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Fundamentals of Computational Fluid            Dynamics        Harvard Lomax and Thomas H. Pulliam            NASA Ames Re...
Contents1 INTRODUCTION                                                                                                    ...
3.4 Constructing Di erencing Schemes of Any Order . . . . .          .   .   .   .   .   .   .   31      3.4.1 Taylor Tabl...
5.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                                          ...
7.2.3 Defective Eigensystems . . . . . . . . . . . . . .         .   .   .   .   .   .   .   .   123 7.3   Numerical Stabi...
9 RELAXATION METHODS                                                                                            163  9.1 F...
12.3.1 Mesh Indexing Convention . . . . . . . . . . .                                      .   .   .   .   .   .   .   .  ...
C THE HOMOGENEOUS PROPERTY OF THE EULER EQUATIONS265
Chapter 1INTRODUCTION1.1     MotivationThe material in this book originated from attempts to understand and systemize nu-m...
2                                                    CHAPTER 1. INTRODUCTIONare the ones actually being solved by the nume...
1.2. BACKGROUND                                                                       3measurements of an existing con gur...
4                                                  CHAPTER 1. INTRODUCTIONMany di erent gridding strategies exist, includi...
1.4. NOTATION                                                                        5numerical approximations are applied...
6   CHAPTER 1. INTRODUCTION
Chapter 2CONSERVATION LAWS ANDTHE MODEL EQUATIONSWe start out by casting our equations in the most general form, the integ...
8      CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONSthe conservation of these quantities in a nite region of space ...
2.2. THE NAVIER-STOKES AND EULER EQUATIONS                                            9where is the uid density, u is the ...
10      CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONSto derive the ux Jacobian matrices, we must rst write the ux v...
2.2. THE NAVIER-STOKES AND EULER EQUATIONS                                                   11                           ...
12       CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS2.3 The Linear Convection Equation2.3.1 Di erential FormThe s...
2.3. THE LINEAR CONVECTION EQUATION                                                13change in the shape of the solution. ...
14     CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONSwhere the wavenumbers are often ordered such that 1 2          ...
2.4. THE DIFFUSION EQUATION                                                       15   In two dimensions, the di usion equ...
16           CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS2.5 Linear Hyperbolic SystemsThe Euler equations, Eq. 2.8...
2.6. PROBLEMS                                                                       17The elements of w are known as chara...
18       CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS        Assume an ideal gas, p = ( ; 1)(e ; u2=2).     2. Fin...
2.6. PROBLEMS                                                               19    where ! = 0 for irrotational ow. For ise...
20   CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS
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Fundamentals of computational_fluid_dynamics_-_h._lomax__t._pulliam__d._zingg

  1. 1. Fundamentals of Computational Fluid Dynamics Harvard Lomax and Thomas H. Pulliam NASA Ames Research Center David W. Zingg University of Toronto Institute for Aerospace Studies August 26, 1999
  2. 2. Contents1 INTRODUCTION 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Problem Speci cation and Geometry Preparation . . . . . . . 2 1.2.2 Selection of Governing Equations and Boundary Conditions . 3 1.2.3 Selection of Gridding Strategy and Numerical Method . . . . 3 1.2.4 Assessment and Interpretation of Results . . . . . . . . . . . . 4 1.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 CONSERVATION LAWS AND THE MODEL EQUATIONS 7 2.1 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 The Navier-Stokes and Euler Equations . . . . . . . . . . . . . . . . . 8 2.3 The Linear Convection Equation . . . . . . . . . . . . . . . . . . . . 12 2.3.1 Di erential Form . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.2 Solution in Wave Space . . . . . . . . . . . . . . . . . . . . . . 13 2.4 The Di usion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4.1 Di erential Form . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4.2 Solution in Wave Space . . . . . . . . . . . . . . . . . . . . . . 15 2.5 Linear Hyperbolic Systems . . . . . . . . . . . . . . . . . . . . . . . . 16 2.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 FINITE-DIFFERENCE APPROXIMATIONS 21 3.1 Meshes and Finite-Di erence Notation . . . . . . . . . . . . . . . . . 21 3.2 Space Derivative Approximations . . . . . . . . . . . . . . . . . . . . 24 3.3 Finite-Di erence Operators . . . . . . . . . . . . . . . . . . . . . . . 25 3.3.1 Point Di erence Operators . . . . . . . . . . . . . . . . . . . . 25 3.3.2 Matrix Di erence Operators . . . . . . . . . . . . . . . . . . . 25 3.3.3 Periodic Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3.4 Circulant Matrices . . . . . . . . . . . . . . . . . . . . . . . . 30 iii
  3. 3. 3.4 Constructing Di erencing Schemes of Any Order . . . . . . . . . . . . 31 3.4.1 Taylor Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.4.2 Generalization of Di erence Formulas . . . . . . . . . . . . . . 34 3.4.3 Lagrange and Hermite Interpolation Polynomials . . . . . . . 35 3.4.4 Practical Application of Pade Formulas . . . . . . . . . . . . . 37 3.4.5 Other Higher-Order Schemes . . . . . . . . . . . . . . . . . . . 38 3.5 Fourier Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.5.1 Application to a Spatial Operator . . . . . . . . . . . . . . . . 39 3.6 Di erence Operators at Boundaries . . . . . . . . . . . . . . . . . . . 43 3.6.1 The Linear Convection Equation . . . . . . . . . . . . . . . . 44 3.6.2 The Di usion Equation . . . . . . . . . . . . . . . . . . . . . . 46 3.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 THE SEMI-DISCRETE APPROACH 51 4.1 Reduction of PDEs to ODEs . . . . . . . . . . . . . . . . . . . . . . 52 4.1.1 The Model ODEs . . . . . . . . . . . . . . . . . . . . . . . . 52 4.1.2 The Generic Matrix Form . . . . . . . . . . . . . . . . . . . . 53 4.2 Exact Solutions of Linear ODEs . . . . . . . . . . . . . . . . . . . . 54 4.2.1 Eigensystems of Semi-Discrete Linear Forms . . . . . . . . . . 54 4.2.2 Single ODEs of First- and Second-Order . . . . . . . . . . . . 55 4.2.3 Coupled First-Order ODEs . . . . . . . . . . . . . . . . . . . 57 4.2.4 General Solution of Coupled ODEs with Complete Eigensystems 59 4.3 Real Space and Eigenspace . . . . . . . . . . . . . . . . . . . . . . . . 61 4.3.1 De nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.3.2 Eigenvalue Spectrums for Model ODEs . . . . . . . . . . . . . 62 4.3.3 Eigenvectors of the Model Equations . . . . . . . . . . . . . . 63 4.3.4 Solutions of the Model ODEs . . . . . . . . . . . . . . . . . . 65 4.4 The Representative Equation . . . . . . . . . . . . . . . . . . . . . . 67 4.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685 FINITE-VOLUME METHODS 71 5.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.2 Model Equations in Integral Form . . . . . . . . . . . . . . . . . . . . 73 5.2.1 The Linear Convection Equation . . . . . . . . . . . . . . . . 73 5.2.2 The Di usion Equation . . . . . . . . . . . . . . . . . . . . . . 74 5.3 One-Dimensional Examples . . . . . . . . . . . . . . . . . . . . . . . 74 5.3.1 A Second-Order Approximation to the Convection Equation . 75 5.3.2 A Fourth-Order Approximation to the Convection Equation . 77 5.3.3 A Second-Order Approximation to the Di usion Equation . . 78 5.4 A Two-Dimensional Example . . . . . . . . . . . . . . . . . . . . . . 80
  4. 4. 5.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836 TIME-MARCHING METHODS FOR ODES 85 6.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.2 Converting Time-Marching Methods to O Es . . . . . . . . . . . . . 87 6.3 Solution of Linear O Es With Constant Coe cients . . . . . . . . . 88 6.3.1 First- and Second-Order Di erence Equations . . . . . . . . . 89 6.3.2 Special Cases of Coupled First-Order Equations . . . . . . . . 90 6.4 Solution of the Representative O Es . . . . . . . . . . . . . . . . . 91 6.4.1 The Operational Form and its Solution . . . . . . . . . . . . . 91 6.4.2 Examples of Solutions to Time-Marching O Es . . . . . . . . 92 6.5 The ; Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.5.1 Establishing the Relation . . . . . . . . . . . . . . . . . . . . . 93 6.5.2 The Principal -Root . . . . . . . . . . . . . . . . . . . . . . . 95 6.5.3 Spurious -Roots . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.5.4 One-Root Time-Marching Methods . . . . . . . . . . . . . . . 96 6.6 Accuracy Measures of Time-Marching Methods . . . . . . . . . . . . 97 6.6.1 Local and Global Error Measures . . . . . . . . . . . . . . . . 97 6.6.2 Local Accuracy of the Transient Solution (er j j er! ) . . . . 98 6.6.3 Local Accuracy of the Particular Solution (er ) . . . . . . . . 99 6.6.4 Time Accuracy For Nonlinear Applications . . . . . . . . . . . 100 6.6.5 Global Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.7 Linear Multistep Methods . . . . . . . . . . . . . . . . . . . . . . . . 102 6.7.1 The General Formulation . . . . . . . . . . . . . . . . . . . . . 102 6.7.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.7.3 Two-Step Linear Multistep Methods . . . . . . . . . . . . . . 105 6.8 Predictor-Corrector Methods . . . . . . . . . . . . . . . . . . . . . . . 106 6.9 Runge-Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.10 Implementation of Implicit Methods . . . . . . . . . . . . . . . . . . . 110 6.10.1 Application to Systems of Equations . . . . . . . . . . . . . . 110 6.10.2 Application to Nonlinear Equations . . . . . . . . . . . . . . . 111 6.10.3 Local Linearization for Scalar Equations . . . . . . . . . . . . 112 6.10.4 Local Linearization for Coupled Sets of Nonlinear Equations . 115 6.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1177 STABILITY OF LINEAR SYSTEMS 121 7.1 Dependence on the Eigensystem . . . . . . . . . . . . . . . . . . . . . 122 7.2 Inherent Stability of ODEs . . . . . . . . . . . . . . . . . . . . . . . 123 7.2.1 The Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 7.2.2 Complete Eigensystems . . . . . . . . . . . . . . . . . . . . . . 123
  5. 5. 7.2.3 Defective Eigensystems . . . . . . . . . . . . . . . . . . . . . . 123 7.3 Numerical Stability of O E s . . . . . . . . . . . . . . . . . . . . . . 124 7.3.1 The Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7.3.2 Complete Eigensystems . . . . . . . . . . . . . . . . . . . . . . 125 7.3.3 Defective Eigensystems . . . . . . . . . . . . . . . . . . . . . . 125 7.4 Time-Space Stability and Convergence of O Es . . . . . . . . . . . . 125 7.5 Numerical Stability Concepts in the Complex -Plane . . . . . . . . . 128 7.5.1 -Root Traces Relative to the Unit Circle . . . . . . . . . . . 128 7.5.2 Stability for Small t . . . . . . . . . . . . . . . . . . . . . . . 132 7.6 Numerical Stability Concepts in the Complex h Plane . . . . . . . . 135 7.6.1 Stability for Large h. . . . . . . . . . . . . . . . . . . . . . . . 135 7.6.2 Unconditional Stability, A-Stable Methods . . . . . . . . . . . 136 7.6.3 Stability Contours in the Complex h Plane. . . . . . . . . . . 137 7.7 Fourier Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . 141 7.7.1 The Basic Procedure . . . . . . . . . . . . . . . . . . . . . . . 141 7.7.2 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.7.3 Relation to Circulant Matrices . . . . . . . . . . . . . . . . . . 143 7.8 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1468 CHOICE OF TIME-MARCHING METHODS 149 8.1 Sti ness De nition for ODEs . . . . . . . . . . . . . . . . . . . . . . 149 8.1.1 Relation to -Eigenvalues . . . . . . . . . . . . . . . . . . . . 149 8.1.2 Driving and Parasitic Eigenvalues . . . . . . . . . . . . . . . . 151 8.1.3 Sti ness Classi cations . . . . . . . . . . . . . . . . . . . . . . 151 8.2 Relation of Sti ness to Space Mesh Size . . . . . . . . . . . . . . . . 152 8.3 Practical Considerations for Comparing Methods . . . . . . . . . . . 153 8.4 Comparing the E ciency of Explicit Methods . . . . . . . . . . . . . 154 8.4.1 Imposed Constraints . . . . . . . . . . . . . . . . . . . . . . . 154 8.4.2 An Example Involving Di usion . . . . . . . . . . . . . . . . . 154 8.4.3 An Example Involving Periodic Convection . . . . . . . . . . . 155 8.5 Coping With Sti ness . . . . . . . . . . . . . . . . . . . . . . . . . . 158 8.5.1 Explicit Methods . . . . . . . . . . . . . . . . . . . . . . . . . 158 8.5.2 Implicit Methods . . . . . . . . . . . . . . . . . . . . . . . . . 159 8.5.3 A Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 8.6 Steady Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 8.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
  6. 6. 9 RELAXATION METHODS 163 9.1 Formulation of the Model Problem . . . . . . . . . . . . . . . . . . . 164 9.1.1 Preconditioning the Basic Matrix . . . . . . . . . . . . . . . . 164 9.1.2 The Model Equations . . . . . . . . . . . . . . . . . . . . . . . 166 9.2 Classical Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 9.2.1 The Delta Form of an Iterative Scheme . . . . . . . . . . . . . 168 9.2.2 The Converged Solution, the Residual, and the Error . . . . . 168 9.2.3 The Classical Methods . . . . . . . . . . . . . . . . . . . . . . 169 9.3 The ODE Approach to Classical Relaxation . . . . . . . . . . . . . . 170 9.3.1 The Ordinary Di erential Equation Formulation . . . . . . . . 170 9.3.2 ODE Form of the Classical Methods . . . . . . . . . . . . . . 172 9.4 Eigensystems of the Classical Methods . . . . . . . . . . . . . . . . . 173 9.4.1 The Point-Jacobi System . . . . . . . . . . . . . . . . . . . . . 174 9.4.2 The Gauss-Seidel System . . . . . . . . . . . . . . . . . . . . . 176 9.4.3 The SOR System . . . . . . . . . . . . . . . . . . . . . . . . . 180 9.5 Nonstationary Processes . . . . . . . . . . . . . . . . . . . . . . . . . 182 9.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18710 MULTIGRID 191 10.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 10.1.1 Eigenvector and Eigenvalue Identi cation with Space Frequencies191 10.1.2 Properties of the Iterative Method . . . . . . . . . . . . . . . 192 10.2 The Basic Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 10.3 A Two-Grid Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 10.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20211 NUMERICAL DISSIPATION 203 11.1 One-Sided First-Derivative Space Di erencing . . . . . . . . . . . . . 204 11.2 The Modi ed Partial Di erential Equation . . . . . . . . . . . . . . . 205 11.3 The Lax-Wendro Method . . . . . . . . . . . . . . . . . . . . . . . . 207 11.4 Upwind Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 11.4.1 Flux-Vector Splitting . . . . . . . . . . . . . . . . . . . . . . . 210 11.4.2 Flux-Di erence Splitting . . . . . . . . . . . . . . . . . . . . . 212 11.5 Arti cial Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 11.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21412 SPLIT AND FACTORED FORMS 217 12.1 The Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 12.2 Factoring Physical Representations | Time Splitting . . . . . . . . . 218 12.3 Factoring Space Matrix Operators in 2{D . . . . . . . . . . . . . . . . 220
  7. 7. 12.3.1 Mesh Indexing Convention . . . . . . . . . . . . . . . . . . . . 220 12.3.2 Data Bases and Space Vectors . . . . . . . . . . . . . . . . . . 221 12.3.3 Data Base Permutations . . . . . . . . . . . . . . . . . . . . . 221 12.3.4 Space Splitting and Factoring . . . . . . . . . . . . . . . . . . 223 12.4 Second-Order Factored Implicit Methods . . . . . . . . . . . . . . . . 226 12.5 Importance of Factored Forms in 2 and 3 Dimensions . . . . . . . . . 226 12.6 The Delta Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 12.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22913 LINEAR ANALYSIS OF SPLIT AND FACTORED FORMS 233 13.1 The Representative Equation for Circulant Operators . . . . . . . . . 233 13.2 Example Analysis of Circulant Systems . . . . . . . . . . . . . . . . . 234 13.2.1 Stability Comparisons of Time-Split Methods . . . . . . . . . 234 13.2.2 Analysis of a Second-Order Time-Split Method . . . . . . . . 237 13.3 The Representative Equation for Space-Split Operators . . . . . . . . 238 13.4 Example Analysis of 2-D Model Equations . . . . . . . . . . . . . . . 242 13.4.1 The Unfactored Implicit Euler Method . . . . . . . . . . . . . 242 13.4.2 The Factored Nondelta Form of the Implicit Euler Method . . 243 13.4.3 The Factored Delta Form of the Implicit Euler Method . . . . 243 13.4.4 The Factored Delta Form of the Trapezoidal Method . . . . . 244 13.5 Example Analysis of the 3-D Model Equation . . . . . . . . . . . . . 245 13.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247A USEFUL RELATIONS AND DEFINITIONS FROM LINEAR AL- GEBRA 249 A.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 A.2 De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 A.3 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 A.4 Eigensystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 A.5 Vector and Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . 254B SOME PROPERTIES OF TRIDIAGONAL MATRICES 257 B.1 Standard Eigensystem for Simple Tridiagonals . . . . . . . . . . . . . 257 B.2 Generalized Eigensystem for Simple Tridiagonals . . . . . . . . . . . . 258 B.3 The Inverse of a Simple Tridiagonal . . . . . . . . . . . . . . . . . . . 259 B.4 Eigensystems of Circulant Matrices . . . . . . . . . . . . . . . . . . . 260 B.4.1 Standard Tridiagonals . . . . . . . . . . . . . . . . . . . . . . 260 B.4.2 General Circulant Systems . . . . . . . . . . . . . . . . . . . . 261 B.5 Special Cases Found From Symmetries . . . . . . . . . . . . . . . . . 262 B.6 Special Cases Involving Boundary Conditions . . . . . . . . . . . . . 263
  8. 8. C THE HOMOGENEOUS PROPERTY OF THE EULER EQUATIONS265
  9. 9. Chapter 1INTRODUCTION1.1 MotivationThe material in this book originated from attempts to understand and systemize nu-merical solution techniques for the partial di erential equations governing the physicsof uid ow. As time went on and these attempts began to crystallize, underlyingconstraints on the nature of the material began to form. The principal such constraintwas the demand for uni cation. Was there one mathematical structure which couldbe used to describe the behavior and results of most numerical methods in commonuse in the eld of uid dynamics? Perhaps the answer is arguable, but the authorsbelieve the answer is a rmative and present this book as justi cation for that be-lief. The mathematical structure is the theory of linear algebra and the attendanteigenanalysis of linear systems. The ultimate goal of the eld of computational uid dynamics (CFD) is to under-stand the physical events that occur in the ow of uids around and within designatedobjects. These events are related to the action and interaction of phenomena suchas dissipation, di usion, convection, shock waves, slip surfaces, boundary layers, andturbulence. In the eld of aerodynamics, all of these phenomena are governed bythe compressible Navier-Stokes equations. Many of the most important aspects ofthese relations are nonlinear and, as a consequence, often have no analytic solution.This, of course, motivates the numerical solution of the associated partial di erentialequations. At the same time it would seem to invalidate the use of linear algebra forthe classi cation of the numerical methods. Experience has shown that such is notthe case. As we shall see in a later chapter, the use of numerical methods to solve partialdi erential equations introduces an approximation that, in e ect, can change theform of the basic partial di erential equations themselves. The new equations, which 1
  10. 10. 2 CHAPTER 1. INTRODUCTIONare the ones actually being solved by the numerical process, are often referred to asthe modi ed partial di erential equations. Since they are not precisely the same asthe original equations, they can, and probably will, simulate the physical phenomenalisted above in ways that are not exactly the same as an exact solution to the basicpartial di erential equation. Mathematically, these di erences are usually referred toas truncation errors. However, the theory associated with the numerical analysis of uid mechanics was developed predominantly by scientists deeply interested in thephysics of uid ow and, as a consequence, these errors are often identi ed with aparticular physical phenomenon on which they have a strong e ect. Thus methods aresaid to have a lot of arti cial viscosity" or said to be highly dispersive. This meansthat the errors caused by the numerical approximation result in a modi ed partialdi erential equation having additional terms that can be identi ed with the physicsof dissipation in the rst case and dispersion in the second. There is nothing wrong,of course, with identifying an error with a physical process, nor with deliberatelydirecting an error to a speci c physical process, as long as the error remains in someengineering sense small". It is safe to say, for example, that most numerical methodsin practical use for solving the nondissipative Euler equations create a modi ed partialdi erential equation that produces some form of dissipation. However, if used andinterpreted properly, these methods give very useful information. Regardless of what the numerical errors are called, if their e ects are not thor-oughly understood and controlled, they can lead to serious di culties, producinganswers that represent little, if any, physical reality. This motivates studying theconcepts of stability, convergence, and consistency. On the other hand, even if theerrors are kept small enough that they can be neglected (for engineering purposes),the resulting simulation can still be of little practical use if ine cient or inappropriatealgorithms are used. This motivates studying the concepts of sti ness, factorization,and algorithm development in general. All of these concepts we hope to clarify inthis book.1.2 BackgroundThe eld of computational uid dynamics has a broad range of applicability. Indepen-dent of the speci c application under study, the following sequence of steps generallymust be followed in order to obtain a satisfactory solution.1.2.1 Problem Speci cation and Geometry PreparationThe rst step involves the speci cation of the problem, including the geometry, owconditions, and the requirements of the simulation. The geometry may result from
  11. 11. 1.2. BACKGROUND 3measurements of an existing con guration or may be associated with a design study.Alternatively, in a design context, no geometry need be supplied. Instead, a setof objectives and constraints must be speci ed. Flow conditions might include, forexample, the Reynolds number and Mach number for the ow over an airfoil. Therequirements of the simulation include issues such as the level of accuracy needed, theturnaround time required, and the solution parameters of interest. The rst two ofthese requirements are often in con ict and compromise is necessary. As an exampleof solution parameters of interest in computing the ow eld about an airfoil, one maybe interested in i) the lift and pitching moment only, ii) the drag as well as the liftand pitching moment, or iii) the details of the ow at some speci c location.1.2.2 Selection of Governing Equations and Boundary Con- ditionsOnce the problem has been speci ed, an appropriate set of governing equations andboundary conditions must be selected. It is generally accepted that the phenomena ofimportance to the eld of continuum uid dynamics are governed by the conservationof mass, momentum, and energy. The partial di erential equations resulting fromthese conservation laws are referred to as the Navier-Stokes equations. However, inthe interest of e ciency, it is always prudent to consider solving simpli ed formsof the Navier-Stokes equations when the simpli cations retain the physics which areessential to the goals of the simulation. Possible simpli ed governing equations includethe potential- ow equations, the Euler equations, and the thin-layer Navier-Stokesequations. These may be steady or unsteady and compressible or incompressible.Boundary types which may be encountered include solid walls, in ow and out owboundaries, periodic boundaries, symmetry boundaries, etc. The boundary conditionswhich must be speci ed depend upon the governing equations. For example, at a solidwall, the Euler equations require ow tangency to be enforced, while the Navier-Stokesequations require the no-slip condition. If necessary, physical models must be chosenfor processes which cannot be simulated within the speci ed constraints. Turbulenceis an example of a physical process which is rarely simulated in a practical context (atthe time of writing) and thus is often modelled. The success of a simulation dependsgreatly on the engineering insight involved in selecting the governing equations andphysical models based on the problem speci cation.1.2.3 Selection of Gridding Strategy and Numerical MethodNext a numerical method and a strategy for dividing the ow domain into cells, orelements, must be selected. We concern ourselves here only with numerical meth-ods requiring such a tessellation of the domain, which is known as a grid, or mesh.
  12. 12. 4 CHAPTER 1. INTRODUCTIONMany di erent gridding strategies exist, including structured, unstructured, hybrid,composite, and overlapping grids. Furthermore, the grid can be altered based onthe solution in an approach known as solution-adaptive gridding. The numericalmethods generally used in CFD can be classi ed as nite-di erence, nite-volume, nite-element, or spectral methods. The choices of a numerical method and a grid-ding strategy are strongly interdependent. For example, the use of nite-di erencemethods is typically restricted to structured grids. Here again, the success of a sim-ulation can depend on appropriate choices for the problem or class of problems ofinterest.1.2.4 Assessment and Interpretation of ResultsFinally, the results of the simulation must be assessed and interpreted. This step canrequire post-processing of the data, for example calculation of forces and moments,and can be aided by sophisticated ow visualization tools and error estimation tech-niques. It is critical that the magnitude of both numerical and physical-model errorsbe well understood.1.3 OverviewIt should be clear that successful simulation of uid ows can involve a wide range ofissues from grid generation to turbulence modelling to the applicability of various sim-pli ed forms of the Navier-Stokes equations. Many of these issues are not addressedin this book. Some of them are presented in the books by Anderson, Tannehill, andPletcher 1] and Hirsch 2]. Instead we focus on numerical methods, with emphasison nite-di erence and nite-volume methods for the Euler and Navier-Stokes equa-tions. Rather than presenting the details of the most advanced methods, which arestill evolving, we present a foundation for developing, analyzing, and understandingsuch methods. Fortunately, to develop, analyze, and understand most numerical methods used to nd solutions for the complete compressible Navier-Stokes equations, we can make useof much simpler expressions, the so-called model" equations. These model equationsisolate certain aspects of the physics contained in the complete set of equations. Hencetheir numerical solution can illustrate the properties of a given numerical methodwhen applied to a more complicated system of equations which governs similar phys-ical phenomena. Although the model equations are extremely simple and easy tosolve, they have been carefully selected to be representative, when used intelligently,of di culties and complexities that arise in realistic two- and three-dimensional uid ow simulations. We believe that a thorough understanding of what happens when
  13. 13. 1.4. NOTATION 5numerical approximations are applied to the model equations is a major rst step inmaking con dent and competent use of numerical approximations to the Euler andNavier-Stokes equations. As a word of caution, however, it should be noted that,although we can learn a great deal by studying numerical methods as applied to themodel equations and can use that information in the design and application of nu-merical methods to practical problems, there are many aspects of practical problemswhich can only be understood in the context of the complete physical systems.1.4 NotationThe notation is generally explained as it is introduced. Bold type is reserved for realphysical vectors, such as velocity. The vector symbol ~ is used for the vectors (orcolumn matrices) which contain the values of the dependent variable at the nodesof a grid. Otherwise, the use of a vector consisting of a collection of scalars shouldbe apparent from the context and is not identi ed by any special notation. Forexample, the variable u can denote a scalar Cartesian velocity component in the Eulerand Navier-Stokes equations, a scalar quantity in the linear convection and di usionequations, and a vector consisting of a collection of scalars in our presentation ofhyperbolic systems. Some of the abbreviations used throughout the text are listedand de ned below. PDE Partial di erential equation ODE Ordinary di erential equation O E Ordinary di erence equation RHS Right-hand side P.S. Particular solution of an ODE or system of ODEs S.S. Fixed (time-invariant) steady-state solution k-D k-dimensional space ~ bc Boundary conditions, usually a vector O( ) A term of order (i.e., proportional to)
  14. 14. 6 CHAPTER 1. INTRODUCTION
  15. 15. Chapter 2CONSERVATION LAWS ANDTHE MODEL EQUATIONSWe start out by casting our equations in the most general form, the integral conserva-tion-law form, which is useful in understanding the concepts involved in nite-volumeschemes. The equations are then recast into divergence form, which is natural for nite-di erence schemes. The Euler and Navier-Stokes equations are brie y discussedin this Chapter. The main focus, though, will be on representative model equations,in particular, the convection and di usion equations. These equations contain manyof the salient mathematical and physical features of the full Navier-Stokes equations.The concepts of convection and di usion are prevalent in our development of nu-merical methods for computational uid dynamics, and the recurring use of thesemodel equations allows us to develop a consistent framework of analysis for consis-tency, accuracy, stability, and convergence. The model equations we study have twoproperties in common. They are linear partial di erential equations (PDEs) withcoe cients that are constant in both space and time, and they represent phenomenaof importance to the analysis of certain aspects of uid dynamic problems.2.1 Conservation LawsConservation laws, such as the Euler and Navier-Stokes equations and our modelequations, can be written in the following integral form: Z Z Z t2 I Z t2 Z QdV ; QdV + n:FdSdt = PdV dt (2.1) V (t2 ) V (t1 ) t1 S (t) t1 V (t)In this equation, Q is a vector containing the set of variables which are conserved,e.g., mass, momentum, and energy, per unit volume. The equation is a statement of 7
  16. 16. 8 CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONSthe conservation of these quantities in a nite region of space with volume V (t) andsurface area S (t) over a nite interval of time t2 ; t1 . In two dimensions, the regionof space, or cell, is an area A(t) bounded by a closed contour C (t). The vector n isa unit vector normal to the surface pointing outward, F is a set of vectors, or tensor,containing the ux of Q per unit area per unit time, and P is the rate of productionof Q per unit volume per unit time. If all variables are continuous in time, then Eq.2.1 can be rewritten as d Z QdV + I n:FdS = Z PdV (2.2) dt V (t) S (t) V (t)Those methods which make various numerical approximations of the integrals in Eqs.2.1 and 2.2 and nd a solution for Q on that basis are referred to as nite-volumemethods. Many of the advanced codes written for CFD applications are based on the nite-volume concept. On the other hand, a partial derivative form of a conservation law can also bederived. The divergence form of Eq. 2.2 is obtained by applying Gausss theorem tothe ux integral, leading to @Q + r:F = P (2.3) @twhere r: is the well-known divergence operator given, in Cartesian coordinates, by ! @ +j @ +k @ : r: i @x @y @z (2.4)and i j, and k are unit vectors in the x y, and z coordinate directions, respectively.Those methods which make various approximations of the derivatives in Eq. 2.3 and nd a solution for Q on that basis are referred to as nite-di erence methods.2.2 The Navier-Stokes and Euler EquationsThe Navier-Stokes equations form a coupled system of nonlinear PDEs describingthe conservation of mass, momentum and energy for a uid. For a Newtonian uidin one dimension, they can be written as @Q + @E = 0 (2.5) @t @xwith 2 3 2 u 3 2 0 3 6 6 7 7 6 6 7 6 7 6 7 7 Q=6 6 u7 7 E =6 6 u 2+p 7;6 7 6 4 @u 7 7 (2.6) 6 7 6 7 6 3 @x 7 4 5 4 5 4 5 e u(e + p) 4 3 u @u + @x @T @x
  17. 17. 2.2. THE NAVIER-STOKES AND EULER EQUATIONS 9where is the uid density, u is the velocity, e is the total energy per unit volume, p isthe pressure, T is the temperature, is the coe cient of viscosity, and is the thermalconductivity. The total energy e includes internal energy per unit volume (where is the internal energy per unit mass) and kinetic energy per unit volume u2=2.These equations must be supplemented by relations between and and the uidstate as well as an equation of state, such as the ideal gas law. Details can be foundin Anderson, Tannehill, and Pletcher 1] and Hirsch 2]. Note that the convective uxes lead to rst derivatives in space, while the viscous and heat conduction termsinvolve second derivatives. This form of the equations is called conservation-law orconservative form. Non-conservative forms can be obtained by expanding derivativesof products using the product rule or by introducing di erent dependent variables,such as u and p. Although non-conservative forms of the equations are analyticallythe same as the above form, they can lead to quite di erent numerical solutions interms of shock strength and shock speed, for example. Thus the conservative form isappropriate for solving ows with features such as shock waves. Many ows of engineering interest are steady (time-invariant), or at least may betreated as such. For such ows, we are often interested in the steady-state solution ofthe Navier-Stokes equations, with no interest in the transient portion of the solution.The steady solution to the one-dimensional Navier-Stokes equations must satisfy @E = 0 (2.7) @x If we neglect viscosity and heat conduction, the Euler equations are obtained. Intwo-dimensional Cartesian coordinates, these can be written as @Q + @E + @F = 0 (2.8) @t @x @ywith 2 3 2 3 2 q1 u 3 2 v 3 Q = 6 q2 7 = 6 u 7 E = 6 u uv p 7 F = 6 v2uv p 7 6 7 6 7 6 2+ 7 6 7 6 7 6 7 4 q3 5 4 v 5 6 4 7 5 6 4 + 7 5 (2.9) q4 e u(e + p) v(e + p)where u and v are the Cartesian velocity components. Later on we will make use ofthe following form of the Euler equations as well: @Q + A @Q + B @Q = 0 (2.10) @t @x @y @E @FThe matrices A = @Q and B = @Q are known as the ux Jacobians. The ux vectorsgiven above are written in terms of the primitive variables, , u, v, and p. In order
  18. 18. 10 CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONSto derive the ux Jacobian matrices, we must rst write the ux vectors E and F interms of the conservative variables, q1 , q2, q3 , and q4 , as follows: 2 3 2 E1 3 6 q2 7 6 6 7 6 7 6 7 7 6 7 6 2 7 3; q2 ; ;1 q3 7 6 6 E2 7 6 7 6 ( ; 1)q4 + 2 q1 2 q1 7 2 E =6 6 7=6 7 6 7 7 (2.11) 6 7 6 7 6 q3 q2 7 7 6 6 E3 7 6 q1 7 6 7 6 7 4 5 6 6 7 7 E4 4 q4 q2 ; ;1 q2 + q3 2 3 2q 5 q1 2 q1 2 q1 2 2 3 2 q3 3 6 F1 7 6 7 6 6 7 6 7 6 6 7 7 7 q3 q2 6 6 F2 7 6 7 6 q1 7 7 F =6 6 6 7=6 7 6 7 7 (2.12) 6 6 F3 7 6 7 6 7 6 ( ; 1)q4 + 3; 2 q3 2 q1 ; 2 7 ;1 q2 7 2 q1 7 6 7 6 7 4 5 6 4 7 5 F4 q4 q 3 ; ;1 q2 2 3 2q + q3 3 q1 2 q1 q1 2We have assumed that the pressure satis es p = ( ; 1) e ; (u2 + v2)=2] from theideal gas law, where is the ratio of speci c heats, cp=cv . From this it follows thatthe ux Jacobian of E can be written in terms of the conservative variables as 2 3 6 0 1 0 0 7 6 6 7 7 6 a21 q (3 ; ) q2 (1 ; ) q3 ;1 7 6 @Ei = 6 1 q1 7 7 A= 6 7 (2.13) @qj 6 6 ; q2 q1 q q q2 0 7 7 6 6 6 q1 3 q3 1 q1 7 7 7 6 7 4 q2 5 a41 a42 a43 q1where ! !2 a21 = ; 1 q3 2 ; 3 ; q2 2 q1 2 q1
  19. 19. 2.2. THE NAVIER-STOKES AND EULER EQUATIONS 11 2 !3 !2 !3 ! ! a41 = ( ; 1)4 q2 + q3 q2 5; q4 q2 q1 q1 q1 q1 q1 ! 2 !2 !2 3 a42 = q4 ; ; 1 43 q2 + q3 5 q1 2 q1 q1 ! ! a43 = ;( ; 1) q q q2 q3 (2.14) 1 1and in terms of the primitive variables as 2 3 6 0 1 0 0 7 6 6 7 7 6 6 a21 (3 ; )u (1 ; )v ( ; 1) 7 7 A=6 6 6 7 7 7 (2.15) 6 6 ;uv v u 0 7 7 6 7 4 5 a41 a42 a43 uwhere a21 = ; 1 v2 ; 3 ; u2 2 2 a41 = ( ; 1)u(u2 + v2) ; ue a42 = e ; ; 1 (3u2 + v2 ) 2 a43 = (1 ; )uv (2.16)Derivation of the two forms of B = @F=@Q is similar. The eigenvalues of the uxJacobian matrices are purely real. This is the de ning feature of hyperbolic systemsof PDEs, which are further discussed in Section 2.5. The homogeneous property ofthe Euler equations is discussed in Appendix C. The Navier-Stokes equations include both convective and di usive uxes. Thismotivates the choice of our two scalar model equations associated with the physicsof convection and di usion. Furthermore, aspects of convective phenomena associ-ated with coupled systems of equations such as the Euler equations are important indeveloping numerical methods and boundary conditions. Thus we also study linearhyperbolic systems of PDEs.
  20. 20. 12 CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS2.3 The Linear Convection Equation2.3.1 Di erential FormThe simplest linear model for convection and wave propagation is the linear convectionequation given by the following PDE: @u + a @u = 0 (2.17) @t @xHere u(x t) is a scalar quantity propagating with speed a, a real constant which maybe positive or negative. The manner in which the boundary conditions are speci edseparates the following two phenomena for which this equation is a model: (1) In one type, the scalar quantity u is given on one boundary, corresponding to a wave entering the domain through this in ow" boundary. No bound- ary condition is speci ed at the opposite side, the out ow" boundary. This is consistent in terms of the well-posedness of a 1st-order PDE. Hence the wave leaves the domain through the out ow boundary without distortion or re ection. This type of phenomenon is referred to, simply, as the convection problem. It represents most of the usual" situations encountered in convect- ing systems. Note that the left-hand boundary is the in ow boundary when a is positive, while the right-hand boundary is the in ow boundary when a is negative. (2) In the other type, the ow being simulated is periodic. At any given time, what enters on one side of the domain must be the same as that which is leaving on the other. This is referred to as the biconvection problem. It is the simplest to study and serves to illustrate many of the basic properties of numerical methods applied to problems involving convection, without special consideration of boundaries. Hence, we pay a great deal of attention to it in the initial chapters. Now let us consider a situation in which the initial condition is given by u(x 0) =u0(x), and the domain is in nite. It is easy to show by substitution that the exactsolution to the linear convection equation is then u(x t) = u0(x ; at) (2.18)The initial waveform propagates unaltered with speed jaj to the right if a is positiveand to the left if a is negative. With periodic boundary conditions, the waveformtravels through one boundary and reappears at the other boundary, eventually re-turning to its initial position. In this case, the process continues forever without any
  21. 21. 2.3. THE LINEAR CONVECTION EQUATION 13change in the shape of the solution. Preserving the shape of the initial conditionu0(x) can be a di cult challenge for a numerical method.2.3.2 Solution in Wave SpaceWe now examine the biconvection problem in more detail. Let the domain be givenby 0 x 2 . We restrict our attention to initial conditions in the form u(x 0) = f (0)ei x (2.19)where f (0) is a complex constant, and is the wavenumber. In order to satisfy theperiodic boundary conditions, must be an integer. It is a measure of the number ofwavelengths within the domain. With such an initial condition, the solution can bewritten as u(x t) = f (t)ei x (2.20)where the time dependence is contained in the complex function f (t). Substitutingthis solution into the linear convection equation, Eq. 2.17, we nd that f (t) satis esthe following ordinary di erential equation (ODE) df = ;ia f (2.21) dtwhich has the solution f (t) = f (0)e;ia t (2.22)Substituting f (t) into Eq. 2.20 gives the following solution u(x t) = f (0)ei (x;at) = f (0)ei( x;!t) (2.23)where the frequency, !, the wavenumber, , and the phase speed, a, are related by != a (2.24) The relation between the frequency and the wavenumber is known as the dispersionrelation. The linear relation given by Eq. 2.24 is characteristic of wave propagationin a nondispersive medium. This means that the phase speed is the same for allwavenumbers. As we shall see later, most numerical methods introduce some disper-sion that is, in a simulation, waves with di erent wavenumbers travel at di erentspeeds. An arbitrary initial waveform can be produced by summing initial conditions ofthe form of Eq. 2.19. For M modes, one obtains M X u(x 0) = fm (0)ei mx (2.25) m=1
  22. 22. 14 CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONSwhere the wavenumbers are often ordered such that 1 2 M . Since thewave equation is linear, the solution is obtained by summing solutions of the form ofEq. 2.23, giving M X u(x t) = fm (0)ei m(x;at) (2.26) m=1Dispersion and dissipation resulting from a numerical approximation will cause theshape of the solution to change from that of the original waveform.2.4 The Di usion Equation2.4.1 Di erential FormDi usive uxes are associated with molecular motion in a continuum uid. A simplelinear model equation for a di usive process is @u = @ 2 u (2.27) @t @x2where is a positive real constant. For example, with u representing the tempera-ture, this parabolic PDE governs the di usion of heat in one dimension. Boundaryconditions can be periodic, Dirichlet (speci ed u), Neumann (speci ed @u=@x), ormixed Dirichlet/Neumann. In contrast to the linear convection equation, the di usion equation has a nontrivialsteady-state solution, which is one that satis es the governing PDE with the partialderivative in time equal to zero. In the case of Eq. 2.27, the steady-state solutionmust satisfy @2u = 0 (2.28) @x2Therefore, u must vary linearly with x at steady state such that the boundary con-ditions are satis ed. Other steady-state solutions are obtained if a source term g(x)is added to Eq. 2.27, as follows: " 2 # @u = @u ; g(x) (2.29) @t @x2giving a steady state-solution which satis es @ 2 u ; g(x) = 0 (2.30) @x2
  23. 23. 2.4. THE DIFFUSION EQUATION 15 In two dimensions, the di usion equation becomes " # @u = @ 2 u + @ 2 u ; g(x y) (2.31) @t @x2 @y2where g(x y) is again a source term. The corresponding steady equation is @ 2 u + @ 2 u ; g(x y) = 0 (2.32) @x2 @y2While Eq. 2.31 is parabolic, Eq. 2.32 is elliptic. The latter is known as the Poissonequation for nonzero g, and as Laplaces equation for zero g.2.4.2 Solution in Wave SpaceWe now consider a series solution to Eq. 2.27. Let the domain be given by 0 xwith boundary conditions u(0) = ua, u( ) = ub. It is clear that the steady-statesolution is given by a linear function which satis es the boundary conditions, i.e.,h(x) = ua + (ub ; ua)x= . Let the initial condition be M X u(x 0) = fm(0) sin m x + h(x) (2.33) m=1where must be an integer in order to satisfy the boundary conditions. A solutionof the form M X u(x t) = fm(t) sin mx + h(x) (2.34) m=1satis es the initial and boundary conditions. Substituting this form into Eq. 2.27gives the following ODE for fm : dfm = ; 2 f (2.35) dt m mand we nd fm(t) = fm(0)e; 2 t m (2.36)Substituting fm(t) into equation 2.34, we obtain M X u(x t) = fm (0)e; 2 t sin m x + h(x) m (2.37) m=1The steady-state solution (t ! 1) is simply h(x). Eq. 2.37 shows that high wavenum-ber components (large m) of the solution decay more rapidly than low wavenumbercomponents, consistent with the physics of di usion.
  24. 24. 16 CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS2.5 Linear Hyperbolic SystemsThe Euler equations, Eq. 2.8, form a hyperbolic system of partial di erential equa-tions. Other systems of equations governing convection and wave propagation phe-nomena, such as the Maxwell equations describing the propagation of electromagneticwaves, are also of hyperbolic type. Many aspects of numerical methods for such sys-tems can be understood by studying a one-dimensional constant-coe cient linearsystem of the form @u + A @u = 0 (2.38) @t @xwhere u = u(x t) is a vector of length m and A is a real m m matrix. Forconservation laws, this equation can also be written in the form @u + @f = 0 (2.39) @t @x @fwhere f is the ux vector and A = @u is the ux Jacobian matrix. The entries in the ux Jacobian are @f aij = @ui (2.40) jThe ux Jacobian for the Euler equations is derived in Section 2.2. Such a system is hyperbolic if A is diagonalizable with real eigenvalues.1 Thus = X ;1AX (2.41)where is a diagonal matrix containing the eigenvalues of A, and X is the matrixof right eigenvectors. Premultiplying Eq. 2.38 by X ;1 , postmultiplying A by theproduct XX ;1, and noting that X and X ;1 are constants, we obtain z }| { @X ;1u @ X ;1AX X ;1u @t + @x =0 (2.42)With w = X ;1u, this can be rewritten as @w + @w = 0 (2.43) @t @xWhen written in this manner, the equations have been decoupled into m scalar equa-tions of the form @wi + @wi = 0 (2.44) i @t @x 1 See Appendix A for a brief review of some basic relations and de nitions from linear algebra.
  25. 25. 2.6. PROBLEMS 17The elements of w are known as characteristic variables. Each characteristic variablesatis es the linear convection equation with the speed given by the correspondingeigenvalue of A. Based on the above, we see that a hyperbolic system in the form of Eq. 2.38 has asolution given by the superposition of waves which can travel in either the positive ornegative directions and at varying speeds. While the scalar linear convection equationis clearly an excellent model equation for hyperbolic systems, we must ensure thatour numerical methods are appropriate for wave speeds of arbitrary sign and possiblywidely varying magnitudes. The one-dimensional Euler equations can also be diagonalized, leading to threeequations in the form of the linear convection equation, although they remain non-linear, of course. The eigenvalues of the ux Jacobian matrix, or wave speeds, are qu u + c, and u ; c, where u is the local uid velocity, and c = p= is the localspeed of sound. The speed u is associated with convection of the uid, while u + cand u ; c are associated with sound waves. Therefore, in a supersonic ow, wherejuj > c, all of the wave speeds have the same sign. In a subsonic ow, where juj < c,wave speeds of both positive and negative sign are present, corresponding to the factthat sound waves can travel upstream in a subsonic ow. The signs of the eigenvalues of the matrix A are also important in determiningsuitable boundary conditions. The characteristic variables each satisfy the linear con-vection equation with the wave speed given by the corresponding eigenvalue. There-fore, the boundary conditions can be speci ed accordingly. That is, characteristicvariables associated with positive eigenvalues can be speci ed at the left boundary,which corresponds to in ow for these variables. Characteristic variables associatedwith negative eigenvalues can be speci ed at the right boundary, which is the in- ow boundary for these variables. While other boundary condition treatments arepossible, they must be consistent with this approach.2.6 Problems 1. Show that the 1-D Euler equations can be written in terms of the primitive variables R = u p]T as follows: @R + M @R = 0 @t @x where 2 3 u 0 M = 6 0 u ;1 7 4 5 0 p u
  26. 26. 18 CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS Assume an ideal gas, p = ( ; 1)(e ; u2=2). 2. Find the eigenvalues and eigenvectors of the matrix M derived in question 1. 3. Derive the ux Jacobian matrix A = @E=@Q for the 1-D Euler equations result- ing from the conservative variable formulation (Eq. 2.5). Find its eigenvalues and compare with those obtained in question 2. 4. Show that the two matrices M and A derived in questions 1 and 3, respectively, are related by a similarity transform. (Hint: make use of the matrix S = @Q=@R.) 5. Write the 2-D di usion equation, Eq. 2.31, in the form of Eq. 2.2. 6. Given the initial condition u(x 0) = sin x de ned on 0 x 2 , write it in the form of Eq. 2.25, that is, nd the necessary values of fm (0). (Hint: use M = 2 with 1 = 1 and 2 = ;1.) Next consider the same initial condition de ned only at x = 2 j=4, j = 0 1 2 3. Find the values of fm(0) required to reproduce the initial condition at these discrete points using M = 4 with m = m ; 1. 7. Plot the rst three basis functions used in constructing the exact solution to the di usion equation in Section 2.4.2. Next consider a solution with boundary conditions ua = ub = 0, and initial conditions from Eq. 2.33 with fm(0) = 1 for 1 m 3, fm (0) = 0 for m > 3. Plot the initial condition on the domain 0 x . Plot the solution at t = 1 with = 1. 8. Write the classical wave equation @ 2 u=@t2 = c2 @ 2 u=@x2 as a rst-order system, i.e., in the form @U + A @U = 0 @t @x where U = @u=@x @u=@t]T . Find the eigenvalues and eigenvectors of A. 9. The Cauchy-Riemann equations are formed from the coupling of the steady compressible continuity (conservation of mass) equation @ u+@ v =0 @x @y and the vorticity de nition ! = ; @x + @u = 0 @v @y
  27. 27. 2.6. PROBLEMS 19 where ! = 0 for irrotational ow. For isentropic and homenthalpic ow, the system is closed by the relation = 1 ; ; 1 u2 + v2 ; 1 1 ;1 2 Note that the variables have been nondimensionalized. Combining the two PDEs, we have @f (q) + @g(q) = 0 @x @y where q= u v f = ;v u g = ;u ; v One approach to solving these equations is to add a time-dependent term and nd the steady solution of the following equation: @q + @f + @g = 0 @t @x @y (a) Find the ux Jacobians of f and g with respect to q. (b) Determine the eigenvalues of the ux Jacobians. (c) Determine the conditions (in terms of and u) under which the system is hyperbolic, i.e., has real eigenvalues. (d) Are the above uxes homogeneous? (See Appendix C.)
  28. 28. 20 CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS
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