Fundamentals of computational fluid dynamics:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2003
|
| Edition: | 1. ed., corr. 2. print. |
| Series: | Scientific computation
Physics and astronomy online library |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 249 S. graph. Darst. |
| ISBN: | 3540416072 |
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MARC
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| adam_text | Titel: Fundamentals of computational fluid dynamics
Autor: Lomax, Harvard
Jahr: 2003
Contents
1. Introduction............................................................................................1
1.1 Motivation ........................................................................................1
1.2 Background........................................................................................2
1.2.1 Problem Specification 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............................................................................................4
2. 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..................................................11
2.3.1 Differential Form..................................................................11
2.3.2 Solution in Wave Space......................................................12
2.4 The Diffusion Equation ..................................................................13
2.4.1 Differential Form..................................................................13
2.4.2 Solution in Wave Space......................................................14
2.5 Linear Hyperbolic Systems ............................................................15
Exercises ....................................................................................................17
3. Finite-Difference Approximations..................................................19
3.1 Meshes and Finite-Difference Notation........................................19
3.2 Space Derivative Approximations..................................................21
3.3 Finite-Difference Operators............................................................22
3.3.1 Point Difference Operators ................................................22
3.3.2 Matrix Difference Operators....................... 23
3.3.3 Periodic Matrices................................. 26
3.3.4 Circulant Matrices................................ 27
3.4 Constructing Differencing Schemes of Any Order........... 28
3.4.1 Taylor Tables.................................... 28
3.4.2 Generalization of Difference Formulas............... 31
X Contents
3.4.3 Lagrange and Herrnite Interpolation Polynomials..........33
3.4.4 Practical Application of Pade Formulas..........................35
3.4.5 Other Higher-Order Schemes..............................................36
3.5 Fourier Error Analysis....................................................................37
3.5.1 Application to a Spatial Operator....................................37
3.G Difference Operators at Boundaries..............................................41
3.6.1 The Linear Convection Equation......................................41
3.6.2 The Diffusion Equation......................................................44
Exercises ....................................................................................................46
4. The Semi-Discrete Approach..........................................................49
4.1 Reduction of PDE s to ODE s........................................................60
4.1.1 The Model ODE s................................................................60
4.1.2 The Generic Matrix Form..................................................51
4.2 Exact Solutions of Linear ODE s..................................................51
4.2.1 Eigensystems of Semi-discrete Linear Forms..................52
4.2.2 Single ODE s of First and Second Order..........................53
4.2.3 Coupled First-Order ODE s................................................54
4.2.4 General Solution of Coupled ODE s
with Complete Eigensystems..............................................56
4.3 Real Space and Eigenspace............................................................58
4.3.1 Definition..............................................................................58
4.3.2 Eigenvalue Spectrums for Model ODE s..........................59
4.3.3 Eigenvectors of the Model Equations................................60
4.3.4 Solutions of the Model ODE s............................................62
4.4 The Representative Equation........................................................64
Exercises ....................................................................................................65
5. Finite-Volume Methods................................... 67
5.1 Basic Concepts......................................... 67
5.2 Model Equations in Integral Form........................ 69
5.2.1 The Linear Convection Equation................... 69
5.2.2 The Diffusion Equation........................... 70
5.3 One-Dimensional Examples.............................. 70
5.3.1 A Second-Order Approximation
to the Convection Equation........................ 71
5.3.2 A Fourth-Order Approximation
to the Convection Equation........................ 72
5.3.3 A Second-Order Approximation
to the Diffusion Equation.......................... 74
5.4 A Two-Dimensional Example............................ 76
Exercises ........................................................................79
Contents XI
6. Time-Marching Methods for ODE S............................................81
6.1 Notation............................................................................................82
6.2 Converting Time-Marching Methods to OAE s..........................83
6.3 Solution of Linear OzlE s with Constant Coefficients................84
6.3.1 First- and Second-Order Difference Equations................84
6.3.2 Special Cases of Coupled First-Order Equations............86
6.4 Solution of the Representative OzlE s ........................................87
6.4.1 The Operational Form and its Solution............................87
6.4.2 Examples of Solutions to Time-Marching CME s..........88
6.5 The A-a Relation ............................................................................89
6.5.1 Establishing the Relation....................................................89
6.5.2 The Principal cr-Root..........................................................90
6.5.3 Spurious (j-Roots..................................................................91
6.5.4 One-Root Time-Marching Methods..................................92
6.6 Accuracy Measures of Time-Marching Methods..........................92
6.6.1 Local and Global Error Measures......................................92
6.6.2 Local Accuracy of the Transient Solution (er , | j|, er^) 93
6.6.3 Local Accuracy of the Particular Solution (er^)............94
6.6.4 Time Accuracy for Nonlinear Applications......................95
6.6.5 Global Accuracy ..................................................................96
6.7 Linear Multistep Methods..............................................................96
6.7.1 The General Formulation....................................................97
6.7.2 Examples ..............................................................................97
6.7.3 Two-Step Linear Multistep Methods................100
6.8 Predictor-Corrector Methods............................101
6.9 Runge-Kutta Methods..................................103
6.10 Implementation of Implicit Methods......................105
6.10.1 Application to Systems of Equations................105
6.10.2 Application to Nonlinear Equations.................106
6.10.3 Local Linearization for Scalar Equations.............107
6.10.4 Local Linearization for Coupled Sets
of Nonlinear Equations............................110
Exercises ..................................................112
7. Stability of Linear Systems...............................115
7.1 Dependence on the Eigensystem..........................115
7.2 Inherent Stability of ODE s..............................116
7.2.1 The Criterion....................................116
7.2.2 Complete Eigensystems...........................117
7.2.3 Defective Eigensystems............................117
7.3 Numerical Stability of OAE s............................118
7.3.1 The Criterion....................................118
7.3.2 Complete Eigensystems...........................118
7.3.3 Defective Eigensystems............................119
7.4 Time-Space Stability and Convergence of OAE s...........119
XII Contents
7.5 Numerical Stability Concepts in the Complex a-Plane.......121
7.5.1 r-Root Traces Relative to the Unit Circle ...........121
7.5.2 Stability for Small At.............................*26
7.6 Numerical Stability Concepts in the Complex Ah Plane......127
7.6.1 Stability for Large h..............................127
7.6.2 Unconditional Stability, A-Stable Methods...........128
7.6.3 Stability Contours in the Complex Ah Plane.........130
7.7 Fourier Stability Analysis................................133
7.7.1 The Basic Procedure..............................I33
7.7.2 Some Examples..................................134
7.7.3 Relation to Circulant Matrices.....................135
7.8 Consistency............................................I33
Exercises ..................................................I33
8. Choosing a Time-Marching Method ......................141
8.1 Stiffness Definition for ODE s............................141
8.1.1 Relation to A-Eigenvalues .........................141
8.1.2 Driving and Parasitic Eigenvalues..................142
8.1.3 Stiffness Classifications............................143
8.2 Relation of Stiffness to Space Mesh Size...................143
8.3 Practical Considerations for Comparing Methods...........144
8.4 Comparing the Efficiency of Explicit Methods..............145
8.4.1 Imposed Constraints..............................145
8.4.2 An Example Involving Diffusion....................146
8.4.3 An Example Involving Periodic Convection..........147
8.5 Coping with Stiffness...................................149
8.5.1 Explicit Methods.................................149
8.5.2 Implicit Methods.................................150
8.5.3 A Perspective....................................151
8.6 Steady Problems.......................................151
Exercises ..................................................152
9. Relaxation Methods......................................153
9.1 Formulation of the Model Problem .......................154
9.1.1 Preconditioning the Basic Matrix...................154
9.1.2 The Model Equations.............................156
9.2 Classical Relaxation ....................................157
9.2.1 The Delta Form of an Iterative Scheme .............157
9.2.2 The Converged Solution, the Residual, and the Error . 158
9.2.3 The Classical Methods............................158
9.3 The ODE Approach to Classical Relaxation................159
9.3.1 The Ordinary Differential Equation Formulation .....159
9.3.2 ODE Form of the Classical Methods................161
9.4 Eigensystems of the Classical Methods....................162
9.4.1 The Point-Jacobi System..........................163
Contents XIII
9.4.2 The Gauss-Seidel System .........................166
9.4.3 The SOR System.................................169
9.5 Nonstationary Processes.................................171
Exercises ..................................................176
10. Multigrid.................................................177
10.1 Motivation ............................................177
10.1.1 Eigenvector and Eigenvalue Identification
with Space Frequencies............................177
10.1.2 Properties of the Iterative Method..................178
10.2 The Basic Process......................................178
10.3 A Two-Grid Process....................................185
Exercises ..................................................187
11. Numerical Dissipation....................................189
11.1 One-Sided First-Derivative Space Differencing..............189
11.2 The Modified Partial Differential Equation.................190
11.3 The Lax-Wendroff Method..............................192
11.4 Upwind Schemes.......................................195
11.4.1 Flux-Vector Splitting .............................196
11.4.2 Flux-Difference Splitting..........................198
11.5 Artificial Dissipation....................................199
Exercises ..................................................200
12. Split and Factored Forms.................................203
12.1 The Concept...........................................203
12.2 Factoring Physical Representations - Time Splitting........204
12.3 Factoring Space Matrix Operators in 2D ..................206
12.3.1 Mesh Indexing Convention ........................206
12.3.2 Data-Bases and Space Vectors .....................206
12.3.3 Data-Base Permutations ..........................207
12.3.4 Space Splitting and Factoring......................207
12.4 Second-Order Factored Implicit Methods..................211
12.5 Importance of Factored Forms
in Two and Three Dimensions ...........................212
12.6 The Delta Form........................................213
Exercises ..................................................214
13. Analysis of Split and Factored Forms.....................217
13.1 The Representative Equation for Circulant Operators.......217
13.2 Example Analysis of Circulant Systems ...................218
13.2.1 Stability Comparisons of Time-Split Methods........218
13.2.2 Analysis of a Second-Order Time-Split Method.......220
13.3 The Representative Equation for Space-Split Operators.....222
13.4 Example Analysis of the 2D Model Equation...............225
XIV Contents
13.4.1 The Unfactored Implicit Euler Method..............225
13.4.2 The Factored Nondelta Form
of the Implicit Euler Method.......................226
13.4.3 The Factored Delta Form of the Implicit Euler Method 227
13.4.4 The Factored Delta Form of the Trapezoidal Method . 227
13.5 Example Analysis of the 3D Model Equation...............228
Exercises ..................................................230
Appendices...................................................231
A. Useful Relations from Linear Algebra.....................231
A.l Notation..............................................231
A.2 Definitions.............................................232
A.3 Algebra...............................................232
A.4 Eigensystems..........................................233
A.5 Vector and Matrix Norms...............................235
B. Some Properties of Tridiagonal Matrices..................237
B.l Standard Eigensystem for Simple Tridiagonal Matrices......237
B.2 Generalized Eigensystem for Simple Tridiagonal Matrices .... 238
B.3 The Inverse of a Simple Tridiagonal Matrix................239
B.4 Eigensystems of Circulant Matrices.......................240
B.4.1 Standard Tridiagonal Matrices.....................240
B.4.2 General Circulant Systems.........................241
B.5 Special Cases Found from Symmetries.....................241
B.G Special Cases Involving Boundary Conditions..............242
C. The Homogeneous Property of the Euler Equations.......245
Index.................................................. 247
|
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| author | Lomax, Harvard 1922-1999 Pulliam, Thomas H. 1951- Zingg, David W. 1958- |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 532 - Fluid mechanics |
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| discipline | Physik |
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| indexdate | 2024-07-09T19:10:15Z |
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| spelling | Lomax, Harvard 1922-1999 Verfasser (DE-588)13051604X aut Fundamentals of computational fluid dynamics H. Lomax ; T. H. Pulliam ; D. W. Zingg 1. ed., corr. 2. print. Berlin [u.a.] Springer 2003 XIV, 249 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Scientific computation Physics and astronomy online library Strömungsmechanik - Numerisches Verfahren - Lehrbuch Strömungsmechanik (DE-588)4077970-1 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Strömungsmechanik (DE-588)4077970-1 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Pulliam, Thomas H. 1951- Verfasser (DE-588)122651006 aut Zingg, David W. 1958- Verfasser (DE-588)122651014 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010154220&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lomax, Harvard 1922-1999 Pulliam, Thomas H. 1951- Zingg, David W. 1958- Fundamentals of computational fluid dynamics Strömungsmechanik - Numerisches Verfahren - Lehrbuch Strömungsmechanik (DE-588)4077970-1 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| subject_GND | (DE-588)4077970-1 (DE-588)4128130-5 |
| title | Fundamentals of computational fluid dynamics |
| title_auth | Fundamentals of computational fluid dynamics |
| title_exact_search | Fundamentals of computational fluid dynamics |
| title_full | Fundamentals of computational fluid dynamics H. Lomax ; T. H. Pulliam ; D. W. Zingg |
| title_fullStr | Fundamentals of computational fluid dynamics H. Lomax ; T. H. Pulliam ; D. W. Zingg |
| title_full_unstemmed | Fundamentals of computational fluid dynamics H. Lomax ; T. H. Pulliam ; D. W. Zingg |
| title_short | Fundamentals of computational fluid dynamics |
| title_sort | fundamentals of computational fluid dynamics |
| topic | Strömungsmechanik - Numerisches Verfahren - Lehrbuch Strömungsmechanik (DE-588)4077970-1 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| topic_facet | Strömungsmechanik - Numerisches Verfahren - Lehrbuch Strömungsmechanik Numerisches Verfahren |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010154220&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT lomaxharvard fundamentalsofcomputationalfluiddynamics AT pulliamthomash fundamentalsofcomputationalfluiddynamics AT zinggdavidw fundamentalsofcomputationalfluiddynamics |