Computational fluid dynamics

Cover ......Page 1
Title page ......Page 2
Date-line ......Page 3
CONTENTS ......Page 4
Preface ......Page 13
10.2.1 Linear and Nonlinear PDEs ......Page 14
10.2.2 Classification Based on Characteristics ......Page 15
10.3 Boundary Conditions ......Page 16
10.4.1.1 One-Space Dimension ......Page 17
10.4.1.2 Multi-Space Dimensions ......Page 20
10.4.2 Elliptic Equations ......Page 21
10.4.3.1 Linear Equations ......Page 23
10.4.3.2 Nonlinear Equations ......Page 26
10.5 Stability Analysis ......Page 27
10.6 Error Analysis ......Page 28
10.7 Grid Generation - Structured Grids ......Page 29
10.8 Transformation of the Equations from the Physical Space to Computational Space ......Page 32
11.1 Introductory Remarks ......Page 34
11.2 Generalized Coordinate Transformation ......Page 35
11.2.1 Equations for the Metrics ......Page 36
11.3 Nondimensionalization of the Equations of Fluid Motion ......Page 38
11.4 Navier-Stokes Equations ......Page 41
11.4.1 Linearization ......Page 44
11.4.2 Inviscid and Viscous Jacobian Matrices ......Page 46
11.5 Thin-Layer Approximation ......Page 70
11.6 Parabolized Navier-Stokes Equations ......Page 73
11.7 Two-Dimensional Planar or Axisymmetric Formulation ......Page 82
11.8 Incompressible Navier-Stokes Equations ......Page 98
11.8.1 Inviscid and Viscous Jacobian Matrices ......Page 102
11.8.2 Two-Dimensional Incompressible Navier-Stokes Equations ......Page 104
11.9 Problems ......Page 107
12.1 Introductory Remarks ......Page 110
12.2 Euler Equations ......Page 111
12.3 Quasi One-Dimensional Euler Equations ......Page 112
12.3.1 Numerical Issues ......Page 113
12.3.2.1 Steger and Warming Flux Vector Splitting ......Page 120
12.3.2.2 Van Leer Flux Vector Splitting ......Page 121
12.3.2.4 Second-Order TVD Formulation ......Page 125
12.3.2.4.1 Harten-Yee Upwind TVD ......Page 126
12.3.2.4.2 Roe-Sweby Upwind TVD ......Page 127
12.3.3 Implicit Formulations ......Page 128
12.3.3.1 Steger and Warming Flux Vector Splitting ......Page 129
12.4 Boundary Conditions ......Page 131
12.5 Application 1: Diverging Nozzle Configuration ......Page 134
12.5.1.1 Analytical Solution ......Page 137
12.5.2.2 Numerical Solutions ......Page 138
12.5.2.1 Analytical Solution ......Page 146
12.5.2.2 Numerical Solutions ......Page 147
12.6 Grid Clustering ......Page 157
12.7 Global Time Step and Local Time Step ......Page 159
12.8.1 Problem Description ......Page 165
12.8.2 Analytical Solution ......Page 166
12.8.3 Numerical Solution ......Page 170
12.9 Two-Dimensional Planar and Axisymmetric Euler Equations ......Page 175
12.9.1 Numerical Considerations ......Page 176
12.9.2.1 Steger and Warming Flux Vector Splitting ......Page 183
12.9.2.1.1 Matrix Manipulations ......Page 184
12.9.2.1.2 Existence of Zero Metrics Within the Domain ......Page 188
12.9.2.1.3 Eigenvector Matrices ......Page 190
12.9.2.2 Van Leer Flux Vector Splitting ......Page 191
12.9.2.3 Modified Runge-Kutta Formulation ......Page 193
12.9.2.4.1 Harten-Yee Upwind TVD ......Page 194
12.9.2.4.2 Roe Sweby Upwind TVD ......Page 196
12.9.2.5 Modified Runge-Kutta Scheme with TVD ......Page 198
12.9.3.1 Body Surface ......Page 199
12.9.3.2 Symmetry ......Page 202
12.9.3.4 Outflow ......Page 203
12.9.3.5 Boundary Conditions Based on Characteristics ......Page 204
12.9.3.5.3 Determination of Flow Variables ......Page 205
12.9.4 Implicit Formulations ......Page 206
12.9.4.1 Steger and Warming Flux Vector Splitting ......Page 207
12.9.4.1.2 Boundary Conditions ......Page 210
12.10.1 Supersonic Channel Flow ......Page 213
12.10.1.1 Grid Generation ......Page 214
12.10.1.2 Numerical Scheme ......Page 216
12.10.1.3 Analytical Solution ......Page 217
12.10.1.4 The Physical Domain and Flow Conditions ......Page 218
12.10.1.6 Results ......Page 219
12.10.2 Axisymmetric Blunt Body ......Page 223
12.11 Concluding Remarks ......Page 228
12.12 Problems ......Page 229
13.1 Introductory Remarks ......Page 231
13.2 Governing Equations of Motion ......Page 234
13.3 Streamwise Pressure Gradient ......Page 236
13.4 Numerical Algorithm ......Page 238
13.5 Boundary Conditions ......Page 248
13.6.1 Numerical Algorithm ......Page 257
13.7 Numerical Damping Terms ......Page 262
13.8 Shock Fitting Procedure ......Page 263
13.8.1 Extension to Three-Dimensions ......Page 270
13.9 Application ......Page 274
13.10 Summary Objectives ......Page 277
13.11 Problems ......Page 278
14.1 Introductory Remarks ......Page 279
14.2 Navier-Stokes Equations ......Page 280
14.3 Thin-Layer Navier-Stokes Equations ......Page 281
14.4.1 Explicit Formulations ......Page 282
14.4.1.1 MacCormack Explicit Formulation ......Page 283
14.4.1.2 Flux Vector Splitting ......Page 285
14.4.1.3 Modified Runge-Kutta Scheme ......Page 288
14.4.2 Boundary Conditions ......Page 289
14.4.3 Implicit Formulations ......Page 290
14.4.3.1 Flux Vector Splitting ......Page 292
14.4.3.2 Higher-Order Flux-Vector Splitting ......Page 301
14.4.3.3 Second-Order Accuracy in Time ......Page 303
14.4.3.4 LU Decomposition ......Page 304
14.5 Extension to Three-Dimensions ......Page 307
14.5.1 Explicit Flux Vector Splitting Scheme ......Page 308
14.5.2 Implicit Formulation ......Page 316
14.6 Concluding Remarks ......Page 317
14.7 Problems ......Page 319
15.1 Introductory Remarks ......Page 320
15.2 Classification of Schemes for Specification of Boundary Conditions ......Page 321
15.3.1 Mathematical Developments ......Page 322
15.3.2 Slip Wall Boundary Condition ......Page 334
15.3.2.1 Nonconservative (Primitive) Variables ......Page 335
15.3.3.1 Nonconservative (Primitive) Variables ......Page 336
15.3.4 Inflow/Outflow Boundary Conditions ......Page 338
15.4 Category Three Boundary Conditions: Addition of Buffer Layer ......Page 340
15.5.1 Application 1: Moving Shock Wave ......Page 341
15.5.2 Application 2: Flow Over a Compression Corner ......Page 342
15.6 Concluding Remarks ......Page 347
16.2 Fundamental Concepts ......Page 349
16.2.1 Real Gas and Perfect Gas ......Page 350
16.2.3 Frozen Flow ......Page 351
16.2.6 Various Modes of Energy ......Page 352
16.2.7 Reaction Rates ......Page 354
16.2.8 Five-Species Model ......Page 356
16.3 Quasi One-Dimensional Flow/Equilibrium Chemistry ......Page 357
16.4 Quasi One-Dimensional Flow/Nonequilibrium Chemistry ......Page 358
16.4.1 Species Continuity Equation ......Page 359
16.4.3 Numerical Procedure for the Loosely Coupled Scheme ......Page 360
16.5.1 Quasi One-Dimensional Flow ......Page 363
16.5.2 Two-Dimensional Axisymmetric Flow ......Page 365
16.6 Concluding Remarks ......Page 367
17.1 Introductory Remarks ......Page 369
17.2 Domain Nodalization ......Page 370
17.3 Domain Triangulation ......Page 373
17.3.1.1 Simply-Connected Domain ......Page 374
17.3.1.2 Multiply-Connected Domain ......Page 378
17.3.2 The Delaunay Method ......Page 379
17.3.2.1 Geometrical Description ......Page 382
17.3.2.2 Outline of the Algorithm ......Page 383
15.3.2.3 An Illustrative Example ......Page 391
17.4 Concluding Remarks ......Page 395
17.5 Problems ......Page 396
18.1 Introductory Remarks ......Page 398
18.2 General Description of the Finite Volume Method ......Page 400
18.2.1 Cell Centered Scheme ......Page 401
18.2.2 Nodal Point Scheme ......Page 402
18.3 Two-Dimensional Heat Conduction Equation ......Page 403
18.3.1 Interior Triangles ......Page 404
18.3.2 Boundary Triangles ......Page 408
18.3.2.2 Neumann Type Boundary Condition ......Page 409
18.4 Flux Vector Splitting Scheme ......Page 415
18.4.1 Interior Triangles ......Page 416
18.4.2 Boundary Triangles ......Page 421
18.5 Concluding Remarks ......Page 423
18.6 Problems ......Page 424
19.1 Introductory Remarks ......Page 431
19.2 Optimization Techniques ......Page 432
19.3 General Description and Development of the Finite Element Method ......Page 433
19.4 Two-Dimensional Heat Conduction Equation ......Page 441
19.5 Construction of the Global Matrix ......Page 442
19.6 Boundary Conditions ......Page 445
19.7 Reduction of the Half-Bandwidth of the Global Matrix ......Page 446
Appendix G: An Introduction to Theory of Characteristics: Euler Equations ......Page 448
Appendix H: Computation of Pressure at the Body Surface ......Page 465
Appendix I: Rate of Formation of Species ......Page 471
References ......Page 474
Index ......Page 477