + b g 9 +2 b g 10 +2 b g 11 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 12 )]TJ/F41 10.9091 Tf 8.555 0 Td [(b g 13 +2 b g 14 ]+ S 11 ; e f 12 = f 12 +[ b g 0 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 1 + b g 2 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 3 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 4 + b g 5 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 6 + b g 9 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 10 +2 b g 11 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 12 )]TJ/F41 10.9091 Tf 8.555 0 Td [(b g 13 +2 b g 14 ]+ S 12 ; e f 13 = f 13 +[ b g 0 + b g 1 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 2 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 3 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 4 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 5 + b g 6 + b g 9 +2 b g 10 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 11 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 12 + b g 13 +2 b g 14 ]+ S 13 ;

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274 e f 14 = f 14 +[ b g 0 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 1 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 2 )]TJ/F41 10.9091 Tf 10.979 0 Td [(b g 3 + b g 4 + b g 5 + b g 6 + b g 9 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 10 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 11 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b g 12 )]TJ/F41 10.9091 Tf 8.555 0 Td [(b g 13 +2 b g 14 ]+ S 14 : .14 Then,afterperformingthestreamingstepasgiveninEq..2b,wegettheupdateddistribution functionfromwhichthevelocityeld u canbecomputedasshowninEq..3.

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275 APPENDIXC 3.1SourceTermsforthe3DCascadedLBEforScalarFieldusingD3Q15Lattice Usingthesourcemomentsprojectedtotheorthogonalbasisvectors b m s; denedinEq..20 andinvertingitbyusing S = K )]TJ/F33 7.9701 Tf 6.586 0 Td [(1 b m s; ,andexploitingtheorthogonalityofthecollisionmatrix K ,wegetfollowingexpressionsforthesourcetermsinthevelocityspacefortheD3Q15lattice usedinthesolutionofthe3DCDEas S 0 = 1 45 h 3 b m s; 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 b m s; 9 + b m s; 14 i ; S 1 = 1 180 h 12 b m s; 0 +18 b m s; 1 +45 b m s; 7 +15 b m s; 8 )]TJ/F15 10.9091 Tf 10.909 0 Td [(10 b m s; 9 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 b m s; 10 )]TJ/F41 10.9091 Tf 12.668 0 Td [(b m s; 14 i ; S 2 = 1 180 h 12 b m s; 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(18 b m s; 1 +45 b m s; 7 +15 b m s; 8 )]TJ/F15 10.9091 Tf 10.909 0 Td [(10 b m s; 9 +9 b m s; 10 )]TJ/F41 10.9091 Tf 12.668 0 Td [(b m s; 14 i ; S 3 = 1 180 h 12 b m s; 0 +18 b m s; 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(45 b m s; 7 +15 b m s; 8 )]TJ/F15 10.9091 Tf 10.909 0 Td [(10 b m s; 9 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 b m s; 11 )]TJ/F41 10.9091 Tf 12.668 0 Td [(b m s; 14 i ; S 4 = 1 180 h 12 b m s; 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(18 b m s; 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(45 b m s; 7 +15 b m s; 8 )]TJ/F15 10.9091 Tf 10.909 0 Td [(10 b m s; 9 +9 b m s; 11 )]TJ/F41 10.9091 Tf 12.668 0 Td [(b m s; 14 i ; S 5 = 1 180 h 12 b m s; 0 +18 b m s; 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(30 b m s; 8 )]TJ/F15 10.9091 Tf 10.909 0 Td [(10 b m s; 9 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 b m s; 12 )]TJ/F41 10.9091 Tf 12.668 0 Td [(b m s; 14 i ; S 6 = 1 180 h 12 b m s; 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(18 b m s; 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(30 b m s; 8 )]TJ/F15 10.9091 Tf 10.909 0 Td [(10 b m s; 9 +9 b m s; 12 )]TJ/F41 10.9091 Tf 12.668 0 Td [(b m s; 14 i ; S 7 = 1 720 h 48 b m s; 0 +72 b m s; 1 +72 b m s; 2 +72 b m s; 3 +90 b m s; 4 +90 b m s; 5 +90 b m s; 6 +40 b m s; 9

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276 +9 b m s; 10 +9 b m s; 11 +9 b m s; 12 +90 b m s; 13 + b m s; 14 i ; S 8 = 1 720 h 48 b m s; 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(72 b m s; 1 +72 b m s; 2 +72 b m s; 3 )]TJ/F15 10.9091 Tf 10.91 0 Td [(90 b m s; 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(90 b m s; 5 +90 b m s; 6 +40 b m s; 9 )]TJ/F15 10.9091 Tf 8.484 0 Td [(9 b m s; 10 +9 b m s; 11 +9 b m s; 12 )]TJ/F15 10.9091 Tf 10.909 0 Td [(90 b m s; 13 + b m s; 14 i ; S 9 = 1 720 h 48 b m s; 0 +72 b m s; 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(72 b m s; 2 +72 b m s; 3 )]TJ/F15 10.9091 Tf 10.91 0 Td [(90 b m s; 4 +90 b m s; 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(90 b m s; 6 +40 b m s; 9 +9 b m s; 10 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 b m s; 11 +9 b m s; 12 )]TJ/F15 10.9091 Tf 10.909 0 Td [(90 b m s; 13 + b m s; 14 i ; S 10 = 1 720 h 48 b m s; 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(72 b m s; 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(72 b m s; 2 +72 b m s; 3 +90 b m s; 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(90 b m s; 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(90 b m s; 6 +40 b m s; 9 )]TJ/F15 10.9091 Tf 8.484 0 Td [(9 b m s; 10 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 b m s; 11 +9 b m s; 12 +90 b m s; 13 + b m s; 14 i ; S 11 = 1 720 h 48 b m s; 0 +72 b m s; 1 +72 b m s; 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(72 b m s; 3 +90 b m s; 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(90 b m s; 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(90 b m s; 6 +40 b m s; 9 +9 b m s; 10 +9 b m s; 11 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 b m s; 12 )]TJ/F15 10.9091 Tf 10.909 0 Td [(90 b m s; 13 + b m s; 14 i ; S 12 = 1 720 h 48 b m s; 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(72 b m s; 1 +72 b m s; 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(72 b m s; 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(90 b m s; 4 +90 b m s; 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(90 b m s; 6 +40 b m s; 9 )]TJ/F15 10.9091 Tf 8.484 0 Td [(9 b m s; 10 +9 b m s; 11 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 b m s; 12 +90 b m s; 13 + b m s; 14 i ; S 13 = 1 720 h 48 b m s; 0 +72 b m s; 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(72 b m s; 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(72 b m s; 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(90 b m s; 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(90 b m s; 5 +90 b m s; 6 +40 b m s; 9 +9 b m s; 10 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 b m s; 11 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 b m s; 12 +90 b m s; 13 + b m s; 14 i ; S 14 = 1 720 h 48 b m s; 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(72 b m s; 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(72 b m s; 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(72 b m s; 3 +90 b m s; 4 +90 b m s; 5 +90 b m s; 6 +40 b m s; 9 )]TJ/F15 10.9091 Tf 8.484 0 Td [(9 b m s; 10 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 b m s; 11 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 b m s; 12 )]TJ/F15 10.9091 Tf 10.909 0 Td [(90 b m s; 13 + b m s; 14 i .1

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277 APPENDIXD 4.13DCascadedLBModelforTransportofTemperatureFieldusingaD3Q7Lattice TheCDEforthescalareld ,suchasthetemperature,giveninEq..4hasthediusion termwithlowerdegreeofsymmetrythanthatoftheviscousstresstensortermintheNSE.As aresult,thelatticesettorepresenttheCDEcanpossiblysatisfylowerdegreeofsymmetryand isotropyrequirementsthanthatfortheNSE.Hence,onecanalsoconstructasimplied3DcascadedLBEfortheCDEusingathree-dimensional,sevenvelocityD3Q7lattice.Inthisregard, thecomponentsoftheparticlevelocityalongwiththeunitvectorforthislatticearegivenby j e x i = ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0 ; 0 ; 0 ; 0 y ; j e y i = ; 0 ; 0 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 0 ; 0 y ; j e z i = ; 0 ; 0 ; 0 ; 0 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 y ; j i = ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 y : .1 Startingfromthenominalbasisvectors T 0 = j i ; T 1 = j e x i ; T 2 = j e y i ; T 3 = j e z i ; T 4 = j e 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(e 2 y i ; T 5 = j e 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(e 2 z i T 6 = j e 2 x + e 2 y + e 2 z i ;

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278 andapplyingtheGram-Schmidtprocedure,thecorrespondinglinearlyindependentorthogonal basisvectorsaregivenby K 0 = j i K 1 = j e x i ; K 2 = j e y i ; K 3 = j e z i ; K 4 = j e 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(e 2 y i ; K 5 =2 j e 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(e 2 z i)-222(j e 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(e 2 y i ; K 6 =7 j e 2 x + e 2 y + e 2 z i)]TJ/F15 10.9091 Tf 17.576 0 Td [(6 j i : Next,theorthogonalcollisionmatrixcanbewrittenas K =[ K 0 ; K 1 ; K 2 ; K 3 ; K 4 ; K 5 ; K 6 ; K 7 ] : .2 Thediscretecentralmomentsofvariousquantitiesandtheircorrespondingrawmomentsare giveninEq..16andEq..19,respectively,where =0 ; 1 ; ; 6isconsidered.Following theoverallprocedurediscussedinSec.2andadoptingitfortheD3Q7lattice,variousresultscan nowbesummarized.First,therawmomentsofthesourcetermatdierentordersaregivenby ^ 0 0 = R; ^ 0 x = u x R; ^ 0 y = u y R;; ^ 0 z = u z R; ^ 0 xx = u 2 x R; ^ 0 yy = u 2 y R;; ^ 0 zz = u 2 z R: .3 Then,asinSec.2transformingthemtothevelocityspace,thesourcetermintheparticlevelocityspacearegivenas S 0 = 1 7 h b m s; 0 )]TJ/F41 10.9091 Tf 12.668 0 Td [(b m s; 6 i ; S 1 = 1 84 h 12 b m s; 0 +42 b m s; 1 +21 b m s; 4 +7 b m s; 5 +2 b m s; 6 i ; S 2 = 1 84 h 12 b m s; 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(42 b m s; 1 +21 b m s; 4 +7 b m s; 5 +2 b m s; 6 i ; S 3 = 1 84 h 12 b m s; 0 +42 b m s; 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(21 b m s; 4 +7 b m s; 5 +2 b m s; 6 i ; S 4 = 1 84 h 12 b m s; 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(42 b m s; 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(21 b m s; 4 +7 b m s; 5 +2 b m s; 6 i ; S 5 = 1 42 h 6 b m s; 0 +21 b m s; 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 b m s; 5 + b m s; 6 i ; S 6 = 1 42 h 6 b m s; 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(21 b m s; 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 b m s; 5 + b m s; 6 i ; .4

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279 where b m s; 0 = R; b m s; 1 = h K 1 j S i = u x R; b m s; 2 = h K 2 j S i = u y R; b m s; 3 = h K 3 j S i = u z R; b m s; 4 = h K 4 j S i = u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y R; b m s; 5 = h K 5 j S i = u 2 x + u 2 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 uz 2 R; b m s; 6 = h K 6 j S i = u 2 x + u 2 y + u 2 z R )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 R: .5 Then,byprescribingcentralmomentrelaxationtotheircorrespondingequilibriaforrstand higherorders,andfollowingtheapproachpresentedinSec.2,wegetthecollisionkernelforthe D3Q7latticeas b h 1 =0 ; b h 1 = ! 1 2 u x )]TJ/F41 10.9091 Tf 11.022 2.182 Td [(b 0 x )]TJ/F35 10.9091 Tf 12.105 7.38 Td [(R 2 u x ; b h 2 = ! 2 2 u y )]TJ/F41 10.9091 Tf 11.022 2.182 Td [(b 0 y )]TJ/F35 10.9091 Tf 12.105 7.38 Td [(R 2 u y ; b h 3 = ! 3 2 u z )]TJ/F41 10.9091 Tf 11.022 2.182 Td [(b 0 z )]TJ/F35 10.9091 Tf 12.105 7.38 Td [(R 2 u z ; b h 4 = ! 4 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [( b 0 xx )]TJ/F41 10.9091 Tf 10.783 2.182 Td [(b 0 yy +2 u x b 0 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y b 0 y )]TJ/F15 10.9091 Tf 10.909 0 Td [( + R 2 u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y + u x b h 1 )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y b h 2 ; b h 5 = ! 5 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [( b 0 xx + b 0 yy )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b 0 zz +2 u x b 0 x + u y b 0 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u z b 0 z )]TJ/F15 10.9091 Tf 10.909 0 Td [( + R 2 u 2 x + u 2 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u 2 z + 1 3 u x b h 1 + u y b h 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u z b h 3 ; b h 6 = ! 6 6 3 c 2 s )]TJ/F15 10.9091 Tf 10.909 0 Td [( b 0 xx + b 0 yy + b 0 zz +2 u x b 0 x + u y b 0 y + u z b 0 z )]TJ/F15 10.9091 Tf 10.909 0 Td [( + R 2 u 2 x + u 2 y + u 2 z + 2 3 u x b h 1 + u y b h 2 + u z b h 3 ; .6 where ! 1 ;! 2 ;:::;! 6 arerelaxationparameters.Thecoecientofdiusivityofthescalareldin theCDE,i.e. D inEq..4isrelatedtotherelaxationparametersfortherstordermoments through D = c 2 s 1 ! j )]TJ/F33 7.9701 Tf 12.491 4.295 Td [(1 2 ,where j =1 ; 2 ; 3.Finally,thepost-collisionvaluesofthedistribution

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280 functionareobtainedfromEq..15aafterexpanding K : b h fortheD3Q7latticeas e g 0 = g 0 + h b h 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 b h 6 i + S 0 ; e g 1 = g 1 + h b h 0 + b h 1 + b h 4 + b h 5 + b h 6 i + S 1 ; e g 2 = g 2 + h b h 0 )]TJ/F41 10.9091 Tf 10.718 2.878 Td [(b h 1 + b h 4 + b h 5 + b h 6 i + S 2 ; e g 3 = g 3 + h b h 0 + b h 2 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 4 + b h 5 + b h 6 i + S 3 ; e g 4 = g 4 + h b h 0 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 2 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 4 + b h 5 + b h 6 i + S 4 ; e g 5 = g 5 + h b h 0 + b h 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b h 5 + b h 6 i + S 5 ; e g 6 = g 6 + h b h 0 )]TJ/F41 10.9091 Tf 10.718 2.879 Td [(b h 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b h 5 + b h 6 i + S 6 : .7

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281 APPENDIXE 5.1Relationbetweennon-equilibriummomentsandspatialderivativesofcomponentsofmomentequilibriaforD2Q9lattice Forbetterclarity,the O momentsystemusinganon-orthogonalmomentbasisgiveninEq..20b inSec.6.2,i.e., @ t 0 + b E i @ i b m = )]TJ/F41 10.9091 Tf 9.122 2.879 Td [(b b m + b S ,whichformsamainelementinthederivation,can beexpandedexplicitlyintermsoftheirvariouscomponentsasfollows: @ t 0 ^ eq 0 0 + @ x ^ eq 0 x + @ y ^ eq 0 y =^ 0 0 ; @ t 0 ^ eq 0 x + @ x ^ eq 0 xx + @ y ^ eq 0 xy =^ 0 x ; @ t 0 ^ eq 0 y + @ x ^ eq 0 xy + @ y ^ eq 0 yy =^ 0 y ; @ t 0 ^ eq 0 xx +^ eq 0 yy + @ x ^ eq 0 x +^ eq 0 xyy + @ y ^ eq 0 y +^ eq 0 xxy = )]TJ/F35 10.9091 Tf 8.485 0 Td [(! 3 ^ m 3 +^ 0 xx +^ 0 yy ; @ t 0 ^ eq 0 xx )]TJ/F15 10.9091 Tf 11.324 0 Td [(^ eq 0 yy + @ x ^ eq 0 x )]TJ/F15 10.9091 Tf 11.324 0 Td [(^ eq 0 xyy + @ y )]TJ/F15 10.9091 Tf 8.901 0 Td [(^ eq 0 y +^ eq 0 xxy = )]TJ/F35 10.9091 Tf 8.485 0 Td [(! 4 ^ m 4 +^ 0 xx )]TJ/F15 10.9091 Tf 11.494 0 Td [(^ 0 yy ; @ t 0 ^ eq 0 xy + @ x ^ eq 0 xxy + @ y ^ eq 0 xyy = )]TJ/F35 10.9091 Tf 8.485 0 Td [(! 5 ^ m 5 +^ 0 xy ; @ t 0 ^ eq 0 xxy + @ x ^ eq 0 xy + @ y ^ eq 0 xxyy = )]TJ/F35 10.9091 Tf 8.485 0 Td [(! 6 ^ m 6 +^ 0 xxy ; @ t 0 ^ eq 0 xyy + @ x ^ eq 0 xxyy + @ y ^ eq 0 xy = )]TJ/F35 10.9091 Tf 8.485 0 Td [(! 7 ^ m 7 +^ 0 xyy ; @ t 0 ^ eq 0 xxyy + @ x ^ eq 0 xyy + @ y ^ eq 0 xyy = )]TJ/F35 10.9091 Tf 8.485 0 Td [(! 8 ^ m 8 +^ 0 xxyy : Ingeneral,itcanbeseenthatanynon-equilibriummomentoforder n dependsonthespatial derivativesofequilibriummomentsoforder n +1and n )]TJ/F15 10.9091 Tf 11.05 0 Td [(1.Inparticular,thediagonalcomponentsofthesecondordermoment^ m 3 and^ m 4 dependonthemomentequilibriaofrst order^ eq 0 x and^ eq 0 y andthirdorder^ eq 0 xxy and^ eq 0 xyy ,whiletheo-diagonalsecondordermoment ^ m 5 dependsonlyonthatofthirdorderequilibriummoments^ eq 0 xxy and^ eq 0 xyy .Theseconsiderationsareimportantinestablishingtherelationshipbetweenthenon-equilibriumsecond-order

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282 momentsandthevelocitygradienttensorcomponents.InthecaseoftheLBEforcomputing uidow,thesymmetryoftheirmomentequilibriatorespecttheisotropyoftheviscousstress tensorlimitsthedependenceofthecorrespondingnon-equilibriumsecondordermomentstoonly onthesymmetricpartofthevelocitygradienttensori.e.,thestrainratetensor.However,the constructionoftheLBEforcomputingthetransportofapassivescalarrepresentedbytheCDE doesnotneedtosatisfytheserestrictiveconstraints,andtheadditionaldegreesoffreedomavailableforthehigherordermomentscanbesuitablyexploited.Indeed,sincethediusionterm oftheCDEneedonlytosatisfyalowerdegreeofisotropythanthatoftheviscoustermofthe NSE,thethirdordermomentequilibriafortheformercasecanbespecicallydesignedtolocally representtheskew-symmetricpartofthevelocitygradienttensorviatherespectiveo-diagonal non-equilibriumsecond-ordermomentbasedonanequationanalogoustothesixthequationin theabovemomentsystemwith^ eq 0 x m y n replacedby^ eq 0 x m y n and^ m j by^ n j {Sec.6.3.1.

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285 6.1.2Surfactantconcentrationevolutionequation Wewillnowpresentaphase-eldbasedevolutionequationforthesurfactantconcentrationeld. Foreaseofpresentation,thesurfactantconcentrationwillbenon-dimensionalizedbyitsmaximumorsaturationvalueforthesurfactantmonolayerontheinterfacegivenby 1 .Hence,the non-dimensionallocalsurfactantconcentration = x ;t isrestricted0 1.Thedistributionofthesurfactantconcentration x ;t isdeterminedunderthecompetingeectsofadsorptionanddiusiveprocessesaswellassolubilityeectsasitundergoesadvectionfromtheuid motion.Ontheotherhand,theinterfacialdynamicsisrepresentedbytheevolutionofthephaseeldvariable x ;t representedbytheconservativeACEEq..1,withthebulkuidineach phaseisidentiedaseither A or B .ItssolutionprocedurebasedonacascadedLBmethodis presentedinSec.7.5.Thediuseinterfacialproleanditsthicknessarethusindependentofthe surfactantconcentration,whichisrequiredtoavoidspuriouseectsandmaintainphysicallyconsistentinterfacialdynamics[250,251]. Then,thetransportofthesurfactantconcentrationisrepresentedbythefollowingevolution equation @ @t + u r = r M r ; .1 where M isthelocalsurfactantmobilitythatcanbeexpressedas M = m )]TJ/F35 10.9091 Tf 12.007 0 Td [( ,with m beingthescaleforthemobilityparameter,and representsthechemicalpotential,whose gradientsdrivethediusion-adsorptiondynamicsoftheadvectingsurfactantconcentrationeld. Deningthefollowingbasedontheorderparameterofthephaseeldvariable usedtocapture theinterfacialdynamics m = A + B = 2 ; 0 = A )]TJ/F35 10.9091 Tf 10.909 0 Td [( B = 2 ; thechemicalpotential inearlierphase-eldformulationscanbewrittenas[247,248] = ln 1 )]TJ/F35 10.9091 Tf 10.909 0 Td [( )]TJ/F35 10.9091 Tf 12.275 7.38 Td [(s 2 j r j 2 + w 2 )]TJ/F35 10.9091 Tf 10.909 0 Td [( m 2 : .2

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286 Inparticular,secondtermontheRHSoftheabovechemicalpotentialrepresentsthepreferentialeectofthesurfactantadsorptionontheinterfaceintermsofadeltafunctionviaasquare gradientoftheorderparameter.However,forwell-posednesswithbetternumericalpropertiesfor widerrangesofparameterchoices,Ref.[249]seealsotherecentstudy[251]suggestedreplacing itbymeansofagradient-freeregularizeddeltafunctionformulationobtainedusingthehyperbolictangentproleacrosstheinterfaceinthenormaldirectionforthephaseeldvariable .By adoptingthislatterstrategy,thechemicalpotentialusedinthisworkcanberewrittenas = ln 1 )]TJ/F35 10.9091 Tf 10.909 0 Td [( )]TJ/F35 10.9091 Tf 12.275 7.38 Td [(s 2 4 2 0 W 2 2 0 )]TJ/F15 10.9091 Tf 10.91 0 Td [( )]TJ/F35 10.9091 Tf 10.909 0 Td [( m 2 2 + w 2 )]TJ/F35 10.9091 Tf 10.909 0 Td [( m 2 : .3 Here, , s and w aremodelparametersthatcharacterizethestrengthsofvariouscompetingprocesses.EachtermontheRHSofthechemicalpotentialappearinginEq..3canbeinterpreted asfollowsseee.g.,[251]:thersttermisreferredtoastheentropytermthatboundsthesurfactantconcentrationbetween0and1anddenotesdecreaseinthesystementropywhenthesurfactantisuniformlydistributedeverywhere,withhigher drivingstrongerdiusionthattends toredistribute moreuniformlythroughoutthedomain;thesecondterm,alsoknownasthe adsorptionterm,expressestheenergeticpreferenceofsurfactanttogetadsorbedoninterfaces, whosestrengthcanbetunedbytheparameter s ;nally,thelasttermisapenaltytermthepenalizesthepresenceofsurfactantinthebulkuidsandexpressessolubilityeects,whosemagnitudeiscontrolledbytheparameter w ,andhenceitcanalsobereferredtoasthebulkterm. Theselasttwotermsopposethediusiveprocessrsttermandsharpenthesurfactantconcentrationprolearoundinterfaces.SubstitutingEq..3inEq..1andrearranging,theevolutionequationforthesurfactantconcentrationeld canberewrittenas @ @t + u r = r m r + r m )]TJ/F35 10.9091 Tf 10.909 0 Td [( P ; .4 wherethesecondtermoftheRHSoftheaboveequationcanbeinterpretedasauxtermrelatedtotheadsorptionandsolubilityofthesurfactant,and P isgivenby P = r )]TJ/F35 10.9091 Tf 9.851 7.38 Td [(s 2 4 2 0 W 2 2 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F35 10.9091 Tf 10.909 0 Td [( m 2 2 + w 2 )]TJ/F35 10.9091 Tf 10.909 0 Td [( m 2 : .5

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287 Followingthenotationsgivenin[249],with U , L and W beingthevelocityscale,lengthscale andinterfacethickness,respectively,thenon-dimensionalformsofvariousmodelparameterscan beexpressedasfollows:Pe = UL=m isthePecletnumber,Ex=4 s= wW 2 isanumberthat characterizestherelativestrengthofadsorptionandsolubilityeects,andPi= W 2 = ~ 0 isa numberdenotingtherelativeroleofsurfactantdiusion. 6.1.3Interfacialforceinthepresenceofsurfactantsviaageometricapproachforthetwo-uid motion Thesolutionofthesurfactant-ladentwo-phaseuidmotionthatcomputesthepressure p andthe velocityelds u willbeobtainedfromthecascadedLBmethoddescribedinSec.7.4,which,in turn,isbasedonthemodiedcontinuousBoltzmannequationMCBEseeSec.7.3.Thesurfacetensionforce F s appearinginEq..8needstobemodiedtoaccountforsurfactanteects. Inthisregard,wewilladoptthegeometricformulationpresentedin[250].Thesmoothedsurface tensionformulationforsurfacant-ladeninterfacialowswithalocalsurfactantconcentration canbewrittenas F s = )]TJ/F15 10.9091 Tf 8.901 0 Td [(~ j r j 2 r n n | {z } Capillaryforce + j r j 2 r s ~ | {z } Marangoniforce ; .6 where r s isthesurfacegradientoperatorgivenby r s r )]TJ/F52 10.9091 Tf 11.885 0 Td [(n n r orinindexnotation @ si = ij )]TJ/F35 10.9091 Tf 11.615 0 Td [(n i n j @ j ,where i;j 2 x;y .ThersttermontheRHSofEq..6representsthe capillaryforce,wheretheloweringofthelocalsurfacetensionbythepresenceofsurfactantisaccountedthroughthedependenceofthesurfacetensionparameter~ on ,i.e.,~ seebelow fordetails.Thesecondtermrepresentstheeectsofthetangentialgradientsofthesurfacetension,ortheMarangoniforce,arisingfromthenon-uniformconcentrationofthesurfactantonthe interface.TheCartesiancomponentsofthesurfacetensionforceforsurfactant-ladeninterfaces

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288 canthenbeexpressedas F sx = )]TJ/F15 10.9091 Tf 8.9 0 Td [(~ j r j 2 r n n x + j r j 2 )]TJ/F35 10.9091 Tf 10.909 0 Td [(n 2 x @ x ~ )]TJ/F35 10.9091 Tf 10.909 0 Td [(n x n y @ y ~ ; .7a F sy = )]TJ/F15 10.9091 Tf 8.9 0 Td [(~ j r j 2 r n n y + j r j 2 )]TJ/F35 10.9091 Tf 10.909 0 Td [(n 2 y @ y ~ )]TJ/F35 10.9091 Tf 10.909 0 Td [(n x n y @ x ~ ; .7b where n x and n y arethecomponentsoftheinterfacialunitnormal n = n x ;n y = r = j r j . Suchageometricstrategyenhancesexibilityastheeectofsurfactantonthesurfacetension forceisnaturallytunablewithanappropriatechoiceoftheinterfacialequationofthestate,when comparedtotheearlierapproaches[247,248].Inthiswork,theinterfaceequationofstateto representtheinuenceofthesurfactantonloweringthelocalsurfacetensionisgivenbythe followingnon-lineardependencebasedontheLangmuirisotherm,i.e., = 0 [1+ ln )]TJ/F35 10.9091 Tf 10.909 0 Td [( ], or,equivalently ~ =~ 0 [1+ ln )]TJ/F35 10.9091 Tf 10.909 0 Td [( ] ; .8 where istheGibbselasticitynumberthatparametrizesthesensitivityofthesurfacetension tothelocalsurfactantconcentration,and 0 and~ 0 correspondtothoseforthecleaninterfaces, i.e.,withoutthepresenceofsurfactant. 6.1.4CascadedLBmethodforthesolutionofthesurfactantconcentrationeld Theorderparameter forthecapturingofinterfacesandthetwo-phaseuidmotionrepresented bythepressureandvelocityeldscanbecomputedusingtherespectivecascadedLBmethods discussedinSecs.7.5and7.4,respectively.Then,inordertosolveforthesurfactantconcentration representedbytheevolutionequation,Eq..4,inwhatfollows,weconstructanother cascadedLBscheme.Thecollisionandstreamingstepsofsuchaschemefortheevolutionofthe discretedistributionfunction h canberespectivelyrepresentedas e h x ;t = h x ;t + K b q ; .9a h x ;t + t = e h x )]TJ/F52 10.9091 Tf 10.909 0 Td [(e t ;t ; .9b

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289 onaD2Q9latticeandbyconsideringthesamemomentbasisasusedintheprevioussections Secs.7.4and7.5. Fortheconstructionofthecollisionoperator,werstdenethefollowingcentralmomentsand rawmomentsof h anditsequilibrium h eq ,respectively,as 0 B @ ^ mn ^ eq mn 1 C A = X 0 B @ h h eq 1 C A e x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x m e y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y n ; .10 0 B @ ^ 0 mn ^ eq 0 mn 1 C A = X 0 B @ h h eq 1 C A e m x e n y : .11 Next,wedenethecontinuouscentralmomentsoftheequilibria b eq; mn = Z 1 Z 1 h eq x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x m y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y n d x d y .12 byexpressingtheequilibriumdistributionfunction h eq inanalogywiththelocalMaxwelldistributionfunctionbyreplacingthedensitywiththesurfactantconcentrationeld : h eq h eq ; u ; = 2 c s exp )]TJ/F33 7.9701 Tf 9.68 5.374 Td [( )]TJ/F53 7.9701 Tf 6.586 0 Td [(u 2 2 c 2 s .Here c s isafreeparameter,whichwillberelatedtothecoecientofmobility m later.Typically,weuse c 2 s = 1 3 .Therelaxationofthecentralmomentstothecorrespondingequilibriagivenaboveonlymodelsadiusionprocess.Inordertoaccountfortheux componentsduetothesurfactantadsorptionontheinterfaceanditssolubilityeectsinthebulk uids,i.e., m )]TJ/F35 10.9091 Tf 11.949 0 Td [( P x and m )]TJ/F35 10.9091 Tf 11.95 0 Td [( P y ,where P = P x ;P y isgiveninEq..5that appearsinthesurfactantconcentrationevolutionequationEq..4,wemodifytherstorder continuouscentralmomentsfrombeingnullto b eq; 10 = m )]TJ/F35 10.9091 Tf 9.508 0 Td [( P x and b eq; 01 = m )]TJ/F35 10.9091 Tf 9.508 0 Td [( P y . Then,bymatchingofthediscreteandcontinuouscentralmomentsoftheequilibria,i.e., b eq mn = b eq; mn forallthenineindependentmomentssupportedbythelattice,wecangetthecomponents of b eq mn ,whicharesummarizedas ^ eq 00 = ; ^ eq 10 = m )]TJ/F35 10.9091 Tf 10.909 0 Td [( P x ; ^ eq 01 = m )]TJ/F35 10.9091 Tf 10.909 0 Td [( P y ; ^ eq 20 = c 2 s ; ^ eq 02 = c 2 s ; ^ eq 11 =0 ; ^ eq 21 =0 ; ^ eq 12 =0 ; ^ eq 22 = c 4 s :

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290 Thecascadedcollisionoperatorcanthenbederivedbyexpressingtherelaxationofcentralmomentsofdierentorderstotheirequilibria,i.e., P K b q e x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x m e y )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y n = ! ^ eq mn )]TJ/F15 10.9091 Tf -453.1 -23.995 Td [(^ mn ,whereonlythezerothmomentbeingconserved^ 00 =^ eq 00 = ,and ! arethevarious relaxationtimes.Theresultingchangesinalltheninecomponentsofmomentsundercollision, i.e., ^q =^ q 0 ; ^ q 1 ; ^ q 2 ; ; ^ q 8 canbewrittenasfollows: ^ q 0 =0 ; ^ q 1 = ! 1 6 h u x )]TJ/F15 10.9091 Tf 12.201 0 Td [(^ 0 10 i ; ^ q 2 = ! 2 6 h u y )]TJ/F15 10.9091 Tf 12.201 0 Td [(^ 0 01 i ; ^ q 3 = ! 3 12 h 2 c 2 s )]TJ/F15 10.9091 Tf 10.909 0 Td [( u 2 x + u 2 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(^ 0 20 +^ 0 02 +2 u x 0 10 + u y ^ 0 01 i + u x ^ q 1 + u y ^ q 2 ; ^ q 4 = ! 4 4 h )]TJ/F15 10.9091 Tf 8.485 0 Td [( u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(^ 0 20 )]TJ/F15 10.9091 Tf 12.201 0 Td [(^ 0 02 +2 u x 0 10 )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y ^ 0 01 i +3 u x ^ q 1 )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y ^ q 2 ; ^ q 5 = ! 5 4 h )]TJ/F35 10.9091 Tf 8.485 0 Td [(u x u y )]TJ/F15 10.9091 Tf 12.201 0 Td [(^ 0 11 + u x 0 01 + u y ^ 0 10 i + 3 2 u x ^ q 2 + u y ^ q 1 ; ^ q 6 = ! 6 4 h )]TJ/F35 10.9091 Tf 8.485 0 Td [(u 2 x u y +^ 0 21 )]TJ/F35 10.9091 Tf 10.909 0 Td [(u y 0 20 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x 0 11 +2 u x u y 0 10 + u 2 x 0 01 i +3 u x u y ^ q 1 + 3 2 u 2 x +1 ^ q 2 )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(3 2 u y ^ q 3 )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(1 2 u y ^ q 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x ^ q 5 ; ^ q 7 = ! 7 4 h )]TJ/F35 10.9091 Tf 8.485 0 Td [(u x u 2 y +^ 0 12 )]TJ/F35 10.9091 Tf 10.909 0 Td [(u x 0 02 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y 0 11 +2 u x u y 0 01 + u 2 y 0 10 i + 3 2 u 2 y +1 ^ q 1 +3 u x u y ^ q 2 )]TJ/F15 10.9091 Tf 12.105 7.38 Td [(3 2 u y ^ q 3 + 1 2 u x ^ q 4 )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 u y ^ q 5 ; ^ q 8 = ! 8 4 h c 4 s )]TJ/F15 10.9091 Tf 12.201 0 Td [(^ 0 22 +2 u x ^ 0 12 + u y ^ 0 21 )]TJ/F15 10.9091 Tf 10.909 0 Td [( u 2 y ^ 0 20 + u 2 x ^ 0 02 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 u x u y ^ 0 11 +2 u x u 2 y ^ 0 10 + u 2 x u y ^ 0 01 )]TJ/F35 10.9091 Tf 8.485 0 Td [(u 2 x u 2 y + u x +3 u x u 2 y ^ q 1 + u y +3 u 2 x u y ^ q 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(+ 3 2 u 2 x + u 2 y ^ q 3 + 1 2 u 2 x )]TJ/F35 10.9091 Tf 10.909 0 Td [(u 2 y ^ q 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 u x u y ^ q 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u y ^ q 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 u x ^ q 7 ; .13 andaccountingfortheadsorptionandsolubilityeects,weprescribethechangesintherstordermomentcomponentsas ^ q 1 =^ q 1 + ! 1 6 m )]TJ/F35 10.9091 Tf 10.909 0 Td [( P x ; ^ q 2 =^ q 2 + ! 2 6 m )]TJ/F35 10.9091 Tf 10.909 0 Td [( P y ; .14 Here,therelaxationtimesoftherstordermoments ! 1 and ! 2 arerelatedtothemobilityparameter m andthediusionparameter inEq..4via m = c 2 s 1 ! j )]TJ/F33 7.9701 Tf 12.105 4.295 Td [(1 2 t ;j =1 ; 2,and therestoftherelaxationtimesaresettounity.Usingtheabove,thepost-collisiondistribution

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291 function e h canbewritteninthecomponentformafterexpanding K b q inEq..9aas e h 0 = h 0 +[^ q 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4^ q 3 )]TJ/F15 10.9091 Tf 11.722 0 Td [(^ q 8 ] ; e h 1 = h 1 +[^ q 0 +^ q 1 )]TJ/F15 10.9091 Tf 11.722 0 Td [(^ q 3 +^ q 4 +2^ q 7 )]TJ/F15 10.9091 Tf 11.722 0 Td [(^ q 8 ] ; e h 2 = h 2 +[^ q 0 +^ q 2 )]TJ/F15 10.9091 Tf 11.722 0 Td [(^ q 3 )]TJ/F15 10.9091 Tf 11.721 0 Td [(^ q 4 +2^ q 6 )]TJ/F15 10.9091 Tf 11.722 0 Td [(^ q 8 ] ; e h 3 = h 3 +[^ q 0 )]TJ/F15 10.9091 Tf 11.721 0 Td [(^ q 1 )]TJ/F15 10.9091 Tf 11.722 0 Td [(^ q 3 +^ q 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2^ q 7 +^ q 8 ] ; e h 4 = h 4 +[^ q 0 )]TJ/F15 10.9091 Tf 11.721 0 Td [(^ q 2 )]TJ/F15 10.9091 Tf 11.722 0 Td [(^ q 3 )]TJ/F15 10.9091 Tf 11.721 0 Td [(^ q 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2^ q 6 +^ q 8 ] ; e h 5 = h 5 +[^ q 0 +^ q 1 +^ q 2 +2^ q 3 +^ q 5 )]TJ/F15 10.9091 Tf 11.722 0 Td [(^ q 6 )]TJ/F15 10.9091 Tf 11.721 0 Td [(^ q 7 +^ q 8 ] ; e h 6 = h 6 +[^ q 0 )]TJ/F15 10.9091 Tf 11.722 0 Td [(^ q 1 +^ q 2 +2^ q 3 )]TJ/F15 10.9091 Tf 11.721 0 Td [(^ q 5 )]TJ/F15 10.9091 Tf 11.722 0 Td [(^ q 6 +^ q 7 +^ q 8 ] ; e h 7 = h 7 +[^ q 0 )]TJ/F15 10.9091 Tf 11.722 0 Td [(^ q 1 )]TJ/F15 10.9091 Tf 11.722 0 Td [(^ q 2 +2^ q 3 +^ q 5 +^ q 6 +^ q 7 +^ q 8 ] ; e h 8 = h 8 +[^ q 0 +^ q 1 )]TJ/F15 10.9091 Tf 11.721 0 Td [(^ q 2 +2^ q 3 )]TJ/F15 10.9091 Tf 11.722 0 Td [(^ q 5 +^ q 6 )]TJ/F15 10.9091 Tf 11.721 0 Td [(^ q 7 +^ q 8 ] : .15 ThisisfollowedbyperformingthestreamingstepshowninEq..9b,whichthennallyupdatesthesurfactantconcentrationeld viatakingthezerothmomentof h as = X h : .16 Itmaybenotedthatfortheimplementationoftheabovescheme,weneedtoinitializethedistributionfunction,whichcanbeexpressedintermsofitsequilibriuminthevelocityspaceas h eq = w " 1+ e u c 2 s + e u 2 2 c 4 s )]TJ/F52 10.9091 Tf 12.105 7.38 Td [(u u 2 c 2 s # + m )]TJ/F35 10.9091 Tf 10.909 0 Td [( w e P c 2 s ; .17 wheretheweightingfactors w aregivenby w 0 =4 = 9, w =1 = 9,where =1 ; 2 ; 3 ; 4and w = 1 = 36,where =5 ; 6 ; 7 ; 8. 6.1.5Numericalresults Young'sproblem:Dropmigrationunderimposedconstantsurfactantconcentrationgradient First,wewillvalidatethecorrectnessoftheimplementationofthesurfacetensionforceforthe surfactant-ladeninterfacialmotion,i.e.,Eqs..7aand.7b,andinparticulartheMarangoni

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292 force,inourformulation.Inthisregard,weconsidertheclassicalYoung'sproblemofthermocapillarymigrationofadrop[252,253]andrecastintotheequivalentsurfactantconcentration gradientdrivenproblem.Accordingtothisproblem,aneutrally-buoyantdropofuid A withdiameter D solelyunderanimposedlinearsurfactantconcentrationprole y = a + G )]TJ/F35 10.9091 Tf 5.786 1.688 Td [(y i.e., G )]TJ/F15 10.9091 Tf -462.213 -22.307 Td [(beingtheconstantgradientofthesurfactantconcentrationeldand y istheverticalcoordinate willself-propelintheambientuid B anditsterminalmigrationvelocityundertheassumption ofcreepingowhasthefollowinganalyticalsolution: V )]TJ/F15 10.9091 Tf 8.817 1.689 Td [(= )]TJ/F35 10.9091 Tf 17.575 7.38 Td [( )]TJ/F35 10.9091 Tf 5.787 1.689 Td [(G )]TJ/F35 10.9091 Tf 5.787 1.689 Td [(D 6 B +9 A ; where )]TJ/F15 10.9091 Tf 9.423 1.689 Td [(isthesensitivityofthesurfacetensionwiththesurfactantconcentration,which,accordingtothelinearizedformoftheLangmuir'sisothermfordilutesurfactantconcentration,can beexpressedas )]TJ/F38 10.9091 Tf 8.817 1.688 Td [( @=@ = )]TJ/F35 10.9091 Tf 8.485 0 Td [( 0 . A and B aretherespectivedynamicviscosities. Weconsideradropwithdiameter D =30initiallylocatednearthebottomofarectangular domainresolvedwith51 201gridnodes.Periodicboundaryconditionsalongthetwoverticalsidesandno-slipboundaryconditionsalongthetwohorizontalsidesareimposed.Byusing adensityratioofunity,weconsiderthesamedynamicviscositiesinboththeuidsbysetting thekinematicviscositiesas A = B =0 : 05.Furthermore,weimposealinearvariationofthe surfactantconcentrationalongtheverticaldirectionbysetting G )]TJ/F15 10.9091 Tf 11.309 1.688 Td [(=9 : 95 10 )]TJ/F33 7.9701 Tf 6.587 0 Td [(5 .Figure6.1 showsthecomputedthedropmigrationvelocitiesforthreedierentsurfacetensionsensitivities 0 =0 : 0048 ; 0 =0 : 0146and 0 =0 : 0244andtheircomparisonsagainsttheavailableanalyticalsolutionfortheterminalvelocity.Itisevidentthataftertheinitialtransients,thecomputed migrationvelocitiesinthelongtimelimitareingoodagreementwiththeanalyticalterminalvelocity.Inaddition,somesnapshotsoftheevolutionofamigratingdropforalltheabovethree casesarepresentedinFig.6.2.Asitcanbeseen,thedropself-propelsundernon-uniformsurface tensioni.e.,Marangoniforcearisingduetoanimposedconstantconcentrationgradientwithout anysmearingeectstotheshapeofthedrop.Thus,theabovenumericalsimulationresultsvalidateourimplementationofthesurfacetensionforceinthepresenceofaspatialdistributionof

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293 a FIGURE6.1:ComparisonofcomputeddropmigrationvelocityunderimposedconstantsurfactantconcentrationgradientinthesimulationofYoung'sproblemsolidlineswiththeanalyticalsolutionfortheterminalvelocitydashedlinesforsurfacetensionsensitivities 0 = 0 : 0048 ; 0 =0 : 0146and 0 =0 : 0244.

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294 surfactantconcentration. Simulationofequilibriumsurfactantconcentrationproleforaplanarinterface Inaddition,wewillnowtesttheimplementationofthecascadedLBschemeforthesolutionof thesurfactantconcentrationequation.Inthisregard,ananalyticalsolutionfortheequilibrium surfactantconcentrationproleforaplanarinterfacecanbeusedtomakecomparison.Suchan analyticalsolutioncanbeobtainedundertheconditionthatthechemicalpotential isuniform throughoutthedomain.Inotherwords,bysetting = ;b ,where ;b isthevalueinthebulk regiongivenby ;b = ln b + w A )]TJ/F35 10.9091 Tf 11.109 0 Td [( B 2 = 8,wegettheanalyticalsolutionoftheequilibrium surfactantconcentrationproleas eq = b b + c ; .18 where b istheprescribedbulksurfactantconcentrationloading,and c inanauxiliaryfunctiongivenby c =exp )]TJ/F35 10.9091 Tf 13.033 7.38 Td [(s 2 2 W m sech 2 2 W 2 + w 2 [ )]TJ/F35 10.9091 Tf 10.909 0 Td [( m ] 2 )]TJ/F35 10.9091 Tf 12.104 7.38 Td [(w 8 A )]TJ/F35 10.9091 Tf 10.909 0 Td [( B 2 : .19 Here, = m + 0 tanh =W isthecorrespondingequilibriumhyperbolicproleofthe phaseeldvariable.Intheabove, isacoordinatealongthenormaldirectionoriginatingatthe interface. Weconsideracomputationaldomainwithgridresolution3 101andchoosetheinterfacewidth W =3,andthenon-dimensionalnumbersaretakenasEx=1,Pi=49100,fromwhichthe valuesoftheparameters s and w canbeobtained.Forthreedierentvaluesofthebulksurfactantconcentration,i.e., b =0 : 0001, b =0 : 0002,and b =0 : 0003,theequilibriumsurfactant concentrationprolesareobtainedthroughnumericalsimulations,whicharepresentedinFig.6.3 alongwiththeanalyticalsolutionforcomparison.Itcanbeseenthatthesurfactantconcentrationpeaksattheinterface.Thisreectsthepreferentialadsorptionofsurfactantaroundthein-

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295 a FIGURE6.2:SnapshotsoftheevolutionofamigratingdropunderimposedconstantsurfactantconcentrationgradientinthesimulationofYoung'sproblemforsurfacetensionsensitivities 0 =0 : 0048 ; 0 =0 : 0146and 0 =0 : 0244.

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296 terface,whichiscounteractedbythediusion.Moreover,thecomputedequilibriumprolesare ingoodagreementwiththeanalyticalsolution,whichvalidatesourimplementationofthecascadedLBschemeforthesurfactantconcentrationequation. a FIGURE6.3:Comparisonofcomputedsymbolsandanalyticallinerequilibriumprolesfor surfactantconcentration foraplanarinterface.Simulationsareperformedfortheimposedsurfactantconcentrationsinthebulkuids b =0 : 0001, b =0 : 0002and b =0 : 0003.