Fundamentals of computational fluid dynamics

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Fundamentals of computational fluid dynamics

  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. Contents 1 INTRODUCTION 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Problem Speci cation and Geometry Preparation . . . . . . 1.2.2 Selection of Governing Equations and Boundary Conditions 1.2.3 Selection of Gridding Strategy and Numerical Method . . . 1.2.4 Assessment and Interpretation of Results . . . . . . . . . . . 1.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 CONSERVATION LAWS AND THE MODEL EQUATIONS 2.1 Conservation Laws . . . . . . . . . . . . 2.2 The Navier-Stokes and Euler Equations . 2.3 The Linear Convection Equation . . . . 2.3.1 Di erential Form . . . . . . . . . 2.3.2 Solution in Wave Space . . . . . . 2.4 The Di usion Equation . . . . . . . . . . 2.4.1 Di erential Form . . . . . . . . . 2.4.2 Solution in Wave Space . . . . . . 2.5 Linear Hyperbolic Systems . . . . . . . . 2.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 FINITE-DIFFERENCE APPROXIMATIONS 3.1 Meshes and Finite-Di erence Notation 3.2 Space Derivative Approximations . . . 3.3 Finite-Di erence Operators . . . . . . 3.3.1 Point Di erence Operators . . . 3.3.2 Matrix Di erence Operators . . 3.3.3 Periodic Matrices . . . . . . . . 3.3.4 Circulant Matrices . . . . . . . iii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2 2 3 3 4 4 5 7 7 8 12 12 13 14 14 15 16 17 21 21 24 25 25 25 29 30
  3. 3. 3.4 Constructing Di erencing Schemes of Any Order . . . . . 3.4.1 Taylor Tables . . . . . . . . . . . . . . . . . . . . 3.4.2 Generalization of Di erence Formulas . . . . . . . 3.4.3 Lagrange and Hermite Interpolation Polynomials 3.4.4 Practical Application of Pade Formulas . . . . . . 3.4.5 Other Higher-Order Schemes . . . . . . . . . . . . 3.5 Fourier Error Analysis . . . . . . . . . . . . . . . . . . . 3.5.1 Application to a Spatial Operator . . . . . . . . . 3.6 Di erence Operators at Boundaries . . . . . . . . . . . . 3.6.1 The Linear Convection Equation . . . . . . . . . 3.6.2 The Di usion Equation . . . . . . . . . . . . . . . 3.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 31 34 35 37 38 39 39 43 44 46 47 4 THE SEMI-DISCRETE APPROACH 51 5 FINITE-VOLUME METHODS 71 4.1 Reduction of PDE's to ODE's . . . . . . . . . . . . . . . . . . . . . . 4.1.1 The Model ODE's . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 The Generic Matrix Form . . . . . . . . . . . . . . . . . . . . 4.2 Exact Solutions of Linear ODE's . . . . . . . . . . . . . . . . . . . . 4.2.1 Eigensystems of Semi-Discrete Linear Forms . . . . . . . . . . 4.2.2 Single ODE's of First- and Second-Order . . . . . . . . . . . . 4.2.3 Coupled First-Order ODE's . . . . . . . . . . . . . . . . . . . 4.2.4 General Solution of Coupled ODE's with Complete Eigensystems 4.3 Real Space and Eigenspace . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 De nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Eigenvalue Spectrums for Model ODE's . . . . . . . . . . . . . 4.3.3 Eigenvectors of the Model Equations . . . . . . . . . . . . . . 4.3.4 Solutions of the Model ODE's . . . . . . . . . . . . . . . . . . 4.4 The Representative Equation . . . . . . . . . . . . . . . . . . . . . . 4.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Model Equations in Integral Form . . . . . . . . . . . . . . . . . . . 5.2.1 The Linear Convection Equation . . . . . . . . . . . . . . . 5.2.2 The Di usion Equation . . . . . . . . . . . . . . . . . . . . . 5.3 One-Dimensional Examples . . . . . . . . . . . . . . . . . . . . . . 5.3.1 A Second-Order Approximation to the Convection Equation 5.3.2 A Fourth-Order Approximation to the Convection Equation 5.3.3 A Second-Order Approximation to the Di usion Equation . 5.4 A Two-Dimensional Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 52 53 54 54 55 57 59 61 61 62 63 65 67 68 72 73 73 74 74 75 77 78 80
  4. 4. 5.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 TIME-MARCHING METHODS FOR ODE'S 6.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Converting Time-Marching Methods to O E's . . . . . . . . . . . . 6.3 Solution of Linear O E's With Constant Coe cients . . . . . . . . 6.3.1 First- and Second-Order Di erence Equations . . . . . . . . 6.3.2 Special Cases of Coupled First-Order Equations . . . . . . . 6.4 Solution of the Representative O E's . . . . . . . . . . . . . . . . 6.4.1 The Operational Form and its Solution . . . . . . . . . . . . 6.4.2 Examples of Solutions to Time-Marching O E's . . . . . . . 6.5 The ; Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Establishing the Relation . . . . . . . . . . . . . . . . . . . . 6.5.2 The Principal -Root . . . . . . . . . . . . . . . . . . . . . . 6.5.3 Spurious -Roots . . . . . . . . . . . . . . . . . . . . . . . . 6.5.4 One-Root Time-Marching Methods . . . . . . . . . . . . . . 6.6 Accuracy Measures of Time-Marching Methods . . . . . . . . . . . 6.6.1 Local and Global Error Measures . . . . . . . . . . . . . . . 6.6.2 Local Accuracy of the Transient Solution (er j j er! ) . . . 6.6.3 Local Accuracy of the Particular Solution (er ) . . . . . . . 6.6.4 Time Accuracy For Nonlinear Applications . . . . . . . . . . 6.6.5 Global Accuracy . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Linear Multistep Methods . . . . . . . . . . . . . . . . . . . . . . . 6.7.1 The General Formulation . . . . . . . . . . . . . . . . . . . . 6.7.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.3 Two-Step Linear Multistep Methods . . . . . . . . . . . . . 6.8 Predictor-Corrector Methods . . . . . . . . . . . . . . . . . . . . . . 6.9 Runge-Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Implementation of Implicit Methods . . . . . . . . . . . . . . . . . . 6.10.1 Application to Systems of Equations . . . . . . . . . . . . . 6.10.2 Application to Nonlinear Equations . . . . . . . . . . . . . . 6.10.3 Local Linearization for Scalar Equations . . . . . . . . . . . 6.10.4 Local Linearization for Coupled Sets of Nonlinear Equations 6.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 STABILITY OF LINEAR SYSTEMS 7.1 Dependence on the Eigensystem 7.2 Inherent Stability of ODE's . . 7.2.1 The Criterion . . . . . . 7.2.2 Complete Eigensystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 85 86 87 88 89 90 91 91 92 93 93 95 95 96 97 97 98 99 100 101 102 102 103 105 106 107 110 110 111 112 115 117 121 122 123 123 123
  5. 5. 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.2.3 Defective Eigensystems . . . . . . . . . . . . . . Numerical Stability of O E 's . . . . . . . . . . . . . . 7.3.1 The Criterion . . . . . . . . . . . . . . . . . . . 7.3.2 Complete Eigensystems . . . . . . . . . . . . . . 7.3.3 Defective Eigensystems . . . . . . . . . . . . . . Time-Space Stability and Convergence of O E's . . . . Numerical Stability Concepts in the Complex -Plane . 7.5.1 -Root Traces Relative to the Unit Circle . . . 7.5.2 Stability for Small t . . . . . . . . . . . . . . . Numerical Stability Concepts in the Complex h Plane 7.6.1 Stability for Large h. . . . . . . . . . . . . . . . 7.6.2 Unconditional Stability, A-Stable Methods . . . 7.6.3 Stability Contours in the Complex h Plane. . . Fourier Stability Analysis . . . . . . . . . . . . . . . . 7.7.1 The Basic Procedure . . . . . . . . . . . . . . . 7.7.2 Some Examples . . . . . . . . . . . . . . . . . . 7.7.3 Relation to Circulant Matrices . . . . . . . . . . Consistency . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 8 CHOICE OF TIME-MARCHING METHODS 8.1 Sti ness De nition for ODE's . . . . . . . . . . . . 8.1.1 Relation to -Eigenvalues . . . . . . . . . . 8.1.2 Driving and Parasitic Eigenvalues . . . . . . 8.1.3 Sti ness Classi cations . . . . . . . . . . . . 8.2 Relation of Sti ness to Space Mesh Size . . . . . . 8.3 Practical Considerations for Comparing Methods . 8.4 Comparing the E ciency of Explicit Methods . . . 8.4.1 Imposed Constraints . . . . . . . . . . . . . 8.4.2 An Example Involving Di usion . . . . . . . 8.4.3 An Example Involving Periodic Convection . 8.5 Coping With Sti ness . . . . . . . . . . . . . . . . 8.5.1 Explicit Methods . . . . . . . . . . . . . . . 8.5.2 Implicit Methods . . . . . . . . . . . . . . . 8.5.3 A Perspective . . . . . . . . . . . . . . . . . 8.6 Steady Problems . . . . . . . . . . . . . . . . . . . 8.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 124 124 125 125 125 128 128 132 135 135 136 137 141 141 142 143 143 146 149 149 149 151 151 152 153 154 154 154 155 158 158 159 160 160 161
  6. 6. 9 RELAXATION METHODS 9.1 Formulation of the Model Problem . . . . . . . . . . . . . . 9.1.1 Preconditioning the Basic Matrix . . . . . . . . . . . 9.1.2 The Model Equations . . . . . . . . . . . . . . . . . . 9.2 Classical Relaxation . . . . . . . . . . . . . . . . . . . . . . 9.2.1 The Delta Form of an Iterative Scheme . . . . . . . . 9.2.2 The Converged Solution, the Residual, and the Error 9.2.3 The Classical Methods . . . . . . . . . . . . . . . . . 9.3 The ODE Approach to Classical Relaxation . . . . . . . . . 9.3.1 The Ordinary Di erential Equation Formulation . . . 9.3.2 ODE Form of the Classical Methods . . . . . . . . . 9.4 Eigensystems of the Classical Methods . . . . . . . . . . . . 9.4.1 The Point-Jacobi System . . . . . . . . . . . . . . . . 9.4.2 The Gauss-Seidel System . . . . . . . . . . . . . . . . 9.4.3 The SOR System . . . . . . . . . . . . . . . . . . . . 9.5 Nonstationary Processes . . . . . . . . . . . . . . . . . . . . 9.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 164 164 166 168 168 168 169 170 170 172 173 174 176 180 182 187 10 MULTIGRID 191 11 NUMERICAL DISSIPATION 203 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 11.1 11.2 11.3 11.4 One-Sided First-Derivative Space Di erencing The Modi ed Partial Di erential Equation . . The Lax-Wendro Method . . . . . . . . . . . Upwind Schemes . . . . . . . . . . . . . . . . 11.4.1 Flux-Vector Splitting . . . . . . . . . . 11.4.2 Flux-Di erence Splitting . . . . . . . . 11.5 Arti cial Dissipation . . . . . . . . . . . . . . 11.6 Problems . . . . . . . . . . . . . . . . . . . . . 12 SPLIT AND FACTORED FORMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 205 207 209 210 212 213 214 217 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. 12.4 12.5 12.6 12.7 12.3.1 Mesh Indexing Convention . . . . . . . . . . . 12.3.2 Data Bases and Space Vectors . . . . . . . . . 12.3.3 Data Base Permutations . . . . . . . . . . . . 12.3.4 Space Splitting and Factoring . . . . . . . . . Second-Order Factored Implicit Methods . . . . . . . Importance of Factored Forms in 2 and 3 Dimensions The Delta Form . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 LINEAR ANALYSIS OF SPLIT AND FACTORED FORMS 13.1 The Representative Equation for Circulant Operators . . . . . . . 13.2 Example Analysis of Circulant Systems . . . . . . . . . . . . . . . 13.2.1 Stability Comparisons of Time-Split Methods . . . . . . . 13.2.2 Analysis of a Second-Order Time-Split Method . . . . . . 13.3 The Representative Equation for Space-Split Operators . . . . . . 13.4 Example Analysis of 2-D Model Equations . . . . . . . . . . . . . 13.4.1 The Unfactored Implicit Euler Method . . . . . . . . . . . 13.4.2 The Factored Nondelta Form of the Implicit Euler Method 13.4.3 The Factored Delta Form of the Implicit Euler Method . . 13.4.4 The Factored Delta Form of the Trapezoidal Method . . . 13.5 Example Analysis of the 3-D Model Equation . . . . . . . . . . . 13.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 221 221 223 226 226 228 229 233 233 234 234 237 238 242 242 243 243 244 245 247 A USEFUL RELATIONS AND DEFINITIONS FROM LINEAR ALGEBRA 249 A.1 A.2 A.3 A.4 A.5 Notation . . . . . . . . . . De nitions . . . . . . . . . Algebra . . . . . . . . . . Eigensystems . . . . . . . Vector and Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B SOME PROPERTIES OF TRIDIAGONAL MATRICES B.1 B.2 B.3 B.4 Standard Eigensystem for Simple Tridiagonals . . Generalized Eigensystem for Simple Tridiagonals . The Inverse of a Simple Tridiagonal . . . . . . . . Eigensystems of Circulant Matrices . . . . . . . . B.4.1 Standard Tridiagonals . . . . . . . . . . . B.4.2 General Circulant Systems . . . . . . . . . B.5 Special Cases Found From Symmetries . . . . . . B.6 Special Cases Involving Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 250 251 251 254 257 257 258 259 260 260 261 262 263
  8. 8. C THE HOMOGENEOUS PROPERTY OF THE EULER EQUATIONS265
  9. 9. Chapter 1 INTRODUCTION 1.1 Motivation The material in this book originated from attempts to understand and systemize numerical solution techniques for the partial di erential equations governing the physics of uid ow. As time went on and these attempts began to crystallize, underlying constraints on the nature of the material began to form. The principal such constraint was the demand for uni cation. Was there one mathematical structure which could be used to describe the behavior and results of most numerical methods in common use in the eld of uid dynamics? Perhaps the answer is arguable, but the authors believe the answer is a rmative and present this book as justi cation for that belief. The mathematical structure is the theory of linear algebra and the attendant eigenanalysis of linear systems. The ultimate goal of the eld of computational uid dynamics (CFD) is to understand the physical events that occur in the ow of uids around and within designated objects. These events are related to the action and interaction of phenomena such as dissipation, di usion, convection, shock waves, slip surfaces, boundary layers, and turbulence. In the eld of aerodynamics, all of these phenomena are governed by the compressible Navier-Stokes equations. Many of the most important aspects of these relations are nonlinear and, as a consequence, often have no analytic solution. This, of course, motivates the numerical solution of the associated partial di erential equations. At the same time it would seem to invalidate the use of linear algebra for the classi cation of the numerical methods. Experience has shown that such is not the case. As we shall see in a later chapter, the use of numerical methods to solve partial di erential equations introduces an approximation that, in e ect, can change the form of the basic partial di erential equations themselves. The new equations, which 1
  10. 10. CHAPTER 1. INTRODUCTION 2 are the ones actually being solved by the numerical process, are often referred to as the modi ed partial di erential equations. Since they are not precisely the same as the original equations, they can, and probably will, simulate the physical phenomena listed above in ways that are not exactly the same as an exact solution to the basic partial di erential equation. Mathematically, these di erences are usually referred to as truncation errors. However, the theory associated with the numerical analysis of uid mechanics was developed predominantly by scientists deeply interested in the physics of uid ow and, as a consequence, these errors are often identi ed with a particular physical phenomenon on which they have a strong e ect. Thus methods are said to have a lot of arti cial viscosity" or said to be highly dispersive. This means that the errors caused by the numerical approximation result in a modi ed partial di erential equation having additional terms that can be identi ed with the physics of 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 deliberately directing an error to a speci c physical process, as long as the error remains in some engineering sense small". It is safe to say, for example, that most numerical methods in practical use for solving the nondissipative Euler equations create a modi ed partial di erential equation that produces some form of dissipation. However, if used and interpreted properly, these methods give very useful information. Regardless of what the numerical errors are called, if their e ects are not thoroughly understood and controlled, they can lead to serious di culties, producing answers that represent little, if any, physical reality. This motivates studying the concepts of stability, convergence, and consistency. On the other hand, even if the errors 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 inappropriate algorithms are used. This motivates studying the concepts of sti ness, factorization, and algorithm development in general. All of these concepts we hope to clarify in this book. 1.2 Background The eld of computational uid dynamics has a broad range of applicability. Independent of the speci c application under study, the following sequence of steps generally must be followed in order to obtain a satisfactory solution. 1.2.1 Problem Speci cation and Geometry Preparation The rst step involves the speci cation of the problem, including the geometry, ow conditions, and the requirements of the simulation. The geometry may result from
  11. 11. 1.2. BACKGROUND 3 measurements 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 set of objectives and constraints must be speci ed. Flow conditions might include, for example, the Reynolds number and Mach number for the ow over an airfoil. The requirements of the simulation include issues such as the level of accuracy needed, the turnaround time required, and the solution parameters of interest. The rst two of these requirements are often in con ict and compromise is necessary. As an example of solution parameters of interest in computing the ow eld about an airfoil, one may be interested in i) the lift and pitching moment only, ii) the drag as well as the lift and pitching moment, or iii) the details of the ow at some speci c location. 1.2.2 Selection of Governing Equations and Boundary Conditions Once the problem has been speci ed, an appropriate set of governing equations and boundary conditions must be selected. It is generally accepted that the phenomena of importance to the eld of continuum uid dynamics are governed by the conservation of mass, momentum, and energy. The partial di erential equations resulting from these conservation laws are referred to as the Navier-Stokes equations. However, in the interest of e ciency, it is always prudent to consider solving simpli ed forms of the Navier-Stokes equations when the simpli cations retain the physics which are essential to the goals of the simulation. Possible simpli ed governing equations include the potential- ow equations, the Euler equations, and the thin-layer Navier-Stokes equations. These may be steady or unsteady and compressible or incompressible. Boundary types which may be encountered include solid walls, in ow and out ow boundaries, periodic boundaries, symmetry boundaries, etc. The boundary conditions which must be speci ed depend upon the governing equations. For example, at a solid wall, the Euler equations require ow tangency to be enforced, while the Navier-Stokes equations require the no-slip condition. If necessary, physical models must be chosen for processes which cannot be simulated within the speci ed constraints. Turbulence is an example of a physical process which is rarely simulated in a practical context (at the time of writing) and thus is often modelled. The success of a simulation depends greatly on the engineering insight involved in selecting the governing equations and physical models based on the problem speci cation. 1.2.3 Selection of Gridding Strategy and Numerical Method Next a numerical method and a strategy for dividing the ow domain into cells, or elements, must be selected. We concern ourselves here only with numerical methods requiring such a tessellation of the domain, which is known as a grid, or mesh.
  12. 12. CHAPTER 1. INTRODUCTION 4 Many di erent gridding strategies exist, including structured, unstructured, hybrid, composite, and overlapping grids. Furthermore, the grid can be altered based on the solution in an approach known as solution-adaptive gridding. The numerical methods 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 gridding strategy are strongly interdependent. For example, the use of nite-di erence methods is typically restricted to structured grids. Here again, the success of a simulation can depend on appropriate choices for the problem or class of problems of interest. 1.2.4 Assessment and Interpretation of Results Finally, the results of the simulation must be assessed and interpreted. This step can require post-processing of the data, for example calculation of forces and moments, and can be aided by sophisticated ow visualization tools and error estimation techniques. It is critical that the magnitude of both numerical and physical-model errors be well understood. 1.3 Overview It should be clear that successful simulation of uid ows can involve a wide range of issues from grid generation to turbulence modelling to the applicability of various simpli ed forms of the Navier-Stokes equations. Many of these issues are not addressed in this book. Some of them are presented in the books by Anderson, Tannehill, and Pletcher 1] and Hirsch 2]. Instead we focus on numerical methods, with emphasis on nite-di erence and nite-volume methods for the Euler and Navier-Stokes equations. Rather than presenting the details of the most advanced methods, which are still evolving, we present a foundation for developing, analyzing, and understanding such methods. Fortunately, to develop, analyze, and understand most numerical methods used to nd solutions for the complete compressible Navier-Stokes equations, we can make use of much simpler expressions, the so-called model" equations. These model equations isolate certain aspects of the physics contained in the complete set of equations. Hence their numerical solution can illustrate the properties of a given numerical method when applied to a more complicated system of equations which governs similar physical phenomena. Although the model equations are extremely simple and easy to solve, 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 5 numerical approximations are applied to the model equations is a major rst step in making con dent and competent use of numerical approximations to the Euler and Navier-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 the model equations and can use that information in the design and application of numerical methods to practical problems, there are many aspects of practical problems which can only be understood in the context of the complete physical systems. 1.4 Notation The notation is generally explained as it is introduced. Bold type is reserved for real physical vectors, such as velocity. The vector symbol ~ is used for the vectors (or column matrices) which contain the values of the dependent variable at the nodes of a grid. Otherwise, the use of a vector consisting of a collection of scalars should be apparent from the context and is not identi ed by any special notation. For example, the variable u can denote a scalar Cartesian velocity component in the Euler and Navier-Stokes equations, a scalar quantity in the linear convection and di usion equations, and a vector consisting of a collection of scalars in our presentation of hyperbolic systems. Some of the abbreviations used throughout the text are listed and de ned below. PDE ODE O E RHS P.S. S.S. k-D ~ bc O( ) Partial di erential equation Ordinary di erential equation Ordinary di erence equation Right-hand side Particular solution of an ODE or system of ODE's Fixed (time-invariant) steady-state solution k-dimensional space Boundary conditions, usually a vector A term of order (i.e., proportional to)
  14. 14. 6 CHAPTER 1. INTRODUCTION
  15. 15. Chapter 2 CONSERVATION LAWS AND THE MODEL EQUATIONS We start out by casting our equations in the most general form, the integral conservation-law form, which is useful in understanding the concepts involved in nite-volume schemes. 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 discussed in this Chapter. The main focus, though, will be on representative model equations, in particular, the convection and di usion equations. These equations contain many of 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 numerical methods for computational uid dynamics, and the recurring use of these model equations allows us to develop a consistent framework of analysis for consistency, accuracy, stability, and convergence. The model equations we study have two properties in common. They are linear partial di erential equations (PDE's) with coe cients that are constant in both space and time, and they represent phenomena of importance to the analysis of certain aspects of uid dynamic problems. 2.1 Conservation Laws Conservation laws, such as the Euler and Navier-Stokes equations and our model equations, can be written in the following integral form: Z V (t2 ) QdV ; Z V (t1 ) QdV + Z t2 I t1 S (t) n:FdSdt = Z t2 Z t1 V (t) PdV dt (2.1) 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 EQUATIONS the conservation of these quantities in a nite region of space with volume V (t) and surface area S (t) over a nite interval of time t2 ; t1 . In two dimensions, the region of space, or cell, is an area A(t) bounded by a closed contour C (t). The vector n is a 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 production of 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-volume methods. 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 be derived. The divergence form of Eq. 2.2 is obtained by applying Gauss's theorem to the ux integral, leading to @Q + r:F = P (2.3) @t where r: is the well-known divergence operator given, in Cartesian coordinates, by ! @ +j @ +k @ : (2.4) r: i @x @y @z 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 Equations The Navier-Stokes equations form a coupled system of nonlinear PDE's describing the conservation of mass, momentum and energy for a uid. For a Newtonian uid in one dimension, they can be written as @Q + @E = 0 (2.5) @t @x with 2 3 2 3 u 3 2 0 6 6 Q=6 6 6 4 7 7 u7 7 7 5 e E 6 6 =6 6 6 4 7 6 7 6 2+p 7;6 u 7 6 7 6 5 4 u(e + p) 4 @u 3 @x 4 3 u @u + @x @T @x 7 7 7 7 7 5 (2.6)
  17. 17. 2.2. THE NAVIER-STOKES AND EULER EQUATIONS 9 where is the uid density, u is the velocity, e is the total energy per unit volume, p is the pressure, T is the temperature, is the coe cient of viscosity, and is the thermal conductivity. 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 uid state as well as an equation of state, such as the ideal gas law. Details can be found in 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 terms involve second derivatives. This form of the equations is called conservation-law or conservative form. Non-conservative forms can be obtained by expanding derivatives of 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 analytically the same as the above form, they can lead to quite di erent numerical solutions in terms of shock strength and shock speed, for example. Thus the conservative form is appropriate for solving ows with features such as shock waves. Many ows of engineering interest are steady (time-invariant), or at least may be treated as such. For such ows, we are often interested in the steady-state solution of the 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. In two-dimensional Cartesian coordinates, these can be written as @Q + @E + @F = 0 (2.8) @t @x @y with 2 3 2 3 2 2 q1 u 3 v 3 6 7 6 7 6 2+ 7 6 7 Q = 6 q2 7 = 6 u 7 E = 6 u uv p 7 F = 6 v2uv p 7 6 7 6 7 6 7 6 (2.9) 4 q3 5 4 v 5 4 5 4 5 + 7 q4 e u(e + p) v ( e + p) where u and v are the Cartesian velocity components. Later on we will make use of the following form of the Euler equations as well: @Q + A @Q + B @Q = 0 (2.10) @t @x @y @E @F The matrices A = @Q and B = @Q are known as the ux Jacobians. The ux vectors given 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 EQUATIONS to derive the ux Jacobian matrices, we must rst write the ux vectors E and F in terms of the conservative variables, q1 , q2, q3 , and q4 , as follows: E F 2 3 6 7 6 7 6 7 6 7 6 7 6 7=6 7 6 7 6 7 6 7 6 7 6 5 6 6 4 2 6 6 6 6 6 =6 6 6 6 6 6 4 E1 2 6 6 6 6 6 =6 6 6 6 6 6 4 3 2 F1 7 6 7 6 7 6 6 F2 7 6 7 6 7=6 7 6 7 6 F3 7 6 7 6 7 6 5 6 4 F4 E2 E3 E4 3 7 7 7 2 2 7 3; q2 ; ;1 q3 7 ; 1)q4 + 2 q1 2 q1 7 7 7 7 q3 q2 7 7 q1 7 7 7 5 3 2q q4 q2 ; ;1 q2 + q3 2 2 2 q1 2 q1 q1 (2.11) 3 7 7 7 7 7 7 7 7 2 7 ;1 q2 7 2 q1 7 7 7 5 3 q3 2 q1 (2.12) q2 ( q3 q3 q2 q1 ( ; 1)q4 + 3; 2 q4 q 3 q1 ; 2 q3 q1 2q ;1 q2 2 3 2 q1 ; + We have assumed that the pressure satis es p = ( ; 1) e ; (u2 + v2)=2] from the ideal gas law, where is the ratio of speci c heats, cp=cv . From this it follows that the ux Jacobian of E can be written in terms of the conservative variables as A= 2 6 6 6 6 6 @Ei = 6 6 6 @qj 6 6 6 6 6 4 0 1 0 0 a21 (1 ; ) q3 q1 q2 q1 ;1 q ; q2 q1 q1 3 (3 ; ) q2 q1 q q3 1 a41 a42 a43 q2 q1 where a21 = ! ; 1 q3 2 ; 3 ; 2 q1 2 q2 q1 !2 0 3 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (2.13)
  19. 19. 2.2. THE NAVIER-STOKES AND EULER EQUATIONS 2 ; 1)4 + q3 q1 !2 q4 ; ; 1 43 q2 a42 = q1 2 q1 ! ! q2 q3 a43 = ;( ; 1) q q 1 1 and in terms of the primitive variables as !2 a41 = ( q2 q1 !3 2 ! 2 6 6 6 6 6 A=6 6 6 6 6 6 4 0 1 q2 q1 !3 5; + q3 q1 !2 3 5 0 0 ;uv v u 0 a41 a42 a43 u a21 = q4 q1 ! q2 q1 ! (2.14) a21 (3 ; )u (1 ; )v ( ; 1) where 11 3 7 7 7 7 7 7 7 7 7 7 7 5 (2.15) ; 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 ux Jacobian matrices are purely real. This is the de ning feature of hyperbolic systems of PDE's, which are further discussed in Section 2.5. The homogeneous property of the Euler equations is discussed in Appendix C. The Navier-Stokes equations include both convective and di usive uxes. This motivates the choice of our two scalar model equations associated with the physics of convection and di usion. Furthermore, aspects of convective phenomena associated with coupled systems of equations such as the Euler equations are important in developing numerical methods and boundary conditions. Thus we also study linear hyperbolic systems of PDE's.
  20. 20. 12 CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS 2.3 The Linear Convection Equation 2.3.1 Di erential Form The simplest linear model for convection and wave propagation is the linear convection equation given by the following PDE: @u + a @u = 0 (2.17) @t @x Here u(x t) is a scalar quantity propagating with speed a, a real constant which may be positive or negative. The manner in which the boundary conditions are speci ed separates 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 boundary 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 convecting 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 exact solution 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 positive and to the left if a is negative. With periodic boundary conditions, the waveform travels through one boundary and reappears at the other boundary, eventually returning to its initial position. In this case, the process continues forever without any
  21. 21. 2.3. THE LINEAR CONVECTION EQUATION 13 change in the shape of the solution. Preserving the shape of the initial condition u0(x) can be a di cult challenge for a numerical method. 2.3.2 Solution in Wave Space We now examine the biconvection problem in more detail. Let the domain be given by 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 the periodic boundary conditions, must be an integer. It is a measure of the number of wavelengths within the domain. With such an initial condition, the solution can be written as u(x t) = f (t)ei x (2.20) where the time dependence is contained in the complex function f (t). Substituting this solution into the linear convection equation, Eq. 2.17, we nd that f (t) satis es the following ordinary di erential equation (ODE) df = ;ia f (2.21) dt which 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 dispersion relation. The linear relation given by Eq. 2.24 is characteristic of wave propagation in a nondispersive medium. This means that the phase speed is the same for all wavenumbers. As we shall see later, most numerical methods introduce some dispersion that is, in a simulation, waves with di erent wavenumbers travel at di erent speeds. An arbitrary initial waveform can be produced by summing initial conditions of the form of Eq. 2.19. For M modes, one obtains u(x 0) = M X m=1 fm (0)ei mx (2.25)
  22. 22. 14 CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS where the wavenumbers are often ordered such that 1 2 M . Since the wave equation is linear, the solution is obtained by summing solutions of the form of Eq. 2.23, giving u(x t) = M X m=1 fm (0)ei m(x;at) (2.26) Dispersion and dissipation resulting from a numerical approximation will cause the shape of the solution to change from that of the original waveform. 2.4 The Di usion Equation 2.4.1 Di erential Form Di usive uxes are associated with molecular motion in a continuum uid. A simple linear model equation for a di usive process is @u = @ 2 u (2.27) @t @x2 where is a positive real constant. For example, with u representing the temperature, this parabolic PDE governs the di usion of heat in one dimension. Boundary conditions can be periodic, Dirichlet (speci ed u), Neumann (speci ed @u=@x), or mixed Dirichlet/Neumann. In contrast to the linear convection equation, the di usion equation has a nontrivial steady-state solution, which is one that satis es the governing PDE with the partial derivative in time equal to zero. In the case of Eq. 2.27, the steady-state solution must satisfy @2u = 0 (2.28) @x2 Therefore, u must vary linearly with x at steady state such that the boundary conditions are satis ed. Other steady-state solutions are obtained if a source term g(x) is added to Eq. 2.27, as follows: @u = @t " 2 @u @x2 ; g(x) # (2.29) giving a steady state-solution which satis es @ 2 u ; g(x) = 0 @x2 (2.30)
  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 @y2 where g(x y) is again a source term. The corresponding steady equation is @ 2 u + @ 2 u ; g(x y) = 0 (2.32) @x2 @y2 While Eq. 2.31 is parabolic, Eq. 2.32 is elliptic. The latter is known as the Poisson equation for nonzero g, and as Laplace's equation for zero g. 2.4.2 Solution in Wave Space We now consider a series solution to Eq. 2.27. Let the domain be given by 0 x with boundary conditions u(0) = ua, u( ) = ub. It is clear that the steady-state solution 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 u(x 0) = M X m=1 fm(0) sin m x + h(x) (2.33) where must be an integer in order to satisfy the boundary conditions. A solution of the form M X u(x t) = fm(t) sin mx + h(x) (2.34) m=1 satis es the initial and boundary conditions. Substituting this form into Eq. 2.27 gives the following ODE for fm : dfm = ; 2 f (2.35) m m dt and we nd m fm(t) = fm(0)e; 2 t (2.36) Substituting fm(t) into equation 2.34, we obtain u(x t) = M X m=1 m fm (0)e; 2 t sin m x + h(x) (2.37) The steady-state solution (t ! 1) is simply h(x). Eq. 2.37 shows that high wavenumber components (large m) of the solution decay more rapidly than low wavenumber components, consistent with the physics of di usion.
  24. 24. CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS 16 2.5 Linear Hyperbolic Systems The Euler equations, Eq. 2.8, form a hyperbolic system of partial di erential equations. Other systems of equations governing convection and wave propagation phenomena, such as the Maxwell equations describing the propagation of electromagnetic waves, are also of hyperbolic type. Many aspects of numerical methods for such systems can be understood by studying a one-dimensional constant-coe cient linear system of the form @u + A @u = 0 (2.38) @t @x where u = u(x t) is a vector of length m and A is a real m m matrix. For conservation laws, this equation can also be written in the form @u + @f = 0 (2.39) @t @x @f where 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) j The 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 matrix of right eigenvectors. Premultiplying Eq. 2.38 by X ;1 , postmultiplying A by the product XX ;1, and noting that X and X ;1 are constants, we obtain @X ;1u z }| { @ X ;1AX X ;1u =0 (2.42) @t + @x With w = X ;1u, this can be rewritten as @w + @w = 0 (2.43) @t @x When written in this manner, the equations have been decoupled into m scalar equations 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 17 The elements of w are known as characteristic variables. Each characteristic variable satis es the linear convection equation with the speed given by the corresponding eigenvalue of A. Based on the above, we see that a hyperbolic system in the form of Eq. 2.38 has a solution given by the superposition of waves which can travel in either the positive or negative directions and at varying speeds. While the scalar linear convection equation is clearly an excellent model equation for hyperbolic systems, we must ensure that our numerical methods are appropriate for wave speeds of arbitrary sign and possibly widely varying magnitudes. The one-dimensional Euler equations can also be diagonalized, leading to three equations in the form of the linear convection equation, although they remain nonlinear, of course. The eigenvalues of the ux Jacobian matrix, or wave speeds, are q u u + c, and u ; c, where u is the local uid velocity, and c = p= is the local speed of sound. The speed u is associated with convection of the uid, while u + c and u ; c are associated with sound waves. Therefore, in a supersonic ow, where juj > 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 fact that sound waves can travel upstream in a subsonic ow. The signs of the eigenvalues of the matrix A are also important in determining suitable boundary conditions. The characteristic variables each satisfy the linear convection equation with the wave speed given by the corresponding eigenvalue. Therefore, the boundary conditions can be speci ed accordingly. That is, characteristic variables associated with positive eigenvalues can be speci ed at the left boundary, which corresponds to in ow for these variables. Characteristic variables associated with negative eigenvalues can be speci ed at the right boundary, which is the inow boundary for these variables. While other boundary condition treatments are possible, 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 resulting 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 @v ! = ; @x + @u = 0 @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 2 1 ;1 Note that the variables have been nondimensionalized. Combining the two PDE's, we have @f (q) + @g(q) = 0 @x @y where v q= u 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
  29. 29. ¡ ¢ 
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