Citation |

- Permanent Link:
- http://digital.auraria.edu/AA00002706/00001
## Material Information- Title:
- Multilevel adaptive methods in computational fluid dynamics
- Creator:
- Liu, Chaoqun
- Publication Date:
- 1989
- Language:
- English
- Physical Description:
- 93 leaves : illustrations ; 29 cm
## Subjects- Subjects / Keywords:
- Fluid dynamics ( lcsh )
Multigrid methods ( lcsh ) Poisson's equation ( lcsh ) Navier-Stokes equations ( lcsh ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Bibliography:
- Includes bibliographical references.
- General Note:
- Submitted in partial fulfillment of the requirements for the degree, Doctor of Philosophy, Department of Mathematical and Statistical Sciences.
- Statement of Responsibility:
- by Chaoqun Liu.
## Record Information- Source Institution:
- |University of Colorado Denver
- Holding Location:
- Auraria Library
- Rights Management:
- All applicable rights reserved by the source institution and holding location.
- Resource Identifier:
- 24655102 ( OCLC )
ocm24655102 - Classification:
- LD1190.L622 1989d .L58 ( lcc )
## Auraria Membership |

Full Text |

Multilevel Adaptive Methods in Computational Fluid Dynamics by Chaoqun Liu Computational Mathematics Group Campus Box 170 University of Colorado at Denver Denver, CO 80204 November,1989 MULTILEVEL ADAPTIVE METHODS IN COMPUTATIONAL FLUID DYNAMICS by Chaoqun Liu B. A., Tsinghua University, Beijing, China, 1967 M. A., Tsinghua University, Beijing, China, 1981 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 Department of Mathematics 1989 This thesis for the Doctor of Philosophy degree by Chaoqun Liu has been approved for the Department of Mathematics Steve McCormick Williams L. Briggs Karl Gustafson Jan Mandel ) Roland Sweet Date November 27, 1989 ) Chaoqun Liu (Ph.D., Applied Mathematics) Multilevel Adaptive Methods in Computational Fluid Dynamics Thesis directed by Professor Steve McCormick We first take Poisson's equation as an example to illustrate the basic principles and schemes for the methods of multigrid (MG), fast adaptive composite grid (FAC) and finite volume element (FVE). The computational results show that local refinement can greatly enhance discretization accuracy. Then we apply full multigrid (FMG) with cubic interpolation to solve the elliptic grid generation (EGG) equations and use FAC to generate a smooth patch grid. We also apply MG and FAC to solve a full potential flow around the NACA0012 airfoil. For this application, discretizations are used in the global grid that are different than in the patches. Computational results show that MG can substantially accelerate convergence and FAC local refinement can greatly improve accuracy, especially for problems with shocks. We apply MG and FAC to driven cavity flow (incompressible Navier-Stokes equations). Reviewing the weakness of classical finite difference theory and conventional notions on artificial viscosity, we introduce the rotated hybrid scheme and FVE scheme for medium and high Reynolds number flows. We establish an equivalence between spatial relaxation and time-marching schemes, then use this interpretation as a basis for the development of an MG7 solver for pseudo-unsteady flows. Finally, we introduce the source term correction (STC) method for incompressible Navier-Stokes equations. The ;form and content of this abstii publication. Sign nd its Dedicated to Mrs. Han, my grandmother I Acknowledgement I am deeply in debt to Professor S. McCormick, the founder of FAC, for his ideas, support, countless discus- sions and helpful guidance. Much of my work wa6 inspired by the dramatic develop- ments in the field of multigrid, I am therefore thankful to Professor A. Brandt for his pioneering work in the field as well as his lectures and helpful discussions. I am also indebted to the United States Air Force Office of Scientific Research and the National Science Fundation for their financial support. CONTENTS CHAPTER I. INTRODUCTION Motivation.........;........................... 1 Arrangement of the thesis...................... 2 II. POISSONS EQUATION Equation and boundary conditions............... 4 FVE discretization........................... 5 Multigrid algorithm (FAS).................... 8 FVE accuracy.................................. 11 ( Nested iteration (FMG).................... 11 FAC for nonuniform (composite) grids..... 12 III. ELLIPTIC GRID GENERATION (EGG) EGG equations................................. 19 Finite difference equations.................... 21 Initial guess.................................. 21 Full approximation scheme (FAS)................ 22 FAC for local refinement....................... 23 Discretization error estimates..........}. 24 IV. TRANSONIC POTENTIAL FLOWS Governing equations.......................... 26 Boundary conditions............................ 27 Discretization ............................... 27 Stability and artificial density............... 35 Multigrid for potential flows................. 36 Computational results......................... 38 V. THE ROTATED HYBRID SCHEME FOR PLANAR CAVITY FLOWS Differential equations and boundary conditions.................................. 43 Conventional difference schemes for 11 the vorticity equation................. 43 Improved difference schemes for 1-D ^ vorticity equation..................... 45 Artificial viscosity revisited........47 Rotated hybrid scheme for the 2-D vorticity equation......................... 50 Collective relaxation........................ 53 Multigrid................................... 55 Alternating direction block-line Gauss-Seidel relaxation.................... 56 FAC for local refinement..................... 57 Computational results...............'.... 57 VI. FVE FOR DRIVEN CAVITY FLOW Equations and boundary conditions............ 59 Difference scheme........................... 59 , Multigrid algorithm............................ 63 c FVE accuracy.............................:.. 64 Nested iteration..................^...... 65 FAC..................'....................... 65 Medium and high Re flow...................... 68 VII. MG FOR PSEUDO TIME-DEPENDENT FLOWS ) Explicit time-marching and Jacobi iteration.................................. 70 Explicit time^marching and Gauss-Seidel iteration.................................. 71 ADI time-marching and line relaxation.... 72 Generalization............................. 73 MG as an explicit Euler solver............... 74 VIII. SOURCE TERM CORRECTION (STC) FOR ' INCOMPRESSIBLE NAVIER-STOKES EQUATIONS Reformulating the governing equations and boundary conditions.................... 79 Discretization................................ 81 The STC method................................ 82 Ill Iteration scheme and STC algorithm....... 84 Solution for pressure......................... 85 Other flow problems.............'........ 86 Preliminary computational results........... 88 REFERENCES............................................. 90 T. ( I CHAPTER I INTRODUCTION 1. Motivation^ Great progress has been achieved during the past two decades in both computational fluid dynamics (CFD) and applied mathematics. Some of the achievements include the development of elliptic grid generation (EGG) [1], multigrid [2-3], and algorithms and difference schemes for solving Euler and Navier-Stokes equations [4-10]. However, there are still several barriers to the efficient and accurate solution of many large scale flow problems, including transonic, high Reynolds number, separated and - turbulent flows. Even with5the current and foreseeable computing machinery, substantial progress must-be made in algorithms in order for the numerical solution of these problems to be practically feasible. It is well known that MG is an optimal order process for solving many partial differential equations (PDEs). In fact, MG schemes have proved effective for a wide number of problems which contribute to application of MG to a variety of flow problems [11-18]. However, there are still many impediments to optimal order accuracy and efficiency for most practical flow models The work presented here repre- sents an attempt to overcome some of these difficulties. Many impediments are caused by local phenomena, such as shocks and boundary layers, where higher resolution and accuracy are seriously needed. To meet this need, FAC is used as an efficient multilevel process for local refinement. FAC implicitly solves nonuniform grid problems by solving a series of problems defined on uniform grids. Therefore, FAC takes advantage of uniform grid properties and it is easily coupled with existing codes and conven- [ -2 tional multigrid methods. Because FAC originates from the multigrid discipline, it can achieve enhanced accuracy at apparently optimal order efficiency. This thesis focuses on the use of FAC in conjunction with MG for solving Poisson equations, EGG equations, potential flows and cavity flows. Accuracy and efficiency of FAC and MG solvers depend critically on the^quality' of discretization, which can be problematic for many highly active flows and locally refined grids. To provide effective discretization schemes for high Reynolds (Re) number flow equations, we developed the rotated hybrid and finite volume element (FVE) schemes. A significant part of this development is the answer to the question of how to treat such aspects as Dirichlet and Neumann boundary conditions, systems, anisotropic problems, singularities, and discretizations and transfer at boun- daries, interfaces and patches. The difficulties involved with solving incompressible Navier-Stokes equations are clarified in [27]. Our source term correction method (STC) represents an attempt to over- come some of these difficulties in a very convenient .way. Several of the methods and results presented here have already appeared in published form. See [23-26]. This thesis summarizes this work, develops several additional techniques (e.g., the time-marching schemes and STC), and presents more recent numerical results. 2. Arrangement of the thesis In Chapter II, we take the Poisson equation as an example to illustrate the basic principles and schemes for the methods of MG, FAC and FVE. The computational results show that local refinement can greatly enhance discretiza- tion accuracy and that these methods can compute an acceptable numerical approximation very efficiently. -3- In Chapter III, we apply full multigrid (FMG) with cubic interpolation to solve the elliptic grid generation (EGG) equation and use FAC "to generate a smooth locally generated patch grid. In Chapter IV, we apply MG and FAC to solve a full potential flow around a NACA0012 airfoil. For this applica- tion, discretizations in the global grid are different than those in the patches. Computational results presented there show that MG can substantially accelerate convergence and the FAC local refinement process can greatly improve accuracy, especially for problems with shocks. In Chapter V and VI, we apply MG and FAC to driven planar cavity flow (incompressible Navier-Stokes equations). Reviewing the weakness of classical finite l difference theory and conventional notions on artificial viscosity, we develop a so-called rotated hybrid scheme (Chapter V) and an FVE scheme (Chapter VI) for medium and high Reynolds number flows. In Chapter VII, we establish an equivalence between spatial relaxation and time-marching schemes, then use this interpretation as a basis for development of MG for pseudo unsteady flow . In Chapter VIII, we develop a so-called source term correction method (STC) for incompressible Navier-Stokes equations. ^ ( CHAPTER II POISSONS EQUATION The purpose of this chapter is to introduce the basic concepts and illustrate some of features of multilevel adaptive methods in a simple setting. For this we restrict discussion to the model Poisson equation. 1. Equation and boundary conditions f=-6(0,0)+6(l,l), where 6(x,y) is the Delta function at point (x,y). This is the so-called five-spot problem, which represents the pressure equation in an idealized oil reservoir. The term -SC0,0) corresponds to an extraction well and 6(1,1) to an injection well. V*Vp=f n. 7
Jn n=o
(8)
suggest the use of local patches for increased resolution
x y
or r=0
For downstream boundary points, we implicitly embed
xx v j j+lJ
(44) and (47), we should compute the density p in advance.
ruK*\ tn
be calculated by (48) and (52). Initially, (48) would be
need to use I?*1, L2^or l\. at the boundaries. At Neumann |