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Cascaded lattice Boltzmann model for isotropic and anisotropic convective thermal flows with local heat sources

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Title:
Cascaded lattice Boltzmann model for isotropic and anisotropic convective thermal flows with local heat sources
Creator:
Elseid, Fatma M.
Place of Publication:
Denver, CO
Publisher:
University of Colorado Denver
Publication Date:
Language:
English

Thesis/Dissertation Information

Degree:
Doctorate ( Doctor of philosophy)
Degree Grantor:
University of Colorado Denver
Degree Divisions:
College of Engineering and Applied Sciences, CU Denver
Degree Disciplines:
Engineering and applied science
Committee Chair:
Jenkins, Peter
Committee Members:
Welch, Samuel
Biringen, Sedat
Ingber, Marc
Mays, David

Notes

Abstract:
A new cascaded central moment based lattice Boltzmann (LB) method for solving low Mach number convective thermal flows with source terms in two-dimensions in a double distribution function framework is presented. For the passive temperature field, which satisfies a convection diffusion equation (CDE) along with a source term to represent internal/external local heat source, a new cascaded collision kernel is presented. Due to the use of a single conserved variable in the thermal energy equation, the cascaded structure in its collision operator begins from the first order moments and evolves to higher order moments. This is markedly different from the collision operator for the fluid flow equations, constructed in previous work, where the cascaded formulation starts at the second order moments in its collision kernel. A consistent implementation of the spatially and temporally varying source terms in the thermal cascaded LB method representing the heat sources in the CDE that maintains second order a accuracy via a variable transformation is discussed. In addition, the first order equilibrium moments in this model are augmented with spatial temperature gradient terms obtained locally and involving a tunable coefficient to maintain additional flexibility in the representation of the transport coefficient for the temperature field. The consistency of the thermal cascaded LB method including a source term with the macroscopic convection-diffusion equation is demonstrated by means of a Chapman-Enskog analysis. The emergent tunable diffusivity is shown to be dependent on the relaxation times of the first order moments as well as the tunable parameter in the additional gradient terms in our cascaded multiple-relaxation-time formulation. The new model is tested on a set of benchmark problems such as the thermal Poiseuille flow, thermal Couette flow with either wall injection or including viscous dissipation and natural convection in a square cavity. The validation studies show that the thermal cascaded LB method with source term is in very good agreement with analytical solutions or numerical results reported for benchmark problems. In addition, the numerical results show that our new thermal cascaded LB model maintains second order spatial accuracy. The new LBE model is modified to simulate anisotropic fluids that are characterized by different diffusion coefficients along different directions. The applicability of the LBE model is validated by numerical simulations including the convection and diffusion of a Gaussian Hill, solving anisotropic convection diffusion equation with variable diffusion tensor and variable source term, and anisotropic natural convection in a Square Cavity. The validation study shows that the anisotropic thermal cascaded LB model with source term is in very good agreement with the analytical solutions or numerical results reported for the benchmark problems. In addition, the numerical results show that our new anisotropic thermal cascaded LB model maintains second order accuracy as does the isotropic model. A stability test of the present cascaded LBE model is conducted to compare our model by single relaxation time (SRT) LB model and multiple relaxation times (MRT) LB model using the diffusion in a Gaussian Hill as a test problem by varying the fluid diffusivity to compare the stability characteristics of the cascaded central moment. This stability study concurs with other work that indicates superior stability characteristics of the cascaded LB method. Finally, the central moment cascaded LB model was adapted to simulate of fluid flow with temperature-dependent viscosity, where the fluid viscosity is exponentially varying with temperature. The simulation results of couette flow with shear heating confirmed the validity of the present central moment LB model to incorporate correctly fluid flow with variable viscosity at different Brinkman numbers.

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Full Text
CASCADED LATTICE BOLTZMANN MODEL FOR ISOTROPIC AND
ANISOTROPIC CONVECTIVE THERMAL FLOWS WITH LOCAL HEAT SOURCES
by
FATMA M. ELSEID
B.S., University of Tripoli, 1994 M.S., University of Belgrade, 2000 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


This thesis for the Doctor of Philosophy degree by
FATMA M. ELSEID has been approved for the Engineering and Applied Science Program by
Peter Jenkins, Chair Samuel Welch, Advisor Sedat Biringen Marc Ingber David Mays
Date:August 3, 2019
n


Elseid M.Fatma (PhD, Engineering and Applied Science Program)
Cascaded Lattice Boltzmann Model for Isotropic and Anisotropic Convective Thermal Flows with Local Heat Sources
Thesis directed by Dr. Samuel Welch
ABSTRACT
A new cascaded central moment based lattice Boltzmann (LB) method for solving low Mach number convective thermal flows with source terms in two-dimensions in a double distribution function framework is presented. For the passive temperature field, which satisfies a convection diffusion equation (CDE) along with a source term to represent internal/external local heat source, a new cascaded collision kernel is presented. Due to the use of a single conserved variable in the thermal energy equation, the cascaded structure in its collision operator begins from the first order moments and evolves to higher order moments. This is markedly different from the collision operator for the fluid flow equations, constructed in previous work, where the cascaded formulation starts at the second order moments in its collision kernel. A consistent implementation of the spatially and temporally varying source terms in the thermal cascaded LB method representing the heat sources in the CDE that maintains second order a accuracy via a variable transformation is discussed. In addition, the first order equilibrium moments in this model are augmented with spatial temperature gradient terms obtained locally and involving a tunable coefficient to maintain additional flexibility in the representation of the transport coefficient for the temperature field. The consistency of the thermal cascaded LB method including a source term with the macroscopic convection-diffusion equation is demonstrated by means of a Chapman-Enskog analysis. The emergent tunable diffusivity is shown to be dependent on the relaxation times of the first order moments as well as the tunable parameter in the additional gradient terms in our cascaded multiple-relaxation-time formulation. The


new model is tested on a set of benchmark problems such as the thermal Poiseuille flow, thermal Couette flow with either wall injection or including viscous dissipation and natural convection in a square cavity. The validation studies show that the thermal cascaded LB method with source term is in very good agreement with analytical solutions or numerical results reported for benchmark problems. In addition, the numerical results show that our new thermal cascaded LB model maintains second order spatial accuracy.
The new LBE model is modified to simulate anisotropic fluids that are characterized by different diffusion coefficients along different directions. The applicability of the LBE model is validated by numerical simulations including the convection and diffusion of a Gaussian Hill, solving anisotropic convection diffusion equation with variable diffusion tensor and variable source term, and anisotropic natural convection in a Square Cavity. The validation study shows that the anisotropic thermal cascaded LB model with source term is in very good agreement with the analytical solutions or numerical results reported for the benchmark problems. In addition, the numerical results show that our new anisotropic thermal cascaded LB model maintains second order accuracy as does the isotropic model.
A stability test of the present cascaded LBE model is conducted to compare our model by single relaxation time (SRT) LB model and multiple relaxation times (MRT) LB model using the diffusion in a Gaussian Hill as a test problem by varying the fluid diffusivity to compare the stability characteristics of the cascaded central moment. This stability study concurs with other work that indicates superior stability characteristics of the cascaded LB method.
Linally, the central moment cascaded LB model was adapted to simulate of fluid flow with temperature-dependent viscosity, where the fluid viscosity is exponentially varying with temperature. The simulation results of couette flow with shear heating confirmed the validity of the present central moment LB model to incorporate correctly fluid flow with variable viscosity at different Brinkman numbers.
IV


The form and content of this abstract are approved. I recommend its publication.
Approved: Samuel Welch
v


ACKNOWLEDGMENTS
I would like to thank my advisor Dr.Sam Welch, for his constant support and guidance, and for providing me an excellent opportunity for my academic and professional development. I am so grateful again to my advisor Dr. Sam Welch for helping and supporting me after my scholarship was ended to continue toward graduation.
I am also so grateful to all my committee members Professor Peter Jenkins, Sedat Biringen, Marc Ingber and Professor David Mays to serve on my advisory committee and for their comments .
vi


CONTENTS
CHAPTER
I THERMAL LATTICE BOLTZMANN MODEL 1
1.1 Introduction..................................................... 1
1.2 Collision Models ................................................ 2
1.2.1 Single Relaxation Time (SRT) LBM......................... 3
1.2.2 Multiple Relaxation Times (MRT) LBM........................ 5
1.2.3 Cascaded LBE............................................... 7
1.3 Thermal flow approaches ......................................... 8
1.3.1 Multispeed approach (MSLBE)................................ 9
1.3.2 Hybrid approach(HybLBE).................................... 9
1.3.3 Double distribution functions(DDfLBE)..................... 10
1.4 Construction of the Cascaded Collision Operator for the Temperature
Field .......................................................... 11
1.5 Anisotropic Cascaded Model for the Temperature Field............ 28
1.6 Boundary Conditions............................................. 29
1.6.1 Halfway bounce-back boundaries............................ 30
1.6.1.1 Velocity Distribution Function.................... 32
1.6.1.2 Temperature Distribution Function................. 32
1.6.2 Periodic Boundary Conditions.............................. 33
II ISOTROPIC THERMAL FLOW 35
2.1 Numerical Results............................................... 35
vii


2.2 Unsteady Reaction-Diffusion Problem: Variable Source Term..... 35
2.3 Thermal Flow in a Channel with Wall Injection............... 38
2.4 Diffusion in 2D Poiseuille Flow............................... 40
2.5 Natural Convection in a Square Cavity......................... 44
2.6 Thermal Couette Flow with Viscous Heat Dissipation: Modeling Heat
Source ....................................................... 48
2.7 Summary and Conclusion........................................ 54
III ANISOTROPIC THERMAL FLOW 58
3.1 Numerical Results............................................. 58
3.1.1 Convection and Diffusion of a Gaussian Hill: No Source Term . . 58
3.1.2 Convection-Diffusion of a Gaussian Hill: Stability Test. 63
3.1.3 Anisotropic Convection Diffusion Equation: Variable Source Term 66
3.2 Natural Convection in a Square Cavity: Anisotropic Case .... 67
3.3 Summary and Conclusion........................................ 74
IV FLOW WITH VARIABLE VISCOSITY 92
4.1 Numerical Results............................................. 92
4.1.1 Couette Flow with shear heating: Variable viscosity..... 92
4.2 Summary and Conclusion........................................ 95
V SUMMARY AND CONCLUSION 96
REFERENCES..........................................................103
APPENDIX
A CHAPMAN-ENSKOG ANALYSIS OF THE ISOTROPIC THERMAL CASCADED LBM......................................................104
B CHAPMAN-ENSKOG ANALYSIS OF THE ANISOTROPIC THERMAL CASCADED LBM......................................................Ill
viii


LIST OF TABLES
TABLE
2.1 Values of gf3 corresponding to each Rayleigh number.................. 46
2.2 Comparison between numerical results obtained using the cascaded LBM and the published results (de Vahl Davis(1983) and Hortmann et
al(1990)) at different Rayleigh numbers (Ra = 103 — 106)............. 57
3.1 Global relative error and stability characteristics after 1000 time increments for SRT, MRT, and Cascaded LBM for different relaxation times for the convection-diffusion of a Gaussian hill problem at Mach number
0.25................................................................... 65
3.2 Values of g[3 corresponding to each Rayleigh number.................. 73
3.3 Numerical results obtained using the cascaded LBM for anisotropic natural convection in a square cavity at different Rayleigh numbers (Ra =
103,104,105,106,107 and 108)........................................... 78
3.4 Comparison between numerical results obtained using the cascaded LBM and the published results Dubois et al(2016) [64] for anisotropic natural convection in a square cavity at different Rayleigh numbers
(Ra = 103,104,105 and 106)............................................. 79
3.6 Comparison between numerical results obtained using the cascaded LBM and the published results (de Vahl Davis(1983) and Hortmann et al(1990)) for isotropic natural convection in a square cavity at different Rayleigh numbers (Ra = 103 — 106)......................................... 81
IX


LIST OF FIGURES
FIGURE
1.1 D2Q9 lattice , Collision (right) and Streaming (left)................... 3
1.2 D2Q9 lattice diagram, showing the 9 velocity directions................. 4
1.3 D2Q13 lattice diagram, showing the 13 velocity directions.............. 10
1.4 Location of boundary nodes............................................. 30
1.5 Boundary Conditions with unknown and known populations................. 31
1.6 Location of boundary nodes............................................. 31
1.7 Schematic illustrating the periodic boundary conditions................ 33
1.8 Periodic boundary condition on the inflow and outflow boundaries. . . 33
2.1 Temperature profile for the unsteady reaction-diffusion problem with a
variable source term at y = 0.5 and diffusion coefficient a = 10-3 at different times. Markers represent the Cascaded LBE results and lines represent the analytical solution...................................... 36
2.2 Temperature profile for the unsteady reaction-diffusion problem with a
variable source term at y = 0.5 and diffusion coefficient a = 10-4 at different times. Markers represent the Cascaded LBE results and lines represent the analytical solutions..................................... 37
2.3 Temperature global relative error at different values of the diffusion co-
efficient a = 10-3, D = 0.397, and a = 10-4, D = 0.3997 for the unsteady reaction-diffusion problem with variable source term.... 38
x


2.4 Velocity profiles for thermal Couette flow in a channel with wall injec-
tion at Reynolds numbers: Re = 5,10,15. Markers represent the Cascaded LBE results and lines represent the analytical solutions.... 40
2.5 Temperature profiles for thermal Couette flow in a channel with wall
injection at Reynolds numbers: Re = 5,10,15. Markers represent the Cascaded LBE results and lines represent the analytical solutions. 41
2.6 Velocity relative global error of Couette flow with wall injection at Re =
10................................................................... 42
2.7 Temperature relative global error of Couette flow with wall injection at
Reynold numbers: Re = 5,10,15........................................ 42
2.8 Velocity and temperature profiles of Poiseuille flow with thermal diffu-
sion at different values of Re and Pe. Markers represent the Cascaded LBE results and lines represent the analytical solution............... 43
2.9 Schematic illustrating the cavity boundary conditions.................. 45
2.10 Temperature profiles along horizontal centerline of the cavity at vari-
ous Rayleigh numbers: Ra = 103,104,105, and 106 computed using the cascaded LBM.......................................................... 47
2.11 Temperature profiles along the vertical centerline of the cavity flow at various Rayleigh numbers Ra = 103,104,105, and 106 computed using
the cascaded LBM...................................................... 47
2.12 Isotherms at different values of Rayleigh numbers Ra = 103,104,105
and 106 for natural convection in a square cavity computed using the cascaded LBM.......................................................... 48
2.13 Streamlines at different values of Rayleigh numbers Ra = 103,104,105
and 106 for natural convection in a square cavity computed using the cascaded LBM.......................................................... 49
2.14 Vorticity contours at various Rayleigh numbers Ra = 103,104,105 and
106 for natural convection in a square cavity computed using the LBM. . 50
xi


2.15 Temperature profiles in Couette flow at various values of Eckert num-
ber. Markers represent the cascaded LBE simulations and lines represent the analytical solutions................................. 52
2.16 Temperature profiles in Couette flow with different values of Prandtl
number. Markers represent the cascaded LBE simulations and lines represent the analytical solutions............................... 52
2.17 Temperature global relative error at different Eckert numbers 7,14, and
28 for thermal Couette flow with viscous heat dissipation........... 53
2.18 Temperature global relative error at different Prandtl numbers 0.25,1.25,
and 2.5 for thermal Couette flow with viscous heat dissipation......53
2.19 Temperature profiles in thermal Couette flow at various values of Peclet
numbers: Pe = 10,102,103,104,105 and 106. Markers represent the Cascaded LBE results and lines represent the analytical solution. 55
2.20 Temperature global relative error at different Peclet numbers: Pe =
10,102, and 103 for thermal Couette flow with viscous heat dissipation. . 55
3.1 Distribution of the scalar variable cp at the time t = tm and u = v = 0.0 . . 60
3.2 Distribution of the scalar variable cp for diffusion u = (0,0) of a Gaussian hill computed using the LBM.......................................... 61
3.3 Concentration contours for diffusion of a Gaussian hill u = (0,0) computed using the LBM....................................................... 62
3.4 Contours of the scaler variable cp at u = (0.05,0.05) computed using LBE 63
3.5 Analytical Contours of the scaler variable cp at u = (0.05,0.05).... 64
3.6 Distribution of the scalar variable cp at the time t = 3 and Pe = 100 .... 68
3.7 Contours of the scalar variable cp at the time t = 3 and Pe = 100.............69
3.8 Temperature global relative errors at different grid sizes for diffusion of
a Gaussian hill.............................................................. 70
3.9 Temperature global relative error with variable diffusion tensor and
source term.................................................................. 70
xii


3.10 Streamlines at different values of Rayleigh numbers Ra = 103,104,105,106,107
and 108 for natural convection in a square cavity computed using the cascaded LBM.(Left) otx = oty/2, (center) olx = oty, (right) olx = 2oty . . 75
3.11 Vorticity at different values of Rayleigh numbers Ra = 103,104,105,106,107
and 108 for natural convection in a square cavity computed using the cascaded LBM.(Left) olx = ay/2, (center) olx = oty, (right) olx = 2oty . . 77
3.12 Isothermals at different values of Rayleigh numbers Ra = 103,104,105
and 106 for natural convection in a square cavity computed using the cascaded LBM.(Left) olx = ay/2, (center) olx = oty, (right) olx = 2oty . . 83
3.13 First component of velocity(u) at different values of Rayleigh numbers Ra = 103,104,105,106,107 and 108 for natural convection in a square cavity computed using the cascaded LBM.(Left) otx = oty/2, (center)
otx = 0Cy, (right) ocx = 2ay........................................... 85
3.14 Second component of velocity(v) at different values of Rayleigh numbers Ra = 103,104,105,106,107 and 108 for natural convection in a square cavity computed using the cascaded LBM.(Left) otx = oty /2, (center)
otx = oty, (right) otx = 2oty.......................................... 87
3.15 Pressure at different values of Rayleigh numbers Ra = 103,104,105,106,107 and 108 for natural convection in a square cavity computed using the cascaded LBM.(Left) otx = oty/2, (center) otx = oty, (right) otx = 2oty . . 89
3.16 Temperature profiles along horizontal centerline of the cavity at various Rayleigh numbers: Ra = 103,104,105,106,107, and 108 computed using
the cascaded LBM....................................................... 90
3.17 Temperature profiles along the vertical centerline of the cavity flow at
various Rayleigh numbers Ra = 103,104,105,106,107, and 108 computed using the cascaded LBM........................................... 91
4.1 Schematic illustrating the problem configuration...................... 93
xiii


4.2 Temperature profiles for velocity driven flow at various Brinkman num-
bers: Br = 3.43,5,10, and 20. Markers represent the Cascaded LBE results and lines represent the analytical solution..................... 94
4.3 Velocity profiles for velocity driven flow at various Brinkman numbers:
Br = 3.43,10, and 20. Markers represent the Cascaded LBE results and lines represent the analytical solution............................... 94
xiv


LIST OF ABBREVIATIONS
LBE Lattice Boltzmann Equation
LBM Lattice Boltzmann Method
LBM Lattice Boltzmann Model
CDE Convection Diffusion Equation
NSE Navier Stokes Equation
SRT Single Relaxation time
MRT Multiple Relaxation time
BGK Bhatnagar- Gross- Krook
DDF Double Distribution Function
MSLB Multi Speed Lattice Boltzmann
HybLB Hybrid Lattice Boltzmann
D2Q9 Discretization of 2 dimention and 9 velocity
xv


LIST OF SYMBOLS
a Discrete diffusion coefficient
a Particle velocity direction
r Non dimensional relaxation time for density rg Non dimensional relaxation time for energy
Tf Non dimensional relaxation time for fluid
t time
T Non dimensional time
p Density
P Pressure
St Time step
Sx Spacing step
A Relaxation time parameter for temperature field
co Relaxation time parameter for velocity field
fa discrete particle density distribution function for velocity field
ga discrete particle density distribution function for temperature field
Equilibrium particle distribution function for velocity field gef Equilibrium particle distribution function for temperature field Sa source term contribution from internal and external forces ea Discrete particle velocity c Particle speed
Cs Speed of sound
£ Particle velocity
xvi


T Temperature
ji Dynamic Viscosity
Ho Typical Dynamic
v kinematic viscosity
e Asymptotic expansion parameter in the Chapman-Enskog analysis A Collision matrix in moment space for velocity field A Collision matrix in moment space for temperature field AT Temperature difference S Strain rate tensor
Ma Mach number
U Fluid velocity
u Cartesian component of velocity in the x-direction
v Cartesian component of velocity in the y-direction
x Cartesian component of position vector
y Cartesian component of position vector
Br Brinkman number
Pe Peclet number
Re Reynold number
I Identity matrix
L Characteristic length
Oa Discrete collision term in the LBE
Woe The Lattice weights
gecl The Lattice
XVII


CHAPTER I
THERMAL LATTICE BOLTZMANN MODEL
1.1 Introduction
Heat and mass transfer coupled with fluid flows is a widespread phenomena has many applications in engineering, energy, chemical reaction, geology etc. These phenomena are mathematically described by convection-diffusion equations and the Navier- Stokes equation. The Lattice Boltzmann method is a particle based method which when compared to traditional computational fluid dynamic methods has some distinct advantages such as ease of implementation of fully parallel algorithms, the capability to handle the complex boundary conditions, flexible and easy to modify the scheme and very fast with hardware acceleration (fast solver for NSE). The Lattice Boltzmann method offers a great potential for including kinetic and atomistic details into the computational models. In fact, this originates from the main purpose of LBM, which is to numerically solve the Lattice Boltzmann equation. This equation can describe the distribution of particles of a system in phase space at any thermodynamic state. The Lattice Boltzmann method does not solve the hydrodynamic and non-hydrodynamic conservation equation but rather models the streaming and collision for particles, i.e. relaxation toward local equilibrium. In other words, the idea of LBM is to construct simplified discrete microscopic dynamics to simulate the macroscopic model described by partial differential equations that model a physical system.
In this chapter, a definition of the Lattice Boltzmann method as a discrete
1


method and the concept of collision and streaming within the method are provided in section (1.2), then a brief description for each LBE collision model is given with the explanation of the main difference among these models in subsections (1.2.1), (1.2.2) and (1.2.3). Next, thermal flow approaches that are used to solve LBE are introduced in section (1.3) with a brief description for each approach.
Finally, the construction of the cascaded LB model for the evolution of the scalar field represented by the convection-diffusion equation (CDE) with a source term is provided in details in Sections (1.4) and (1.5)
1.2 Collision Models
The micro-dynamics of gases and fluids consists of the repetition of two processes collision and propagation (streaming). The macroscopic values of temperature, mass and momentum density are then calculated by a mean values over large spatial regions with thousands of nodes. The Lattice Boltzmann method as a kinetic based numerical method is designed to idealize the microscopic description that allows the recovery of the desired macroscopic equations throughout the artificial lattice (Boltzmann lattice). The movement of particles in the Lattice Boltzmann Method is assumed to be discrete in time and space, i.e. a set of directions with given velocities within the method is considered. Here, the collision of particles happens locally at a single node after a given time step. During the collision the particles are assumed to conserve density and momentum for isothermal flow and also the concentration or temperature for the convection diffusion flow then the particles are strictly streamed to their neighbors according to the allowed directions. Collisions in the LB models are typically represented by Cl{x, f), in which the particle distribution or moment approaches its equilibrium values over a characteristic time (relaxation time r). In other words, as the relaxation time constants determine the microscopic dynamics towards the local equilibrium, it also determines the macroscopic transport coefficients. The effectiveness of the collision operator determines the time to reach
2


i


â–  V j
FIGURE 1.1: D2Q9 lattice , Collision (right) and Streaming (left).
the equilibrium configuration by choosing a proper set of relaxation time constants and at the same time it keeps the numerical scheme stable. The LBE has several models, which differ from each other by the way they handle the collision step. In the following, a brief description of the most popular Lattice Boltzmann collision models is provided.
1.2.1 Single Relaxation Time (SRT) LBM
SRT LBM is a single relaxation time Lattice Boltzmann model contains with a simplified collision term. In fact, the Bhatragar-Gross-Krook (BGK) is a first form of the collision term introduced in [1] then extended to Lattice Boltzmann in [2, 3] which resulted in a considerable simplification of the LBE . The method has since been applied in the context of LBE by several authors for isothermal flow [4, 6, 7, 9,17].
The SRT Lattice Boltzmann model was first introduced to simulate incompressible fluid flow more than two decades ago, then it has been widely employed to study thermal flow [14, 20, 21, 23, 51] because of its simplicity in structure. In this model, both collision and streaming processes are executed in phase space. We consider the D2Q9 SRT Lattice Boltzmann model. The evaluation equation of the SRT model for thermal flow is
3


?
5
FIGURE 1.2: D2Q9 lattice diagram, showing the 9 velocity directions.
ga{x + elX6t,t + 5t) ~ ga{x,t + 6t) = O(x,t) + Sa{x,t). (1.1)
where giX is the distribution function, O(x,t) = — ^ \gK(x,t) — geq(x,t)] is the collision operator, SlX is the discrete source term. The equilibrium distribution function gq is given by
g* = wlXT
l + ^ +
csz
elX.u
u.u
2 cs4 2 cs2
(1.2)
The Lattice weights oolX and the nine discrete velocities are given as
CVix
| ~et = (0,0), a = 0 < l ~e^ = (±1,0), (0,±l),a = 1,2,3,4 ^ i ^ = (±h±l),a = 5,6,7,8
(1.3)
where cs is the lattice speed of sound is given by cl = y. Typically for D2Q9 model, cl = 1/3, c is the lattice speed and is given by c = Here, Ax = elX At = 1, At = 1. The temperature can be obtained in terms of transformed distribution function as
4


following
8
T = E *<'
(1.4)
Through the Chapman-Enskog multi-scale expansion, the thermal diffusivity is found as:
where A = 1 / r.
The shortcomings of BGK model are apparent in that the model suffers numerical instability due to the single relaxation time for all modes, .i.e. All modes are relaxed on the same rate which leads to numerical instability at relatively small thermal diffusivity or at small viscosity for fluid flow. In addition, the Prandtl number is constant when we apply it to thermal flow and this model does not have sufficient parameters (one relaxation time r only) to describe anisotropic diffusion.
1.2.2 Multiple Relaxation Times (MRT) LBM
Here, different moments can relax at different rates, the collision process is mapped onto the raw moment space ( fixed frame of reference) throughout an orthogonal transformation matrix while the streaming process is still executed in phase space. Certain Relaxation times can represent the hydrodynamic and non-hydrodynamic moments i.e. for temperature field, A3 and A5 are represent the thermal diffusivity
(1.5)
(diffusion coefficient). [41,46, 50, 51, 53, 55, 56].
(1.6)
The governing equation of the MRT model
gx(x + ex5t,t + St) -gx(x,t + 5t) = Ci(x,t) + Sa(x,t).
(1.7)
5


Where Cl(x, t) = —M 1 AM [ga(x, t) — gSa(x, f)] , A is a diagonal collision matrix
given by
A = diag( Ao, Ai, A2, A3,., A§). (1.8)
Where M is the orthogonal matrix that is used to transform from velocity space to moment space as follows
| = Mg, g^ = Mg^, S = MS,
(1.9)
The MRT LBE evaluation equation for temperature field Eq. (1.7) can be written in terms of moment space as
ix (* + *«st, t +St)~ ga {x, t)=M
-l
-A(g-r) + (I-^A)S
(1.10)
The equilibrium distribution function for temperature g£ is taken as
gx =waT
e^u (ea.u)2 _ u.u
cs2 2cs4 2 cs2
(1.11)
M is the orthogonal set of basis vectors matrix given by
1 1 1 1 1 1 1 1 1
4 -1 -1 -1 -1 2 2 2 2
4 -2 -2 -2 -2 1 1 1 1
0 1 0 -1 0 1 -1 -1 1
0 -2 0 2 0 1 -1 -1 1
0 0 1 0 -1 1 1 -1 -1
0 0 -2 0 2 1 1 -1 -1
0 1 -1 1 -1 0 0 0 0
0 0 0 0 0 1 -1 1 -1
(1.12)
6


It has been widely proved and accepted that the MRT LBM significantly improved the accuracy and numerical stability of Lattice Boltzmann schemes. In addition, the different relaxation time constants in MRT model are sufficient to cover the anisotropic diffusion coefficient tensor.
1.2.3 Cascaded LBE
The cascaded Lattice Boltzmann model is a multiple relaxation time model based on using central moments. Central moments are the moments that are obtained by shifting the partial velocity by the local fluid velocity i.e the moments are calculated in a moving frame of reference while the moments in MRT LBM are computed in the frame of reference at rest and the collision process is mapped onto the raw moment. Here, no regular equilibrium distribution function is used, the equilibrium distribution function gMT is obtained by making an analogy with Maxwell-Boltzmann distribution function fM (p, u, £) in the continous velocity space if by replacing the density p with temperature T in our DDF formulation. That is,
gMT = lfM (p, u, t) where fM (P' u, £) = ^2 exp
(£-«)
2c?
, The collision
process is also mapped onto the central moment space throughout an orthogonal transformation matrix T. The governing equation of the cascaded model
gx(x + exSt/t + St) -gx(x,t + 5t) = Cl(x,t) + Sx(x,t). (1.13)
Where Oa can be represented as Oa = (/C • h)a, h the collision kernel A is a diagonal collision matrix given by
A = diag( Aq, M, M, A3,....., As). (1-14)
7


/C is the orthogonal set of basis vectors matrix given by
1 0 0 -4 0 0 0 0 4
1 1 0 -1 1 0 0 2 -2
1 0 1 -1 -1 0 2 0 -2
1 -1 0 -1 1 0 0 -2 -2
1 0 -1 -1 -1 0 -2 0 -2
1 1 1 2 0 1 -1 -1 1
0 0 -2 0 2 1 1 -1 -1
1 -1 -1 2 0 1 1 1 1
1 1 -1 2 0 -1 1 -1 1
Here, and A2 represent the thermal diffusivity
1
M,2
(1.15)
(1.16)
The cascaded Lattice Boltzmann model is More stable than both SRT and MRT models and More Galilean invariant than the rest frame of reference models All the steps for the construction of the cascaded collision operator for the temperature field are provided in sec (1.4)
1.3 Thermal flow approaches
To capture or simulate thermal flow, the solution of the temperature field, whose evolution is represented by a convection-diffusion equation (CDE), for energy transport, coupled to the fluid velocity, which is represented by the Navier-Stokes (NS) equations. Classical numerical methods can become challenging to apply for the simulation of such flows, especially in the complex geometries such as thermal flows in pourous media. Within the LB framework, broadly, three diferent approaches that have been developed to construct the thermal (LBE ) equation models. In all of these
8


methods, a distribution function (/) is applied to simulate the velocity field. The energy field is then simulated via either:
1) Multispeed approach (MS-LBE)
2) Hybrid approach
3) Double distribution functions(DDF)
1.3.1 Multispeed approach (MSLBE)
In this method, a distribution function (/) is applied to simulate both the velocity and energy field, i.e. using the same distribution function to solve both the NSE equations and CDE, this requires a large velocity set for the Lattice in order to recover the energy equation. The multi-speed Lattice Boltzmann is first introduced by [32], The authors used thirteen velocities for two dimensions, LBE for fluid needs nine velocites and the extra four velocites to recover the energy equation, this leads to higher order moments than the regular LBE to account for the additional equation for the conservation of energy. In addition, complex terms and calculations due to the large set of velocities considered lead to numerical instability. Different authors later tried to implement this model with some modification to stabilize the method [33, 34, 35, 36,47], All these models are multi-speed models and suffer from numerical instability, are restricted to a limited temperature range and the treatment of boundary conditions with large number of discrete velocities sets is difficult
1.3.2 Hybrid approach(HybLBE)
In the Hybrid approach, the LBE method is used to solve the flow field while any conventional numerical method such as finite difference or finite element scheme is used to solve the energy equation [31, 37, 38, 39]. The hybrid approach is more stable than multi-speed approch. However, this approach has the disadvantage that the simplicity of LBE is lost.
9


FIGURE 1.3: D2Q13 lattice diagram, showing the 13 velocity directions.
1.3.3 Double distribution functions(DDfLBE)
In the DDF approach, two separate distributions functions are employed, one for the flow field and the other one for the temperature field [28,17,45, 56]. Unlike the above two models, many of such limitations can be overcome by the DDF models [30,49] and they have, hence, received significantly more attention recently. Most of the developments related to the DDF-LBE methods considered single relaxation time (SRT) collision [65] .models [14, 21, 20, 23, 51]. Various aspects relevant to the LBE for correct representation of the CDE for the temperature field were identified. For example, the choice of the equilibrium distribution function with nonlinear velocity terms was important in this regard [23]. Concurrently, various boundary condition schemes for the DDF-LBE were developed [18,46,48,53, 54, 60]. However, the use of SRT collision models, though simple in structure and characterized by the relaxation of all models at the same rate, is known to suffer from instability issues, particularly where the transport coefficients such as the fluid viscosity and thermal diffusivity become relatively small. This limits the ability to reach higher Reynolds or Peclet numbers. One possibility to address this issue is to consider using multi-relaxation time (MRT) models for DDF-LBE approach [50,41,46, 53, 55, 56]. In the MRT model,
10


the collision process is mapped onto the raw moment space through an orthogonal transformation matrix, where different moments can relax at different rates. By representing the relaxation times of the hydrodynamic and non-hydrodynamic moments, the stability of the MRT model can be significantly improved. A further improvement is to consider another type of MRT model, in which the collision step is executed in terms of the relaxation of central moments, in which the particle velocities are shifted by the local fluid velocity. Such type of collision model in a moving frame of reference leads to a cascaded structure of the higher order moments in terms of those at lower order following collision and hence is referred to as the cascaded LB method [22],
1.4 Construction of the Cascaded Collision Operator for the Temperature Field
Our main goal in this investigation is to construct a cascaded LB model for the evolution of the temperature field represented by the following convection-diffusion equation (CDE) with a source term
— + U- VT = V- (aVT) + G (1.17)
at
where a is the thermal diffusivity coefficient, T = T(x,y, t) and u = u(x,y, t) are local temperature and velocity field, respectively. In addition G = G{x,y, t) is the local source term, for example, due to internal heat generation or viscous dissipation. In general, thermal transport can be significantly influenced by the presence of internal heat generation, such as those related to nuclear or chemical reactions generating local heating. Viscous heating effects due to shear stresses is another example. All of these effects can be represented as a prescribed local source term G = G(x,y,t) in the thermal transport equation. To handle such a general case, here we develop a new cascaded LB model with a source term, which can recover the macroscopic equation represented by CDE given in Eq. (1.17) above with second order accuracy. In
11


Eq. (1.17), the local velocity field u = u(x,y, t) satisfied the Navies-Stokes equations (NSE) given by
V • u = 0, (1.18a)
dtl 1 7
— + m-Vm = —VP + vV2m + F (1.18b)
at p
Where p is the pressure of the fluid flow, v is the kinematic viscosity of the fluid, p is the reference density and F = pais the local external force field. The velocity field u to be used in Eq. (1.17) is considered to be known, and can be obtained by solving another cascaded LBE constructed in previous work [22,44], In particular, the specific cascaded LBE with forcing term for obtaining the velocity field u can be coupled to the new cascaded for the CDE to be developed in this work. In such a a double distribution function DDF formulation, we refer the reader to the cascaded LBE with forcing term for the flow field presented in to [44] to maintain brevity and focus here on the construction of the cascaded LBE to solve for the temperature field T = T(x, y, t), whose evolution is represented by Eq. (1.17)
The overall procedure to develop a thermal cascaded LBE involves the following: (i) prescribe a suitable choice of an orthogonal moment basis for the lattice velocity sets, (ii) specify formulations for the continuous central moments of equilibrium and source term and equate them to the corresponding discrete central moments involved in the cascaded LBE for the CDE, (iii) Transform the various discrete central moments in terms of various corresponding raw moments by using the binomial theorem, (iv) construct the collision kernel appearing in the cascaded collision operator for solving the CDE and the source term in the LB model.
First, we select a sutible moment basis for the two-dimensional, nine velocity (D2Q9) lattice. We consider the usual "bra" and the "ket" notations, i.e. (.| and |.) to denote 9-dimensional row and column vectors, respectively. Then, we obtain
12


the following nine non-orthogonal basis vectors obtained from monomials e™xe”y at successively increasing orders:
(1.19)
where
IT) = (1,1, l,b l,b 1,1,1 )T,
\e*x) = (0,1,0, —1,0,1, —1, —1,1)T,
M = (0,04,0,-1,1,1,-1,-1)T
The above nominal set of basis vectors is then transformed into an equivalent orthogonal set of basis vectors by means of the standard Gram-Schmidt procedure arranged in the increasing order of moments:
\Kq) = |1), \K\) = \eax), 1^2) = | Gy),
I^3) 0 1Gx T" Gy) ^ 11) , l-^l) |Gx Gy), I^G) |GxGy ),
I-IG) = 3 l^axGy) “I- 2 |Gy), I-K7) = 3 |Gx^y) T- 2 |Gx),
1^8) 9 |GxGy) ^ iGx 3” Gy) "b 4 |l).
By grouping the above, set of vectors, we obtain an orthogonal transformation matrix /C as
/C = [\K0), IKi) , |K2), |^3), |^4), l^s), |K6),\K7), |1C8)], (1.20)
13


which can be explicitly written as
1 0 0 -4 0 0 0 0 4
1 1 0 -1 1 0 0 2 -2
1 0 1 -1 -1 0 2 0 -2
1 -1 0 -1 1 0 0 -2 -2
1 0 -1 -1 -1 0 -2 0 -2
1 1 1 2 0 1 -1 -1 1
0 0 -2 0 2 1 1 -1 -1
1 -1 -1 2 0 1 1 1 1
1 1 -1 2 0 -1 1 -1 1
(1.21)
Next, in order to construct a cascaded LB collision operator for representing the evaluation of the temperature field, we need to present the continous moments of the equilibrium state and the source term. The continuous equilibrium central moments of order (m + n) can be defined as
n= fZo I-ooSMT^x - ux)m{£y - Uy)nd£xdZy, which yields
in'
\ _
__(f\ec! friT^ rb1* fre^ iTK^ tr'
— V1 Ao '1 *-x /1 Ly /1 *-xxr1 Lyy>1 Lxyr1 Wry/1 Wyy/1L.

'fP

-eq
yy
Jecl
ixy/
Wry/
ixyy/
Ie Wryy)
= (T, 0,0, c2s T, c\ T, 0,0,0,4 T)T
(1.22)
Here, the equilibrium distribution function gMT is obtained by making an analogy with Maxwell-Boltzmann distribution function fM (p, u, £) in the continous velocity space £ by replacing the density p with temperature T in our DDF formulation. That
is,
gMT = (p, u, £) where fM (p, u, £) = ^ exp
â– M,
_ P
(£-«)
2 c?
, where cs is the
lattice speed of sound. Typically for D2Q9 model, cj = 1/3.
14


Similarly, defining the continous central moments of the source term of order (m + n) due to G = G{x,y, t) appearing in Eq. (1.17) as
/CO rOQ
/ A gg(^-UX)m(^-Uy)nd^y (1.23)
-00 J —CO
where Ais the change in the distribution function due to source G. Since the source G is expected influnce only the lowest order (zeroth) moment, we can prescribed the following ansatz:
xrnyn
= (r0, hr / Ty, Txx/ ryy, EXy, Txxi// Txyy/ Txxyy ) /
= (G,0,0,0,0,0,0,0,0)t.
(1.24)
By using the above central moments our goal is to develop the collision operator and the source term of the cascaded LBE. For representing the transport of the temperature field the corresponding cascaded LBE using the trapezoidal rule to evaluate the source term of cascaded LBE to maintain the second order accuracy can be written as:
l
ga(x + 6aSt, t + St) = ga{x, t) + Cla(x, t) + — [Sa(x, f) + Sa(x + ga, t + Af)]. (1-25)
Here, the collision term Oa can be represented as Oa = Oa(g, h) = (/C • h)ff, where g = \got) = (go,gi, • • • / ys)T is the vector of distribution functions and h = |hx) = {ho,h\, • • • ,hs)T is the vector of unknown collision kernel to be obtained later. The discrete form of the source term Sa in the cascaded LBE given above represents the influence of the source G in the velocity space and is defined as S = |Sa) = (So, Si, S2, • • • , Ss)T. Noting that Eq. (1.25) is semi-implicit, by using the standard variable transformation g = ga ~ [13], the implicitness can be
effectively removed. This yields
ga(x + ea5t,t + 5t) = gx(x,t) + Clx(x,t) + Sx(x,t).
(1.26)
15


which can maintain second order accuracy in an effectively time explicit method.
In order to obtain the expressions for the structure of the cascaded collision operator h and the source terms Sa in the presence of a spatially and/or temporally source G, i.e. G = G(x, y, t), we define the following set of discrete central moments.
Kxmytt = YlS0c(e0CX-Ux)m(e0iy-Uy)n, X (1.27a)
Kq Kxmyn = YjS^^X ~ UX)m(eay ~ UyY, X (1.27b)
Kxniyn = ~ ux)m(e0cy ~ uv)H• X (1.27c)
GG€L} Kxniyn = Y.fxiexx ~ ux)m(exy - Uy)n, X (1.27d)
(Jxmyn = J^Sa(eax-Ux)m(eay-Uy)n. (1.27e)
X
We then match the discrete central moments of the distribution functions and source terms with the corresponding continuous central moments at each order, i.e.
— fr-M
Kxmyn — LLxmyn'
(1.28a)
0~xmyn —
yF
1 xmyn.
(1.28b)
Thus, we obtain
\^~eq \ ,^~eq ^~eq ^~eq ^~eq ^~eq ^~eq ^~eq ^~eq ^~eq \T
v^xmynl — v^o r^x '^y '^xx'^yy'^xy'^xxy'^xyy'^xxyy) >
= (r,o,o,c2r,c2r,o,o,o,c^r)T.
(1.29)
|(Fxmyn) — \(F0/ 0~x/ (Fy/ 0~xx/ (Fyy/ 0~xy/ 0~xxy/ 0~xyy/ (Fxxyy)
= (G,0,0,0,0,0,0,0,0)t.
(1.30)
Since the actual computations in the cascaded LBE are performed in terms of the various raw moments, we define the following set of discrete raw moments (denoted
16


with a prime symbol):
KXmyn X (1.31a)
X ect Kxniyn X (1.31b)
Kxniyn = YjSccMm{eay)n- X (1.31c)
Kxniyn X (1.31d)
(Jxniyn (1.31e)
X
By using Eqs. (1.27e), (1.30), (1.31e) and the binomial theorm, we obtain the following sets of discrete raw moments for the source term at different orders:
a0 = = (S, x\T) CO II = G, (1.32a)
Ex = (Sx\ Sax) co II Sax = uxG, (1.32b)
Vy = (Sa\ Say) co II Say = UyG, (1.32c)
^J = (Sa\ Sax) co II Sax u\G, (1.32d)
d'yy = (Sx\ e2 ) ^txy / co II g2 = eay UyG, (1.32e)
axy = (Sc 11 Sax â– ^ay) co II Sax Say = UXUyG, (1.32f)
®xxy (Sc \e2 t l^ax ■Say) co II 2 ^axhxy U^ttyCjf (l-32g)
Jxyy = (Sc 11 Sax 'Say) co II SaxS2y = UXUyG, (1.32h)
®xxyy (Sc \e2 t l^ax s2 ) ■^ay/ = Es* 2 2 g g ^ax^ay = UXUyG. (1.32i)
X
In order to obtain the source terms in the velocity space, we first compute the source moments projected to the orthogonal moment space, i.e. rhp = (Kp \SK)
17


, j6 = 0,l,2,---,8
m0 = (K0\S0i) = G, (1.33a)
mi = {Ki\Sx) = uxG, (1.33b)
m2 = (K2\ Set) = uyG, (1.33c)
m3 = (K3\Sa) = {3u2x + 3Uy — 4)G, (1.33d)
ra4 = (K^Sa) = (ux — iiy)G, (1.33e)
7^5 = (^5|Sa) = UxtlyG, (l-33f)
m6 = (K6\Sa) = (—3uxuy + 2uy)G, (1.33g)
m7 = (K7\Scc) = (-3uxiiy + 2ux)G, (1.33h)
m3 = (Kg\Sa) = (9uxiiy — 6(ux + u2) + 4)G. (1.33i)
Since there is only one conserved scalar for the thermal transport equation, the components of raw moments of sources are different from those appearing in the cascaded LBE with forcing terms [44], Then using
(/C • S)a = (mo, mi, m2,..., m$)T and inverting it by exploiting the orthogonality of /C,
18


we get the following expressions for the source terms Sa in velocity space:
So = ^ [mo - nx3 + nx8], (1.34a)
Si = 1 — [4mo + 611x1 — 11x3 + 9nx4 + 6JW7 — 2nx8], 36 (1.34b)
s2 = 1 — [4mo + 611x2 — m3 — 9nx4 + 6ms ~ 2nx8], 36 (1.34c)
S3 = 1 — [4mo — 611x1 —m3 + 9ra4 — 61W7 — 2m8], 36 (1.34d)
s4 = 1 — [4mo — 61W2 — m3 — 9m4 — 6m6 — 2m8], 36 (1.34e)
S5 = 1 — [4mo + 6m4 + 6m2 + 2m3 + 9ms — 3ms ~ 311x7 + m8], 36 (1.34f)
S6 = 1 — [4xixo — 6m4 + 6m2 + 211x3 — 9ms — 3ms + 311x7 + nx8], 36 (l-34g)
S7 = 1 — [4nxo — 611x1 — 611x2 + 211x3 + 911x5 + 3ms + 311x7 + nx8], 36 (1.34h)
S8 = 1 — [4nxo + 611x1 — 611x2 + 211x3 — 911x5 + 3ms ~ 311x7 + nx8]. 36 (1.34i)
In addition, in order to construct the cascaded collision operator for the solution of the temperature field, we need the raw moments of the collision kernel of different order, i.e. £a(/C • h)aefxexy = (Kp\efxexy) hp. Since the temperature field T is a collision invariant, it follows that ho = 0. Using this and considering the orthogonal
19


basis vector kp in Eq. (1.20), we get
YJ(lC-h)a = YJ(Kp\T)hp = °' (L35a)
a p
Y,(£-b)ccexx = 'E(Kfi\exx)hfi = 6/zi, (1.35b)
a p
Y^{lC-h)xexy = '£(Kp\exy)hp = (1.35c)
a p
E(^-h)a^x = E^kL)^ = 6/z3 + 2/z4, (l-35d)
a p
E(^-h)«4y = E(^l^y)^ = 6/z3 — 2/z4, (1.35e)
a p
• h)aeaxeay = |eaxeay) /z^g = 4/25, (1.35f)
a p
E(^C • h)aejxeay = E kL^y) hp = 4fe2 - 4/z6, (l-35g)
a p
YJK • h)aeax^y = E (^j6 kaz^y) hp = Ahi - 4Ji7, (1.35h)
a p
E(^-h)ae^^y = E^kL^y)^ = 8/z3 + 4/z8. (1.35i)
a p
Now, we are in a position to determine the structure of the cascaded collision operator with source terms to solve for the thermal transport equation represented by the CDE. The procedure can be briefly summarized as follows: Starting from the lowest order non-conserved post-collision central moments (i.e. the first order components in the present case), we tentatively set them equal to their corresponding equilibrium states. Once the expressions for the collision kernel (hp) (fi > 1) are determined, we discard these equilibrium assumption and multiply it with a corresponding relaxation parameters (A^) to allow for a relaxation process during collision [22,44], Here, only those terms that are not in the post-collision states for the lower order moments are multiplied by the relaxation parameters. Thus, we start from the first order post-collision central moment, i.e. (g%\(eax — ux)) and {ga\ (Ay — %)) and tentatively set to kx and Ky , respectively:
= 0 = (gll(e« - «*) = (gl\e«x) - ux (gl|r) (i.36)
20


Where from Eq. (1.54b) raw moments of the post-collision distribution function in terms of its pre-collision value, collision kernel, and source term are used to obtain the right hand sides of the above equation. That is,
(ga |htx) = (%oc\htx) 4“ ((^ ' h)a|^ax) 4“ (Sa|ht:x) = Kx "h 6h\ + d x (f«\T) = (£„|T) + <(/C • h)«|T> + (S«|T) = T+ ic
Substituting the above expressions in Eq. (1.36) and rearranging, and solve for collision kernel we get the following tentative expression
h = ^ | fx -kx- lwx | (1.37)
'~' P Q /
Where kx = uxT. In order to provide further flexibility in adjusting the transport
/\pn /
coefficient in the CDE, the raw moment equilibrium kx will be agumented with an extended moment equilibrium \DSt(dxT), where D is the adjustable parameter. See Appendix B for the analysis of such a scheme. In addition, we apply a relaxation parameter Ai in the equation above Eq. (1.37) to reflect the collision as a relaxation process. Thus, we get
h = y fe*' -kx - ^dx + ^DSt(dxT)
(1.38)
Similarly, setting (g%| (exy — uy)) to Ky = 0 and using
^eq __
___ __/ __ ______i
{§a\Ctxy) = (^a|Ctxy) A ((^- ’ h)a|gay) + (Sa|gay) = Ky + 6/?2 T- dy
and following the same procedure as above, we obtain
hi = J- 1*7 -% - \^ + lDSt(d,T)\ (1.39)
In the above, the temperature gradient needed in the extended moment equilibria can
21


be locally computed in terms of the first order non-equilibrium moments (see Appendix B for details). Next, consider the second order diagonal central moments and tentativly set them to their corresponding equilibrium states, i.e.
S = clT = (gl\(eccx ~ ux)2) = (ga\(elx) - 2ux (gl\exx) + u2x (gl\T). (1.40)
and
N/y = c2sT = (gl\(eay - uyf) = (,gl\{ely) - 2 uy (gl \exy) + u) (gl\T). (1.41)
Then, using
(gct\Ctxx) l^ax) A ((A^ • h)a |eax) + (Sa |cax) Kxx + 6/z3 + 2/Z4 + dxx
(g0i\^O.y) l^y) A ((^1 • ll)a |Cay) + (Sfl; |Cay) Kyy + 6/Z3 + dyy
substituting the above two expressions in Eqs. (1.40) and (1.41), respectively and rearranging, we get
6/z3 + 2/z4 = — u2T — kxx + 2uxkx - axx + 2uxax + ^u2xG + 12uxhi. (1.42)
6^3 - 2/z4 = - u2T - tyy + lUyty - ayy + 2uyUy + ^uyG + 12uy/z2- (1.43)
Solving for /z3 and /q from the above two equations and then applying the relaxation parameters A3 and A4, respectively, for /z3 and /q, while excluding the lower order collision kernel terms (i.e. h\ and /q) as they are already in the post-collision state, we finally get
h = U 3T“
A + uy)T ~ (Kxx + Ki/i/l + + -'Mi/ + + (7i/i/)
+Ux/?1 + Uyh-2,
(1.44)
22


/z4 = — l ~(u2x - u2)T - (kxx -Kyy)+ 2UXKX - 2uyKy + -(axx -Vyy)>
+3uxh\ — 3uyh2, (1-45)
Clearly, the cascaded structure is already evident in the collision kernels of the second order moments, which is unlike that for the fluid flow LBE solver, where the cascaded structure starts to appear only at the third order moment collision kernels. This arises due to differences in the number of collision invariants between the two cascaded LBE models. Next, considering the post-collision state of the off-diagonal central moment as
Kxy — 0 — {gx\(,£ax Ux)(say Wy)) — {gx\htxhty) Uy (gx\sax) Ux (gx\eay)
+uxuy {gx\T) (1.46)
Using (gx\eaxeay) = Kxy + 4115 + crxy in the above equation and simplify it as a tentative expression for h5 and then applying the relaxation parameter A5 to those terms that are not yet in the post-collision states, we get
V A5 ( xxeq id d d 1±' 1 3, <> <> . , .
h5 = — j-Kxy ~ Kxy + uxKy + uyKx +-axy \ + -{uxh2duyhi) (1.47)
Now, consider the determination of the third order moment collision kernel. Setting tentatively
Kxxy = 0 = (gZ\(exx ~ ux)2(eay - uy)) (1.48)
^xyy = 0 = (gZ\(exx - Ux)(eay - uy)2) (1.49)
and using
(gZ\(e*x)2(eccV)) = Kxxy+4h2-4h6 + axxy/ i$oc I i^ocx) ay) ) ^XVV ^ xyyf
23


in Eqs. (1.48) and (1.49), respectively, and simplifying to obtain the tentative expressions for As and A7, respectively, to those terms that are not yet post-collision states, we obtain

h() •( llxllyT KXXy llyKxx 2llxKXy llxKy 2llxllyKx C7".
1,
ixuy -
1
'â– xxy w3/,vxx 1
xy
"x"y
r 2 xxy
+—(2/z2 + 3uxh2) - -uy(6h3 + 2h4) + 3uxuyh\ - 2uxh5,
(1.50)
fc7 =
Y \ ~UxllyT + *xyy ~ UxKyy ~ 2uy^xy + 2uxiiyKy + u2^kx + ^
+ -{2h\ + 3u.yh\) - -ux(6h3 - 2h4) + 3uxuyh2 - 2uyh5
xyy
(1.51)
Finally, by setting the post-collision state of the fourth order central moment to its corresponding equilibrium as a tentative step as
Syy = lT = {gx\{exx - ux)\eXy - u2y) (1.52)
and applying K^xyy = ±T = (gp\eax)2ely) = Kxxyy + 8h3 + 4Jt8 + fxxyy and simplfiying Eq. (1.52) and following the same procedure as above by applying the relaxation parameter As, we get
h = ^{ir-«2^T +
xxyy
xyy + 2uyKxxy ~ ulKyy ~ uyKxx ~ 4llxUyKxy
+2 UxUyKx + 2 uiuyK
2XUyKy
2 UXKXyy ~T ^-tA.yt^XXy
+ x^rn/1 , \ - 2h3 - ^u2(3h3 + h4) + 2ux(3h\ - h7)
1
2" xxvv
--ii2(3h3 — h4) + 2u.y (3/?2 — he) — Auxuygs + 3u2uyh2 + 3uxu2h\. (1.53)
In addition, in order to maintain additional flexibility in the representation of the emergent transport coefficient (i.e. the thermal diffusivity of the CDE), we also
24


introduce extended moment equilibria involving temperature gradient terms with an adjustable coefficient D in the first order equilibrium moments. The resulting final expressions of the collision kernel are given as follows:
ho
hi
hi
%
h^
h5
h
h?
hg
0,
A2 heyq> - ty - + ^DSt(dyT) |,
A3 f 2 w w w .______/ ._./ ._./
uy^ ~ (%xx Kyy) + 2uxkx + 2uyKy + -(crxx + ayy)
(^xx Kyy) 2UyKy + — (^"xx ^"yy) j
12 \ 3
-\-UXh\ + Uy/i2.
^4 ( / 2 2
+3ux/zi — 3uyh2
A5 f /X X X 1N 1 3 , <> y \
-J- j “No/ - Kxy + + WyKx + ~CTxy ^ + ~{uxh2 + UyhX),
X 2-' ~
xKxy + wxKy + 2 UxUyKx + ~(J xxy
^6 / 2 t 1 —
— - 2uxk
xxy ^y^xx AUxK-Xy ' uxKy
1^. \ ^ ^ +—(2/r2 + 3uxh2) - ~uy(6h3 + 2hi) + 3uxuyh\ - 2uxh5/
(1.54b)
(1.54c)
>
(1.54d)
(1.54e)
(1.54f)
(1.54g)
A7 I 2t - -
UxUyl 3~ K-xyy UxK-yy
X X X 2X
'Kyy 2UyKxy T" 2tlxllyKy T" tlyKx T" ~e“xyy
4
+
^ ^
2(2/11 + 3Uy/zi) - -ux(6h3 - 2hi) + 3uxuyh2 - 2uyh5, (1.54h)
' " r2^' 2^ ^
^xxyy ^-^xKxyy ^^y^xxy ^x^yy ^y^xx 4Uxt£y7CXy
+ 2^**yy ? - 2/13 - 2Uy(3h3 + hi) + 2ux(3h\ - h7) -u2x(3h3 - hi) + 2uy{3h2 - he) - 4uxiiyg5 + 3u2xuyh2 + 3uxu2h\. (1.54i)
'y{It~uHt+
2 X '
+ 2UXWyKX + 2 UxUyKy
Notice that h\ 7^ h2 7^ 0,in the present case, which is unlike that for the cascaded LBE for fluid flow [44], This difference arises from the fact that the cascaded LBE for the flow field has three collision invarients, i.e. mass and momentum, and hence its corresponding zeroth and first order collision kernels are zero; On the other hand, in the case of cascaded LBE for the thermal transport equation, there is only one
25


collision invariant, i.e. temperature field, and therefore only its zeroth order collision kernel is (ho) is zero. As a result of these differences the cascaded collision operator for the temperature field is markedly different from that for the flow field.
note that, A^, ft = 1,2,3,..., 8, are the relaxation parameters, satisfying the bounds 0 < A^ < 2. Notice the cascaded structure for the second and higher order moment kernels, i.e. their dependence on the lower order moments for our thermal cascaded LBE. By contrast, the cascaded LBE for the fluid flow is significantly different, with the cascaded structure appearing only for third and higher order moments. When a Chapman-Enskog expansion (C-E) is performed on the above cascaded LB model (see Appendix B for details), it can be shown to recover the convection-diffusion thermal transport equation, with the relaxation parameters for the first order moments Ai and A2 and the adjustable coefficient D controlling the thermal diffusivity coefficient a (see Eq. (1.17)):
1/1 1 \
a = 3 ( Yj~2~D)3t' j = 1'Z (L55)
The rest of the parameters can be adjusted independently to improve numerical stability. In this work, Ai = A2 = (1/t> ) is selected based on the specified diffusivity, while the remaining relaxation parameters are set to be unity.
Moreover, the temperature gradients dxT and dyT appearing in the above (see Eqs. (1.54b) and (1.54c)) can be calculated locally in terms of the first order non-equilibrium moments. See the C-E analysis given in Appendix B for details. Thus, we have
3T
dx
—3Ai

xxea'
~Kx
(1-DAi)
dT
dy
—3A2
A xxeq'
Ky ~ Ky
(1 - DA2)
(1.56a)
(1.56b)
26


Where kx = X^=0 eaxgK, Ky = Zl=o e«.ygx, Kxq = uxT, and Kyq = uyT. Then, the thermal cascaded LBE given in Eq. (1.26) can be written in terms of the following collision and streaming steps:
t) = ga(lt/t) + n0i(lt/t) + S0i(lt/t)/ (1.57a)
gx{~£ + ~ By expanding in Eq. (1.57a), the explicit expressions for the post collision
distribution functions are given as follows:
—V So = = £o + ho -4(/j3-' hs) + Sq, (1.58a)
—V Si = = 8i + h0 + h\ -h + 1*4 + 2(1*7 -hs) + Si, (1.58b)
—V S 2 = = S2 + h0 + h2 -h — 1*4 + 2(1*6 — hg) + s2/ (1.58c)
—V S3 = = S3 + h0 — h\ -h + 1*4 — 2(1*7 + h8) + S3, (1.58d)
—V S 4 = = Sa + h0 -h2 -h -hi -2(1*6 + h8) + S4, (1.58e)
—V S 5 = = S5 + ho + h\ + 1*2 + 2h3 + h5~ 1*6 1*7 + 1*8 + S5, (1.58f)
—V Se = S6 + ho — h\ + 1*2 + 21*3 ' 1*6 + 1*7 + 1*8 + §6, (1.58g)
—p S? = s? + ho — h\ -h2 + 21*3 + 1*5 + 1*6 + 1*7 + 1*8 + S7, (1.58h)
—p Ss = = Ss + ho + h\ -h2 + 21*3 — 1*5 + 1*6 — 1*7 + 1*8 + s8. (1.58i)
Finally, basesd on the solution of the thermal cascaded LBE given in Eqs. (1.57a) and (1.57b), the temperature field T can be obtained as
T = (1-59)
X X ^
27


1.5 Anisotropic Cascaded Model for the Temperature Field
Anisotropic diffusion is a common physical phenomenon and describe processes where the diffusion of some location is direction dependent, i.e diffusion coefficients are location and / or direction dependent. In addition, many practical problems such as fluid flows in anisotropic media arising in environmental science and geophysics do have anisotropic diffusion coefficients. Most of the LBE models for CDE are limited to the description of the isotropic diffusion problems [5,25, 26]
To incorporate full anisotropy with off-diagonal components of the thermal diffusion-coefficient tensor, we consider
h = (*7 -i, - \9X + ^DSt(dxT) ) + + 5DSt(dyT)
1
1
3
A-12 I ^.eqt
~M\Ky
i,
2l
1
3
(1.60)
M
Mi
6
C£q> d Kx — Kx
2ax + 3DSt(dxT) > + —
aeq
Ky
~\°y + \D5t{dyT)
(1.61)
where the off-diagonal components of the relaxation time matrix , as a result of the rotation of the principal axes, enable us to simulate the full anisotropy convection-diffusion.
Here, A is a collision matrix given by
A = diag( Aq,
An
Mi
A12
A22
, A3, A4,..., Ag)
(1.62)
28


0
Ao 0 0 0 0 0 0 0
0 An Al2 0 0 0 0 0
0 A2i A22 0 0 0 0 0
0 0 0 A3 0 0 0 0
0 0 0 0 A4 0 0 0
0 0 0 0 0 A5 0 0
0 0 0 0 0 0 A(3 0
0 0 0 0 0 0 0 a7
0 0 0 0 0 0 0 0
1.6 Boundary Conditions
(1.63)
As mentioned in the previous sections, the Lattice Boltzmann method has many important advantages, one of these features is it's capability to handle the complex boundary. Boundary conditions (BCs) have very important role for stability and the accuracy of any numerical solution for the lattice Boltzmann method, the discrete distribution functions on the boundary have to be taken care of to reflect the macroscopic BCs of the fluid. There are many types of boundary conditions that have been handled by LBE. There have been many types of thermal boundary conditions applied to CDE. [29]. The boundary conditions used in this study for simulating isotropic and anisotropic thermal convective flow consists of periodic boundary conditions and bounce-back boundary conditions. In this thesis, the halfway bounce-back scheme is employed to treat velocity boundary conditions while the general anti-bounce-back scheme [48] is adopted to deal with temperature boundary conditions and periodic boundary condition for inflow and outflow boundaries.
29


fluid
O
O
ox
(). DO.V
ill
FIGURE 1.4: Location of boundary nodes.
1.6.1 Halfway bounce-back boundaries
In the Lattice Boltzmann method the interaction of a fluid particle with a solid
functions, bounce-back boundary conditions were applied on all solid boundaries, which means that incoming boundary population equal to out-going populations after the collision. In LBE distribution functions out of the domain are known from streaming process, Fig. 1.5 shows the unknown distribution functions (velocity and temperature) which are needed to be determined as dotted lines, which means that incoming boundary population equal to out-going populations after the collision. For example for east boundary, f5, fl, and f8 stream into the wall, and are bounced back by setting f3=fl, f7=f5, and f6=f8 as it can be seen in Fig. 1.5. The bounce-back method has been discussed extensively in the literature and a few versions have been developed. The most simple scheme is to place a wall halfway between a wall grid point and a fluid grid point and then "bounce-back" particles that stream into the wall. It has been found that the halfway bounce-back scheme is second order accurate with respect to grid spacing for regular boundary [27].
particle is performed the using bounce-back method. For the density distribution
30


» 6V t *5 •*. :
TLx ' » I ^ E Ay/2
Solid boundary
FIGURE 1.6: Location of boundary nodes.
31


1.6.1.1 Velocity Distribution Function
in order to obtain a no-slip boundary condition for a specified boundary, we use the "Bounce Back Method. The most simple scheme is to place a wall halfway between a
into the wall. For instance, f4, f7, and f8 stream into the wall, and are bounced back by setting f5=f7, f2=f4, and f6=f8 (Fig. 1.6)
1.6.1.2 Temperature Distribution Function
The general anti-bounce-back scheme [48] is adopted here to deal with temperature boundary conditions, the thermal or concentration boundary conditions can be classified into three types, Dirichlet, Neumann and mixed boundary conditions as
is constant at the boundary. To implement these boundary conditions with LBE, the distribution function is constructed as
wall grid point and a fluid grid point and then "bounce-back" particles that stream
bi— + b2T = &3
(1.64)
First: for Dirichlet boundary conditions: for static boundary, i.e. uw = 0
(1.65)
£«(*//*) = ~gp(xf,t)+2cvaTw
(1.66)
for moving boundary, i.e. uw ^ 0
gAxf>t) = -gPAxf>t)+2a’ocTwx 1 + 4.5-^^
21
(1.67)
Second: for Neumann boundary conditions:
32


FIGURE 1.7: Schematic illustrating the periodic boundary conditions.
Boundary
Boundary
FIGURE 1.8: Periodic boundary condition on the inflow and outflow
boundaries.
ar
Tf - Tu
dn —0.5n • elXSx
Where Tf is the temperature at the fluid node neighboring the interface, oolX is the Lattice weight and eLX is the discrete velocity
(1.68)
1.6.2 Periodic Boundary Conditions
The periodic boundary is the simplest boundary conditions. In this type of boundary, the unknowns of one boundary can be directly related to the knowns of the other boundary, i.e, it is applied directly to calculate the unknown components of the particle distribution function of the nodes located at the boundaries of the domain,e.g. inlet and outlet as shown in Fig. 1.7
For example, a periodic boundary condition is required on the inflow and
33


outflow boundaries as shown in Fig. 1.8. The periodic boundary condition on the inflow boundary
gx{xi,t + St) = gx{xn,t), a = 1,5,8
(1.69)
and on the outflow boundary
gec(xn,t + St) = gx{Xi,t), IX = 3,6,7
(1.70)
where X{ and xn are the coordinates of the inflow and outflow boundaries
34


CHAPTER II
ISOTROPIC THERMAL FLOW
2.1 Numerical Results
Here, numerical simulations of some benchmark problems are conducted to validate the accuracy of our proposed cascaded LBE model for convective thermal flows. The test problems without source terms in the energy equation are thermal Poiseuille flow, thermal flow in a channel with wall injection, and natural convection in square cavity Also, problems considered with variable source terms in the energy equation are a reaction-diffusion problem, and couette flow with temperature gradients (i.e. thermal Couette flow with viscous heat dissipation). In this study, the halfway bounce-back scheme is employed to treat velocity boundary conditions while the general anti-bounce-back scheme [48] is adopted to deal with temperature boundary conditions. In problems involving LBM solution of fluid flow all relaxation parameters are set to 1.0 except oq and 005 (the relaxation rates for the first order moments) which are both equal to rIn the thermal model all relaxation parameters are set to 1.0 except Ai and A2 which are both equal to r”1.
2.2 Unsteady Reaction-Diffusion Problem: Variable Source Term
The unsteady reaction diffusion problem is a good problem to test the accuracy of the present LBE cascaded model for the equivalent energy equation with a variable source term. Such a system defined in the region 0 < x, y < /, with macroscopic
35


FIGURE 2.1: Temperature profile for the unsteady reaction-diffusion problem with a variable source term at y = 0.5 and diffusion coefficient a = 10 3 at different times. Markers represent the Cascaded LBE results and lines represent the analytical solution.
governing equations written as in [59]:
dT
dt
ocV2T + 2C sm(nx/l)sm(ny/l)
(2.1)
Where G = 2C sm(nx/l) sm(ny/l) is the variable source term, l is the width of the region, C is a constant, and a is the diffusion coefficient. The initial and boundary conditions of this system are: T(x,y, 0) = 0, T(0,y,t) = T(l,y,t) = 0,
T(x,0,t) = T(x,l,t) = 0.
The general bounce-back scheme [48] is employed to represent these boundary conditions.
The analytical solution of this problem is given by
T(x,y,t) =
n20L
C
1 — exp
2n2ia.
sin(nx/l) sin(ny/l)
(2.2)
We conduct our numerical simulation with a grid resolution of 61 x 61, C = 10
36


FIGURE 2.2: Temperature profile for the unsteady reaction-diffusion problem with a variable source term at y = 0.5 and diffusion coefficient a = 10 4 at different times. Markers represent the Cascaded LBE results and lines represent the analytical solutions.
and with the thermal diffusivity coefficients a = 10-3 and a = 10-4. The simulation results and the analytical solutions are compared at three different times t = 50, t = 100, and t = 150 as used in [59]. The relaxation time is set to rg = 0.503 for a = 10-3, and Tg = 0.5003 for a = 10-4. Fig. 2.1, and Fig. 2.2 show the temperature profiles for both the values of diffusivity coefficients which indicate very good agreement with the analytical solutions for both thermal diffusion coefficients. We also examine the spatial accuracy of the present model. In this regard, a set of simulations are performed at four different grid sizes, i.e., 25 x 25,51 x 51,101 x 101, and 201 x 201 for both values of the diffusion coefficient, i.e. a = 10-3, and a = 10-4. The global relative error of temperature (Ej) used to measure the accuracy of the model is calculated as
Et
\\(Tc — Ta) 2
|| Ta 2
Where 11.112 is the Euclidean norm, 11 (Tc — Ta) \ \2 = \/I^ (Tcj — Taj)2,
(2.3)
37


FIGURE 2.3: Temperature global relative error at different values of the diffusion coefficient a = 10~3, D = 0.397, and a = 10~4, D = 0.3997 for the unsteady reaction-diffusion problem with variable source term.
11 (Ta) 112 = \/Li (Ta,i)2, Tc and Tn are the computed and the analytical solutions respectively. The relative global error of temperature for each value of diffusivity coefficient are plotted in Fig. 2.3. It can be seen that the temperature global error decreases with increase in grid resolution with a slope of —2 in the log-log plot. Hence, our present cascaded LBM model with source term is second order accurate.
2.3 Thermal Flow in a Channel with Wall Injection
In this section, the present cascaded LBE model for convective thermal flow is employed to simulate the fully developed thermal flow in a channel, where the upper plate moves along the x-direction with velocity Up, and a fluid is injected in the positive y-direction with a constant velocity Vo through the stationary bottom wall. The upper wall is maintained at a higher temperature (Tjt) and the bottom wall is fixed at a lower temperature (Tc). The computational domain of the problem is 0 < x, y < L. In the steady state case, the analytical solutions for both velocity and
38


temperature fields are, respectively, given by.
(2.4)
T=T ATexp[PrRe(y/L)] -1 c exp(PrRe) — 1
(2.5)
where Re is the Reynolds number defined by Re = [Lvo /v] , L is the width of the channel and AT is the temperature difference. In our numerical test, we set Up = vo = 0.01, Tfj = 1, Tc = 0, Pr = 0.71, with a grid size 31 x 61 at different Reynolds numbers Re = 5,10, and 15. The relaxation rates for the flow and thermal equation cascaded LB solvers are obtained based on the value of Pr and Re at each case where v = [Lvo/Re], and a = [v/Pr]. The rest of the relaxation rates are set to be 1.0. The general bounce-back scheme [48] is implemented at the upper and bottom plates for the temperature boundary conditions, a halfway bounce-back scheme is employed for the velocity boundary conditions, and periodic boundary conditions are imposed at the inlet and outlet of the channel. The profiles of velocity and temperature along the y direction at different Reynolds numbers and Pr = 0.71 are plotted in Figs. 2.4 and 2.5 respectively. It is found that the numerical results agree well with the analytical solutions for this test case. We also study the convergence rate by considering the following grid resolutions Ny = 31,61,91, and 121. In these simulations, we conduct the convergence study at Reynolds numbers: Re = 5,10, and 15 for the above set of grid resolutions using oq = oq = Ai = A2 = and all other relaxation parameters are set to be 1 with the corresponding values of the tunable parameter D as 0.05, 0.1, and 0.15 respectively. For the velocity field, the Reynolds number is set to be Re = 10 and relaxation rates are (ty = rg = 0.8). The relative global errors of velocity and temperature are plotted in Figs. 2.6 and 2.7. It can be seen that the relative errors have slopes of almost equal to 2.00, which a gain confirms that the present cascaded LBM model for thermal flow is second order accurate. In the above, the relative global error of temperature and velocity are defined,
39


FIGURE 2.4: Velocity profiles for thermal Couette flow in a channel with wall injection at Reynolds numbers: Re = 5,10,15. Markers represent the Cascaded LBE results and lines represent the analytical solutions.
respectively, by
c ll(E-T„)| Ej= IITII IMfll\2 2 (2.6)
r 11 {lie Ua) | — II || 2 (2.7)
| \ Ua \ \ 2
where . 2 is the Euclidean norm, \ \(TC — Tn) II2 = VLiiTci-Taj)2,
- Ua) 112 = y/Li (Uc,i - ua,iY, 11 {Ta) \ |2 = y/Li (Tn,i)2 , \ \{Ua) 2 = y/Td ('Un,i)2-
Here, Tc, u<- and Ta, ua are the computed and the analytical solutions respectively.
2.4 Diffusion in 2D Poiseuille Flow
Next, we consider a 2D Poiseuille flow between two parallel plates in the streamwise direction driven by a constant body force Fx. Both the upper and bottom walls are stationary and subjected to higher (7)t) and lower (Tc) uniform temperature respectively. The computational domain is 0 < x, y < L. Where L is the channel width. A periodic boundary condition is applied at the entrance and the exit for both velocity and temperature fields, while the halfway bounce back scheme is
40


1
0.8 -
0.6 -
0.4 - :
0.2 - :
0.2 0.4 0.6 0.8 1
FIGURE 2.5: Temperature profiles for thermal Couette flow in a channel with wall injection at Reynolds numbers: Re = 5,10,15. Markers represent the Cascaded LBE results and lines represent the analytical solutions.
implemented at the solid boundaries (upper and bottom walls) for the velocity field to represent the no-slip boundary condition. The general bounce-back scheme [48] is employed to the solid boundaries for the temperature Dirichlet boundary conditions. The analytical solution for the velocity in Poiseuille flow (parabolic profiles) is given by
u(y) = Umax(1 - (}//L)2) (2.8)
where umax = FxL2/2v is the maximum velocity occurring halfway between the plates, v is the kinematic viscosity related the to relaxation time r. Here, L is the half distance between the two parallel plates. The analytical solution for the temperature in Poiseuille flow is given by
T = Tc + AT{y/L)
(2.9)
41


FIGURE 2.6: Velocity relative global error of Couette flow with wall injection at Re = 10 .
FIGURE 2.7: Temperature relative global error of Couette flow with wall injection at Reynold numbers: Re = 5,10,15 .
42


where AT = Tjj — Tc is the temperature difference. In our simulation, a grid size of 30 x 60 is employed. We consider two cases corresponding to different sets of Reynolds numbers Re = umaxL/v, Peclet numbers Pe = umaxL/a and Prandtl number Pr = v/oc. In the first case, we set Pr = 0.71, Re = 10 and Pe = 7. In the second case, we consider Re = Pe = 10, i.e. Pr = 1. Where, we consider 7)t = 1.1,
Tc = 1, and iy = 0.674 in both cases. Fig. 2.8 presents a comparison of the velocity and temperature profiles for these cases. Excellent agreement with the analytical solution is seen.
(a) Velocity Profile Re = 10,Pe = 7 (b) Temperature Profile Re = 10,Pe = 7
(c) Velocity Profile Re = Pe = 10 (d) Temperature Profile Re = Pe = 10
FIGURE 2.8: Velocity and temperature profiles of Poiseuille flow with thermal diffusion at different values of Re and Pe. Markers represent the Cascaded LBE results and lines represent the analytical solution.
43


2.5 Natural Convection in a Square Cavity
We now present a validation study involving coupled thermal convective flow. In this regard, our cascaded LBE model is employed to simulate natural convection in a square cavity. Here, the flow is driven by the buoyancy force due to the local temperature difference against a reference temperature in the present of gravity. The left wall is maintained at higher temperature 7)t and the right wall at lower temperature Tc, while the top and bottom walls are considered to be adiabatic.The macroscopic governing equations can be expressed as follows:
V • u = 0,
——b u • Vm = —VP + vV2m + F,
dt p
r)!-1
— + u- VT = V- (aVT).
(2.10a)
(2.10b)
(2.10c)
where F is the body force which is based on the Boussinesq approximation and is given by
F = gfi(T - T0)T (2.11)
Here, [5 is the thermal expansion coefficient, g is the acceleration due to gravity,
To = (T/, + P,-)/2 is the reference temperature, j is the unit vector in positive y-direction. This classical natural convection problem is governed by two non-dimensional parameters: The Prandtl number Pr and the Rayleigh number Ra, which are given by
Pr = — a
Ra = (2.1:
VOL
where, AT = — Tc is the temperature difference between hot and cold walls, and
H is the height of the square cavity.
44


T=T„ u =0 v = 0
T=T, u =0 v = 0
r)T â– ' '
II =
I.
v = 0
X
FIGURE 2.9: Schematic illustrating the cavity boundary conditions.
The boundary conditions on the cavity walls can then be summarized as:
On the left wall:
ux = Uy = 0, T = Th = 21, (2.14)
On the right wall:
Ux - Uy -- 0,
T=TC = 1,
(2.15)
On the top wall:
Ux — Uy — 0,
(2.16)
On the bottom wall:
ST
ux = Uy = 0, — = 0. (2.17)
The general bounce-back scheme [48] is adopted to treat the thermal Dirichlet and Neumann boundary conditions. In this simulations, Pr = 0.71, the relaxation rates for fluid flow and temperature are set as 0.55, 0.57 respectively. The streamlines and the isotherms for the ranges of Rayleigh number Ra between 103 to 106 are shown in Fig. 2.12. Also, the vorticity contours for Ra = 103 — 106 are shown in Fig. 2.14. The streamlines, isotherms and vorticity contours are in very good correspondence and consistent with prior benchmark solution results [15,16]. The natural convection flow patterns become more complex as Ra increases. In order to characterized this in more
45


detail, the temperature at the vertical and horizontal mid-planes of the square cavity,
i.e. x/H = 0.5 and y/H = 0.5, respectively for various Rayleigh numbers
(Ra = 103 — 106) are presented in Figs. 2.10 and 2.11. In these cases, the value of the
factor gfi reeded in the simulation is obtained as a function of Rayleigh number Ra
using
— vix^a SP ~ A TH3
(2.18)
The representative values of gfi corresponding to each Ra is shown in Table 3.2. From Figs. 2.10 and 2.11, it is seen that the temperature contour lines become almost horizontal around the center of the cavity as the Rayleigh number Ra increases. The streamlines become more packed next to the side wall as Ra increases, i.e. the flow moves faster as natural convection is intensified. Finally, Table 3.6 shows quantitative comparison between the key parameters for this problem (average Nusselt number, maximum velocities magnitudes and their locations) between the present cascaded LBE results and benchmark data [15,16]. Excellent agreement is seen (within 0.01 percent). The streamlines and isotherms for all Rayleigh numbers Ra = (103 — 106) as TABLE 2.1: Values of gf5 corresponding to each Rayleigh number.
Rayleigh number Ra Grid Size (Nx x Ny) op, — SP a TH3
103 128 x 128 9.44 x 10“9
104 128 x 128 9.44 x 10“8
105 128 x 128 9.44 x 10“7
106 128 x 128 9.44 x 10“6
shown in Fig.( 2.12) show that the computed results using the present cascaded LBM are in excellent agreement (within 0.01 percent) with the data by [15] and [16]. These numerical results for the comparison are presented in Table ( 3.6). In addition, the temperature contour lines become almost horizontal lines around the center of the cavity as the Rayleigh number increases and the stream lines become more packed next to the side wall as the Rayleigh number increases, i.e. the flow moves faster as natural convection is intensified.
46


FIGURE 2.10: Temperature profiles along horizontal centerline of the cavity at various Rayleigh numbers: Ra = 103,104,105, and 106 computed
using the cascaded LBM.
FIGURE 2.11: Temperature profiles along the vertical centerline of the cavity flow at various Rayleigh numbers Ra = 103,104,105, and 106 computed using the cascaded LBM.
47


FIGURE 2.12: Isotherms at different values of Rayleigh numbers Ra = lOy 104,105 and 106 for natural convection in a square cavity computed
using the cascaded LBM.
2.6 Thermal Couette Flow with Viscous Heat Dissipation: Modeling Heat Source
Finally, we consider the simulation of Couette flow with temperature gradient to test the ability of the present thermal cascaded LB model with a source term to describe the viscous heat dissipation. We consider 2D thermal couette flow between two parallel plates, where the upper plate moves along x-direction with a velocity U, and at higher temperature Tju whereas the bottom wall is stationary and maintained at lower temperature Tc; L is the distance between the two plates. In this case the source term G in the thermal energy equation, Eq. (1.17) represents the viscous heat dissipation and is given by
2v
G = —(S:S) (2.19)
48


(b) Streamlines Ra = 104
FIGURE 2.13: Streamlines at different values of Rayleigh numbers Ra = lOy 104,105 and 106 for natural convection in a square cavity computed
using the cascaded LBM.
S = 1(V«+(Vk)t)
(2.20)
Where S is strain rate tensor, and Cp is the specific heat at constant pressure. The macroscopic governing equations for momentum and energy can be written, respectively, as:
d2ll
= 0/
at/2
d2T v ldll
adf + c'p [ay.
n 2
= 0.
The analytical solutions of the velocity and temperature are then given by:
(2.21)
(2.22)
u{y)
ii|, v = 0
(2.23)
49


FIGURE 2.14: Vorticity contours at various Rayleigh numbers Ra = lOy 104,105 and 106 for natural convection in a square cavity computed
using the LBM.
P_
Tc
| + 0.5Br| (l
(2.24)
Where Ec = ^ t v Eckert number, and Pr = v /a is the Prandtl number. The effect of viscous heat dissipation is controlled by the Brinkman number Br = Ec.Pr. In this simulation, we set iy = 0.9, Tc = 1 with a grid resolution of 5 x 41, Re = 10, and Pr = 0.71 at Eckert numbers 7,14, and 28. For the solution of this problem, it is important to note that the convection diffusion equation with a source term (1.17) is coupled with Navier-Stokes equations (1.18b) and (1.18b). This is another cascaded LB model [44] is used to solve the N-S equations. The source term G in Eq. (2.19) can be written as
G
2i/
cp
C 2 _i_ C 2 , 2 ^xx i yy i t-^xy
(2.25)
50


In this case, we have
(2.26)
These strain rate components can be computed locally using the non-equilibrium moments in the cascaded LBE for the fluid flow as mentioned in [44], Figure (2.15) shows the temperature profiles for different Eckert numbers. In addition, we carried out simulation for different values of Prandtl numbers, 0.25,1.25, and 2.5 with the Eckert number being fixed at 8 with a grid size 5 x 41. Figure (2.16) shows the temperature profiles for different Prandtl numbers. In both these cases, very good agreement between the cascaded LBM results and analytical solutions seen.
Next, we study the convergence rate of the case with a fix Ec = 8 at different values of the Prandtl number: 0.25,1.25, and 2.5. Here, we set U = 0.07,
004 = 0J5 = Ai = A2 = 0^9, and all other relaxation parameters are set to be 1. The values of tunable parameter D at each Prandtl number are -1.2, 0.08, and 0.24 respectively. We also conduct the convergence study for the case where the Prandtl number is fixed at Pr = 0.71 while the Eckert number is changed as 7,14, and 28,
004 = CO5 = 0.9, Ai = A2 = 0.94, D = 0 and all other relaxation parameters are set to be 1. Three grid resolutions in y-direction (Ny=41,81, and 161) are employed to both cases of the convergence study. Figs. (2.17) and (2.18) show that the slope of the temperature relative global error is about 2, i.e. the present thermal cascaded LB model is of second order accuracy in space.
Couette flow with viscous heat dissipation was also used to test the ability of the present thermal cascaded Lattice Boltzmann model to simulate relatively high Peclet numbers at low grid resolution 5 x 23. The analytical solutions of temperature is given by Eq. (2.24), Here, the Brinkman number Br is rewritten as Br =
(2.27)
51


FIGURE 2.15: Temperature profiles in Couette flow at various values of Eckert number. Markers represent the cascaded LBE simulations and lines represent the analytical solutions.
FIGURE 2.16: Temperature profiles in Couette flow with different values of Prandtl number. Markers represent the cascaded LBE simulations and lines represent the analytical solutions
52


FIGURE 2.17: Temperature global relative error at different Eckert numbers 7,14, and 28 for thermal Couette flow with viscous heat dissipation.
Log(A x)
FIGURE 2.18: Temperature global relative error at different Prandtl numbers 0.25,1.25, and 2.5 for thermal Couette flow with viscous heat dissipation.
53


Where Pe is Peclet number. In this case study, the temperature profiles for fixed Reynolds number (Re = 10), and Eckert number (Ec = 0.1) at different values of Peclet numbers 10,102,103, and 104 are shown in Fig. 2.19. Here, we set Th = 1,
Tc = 0, (U4 = (U5 = Ai = A2 = 1.063, and all other relaxation parameters set to be 1. The values of tunable parameter D at each Peclet number are 0, 0.397,0.4366, and 0.4406 respectively. Excellent agreement between cascaded LBM results and analytical solution is seen for relatively high Peclet number results. This result is indicative of the improved stability properties of the cascaded LBM as researchers utilizing SRT LBM [17] only presented results for Brinkman numbers up to Br = 100. The convergence study was done for this problem at different values of Peclet numbers 10,102, and 103 for grid resolutions Ny=81,161, and 321,
CV4 = CO5 = Ai = A2 = 1.11, and all other relaxation parameters are set to be 1; the values of tunable parameter D at each Peclet number are set to be 0, 0.36, and 0.696, respectively. The relative global error of temperature against the grid resolutions is shown in Fig. 2.20. It is evident that the slope of the temperature relative global error is about —2, .i.e the present thermal cascaded LB model is again of second order accuracy in space at relatively high Peclet numbers 10,102,103,104,105 and 106 (corresponding to Brinkman numbers Br = 10-4,1,104,102,103 and 104.
2.7 Summary and Conclusion
In this chapter, we presented the isotropic thermal cascaded (LB) MRT model based on central moments and including a source term. This model solves the convection-difusion equation (CDE) for the temperature field within the double distribution function framework for the D2Q9 lattice, where the fluid motion is represented by another cascaded LB model constructed in in [44], The collision operator for the thermal field has significantly different cascaded structure for its collision kernel where compared to that for the flow field due to the differences in the number of collision invariants between them. A consistent second order scheme to
54


FIGURE 2.19: Temperature profiles in thermal Couette flow at various values of Peclet numbers: Pe = 10,102,103,104,105 and 106. Markers represent the Cascaded LBE results and lines represent the analytical solution.
FIGURE 2.20: Temperature global relative error at different Peclet numbers: Pe = 10,102, and 103 for thermal Couette flow with viscous heat
dissipation.
55


incorporate the effect of locally varying heat sources by means of a variable transformation for the thermal cascaded LB model is also discussed. A Chapman-Enskog analysis of the thermal cascaded LB model shows its consistency with the CDE including a source term. It also provides expressions for temperature gradients in the augmented moment equilibria in terms of locally known non-equilibrium moments. The new thermal cascaded LBE is validated for a number of benchmark problems, including thermal Poiseuille flow, thermal Couette flow and natural convection in a square cavity. Comparison of the temperature profiles under different conditions for these problems, as well as the average Nusselt number at different Rayleigh numbers in the case of the natural convection within a square cavity, with prior benchmark results demonstrate high accuracy of the isotropic thermal cascaded LBE model. Furthermore, it is shown numerically that the model is second order accurate in space for a range of thermal convective flow problems.
56


TABLE 2.2: Comparison between numerical results obtained using the cascaded LBM and the published results (de Vahl Davis(1983) and Hort-mann et al(1990)) at different Rayleigh numbers (Ra = 103 — 106).
Ra Parameter Present Cascaded LBM de Vahl Davis [15] Hortmann et al(1990) [16]
Nu 1.117 1.116 NA
Umax 3.605 3.634 NA
CD o ymax 0.816 0.813 NA
rH ^max 3.654 3.679 NA
%max 0.176 0.179 NA
l^lmax 1.16 1.174 NA
Nu 2.237 2.234 2.24475
Umax 16.182 16.182 16.1759
ymax 0.824 0.823 0.8255

O rH "Umax 19.551 19.509 19.6242
%max 0.12 0.12 0.12
l^lmax 5.09 5.098 NA
Nu 4.509 4.51 4.521
Umax 35.137 34.81 34.7398
ymax 0.856 0.855 0.85312
ID
o rH "Umax 68.511 68.22 68.6465
%max 0.064 0.066 0.656
l^lmax 9.189 9.144 NA
Nu 8.797 8.798 8.825
Umax 65.57 65.33 64.865
ymax 0.856 0.851 0.8532
o
O rH "Umax 219.95 216.75 219.861
%max 0.032 0.0387 0.0406
l^lmax 16.519 16.53 NA
57


CHAPTER III
ANISOTROPIC THERMAL FLOW
3.1 Numerical Results
In this section, numerical tests are carried out to validate the accuracy of our
The applicability of the anisotropic LBE model is validated by numerical simulations including the convection and diffusion of a Gaussian Hill, solving anisotropic convection diffusion equation with variable diffusion tensor and variable source term, and anisotropic natural convection in a square cavity In this study, the periodic boundary conditions and bounce-back are employed for temperature boundary conditions.
3.1.1 Convection and Diffusion of a Gaussian Hill: No Source Term
The following convection-diffusion equation (CDE) without a source term
proposed cascaded LBE model for anisotropic convective diffusion thermal flows.
+ u • V(p = V • (txVtp)
(3.1)
Here, the initial distribution of the scalar variable is given as
(3.2)
58


Where do2 is the initial variance, (po = 2ndo2 is the total concentration. The analytical solution of this problem is given by
(3.3)
Where crt = (Jq21 + 2toc. | |o>| | is the absolute value of the determinant of crt, while df1 is the inverse of crt The computational domain is chosen to be [—1,1] x [—1,1], Three types of diffusion tensors are considered
Which represent the isotropic convection diffusion, diagonal anisotropic convection diffusion, and full anisotropic convection diffusion problems respectively. The periodic boundary condition is employed to this problem. We conduct our numerical simulation with a grid resolution of 151 x 151, do = 0.05 . First, we examine the pure diffusion for the three diffusion tensors given above by setting u = (0,0) at time t,„ as shown in Fig. 3.2 and Fig. 3.3 then we test the convection diffusion case by set u = (0.05,0.05) at time tm, and 0.5tm as shown in Fig. 3.4 . The time tm is determined by Where | |a| | = |an^22 ~ an^2i |/ tm is equal to 6.25, 5.5375, 5.8547 for the isotropic, diagonal anisotropic, and full anisotropic case respectively.
different diffusion tensors as in [59]. The relaxation rates are taken An = A22 = 1-96, A12 = A21 = 0 for isotropic case, An = 1.96, A22 = 1-85, An = A21 = 0 for diagonal anisotropic case, and An = 1-96, A22 = 1-85, An = A21 = 0.036 for full anisotropic
(3.4)
The simulation results and the analytical solutions are compared at the three
case, with 3x2 /5t = 0.03. The good agreement with the analytical solutions is shown for all thermal diffusion coefficients.
59


(a) Isotropic diffusion, t=tm
(b) Diagonally anisotropic diffusion, t=tm
FIGURE 3.1: Distribution of the scalar variable (p at the time t = tm and
n = v = 0.0 .
60


(a) Fully anisotropic diffusion, t=tm
(b) Analytical Solution
FIGURE 3.2: Distribution of the scalar variable (p for diffusion u = (0,0) of a Gaussian hill computed using the LBM.
61


(c) Diagonally anisotropic diffusion, t=tm (d) Fully anisotropic diffusion, t=tm
FIGURE 3.3: Concentration contours for diffusion of a Gaussian hill u =
(0,0) computed using the LBM.
We also test the convergence rate of the present model. For this purpose, a set of simulations are conducted at four different grid sizes, i.e., 101 x 101,151 x 151,
201 x 201, and 301 x 301. The global relative error of temperature (Ej) used to measure the accuracy of the model is calculated as
ET = milTln)lh (3-5)
Where 11.112 is the Euclidean norm, 11 (Tc — Tn) \ |2 = \/X^ {Tc,i — Tn,i)2,
11 {Ta) 112 = y/Li (Ta,i)2, Tc and Tn are the computed and the analytical solutions respectively. The relative global error of temperature for the isotropic, diagonal anisotropic, and full anisotropic case are plotted in Fig. 3.8. It can be seen that the temperature global error decreases with increase in grid resolution with a slope of 2 in the log-log plot. Hence, our present cascaded LBM model with source term is
62


second order accurate.
(a) Isotropic diffusion
(b) Diagonally anisotropic
(c) Fully anisotropic diffusion
FIGURE 3.4: Contours of the scaler variable (p at a = (0.05,0.05) computed using LBE
3.1.2 Convection-Diffusion of a Gaussian Hill: Stability Test
We consider the convection diffusion equation, Eq. (1.17), where u = u0i + v0j is a prescribed 2-D uniform velocity field and subjected to the Gaussian hill initial condition.
T(x, i/,0)
To 2 n do
2 exP
f-[x2 + y2}\
V 2u02 ) '
(3.6)
63


(a) Isotropic diffusion
(b) Diagonally anisotropic
(c) Fully anisotropic diffusion
FIGURE 3.5: Analytical Contours of the scaler variable ^ at u =
(0.05,0.05)
where the parameter j0 controls the width of the profile. The analytical solution of this problem is given by
T{x,y,t) =
2n((T02 + led)
exp
-[(x - ii0t)2 + (y-v0t)2} l(j02 + Ixt)
(3.7)
We set J0 = 0.05 and advect the profile with the diagonal velocity vector u0 = v0 = 0.25cs. We choose T0 = Inj2 so that the initial profile has a peak magnitude of 1.0. Periodic boundary conditions for the temperature are employed. In what follows we vary the fluid diffusivity to compare the stability characteristics of the cascaded centeral moment LBM with the SRT and MRT implementations of the LBM. We consider the MRT method in [59] and we set the tunable parameters in both methods to be zero. In the three methods the fluid diffusivity is given by
64


ol = cs2(j — |) where A is the relaxation rate of the first order moments corresponding to the equilibrium moments uxT and UyT in the cascaded and MRT LBM. We use a 521 x 521 grid and vary the diffusivity by varying the relaxation time rg = j. Comparison of the cascaded and MRT LBM methods is complicated by the large number of relaxation parameters associated with each method. We set the relaxation time for the first order moments to Tg. We then set all other relaxation times to 1.0. With this choice of parameters we are relaxing the energy fluxes (first order moments) at the same rate in both methods. The higher order moments are also relaxed at the same rate. This choice of parameters is not necessarily optimal for either method but it does give us a rational basis for comparison.
Finally, we consider numerical stability results from the SRT, MRT, and cascaded LBM at various values of the relaxation time rg. Table 3.1 provides the numerical stability and global relative error (given by Eq. (3.5) indicates the accuracy of the three methods is similar. The table also indicates that the SRT LBM is not stable at the smaller diffusivities and that the MRT is eventually not stable at the still smaller diffusivity for which the cascaded LBM is stable. It is likely that relaxation rates for the higher moments of the MRT and cascaded LB methods may be found that result in more stable behavior but this short study concurs with other work that indicate superior stability characteristics of the cascaded LBM [61, 63].
TABLE 3.1: Global relative error and stability characteristics after 1000 time increments for SRT, MRT, and Cascaded LBM for different relaxation times for the convection-diffusion of a Gaussian hill problem at Mach
number 0.25.
T* 0.55 0.51 0.501
a 1.67 x 10~2 3.33 x 10“3 3.33 x 10“4
SRT 0.0097 unstable unstable
MRT 0.0086 0.0096 unstable
Cascaded 0.0101 0.0108 0.0110
65


3.1.3 Anisotropic Convection Diffusion Equation: Variable Source Term
The following anisotropic convection-diffusion equation (CDE) with a constant velocity and variable diffusion tensor oc and with a distributed source term
^ + u • V(p = V • (a • V(p) + G (3.8)
Where u = (uX/ uy) and G = G(x,y, t) is the source term. The initial distribution of the scalar variable is given by
cp(x/y,0) = sin(2nx)sin(2ny). (3.9)
The analytical solution of this problem is given by
sin(2nx)sin(2ny).
The source term is defined as
= exp
1 — 12n2tx) t
(3.10)
G = exp
1 — 127T2x] t
sin(2nx)sin(27Ty) + 4oicos(4nx)sin2 (2ny)
+2n \uxcos(2nx)sin(2ny) + UySin(2nx)cos(2ny)] } .
(3.11)
The computational domain is chosen to be [0,1] x [0,1]. Here, the diffusion tensor is a function of space x,y. The diffusion tensor oc is given by a diagonal matrix
oc = a x

\
sin(2nx)sin(2ny)
0
(3.12)
Where a is constant and is given to be a = 1.0 x 10-3. Here, the diffusion tensor oc represents the diagonal anisotropic convection diffusion. The periodic boundary condition is employed to this problem. We conduct our numerical simulation with a grid resolution of 101 x 101 as shown in Fig. 3.7, we test the convection diffusion case
66


by set u = (0.1,0.1) at time t = 3 and Pe = 100. The relaxation time is set to Tn = 3 x 10_3(2 — sin(2nx)sin(2ny)) + 0.5, T22 = 0.54, T12 = T21 = 0. Fig. 3.7 shows the temperature distribution for the given values of diffusivity coefficients which show very good agreement with the analytical solutions. We also examine the spatial accuracy of the present model. In this regard, a set of simulations are performed at five different grid sizes, i.e., 201 x 201, 301 x 301,401 x 401, 501 x 501 and 521 x 521 for given value of the diffusion tensor. The relative global error of temperature for the given value of diffusivity coefficient are plotted in Fig. 3.9. It can be seen that the temperature global error decreases with increase in grid resolution with a slope of —2 in the log-log plot. Hence, our present cascaded LBM model with source term is second order accurate.
3.2 Natural Convection in a Square Cavity: Anisotropic Case
Natural convection heat transfer and fluid flow in cavities are important subjects of investigation due to their effect on many engineering applications and nature phenomena, such as thermal power, petrochemical industries, aerospace, construction and solar collectors. In this section, the effect of anisotropy is performed for a heated cavity for the following values of Rayleigh number (Ra = 103,104,105,106,107 and 108). In this regard, we define the x thermal diffusivity as ocx and y thermal diffusivity as oty cascaded LBE model is employed to simulate natural convection in a square cavity. Here, the flow is driven by the buoyancy force due to the local temperature difference against a reference temperature in the present of gravity. The left wall is maintained at higher temperature 7)t and the right wall at lower temperature Tc, while the top and bottom walls are considered to be adiabatic.The macroscopic governing equations can be
67


(a) Numerical Solution
(b) Analytical solution
FIGURE 3.6: Distribution of the scalar variable (p at the time t = 3 and
Pe = 100 .
68


(a) Numerical Solution
(b) Analytical solution
FIGURE 3.7: Contours of the scalar variable (p at the time t 3 and Pc =
100.
69


Log(GRE)
FIGURE 3.8: Temperature global relative errors at different grid sizes for
diffusion of a Gaussian hill.
FIGURE 3.9: Temperature global relative error with variable diffusion tensor and source term.
70


expressed as follows:
dx dy = o,
du du du Tt+Ud~x + Vfy 1 dp = b V p dx / d2U \3x2
dv dv dv di + uTx + '% 1 dp = +v pdy / d2V

dT dT dT b U b V dt dx dy d2T d2T dy2'
d2u
dy2
d^y
dy
+
+ ^2 1 + F'
(3.13a)
(3.13b)
(3.13c)
(3.13d)
(3.13e)
where F is the body force which is based on the Boussinesq approximation and is given by
F = gp(T - T0)T (3.14)
Here, is the thermal expansion coefficient, g is the acceleration due to gravity,
To = (7), + T<-)/2 is the reference temperature, j is the unit vector in positive y-direction. This classical natural convection problem is governed by two non-dimensional parameters: The Prandtl number Pr
„ v Pr = — a
and the Rayleigh number Ra, which are given by
Tcix
gpATH3
V0Lx
(3.16)
= gjAnr
y vcty
where, AT = — Tc is the temperature difference between hot and cold walls, and
H is the height of the square cavity.
71


otx is x thermal diffusivity and oty is y thermal diffusivity. The choice for the anisotropic as follows.
x thermal diffusivity given by ctx = 0.5cty and x thermal diffusivity given by
olx = loiy for different Rayleigh numbers (Ray = 103,104,.......108). The average
of Nusselt number for the right wall is given by
1 fH
Nu = wvr / x'V)dV (3-18)
XxlXl JO
where qx{x,y) = uT{x,y) — otx^ is the local heat flux in x-direction.
The boundary conditions on the cavity walls can then be summarized as:
On the left wall:
ux = Uy = 0, T = Th = 21, (3.19)
On the right wall:
UX — Uy — 0,
T = Tc = 1,
(3.20)
On the top wall:
UX — lly — 0,
(3.21)
On the bottom wall:
Ux = Uy = 0, — = 0. (3.22)
The general bounce-back scheme [48] is adopted to treat the thermal Dirichlet and Neumann boundary conditions. In this simulations, Pr = 0.71, the relaxation rates for fluid flow and temperature are set as 0.55, 0.57 respectively. The streamlines and the isotherms for the ranges of Rayleigh number Ray between 103 to 106 are shown in Fig. 2.12. Also, the vorticity contours for Ray = 103 — 108 are shown in Fig. 2.14. The streamlines, isotherms and vorticity contours are in very good correspondence and consistent with prior benchmark solution results [15,16]. The natural convection flow patterns become more complex as Ra increases. In order to characterized this in more detail, the temperature at the vertical and horizontal mid-planes of the square cavity,
72


i.e. x/H = 0.5 and y/H = 0.5, respectively for various Rayleigh numbers
(Ray = 103 — 108) are presented in Figs. 2.10 and 2.11. In these cases, the value of the
factor gfi reeded in the simulation is obtained as a function of Rayleigh number Ra
using
VOLyRtty
8^~ A TH3
(3.23)
The representative values of gfi corresponding to each Ra is shown in Table 3.2. From Figs. 2.10 and 2.11, it is seen that the temperature contour lines become almost horizontal around the center of the cavity as the Rayleigh number Ra increases. The streamlines become more packed next to the side wall as Ra increases, i.e. the flow moves faster as natural convection is intensified. Finally, Table 3.6 shows quantitative comparison between the key parameters for this problem (average Nusselt number, maximum velocities magnitudes and their locations) between the present cascaded LBE results and benchmark data [15,16]. Excellent agreement is seen (within 0.01 percent). The streamlines and isotherms for all Rayleigh numbers Ray = (103 — 108)
TABLE 3.2: Values of gf5 corresponding to each Rayleigh number.
Rayleigh number Ra Grid Size (Nx x Ny) _ KUy.VMy SP ATH3
103 128 x 128 9.44 x 10“9
104 128 x 128 9.44 x 10“8
105 128 x 128 9.44 x 10“7
106 128 x 128 9.44 x 10“6
107 128 x 128 9.44 x 10“5
108 521 x 521 1.37 x 10“5
as shown in Fig.( 2.12) show that the computed results using the present cascaded LBM are in excellent agreement (within 0.01 percent) with the data by [15] and [16]. These numerical results for the comparison are presented in Table ( 3.6). In addition, the temperature contour lines become almost horizontal lines around the center of the cavity as the Rayleigh number increases and the stream lines become more packed next to the side wall as the Rayleigh number increases, i.e. the flow moves faster as natural convection is intensified.
73


0 0.2
0.8 - os:
m 0.6 - 0.4 - 0.6 - [((( o)))
0.2 - 0.2 - =^J
0 0.2
0 0.2
0.6 0.8
3.3 Summary and Conclusion
In this chapter, we presented the numerical results for simulation of anisotropic thermal cascaded (LB) MRT model based on central moments and including a source term. This model solves the anisotropic convection-difusion equation (CDE) for the temperature field within the double distribution function framework for the D2Q9 lattice, where the fluid motion is represented by another cascaded LB model constructed in [44], A Chapman-Enskog analysis of the anisotropic thermal cascaded
74


FIGURE 3.10: Streamlines at different values of Rayleigh numbers Ra = lOh 104, 105,106,107 and 108 for natural convection in a square cavity computed using the cascaded LBM.(Left) txx = (Xy /2, (center) ax = (Xy, (right)
Oi x = 2 (Xy
LB model shows its consistency with the CDE including a source term. It also provides expressions for temperature gradients in the augmented moment equilibria in terms of locally known non-equilibrium moments.
The new thermal cascaded LBE is validated for a number of benchmark problems, including convection-diffusion of a Gaussian Hill, solving anisotropic convection diffusion equation with variable diffusion tensor and variable source term, and anisotropic natural convection in a square cavity. Comparison of the temperature profiles under different conditions for these problems, as well as the average Nusselt number at different Rayleigh numbers in the case of the anisotropic natural convection within a square cavity, with prior benchmark results demonstrate high accuracy of the anisotropic thermal cascaded LBE model. Furthermore, it is shown numerically that the model is second order accurate in space for a range of anisotropic thermal convective flow problems.
Finally, a stability test of the present cascaded LBE model is conducted to compare our model by single relaxation time (SRT) LB model and multiple relaxation
75


times (MRT) LB model using the diffusion in a Gaussian Hill as a test problem by varying the fluid diffusivity to compare the stability characteristics of the cascaded central moment. This stability study concurs with other work that indicates superior stability characteristics of the cascaded LB method. The present anisotropic thermal cascaded LB model exhibits improved stability characteristics over the SRT LB model and the conventional multiple MRT LB model.
76


FIGURE 3.11: Vorticity at different values of Rayleigh numbers Ra = lOh 104, 105,106,107 and 108 for natural convection in a square cavity computed using the cascaded LBM.(Left) txx = (Xy /2, (center) ax = (Xy, (right)
Oi x = 2 (Xy
77


TABLE 3.3: Numerical results obtained using the cascaded LBM for anisotropic natural convection in a square cavity at different Rayleigh numbers (Ra = 103,104,105,106,107 and 108).
Ray Parameter ax = 0.5 dy %-x — ^y CCX 2(Xy
Nu 1.247 1.1171 1.045
Umax 3.3417 3.6046 3.7635
CD o ymax 0.816 0.8160 0.816
rH ^max 3.429 3.6540 3.78
%max 0.168 0.1760 .168
l^lmax 1.059 1.16 1.195
Nu 2.681 2.2374 1.8456
Umax 12.408 16.1827 21.058
ymax 0.832 0.8240 0.824

O rH "Umax 15.98 19.5519 23.879
%max 0.104 0.1120 0.128
l^lmax 3.794 5.09 6.62
Nu 5.458 4.5090 3.645
Umax 23.825 35.1375 56.294
ymax 0.856 0.8560 0.864
ID
o rH "Umax 53.03 68.5111 85.669
%max 0.0560 0.0640 0.072
l^lmax 6.475 9.189 13.385
Nu 10.636 8.7978 7.031
Umax 51.03 65.5747 121.825
ymax 0.8960 0.8560 0.8720
o
O rH "Umax 166.44 219.9522 281.8181
%max 0.032 0.0320 0.04
l^lmax 10.884 16.519 22.276
Nu 20.1367 16.5112 13.2339
Umax 116.933 150.194 284.727
ymax 0.936 0.8782 0.856
bs
O rH "Umax 523.749 703.414 904.581
%max 0.016 0.016 0.024
l^lmax 19.209 29.72 42.39
Nu 36.7660 30.1552 24.1777
Umax 245.329 427.56 851.476
ymax 0.9534 0.9496 0.778
00
O rH "Umax 1659.536 2259.297 2951.734
%max 0.0097 0.0116 0.0116
l^lmax 33.89 52.9 76.18
78


TABLE 3.4: Comparison between numerical results obtained using the cascaded LBM and the published results Dubois et aZ(2016) [64] for anisotropic natural convection in a square cavity at different Rayleigh numbers (Ra = 103,104,105 and 106).
Ray Parameter Model % = 0.5 dy %-x — ^y CCX 2(Xy
Nu Dubois et al [64] 2.4957 1.1179 0.5226
present 1.2474 1.1171 1.0447
CD o
rH Umax Dubois et al [64] 3.3705 3.6496 3.8185
present 3.3417 3.6046 3.7635
ymax Dubois et al [64] 0.8142 0.8142 0.8142
present 0.8160 0.8160 0.8160
^max Dubois et al [64] 3.4515 3.6973 3.8428
present 3.4293 3.6540 3.7909
%max Dubois et al [64] 0.1761 0.1761 0.1857
present 0.1680 0.1760 0.1760
Nu Dubois et al [64] 5.3711 2.2438 0.9261
present 2.6811 2.2374 1.8456

o
rH Umax Dubois et al [64] 12.3628 16.1881 21.1512
present 12.4082 16.1827 21.0584
ymax Dubois et al [64] 0.8290 0.8225 0.8225
present 0.8320 0.8240 0.8240
"Umax Dubois et al [64] 16.0159 19.6323 24.0035
present 15.9800 19.5519 23.8797
%max Dubois et al [64] 0.1064 0.1193 0.1322
present 0.1040 0.1120 0.1280
Nu Dubois et al [64] 10.9325 4.5177 1.8262
present 5.4584 4.5090 3.6451
ID o
rH Umax Dubois et al [64] 23.5783 34.7486 56.0032
present 23.8250 35.1375 56.2946
ymax Dubois et al [64] 0.8560 0.8560 0.8609
present 0.8560 0.8560 0.8640
"Umax Dubois et al [64] 53.5863 68.6527 86.0559
present 53.5500 68.5111 85.6690
%max Dubois et al [64] 0.0609 0.0658 0.0707
present 0.0560 0.0640 0.0720
79


Ray Parameter Model ax = 0.5 dy %-x — ^y CCX 2(Xy
Nu Dubois et al [64] 21.2805 8.8062 3.5197
O O rH present 10.6367 8.7978 7.0308
Umax Dubois et al [64] present 50.6999 51.0300 64.8428 65.5747 120.0525 121.8256
ymax Dubois et al [64] present 0.9000 0.8960 0.8490 0.8560 0.8647 0.8720
^max Dubois et al [64] present 166.5744 166.4405 220.6695 219.9522 282.8172 281.8181
%max Dubois et al [64] present 0.0333 0.0320 0.0372 0.0320 0.0411 0.0400
80


TABLE 3.6: Comparison between numerical results obtained using the cascaded LBM and the published results (de Vahl Davis(1983) and Hort-mann et al{1990)) for isotropic natural convection in a square cavity at different Rayleigh numbers (Ra = 103 — 106).
Ray Parameter Present Cascaded LBM de Vahl Davis [15] Hortmann et al(1990) [16]
Nu 1.117 1.116 NA
Umax 3.605 3.634 NA
CD o ymax 0.816 0.813 NA
rH ^max 3.654 3.679 NA
%max 0.176 0.179 NA
y¥\max 1.16 1.174 NA
Nu 2.237 2.234 2.24475
Umax 16.182 16.182 16.1759
ymax 0.824 0.823 0.8255

O rH "Umax 19.551 19.509 19.6242
%max 0.12 0.12 0.12
\^\max 5.09 5.098 NA
Nu 4.509 4.51 4.521
Umax 35.137 34.81 34.7398
ymax 0.856 0.855 0.85312
ID
o rH "Umax 68.511 68.22 68.6465
%max 0.064 0.066 0.656
\^\max 9.189 9.144 NA
Nu 8.797 8.798 8.825
Umax 65.57 65.33 64.865
ymax 0.856 0.851 0.8532
o
O rH "Umax 219.95 216.75 219.861
%max 0.032 0.0387 0.0406
\^\max 16.519 16.53 NA
Nu 8.797 8.798 16.52
Umax 65.57 65.33 64.865
ymax 0.856 0.851 0.8532
bs
O rH "Umax 219.95 216.75 219.861
%max 0.032 0.0387 0.0406
\^\max 16.519 16.53 30.24
Nu 8.797 8.798 30.48
Umax 65.57 65.33 64.865
ymax 0.856 0.851 0.8532
00
O rH "Umax 219.95 216.75 219.861
%max 0.032 0.0387 0.0406
\^\max 16.519 16.53 54.32
81


82


FIGURE 3.12: Isothermals at different values of Rayleigh numbers Ra = lOy 104,105 and 106 for natural convection in a square cavity computed using the cascaded LBM.(Left) ax = cty /2, (center) ax = (Xy, (right)
Kx = 2IX y
83


Full Text

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CASCADEDLATTICEBOLTZMANNMODELFORISOTROPICAND ANISOTROPICCONVECTIVETHERMALFLOWSWITHLOCALHEATSOURCES by FATMAM.ELSEID B.S.,UniversityofTripoli,1994 M.S.,UniversityofBelgrade,2000 Athesissubmittedtothe FacultyoftheGraduateSchoolofthe UniversityofColoradoinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy EngineeringandAppliedScienceProgram 2019

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ThisthesisfortheDoctorofPhilosophydegreeby FATMAM.ELSEID hasbeenapprovedforthe EngineeringandAppliedScienceProgram by PeterJenkins,Chair SamuelWelch,Advisor SedatBiringen MarcIngber DavidMays Date:August3,2019 ii

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ElseidM.FatmaPhD,EngineeringandAppliedScienceProgram CascadedLatticeBoltzmannModelforIsotropicandAnisotropicConvective ThermalFlowswithLocalHeatSources ThesisdirectedbyDr.SamuelWelch ABSTRACT AnewcascadedcentralmomentbasedlatticeBoltzmannLBmethodfor solvinglowMachnumberconvectivethermalowswithsourcetermsin two-dimensionsinadoubledistributionfunctionframeworkispresented.Forthe passivetemperatureeld,whichsatisesaconvectiondiffusionequationCDE alongwithasourcetermtorepresentinternal/externallocalheatsource,anew cascadedcollisionkernelispresented.Duetotheuseofasingleconservedvariable inthethermalenergyequation,thecascadedstructureinitscollisionoperatorbegins fromtherstordermomentsandevolvestohigherordermoments.Thisismarkedly differentfromthecollisionoperatorfortheuidowequations,constructedin previouswork,wherethecascadedformulationstartsatthesecondordermoments initscollisionkernel.Aconsistentimplementationofthespatiallyandtemporally varyingsourcetermsinthethermalcascadedLBmethodrepresentingtheheat sourcesintheCDEthatmaintainssecondorderaaccuracyviaavariable transformationisdiscussed.Inaddition,therstorderequilibriummomentsinthis modelareaugmentedwithspatialtemperaturegradienttermsobtainedlocallyand involvingatunablecoefcienttomaintainadditionalexibilityintherepresentation ofthetransportcoefcientforthetemperatureeld.Theconsistencyofthethermal cascadedLBmethodincludingasourcetermwiththemacroscopic convection-diffusionequationisdemonstratedbymeansofaChapman-Enskog analysis.Theemergenttunablediffusivityisshowntobedependentonthe relaxationtimesoftherstordermomentsaswellasthetunableparameterinthe additionalgradienttermsinourcascadedmultiple-relaxation-timeformulation.The iii

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newmodelistestedonasetofbenchmarkproblemssuchasthethermalPoiseuille ow,thermalCouetteowwitheitherwallinjectionorincludingviscousdissipation andnaturalconvectioninasquarecavity.Thevalidationstudiesshowthatthe thermalcascadedLBmethodwithsourcetermisinverygoodagreementwith analyticalsolutionsornumericalresultsreportedforbenchmarkproblems.In addition,thenumericalresultsshowthatournewthermalcascadedLBmodel maintainssecondorderspatialaccuracy. ThenewLBEmodelismodiedtosimulateanisotropicuidsthatare characterizedbydifferentdiffusioncoefcientsalongdifferentdirections.The applicabilityoftheLBEmodelisvalidatedbynumericalsimulationsincludingthe convectionanddiffusionofaGaussianHill,solvinganisotropicconvectiondiffusion equationwithvariablediffusiontensorandvariablesourceterm,andanisotropic naturalconvectioninaSquareCavity.Thevalidationstudyshowsthatthe anisotropicthermalcascadedLBmodelwithsourcetermisinverygoodagreement withtheanalyticalsolutionsornumericalresultsreportedforthebenchmark problems.Inaddition,thenumericalresultsshowthatournewanisotropicthermal cascadedLBmodelmaintainssecondorderaccuracyasdoestheisotropicmodel. AstabilitytestofthepresentcascadedLBEmodelisconductedtocompareour modelbysinglerelaxationtimeSRTLBmodelandmultiplerelaxationtimesMRT LBmodelusingthediffusioninaGaussianHillasatestproblembyvaryingtheuid diffusivitytocomparethestabilitycharacteristicsofthecascadedcentralmoment. Thisstabilitystudyconcurswithotherworkthatindicatessuperiorstability characteristicsofthecascadedLBmethod. Finally,thecentralmomentcascadedLBmodelwasadaptedtosimulateofuid owwithtemperature-dependentviscosity,wheretheuidviscosityisexponentially varyingwithtemperature.Thesimulationresultsofcouetteowwithshearheating conrmedthevalidityofthepresentcentralmomentLBmodeltoincorporate correctlyuidowwithvariableviscosityatdifferentBrinkmannumbers. iv

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Theformandcontentofthisabstractareapproved.Irecommenditspublication. Approved:SamuelWelch v

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ACKNOWLEDGMENTS IwouldliketothankmyadvisorDr.SamWelch,forhisconstantsupportand guidance,andforprovidingmeanexcellentopportunityformyacademicand professionaldevelopment.IamsogratefulagaintomyadvisorDr.SamWelchfor helpingandsupportingmeaftermyscholarshipwasendedtocontinuetoward graduation. IamalsosogratefultoallmycommitteemembersProfessorPeterJenkins, SedatBiringen,MarcIngberandProfessorDavidMaystoserveonmyadvisory committeeandfortheircomments. vi

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CONTENTS CHAPTER ITHERMALLATTICEBOLTZMANNMODEL1 1.1Introduction...................................1 1.2CollisionModels................................2 1.2.1SingleRelaxationTimeSRTLBM..................3 1.2.2MultipleRelaxationTimesMRTLBM...............5 1.2.3CascadedLBE..............................7 1.3Thermalowapproaches...........................8 1.3.1MultispeedapproachMSLBE....................9 1.3.2HybridapproachHybLBE......................9 1.3.3DoubledistributionfunctionsDDfLBE...............10 1.4ConstructionoftheCascadedCollisionOperatorfortheTemperature Field.......................................11 1.5AnisotropicCascadedModelfortheTemperatureField..........28 1.6BoundaryConditions..............................29 1.6.1Halfwaybounce-backboundaries..................30 1.6.1.1VelocityDistributionFunction...............32 1.6.1.2TemperatureDistributionFunction............32 1.6.2PeriodicBoundaryConditions....................33 IIISOTROPICTHERMALFLOW35 2.1NumericalResults................................35 vii

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2.2UnsteadyReaction-DiffusionProblem:VariableSourceTerm.......35 2.3ThermalFlowinaChannelwithWallInjection...............38 2.4Diffusionin2DPoiseuilleFlow........................40 2.5NaturalConvectioninaSquareCavity....................44 2.6ThermalCouetteFlowwithViscousHeatDissipation:ModelingHeat Source......................................48 2.7SummaryandConclusion...........................54 IIIANISOTROPICTHERMALFLOW58 3.1NumericalResults................................58 3.1.1ConvectionandDiffusionofaGaussianHill:NoSourceTerm..58 3.1.2Convection-DiffusionofaGaussianHill:StabilityTest......63 3.1.3AnisotropicConvectionDiffusionEquation:VariableSourceTerm66 3.2NaturalConvectioninaSquareCavity:AnisotropicCase........67 3.3SummaryandConclusion...........................74 IVFLOWWITHVARIABLEVISCOSITY92 4.1NumericalResults................................92 4.1.1CouetteFlowwithshearheating:Variableviscosity........92 4.2SummaryandConclusion...........................95 VSUMMARYANDCONCLUSION96 REFERENCES..............................103 APPENDIX ACHAPMAN-ENSKOGANALYSISOFTHEISOTROPICTHERMALCASCADEDLBM............................104 BCHAPMAN-ENSKOGANALYSISOFTHEANISOTROPICTHERMALCASCADEDLBM............................111 viii

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LISTOFTABLES TABLE 2.1Valuesof g b correspondingtoeachRayleighnumber............46 2.2Comparisonbetweennumericalresultsobtainedusingthecascaded LBMandthepublishedresultsdeVahlDavisandHortmann et al atdifferentRayleighnumbers Ra = 10 3 )]TJ/F59 11.9552 Tf 12.157 0 Td [(10 6 ..........57 3.1Globalrelativeerrorandstabilitycharacteristicsafter1000timeincrementsforSRT,MRT,andCascadedLBMfordifferentrelaxationtimes fortheconvection-diffusionofaGaussianhillproblematMachnumber 0.25.........................................65 3.2Valuesof g b correspondingtoeachRayleighnumber............73 3.3NumericalresultsobtainedusingthecascadedLBMforanisotropicnaturalconvectioninasquarecavityatdifferentRayleighnumbers Ra = 10 3 ,10 4 ,10 5 ,10 6 ,10 7 and10 8 ..........................78 3.4Comparisonbetweennumericalresultsobtainedusingthecascaded LBMandthepublishedresults Dubois etal [64] foranisotropic naturalconvectioninasquarecavityatdifferentRayleighnumbers Ra = 10 3 ,10 4 ,10 5 and10 6 ...........................79 3.6Comparisonbetweennumericalresultsobtainedusingthecascaded LBMandthepublishedresultsdeVahlDavisandHortmann et al forisotropicnaturalconvectioninasquarecavityatdifferent Rayleighnumbers Ra = 10 3 )]TJ/F59 11.9552 Tf 12.157 0 Td [(10 6 ......................81 ix

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LISTOFFIGURES FIGURE 1.1D2Q9lattice,CollisionrightandStreamingleft.............3 1.2D2Q9latticediagram,showingthe9velocitydirections..........4 1.3D2Q13latticediagram,showingthe13velocitydirections.........10 1.4Locationofboundarynodes..........................30 1.5BoundaryConditionswithunknownandknownpopulations.......31 1.6Locationofboundarynodes..........................31 1.7Schematicillustratingtheperiodicboundaryconditions..........33 1.8Periodicboundaryconditionontheinowandoutowboundaries...33 2.1Temperatureprolefortheunsteadyreaction-diffusionproblemwitha variablesourcetermat y = 0.5anddiffusioncoefcient a = 10 )]TJ/F59 8.9664 Tf 7.375 0 Td [(3 at differenttimes.MarkersrepresenttheCascadedLBEresultsandlines representtheanalyticalsolution........................36 2.2Temperatureprolefortheunsteadyreaction-diffusionproblemwitha variablesourcetermat y = 0.5anddiffusioncoefcient a = 10 )]TJ/F59 8.9664 Tf 7.375 0 Td [(4 at differenttimes.MarkersrepresenttheCascadedLBEresultsandlines representtheanalyticalsolutions........................37 2.3Temperatureglobalrelativeerroratdifferentvaluesofthediffusioncoefcient a = 10 )]TJ/F59 8.9664 Tf 7.375 0 Td [(3 , D = 0.397,and a = 10 )]TJ/F59 8.9664 Tf 7.375 0 Td [(4 , D = 0.3997fortheunsteadyreaction-diffusionproblemwithvariablesourceterm........38 x

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2.4VelocityprolesforthermalCouetteowinachannelwithwallinjectionatReynoldsnumbers: Re = 5,10,15.MarkersrepresenttheCascadedLBEresultsandlinesrepresenttheanalyticalsolutions.......40 2.5TemperatureprolesforthermalCouetteowinachannelwithwall injectionatReynoldsnumbers: Re = 5,10,15.Markersrepresentthe CascadedLBEresultsandlinesrepresenttheanalyticalsolutions.....41 2.6VelocityrelativeglobalerrorofCouetteowwithwallinjectionat Re = 10.........................................42 2.7TemperaturerelativeglobalerrorofCouetteowwithwallinjectionat Reynoldnumbers: Re = 5,10,15........................42 2.8VelocityandtemperatureprolesofPoiseuilleowwiththermaldiffusionatdifferentvaluesof Re and Pe .MarkersrepresenttheCascaded LBEresultsandlinesrepresenttheanalyticalsolution............43 2.9Schematicillustratingthecavityboundaryconditions............45 2.10TemperatureprolesalonghorizontalcenterlineofthecavityatvariousRayleighnumbers: Ra = 10 3 ,10 4 ,10 5 ,and10 6 computedusingthe cascadedLBM...................................47 2.11Temperatureprolesalongtheverticalcenterlineofthecavityowat variousRayleighnumbers Ra = 10 3 ,10 4 ,10 5 ,and10 6 computedusing thecascadedLBM................................47 2.12IsothermsatdifferentvaluesofRayleighnumbers Ra = 10 3 ,10 4 ,10 5 and10 6 fornaturalconvectioninasquarecavitycomputedusingthe cascadedLBM...................................48 2.13StreamlinesatdifferentvaluesofRayleighnumbers Ra = 10 3 ,10 4 ,10 5 and10 6 fornaturalconvectioninasquarecavitycomputedusingthe cascadedLBM...................................49 2.14VorticitycontoursatvariousRayleighnumbers Ra = 10 3 ,10 4 ,10 5 and 10 6 fornaturalconvectioninasquarecavitycomputedusingtheLBM..50 xi

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2.15TemperatureprolesinCouetteowatvariousvaluesofEckertnumber.MarkersrepresentthecascadedLBEsimulationsandlinesrepresenttheanalyticalsolutions...........................52 2.16TemperatureprolesinCouetteowwithdifferentvaluesofPrandtl number.MarkersrepresentthecascadedLBEsimulationsandlinesrepresenttheanalyticalsolutions.........................52 2.17TemperatureglobalrelativeerroratdifferentEckertnumbers7,14,and 28forthermalCouetteowwithviscousheatdissipation.........53 2.18TemperatureglobalrelativeerroratdifferentPrandtlnumbers0.25,1.25, and2.5forthermalCouetteowwithviscousheatdissipation......53 2.19TemperatureprolesinthermalCouetteowatvariousvaluesofPeclet numbers: Pe = 10,10 2 ,10 3 ,10 4 ,10 5 and10 6 .MarkersrepresenttheCascadedLBEresultsandlinesrepresenttheanalyticalsolution........55 2.20TemperatureglobalrelativeerroratdifferentPecletnumbers: Pe = 10,10 2 ,and10 3 forthermalCouetteowwithviscousheatdissipation..55 3.1Distributionofthescalarvariable f atthetime t = t m and u = v = 0.0..60 3.2Distributionofthescalarvariable f fordiffusion u = 0,0 ofaGaussianhillcomputedusingtheLBM.......................61 3.3ConcentrationcontoursfordiffusionofaGaussianhill u = 0,0 computedusingtheLBM...............................62 3.4Contoursofthescalervariable f at u = 0.05,0.05 computedusingLBE63 3.5AnalyticalContoursofthescalervariable f at u = 0.05,0.05 ......64 3.6Distributionofthescalarvariable f atthetime t = 3and Pe = 100....68 3.7Contoursofthescalarvariable f atthetime t = 3and Pe = 100.....69 3.8Temperatureglobalrelativeerrorsatdifferentgridsizesfordiffusionof aGaussianhill..................................70 3.9Temperatureglobalrelativeerrorwithvariablediffusiontensorand sourceterm....................................70 xii

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3.10StreamlinesatdifferentvaluesofRayleighnumbers Ra = 10 3 ,10 4 ,10 5 ,10 6 ,10 7 and10 8 fornaturalconvectioninasquarecavitycomputedusingthe cascadedLBM.Left a x = a y /2,center a x = a y ,right a x = 2 a y ..75 3.11VorticityatdifferentvaluesofRayleighnumbers Ra = 10 3 ,10 4 ,10 5 ,10 6 ,10 7 and10 8 fornaturalconvectioninasquarecavitycomputedusingthe cascadedLBM.Left a x = a y /2,center a x = a y ,right a x = 2 a y ..77 3.12IsothermalsatdifferentvaluesofRayleighnumbers Ra = 10 3 ,10 4 ,10 5 and10 6 fornaturalconvectioninasquarecavitycomputedusingthe cascadedLBM.Left a x = a y /2,center a x = a y ,right a x = 2 a y ..83 3.13FirstcomponentofvelocityuatdifferentvaluesofRayleighnumbers Ra = 10 3 ,10 4 ,10 5 ,10 6 ,10 7 and10 8 fornaturalconvectioninasquare cavitycomputedusingthecascadedLBM.Left a x = a y /2,center a x = a y ,right a x = 2 a y ...........................85 3.14SecondcomponentofvelocityvatdifferentvaluesofRayleighnumbers Ra = 10 3 ,10 4 ,10 5 ,10 6 ,10 7 and10 8 fornaturalconvectioninasquare cavitycomputedusingthecascadedLBM.Left a x = a y /2,center a x = a y ,right a x = 2 a y ...........................87 3.15PressureatdifferentvaluesofRayleighnumbers Ra = 10 3 ,10 4 ,10 5 ,10 6 ,10 7 and10 8 fornaturalconvectioninasquarecavitycomputedusingthe cascadedLBM.Left a x = a y /2,center a x = a y ,right a x = 2 a y ..89 3.16Temperatureprolesalonghorizontalcenterlineofthecavityatvarious Rayleighnumbers: Ra = 10 3 ,10 4 ,10 5 ,10 6 ,10 7 ,and10 8 computedusing thecascadedLBM................................90 3.17Temperatureprolesalongtheverticalcenterlineofthecavityowat variousRayleighnumbers Ra = 10 3 ,10 4 ,10 5 ,10 6 ,10 7 ,and10 8 computedusingthecascadedLBM.........................91 4.1Schematicillustratingtheproblemconguration...............93 xiii

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4.2TemperatureprolesforvelocitydrivenowatvariousBrinkmannumbers: Br = 3.43,5,10,and20.MarkersrepresenttheCascadedLBEresultsandlinesrepresenttheanalyticalsolution...............94 4.3VelocityprolesforvelocitydrivenowatvariousBrinkmannumbers: Br = 3.43,10,and20.MarkersrepresenttheCascadedLBEresultsand linesrepresenttheanalyticalsolution.....................94 xiv

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LISTOFABBREVIATIONS LBEL attice B oltzmann E quation LBML attice B oltzmann M ethod LBML attice B oltzmann M odel CDEC onvection D iffusion E quation NSEN avier S tokes E quation SRTS ingle R elaxation t ime MRTM ultiple R elaxation t ime BGKB hatnagarG rossK rook DDFD ouble D istribution F unction MSLBM ulti S peed L attice B oltzmann HybLBHy brid L attice B oltzmann D2Q9D iscretizationof 2 dimentionand 9 velocity xv

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LISTOFSYMBOLS a Discretediffusioncoefcient a Particlevelocitydirection t Nondimensionalrelaxationtimefordensity t g Nondimensionalrelaxationtimeforenergy t f Nondimensionalrelaxationtimeforuid t time T Nondimensionaltime r Density P Pressure d t Timestep d x Spacingstep l Relaxationtimeparameterfortemperatureeld w Relaxationtimeparameterforvelocityeld f a discreteparticledensitydistributionfunctionforvelocityeld g a discreteparticledensitydistributionfunctionfortemperatureeld f eq a Equilibriumparticledistributionfunctionforvelocityeld g eq a Equilibriumparticledistributionfunctionfortemperatureeld S a sourcetermcontributionfrominternalandexternalforces e a Discreteparticlevelocity c Particlespeed C s Speedofsound x Particlevelocity xvi

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T Temperature m DynamicViscosity m 0 TypicalDynamic n kinematicviscosity e AsymptoticexpansionparameterintheChapman-Enskoganalysis L Collisionmatrixinmomentspaceforvelocityeld b L Collisionmatrixinmomentspacefortemperatureeld 4 T Temperaturedifference S Strainratetensor Ma Machnumber U Fluidvelocity u Cartesiancomponentofvelocityinthex-direction v Cartesiancomponentofvelocityinthey-direction x Cartesiancomponentofpositionvector y Cartesiancomponentofpositionvector B r Brinkmannumber P e Pecletnumber R e Reynoldnumber I Identitymatrix L Characteristiclength W a DiscretecollisiontermintheLBE w a TheLatticeweights b g eq TheLattice xvii

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CHAPTERI THERMALLATTICEBOLTZMANNMODEL 1.1Introduction Heatandmasstransfercoupledwithuidowsisawidespreadphenomenahas manyapplicationsinengineering,energy,chemicalreaction,geologyetc.These phenomenaaremathematicallydescribedbyconvection-diffusionequationsandthe Navier-Stokesequation.TheLatticeBoltzmannmethodisaparticlebasedmethod whichwhencomparedtotraditionalcomputationaluiddynamicmethodshassome distinctadvantagessuchaseaseofimplementationoffullyparallelalgorithms,the capabilitytohandlethecomplexboundaryconditions,exibleandeasytomodify theschemeandveryfastwithhardwareaccelerationfastsolverforNSE.TheLattice Boltzmannmethodoffersagreatpotentialforincludingkineticandatomisticdetails intothecomputationalmodels.Infact,thisoriginatesfromthemainpurposeof LBM,whichistonumericallysolvetheLatticeBoltzmannequation.Thisequation candescribethedistributionofparticlesofasysteminphasespaceatany thermodynamicstate.TheLatticeBoltzmannmethoddoesnotsolvethe hydrodynamicandnon-hydrodynamicconservationequationbutrathermodelsthe streamingandcollisionforparticles,i.e.relaxationtowardlocalequilibrium.Inother words,theideaofLBMistoconstructsimplieddiscretemicroscopicdynamicsto simulatethemacroscopicmodeldescribedbypartialdifferentialequationsthat modelaphysicalsystem. Inthischapter,adenitionoftheLatticeBoltzmannmethodasadiscrete 1

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methodandtheconceptofcollisionandstreamingwithinthemethodareprovided insection.2,thenabriefdescriptionforeachLBEcollisionmodelisgivenwiththe explanationofthemaindifferenceamongthesemodelsinsubsections.2.1,.2.2 and.2.3.Next,thermalowapproachesthatareusedtosolveLBEareintroduced insection.3withabriefdescriptionforeachapproach. Finally,theconstructionofthecascadedLBmodelfortheevolutionofthescalar eldrepresentedbytheconvection-diffusionequationCDEwithasourcetermis providedindetailsinSections.4and.5 1.2CollisionModels Themicro-dynamicsofgasesanduidsconsistsoftherepetitionoftwoprocesses collisionandpropagationstreaming.Themacroscopicvaluesoftemperature,mass andmomentumdensityarethencalculatedbyameanvaluesoverlargespatial regionswiththousandsofnodes.TheLatticeBoltzmannmethodasakineticbased numericalmethodisdesignedtoidealizethemicroscopicdescriptionthatallowsthe recoveryofthedesiredmacroscopicequationsthroughoutthearticiallattice Boltzmannlattice.ThemovementofparticlesintheLatticeBoltzmannMethodis assumedtobediscreteintimeandspace,i.e.asetofdirectionswithgivenvelocities withinthemethodisconsidered.Here,thecollisionofparticleshappenslocallyata singlenodeafteragiventimestep.Duringthecollisiontheparticlesareassumedto conservedensityandmomentumforisothermalowandalsotheconcentrationor temperaturefortheconvectiondiffusionowthentheparticlesarestrictlystreamed totheirneighborsaccordingtothealloweddirections.CollisionsintheLBmodels aretypicallyrepresentedby W x , t ,inwhichtheparticledistributionormoment approachesitsequilibriumvaluesoveracharacteristictimerelaxationtime t .In otherwords,astherelaxationtimeconstantsdeterminethemicroscopicdynamics towardsthelocalequilibrium,italsodeterminesthemacroscopictransport coefcients.Theeffectivenessofthecollisionoperatordeterminesthetimetoreach 2

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F IGURE 1.1:D2Q9lattice,CollisionrightandStreamingleft. theequilibriumcongurationbychoosingapropersetofrelaxationtimeconstants andatthesametimeitkeepsthenumericalschemestable.TheLBEhasseveral models,whichdifferfromeachotherbythewaytheyhandlethecollisionstep.Inthe following,abriefdescriptionofthemostpopularLatticeBoltzmanncollisionmodels isprovided. 1.2.1SingleRelaxationTimeSRTLBM SRTLBMisasinglerelaxationtimeLatticeBoltzmannmodelcontainswitha simpliedcollisionterm.Infact,theBhatragar-Gross-KrookBGKisarstformof thecollisiontermintroducedin[1]thenextendedtoLatticeBoltzmannin[2,3]which resultedinaconsiderablesimplicationoftheLBE.Themethodhassincebeen appliedinthecontextofLBEbyseveralauthorsforisothermalow[4,6,7,9,17]. TheSRTLatticeBoltzmannmodelwasrstintroducedtosimulateincompressible uidowmorethantwodecadesago,thenithasbeenwidelyemployedtostudy thermalow[14,20,21,23,51]becauseofitssimplicityinstructure.Inthismodel, bothcollisionandstreamingprocessesareexecutedinphasespace.Weconsiderthe D2Q9SRTLatticeBoltzmannmodel.TheevaluationequationoftheSRTmodelfor thermalowis 3

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F IGURE 1.2:D2Q9latticediagram,showingthe9velocitydirections. g a ~ x + ~ e a d t , t + d t )]TJETq1 0 0 1 267.476 481.638 cm[]0 d 0 J 0.478 w 0 0 m 6.539 0 l SQBT/F68 11.9552 Tf 267.889 474.184 Td [(g a ~ x , t + d t = W x , t + S a ~ x , t ..1 where g a isthedistributionfunction, W x , t = )]TJ/F59 8.9664 Tf 11.36 4.71 Td [(1 t g a ~ x , t )]TJ/F68 11.9552 Tf 12.57 0 Td [(g eq a ~ x , t isthe collisionoperator, S a isthediscretesourceterm.Theequilibriumdistribution function g eq a isgivenby g eq a = w a T 1 + e a . u cs 2 + e a . u 2 2 cs 4 )]TJ/F68 11.9552 Tf 15.599 8.094 Td [(u . u 2 cs 2 .2 TheLatticeweights w a andtheninediscretevelocitiesaregivenas w a = 8 > > > > < > > > > : 4 9 )723(! e a = 0,0 , a = 0 1 9 )723(! e a = 1,0 , 0, 1 , a = 1,2,3,4 1 36 )723(! e a = 1, 1 , a = 5,6,7,8 .3 where c s isthelatticespeedofsoundisgivenby c 2 s = c 2 3 .TypicallyforD2Q9model, c 2 s = 1/3, c isthelatticespeedandisgivenby c = D x D t .Here, D x = e a D t = 1, D t = 1. Thetemperaturecanbeobtainedintermsoftransformeddistributionfunctionas 4

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following T = 8 a = 0 g a ,.4 ThroughtheChapman-Enskogmulti-scaleexpansion,thethermaldiffusivityis foundas: a = 1 3 1 l )]TJ/F59 11.9552 Tf 13.353 8.094 Td [(1 2 .5 where l = 1/ t . TheshortcomingsofBGKmodelareapparentinthatthemodelsuffers numericalinstabilityduetothesinglerelaxationtimeforallmodes,.i.e.Allmodes arerelaxedonthesameratewhichleadstonumericalinstabilityatrelativelysmall thermaldiffusivityoratsmallviscosityforuidow.Inaddition,thePrandtl numberisconstantwhenweapplyittothermalowandthismodeldoesnothave sufcientparametersonerelaxationtime t onlytodescribeanisotropicdiffusion. 1.2.2MultipleRelaxationTimesMRTLBM Here,differentmomentscanrelaxatdifferentrates,thecollisionprocessismapped ontotherawmomentspacexedframeofreferencethroughoutanorthogonal transformationmatrixwhilethestreamingprocessisstillexecutedinphasespace. CertainRelaxationtimescanrepresentthehydrodynamicandnon-hydrodynamic momentsi.e.fortemperatureeld, l 3 and l 5 arerepresentthethermaldiffusivity diffusioncoefcient.[41,46,50,51,53,55,56]. a = 1 3 1 l 3,5 )]TJ/F59 11.9552 Tf 13.353 8.093 Td [(1 2 .6 ThegoverningequationoftheMRTmodel g a ~ x + ~ e a d t , t + d t )]TJETq1 0 0 1 267.476 132.504 cm[]0 d 0 J 0.478 w 0 0 m 6.539 0 l SQBT/F68 11.9552 Tf 267.889 125.05 Td [(g a ~ x , t + d t = W x , t + S a ~ x , t ..7 5

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Where W x , t = )]TJ/F68 11.9552 Tf 10.52 0 Td [(M )]TJ/F59 8.9664 Tf 7.375 0 Td [(1 b L M g a ~ x , t )]TJ/F68 11.9552 Tf 12.57 0 Td [(g eq a ~ x , t , b L isadiagonalcollisionmatrix givenby b L = diag l 0 , l 1 , l 2 , l 3 ,......., l 8 ..8 Where M istheorthogonalmatrixthatisusedtotransformfromvelocityspaceto momentspaceasfollows b g = M g , b g eq = M g eq , b S = M S ,.9 TheMRTLBEevaluationequationfortemperatureeldEq..7canbewrittenin termsofmomentspaceas b g a ~ x + ~ e a d t , t + d t )]TJ/F85 11.9552 Tf 12.163 2.415 Td [(b g a ~ x , t = M )]TJ/F59 8.9664 Tf 7.375 0 Td [(1 )]TJ/F85 11.9552 Tf 11.316 2.815 Td [(b L b g )]TJ/F85 11.9552 Tf 12.163 0.024 Td [(b g eq + I)]TJ/F59 11.9552 Tf 23.664 8.094 Td [(1 2 b L b S .10 Theequilibriumdistributionfunctionfortemperature g eq a istakenas g eq a = w a T 1 + e a . u cs 2 + e a . u 2 2 cs 4 )]TJ/F68 11.9552 Tf 15.599 8.093 Td [(u . u 2 cs 2 .11 M istheorthogonalsetofbasisvectorsmatrixgivenby M = 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 4 111111111 )]TJ/F59 11.9552 Tf 9.833 0 Td [(4 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1 )]TJ/F59 11.9552 Tf 9.833 0 Td [(12222 4 )]TJ/F59 11.9552 Tf 9.833 0 Td [(2 )]TJ/F59 11.9552 Tf 9.833 0 Td [(2 )]TJ/F59 11.9552 Tf 9.833 0 Td [(2 )]TJ/F59 11.9552 Tf 9.833 0 Td [(21111 010 )]TJ/F59 11.9552 Tf 9.833 0 Td [(101 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1 )]TJ/F59 11.9552 Tf 9.833 0 Td [(11 0 )]TJ/F59 11.9552 Tf 9.833 0 Td [(20201 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1 )]TJ/F59 11.9552 Tf 9.833 0 Td [(11 0010 )]TJ/F59 11.9552 Tf 9.833 0 Td [(111 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1 00 )]TJ/F59 11.9552 Tf 9.833 0 Td [(20211 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1 01 )]TJ/F59 11.9552 Tf 9.833 0 Td [(11 )]TJ/F59 11.9552 Tf 9.833 0 Td [(10000 000001 )]TJ/F59 11.9552 Tf 9.833 0 Td [(11 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1 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 5 ..12 6

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IthasbeenwidelyprovedandacceptedthattheMRTLBMsignicantly improvedtheaccuracyandnumericalstabilityofLatticeBoltzmannschemes.In addition,thedifferentrelaxationtimeconstantsinMRTmodelaresufcienttocover theanisotropicdiffusioncoefcienttensor. 1.2.3CascadedLBE ThecascadedLatticeBoltzmannmodelisamultiplerelaxationtimemodelbasedon usingcentralmoments.Centralmomentsarethemomentsthatareobtainedby shiftingthepartialvelocitybythelocaluidvelocityi.ethemomentsarecalculated inamovingframeofreferencewhilethemomentsinMRTLBMarecomputedinthe frameofreferenceatrestandthecollisionprocessismappedontotherawmoment. Here,noregularequilibriumdistributionfunctionisused,theequilibrium distributionfunction g M T isobtainedbymakingananalogywith Maxwell-Boltzmanndistributionfunction f M r , ~ u , ~ x inthecontinousvelocityspace ~ x byreplacingthedensity r withtemperature T inourDDFformulation.Thatis, g M T = T r f M r , ~ u , ~ x where f M r , ~ u , ~ x = r 2 p c 2 s exp h )]TJ/F70 9.343 Tf 11.141 6.046 Td [( ~ x )]TJ/F67 9.343 Tf 6.931 0.013 Td [(~ u 2 2 c 2 s i ,Thecollision processisalsomappedontothecentralmomentspacethroughoutanorthogonal transformationmatrix T .Thegoverningequationofthecascadedmodel g a ~ x + ~ e a d t , t + d t )]TJETq1 0 0 1 267.476 317.995 cm[]0 d 0 J 0.478 w 0 0 m 6.539 0 l SQBT/F68 11.9552 Tf 267.889 310.541 Td [(g a ~ x , t + d t = W x , t + S a ~ x , t ..13 Where W a canberepresentedas W a = K b h a ,hthecollisionkernel b L isa diagonalcollisionmatrixgivenby b L = diag l 0 , l 1 , l 2 , l 3 ,......., l 8 ..14 7

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K istheorthogonalsetofbasisvectorsmatrixgivenby K = 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 4 100 )]TJ/F59 11.9552 Tf 9.833 0 Td [(400004 110 )]TJ/F59 11.9552 Tf 9.833 0 Td [(11002 )]TJ/F59 11.9552 Tf 9.833 0 Td [(2 101 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1020 )]TJ/F59 11.9552 Tf 9.833 0 Td [(2 1 )]TJ/F59 11.9552 Tf 9.833 0 Td [(10 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1100 )]TJ/F59 11.9552 Tf 9.833 0 Td [(2 )]TJ/F59 11.9552 Tf 9.833 0 Td [(2 10 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1 )]TJ/F59 11.9552 Tf 9.833 0 Td [(10 )]TJ/F59 11.9552 Tf 9.833 0 Td [(20 )]TJ/F59 11.9552 Tf 9.833 0 Td [(2 111201 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1 )]TJ/F59 11.9552 Tf 9.833 0 Td [(11 00 )]TJ/F59 11.9552 Tf 9.833 0 Td [(20211 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1 1 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1201111 11 )]TJ/F59 11.9552 Tf 9.833 0 Td [(120 )]TJ/F59 11.9552 Tf 9.833 0 Td [(11 )]TJ/F59 11.9552 Tf 9.833 0 Td [(11 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 5 ..15 Here, l 1 and l 2 representthethermaldiffusivity a = 1 3 1 l 1,2 )]TJ/F59 11.9552 Tf 13.353 8.094 Td [(1 2 .16 ThecascadedLatticeBoltzmannmodelisMorestablethanbothSRTandMRT modelsandMoreGalileaninvariantthantherestframeofreferencemodels Allthestepsfortheconstructionofthecascadedcollisionoperatorforthe temperatureeldareprovidedinsec.4 1.3Thermalowapproaches Tocaptureorsimulatethermalow,thesolutionofthetemperatureeld,whose evolutionisrepresentedbyaconvection-diffusionequationCDE,forenergy transport,coupledtotheuidvelocity,whichisrepresentedbytheNavier-Stokes NSequations.Classicalnumericalmethodscanbecomechallengingtoapplyforthe simulationofsuchows,especiallyinthecomplexgeometricssuchasthermalows inpourousmedia.WithintheLBframework,broadly,threediferentapproachesthat havebeendevelopedtoconstructthethermalLBEequationmodels.Inallofthese 8

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methods,adistributionfunction f isappliedtosimulatethevelocityeld.The energyeldisthensimulatedviaeither: 1MultispeedapproachMS-LBE 2Hybridapproach 3DoubledistributionfunctionsDDF 1.3.1MultispeedapproachMSLBE Inthismethod,adistributionfunction f isappliedtosimulateboththevelocity andenergyeld,i.e.usingthesamedistributionfunctiontosolveboththeNSE equationsandCDE,thisrequiresalargevelocitysetfortheLatticeinordertorecover theenergyequation.Themulti-speedLatticeBoltzmannisrstintroducedby[32]. Theauthorsusedthirteenvelocitiesfortwodimensions,LBEforuidneedsnine velocitesandtheextrafourvelocitestorecovertheenergyequation,thisleadsto higherordermomentsthantheregularLBEtoaccountfortheadditionalequationfor theconservationofenergy.Inaddition,complextermsandcalculationsduetothe largesetofvelocitiesconsideredleadtonumericalinstability.Differentauthorslater triedtoimplementthismodelwithsomemodicationtostabilizethe method[33,34,35,36,47].Allthesemodelsaremulti-speedmodelsandsufferfrom numericalinstability,arerestrictedtoalimitedtemperaturerangeandthetreatment ofboundaryconditionswithlargenumberofdiscretevelocitiessetsisdifcult . 1.3.2HybridapproachHybLBE IntheHybridapproach,theLBEmethodisusedtosolvetheoweldwhileany conventionalnumericalmethodsuchasnitedifferenceorniteelementschemeis usedtosolvetheenergyequation[31,37,38,39].Thehybridapproachismorestable thanmulti-speedapproch.However,thisapproachhasthedisadvantagethatthe simplicityofLBEislost. 9

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F IGURE 1.3:D2Q13latticediagram,showingthe13velocitydirections. 1.3.3DoubledistributionfunctionsDDfLBE IntheDDFapproach,twoseparatedistributionsfunctionsareemployed,oneforthe oweldandtheotheroneforthetemperatureeld[28,17,45,56].Unliketheabove twomodels,manyofsuchlimitationscanbeovercomebytheDDFmodels[30,49] andtheyhave,hence,receivedsignicantlymoreattentionrecently.Mostofthe developmentsrelatedtotheDDF-LBEmethodsconsideredsinglerelaxationtime SRTcollision[65].models[14,21,20,23,51].VariousaspectsrelevanttotheLBEfor correctrepresentationoftheCDEforthetemperatureeldwereidentied.For example,thechoiceoftheequilibriumdistributionfunctionwithnonlinearvelocity termswasimportantinthisregard[23].Concurrently,variousboundarycondition schemesfortheDDF-LBEweredeveloped[18,46,48,53,54,60].However,theuseof SRTcollisionmodels,thoughsimpleinstructureandcharacterizedbytherelaxation ofallmodelsatthesamerate,isknowntosufferfrominstabilityissues,particularly wherethetransportcoefcientssuchastheuidviscosityandthermaldiffusivity becomerelativelysmall.ThislimitstheabilitytoreachhigherReynoldsorPeclet numbers.Onepossibilitytoaddressthisissueistoconsiderusingmulti-relaxation timeMRTmodelsforDDF-LBEapproach[50,41,46,53,55,56].IntheMRTmodel, 10

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thecollisionprocessismappedontotherawmomentspacethroughanorthogonal transformationmatrix,wheredifferentmomentscanrelaxatdifferentrates.By representingtherelaxationtimesofthehydrodynamicandnon-hydrodynamic moments,thestabilityoftheMRTmodelcanbesignicantlyimproved.Afurther improvementistoconsideranothertypeofMRTmodel,inwhichthecollisionstepis executedintermsoftherelaxationofcentralmoments,inwhichtheparticle velocitiesareshiftedbythelocaluidvelocity.Suchtypeofcollisionmodelina movingframeofreferenceleadstoacascadedstructureofthehigherordermoments intermsofthoseatlowerorderfollowingcollisionandhenceisreferredtoasthe cascadedLBmethod[22]. 1.4ConstructionoftheCascadedCollisionOperatorfortheTemperatureField OurmaingoalinthisinvestigationistoconstructacascadedLBmodelforthe evolutionofthetemperatureeldrepresentedbythefollowingconvection-diffusion equationCDEwithasourceterm T t + u r T = r a r T + G .17 where a isthethermaldiffusivitycoefcient, T = T x , y , t and u = u x , y , t arelocal temperatureandvelocityeld,respectively.Inaddition G = G x , y , t isthelocal sourceterm,forexample,duetointernalheatgenerationorviscousdissipation.In general,thermaltransportcanbesignicantlyinuencedbythepresenceofinternal heatgeneration,suchasthoserelatedtonuclearorchemicalreactionsgenerating localheating.Viscousheatingeffectsduetoshearstressesisanotherexample.Allof theseeffectscanberepresentedasaprescribedlocalsourceterm G = G x , y , t inthe thermaltransportequation.Tohandlesuchageneralcase,herewedevelopanew cascadedLBmodelwithasourceterm,whichcanrecoverthemacroscopicequation representedbyCDEgiveninEq..17abovewithsecondorderaccuracy.In 11

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Eq..17,thelocalvelocityeld u = u x , y , t satisedtheNavies-Stokesequations NSEgivenby r u = 0,.18a u t + u r u = )]TJ/F59 11.9552 Tf 11.178 8.094 Td [(1 r r P + n r 2 u + F .18b Where p isthepressureoftheuidow, n isthekinematicviscosityoftheuid, r isthereferencedensityand F = r a isthelocalexternalforceeld.Thevelocity eld u tobeusedinEq..17isconsideredtobeknown,andcanbeobtainedby solvinganothercascadedLBEconstructedinpreviouswork[22,44].Inparticular,the speciccascadedLBEwithforcingtermforobtainingthevelocityeld u canbe coupledtothenewcascadedfortheCDEtobedevelopedinthiswork.Insuchaa doubledistributionfunctionDDFformulation,wereferthereadertothecascaded LBEwithforcingtermfortheoweldpresentedinto[44]tomaintainbrevityand focushereontheconstructionofthecascadedLBEtosolveforthetemperatureeld T = T x , y , t ,whoseevolutionisrepresentedbyEq..17 TheoverallproceduretodevelopathermalcascadedLBEinvolvesthe following:iprescribeasuitablechoiceofanorthogonalmomentbasisforthelattice velocitysets,iispecifyformulationsforthecontinuouscentralmomentsof equilibriumandsourcetermandequatethemtothecorrespondingdiscretecentral momentsinvolvedinthecascadedLBEfortheCDE,iiiTransformthevarious discretecentralmomentsintermsofvariouscorrespondingrawmomentsbyusing thebinomialtheorem,ivconstructthecollisionkernelappearinginthecascaded collisionoperatorforsolvingtheCDEandthesourcetermintheLBmodel. First,weselectasutiblemomentbasisforthetwo-dimensional,ninevelocity D2Q9lattice.Weconsidertheusual“bra"andthe“ket"notations,i.e. h . j and j . i todenote9-dimensionalrowandcolumnvectors,respectively.Then,weobtain 12

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thefollowingninenon-orthogonalbasisvectorsobtainedfrommonomials e m a x e n a y at successivelyincreasingorders: T = h j 1 i , j e a x i , j e a y i , j e 2 a x + e 2 a y i , j e 2 a x )]TJ/F68 11.9552 Tf 12.307 0 Td [(e 2 a y i , j e a x e a y i , j e 2 a x e a y i , j e a x e 2 a y i , j e 2 a x e 2 a y i i , .19 where j T i = 1,1,1,1,1,1,1,1,1 T , j e a x i = 0,1,0, )]TJ/F59 11.9552 Tf 9.833 0 Td [(1,0,1, )]TJ/F59 11.9552 Tf 9.833 0 Td [(1, )]TJ/F59 11.9552 Tf 9.833 0 Td [(1,1 T , j e a y i = 0,0,1,0, )]TJ/F59 11.9552 Tf 9.833 0 Td [(1,1,1, )]TJ/F59 11.9552 Tf 9.833 0 Td [(1, )]TJ/F59 11.9552 Tf 9.833 0 Td [(1 T . Theabovenominalsetofbasisvectorsisthentransformedintoanequivalent orthogonal setofbasisvectorsbymeansofthestandardGram-Schmidtprocedure arrangedintheincreasingorderofmoments: j K 0 i = j 1 i , j K 1 i = j e a x i , j K 2 i = j e a y i , j K 3 i = 3 j e 2 a x + e 2 a y i)]TJ/F59 11.9552 Tf 19.61 0 Td [(4 j 1 i , j K 4 i = j e 2 a x )]TJ/F68 11.9552 Tf 12.307 0 Td [(e 2 a y i , j K 5 i = j e a x e a y i , j K 6 i = )]TJ/F59 11.9552 Tf 9.833 0 Td [(3 j e 2 a x e a y i + 2 j e a y i , j K 7 i = )]TJ/F59 11.9552 Tf 9.833 0 Td [(3 j e a x e 2 a y i + 2 j e a x i , j K 8 i = 9 j e 2 a x e 2 a y i)]TJ/F59 11.9552 Tf 19.611 0 Td [(6 j e 2 a x + e 2 a y i + 4 j 1 i . Bygroupingtheabove,setofvectors,weobtainanorthogonaltransformation matrix K as 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 ] ,.20 13

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whichcanbeexplicitlywrittenas K = 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 4 100 )]TJ/F59 11.9552 Tf 9.833 0 Td [(400004 110 )]TJ/F59 11.9552 Tf 9.833 0 Td [(11002 )]TJ/F59 11.9552 Tf 9.833 0 Td [(2 101 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1020 )]TJ/F59 11.9552 Tf 9.833 0 Td [(2 1 )]TJ/F59 11.9552 Tf 9.833 0 Td [(10 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1100 )]TJ/F59 11.9552 Tf 9.833 0 Td [(2 )]TJ/F59 11.9552 Tf 9.833 0 Td [(2 10 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1 )]TJ/F59 11.9552 Tf 9.833 0 Td [(10 )]TJ/F59 11.9552 Tf 9.833 0 Td [(20 )]TJ/F59 11.9552 Tf 9.833 0 Td [(2 111201 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1 )]TJ/F59 11.9552 Tf 9.833 0 Td [(11 00 )]TJ/F59 11.9552 Tf 9.833 0 Td [(20211 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1 1 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1 )]TJ/F59 11.9552 Tf 9.833 0 Td [(1201111 11 )]TJ/F59 11.9552 Tf 9.833 0 Td [(120 )]TJ/F59 11.9552 Tf 9.833 0 Td [(11 )]TJ/F59 11.9552 Tf 9.833 0 Td [(11 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 5 ..21 Next,inordertoconstructacascadedLBcollisionoperatorforrepresentingthe evaluationofthetemperatureeld,weneedtopresentthecontinousmomentsofthe equilibriumstateandthesourceterm.Thecontinuousequilibriumcentralmoments oforder m + n canbedenedas b P eq x m y n = R )]TJ/F69 8.9664 Tf 7.487 0 Td [( R )]TJ/F69 8.9664 Tf 7.487 0 Td [( g M T x x )]TJ/F68 11.9552 Tf 12.307 0 Td [(u x m x y )]TJ/F68 11.9552 Tf 12.307 0 Td [(u y n d x x d x y ,whichyields j b P eq x m y n i = b P eq 0 , b P eq x , b P eq y , b P eq xx , b P eq yy , b P eq xy , b P eq xxy , b P eq xyy , b P eq xxyy T , = T ,0,0, c 2 s T , c 2 s T ,0,0,0, c 4 s T T . .22 Here,theequilibriumdistributionfunction g M T isobtainedbymakingananalogy withMaxwell-Boltzmanndistributionfunction f M r , ~ u , ~ x inthecontinousvelocity space ~ x byreplacingthedensity r withtemperature T inourDDFformulation.That is, g M T = T r f M r , ~ u , ~ x where f M r , ~ u , ~ x = r 2 p c 2 s exp h )]TJ/F70 9.343 Tf 11.141 6.046 Td [( ~ x )]TJ/F67 9.343 Tf 6.931 0.013 Td [(~ u 2 2 c 2 s i ,where c s isthe latticespeedofsound.TypicallyforD2Q9model, c 2 s = 1/3. 14

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Similarly,deningthecontinouscentralmomentsofthesourcetermoforder m + n dueto G = G x , y , t appearinginEq..17as b G x m y n = Z )]TJ/F69 8.9664 Tf 7.487 0 Td [( Z )]TJ/F69 8.9664 Tf 7.486 0 Td [( D g G x x )]TJ/F68 11.9552 Tf 12.307 0 Td [(u x m x y )]TJ/F68 11.9552 Tf 12.307 0 Td [(u y n d x x d x y .23 where D g G isthechangeinthedistributionfunctionduetosource G .Sincethesource G isexpectedinunceonlythelowestorderzerothmoment,wecanprescribedthe followingansatz: j b G x m y n i = b G 0 , b G x , b G y , b G xx , b G yy , b G xy , b G xxy , b G xyy , b G xxyy T , = G ,0,0,0,0,0,0,0,0 T . .24 Byusingtheabovecentralmomentsourgoalistodevelopthecollisionoperatorand thesourcetermofthecascadedLBE.Forrepresentingthetransportofthe temperatureeldthecorrespondingcascadedLBEusingthetrapezoidalruleto evaluatethesourcetermofcascadedLBEtomaintainthesecondorderaccuracycan bewrittenas: g a ~ x + ~ e a d t , t + d t = g a ~ x , t + W a ~ x , t + 1 2 S a ~ x , t + S a ~ x + ~ e a , t + d t ..25 Here,thecollisionterm W a canberepresentedas W a W a g , b h = K b h ff ,where g j g a i = g 0 , g 1 , , g 8 T isthevectorofdistributionfunctionsand b h j b h a i = b h 0 , b h 1 , , b h 8 T isthevectorofunknowncollisionkerneltobeobtained later.Thediscreteformofthesourceterm S a inthecascadedLBEgivenabove representstheinuenceofthesource G inthevelocityspaceandisdenedas S j S a i = S 0 , S 1 , S 2 , , S 8 T .NotingthatEq..25issemi-implicit,byusingthe standardvariabletransformation g = g a )]TJ/F59 8.9664 Tf 13.353 4.71 Td [(1 2 S a [13],theimplicitnesscanbe effectivelyremoved.Thisyields g a ~ x + ~ e a d t , t + d t = g a ~ x , t + W a ~ x , t + S a ~ x , t ..26 15

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whichcanmaintainsecondorderaccuracyinaneffectivelytimeexplicitmethod. Inordertoobtaintheexpressionsforthestructureofthecascadedcollision operator b h andthesourceterms S a inthepresenceofaspatiallyand/ortemporally source G ,i.e. G = G x , y , t ,wedenethefollowingsetofdiscretecentralmoments. b k x m y n = a g a e a x )]TJ/F68 11.9552 Tf 12.307 0 Td [(u x m e a y )]TJ/F68 11.9552 Tf 12.307 0 Td [(u y n ,.27a b k eq x m y n = a g eq a e a x )]TJ/F68 11.9552 Tf 12.307 0 Td [(u x m e a y )]TJ/F68 11.9552 Tf 12.307 0 Td [(u y n ,.27b b k x m y n = a g a e a x )]TJ/F68 11.9552 Tf 12.307 0 Td [(u x m e a y )]TJ/F68 11.9552 Tf 12.307 0 Td [(u y n ..27c b k eq x m y n = a g eq a e a x )]TJ/F68 11.9552 Tf 12.307 0 Td [(u x m e a y )]TJ/F68 11.9552 Tf 12.307 0 Td [(u y n ,.27d b s x m y n = a S a e a x )]TJ/F68 11.9552 Tf 12.307 0 Td [(u x m e a y )]TJ/F68 11.9552 Tf 12.307 0 Td [(u y n ..27e Wethenmatchthediscretecentralmomentsofthedistributionfunctionsandsource termswiththecorrespondingcontinuouscentralmomentsateachorder,i.e. b k eq x m y n = b P M x m y n ,.28a b s x m y n = b G F x m y n ..28b Thus,weobtain j b k eq x m y n i = b k eq 0 , b k eq x , b k eq y , b k eq xx , b k eq yy , b k eq xy , b k eq xxy , b k eq xyy , b k eq xxyy T , = T ,0,0, c 2 s T , c 2 s T ,0,0,0, c 4 s T T . .29 j b s x m y n i = b s 0 , b s x , b s y , b s xx , b s yy , b s xy , b s xxy , b s xyy , b s xxyy T , = G ,0,0,0,0,0,0,0,0 T . .30 SincetheactualcomputationsinthecascadedLBEareperformedintermsofthe variousrawmoments,wedenethefollowingsetofdiscreterawmomentsdenoted 16

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withaprimesymbol: b k 0 x m y n = a g a e a x m e a y n ,.31a b k 0 x m y n eq = a g eq a e a x m e a y n ,.31b b k 0 x m y n = a g a e a x m e a y n ..31c b k 0 x m y n eq = a g eq a e a x m e a y n ,.31d b s 0 x m y n = a S a e a x m e a y n ..31e ByusingEqs..27e,.30,.31eandthebinomialtheorm,weobtainthefollowing setsofdiscreterawmomentsforthesourcetermatdifferentorders: b s 0 0 = h S a j T i = a S a = G ,.32a b s 0 x = h S a j e a x i = a S a e a x = u x G ,.32b b s 0 y = h S a j e a y i = a S a e a y = u y G ,.32c b s 0 xx = h S a j e 2 a x i = a S a e 2 a x = u 2 x G ,.32d b s 0 yy = h S a j e 2 a y i = a S a e 2 a y = u 2 y G ,.32e b s 0 xy = h S a j e a x e a y i = a S a e a x e a y = u x u y G ,.32f b s 0 xxy = h S a j e 2 a x e a y i = a S a e 2 a x e a y = u 2 x u y G ,.32g b s 0 xyy = h S a j e a x e 2 a y i = a S a e a x e 2 a y = u x u 2 y G ,.32h b s 0 xxyy = h S a j e 2 a x e 2 a y i = a S a e 2 a x e 2 a y = u 2 x u 2 y G ..32i Inordertoobtainthesourcetermsinthevelocityspace,werstcomputethesource momentsprojectedtotheorthogonalmomentspace,i.e. b m b = h K b j S a i 17

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, b = 0,1,2, ,8 m 0 = h K 0 j S a i = G ,.33a m 1 = h K 1 j S a i = u x G ,.33b m 2 = h K 2 j S a i = u y G ,.33c m 3 = h K 3 j S a i = 3 u 2 x + 3 u 2 y )]TJ/F59 11.9552 Tf 12.157 0 Td [(4 G ,.33d m 4 = h K 4 j S a i = u 2 x )]TJ/F68 11.9552 Tf 12.307 0 Td [(u 2 y G ,.33e m 5 = h K 5 j S a i = u x u y G ,.33f m 6 = h K 6 j S a i = )]TJ/F59 11.9552 Tf 9.833 0 Td [(3 u 2 x u y + 2 u y G ,.33g m 7 = h K 7 j S a i = )]TJ/F59 11.9552 Tf 9.833 0 Td [(3 u x u 2 y + 2 u x G ,.33h m 8 = h K 8 j S a i = 9 u 2 x u 2 y )]TJ/F59 11.9552 Tf 12.158 0 Td [(6 u 2 x + u 2 y + 4 G ..33i Sincethereisonlyoneconservedscalarforthethermaltransportequation,the componentsofrawmomentsofsources b s 0 x m y n aredifferentfromthoseappearingin thecascadedLBEwithforcingterms[44].Thenusing K S a = b m 0 , b m 1 , b m 2 ,..., b m 8 T andinvertingitbyexploitingtheorthogonalityof K , 18

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wegetthefollowingexpressionsforthesourceterms S a invelocityspace: S 0 = 1 9 [ m 0 )]TJ/F68 11.9552 Tf 12.307 0 Td [(m 3 + m 8 ] ,.34a S 1 = 1 36 [ 4 m 0 + 6 m 1 )]TJ/F68 11.9552 Tf 12.307 0 Td [(m 3 + 9 m 4 + 6 m 7 )]TJ/F59 11.9552 Tf 12.157 0 Td [(2 m 8 ] ,.34b S 2 = 1 36 [ 4 m 0 + 6 m 2 )]TJ/F68 11.9552 Tf 12.307 0 Td [(m 3 )]TJ/F59 11.9552 Tf 12.157 0 Td [(9 m 4 + 6 m 6 )]TJ/F59 11.9552 Tf 12.157 0 Td [(2 m 8 ] ,.34c S 3 = 1 36 [ 4 m 0 )]TJ/F59 11.9552 Tf 12.158 0 Td [(6 m 1 )]TJ/F68 11.9552 Tf 12.307 0 Td [(m 3 + 9 m 4 )]TJ/F59 11.9552 Tf 12.158 0 Td [(6 m 7 )]TJ/F59 11.9552 Tf 12.157 0 Td [(2 m 8 ] ,.34d S 4 = 1 36 [ 4 m 0 )]TJ/F59 11.9552 Tf 12.158 0 Td [(6 m 2 )]TJ/F68 11.9552 Tf 12.307 0 Td [(m 3 )]TJ/F59 11.9552 Tf 12.157 0 Td [(9 m 4 )]TJ/F59 11.9552 Tf 12.158 0 Td [(6 m 6 )]TJ/F59 11.9552 Tf 12.157 0 Td [(2 m 8 ] ,.34e S 5 = 1 36 [ 4 m 0 + 6 m 1 + 6 m 2 + 2 m 3 + 9 m 5 )]TJ/F59 11.9552 Tf 12.158 0 Td [(3 m 6 )]TJ/F59 11.9552 Tf 12.158 0 Td [(3 m 7 + m 8 ] ,.34f S 6 = 1 36 [ 4 m 0 )]TJ/F59 11.9552 Tf 12.158 0 Td [(6 m 1 + 6 m 2 + 2 m 3 )]TJ/F59 11.9552 Tf 12.158 0 Td [(9 m 5 )]TJ/F59 11.9552 Tf 12.158 0 Td [(3 m 6 + 3 m 7 + m 8 ] ,.34g S 7 = 1 36 [ 4 m 0 )]TJ/F59 11.9552 Tf 12.158 0 Td [(6 m 1 )]TJ/F59 11.9552 Tf 12.158 0 Td [(6 m 2 + 2 m 3 + 9 m 5 + 3 m 6 + 3 m 7 + m 8 ] ,.34h S 8 = 1 36 [ 4 m 0 + 6 m 1 )]TJ/F59 11.9552 Tf 12.158 0 Td [(6 m 2 + 2 m 3 )]TJ/F59 11.9552 Tf 12.158 0 Td [(9 m 5 + 3 m 6 )]TJ/F59 11.9552 Tf 12.158 0 Td [(3 m 7 + m 8 ] ..34i Inaddition,inordertoconstructthecascadedcollisionoperatorforthesolutionof thetemperatureeld,weneedtherawmomentsofthecollisionkernelofdifferent order,i.e. a K b h a e m a x e n a y = b h K b j e m a x e n a y i b h b .Sincethetemperatureeld T isa collisioninvariant,itfollowsthat b h 0 = 0.Usingthisandconsideringtheorthogonal 19

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basisvector k b inEq..20,weget a K b h a = b h K b j T i b h b = 0,.35a a K b h a e a x = b h K b j e a x i b h b = 6 b h 1 ,.35b a K b h a e a y = b h K b j e a y i b h b = 6 b h 2 ,.35c a K b h a e 2 a x = b h K b j e 2 a x i b h b = 6 b h 3 + 2 b h 4 ,.35d a K b h a e 2 a y = b h K b j e 2 a y i b h b = 6 b h 3 )]TJ/F59 11.9552 Tf 12.158 0 Td [(2 b h 4 ,.35e a K b h a e a x e a y = b h K b j e a x e a y i b h b = 4 b h 5 ,.35f a K b h a e 2 a x e a y = b h K b j e 2 a x e a y i b h b = 4 b h 2 )]TJ/F59 11.9552 Tf 12.158 0 Td [(4 b h 6 ,.35g a K b h a e a x e 2 a y = b h K b j e a x e 2 a y i b h b = 4 b h 1 )]TJ/F59 11.9552 Tf 12.157 0 Td [(4 b h 7 ,.35h a K b h a e 2 a x e 2 a y = b h K b j e 2 a x e 2 a y i b h b = 8 b h 3 + 4 b h 8 ..35i Now,weareinapositiontodeterminethestructureofthecascadedcollision operatorwithsourcetermstosolveforthethermaltransportequationrepresentedby theCDE.Theprocedurecanbebrieysummarizedasfollows:Startingfromthe lowestordernon-conservedpost-collisioncentralmomentsi.e.therstorder componentsinthepresentcase,wetentativelysetthemequaltotheircorresponding equilibriumstates.Oncetheexpressionsforthecollisionkernel b h b b 1 are determined,wediscardtheseequilibriumassumptionandmultiplyitwitha correspondingrelaxationparameters b l b toallowforarelaxationprocessduring collision[22,44].Here,onlythosetermsthatarenotinthepost-collisionstatesfor thelowerordermomentsaremultipliedbytherelaxationparameters.Thus,westart fromtherstorderpost-collisioncentralmoment,i.e. h g p a j e a x )]TJ/F68 11.9552 Tf 12.307 0 Td [(u x i and h g p a j e a y )]TJ/F68 11.9552 Tf 12.307 0 Td [(u y i andtentativelysetto b k eq x and b k eq y ,respectively: b k eq x = 0 = h g p a j e a x )]TJ/F68 11.9552 Tf 12.307 0 Td [(u x i = h g p a j e a x i)]TJ/F68 11.9552 Tf 19.76 0 Td [(u x h g p a j T i .36 20

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WherefromEq..54brawmomentsofthepost-collisiondistributionfunctionin termsofitspre-collisionvalue,collisionkernel,andsourcetermareusedtoobtain therighthandsidesoftheaboveequation.Thatis, h g p a j e a x i = h g a j e a x i + h K b h a j e a x i + h S a j e a x i = b k 0 x + 6 b h 1 + b s 0 x h g p a j T i = h g a j T i + h K b h a j T i + h S a j T i = T + 1 2 G SubstitutingtheaboveexpressionsinEq..36andrearranging,andsolvefor collisionkernelwegetthefollowingtentativeexpression b h 1 = 1 6 b k eq x 0 )]TJ/F85 11.9552 Tf 11.966 2.565 Td [(b k 0 x )]TJ/F59 11.9552 Tf 13.354 8.094 Td [(1 2 b s 0 x .37 Where b k eq x 0 = u x T .Inordertoprovidefurtherexibilityinadjustingthetransport coefcientintheCDE,therawmomentequilibrium b k eq x 0 willbeagumentedwithan extendedmomentequilibrium 1 3 D d t )]TJ/F66 11.9552 Tf 5.774 -9.69 Td [( x T ,whereDistheadjustableparameter.See AppendixBfortheanalysisofsuchascheme.Inaddition,weapplyarelaxation parameter l 1 intheequationaboveEq..37toreectthecollisionasarelaxation process.Thus,weget b h 1 = l 1 6 b k eq x 0 )]TJ/F85 11.9552 Tf 11.966 2.565 Td [(b k 0 x )]TJ/F59 11.9552 Tf 13.353 8.094 Td [(1 2 b s 0 x + 1 3 D d t )]TJ/F66 11.9552 Tf 5.774 -9.69 Td [( x T .38 Similarly,setting h g p a j e a y )]TJ/F68 11.9552 Tf 12.307 0 Td [(u y i to b k eq y = 0andusing h g p a j e a y i = h g a j e a y i + h K b h a j e a y i + h S a j e a y i = b k 0 y + 6 b h 2 + b s 0 y andfollowingthesameprocedureasabove,weobtain b h 2 = l 2 6 b k eq y 0 )]TJ/F85 11.9552 Tf 11.966 2.564 Td [(b k 0 y )]TJ/F59 11.9552 Tf 13.353 8.093 Td [(1 2 b s 0 y + 1 3 D d t )]TJ/F66 11.9552 Tf 5.774 -9.689 Td [( y T .39 Intheabove,thetemperaturegradientneededintheextendedmomentequilibriacan 21

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belocallycomputedintermsoftherstordernon-equilibriummomentssee AppendixBfordetails.Next,considerthesecondorderdiagonalcentralmoments andtentativlysetthemtotheircorrespondingequilibriumstates,i.e. b k eq xx = c 2 s T = h g p a j e a x )]TJ/F68 11.9552 Tf 12.307 0 Td [(u x 2 i = h g p a j e 2 a x i)]TJ/F59 11.9552 Tf 19.611 0 Td [(2 u x h g p a j e a x i + u 2 x h g p a j T i ..40 and b k eq yy = c 2 s T = h g p a j e a y )]TJ/F68 11.9552 Tf 12.307 0 Td [(u y 2 i = h g p a j e 2 a y i)]TJ/F59 11.9552 Tf 19.611 0 Td [(2 u y h g p a j e a y i + u 2 y h g p a j T i ..41 Then,using h g p a j e 2 a x i = h g a j e 2 a x i + h K b h a j e 2 a x i + h S a j e 2 a x i = b k 0 xx + 6 b h 3 + 2 b h 4 + b s 0 xx h g p a j e 2 a y i = h g a j e 2 a y i + h K b h a j e 2 a y i + h S a j e 2 a y i = b k 0 yy + 6 b h 3 )]TJ/F59 11.9552 Tf 12.157 0 Td [(2 b h 4 + b s 0 yy substitutingtheabovetwoexpressionsinEqs..40and.41,respectivelyand rearranging,weget 6 b h 3 + 2 b h 4 = 1 3 T )]TJ/F68 11.9552 Tf 12.307 0 Td [(u 2 x T )]TJ/F85 11.9552 Tf 11.967 2.564 Td [(b k 0 xx + 2 u x b k 0 x )]TJ/F85 11.9552 Tf 12.384 2.684 Td [(b s 0 xx + 2 u x b s 0 x + 1 2 u 2 x G + 12 u x b h 1 ..42 6 b h 3 )]TJ/F59 11.9552 Tf 12.157 0 Td [(2 b h 4 = 1 3 T )]TJ/F68 11.9552 Tf 12.307 0 Td [(u 2 y T )]TJ/F85 11.9552 Tf 11.967 2.564 Td [(b k 0 yy + 2 u y b k 0 y )]TJ/F85 11.9552 Tf 12.384 2.684 Td [(b s 0 yy + 2 u y b s 0 y + 1 2 u 2 y G + 12 u y b h 2 ..43 Solvingfor b h 3 and b h 4 fromtheabovetwoequationsandthenapplyingtherelaxation parameters l 3 and l 4 ,respectively,for b h 3 and b h 4 ,whileexcludingthelowerorder collisionkerneltermsi.e. b h 1 and b h 2 astheyarealreadyinthepost-collisionstate,we nallyget b h 3 = l 3 12 2 3 T )]TJ/F70 12.4573 Tf 12.307 0 Td [( u 2 x + u 2 y T )]TJ/F70 12.4573 Tf 12.307 0 Td [( b k 0 xx + b k 0 yy + 2 u x b k 0 x + 2 u y b k 0 y + 1 2 b s 0 xx + b s 0 yy + u x b h 1 + u y b h 2 ,.44 22

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b h 4 = l 4 4 )]TJ/F70 12.4573 Tf 9.982 0 Td [( u 2 x )]TJ/F68 11.9552 Tf 12.307 0 Td [(u 2 y T )]TJ/F70 12.4573 Tf 12.307 0 Td [( b k 0 xx )]TJ/F85 11.9552 Tf 11.966 2.565 Td [(b k 0 yy + 2 u x b k 0 x )]TJ/F59 11.9552 Tf 12.158 0 Td [(2 u y b k 0 y + 1 2 b s 0 xx )]TJ/F85 11.9552 Tf 12.385 2.684 Td [(b s 0 yy + 3 u x b h 1 )]TJ/F59 11.9552 Tf 12.157 0 Td [(3 u y b h 2 ,.45 Clearly,thecascadedstructureisalreadyevidentinthecollisionkernelsofthesecond ordermoments,whichisunlikethatfortheuidowLBEsolver,wherethecascaded structurestartstoappearonlyatthethirdordermomentcollisionkernels.Thisarises duetodifferencesinthenumberofcollisioninvariantsbetweenthetwocascaded LBEmodels.Next,consideringthepost-collisionstateoftheoff-diagonalcentral momentas b k eq xy = 0 = h g p a j e a x )]TJ/F68 11.9552 Tf 12.307 0 Td [(u x e a y )]TJ/F68 11.9552 Tf 12.307 0 Td [(u y i = h g p a j e a x e a y i)]TJ/F68 11.9552 Tf 19.76 0 Td [(u y h g p a j e a x i)]TJ/F68 11.9552 Tf 19.761 0 Td [(u x h g p a j e a y i + u x u y h g p a j T i .46 Using h g p a j e a x e a y i = b k eq xy + 4 b h 5 + b s 0 xy intheaboveequationandsimplifyitasa tentativeexpressionfor b h 5 andthenapplyingtherelaxationparameter l 5 tothose termsthatarenotyetinthepost-collisionstates,weget b h 5 = l 5 4 )]TJ/F85 11.9552 Tf 9.642 2.564 Td [(b k eq xy 0 )]TJ/F85 11.9552 Tf 11.967 2.564 Td [(b k 0 xy + u x b k 0 y + u y b k 0 x + 1 2 b s 0 xy + 3 2 u x b h 2 + u y b h 1 .47 Now,considerthedeterminationofthethirdordermomentcollisionkernel.Setting tentatively b k eq xxy = 0 = h g p a j e a x )]TJ/F68 11.9552 Tf 12.307 0 Td [(u x 2 e a y )]TJ/F68 11.9552 Tf 12.307 0 Td [(u y i .48 b k eq xyy = 0 = h g p a j e a x )]TJ/F68 11.9552 Tf 12.307 0 Td [(u x e a y )]TJ/F68 11.9552 Tf 12.307 0 Td [(u y 2 i .49 andusing h g p a j e a x 2 e a y i = b k 0 xxy + 4 b h 2 )]TJ/F59 11.9552 Tf 12.158 0 Td [(4 b h 6 + b s 0 xxy , h g p a j e a x e 2 a y i = b k 0 xyy + 4 b h 1 )]TJ/F59 11.9552 Tf 12.158 0 Td [(4 b h 7 + b s 0 xyy , 23

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inEqs..48and.49,respectively,andsimplifyingtoobtainthetentative expressionsfor l 6 and l 7 ,respectively,tothosetermsthatarenotyetpost-collision states,weobtain b h 6 = l 6 4 )]TJ/F68 11.9552 Tf 9.982 0 Td [(u 2 x u y T + b k 0 xxy )]TJ/F68 11.9552 Tf 12.307 0 Td [(u y b k 0 xx )]TJ/F59 11.9552 Tf 12.158 0 Td [(2 u x b k 0 xy + u 2 x b k 0 y + 2 u x u y b k 0 x + 1 2 b s 0 xxy + 1 2 2 b h 2 + 3 u 2 x b h 2 )]TJ/F59 11.9552 Tf 13.353 8.094 Td [(1 4 u y 6 b h 3 + 2 b h 4 + 3 u x u y b h 1 )]TJ/F59 11.9552 Tf 12.158 0 Td [(2 u x b h 5 ,.50 b h 7 = l 7 4 )]TJ/F68 11.9552 Tf 9.982 0 Td [(u x u 2 y T + b k 0 xyy )]TJ/F68 11.9552 Tf 12.307 0 Td [(u x b k 0 yy )]TJ/F59 11.9552 Tf 12.157 0 Td [(2 u y b k 0 xy + 2 u x u y b k 0 y + u 2 y b k 0 x + 1 2 b s xyy + 1 2 2 b h 1 + 3 u 2 y b h 1 )]TJ/F59 11.9552 Tf 13.353 8.094 Td [(1 4 u x 6 b h 3 )]TJ/F59 11.9552 Tf 12.158 0 Td [(2 b h 4 + 3 u x u y b h 2 )]TJ/F59 11.9552 Tf 12.158 0 Td [(2 u y b h 5 .51 Finally,bysettingthepost-collisionstateofthefourthordercentralmomenttoits correspondingequilibriumasatentativestepas b k eq xxyy = 1 9 T = h g p a j e a x )]TJ/F68 11.9552 Tf 12.307 0 Td [(u x 2 e a y )]TJ/F68 11.9552 Tf 12.307 0 Td [(u 2 y i .52 andapplying b k eq xxyy = 1 9 T = h g p a j e a x 2 e 2 a y i = b k eq xxyy + 8 b h 3 + 4 b h 8 + b s 0 xxyy and simplyingEq..52andfollowingthesameprocedureasabovebyapplyingthe relaxationparameter l 8 ,weget b h 8 = l 8 4 1 9 T )]TJ/F68 11.9552 Tf 12.307 0 Td [(u 2 x u 2 y T + b k 0 xxyy + 2 u x b k 0 xyy + 2 u y b k 0 xxy )]TJ/F68 11.9552 Tf 12.307 0 Td [(u 2 x b k 0 yy )]TJ/F68 11.9552 Tf 12.307 0 Td [(u 2 y b k 0 xx )]TJ/F59 11.9552 Tf 12.157 0 Td [(4 u x u y b k 0 xy + 2 u x u 2 y b k 0 x + 2 u 2 x u y b k 0 y + 1 2 b s 0 xxyy )]TJ/F59 11.9552 Tf 12.158 0 Td [(2 b h 3 )]TJ/F59 11.9552 Tf 13.353 8.093 Td [(1 2 u 2 y 3 b h 3 + b h 4 + 2 u x 3 b h 1 )]TJ/F85 11.9552 Tf 12.008 3.096 Td [(b h 7 )]TJ/F59 11.9552 Tf 11.028 8.094 Td [(1 2 u 2 x 3 b h 3 )]TJ/F85 11.9552 Tf 12.008 3.096 Td [(b h 4 + 2 u y 3 b h 2 )]TJ/F85 11.9552 Tf 12.008 3.096 Td [(b h 6 )]TJ/F59 11.9552 Tf 12.157 0 Td [(4 u x u y b g 5 + 3 u 2 x u y b h 2 + 3 u x u 2 y b h 1 ..53 Inaddition,inordertomaintainadditionalexibilityintherepresentationofthe emergenttransportcoefcienti.e.thethermaldiffusivityoftheCDE,wealso 24

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introduceextendedmomentequilibriainvolvingtemperaturegradienttermswithan adjustablecoefcientDintherstorderequilibriummoments.Theresultingnal expressionsofthecollisionkernelaregivenasfollows: b h 0 = 0,.54a b h 1 = l 1 6 b k eq x 0 )]TJ/F85 11.9552 Tf 11.966 2.565 Td [(b k 0 x )]TJ/F59 11.9552 Tf 13.353 8.094 Td [(1 2 b s 0 x + 1 3 D d t )]TJ/F66 11.9552 Tf 5.774 -9.69 Td [( x T ,.54b b h 2 = l 2 6 b k eq y 0 )]TJ/F85 11.9552 Tf 11.966 2.564 Td [(b k 0 y )]TJ/F59 11.9552 Tf 13.353 8.093 Td [(1 2 b s 0 y + 1 3 D d t )]TJ/F66 11.9552 Tf 5.774 -9.689 Td [( y T ,.54c b h 3 = l 3 12 2 3 T )]TJ/F70 12.4573 Tf 12.307 0 Td [( u 2 x + u 2 y T )]TJ/F70 12.4573 Tf 12.307 0 Td [( b k 0 xx + b k 0 yy + 2 u x b k 0 x + 2 u y b k 0 y + 1 2 b s 0 xx + b s 0 yy + u x b h 1 + u y b h 2 ,.54d b h 4 = l 4 4 )]TJ/F70 12.4573 Tf 9.983 0 Td [( u 2 x )]TJ/F68 11.9552 Tf 12.307 0 Td [(u 2 y T )]TJ/F70 12.4573 Tf 12.307 0 Td [( b k 0 xx )]TJ/F85 11.9552 Tf 11.966 2.564 Td [(b k 0 yy + 2 u x b k 0 x )]TJ/F59 11.9552 Tf 12.157 0 Td [(2 u y b k 0 y + 1 2 b s 0 xx )]TJ/F85 11.9552 Tf 12.385 2.684 Td [(b s 0 yy + 3 u x b h 1 )]TJ/F59 11.9552 Tf 12.158 0 Td [(3 u y b h 2 ,.54e b h 5 = l 5 4 )]TJ/F85 11.9552 Tf 9.642 2.564 Td [(b k eq xy 0 )]TJ/F85 11.9552 Tf 11.966 2.564 Td [(b k 0 xy + u x b k 0 y + u y b k 0 x + 1 2 b s 0 xy + 3 2 u x b h 2 + u y b h 1 ,.54f b h 6 = l 6 4 )]TJ/F68 11.9552 Tf 9.983 0 Td [(u 2 x u y T + b k 0 xxy )]TJ/F68 11.9552 Tf 12.307 0 Td [(u y b k 0 xx )]TJ/F59 11.9552 Tf 12.158 0 Td [(2 u x b k 0 xy + u 2 x b k 0 y + 2 u x u y b k 0 x + 1 2 b s 0 xxy + 1 2 2 b h 2 + 3 u 2 x b h 2 )]TJ/F59 11.9552 Tf 13.353 8.093 Td [(1 4 u y 6 b h 3 + 2 b h 4 + 3 u x u y b h 1 )]TJ/F59 11.9552 Tf 12.157 0 Td [(2 u x b h 5 ,.54g b h 7 = l 7 4 )]TJ/F68 11.9552 Tf 9.983 0 Td [(u x u 2 y T + b k 0 xyy )]TJ/F68 11.9552 Tf 12.307 0 Td [(u x b k 0 yy )]TJ/F59 11.9552 Tf 12.158 0 Td [(2 u y b k 0 xy + 2 u x u y b k 0 y + u 2 y b k 0 x + 1 2 b s xyy + 1 2 2 b h 1 + 3 u 2 y b h 1 )]TJ/F59 11.9552 Tf 13.353 8.094 Td [(1 4 u x 6 b h 3 )]TJ/F59 11.9552 Tf 12.157 0 Td [(2 b h 4 + 3 u x u y b h 2 )]TJ/F59 11.9552 Tf 12.157 0 Td [(2 u y b h 5 ,.54h b h 8 = l 8 4 1 9 T )]TJ/F68 11.9552 Tf 12.307 0 Td [(u 2 x u 2 y T + b k 0 xxyy + 2 u x b k 0 xyy + 2 u y b k 0 xxy )]TJ/F68 11.9552 Tf 12.307 0 Td [(u 2 x b k 0 yy )]TJ/F68 11.9552 Tf 12.307 0 Td [(u 2 y b k 0 xx )]TJ/F59 11.9552 Tf 12.157 0 Td [(4 u x u y b k 0 xy + 2 u x u 2 y b k 0 x + 2 u 2 x u y b k 0 y + 1 2 b s 0 xxyy )]TJ/F59 11.9552 Tf 12.158 0 Td [(2 b h 3 )]TJ/F59 11.9552 Tf 13.353 8.093 Td [(1 2 u 2 y 3 b h 3 + b h 4 + 2 u x 3 b h 1 )]TJ/F85 11.9552 Tf 12.008 3.096 Td [(b h 7 )]TJ/F59 11.9552 Tf 11.028 8.094 Td [(1 2 u 2 x 3 b h 3 )]TJ/F85 11.9552 Tf 12.008 3.096 Td [(b h 4 + 2 u y 3 b h 2 )]TJ/F85 11.9552 Tf 12.008 3.096 Td [(b h 6 )]TJ/F59 11.9552 Tf 12.157 0 Td [(4 u x u y b g 5 + 3 u 2 x u y b h 2 + 3 u x u 2 y b h 1 ..54i Noticethat b h 1 6 = b h 2 6 = 0,inthepresentcase,whichisunlikethatforthecascadedLBE foruidow[44].ThisdifferencearisesfromthefactthatthecascadedLBEforthe oweldhasthreecollisioninvarients,i.e.massandmomentum,andhenceits correspondingzerothandrstordercollisionkernelsarezero;Ontheotherhand,in thecaseofcascadedLBEforthethermaltransportequation,thereisonlyone 25

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collisioninvariant,i.e.temperatureeld,andthereforeonlyitszerothordercollision kernelis b h 0 iszero.Asaresultofthesedifferencesthecascadedcollisionoperator forthetemperatureeldismarkedlydifferentfromthatfortheoweld. notethat, l b , b = 1,2,3,...,8,aretherelaxationparameters,satisfyingthe bounds0 < l b < 2.Noticethecascadedstructureforthesecondandhigherorder momentkernels,i.e.theirdependenceonthelowerordermomentsforourthermal cascadedLBE.Bycontrast,thecascadedLBEfortheuidowissignicantly different,withthecascadedstructureappearingonlyforthirdandhigherorder moments.WhenaChapman-EnskogexpansionC-Eisperformedontheabove cascadedLBmodelseeAppendixBfordetails,itcanbeshowntorecoverthe convection-diffusionthermaltransportequation,withtherelaxationparametersfor therstordermoments l 1 and l 2 andtheadjustablecoefcientDcontrollingthe thermaldiffusivitycoefcient a seeEq..17: a = 1 3 1 l j )]TJ/F59 11.9552 Tf 13.353 8.093 Td [(1 2 )]TJ/F68 11.9552 Tf 12.546 0 Td [(D ! d t , j = 1,2..55 Therestoftheparameterscanbeadjustedindependentlytoimprovenumerical stability.Inthiswork, l 1 = l 2 = 1/ t t isselectedbasedonthespecieddiffusivity, whiletheremainingrelaxationparametersaresettobeunity. Moreover,thetemperaturegradients x T and y T appearingintheabovesee Eqs..54band.54ccanbecalculatedlocallyintermsoftherstorder non-equilibriummoments.SeetheC-EanalysisgiveninAppendixBfordetails. Thus,wehave T x = )]TJ/F59 11.9552 Tf 9.833 0 Td [(3 l 1 b k 0 x )]TJ/F85 11.9552 Tf 11.966 2.564 Td [(b k eq 0 x 1 )]TJ/F68 11.9552 Tf 12.546 0 Td [(D l 1 ,.56a T y = )]TJ/F59 11.9552 Tf 9.833 0 Td [(3 l 2 b k 0 y )]TJ/F85 11.9552 Tf 11.966 2.565 Td [(b k eq 0 y 1 )]TJ/F68 11.9552 Tf 12.546 0 Td [(D l 2 .56b 26

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Where b k 0 x = 8 a = 0 e a x g a , b k 0 y = 8 a = 0 e a y g a , b k eq 0 x = u x T ,and b k eq 0 y = u y T .Then, thethermalcascadedLBEgiveninEq..26canbewrittenintermsofthefollowing collisionandstreamingsteps: g p a )723(! x , t = g a )723(! x , t + W a )723(! x , t + S a )723(! x , t ,.57a g a )723(! x + )723(! e a , t + 1 = g p a )723(! x , t ..57b Byexpanding K . b h a inEq..57a,theexplicitexpressionsforthepostcollision distributionfunctionsaregivenasfollows: g p 0 = g 0 + h b h 0 )]TJ/F59 11.9552 Tf 12.158 0 Td [(4 b h 3 )]TJ/F85 11.9552 Tf 12.008 3.096 Td [(b h 8 i + S 0 ,.58a g p 1 = g 1 + h b h 0 + b h 1 )]TJ/F85 11.9552 Tf 12.008 3.097 Td [(b h 3 + b h 4 + 2 b h 7 )]TJ/F85 11.9552 Tf 12.009 3.097 Td [(b h 8 i + S 1 ,.58b g p 2 = g 2 + h b h 0 + b h 2 )]TJ/F85 11.9552 Tf 12.008 3.096 Td [(b h 3 )]TJ/F85 11.9552 Tf 12.008 3.096 Td [(b h 4 + 2 b h 6 )]TJ/F85 11.9552 Tf 12.009 3.096 Td [(b h 8 i + S 2 ,.58c g p 3 = g 3 + h b h 0 )]TJ/F85 11.9552 Tf 12.009 3.097 Td [(b h 1 )]TJ/F85 11.9552 Tf 12.008 3.097 Td [(b h 3 + b h 4 )]TJ/F59 11.9552 Tf 12.157 0 Td [(2 b h 7 + b h 8 i + S 3 ,.58d g p 4 = g 4 + h b h 0 )]TJ/F85 11.9552 Tf 12.009 3.096 Td [(b h 2 )]TJ/F85 11.9552 Tf 12.008 3.096 Td [(b h 3 )]TJ/F85 11.9552 Tf 12.008 3.096 Td [(b h 4 )]TJ/F59 11.9552 Tf 12.157 0 Td [(2 b h 6 + b h 8 i + S 4 ,.58e g p 5 = g 5 + h b h 0 + b h 1 + b h 2 + 2 b h 3 + b h 5 )]TJ/F85 11.9552 Tf 12.008 3.096 Td [(b h 6 )]TJ/F85 11.9552 Tf 12.008 3.096 Td [(b h 7 + b h 8 i + S 5 ,.58f g p 6 = g 6 + h b h 0 )]TJ/F85 11.9552 Tf 12.009 3.097 Td [(b h 1 + b h 2 + 2 b h 3 )]TJ/F85 11.9552 Tf 12.009 3.097 Td [(b h 5 )]TJ/F85 11.9552 Tf 12.008 3.097 Td [(b h 6 + b h 7 + b h 8 i + S 6 ,.58g g p 7 = g 7 + h b h 0 )]TJ/F85 11.9552 Tf 12.009 3.096 Td [(b h 1 )]TJ/F85 11.9552 Tf 12.008 3.096 Td [(b h 2 + 2 b h 3 + b h 5 + b h 6 + b h 7 + b h 8 i + S 7 ,.58h g p 8 = g 8 + h b h 0 + b h 1 )]TJ/F85 11.9552 Tf 12.008 3.097 Td [(b h 2 + 2 b h 3 )]TJ/F85 11.9552 Tf 12.009 3.097 Td [(b h 5 + b h 6 )]TJ/F85 11.9552 Tf 12.008 3.097 Td [(b h 7 + b h 8 i + S 8 ..58i Finally,basesdonthesolutionofthethermalcascadedLBEgiveninEqs..57a and.57b,thetemperatureeldTcanbeobtainedas T = a g a = a g a + 1 2 G d t ..59 27

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1.5AnisotropicCascadedModelfortheTemperatureField Anisotropicdiffusionisacommonphysicalphenomenonanddescribeprocesses wherethediffusionofsomelocationisdirectiondependent,i.ediffusioncoefcients arelocationand/ordirectiondependent.Inaddition,manypracticalproblemssuch asuidowsinanisotropicmediaarisinginenvironmentalscienceandgeophysics dohaveanisotropicdiffusioncoefcients.MostoftheLBEmodelsforCDEare limitedtothedescriptionoftheisotropicdiffusionproblems[5,25,26] Toincorporatefullanisotropywithoff-diagonalcomponentsofthethermal diffusion-coefcienttensor,weconsider b h 1 = l 11 6 b k eq x 0 )]TJ/F85 11.9552 Tf 11.966 2.564 Td [(b k 0 x )]TJ/F59 11.9552 Tf 13.353 8.093 Td [(1 2 b s 0 x + 1 3 D d t )]TJ/F66 11.9552 Tf 5.774 -9.689 Td [( x T + l 12 6 b k eq y 0 )]TJ/F85 11.9552 Tf 11.966 2.564 Td [(b k 0 y )]TJ/F59 11.9552 Tf 13.353 8.093 Td [(1 2 b s 0 y + 1 3 D d t )]TJ/F66 11.9552 Tf 5.774 -9.689 Td [( y T .60 b h 2 = l 21 6 b k eq x 0 )]TJ/F85 11.9552 Tf 11.966 2.565 Td [(b k 0 x )]TJ/F59 11.9552 Tf 13.353 8.094 Td [(1 2 b s 0 x + 1 3 D d t )]TJ/F66 11.9552 Tf 5.774 -9.69 Td [( x T + l 22 6 b k eq y 0 )]TJ/F85 11.9552 Tf 11.966 2.565 Td [(b k 0 y )]TJ/F59 11.9552 Tf 13.353 8.094 Td [(1 2 b s 0 y + 1 3 D d t )]TJ/F66 11.9552 Tf 5.774 -9.69 Td [( y T .61 wheretheoff-diagonalcomponentsoftherelaxationtimematrix,asaresultofthe rotationoftheprincipalaxes,enableustosimulatethefullanisotropy convection-diffusion. Here, L isacollisionmatrixgivenby L = diag l 0 , 2 6 4 l 11 l 12 l 21 l 22 3 7 5 , l 3 , l 4 ,......., l 8 .62 28

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= 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 4 l 0 00000000 0 l 11 l 12 000000 0 l 21 l 22 000000 000 l 3 00000 0000 l 4 0000 00000 l 5 000 000000 l 6 00 0000000 l 7 0 00000000 l 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 5 ..63 1.6BoundaryConditions Asmentionedintheprevioussections,theLatticeBoltzmannmethodhasmany importantadvantages,oneofthesefeaturesisit'scapabilitytohandlethecomplex boundary.BoundaryconditionsBCshaveveryimportantroleforstabilityandthe accuracyofanynumericalsolutionforthelatticeBoltzmannmethod,thediscrete distributionfunctionsontheboundaryhavetobetakencareoftoreectthe macroscopicBCsoftheuid.Therearemanytypesofboundaryconditionsthathave beenhandledbyLBE.Therehavebeenmanytypesofthermalboundaryconditions appliedtoCDE.[29].Theboundaryconditionsusedinthisstudyforsimulating isotropicandanisotropicthermalconvectiveowconsistsofperiodicboundary conditionsandbounce-backboundaryconditions.Inthisthesis,thehalfway bounce-backschemeisemployedtotreatvelocityboundaryconditionswhilethe generalanti-bounce-backscheme[48]isadoptedtodealwithtemperatureboundary conditionsandperiodicboundaryconditionforinowandoutowboundaries. 29

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F IGURE 1.4:Locationofboundarynodes. 1.6.1Halfwaybounce-backboundaries IntheLatticeBoltzmannmethodtheinteractionofauidparticlewithasolid particleisperformedtheusingbounce-backmethod.Forthedensitydistribution functions,bounce-backboundaryconditionswereappliedonallsolidboundaries, whichmeansthatincomingboundarypopulationequaltoout-goingpopulations afterthecollision.InLBEdistributionfunctionsoutofthedomainareknownfrom streamingprocess,Fig.1.5showstheunknowndistributionfunctionsvelocityand temperaturewhichareneededtobedeterminedasdottedlines,whichmeansthat incomingboundarypopulationequaltoout-goingpopulationsafterthecollision.For exampleforeastboundary,f5,f1,andf8streamintothewall,andarebouncedback bysettingf3=f1,f7=f5,andf6=f8asitcanbeseeninFig.1.5.Thebounce-back methodhasbeendiscussedextensivelyintheliteratureandafewversionshavebeen developed.Themostsimpleschemeistoplaceawallhalfwaybetweenawallgrid pointandauidgridpointandthen"bounce-back"particlesthatstreamintothe wall.Ithasbeenfoundthatthehalfwaybounce-backschemeissecondorder accuratewithrespecttogridspacingforregularboundary[27]. 30

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F IGURE 1.5:BoundaryConditionswithunknownandknownpopulations. F IGURE 1.6:Locationofboundarynodes. 31

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1.6.1.1VelocityDistributionFunction inordertoobtainano-slipboundaryconditionforaspeciedboundary,weusethe "BounceBackMethod.Themostsimpleschemeistoplaceawallhalfwaybetweena wallgridpointandauidgridpointandthen"bounce-back"particlesthatstream intothewall.Forinstance,f4,f7,andf8streamintothewall,andarebouncedback bysettingf5=f7,f2=f4,andf6=f8Fig.1.6 1.6.1.2TemperatureDistributionFunction Thegeneralanti-bounce-backscheme[48]isadoptedheretodealwithtemperature boundaryconditions,thethermalorconcentrationboundaryconditionscanbe classiedintothreetypes,Dirichlet,Neumannandmixedboundaryconditionsas b 1 T n + b 2 T = b 3 .64 First:forDirichletboundaryconditions: forstaticboundary,i.e. u w = 0 T = T w = b 3 b 2 .65 isconstantattheboundary.ToimplementtheseboundaryconditionswithLBE,the distributionfunctionisconstructedas g a xf , t = )]TJETq1 0 0 1 286.191 250.261 cm[]0 d 0 J 0.478 w 0 0 m 6.539 0 l SQBT/F68 11.9552 Tf 286.604 242.807 Td [(g p a xf , t + 2 w a T w .66 formovingboundary,i.e. u w 6 = 0 g a xf , t = )]TJETq1 0 0 1 204.236 158.751 cm[]0 d 0 J 0.478 w 0 0 m 6.539 0 l SQBT/F68 11.9552 Tf 204.648 151.297 Td [(g p a xf , t + 2 w a T w 1 + 4.5 e a . u w c 2 )]TJ/F59 11.9552 Tf 12.158 0 Td [(1.5 j u w j 2 c 2 .67 Second:forNeumannboundaryconditions: 32

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F IGURE 1.7:Schematicillustratingtheperiodicboundaryconditions. F IGURE 1.8:Periodicboundaryconditionontheinowandoutow boundaries. T n = T f )]TJ/F68 11.9552 Tf 12.463 0 Td [(T w )]TJ/F59 11.9552 Tf 9.833 0 Td [(0.5 n e a d x .68 Where T f isthetemperatureattheuidnodeneighboringtheinterface, w a is theLatticeweightand e a isthediscretevelocity 1.6.2PeriodicBoundaryConditions Theperiodicboundaryisthesimplestboundaryconditions.Inthistypeofboundary, theunknownsofoneboundarycanbedirectlyrelatedtotheknownsoftheother boundary.i.e,itisapplieddirectlytocalculatetheunknowncomponentsofthe particledistributionfunctionofthenodeslocatedattheboundariesofthe domain,e.g.inletandoutletasshowninFig.1.7 Forexample,aperiodicboundaryconditionisrequiredontheinowand 33

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outowboundariesasshowninFig.1.8.Theperiodicboundaryconditiononthe inowboundary g a x i , t + d t = g a x n , t , a = 1,5,8.69 andontheoutowboundary g a x n , t + d t = g a x i , t , a = 3,6,7.70 where x i and x n arethecoordinatesoftheinowandoutowboundaries 34

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CHAPTERII ISOTROPICTHERMALFLOW 2.1NumericalResults Here,numericalsimulationsofsomebenchmarkproblemsareconductedtovalidate theaccuracyofourproposedcascadedLBEmodelforconvectivethermalows.The testproblemswithoutsourcetermsintheenergyequationarethermalPoiseuille ow,thermalowinachannelwithwallinjection,andnaturalconvectioninsquare cavity.Also,problemsconsideredwithvariablesourcetermsintheenergyequation areareaction-diffusionproblem,andcouetteowwithtemperaturegradientsi.e. thermalCouetteowwithviscousheatdissipation.Inthisstudy,thehalfway bounce-backschemeisemployedtotreatvelocityboundaryconditionswhilethe generalanti-bounce-backscheme[48]isadoptedtodealwithtemperatureboundary conditions.InproblemsinvolvingLBMsolutionofuidowallrelaxation parametersaresetto1.0except w 4 and w 5 therelaxationratesfortherstorder momentswhicharebothequalto t )]TJ/F59 8.9664 Tf 7.375 0 Td [(1 f .Inthethermalmodelallrelaxation parametersaresetto1.0except l 1 and l 2 whicharebothequalto t )]TJ/F59 8.9664 Tf 7.374 0 Td [(1 g . 2.2UnsteadyReaction-DiffusionProblem:VariableSourceTerm Theunsteadyreactiondiffusionproblemisagoodproblemtotesttheaccuracyofthe presentLBEcascadedmodelfortheequivalentenergyequationwithavariable sourceterm.Suchasystemdenedintheregion0 x , y l ,withmacroscopic 35

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F IGURE 2.1:Temperatureprolefortheunsteadyreaction-diffusion problemwithavariablesourcetermat y = 0.5anddiffusioncoefcient a = 10 )]TJ/F59 7.9701 Tf 6.556 0 Td [(3 atdifferenttimes.MarkersrepresenttheCascadedLBEresults andlinesrepresenttheanalyticalsolution. governingequationswrittenasin[59]: T t = a r 2 T + 2 C sin p x / l sin p y / l .1 Where G = 2 C sin p x / l sin p y / l isthevariablesourceterm, l isthewidthofthe region, C isaconstant,and a isthediffusioncoefcient.Theinitialandboundary conditionsofthissystemare: T x , y ,0 = 0, T 0, y , t = T l , y , t = 0, T x ,0, t = T x , l , t = 0. Thegeneralbounce-backscheme[48]isemployedtorepresenttheseboundary conditions. Theanalyticalsolutionofthisproblemisgivenby T x , y , t = l 2 p 2 a C 1 )]TJ/F59 11.9552 Tf 12.157 0 Td [(exp )]TJ/F59 11.9552 Tf 11.029 8.094 Td [(2 p 2 t a l 2 sin p x / l sin p y / l .2 Weconductournumericalsimulationwithagridresolutionof61 61, C = 10 36

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F IGURE 2.2:Temperatureprolefortheunsteadyreaction-diffusion problemwithavariablesourcetermat y = 0.5anddiffusioncoefcient a = 10 )]TJ/F59 7.9701 Tf 6.556 0 Td [(4 atdifferenttimes.MarkersrepresenttheCascadedLBEresults andlinesrepresenttheanalyticalsolutions. andwiththethermaldiffusivitycoefcients a = 10 )]TJ/F59 8.9664 Tf 7.375 0 Td [(3 and a = 10 )]TJ/F59 8.9664 Tf 7.374 0 Td [(4 .Thesimulation resultsandtheanalyticalsolutionsarecomparedatthreedifferenttimes t = 50, t = 100,and t = 150asusedin[59].Therelaxationtimeissetto t g = 0.503for a = 10 )]TJ/F59 8.9664 Tf 7.375 0 Td [(3 ,and t g = 0.5003for a = 10 )]TJ/F59 8.9664 Tf 7.375 0 Td [(4 .Fig.2.1,andFig.2.2showthetemperature prolesforboththevaluesofdiffusivitycoefcientswhichindicateverygood agreementwiththeanalyticalsolutionsforboththermaldiffusioncoefcients.We alsoexaminethespatialaccuracyofthepresentmodel.Inthisregard,asetof simulationsareperformedatfourdifferentgridsizes,i.e.,25 25,51 51,101 101, and201 201forbothvaluesofthediffusioncoefcient,i.e. a = 10 )]TJ/F59 8.9664 Tf 7.375 0 Td [(3 ,and a = 10 )]TJ/F59 8.9664 Tf 7.375 0 Td [(4 . Theglobalrelativeerroroftemperature E T usedtomeasuretheaccuracyofthe modeliscalculatedas E T = jj T c )]TJ/F68 11.9552 Tf 12.463 0 Td [(T a jj 2 jj T a jj 2 .3 Where jj . jj 2 istheEuclideannorm, jj T c )]TJ/F68 11.9552 Tf 12.463 0 Td [(T a jj 2 = p i T c , i )]TJ/F68 11.9552 Tf 12.463 0 Td [(T a , i 2 , 37

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F IGURE 2.3:Temperatureglobalrelativeerroratdifferentvaluesofthe diffusioncoefcient a = 10 )]TJ/F59 7.9701 Tf 6.555 0 Td [(3 , D = 0.397,and a = 10 )]TJ/F59 7.9701 Tf 6.556 0 Td [(4 , D = 0.3997for theunsteadyreaction-diffusionproblemwithvariablesourceterm. jj T a jj 2 = p i T a , i 2 , T c and T a arethecomputedandtheanalyticalsolutions respectively.Therelativeglobalerroroftemperatureforeachvalueofdiffusivity coefcientareplottedinFig.2.3.Itcanbeseenthatthetemperatureglobalerror decreaseswithincreaseingridresolutionwithaslopeof )]TJ/F59 11.9552 Tf 9.833 0 Td [(2inthelog-logplot. Hence,ourpresentcascadedLBMmodelwithsourcetermissecondorderaccurate. 2.3ThermalFlowinaChannelwithWallInjection Inthissection,thepresentcascadedLBEmodelforconvectivethermalowis employedtosimulatethefullydevelopedthermalowinachannel,wheretheupper platemovesalongthex-directionwithvelocity U p ,andauidisinjectedinthe positivey-directionwithaconstantvelocity v 0 throughthestationarybottomwall. Theupperwallismaintainedatahighertemperature T h andthebottomwallis xedatalowertemperature T c .Thecomputationaldomainoftheproblemis 0 x , y L .Inthesteadystatecase,theanalyticalsolutionsforbothvelocityand 38

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temperatureeldsare,respectively,givenby. u x y = exp [ Re y / L ] )]TJ/F59 11.9552 Tf 12.158 0 Td [(1 exp Re )]TJ/F59 11.9552 Tf 12.158 0 Td [(1 ,.4 T = T c + 4 T exp [ PrRe y / L ] )]TJ/F59 11.9552 Tf 12.158 0 Td [(1 exp PrRe )]TJ/F59 11.9552 Tf 12.158 0 Td [(1 .5 where Re istheReynoldsnumberdenedby Re = [ Lv 0 / n ] , L isthewidthof thechanneland 4 T isthetemperaturedifference.Inournumericaltest,weset U p = v 0 = 0.01, T h = 1, T c = 0, Pr = 0.71,withagridsize31 61atdifferent Reynoldsnumbers Re = 5,10,and15.Therelaxationratesfortheowandthermal equationcascadedLBsolversareobtainedbasedonthevalueof Pr and Re ateach casewhere n = [ Lv 0 / Re ] ,and a = [ n / Pr ] .Therestoftherelaxationratesaresettobe 1.0.Thegeneralbounce-backscheme[48]isimplementedattheupperandbottom platesforthetemperatureboundaryconditions,ahalfwaybounce-backschemeis employedforthevelocityboundaryconditions,andperiodicboundaryconditions areimposedattheinletandoutletofthechannel.Theprolesofvelocityand temperaturealongtheydirectionatdifferentReynoldsnumbersand Pr = 0.71are plottedinFigs.2.4and2.5respectively.Itisfoundthatthenumericalresultsagree wellwiththeanalyticalsolutionsforthistestcase.Wealsostudytheconvergence ratebyconsideringthefollowinggridresolutions Ny = 31,61,91,and121.Inthese simulations,weconducttheconvergencestudyatReynoldsnumbers: Re = 5,10,and 15fortheabovesetofgridresolutionsusing w 4 = w 5 = l 1 = l 2 = 1 0.8 ,andallother relaxationparametersaresettobe1withthecorrespondingvaluesofthetunable parameterDas0.05,0.1,and0.15respectively.Forthevelocityeld,theReynolds numberissettobe Re = 10andrelaxationratesare t f = t g = 0.8 .Therelative globalerrorsofvelocityandtemperatureareplottedinFigs.2.6and2.7.Itcanbe seenthattherelativeerrorshaveslopesofalmostequalto2.00,whichagainconrms thatthepresentcascadedLBMmodelforthermalowissecondorderaccurate.In theabove,therelativeglobalerroroftemperatureandvelocityaredened, 39

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F IGURE 2.4:VelocityprolesforthermalCouetteowinachannelwith wallinjectionatReynoldsnumbers: Re = 5,10,15.Markersrepresentthe CascadedLBEresultsandlinesrepresenttheanalyticalsolutions. respectively,by E T = jj T c )]TJ/F68 11.9552 Tf 12.463 0 Td [(T a jj 2 jj T a jj 2 .6 E u = jj u c )]TJ/F68 11.9552 Tf 12.307 0 Td [(u a jj 2 jj u a jj 2 .7 where jj . jj 2 istheEuclideannorm, jj T c )]TJ/F68 11.9552 Tf 12.463 0 Td [(T a jj 2 = p i T c , i )]TJ/F68 11.9552 Tf 12.463 0 Td [(T a , i 2 , jj u c )]TJ/F68 11.9552 Tf 12.307 0 Td [(u a jj 2 = p i u c , i )]TJ/F68 11.9552 Tf 12.307 0 Td [(u a , i 2 , jj T a jj 2 = p i T a , i 2 , jj u a jj 2 = p i u a , i 2 . Here, T c , u c and T a , u a arethecomputedandtheanalyticalsolutionsrespectively. 2.4Diffusionin2DPoiseuilleFlow Next,weconsidera2DPoiseuilleowbetweentwoparallelplatesinthestreamwise directiondrivenbyaconstantbodyforce F x .Boththeupperandbottomwallsare stationaryandsubjectedtohigher T h andlower T c uniformtemperature respectively.Thecomputationaldomainis0 x , y L .Where L isthechannel width.Aperiodicboundaryconditionisappliedattheentranceandtheexitforboth velocityandtemperatureelds,whilethehalfwaybouncebackschemeis 40

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F IGURE 2.5:TemperatureprolesforthermalCouetteowinachannel withwallinjectionatReynoldsnumbers: Re = 5,10,15.MarkersrepresenttheCascadedLBEresultsandlinesrepresenttheanalyticalsolutions. implementedatthesolidboundariesupperandbottomwallsforthevelocityeld torepresenttheno-slipboundarycondition.Thegeneralbounce-backscheme[48]is employedtothesolidboundariesforthetemperatureDirichletboundaryconditions. TheanalyticalsolutionforthevelocityinPoiseuilleowparabolicprolesisgiven by u y = u max 1 )]TJ/F70 12.4573 Tf 12.307 0 Td [( y / L 2 .8 , where u max = F x L 2 /2 n isthemaximumvelocityoccurringhalfwaybetweenthe plates, n isthekinematicviscosityrelatedthetorelaxationtime t .Here, L isthehalf distancebetweenthetwoparallelplates.Theanalyticalsolutionforthetemperature inPoiseuilleowisgivenby T = T c + 4 T y / L .9 , 41

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F IGURE 2.6:VelocityrelativeglobalerrorofCouetteowwithwallinjectionat Re = 10. F IGURE 2.7:TemperaturerelativeglobalerrorofCouetteowwithwall injectionatReynoldnumbers: Re = 5,10,15. 42

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where 4 T = T h )]TJ/F68 11.9552 Tf 12.405 0 Td [(T c isthetemperaturedifference.Inoursimulation,agridsize of30 60isemployed.Weconsidertwocasescorrespondingtodifferentsetsof Reynoldsnumbers Re = u max L / n ,Pecletnumbers Pe = u max L / a andPrandtl number Pr = n / a .Intherstcase,weset Pr = 0.71, Re = 10and Pe = 7.Inthe secondcase,weconsider Re = Pe = 10,i.e. Pr = 1.Where,weconsider T h = 1.1, T c = 1,and t f = 0.674inbothcases.Fig.2.8presentsacomparisonofthevelocity andtemperatureprolesforthesecases.Excellentagreementwiththeanalytical solutionisseen. aVelocityProle Re = 10, Pe = 7 bTemperatureProle Re = 10, Pe = 7 cVelocityProle Re = Pe = 10 dTemperatureProle Re = Pe = 10 F IGURE 2.8:VelocityandtemperatureprolesofPoiseuilleowwith thermaldiffusionatdifferentvaluesof Re and Pe .Markersrepresentthe CascadedLBEresultsandlinesrepresenttheanalyticalsolution. 43

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2.5NaturalConvectioninaSquareCavity Wenowpresentavalidationstudyinvolvingcoupledthermalconvectiveow.In thisregard,ourcascadedLBEmodelisemployedtosimulatenaturalconvectionina squarecavity.Here,theowisdrivenbythebuoyancyforceduetothelocal temperaturedifferenceagainstareferencetemperatureinthepresentofgravity.The leftwallismaintainedathighertemperature T h andtherightwallatlower temperature T c ,whilethetopandbottomwallsareconsideredtobeadiabatic.The macroscopicgoverningequationscanbeexpressedasfollows: r u = 0,.10a u t + u r u = )]TJ/F59 11.9552 Tf 11.178 8.093 Td [(1 r r P + n r 2 u + F ,.10b T t + u r T = r a r T ..10c where F isthebodyforcewhichisbasedontheBoussinesqapproximationand isgivenby F = g b T )]TJ/F68 11.9552 Tf 12.462 0 Td [(T 0 b j .11 Here, b isthethermalexpansioncoefcient, g istheaccelerationduetogravity, T 0 = T h + T c /2isthereferencetemperature, b j istheunitvectorinpositive y-direction.Thisclassicalnaturalconvectionproblemisgovernedbytwo non-dimensionalparameters:ThePrandtlnumber Pr andtheRayleighnumber Ra , whicharegivenby Pr = n a Ra = g b 4 TH 3 na .13 where, 4 T = T h )]TJ/F68 11.9552 Tf 12.462 0 Td [(T c isthetemperaturedifferencebetweenhotandcoldwalls,and H istheheightofthesquarecavity. 44

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F IGURE 2.9:Schematicillustratingthecavityboundaryconditions. Theboundaryconditionsonthecavitywallscanthenbesummarizedas: Ontheleftwall: u x = u y = 0, T = T h = 21,.14 Ontherightwall: u x = u y = 0, T = T c = 1,.15 Onthetopwall: u x = u y = 0, T y = 0,.16 Onthebottomwall: u x = u y = 0, T y = 0..17 Thegeneralbounce-backscheme[48]isadoptedtotreatthethermalDirichletand Neumannboundaryconditions.Inthissimulations, Pr = 0.71,therelaxationrates foruidowandtemperaturearesetas0.55,0.57respectively.Thestreamlinesand theisothermsfortherangesofRayleighnumber Ra between10 3 to10 6 areshownin Fig.2.12.Also,thevorticitycontoursfor Ra = 10 3 )]TJ/F59 11.9552 Tf 12.158 0 Td [(10 6 areshowninFig.2.14.The streamlines,isothermsandvorticitycontoursareinverygoodcorrespondenceand consistentwithpriorbenchmarksolutionresults[15,16].Thenaturalconvectionow patternsbecomemorecomplexas Ra increases.Inordertocharacterizedthisinmore 45

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detail,thetemperatureattheverticalandhorizontalmid-planesofthesquarecavity, i.e. x / H = 0.5and y / H = 0.5,respectivelyforvariousRayleighnumbers Ra = 10 3 )]TJ/F59 11.9552 Tf 12.157 0 Td [(10 6 arepresentedinFigs.2.10and2.11.Inthesecases,thevalueofthe factor g b reededinthesimulationisobtainedasafunctionofRayleighnumber Ra using g b = na Ra 4 TH 3 .18 Therepresentativevaluesof g b correspondingtoeach Ra isshowninTable3.2.From Figs.2.10and2.11,itisseenthatthetemperaturecontourlinesbecomealmost horizontalaroundthecenterofthecavityastheRayleighnumber Ra increases.The streamlinesbecomemorepackednexttothesidewallas Ra increases,i.e.theow movesfasterasnaturalconvectionisintensied.Finally,Table3.6showsquantitative comparisonbetweenthekeyparametersforthisproblemaverageNusseltnumber, maximumvelocitiesmagnitudesandtheirlocationsbetweenthepresentcascaded LBEresultsandbenchmarkdata[15,16].Excellentagreementisseenwithin0.01 percent.ThestreamlinesandisothermsforallRayleighnumbers Ra = 10 3 )]TJ/F59 11.9552 Tf 12.079 0 Td [(10 6 as T ABLE 2.1:Valuesof g b correspondingtoeachRayleighnumber. Rayleighnumber Ra GridSizeN x N y g b = Ra . n . a 4 TH 3 10 3 128 128 9.44 10 )]TJ/F59 8.9664 Tf 7.375 0 Td [(9 10 4 128 128 9.44 10 )]TJ/F59 8.9664 Tf 7.375 0 Td [(8 10 5 128 128 9.44 10 )]TJ/F59 8.9664 Tf 7.375 0 Td [(7 10 6 128 128 9.44 10 )]TJ/F59 8.9664 Tf 7.375 0 Td [(6 showninFig.2.12showthatthecomputedresultsusingthepresentcascadedLBM areinexcellentagreementwithin0.01percentwiththedataby[15]and[16].These numericalresultsforthecomparisonarepresentedinTable3.6.Inaddition,the temperaturecontourlinesbecomealmosthorizontallinesaroundthecenterofthe cavityastheRayleighnumberincreasesandthestreamlinesbecomemorepacked nexttothesidewallastheRayleighnumberincreases,i.e.theowmovesfasteras naturalconvectionisintensied. 46

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F IGURE 2.10:TemperatureprolesalonghorizontalcenterlineofthecavityatvariousRayleighnumbers: Ra = 10 3 ,10 4 ,10 5 ,and10 6 computed usingthecascadedLBM. F IGURE 2.11:Temperatureprolesalongtheverticalcenterlineofthe cavityowatvariousRayleighnumbers Ra = 10 3 ,10 4 ,10 5 ,and10 6 computedusingthecascadedLBM. 47

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aIsotherms Ra = 10 3 bIsotherms Ra = 10 4 cIsotherms Ra = 10 5 dIsotherms Ra = 10 6 F IGURE 2.12:IsothermsatdifferentvaluesofRayleighnumbers Ra = 10 3 ,10 4 ,10 5 and10 6 fornaturalconvectioninasquarecavitycomputed usingthecascadedLBM. 2.6ThermalCouetteFlowwithViscousHeatDissipation:ModelingHeatSource Finally,weconsiderthesimulationofCouetteowwithtemperaturegradienttotest theabilityofthepresentthermalcascadedLBmodelwithasourcetermtodescribe theviscousheatdissipation.Weconsider2Dthermalcouetteowbetweentwo parallelplates,wheretheupperplatemovesalong x -directionwithavelocity U ,and athighertemperature T h ,whereasthebottomwallisstationaryandmaintainedat lowertemperature T c ;Listhedistancebetweenthetwoplates.Inthiscasethesource term G inthethermalenergyequation,Eq..17representstheviscousheat dissipationandisgivenby G = 2 n C p S : S .19 48

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aStreamlines Ra = 10 3 bStreamlines Ra = 10 4 cStreamlines Ra = 10 5 dStreamlines Ra = 10 6 F IGURE 2.13:StreamlinesatdifferentvaluesofRayleighnumbers Ra = 10 3 ,10 4 ,10 5 and10 6 fornaturalconvectioninasquarecavitycomputed usingthecascadedLBM. S = 1 2 r u + r u T .20 Where S isstrainratetensor,and C p isthespecicheatatconstantpressure.The macroscopicgoverningequationsformomentumandenergycanbewritten, respectively,as: 2 u y 2 = 0,.21 a 2 T y 2 + n C p u y 2 = 0..22 Theanalyticalsolutionsofthevelocityandtemperaturearethengivenby: u y = U y L , v = 0.23 49

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aVorticity Ra = 10 3 bVorticity Ra = 10 4 cVorticity Ra = 10 5 dVorticity Ra = 10 6 F IGURE 2.14:VorticitycontoursatvariousRayleighnumbers Ra = 10 3 ,10 4 ,10 5 and10 6 fornaturalconvectioninasquarecavitycomputed usingtheLBM. T )]TJ/F68 11.9552 Tf 12.462 0 Td [(T c T h )]TJ/F68 11.9552 Tf 12.463 0 Td [(T c = y L + 0.5 Br y L 1 )]TJ/F68 11.9552 Tf 14.017 8.094 Td [(y L .24 Where Ec = U 2 Cp T h )]TJ/F68 8.9664 Tf 7.603 0 Td [(T c Eckertnumber,and Pr = n / a isthePrandtlnumber.Theeffect ofviscousheatdissipationiscontrolledbytheBrinkmannumber Br = Ec . Pr .Inthis simulation,weset t f = 0.9, T c = 1withagridresolutionof5 41, Re = 10,and Pr = 0.71atEckertnumbers7,14,and28.Forthesolutionofthisproblem,itis importanttonotethattheconvectiondiffusionequationwithasourceterm.17is coupledwithNavier-Stokesequations.18band.18b.Thisisanothercascaded LBmodel[44]isusedtosolvetheN-Sequations.ThesourcetermGinEq..19can bewrittenas G = 2 n cp h S xx 2 + S yy 2 + 2 S xy 2 i .25 50

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Inthiscase,wehave S xx = u x = 0, S yy = v y = 0, S xy = 1 2 u y + v x .26 Thesestrainratecomponentscanbecomputedlocallyusingthenon-equilibrium momentsinthecascadedLBEfortheuidowasmentionedin[44].Figure.15 showsthetemperatureprolesfordifferentEckertnumbers.Inaddition,wecarried outsimulationfordifferentvaluesofPrandtlnumbers,0.25,1.25,and2.5withthe Eckertnumberbeingxedat8withagridsize5 41.Figure.16showsthe temperatureprolesfordifferentPrandtlnumbers.Inboththesecases,verygood agreementbetweenthecascadedLBMresultsandanalyticalsolutionsseen. Next,westudytheconvergencerateofthecasewithax Ec = 8atdifferent valuesofthePrandtlnumber:0.25,1.25,and2.5.Here,weset U = 0.07, w 4 = w 5 = l 1 = l 2 = 1 0.9 ,andallotherrelaxationparametersaresettobe1.The valuesoftunableparameterDateachPrandtlnumberare-1.2,0.08,and0.24 respectively.WealsoconducttheconvergencestudyforthecasewherethePrandtl numberisxedat Pr = 0.71whiletheEckertnumberischangedas7,14,and28, w 4 = w 5 = 0.9, l 1 = l 2 = 0.94, D = 0andallotherrelaxationparametersaresetto be1.Threegridresolutionsiny-directionNy=41,81,and161areemployedtoboth casesoftheconvergencestudy.Figs..17and.18showthattheslopeofthe temperaturerelativeglobalerrorisabout2,i.e.thepresentthermalcascadedLB modelisofsecondorderaccuracyinspace. Couetteowwithviscousheatdissipationwasalsousedtotesttheabilityof thepresentthermalcascadedLatticeBoltzmannmodeltosimulaterelativelyhigh Pecletnumbersatlowgridresolution5 23.Theanalyticalsolutionsoftemperature isgivenbyEq..24,Here,theBrinkmannumber Br isrewrittenas Br = PeEc Re T )]TJ/F68 11.9552 Tf 12.463 0 Td [(T c T h )]TJ/F68 11.9552 Tf 12.462 0 Td [(T c = y L 1 + 0.5 PeEc Re 1 )]TJ/F68 11.9552 Tf 14.017 8.094 Td [(y L .27 51

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F IGURE 2.15:TemperatureprolesinCouetteowatvariousvaluesof Eckertnumber.MarkersrepresentthecascadedLBEsimulationsand linesrepresenttheanalyticalsolutions. F IGURE 2.16:TemperatureprolesinCouetteowwithdifferentvalues ofPrandtlnumber.MarkersrepresentthecascadedLBEsimulationsand linesrepresenttheanalyticalsolutions 52

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F IGURE 2.17:TemperatureglobalrelativeerroratdifferentEckertnumbers7,14,and28forthermalCouetteowwithviscousheatdissipation. F IGURE 2.18:TemperatureglobalrelativeerroratdifferentPrandtlnumbers0.25,1.25,and2.5forthermalCouetteowwithviscousheatdissipation. 53

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Where Pe isPecletnumber.Inthiscasestudy,thetemperatureprolesforxed Reynoldsnumber Re = 10 ,andEckertnumber Ec = 0.1 atdifferentvaluesof Pecletnumbers10,10 2 ,10 3 ,and10 4 areshowninFig.2.19.Here,weset T h = 1, T c = 0, w 4 = w 5 = l 1 = l 2 = 1.063,andallotherrelaxationparameterssettobe1. ThevaluesoftunableparameterDateachPecletnumberare0,0.397,0.4366,and 0.4406respectively.ExcellentagreementbetweencascadedLBMresultsand analyticalsolutionisseenforrelativelyhighPecletnumberresults.Thisresultis indicativeoftheimprovedstabilitypropertiesofthecascadedLBMasresearchers utilizingSRTLBM[17]onlypresentedresultsforBrinkmannumbersupto Br = 100. TheconvergencestudywasdoneforthisproblematdifferentvaluesofPeclet numbers10,10 2 ,and10 3 forgridresolutionsNy=81,161,and321, w 4 = w 5 = l 1 = l 2 = 1.11,andallotherrelaxationparametersaresettobe1;the valuesoftunableparameterDateachPecletnumberaresettobe0,0.36,and0.696, respectively.Therelativeglobalerroroftemperatureagainstthegridresolutionsis showninFig.2.20.Itisevidentthattheslopeofthetemperaturerelativeglobalerror isabout )]TJ/F59 11.9552 Tf 9.833 0 Td [(2,.i.ethepresentthermalcascadedLBmodelisagainofsecondorder accuracyinspaceatrelativelyhighPecletnumbers10,10 2 ,10 3 ,10 4 ,10 5 and10 6 correspondingtoBrinkmannumbers Br = 10 )]TJ/F59 8.9664 Tf 7.375 0 Td [(1 ,1,10 1 ,10 2 ,10 3 and10 4 . 2.7SummaryandConclusion Inthischapter,wepresentedtheisotropicthermalcascadedLBMRTmodelbased oncentralmomentsandincludingasourceterm.Thismodelsolvesthe convection-difusionequationCDEforthetemperatureeldwithinthedouble distributionfunctionframeworkfortheD2Q9lattice,wheretheuidmotionis representedbyanothercascadedLBmodelconstructedinin[44].Thecollision operatorforthethermaleldhassignicantlydifferentcascadedstructureforits collisionkernelwherecomparedtothatfortheoweldduetothedifferencesinthe numberofcollisioninvariantsbetweenthem.Aconsistentsecondorderschemeto 54

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F IGURE 2.19:TemperatureprolesinthermalCouetteowatvarious valuesofPecletnumbers: Pe = 10,10 2 ,10 3 ,10 4 ,10 5 and10 6 .Markers representtheCascadedLBEresultsandlinesrepresenttheanalyticalsolution. F IGURE 2.20:TemperatureglobalrelativeerroratdifferentPecletnumbers: Pe = 10,10 2 ,and10 3 forthermalCouetteowwithviscousheat dissipation. 55

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incorporatetheeffectoflocallyvaryingheatsourcesbymeansofavariable transformationforthethermalcascadedLBmodelisalsodiscussed.A Chapman-EnskoganalysisofthethermalcascadedLBmodelshowsitsconsistency withtheCDEincludingasourceterm.Italsoprovidesexpressionsfortemperature gradientsintheaugmentedmomentequilibriaintermsoflocallyknown non-equilibriummoments.ThenewthermalcascadedLBEisvalidatedforanumber ofbenchmarkproblems,includingthermalPoiseuilleow,thermalCouetteowand naturalconvectioninasquarecavity.Comparisonofthetemperatureprolesunder differentconditionsfortheseproblems,aswellastheaverageNusseltnumberat differentRayleighnumbersinthecaseofthenaturalconvectionwithinasquare cavity,withpriorbenchmarkresultsdemonstratehighaccuracyoftheisotropic thermalcascadedLBEmodel.Furthermore,itisshownnumericallythatthemodelis secondorderaccurateinspaceforarangeofthermalconvectiveowproblems. 56

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T ABLE 2.2:Comparisonbetweennumericalresultsobtainedusingthe cascadedLBMandthepublishedresultsdeVahlDavisandHortmann etal atdifferentRayleighnumbers Ra = 10 3 )]TJ/F59 10.9091 Tf 11.093 0 Td [(10 6 . RaParameterPresentCascadedLBMdeVahlDavis[15]Hortmann etal [16] 10 3 Nu 1.1171.116NA u max 3.6053.634NA y max 0.8160.813NA v max 3.6543.679NA x max 0.1760.179NA j Y j max 1.161.174NA 10 4 Nu 2.2372.2342.24475 u max 16.18216.18216.1759 y max 0.8240.8230.8255 v max 19.55119.50919.6242 x max 0.120.120.12 j Y j max 5.095.098NA 10 5 Nu 4.5094.514.521 u max 35.13734.8134.7398 y max 0.8560.8550.85312 v max 68.51168.2268.6465 x max 0.0640.0660.656 j Y j max 9.1899.144NA 10 6 Nu 8.7978.7988.825 u max 65.5765.3364.865 y max 0.8560.8510.8532 v max 219.95216.75219.861 x max 0.0320.03870.0406 j Y j max 16.51916.53NA 57

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CHAPTERIII ANISOTROPICTHERMALFLOW 3.1NumericalResults Inthissection,numericaltestsarecarriedouttovalidatetheaccuracyofour proposedcascadedLBEmodelforanisotropicconvectivediffusionthermalows. TheapplicabilityoftheanisotropicLBEmodelisvalidatedbynumericalsimulations includingtheconvectionanddiffusionofaGaussianHill,solvinganisotropic convectiondiffusionequationwithvariablediffusiontensorandvariablesource term,andanisotropicnaturalconvectioninasquarecavity.Inthisstudy,theperiodic boundaryconditionsandbounce-backareemployedfortemperatureboundary conditions. 3.1.1ConvectionandDiffusionofaGaussianHill:NoSourceTerm Thefollowingconvection-diffusionequationCDEwithoutasourceterm f t + u r f = r a r f .1 Here,theinitialdistributionofthescalarvariableisgivenas f x ,0 = f 0 2 ps 0 2 exp )]TJ/F63 11.9552 Tf 20.937 8.094 Td [(x 2 2 ps 0 2 .2 58

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Where s 0 2 istheinitialvariance, f 0 = 2 ps 0 2 isthetotalconcentration.Theanalytical solutionofthisproblemisgivenby f x , t = f 0 2 p p jj s t jj exp )]TJ/F59 11.9552 Tf 11.029 8.093 Td [(1 2 s )]TJ/F59 8.9664 Tf 7.375 0 Td [(1 t : [ x )]TJ/F63 11.9552 Tf 12.158 0 Td [(u t x )]TJ/F63 11.9552 Tf 12.158 0 Td [(u t ] .3 Where s t = s 0 2 I + 2 t a . jj s t jj istheabsolutevalueofthedeterminantof s t ,while s )]TJ/F59 8.9664 Tf 7.375 0 Td [(1 t istheinverseof s t Thecomputationaldomainischosentobe [ )]TJ/F59 11.9552 Tf 9.833 0 Td [(1,1 ] [ )]TJ/F59 11.9552 Tf 9.833 0 Td [(1,1 ] . Threetypesofdiffusiontensorsareconsidered a = 0 B @ 20 02 1 C A 10 )]TJ/F59 8.9664 Tf 7.374 0 Td [(4 , 0 B @ 10 04 1 C A 10 )]TJ/F59 8.9664 Tf 7.375 0 Td [(4 , 0 B @ 11 14 1 C A 10 )]TJ/F59 8.9664 Tf 7.375 0 Td [(4 Whichrepresenttheisotropicconvectiondiffusion,diagonalanisotropic convectiondiffusion,andfullanisotropicconvectiondiffusionproblemsrespectively. Theperiodicboundaryconditionisemployedtothisproblem.Weconductour numericalsimulationwithagridresolutionof151 151, s 0 = 0.05.First,we examinethepurediffusionforthethreediffusiontensorsgivenabovebysetting u = 0,0 attime t m asshowninFig.3.2andFig.3.3thenwetesttheconvection diffusioncasebyset u = 0.05,0.05 attime t m ,and0.5 t m asshowninFig.3.4.The time t m isdeterminedby f max x , t m = 0.5 f max x ,0 as t m = s 0 2 4 0 @ q a 11 + a 22 2 + 12 jj a jj)]TJ/F66 11.9552 Tf 22.139 0 Td [(a 11 )]TJ/F66 11.9552 Tf 12.307 0 Td [(a 22 jj a jj 1 A .4 Where jj a jj = j a 11 a 22 )]TJ/F66 11.9552 Tf 12.307 0 Td [(a 12 a 21 j , t m isequalto6.25,5.5375,5.8547fortheisotropic, diagonalanisotropic,andfullanisotropiccaserespectively. Thesimulationresultsandtheanalyticalsolutionsarecomparedatthethree differentdiffusiontensorsasin[59].Therelaxationratesaretaken l 11 = l 22 = 1.96, l 12 = l 21 = 0forisotropiccase, l 11 = 1.96, l 22 = 1.85, l 12 = l 21 = 0fordiagonal anisotropiccase,and l 11 = 1.96, l 22 = 1.85, l 12 = l 21 = 0.036forfullanisotropic case,with d x 2 / d t = 0.03.Thegoodagreementwiththeanalyticalsolutionsisshown forallthermaldiffusioncoefcients. 59

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aIsotropicdiffusion,t=tm bDiagonallyanisotropicdiffusion,t=tm F IGURE 3.1:Distributionofthescalarvariable f atthetime t = t m and u = v = 0.0. 60

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aFullyanisotropicdiffusion,t=tm bAnalyticalSolution F IGURE 3.2:Distributionofthescalarvariable f fordiffusion u = 0,0 ofaGaussianhillcomputedusingtheLBM. 61

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at=0 bIsotropicdiffusion,t=tm cDiagonallyanisotropicdiffusion,t=tm dFullyanisotropicdiffusion,t=tm F IGURE 3.3:ConcentrationcontoursfordiffusionofaGaussianhill u = 0,0 computedusingtheLBM. Wealsotesttheconvergencerateofthepresentmodel.Forthispurpose,asetof simulationsareconductedatfourdifferentgridsizes,i.e.,101 101,151 151, 201 201,and301 301.Theglobalrelativeerroroftemperature E T usedto measuretheaccuracyofthemodeliscalculatedas E T = jj T c )]TJ/F68 11.9552 Tf 12.463 0 Td [(T a jj 2 jj T a jj 2 .5 Where jj . jj 2 istheEuclideannorm, jj T c )]TJ/F68 11.9552 Tf 12.463 0 Td [(T a jj 2 = p i T c , i )]TJ/F68 11.9552 Tf 12.463 0 Td [(T a , i 2 , jj T a jj 2 = p i T a , i 2 , T c and T a arethecomputedandtheanalyticalsolutions respectively.Therelativeglobalerroroftemperaturefortheisotropic,diagonal anisotropic,andfullanisotropiccaseareplottedinFig.3.8.Itcanbeseenthatthe temperatureglobalerrordecreaseswithincreaseingridresolutionwithaslopeof2 inthelog-logplot.Hence,ourpresentcascadedLBMmodelwithsourcetermis 62

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secondorderaccurate. aIsotropicdiffusion bDiagonallyanisotropic cFullyanisotropicdiffusion F IGURE 3.4:Contoursofthescalervariable f at u = 0.05,0.05 computedusingLBE 3.1.2Convection-DiffusionofaGaussianHill:StabilityTest Weconsidertheconvectiondiffusionequation,Eq..17,where u = u o i + v o j isa prescribed2-DuniformvelocityeldandsubjectedtotheGaussianhillinitial condition. T x , y ,0 = T o 2 ps o 2 exp )]TJ/F70 12.4573 Tf 9.983 0 Td [([ x 2 + y 2 ] 2 s o 2 ,.6 63

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aIsotropicdiffusion bDiagonallyanisotropic cFullyanisotropicdiffusion F IGURE 3.5:AnalyticalContoursofthescalervariable f at u = 0.05,0.05 wheretheparameter s o controlsthewidthoftheprole.Theanalyticalsolutionof thisproblemisgivenby T x , y , t = T o 2 p s o 2 + 2 a t exp )]TJ/F70 12.4573 Tf 9.982 0 Td [([ x )]TJ/F68 11.9552 Tf 12.307 0 Td [(u o t 2 + y )]TJ/F68 11.9552 Tf 12.307 0 Td [(v o t 2 ] 2 s o 2 + 2 a t ..7 Weset s o = 0.05andadvecttheprolewiththediagonalvelocityvector u o = v o = 0.25 c s .Wechoose T o = 2 ps o 2 sothattheinitialprolehasapeak magnitudeof1.0.Periodicboundaryconditionsforthetemperatureareemployed.In whatfollowswevarytheuiddiffusivitytocomparethestabilitycharacteristicsof thecascadedcenteralmomentLBMwiththeSRTandMRTimplementationsofthe LBM.WeconsidertheMRTmethodin[59]andwesetthetunableparametersin bothmethodstobezero.Inthethreemethodstheuiddiffusivityisgivenby 64

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a = c s 2 1 l )]TJ/F59 8.9664 Tf 13.353 4.711 Td [(1 2 where l istherelaxationrateoftherstordermoments correspondingtotheequilibriummoments u x T and u y T inthecascadedandMRT LBM.Weusea521 521gridandvarythediffusivitybyvaryingtherelaxationtime t g = 1 l .ComparisonofthecascadedandMRTLBMmethodsiscomplicatedbythe largenumberofrelaxationparametersassociatedwitheachmethod.Wesetthe relaxationtimefortherstordermomentsto t g .Wethensetallotherrelaxation timesto1.0.Withthischoiceofparameterswearerelaxingtheenergyuxesrst ordermomentsatthesamerateinbothmethods.Thehigherordermomentsarealso relaxedatthesamerate.Thischoiceofparametersisnotnecessarilyoptimalfor eithermethodbutitdoesgiveusarationalbasisforcomparison. Finally,weconsidernumericalstabilityresultsfromtheSRT,MRT,and cascadedLBMatvariousvaluesoftherelaxationtime t g .Table3.1providesthe numericalstabilityandglobalrelativeerrorgivenbyEq..5indicatestheaccuracy ofthethreemethodsissimilar.ThetablealsoindicatesthattheSRTLBMisnotstable atthesmallerdiffusivitiesandthattheMRTiseventuallynotstableatthestill smallerdiffusivityforwhichthecascadedLBMisstable.Itislikelythatrelaxation ratesforthehighermomentsoftheMRTandcascadedLBmethodsmaybefound thatresultinmorestablebehaviorbutthisshortstudyconcurswithotherworkthat indicatesuperiorstabilitycharacteristicsofthecascadedLBM[61,63]. T ABLE 3.1:Globalrelativeerrorandstabilitycharacteristicsafter1000 timeincrementsforSRT,MRT,andCascadedLBMfordifferentrelaxation timesfortheconvection-diffusionofaGaussianhillproblematMach number0.25. t g 0.55 0.51 0.501 a 1.67 10 )]TJ/F59 8.9664 Tf 7.375 0 Td [(2 3.33 10 )]TJ/F59 8.9664 Tf 7.375 0 Td [(3 3.33 10 )]TJ/F59 8.9664 Tf 7.375 0 Td [(4 SRT 0.0097 unstable unstable MRT 0.0086 0.0096 unstable Cascaded 0.0101 0.0108 0.0110 65

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3.1.3AnisotropicConvectionDiffusionEquation:VariableSourceTerm Thefollowinganisotropicconvection-diffusionequationCDEwithaconstant velocityandvariablediffusiontensor a andwithadistributedsourceterm f t + u r f = r a r f + G .8 Where u = u x , u y and G = G x , y , t isthesourceterm.Theinitialdistributionof thescalarvariableisgivenby f x , y ,0 = sin 2 p x sin 2 p y ..9 Theanalyticalsolutionofthisproblemisgivenby f = exp h 1 )]TJ/F59 11.9552 Tf 12.158 0 Td [(12 p 2 a t i sin 2 p x sin 2 p y ..10 Thesourcetermisdenedas G = exp h 1 )]TJ/F59 11.9552 Tf 12.158 0 Td [(12 p 2 a t in sin 2 p x sin 2 p y + 4 a cos 4 p x sin 2 2 p y + 2 p u x cos 2 p x sin 2 p y + u y sin 2 p x cos 2 p y ..11 Thecomputationaldomainischosentobe [ 0,1 ] [ 0,1 ] .Here,thediffusiontensorisa functionofspacex,y.Thediffusiontensor a isgivenbyadiagonalmatrix a = a 0 B @ 2 )]TJ/F68 11.9552 Tf 12.307 0 Td [(sin 2 p x sin 2 p y 0 01 1 C A .12 Where a isconstantandisgiventobe a = 1.0 10 )]TJ/F59 8.9664 Tf 7.375 0 Td [(3 .Here,thediffusiontensor a representsthediagonalanisotropicconvectiondiffusion.Theperiodicboundary conditionisemployedtothisproblem.Weconductournumericalsimulationwitha gridresolutionof101 101asshowninFig.3.7,wetesttheconvectiondiffusioncase 66

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byset u = 0.1,0.1 attime t = 3and Pe = 100.Therelaxationtimeissetto t 11 = 3 10 )]TJ/F59 8.9664 Tf 7.375 0 Td [(3 2 )]TJ/F68 11.9552 Tf 12.307 0 Td [(sin 2 p x sin 2 p y + 0.5, t 22 = 0.54, t 12 = t 21 = 0.Fig.3.7shows thetemperaturedistributionforthegivenvaluesofdiffusivitycoefcientswhich showverygoodagreementwiththeanalyticalsolutions.Wealsoexaminethespatial accuracyofthepresentmodel.Inthisregard,asetofsimulationsareperformedat vedifferentgridsizes,i.e.,201 201,301 301,401 401,501 501and521 521 forgivenvalueofthediffusiontensor.Therelativeglobalerroroftemperatureforthe givenvalueofdiffusivitycoefcientareplottedinFig.3.9.Itcanbeseenthatthe temperatureglobalerrordecreaseswithincreaseingridresolutionwithaslopeof )]TJ/F59 11.9552 Tf 9.833 0 Td [(2 inthelog-logplot.Hence,ourpresentcascadedLBMmodelwithsourcetermis secondorderaccurate. 3.2NaturalConvectioninaSquareCavity:AnisotropicCase Naturalconvectionheattransferanduidowincavitiesareimportantsubjectsof investigationduetotheireffectonmanyengineeringapplicationsandnature phenomena,suchasthermalpower,petrochemicalindustries,aerospace, constructionandsolarcollectors.Inthissection,theeffectofanisotropyisperformed foraheatedcavityforthefollowingvaluesofRayleighnumber Ra = 10 3 ,10 4 ,10 5 ,10 6 ,10 7 and10 8 .Inthisregard,wedenethexthermal diffusivityas a x andythermaldiffusivityas a y cascadedLBEmodelisemployedto simulatenaturalconvectioninasquarecavity.Here,theowisdrivenbythe buoyancyforceduetothelocaltemperaturedifferenceagainstareference temperatureinthepresentofgravity.Theleftwallismaintainedathigher temperature T h andtherightwallatlowertemperature T c ,whilethetopandbottom wallsareconsideredtobeadiabatic.Themacroscopicgoverningequationscanbe 67

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aNumericalSolution bAnalyticalsolution F IGURE 3.6:Distributionofthescalarvariable f atthetime t = 3and Pe = 100. 68

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aNumericalSolution bAnalyticalsolution F IGURE 3.7:Contoursofthescalarvariable f atthetime t = 3and Pe = 100. 69

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F IGURE 3.8:Temperatureglobalrelativeerrorsatdifferentgridsizesfor diffusionofaGaussianhill. F IGURE 3.9:Temperatureglobalrelativeerrorwithvariablediffusiontensorandsourceterm. 70

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expressedasfollows: u x + v y = 0,.13a u t + u u x + v u y = )]TJ/F59 11.9552 Tf 11.178 8.094 Td [(1 r p x + n 2 u x 2 + 2 u y 2 ,.13b v t + u v x + v v y = )]TJ/F59 11.9552 Tf 11.178 8.093 Td [(1 r p y + n 2 v x 2 + 2 v y 2 + F ,.13c T t + u T x + v T y = a x 2 T x 2 + a y 2 T y 2 ..13d .13e where F isthebodyforcewhichisbasedontheBoussinesqapproximationand isgivenby F = g b T )]TJ/F68 11.9552 Tf 12.462 0 Td [(T 0 b j .14 Here, b isthethermalexpansioncoefcient, g istheaccelerationduetogravity, T 0 = T h + T c /2isthereferencetemperature, b j istheunitvectorinpositive y-direction.Thisclassicalnaturalconvectionproblemisgovernedbytwo non-dimensionalparameters:ThePrandtlnumber Pr Pr = n a andtheRayleighnumber Ra ,whicharegivenby Ra x = g b 4 TH 3 na x .16 Ra y = g b 4 TH 3 na y .17 where, 4 T = T h )]TJ/F68 11.9552 Tf 12.462 0 Td [(T c isthetemperaturedifferencebetweenhotandcoldwalls,and H istheheightofthesquarecavity. 71

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a x isxthermaldiffusivityand a y isythermaldiffusivity.Thechoiceforthe anisotropicasfollows. xthermaldiffusivitygivenby a x = 0.5 a y andxthermaldiffusivitygivenby a x = 2 a y fordifferentRayleighnumbers Ra y = 10 3 ,10 4 ,......10 8 .Theaverage ofNusseltnumberfortherightwallisgivenby Nu = 1 a x 4 T Z H 0 q x x , y dy .18 where q x x , y = uT x , y )]TJ/F66 11.9552 Tf 12.307 0 Td [(a x T x isthelocalheatuxinx-direction. Theboundaryconditionsonthecavitywallscanthenbesummarizedas: Ontheleftwall: u x = u y = 0, T = T h = 21,.19 Ontherightwall: u x = u y = 0, T = T c = 1,.20 Onthetopwall: u x = u y = 0, T y = 0,.21 Onthebottomwall: u x = u y = 0, T y = 0..22 Thegeneralbounce-backscheme[48]isadoptedtotreatthethermalDirichletand Neumannboundaryconditions.Inthissimulations, Pr = 0.71,therelaxationrates foruidowandtemperaturearesetas0.55,0.57respectively.Thestreamlinesand theisothermsfortherangesofRayleighnumber Ra y between10 3 to10 6 areshownin Fig.2.12.Also,thevorticitycontoursfor Ra y = 10 3 )]TJ/F59 11.9552 Tf 12.158 0 Td [(10 8 areshowninFig.2.14.The streamlines,isothermsandvorticitycontoursareinverygoodcorrespondenceand consistentwithpriorbenchmarksolutionresults[15,16].Thenaturalconvectionow patternsbecomemorecomplexas Ra increases.Inordertocharacterizedthisinmore detail,thetemperatureattheverticalandhorizontalmid-planesofthesquarecavity, 72

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i.e. x / H = 0.5and y / H = 0.5,respectivelyforvariousRayleighnumbers Ra y = 10 3 )]TJ/F59 11.9552 Tf 12.158 0 Td [(10 8 arepresentedinFigs.2.10and2.11.Inthesecases,thevalueofthe factor g b reededinthesimulationisobtainedasafunctionofRayleighnumber Ra using g b = na y Ra y 4 TH 3 .23 Therepresentativevaluesof g b correspondingtoeach Ra isshowninTable3.2.From Figs.2.10and2.11,itisseenthatthetemperaturecontourlinesbecomealmost horizontalaroundthecenterofthecavityastheRayleighnumber Ra increases.The streamlinesbecomemorepackednexttothesidewallas Ra increases,i.e.theow movesfasterasnaturalconvectionisintensied.Finally,Table3.6showsquantitative comparisonbetweenthekeyparametersforthisproblemaverageNusseltnumber, maximumvelocitiesmagnitudesandtheirlocationsbetweenthepresentcascaded LBEresultsandbenchmarkdata[15,16].Excellentagreementisseenwithin0.01 percent.ThestreamlinesandisothermsforallRayleighnumbers Ra y = 10 3 )]TJ/F59 11.9552 Tf 12.158 0 Td [(10 8 T ABLE 3.2:Valuesof g b correspondingtoeachRayleighnumber. Rayleighnumber Ra GridSizeN x N y g b = Ra y . n . a y 4 TH 3 10 3 128 128 9.44 10 )]TJ/F59 8.9664 Tf 7.375 0 Td [(9 10 4 128 128 9.44 10 )]TJ/F59 8.9664 Tf 7.375 0 Td [(8 10 5 128 128 9.44 10 )]TJ/F59 8.9664 Tf 7.375 0 Td [(7 10 6 128 128 9.44 10 )]TJ/F59 8.9664 Tf 7.375 0 Td [(6 10 7 128 128 9.44 10 )]TJ/F59 8.9664 Tf 7.375 0 Td [(5 10 8 521 521 1.37 10 )]TJ/F59 8.9664 Tf 7.375 0 Td [(5 asshowninFig.2.12showthatthecomputedresultsusingthepresentcascaded LBMareinexcellentagreementwithin0.01percentwiththedataby[15]and[16]. ThesenumericalresultsforthecomparisonarepresentedinTable3.6.Inaddition, thetemperaturecontourlinesbecomealmosthorizontallinesaroundthecenterof thecavityastheRayleighnumberincreasesandthestreamlinesbecomemore packednexttothesidewallastheRayleighnumberincreases,i.e.theowmoves fasterasnaturalconvectionisintensied. 73

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3.3SummaryandConclusion Inthischapter,wepresentedthenumericalresultsforsimulationofanisotropic thermalcascadedLBMRTmodelbasedoncentralmomentsandincludingasource term.Thismodelsolvestheanisotropicconvection-difusionequationCDEforthe temperatureeldwithinthedoubledistributionfunctionframeworkfortheD2Q9 lattice,wheretheuidmotionisrepresentedbyanothercascadedLBmodel constructedin[44].AChapman-Enskoganalysisoftheanisotropicthermalcascaded 74

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F IGURE 3.10:StreamlinesatdifferentvaluesofRayleighnumbers Ra = 10 3 ,10 4 ,10 5 ,10 6 ,10 7 and10 8 fornaturalconvectioninasquarecavitycomputedusingthecascadedLBM.Left a x = a y /2,center a x = a y ,right a x = 2 a y LBmodelshowsitsconsistencywiththeCDEincludingasourceterm.Italso providesexpressionsfortemperaturegradientsintheaugmentedmomentequilibria intermsoflocallyknownnon-equilibriummoments. ThenewthermalcascadedLBEisvalidatedforanumberofbenchmark problems,includingconvection-diffusionofaGaussianHill,solvinganisotropic convectiondiffusionequationwithvariablediffusiontensorandvariablesource term,andanisotropicnaturalconvectioninasquarecavity.Comparisonofthe temperatureprolesunderdifferentconditionsfortheseproblems,aswellasthe averageNusseltnumberatdifferentRayleighnumbersinthecaseoftheanisotropic naturalconvectionwithinasquarecavity,withpriorbenchmarkresultsdemonstrate highaccuracyoftheanisotropicthermalcascadedLBEmodel.Furthermore,itis shownnumericallythatthemodelissecondorderaccurateinspaceforarangeof anisotropicthermalconvectiveowproblems. Finally,astabilitytestofthepresentcascadedLBEmodelisconductedto compareourmodelbysinglerelaxationtimeSRTLBmodelandmultiplerelaxation 75

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timesMRTLBmodelusingthediffusioninaGaussianHillasatestproblemby varyingtheuiddiffusivitytocomparethestabilitycharacteristicsofthecascaded centralmoment.Thisstabilitystudyconcurswithotherworkthatindicatessuperior stabilitycharacteristicsofthecascadedLBmethod.Thepresentanisotropicthermal cascadedLBmodelexhibitsimprovedstabilitycharacteristicsovertheSRTLBmodel andtheconventionalmultipleMRTLBmodel. 76

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F IGURE 3.11:VorticityatdifferentvaluesofRayleighnumbers Ra = 10 3 ,10 4 ,10 5 ,10 6 ,10 7 and10 8 fornaturalconvectioninasquarecavitycomputedusingthecascadedLBM.Left a x = a y /2,center a x = a y ,right a x = 2 a y 77

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T ABLE 3.3:NumericalresultsobtainedusingthecascadedLBMfor anisotropicnaturalconvectioninasquarecavityatdifferentRayleigh numbers Ra = 10 3 ,10 4 ,10 5 ,10 6 ,10 7 and10 8 . Ra y Parameter a x = 0.5 a y a x = a y a x = 2 a y 10 3 Nu 1.2471.11711.045 u max 3.34173.60463.7635 y max 0.8160.81600.816 v max 3.4293.65403.78 x max 0.1680.1760.168 j Y j max 1.0591.161.195 10 4 Nu 2.6812.23741.8456 u max 12.40816.182721.058 y max 0.8320.82400.824 v max 15.9819.551923.879 x max 0.1040.11200.128 j Y j max 3.7945.096.62 10 5 Nu 5.4584.50903.645 u max 23.82535.137556.294 y max 0.8560.85600.864 v max 53.0368.511185.669 x max 0.05600.06400.072 j Y j max 6.4759.18913.385 10 6 Nu 10.6368.79787.031 u max 51.0365.5747121.825 y max 0.89600.85600.8720 v max 166.44219.9522281.8181 x max 0.0320.03200.04 j Y j max 10.88416.51922.276 10 7 Nu 20.136716.511213.2339 u max 116.933150.194284.727 y max 0.9360.87820.856 v max 523.749703.414904.581 x max 0.0160.0160.024 j Y j max 19.20929.7242.39 10 8 Nu 36.766030.155224.1777 u max 245.329427.56851.476 y max 0.95340.94960.778 v max 1659.5362259.2972951.734 x max 0.00970.01160.0116 j Y j max 33.8952.976.18 78

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T ABLE 3.4:Comparisonbetweennumericalresultsobtainedusingthe cascadedLBMandthepublishedresults Dubois etal [64] for anisotropicnaturalconvectioninasquarecavityatdifferentRayleigh numbers Ra = 10 3 ,10 4 ,10 5 and10 6 . Ra y ParameterModel a x = 0.5 a y a x = a y a x = 2 a y 10 3 Nu Duboisetal[64] 2.49571.11790.5226 present 1.24741.11711.0447 u max Duboisetal[64] 3.37053.64963.8185 present 3.34173.60463.7635 y max Duboisetal[64] 0.81420.81420.8142 present 0.81600.81600.8160 v max Duboisetal[64] 3.45153.69733.8428 present 3.42933.65403.7909 x max Duboisetal[64] 0.17610.17610.1857 present 0.16800.17600.1760 10 4 Nu Duboisetal[64] 5.37112.24380.9261 present 2.68112.23741.8456 u max Duboisetal[64] 12.362816.188121.1512 present 12.408216.182721.0584 y max Duboisetal[64] 0.82900.82250.8225 present 0.83200.82400.8240 v max Duboisetal[64] 16.015919.632324.0035 present 15.980019.551923.8797 x max Duboisetal[64] 0.10640.11930.1322 present 0.10400.11200.1280 10 5 Nu Duboisetal[64] 10.93254.51771.8262 present 5.45844.50903.6451 u max Duboisetal[64] 23.578334.748656.0032 present 23.825035.137556.2946 y max Duboisetal[64] 0.85600.85600.8609 present 0.85600.85600.8640 v max Duboisetal[64] 53.586368.652786.0559 present 53.550068.511185.6690 x max Duboisetal[64] 0.06090.06580.0707 present 0.05600.06400.0720 79

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Ra y ParameterModel a x = 0.5 a y a x = a y a x = 2 a y 10 6 Nu Duboisetal[64] 21.28058.80623.5197 present 10.63678.79787.0308 u max Duboisetal[64] 50.699964.8428120.0525 present 51.030065.5747121.8256 y max Duboisetal[64] 0.90000.84900.8647 present 0.89600.85600.8720 v max Duboisetal[64] 166.5744220.6695282.8172 present 166.4405219.9522281.8181 x max Duboisetal[64] 0.03330.03720.0411 present 0.03200.03200.0400 80

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T ABLE 3.6:Comparisonbetweennumericalresultsobtainedusingthe cascadedLBMandthepublishedresultsdeVahlDavisandHortmann etal forisotropicnaturalconvectioninasquarecavityat differentRayleighnumbers Ra = 10 3 )]TJ/F59 10.9091 Tf 11.094 0 Td [(10 6 . Ra y ParameterPresentCascadedLBMdeVahlDavis[15]Hortmann etal [16] 10 3 Nu 1.1171.116NA u max 3.6053.634NA y max 0.8160.813NA v max 3.6543.679NA x max 0.1760.179NA j Y j max 1.161.174NA 10 4 Nu 2.2372.2342.24475 u max 16.18216.18216.1759 y max 0.8240.8230.8255 v max 19.55119.50919.6242 x max 0.120.120.12 j Y j max 5.095.098NA 10 5 Nu 4.5094.514.521 u max 35.13734.8134.7398 y max 0.8560.8550.85312 v max 68.51168.2268.6465 x max 0.0640.0660.656 j Y j max 9.1899.144NA 10 6 Nu 8.7978.7988.825 u max 65.5765.3364.865 y max 0.8560.8510.8532 v max 219.95216.75219.861 x max 0.0320.03870.0406 j Y j max 16.51916.53NA 10 7 Nu 8.7978.79816.52 u max 65.5765.3364.865 y max 0.8560.8510.8532 v max 219.95216.75219.861 x max 0.0320.03870.0406 j Y j max 16.51916.5330.24 10 8 Nu 8.7978.79830.48 u max 65.5765.3364.865 y max 0.8560.8510.8532 v max 219.95216.75219.861 x max 0.0320.03870.0406 j Y j max 16.51916.5354.32 81

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82

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F IGURE 3.12:IsothermalsatdifferentvaluesofRayleighnumbers Ra = 10 3 ,10 4 ,10 5 and10 6 fornaturalconvectioninasquarecavitycomputed usingthecascadedLBM.Left a x = a y /2,center a x = a y ,right a x = 2 a y 83

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84

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F IGURE 3.13:Firstcomponentofvelocityuatdifferentvaluesof Rayleighnumbers Ra = 10 3 ,10 4 ,10 5 ,10 6 ,10 7 and10 8 fornaturalconvectioninasquarecavitycomputedusingthecascadedLBM.Left a x = a y /2,center a x = a y ,right a x = 2 a y 85

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86

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F IGURE 3.14:Secondcomponentofvelocityvatdifferentvaluesof Rayleighnumbers Ra = 10 3 ,10 4 ,10 5 ,10 6 ,10 7 and10 8 fornaturalconvectioninasquarecavitycomputedusingthecascadedLBM.Left a x = a y /2,center a x = a y ,right a x = 2 a y 87

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88

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F IGURE 3.15:PressureatdifferentvaluesofRayleighnumbers Ra = 10 3 ,10 4 ,10 5 ,10 6 ,10 7 and10 8 fornaturalconvectioninasquarecavitycomputedusingthecascadedLBM.Left a x = a y /2,center a x = a y ,right a x = 2 a y 89

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akx=0.5ky bIsotropickx=ky ckx=2ky F IGURE 3.16:TemperatureprolesalonghorizontalcenterlineofthecavityatvariousRayleighnumbers: Ra = 10 3 ,10 4 ,10 5 ,10 6 ,10 7 ,and10 8 computedusingthecascadedLBM. 90

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akx=0.5ky bIsotropickx=ky ckx=2ky F IGURE 3.17:TemperatureprolesalongtheverticalcenterlineofthecavityowatvariousRayleighnumbers Ra = 10 3 ,10 4 ,10 5 ,10 6 ,10 7 ,and10 8 computedusingthecascadedLBM. 91

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CHAPTERIV FLOWWITHVARIABLEVISCOSITY 4.1NumericalResults Inthissection,numericaltestiscarriedouttovalidatetheaccuracyofourproposed cascadedLBEmodelforvariableviscositythermalows.Weconsideraverybasic owcongurationtoisolatetheeffectofviscosityvariation,wheretheuidows betweenparallelplates,i.etheowsisthroughathinchannel.Inthisstudy,the periodicboundaryconditionisemployedfortemperatureboundaryconditions. 4.1.1CouetteFlowwithshearheating:Variableviscosity Here,Weconsiderincompressibleuidwithanexponentialviscosityvariationows throughthenarrowchannelofwithHandlengthLinthex-direction,theowis drivenbyshearorvelocityonsurface z = H .Theproblemcongurationisshownin FIG..1.Thetopwallismaintainedattemperature T u andthebottomwallat temperature T 1 .Themacroscopicgoverningequationscanbeexpressedasfollows: m = e )]TJ/F66 8.9664 Tf 7.711 0 Td [(b T .1 u z = Ae b T .2 2 T z 2 + B r A 2 e b T = 0..3 92

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F IGURE 4.1:Schematicillustratingtheproblemconguration. Where B r = m o U 2 a 4 T istheBrinkmannumber, 4 T = T u )]TJ/F59 11.9552 Tf 9.833 0 Td [(T l , A istheshearstress, b and m o areconstant Theanalyticalsolutionsofthevelocityandtemperaturearethengivenby: u z = p 2 e 0.5 b T m p b B r tanh " A p b B r p 2 e )]TJ/F59 11.9552 Tf 7.873 -3.456 Td [(0.5 b T m z )]TJ/F68 11.9552 Tf 12.307 0 Td [(tanh )]TJ/F59 8.9664 Tf 7.375 0 Td [(1 p 1 )]TJ/F68 11.9552 Tf 12.307 0 Td [(e )]TJ/F66 8.9664 Tf 7.711 0 Td [(b T m # + p 1 )]TJ/F68 11.9552 Tf 12.307 0 Td [(e )]TJ/F66 8.9664 Tf 7.711 0 Td [(b T m ! .4 T = T m + 1 b ln 1 )]TJ/F68 11.9552 Tf 12.307 0 Td [(tanh 2 " tanh )]TJ/F59 8.9664 Tf 7.375 0 Td [(1 p 1 )]TJ/F68 11.9552 Tf 12.307 0 Td [(e )]TJ/F66 8.9664 Tf 7.71 0 Td [(b T m )]TJ/F68 11.9552 Tf 26.878 8.697 Td [(A p b B r p 2 e )]TJ/F59 11.9552 Tf 7.873 -3.455 Td [(0.5 b T m z #! .5 Where T m isthemaximumtemperature,i.e.Thetemperaturewherethetemperature gradientiszero. Weconductournumericalsimulationwithagridresolutionof5 23 T u = 1,T l = 0, t t = 0.65, U = 1,and t f = 3 e )]TJ/F68 8.9664 Tf 7.603 0 Td [(T + 0.5fordifferentBrinkman numbers B r = B r 0 ,5,10,and20.Thesimulationresultsandtheanalyticalsolutionsfor temperatureeldarecomparedatthefourdifferentBrinkmannumbers Br = 3.43,5,10,and20,whilethesimulationresultsandtheanalyticalsolutionsfor temperatureeldarecomparedatthethreedifferentBrinkmannumbers 93

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F IGURE 4.2:Temperatureprolesforvelocitydrivenowatvarious Brinkmannumbers: Br = 3.43,5,10,and20.MarkersrepresenttheCascadedLBEresultsandlinesrepresenttheanalyticalsolution. F IGURE 4.3:VelocityprolesforvelocitydrivenowatvariousBrinkman numbers: Br = 3.43,10,and20.MarkersrepresenttheCascadedLBE resultsandlinesrepresenttheanalyticalsolution. 94

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Br = 3.43,10,and20.Thegoodagreementwiththeanalyticalsolutionsisshownfor allcasesfortemperatureandvelocityinFIG..2andFIG..3respectively. 4.2SummaryandConclusion ThenewcentralmomentcascadedLBmodelwasadaptedtosimulateofuidow withtemperature-dependentviscosityforisotropiccase,wheretheuidviscosityis exponentiallyvaryingwithtemperature.ThesimulationresultsofCouetteowwith shearheatingconrmedthevalidityofthepresentcentralmomentLBmodelto incorporatecorrectlyuidowwithvariableviscosityatdifferentBrinkman numbers. 95

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CHAPTERV SUMMARYANDCONCLUSION AnewcascadedcentralmomentbasedlatticeBoltzmannLBmethodforsolving lowMachnumberconvectivethermalowswithsourcetermsintwo-dimensionsin adoubledistributionfunctionframeworkispresented.Forthepassivetemperature eld,whichsatisesaconvectiondiffusionequationCDEalongwithasourceterm torepresentinternal/externallocalheatsource,anewcascadedcollisionkernelis presented.Duetotheuseofasingleconservedvariableinthethermalenergy equation,thecascadedstructureinitscollisionoperatorbeginsfromtherstorder momentsandevolvestohigherordermoments.Thisismarkedlydifferentfromthe collisionoperatorfortheuidowequations,constructedinpreviouswork,where thecascadedformulationstartsatthesecondordermomentsinitscollisionkernel.A consistentimplementationofthespatiallyandtemporallyvaryingsourcetermsin thethermalcascadedLBmethodrepresentingtheheatsourcesintheCDEthat maintainssecondorderaaccuracyviaavariabletransformationisdiscussed.In addition,therstorderequilibriummomentsinthismodelareaugmentedwith spatialtemperaturegradienttermsobtainedlocallyandinvolvingatunable coefcienttomaintainadditionalexibilityintherepresentationofthetransport coefcientforthetemperatureeld.TheconsistencyofthethermalcascadedLB methodincludingasourcetermwiththemacroscopicconvection-diffusionequation isdemonstratedbymeansofaChapman-Enskoganalysis.Theemergenttunable diffusivityisshowntobedependentontherelaxationtimesoftherstorder momentsaswellasthetunableparameterintheadditionalgradienttermsinour 96

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cascadedmultiple-relaxation-timeformulation.Thenewmodelistestedonasetof benchmarkproblemssuchasthethermalPoiseuilleow,thermalCouetteowwith eitherwallinjectionorincludingviscousdissipationandnaturalconvectionina squarecavity.ThevalidationstudiesshowthatthethermalcascadedLBmethodwith sourcetermisinverygoodagreementwithanalyticalsolutionsornumericalresults reportedforbenchmarkproblems.Inaddition,thenumericalresultsshowthatour newthermalcascadedLBmodelmaintainssecondorderspatialaccuracy. ThenewLBEmodelismodiedtosimulateanisotropicuidsthatare characterizedbydifferentdiffusioncoefcientsalongdifferentdirections.The applicabilityoftheLBEmodelisvalidatedbynumericalsimulationsincludingthe convectionanddiffusionofaGaussianHill,solvinganisotropicconvectiondiffusion equationwithvariablediffusiontensorandvariablesourceterm,andanisotropic naturalconvectioninaSquareCavity.Thevalidationstudyshowsthatthe anisotropicthermalcascadedLBmodelwithsourcetermisinverygoodagreement withtheanalyticalsolutionsornumericalresultsreportedforthebenchmark problems.Inaddition,thenumericalresultsshowthatournewanisotropicthermal cascadedLBmodelmaintainssecondorderaccuracyasdoestheisotropicmodel. AstabilitytestofthepresentcascadedLBEmodelisconductedtocompareour modelbysinglerelaxationtimeSRTLBmodelandmultiplerelaxationtimesMRT LBmodelusingthediffusioninaGaussianHillasatestproblembyvaryingtheuid diffusivitytocomparethestabilitycharacteristicsofthecascadedcentralmoment. Thisstabilitystudyconcurswithotherworkthatindicatessuperiorstability characteristicsofthecascadedLBmethod. Finally,thecentralmomentcascadedLBmodelwasadaptedtosimulateofuid owwithtemperature-dependentviscosity,wheretheuidviscosityisexponentially varyingwithtemperature.Thesimulationresultsofcouetteowwithshearheating conrmedthevalidityofthepresentcentralmomentLBmodeltoincorporate correctlyuidowwithvariableviscosityatdifferentBrinkmannumbers. 97

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[64][Duboisetal.]F.Dubois,C.AnLin,andM.Tekitek,J.Computers.and. Fluids 124 ,278-287. [65][Brownetal.]D.Brown,K.Luo,R.Liu,andP.Lv,Phys.Flu. 26 ,023303 . 103

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APPENDIXI CHAPMAN-ENSKOGANALYSISOFTHEISOTROPICTHERMALCASCADED LBM Inthissection,aChapman-EnskogC-EanalysisofthethermalcascadedLBMis presentedandtheresultsoftheanalysisprovidethemacroscopicemergent equations,vis;theconvection-diffusionequationCDEwithasourcetermgivenin Eq..17earlier.Inthisregard,weconsiderthestrategyofrewritingthecentral momentLBMintermsoftherelaxationtoageneralizedequilibriumintherestframe ofreference.TofacilitateanalysisanditsestablishconsistencytotheCDE,itis sufcienttoconsidertermsonlyuptosecondorderinMachnumberinsuchan equivalentformulation[24,44].Inthisregard,weconsiderperformingcalculations intermsofvariousrawmomentsdesignnedwith"primesymbols"withrespectto thenon-orthogonalmomentbasisvectorscollectedinthematrix T andgivenin Eq..19.First,thebaserawmomentequilibrium b k eq 0 x m y n canbeobtainedfromthe correspondingcentralmomentequilibria b k eq x m y n giveninequationEq..29.This 104

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readsas b k eq 0 = T ,.1a b k eq x = u x T ,.1b b k eq y = u y T ,.1c b k eq xx = c 2 s T + u 2 x T ,.1d b k eq yy = c 2 s T + u 2 y T ,.1e b k eq xy = u x u y T ,.1f b k eq xxy = c 2 s u y T + u 2 x u y T ,.1g b k eq xyy = c 2 s u x T + u 2 y u x T ,1.1h b k eq xxyy = c 4 s T + c 2 s u 2 x + u 2 y T + u 2 x u 2 y T .1i Similarly,therawmomentforthesourcetermscanbeobtainedfromtheir correspondingcentralmomentsEq..30,whicharepresentedinEq..32a.Now, forconvenience,thevariousrawmomentscanberelatedtotheircorresponding startsinthevelocityspaceviathenon-orthogonaltransformationmatrix T .Wenow denerawmomentsofdistributionfunctionsincludingthetransformedone, equilibriumandsourcesforconvenienceas b g = T g , b g = T g , b S = T S ,.2 where c representscolumnvectorsinrawmomentspaceandthetransformation matrix T isgiveninEq..19.Thatis, b g = b g 0 , b g 1 , b g 2 ,..., b g 8 = b k 0 0 , b k 0 x , b k 0 y , b k 0 xx + b k 0 yy , b k 0 xx )]TJ/F85 11.9552 Tf 12.265 0.173 Td [(b k 0 yy , b k 0 xy , b k 0 xxy , b k 0 xyy , b k 0 xxyy , b g = b g 0 , b g 1 , b g 2 ,..., b g 8 = b k 0 0 , b k 0 x , b k 0 y , b k 0 xx + b k 0 yy , b k 0 xx )]TJ/F85 11.9552 Tf 11.967 2.564 Td [(b k 0 yy , b k 0 xy , b k 0 xxy , b k 0 xyy , b k 0 xxyy , b S = b S 0 , b S 1 , b S 2 ,..., b S 8 = b s 0 0 , b s 0 x , b s 0 y , b s 0 xx + b s 0 yy , b s 0 xx )]TJ/F85 11.9552 Tf 12.146 0.293 Td [(b s 0 yy , b s 0 xy , b s 0 xxy , b s 0 xyy , b s 0 xxyy . 105

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Inddition,inordertomaintainexibilityinthespecicationofthetransport coefcienti.e.thethermaldiffusivityappearingintheemergentCDE,wespecifythe rawmomentequilibrium b g eq byaugmentingthebasemomentequilibriagivenin Eq..1iwithanextendedrstordermomentequilibriainvolvingthecomponents ofthetemperaturegradientswithanadjustablecoefcientdesignnedasDbelow. Thatis b g eq = T g eq .3 Here b g eq = b g eq 0 + b g eq 1 .4 Where, b g eq 0 and b g eq 1 arethebaseandextendedmomentequilibria,respectivly,and aregivenasfollows: b g eq 0 = b g eq 0 0 , b g eq 0 1 ,..., b g eq 0 8 = b k eq 0 0 , b k eq 0 x , b k eq 0 y , b k eq 0 xx + b k eq 0 yy , b k eq 0 xx )]TJ/F85 11.9552 Tf 12.265 0.173 Td [(b k eq 0 yy , b k eq 0 xy , b k eq 0 xxy , b k eq 0 xyy , b k eq 0 xxyy .5 b g eq 1 = b g eq 1 0 , b g eq 1 1 , b g eq 1 2 , b g eq 1 3 ,..., b g eq 1 8 = 0, c 2 s D x T , c 2 s D y T ,0,0,0,0,0,0 .6 Wecanthenrewritepost-collisionstateofthethermalcascadedLBEinEq. ?? in termsofthefollowing: g p = g + T )]TJ/F59 8.9664 Tf 7.375 0 Td [(1 )]TJ/F69 11.9552 Tf 9.982 0 Td [(L b g )]TJ/F85 11.9552 Tf 12.163 0.023 Td [(b g eq + I)]TJ/F59 11.9552 Tf 23.663 8.093 Td [(1 2 L b S .7 106

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where L isadiagonalcollisionmatrixgivenby L = diag l 0 , l 1 , l 2 , l 3 ,......., l 8 ..8 Now,applyingaChapman-Enskogmultiscaleexpansionbyexpandingtheraw moments b g andthetimederivativeintermsofasmallperturbationparameter e = d t whichwillbesetto1attheendoftheanalysisusingthefollowingmulti-scale expansions: b g = n = 0 e n b g n = b g 0 + e b g 1 + e 2 b g 2 ,.9a b g eq = n = 0 e n b g eq n = b g eq 0 + e b g eq 1 ,.9b t = n = 0 e n t n = t 0 + e t 1 + e 2 t 2 , r = e r .9c Noticethatintheabovewehaveusedthemomentequilibria b g eq intermsofthesum ofthebasemomentequilibria b g eq 0 andtheextendedmomentequilibria b g eq 1 .In addition,aTaylorexpansionisusedfortherepresentationofthestreamingoperator, whichiscarriedoutinitsnaturalvelocityspace: g )723(! x + )723(! e a e , t + e = n n = 0 e n n ! t + )723(! e a )723(! r n g )723(! x , t ..10 O e 0 : b g 0 = b g eq ,.11a O e 1 : t 0 + b E i i b g 0 = )]TJ/F69 11.9552 Tf 9.983 0 Td [(L h b g 1 )]TJ/F85 11.9552 Tf 12.163 0.024 Td [(b g eq 1 i + b S a ,.11b O e 2 : t 1 b g 0 + t 0 + b E i i I)]TJ/F69 11.9552 Tf 23.813 8.094 Td [(L 2 b g 1 = )]TJ/F69 11.9552 Tf 9.983 0 Td [(L b g 2 ,.11c where b E i = T e a i I T )]TJ/F59 8.9664 Tf 7.374 0 Td [(1 , i 2 x , y .InorderderivethemacroscopicCDE,uptothe rstordermomentcomponentsin O e Eq.2.11barerelevant,whicharelistedas 107

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follows: t 0 T + x Tu x + y Tu y = G ,.12a t 0 Tu x + x 1 3 T + Tu 2 x + y )]TJ/F68 11.9552 Tf 5.929 -9.69 Td [(Tu x u y = )]TJ/F66 11.9552 Tf 9.983 0 Td [(l 1 b g 1 1 + 1 3 D l 1 x T + u x G ,.12b t 0 )]TJ/F68 11.9552 Tf 5.93 -9.69 Td [(Tu y + x )]TJ/F68 11.9552 Tf 5.929 -9.69 Td [(Tu x u y + y 1 3 T + Tu 2 y = )]TJ/F66 11.9552 Tf 9.983 0 Td [(l 2 b g 1 2 + 1 3 D l 2 y T + u y G ,.12c Similarly,thesecond-ordermomentequationscanbederivedfromEq..11c,which canbewrittenas t 1 T + x 1 )]TJ/F66 11.9552 Tf 13.503 8.094 Td [(l 1 2 b g 1 1 + y 1 )]TJ/F66 11.9552 Tf 13.502 8.094 Td [(l 2 2 b g 1 2 + x l 1 6 D x T + y l 2 6 D y T = 0, .13 Now,combiningEqs..12awith e timesEq..13andsetting t = t 0 + e t 1 ,weget thedynamicalequationsfortheconservedorhydrodynamicmomentsaftersetting theparameter e tounity.Thatis, t T + x Tu x + y Tu y = )]TJ/F66 11.9552 Tf 9.982 0 Td [(e x 1 )]TJ/F66 11.9552 Tf 13.503 8.094 Td [(l 1 2 b g 1 1 )]TJ/F66 11.9552 Tf 12.307 0 Td [(e y 1 )]TJ/F66 11.9552 Tf 13.502 8.094 Td [(l 2 2 b g 1 2 )]TJ/F66 11.9552 Tf 9.982 0 Td [(e x l 1 6 D x T )]TJ/F66 11.9552 Tf 12.307 0 Td [(e y l 2 6 D y T + G ,.14 Intheaboveequation,Eq..14,weneedtherstordernon-equilibriumraw moments b g 1 1 and b g 1 2 .TheycanbeobtainedfromEqs..12band.12c, respectively.Thus, b g 1 1 = 1 l 1 l 1 3 D x T )]TJ/F66 11.9552 Tf 12.307 0 Td [( t 0 u x T )]TJ/F66 11.9552 Tf 12.307 0 Td [( x 1 3 T + u 2 x T )]TJ/F66 11.9552 Tf 12.307 0 Td [( y )]TJ/F68 11.9552 Tf 5.774 -9.69 Td [(u x u y T + u x G ] ,.15 b g 1 2 = 1 l 2 l 2 3 D y T )]TJ/F66 11.9552 Tf 12.307 0 Td [( t 0 )]TJ/F68 11.9552 Tf 5.774 -9.69 Td [(u y T )]TJ/F66 11.9552 Tf 12.307 0 Td [( x )]TJ/F68 11.9552 Tf 5.774 -9.69 Td [(u x u y T )]TJ/F66 11.9552 Tf 12.307 0 Td [( y 1 3 T + u 2 y T + u y G ,.16 108

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Withthehelpoftherst-orderincompressibleNavierStokesequationrst-order hydrodynamicmomentequationsandcontinouty[56,44] t u + u . r u = )-24(r P + a wehave t 0 r u 2 x 2 F x u x , t 0 r u 2 y 2 F y u y and t 0 r u x u y F x u y + F y u x . SubstitutingforthesetermsinEqs..15-.16,andrepresentingthecomponentsof momentumforbrevityasThetwonon-equilibriummomentscanbesimplied.we get b g 1 1 = 1 3 D )]TJ/F59 11.9552 Tf 16.621 8.094 Td [(1 l 1 x T ,.17 b g 1 2 = 1 3 D )]TJ/F59 11.9552 Tf 16.621 8.093 Td [(1 l 2 y T .18 Now,bysubstitutingthesesimpliedexpressionsforthenon-equilibriummoments, Eq..17andEq..18intoEq..14itfollowsthat t T + x Tu x + y Tu y = x [ a 1 x T ] + y a 2 y T + G .19 Whichrepresentstheconvection-diffusionequationCDEwithsourceterm.The coefcients a 1 and a 2 representthethermaldiffusivityandarerelatedtotheir relaxationparameters l 1 and l 2 ,andtheadjustableparameterDintheextended momentequilibria: a 1 = 1 3 1 l 1 )]TJ/F59 11.9552 Tf 13.353 8.094 Td [(1 2 )]TJ/F68 11.9552 Tf 12.546 0 Td [(D , a 2 = 1 3 1 l 2 )]TJ/F59 11.9552 Tf 13.353 8.094 Td [(1 2 )]TJ/F68 11.9552 Tf 12.546 0 Td [(D .20 Forisotropyofthethermaldiffusion a = a 1 = a 2 ,itfollowsthat l 1 = l 2 ,andthe restofthehigherorderrelaxationparameterscanbetunedtoimprovethenumerical 109

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stability.Finally,thetemperaturegradients x T and y T neededintheextended momentequilibriaEq..6canbecomputedlocallyfromEq..17andEq..18,i.e. therstordernon-equilibriummoments.Using b g 1 1 = b g 1 )]TJ/F85 11.9552 Tf 13.007 0.173 Td [(b g eq 0 1 = b k 0 x )]TJ/F85 11.9552 Tf 11.967 2.564 Td [(b k eq 0 x and b g 1 2 = b g 2 )]TJ/F85 11.9552 Tf 13.006 0.173 Td [(b g eq 0 2 = b k 0 y )]TJ/F85 11.9552 Tf 11.966 2.564 Td [(b k eq 0 y inEqs..17and.18,andrearranging,weget x T = 3 l 1 b k 0 x )]TJ/F85 11.9552 Tf 11.967 2.564 Td [(b k eq 0 x D l 1 )]TJ/F59 11.9552 Tf 12.157 0 Td [(1 ,.21a y T = 3 l 2 b k 0 y )]TJ/F85 11.9552 Tf 11.966 2.564 Td [(b k eq 0 y D l 2 )]TJ/F59 11.9552 Tf 12.158 0 Td [(1 .21b 110

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APPENDIXII CHAPMAN-ENSKOGANALYSISOFTHEANISOTROPICTHERMAL CASCADEDLBM Inthissection,aChapman-EnskogC-Eanalysisoftheanisotropicthermalcascaded LBMispresentedandtheresultsoftheanalysisprovidethemacroscopicemergent equations,vis;theconvection-diffusionequationCDEwithasourcetermgivenin Eq..17earlier.Inthisregard,weconsiderthestrategyofrewritingthecentral momentLBMintermsoftherelaxationtoageneralizedequilibriumintherestframe ofreference.TofacilitateanalysisanditsestablishconsistencytotheCDE,itis sufcienttoconsidertermsonlyuptosecondorderinMachnumberinsuchan equivalentformulation[24,44].Inthisregard,weconsiderperformingcalculations intermsofvariousrawmomentsdesignnedwith"primesymbols"withrespectto thenon-orthogonalmomentbasisvectorscollectedinthematrix T andgivenin Eq..19.First,thebaserawmomentequilibrium b k eq 0 x m y n canbeobtainedfromthe correspondingcentralmomentequilibria b k eq x m y n giveninequationEq..29.This 111

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readsas b k eq 0 = T ,.1a b k eq x = u x T ,.1b b k eq y = u y T ,.1c b k eq xx = c 2 s T + u 2 x T ,.1d b k eq yy = c 2 s T + u 2 y T ,.1e b k eq xy = u x u y T ,.1f b k eq xxy = c 2 s u y T + u 2 x u y T ,.1g b k eq xyy = c 2 s u x T + u 2 y u x T ,2.1h b k eq xxyy = c 4 s T + c 2 s u 2 x + u 2 y T + u 2 x u 2 y T .1i Similarly,therawmomentforthesourcetermscanbeobtainedfromtheir correspondingcentralmomentsEq..30,whicharepresentedinEq..32a.Now, forconvenience,thevariousrawmomentscanberelatedtotheircorresponding startsinthevelocityspaceviathenon-orthogonaltransformationmatrix T .Wenow denerawmomentsofdistributionfunctionsincludingthetransformedone, equilibriumandsourcesforconvenienceas b g = T g , b g = T g , b S = T S ,.2 where c representscolumnvectorsinrawmomentspaceandthetransformation matrix T isgiveninEq..19.Thatis, b g = b g 0 , b g 1 , b g 2 ,..., b g 8 = b k 0 0 , b k 0 x , b k 0 y , b k 0 xx + b k 0 yy , b k 0 xx )]TJ/F85 11.9552 Tf 12.265 0.173 Td [(b k 0 yy , b k 0 xy , b k 0 xxy , b k 0 xyy , b k 0 xxyy , b g = b g 0 , b g 1 , b g 2 ,..., b g 8 = b k 0 0 , b k 0 x , b k 0 y , b k 0 xx + b k 0 yy , b k 0 xx )]TJ/F85 11.9552 Tf 11.967 2.564 Td [(b k 0 yy , b k 0 xy , b k 0 xxy , b k 0 xyy , b k 0 xxyy , b S = b S 0 , b S 1 , b S 2 ,..., b S 8 = b s 0 0 , b s 0 x , b s 0 y , b s 0 xx + b s 0 yy , b s 0 xx )]TJ/F85 11.9552 Tf 12.146 0.293 Td [(b s 0 yy , b s 0 xy , b s 0 xxy , b s 0 xyy , b s 0 xxyy . 112

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Inddition,inordertomaintainexibilityinthespecicationofthetransport coefcienti.e.thethermaldiffusivityappearingintheemergentCDE,wespecifythe rawmomentequilibrium b g eq byaugmentingthebasemomentequilibriagivenin Eq..1iwithanextendedrstordermomentequilibriainvolvingthecomponents ofthetemperaturegradientswithanadjustablecoefcientdesignnedasDbelow. Thatis b g eq = T g eq .3 Here b g eq = b g eq 0 + b g eq 1 .4 Where, b g eq 0 and b g eq 1 arethebaseandextendedmomentequilibria,respectivly,and aregivenasfollows: b g eq 0 = b g eq 0 0 , b g eq 0 1 ,..., b g eq 0 8 = b k eq 0 0 , b k eq 0 x , b k eq 0 y , b k eq 0 xx + b k eq 0 yy , b k eq 0 xx )]TJ/F85 11.9552 Tf 12.265 0.173 Td [(b k eq 0 yy , b k eq 0 xy , b k eq 0 xxy , b k eq 0 xyy , b k eq 0 xxyy .5 b g eq 1 = b g eq 1 0 , b g eq 1 1 , b g eq 1 2 , b g eq 1 3 ,..., b g eq 1 8 = 0, c 2 s D x T , c 2 s D y T ,0,0,0,0,0,0 .6 Wecanthenrewritepost-collisionstateofthethermalcascadedLBEinEq. ?? in termsofthefollowing: g p = g + T )]TJ/F59 8.9664 Tf 7.375 0 Td [(1 )]TJ/F69 11.9552 Tf 9.982 0 Td [(L b g )]TJ/F85 11.9552 Tf 12.163 0.023 Td [(b g eq + I)]TJ/F59 11.9552 Tf 23.663 8.093 Td [(1 2 L b S .7 113

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where L isadiagonalcollisionmatrixgivenby L = diag l 0 , 2 6 4 l 11 l 12 l 21 l 22 3 7 5 , l 3 , l 4 ,......., l 8 ..8 Now,applyingaChapman-Enskogmultiscaleexpansionbyexpandingtheraw moments b g andthetimederivativeintermsofasmallperturbationparameter e = d t whichwillbesetto1attheendoftheanalysisusingthefollowingmulti-scale expansions: b g = n = 0 e n b g n = b g 0 + e b g 1 + e 2 b g 2 ,.9a b g eq = n = 0 e n b g eq n = b g eq 0 + e b g eq 1 ,.9b t = n = 0 e n t n = t 0 + e t 1 + e 2 t 2 , r = e r .9c Noticethatintheabovewehaveusedthemomentequilibria b g eq intermsofthesum ofthebasemomentequilibria b g eq 0 andtheextendedmomentequilibria b g eq 1 .In addition,aTaylorexpansionisusedfortherepresentationofthestreamingoperator, whichiscarriedoutinitsnaturalvelocityspace: g )723(! x + )723(! e a e , t + e = n n = 0 e n n ! t + )723(! e a )723(! r n g )723(! x , t ..10 O e 0 : b g 0 = b g eq ,.11a O e 1 : t 0 + b E i i b g 0 = )]TJ/F69 11.9552 Tf 9.983 0 Td [(L h b g 1 )]TJ/F85 11.9552 Tf 12.163 0.024 Td [(b g eq 1 i + b S a ,.11b O e 2 : t 1 b g 0 + t 0 + b E i i I)]TJ/F69 11.9552 Tf 23.813 8.094 Td [(L 2 b g 1 = )]TJ/F69 11.9552 Tf 9.983 0 Td [(L b g 2 ,.11c 114

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where b E i = T e a i I T )]TJ/F59 8.9664 Tf 7.374 0 Td [(1 , i 2 x , y .InorderderivethemacroscopicCDE,uptothe rstordermomentcomponentsin O e Eq.2.11barerelevant,whicharelistedas follows: t 0 T + x Tu x + y Tu y = G ,.12a t 0 Tu x + x 1 3 T + Tu 2 x + y )]TJ/F68 11.9552 Tf 5.93 -9.69 Td [(Tu x u y = )]TJ/F66 11.9552 Tf 9.982 0 Td [(l 1 b g 1 1 )]TJ/F66 11.9552 Tf 12.307 0 Td [(l 12 b g 1 2 + 1 3 D l 1 x T + 1 3 D l 12 y T + u x G ,.12b t 0 )]TJ/F68 11.9552 Tf 5.929 -9.69 Td [(Tu y + x )]TJ/F68 11.9552 Tf 5.929 -9.69 Td [(Tu x u y + y 1 3 T + Tu 2 y = )]TJ/F66 11.9552 Tf 9.982 0 Td [(l 2 b g 1 2 )]TJ/F66 11.9552 Tf 12.307 0 Td [(l 21 b g 1 1 + 1 3 D l 2 y T + 1 3 D l 21 x T + u y G .12c Similarly,thesecond-ordermomentequationscanbederivedfromEq..11c,which canbewrittenas t 1 T + x 1 )]TJ/F66 11.9552 Tf 13.502 8.094 Td [(l 1 2 b g 1 1 )]TJ/F66 11.9552 Tf 13.503 8.094 Td [(l 12 2 b g 1 2 + y 1 )]TJ/F66 11.9552 Tf 13.502 8.094 Td [(l 2 2 b g 1 2 )]TJ/F66 11.9552 Tf 13.502 8.094 Td [(l 21 2 b g 1 1 + x l 1 6 D x T + l 12 6 D y T + y l 2 6 D y T + l 21 6 D x T = 0.13 Now,combiningEqs..12awith e timesEq..13andsetting t = t 0 + e t 1 , wegetthedynamicalequationsfortheconservedorhydrodynamicmomentsafter settingtheparameter e tounity.Thatis, t T + x Tu x + y Tu y = )]TJ/F66 11.9552 Tf 9.983 0 Td [(e x 1 )]TJ/F66 11.9552 Tf 13.502 8.094 Td [(l 1 2 b g 1 1 )]TJ/F66 11.9552 Tf 13.503 8.094 Td [(l 12 2 b g 1 2 )]TJ/F66 11.9552 Tf 12.307 0 Td [(e y 1 )]TJ/F66 11.9552 Tf 13.502 8.094 Td [(l 2 2 b g 1 2 )]TJ/F66 11.9552 Tf 13.502 8.094 Td [(l 21 2 b g 1 1 )]TJ/F66 11.9552 Tf 9.983 0 Td [(e x l 1 6 D x T + l 12 6 D y T )]TJ/F66 11.9552 Tf 12.307 0 Td [(e y l 2 6 D y T + l 21 6 D x T + G .14 Intheaboveequation,Eq..14,weneedtherstordernon-equilibriumraw moments b g 1 1 and b g 1 2 .TheycanbeobtainedfromEqs..12band.12c, 115

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respectively.Thus, b g 1 1 = 1 l 1 l 1 3 D x T + l 12 3 D y T )]TJ/F66 11.9552 Tf 12.307 0 Td [( t 0 u x T )]TJ/F66 11.9552 Tf 12.307 0 Td [( x 1 3 T + u 2 x T )]TJ/F66 11.9552 Tf 12.307 0 Td [( y )]TJ/F68 11.9552 Tf 5.774 -9.689 Td [(u x u y T + u x G )]TJ/F66 11.9552 Tf 12.307 0 Td [(l 12 b g 1 2 i ,.15 b g 1 2 = 1 l 2 l 2 3 D y T + l 21 3 D x T )]TJ/F66 11.9552 Tf 12.307 0 Td [( t 0 )]TJ/F68 11.9552 Tf 5.774 -9.69 Td [(u y T )]TJ/F66 11.9552 Tf 12.307 0 Td [( x )]TJ/F68 11.9552 Tf 5.774 -9.69 Td [(u x u y T )]TJ/F66 11.9552 Tf 12.307 0 Td [( y 1 3 T + u 2 y T + u y G )]TJ/F66 11.9552 Tf 12.307 0 Td [(l 21 b g 1 1 i ,.16 Withthehelpoftherst-orderincompressibleNavierStokesequationrst-order hydrodynamicmomentequationsandcontinouty[56,44] t u + u . r u = )-24(r P + a wehave t 0 r u 2 x 2 F x u x , t 0 r u 2 y 2 F y u y and t 0 r u x u y F x u y + F y u x . SubstitutingforthesetermsinEqs..15-.16,andsolvingfor b g 1 1 and b g 1 2 .The twonon-equilibriummomentscanbesimpliedas b g 1 1 = 1 3 D )]TJ/F66 11.9552 Tf 44.071 8.094 Td [(l 2 l 1 l 2 )]TJ/F66 11.9552 Tf 12.307 0 Td [(l 12 l 21 x T + 1 3 l 12 l 1 l 2 )]TJ/F66 11.9552 Tf 12.307 0 Td [(l 12 l 21 y T ,.17 b g 1 2 = 1 3 l 21 l 1 l 2 )]TJ/F66 11.9552 Tf 12.307 0 Td [(l 12 l 21 x T + 1 3 D )]TJ/F66 11.9552 Tf 44.071 8.093 Td [(l 1 l 1 l 2 )]TJ/F66 11.9552 Tf 12.307 0 Td [(l 12 l 21 y T .18 Now,bysubstitutingthesesimpliedexpressionsforthenon-equilibriummoments, Eq..17andEq..18intoEq..14itfollowsthat t T + x Tu x + y Tu y = x [ a 11 x T ] + x a 12 y T + y [ a 21 x T ] + y a 22 y T + G .19 116

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Whichrepresentstheconvection-diffusionequationCDEwithsourceterm.The coefcients a 11 , a 12 , a 21 and a 22 representthethermaldiffusivityandarerelatedto theirrelaxationparameters l 1 , l 12 , l 21 and l 22 ,andtheadjustableparameterDin theextendedmomentequilibria: a 11 = 1 3 l 2 j A j )]TJ/F59 11.9552 Tf 13.353 8.094 Td [(1 2 )]TJ/F68 11.9552 Tf 12.546 0 Td [(D , a 12 = 1 3 )]TJ/F66 11.9552 Tf 9.983 0 Td [(l 12 j A j , a 21 = 1 3 )]TJ/F66 11.9552 Tf 9.983 0 Td [(l 21 j A j , a 22 = 1 3 l 1 j A j )]TJ/F59 11.9552 Tf 13.353 8.093 Td [(1 2 )]TJ/F68 11.9552 Tf 12.546 0 Td [(D .20 Where A = 2 6 4 l 11 l 12 l 21 l 22 3 7 5 , j A j = j l 1 l 2 )]TJ/F66 11.9552 Tf 12.307 0 Td [(l 12 l 21 j .Fordiagonalanisotropiccase wherethediffusiontensorisdiagonalmatrix i . e . l 12 = l 21 = 0 ,therelaxation parameterscanbesimpliedas a 11 = 1 3 1 l 1 )]TJ/F59 11.9552 Tf 13.354 8.093 Td [(1 2 )]TJ/F68 11.9552 Tf 12.546 0 Td [(D , a 22 = 1 3 1 l 2 )]TJ/F59 11.9552 Tf 13.353 8.093 Td [(1 2 )]TJ/F68 11.9552 Tf 12.546 0 Td [(D .21 117