Computational fluid dynamics

Title page  ......Page 1
Date-line  ......Page 2
CONTENTS  ......Page 3
Preface  ......Page 11
Introduction  ......Page 15
1.2 Linear and Nonlinear Partial Differential Equations  ......Page 17
1.3 Second-Order Partial Differential Equations  ......Page 18
1.5 Parabolic Equations  ......Page 20
1.6 Hyperbolic Equations  ......Page 22
1.7 Model Equations  ......Page 24
1.8 System of First-Order Partial Differential Equations  ......Page 25
1.9 System of Second-Order Partial Differential Equations  ......Page 30
1.10 Initial and Boundary Conditions  ......Page 34
1.11 Remarks and Definitions  ......Page 36
1.12 Summary Objectives  ......Page 38
1.13 Problems  ......Page 39
2.2 Taylor Series Expansion  ......Page 43
2.3 Finite Difference by Polynomials  ......Page 49
2.4 Finite Difference Equations  ......Page 51
2.5 Applications  ......Page 54
2.6.1 Taylor Series Expansion  ......Page 65
2.6.2 The Use of Partial Derivatives with Respect to One Independent Variable  ......Page 66
2.7 Summary Objectives  ......Page 67
2.8 Problems  ......Page 69
3.2 Finite Difference Formulations  ......Page 74
3.3.3 The DuFort-Frankel Method  ......Page 78
3.4 Implicit Methods  ......Page 79
3.4.2 The Crank-Nicolson Method  ......Page 80
3.5 Applications  ......Page 81
3.6 Analysis  ......Page 86
3.7 Parabolic Equations in Two-Space Dimensions  ......Page 90
3.8 Approximate Factorization  ......Page 99
3.10 Extension to Three-Space Dimensions  ......Page 101
3.11 Consistency Analysis of Finite Difference Equations  ......Page 102
3.12 Linearization  ......Page 104
3.13 Irregular Boundaries  ......Page 106
3.14 Summary Objectives  ......Page 108
3.15 Problems  ......Page 110
4.1 Introductory Remarks  ......Page 127
4.2 Discrete Perturbation Stability Analysis  ......Page 128
4.3 Von Neumann Stability Analysis  ......Page 138
4.4 Multidimensional Problems  ......Page 151
4.5 Error Analysis  ......Page 155
4.6 Modified Equation  ......Page 157
4.7 Artificial Viscosity  ......Page 160
4.8 Summary Objectives  ......Page 162
4.9 Problems  ......Page 163
5.2 Finite Difference Formulations  ......Page 166
5.3 Solution Algorithms  ......Page 170
5.3.1 The Jacobi Iteration Method  ......Page 171
5.3.2 The Point Gauss-Seidel Iteration Method  ......Page 174
5.3.3 The Line Gauss-Seidel Iteration Method  ......Page 176
5.3.4 Point Successive Over-Relaxation Method (PSOR)  ......Page 178
5.3.6 The Alternating Direction Implicit Method (ADI)  ......Page 179
5.4 Applications  ......Page 181
5.5 Summary Objectives  ......Page 188
5.6 Problems  ......Page 189
6.2 Finite Difference Formulations  ......Page 199
6.2.1.5 Midpoint Leapfrog Method  ......Page 200
6.2.1.6 The Lax-Wendroff Method  ......Page 201
6.2.2.3 Crank-Nicolson Method  ......Page 202
6.4.1 Richtmyer/Lax-Wendroff Multi-Step Method  ......Page 203
6.4.2 The MacCormack Method  ......Page 204
6.5 Applications to a Linear Problem  ......Page 205
6.6 Nonlinear Problem  ......Page 220
6.6.1 The Lax Method  ......Page 221
6.6.2 The Lax-Wendroff Method  ......Page 222
6.6.3 The MacCormack Method  ......Page 225
6.6.4 The Beam and Warming Implicit Method  ......Page 227
6.6.6 Implicit First-Order Upwind Scheme  ......Page 232
6.6.7 Runge-Kutta Method  ......Page 233
6.6.8 Modified Runge-Kutta Method  ......Page 239
6.7 Linear Damping  ......Page 242
6.7.1 Application  ......Page 245
6.8 Flux Corrected Transport  ......Page 247
6.8.1 Application  ......Page 249
6.9.1 Monotone Schemes  ......Page 250
6.9.2 Total Variation Diminishing Schemes  ......Page 251
6.10 TVD Formulations  ......Page 252
6.10.1 First-Order TVD Schemes  ......Page 253
6.10.3 Application  ......Page 257
6.10.4 Second-Order TVD Schemes  ......Page 258
6.10.4.1 Harten-Yee Upwind TVD Limiters  ......Page 259
6.10.4.2 Roe-Sweby Upwind TVD Limiters  ......Page 261
6.10.4.3 Davis-Yee Symmetric TVD Limiters  ......Page 264
6.11 Modified Runge-Kutta Method with TVD  ......Page 265
6.12 Summary Objectives  ......Page 267
6.13 Problems  ......Page 268
7.1 Introductory Remarks  ......Page 286
7.2 Model Equation  ......Page 287
7.3 Equations of Fluid Motion  ......Page 288
7.4.1 FTCS Explicit  ......Page 290
7.4.3 DuFort-Frankel Explicit  ......Page 291
7.4.5 MacCormack Implicit  ......Page 292
7.4.7 BTBCS Implicit  ......Page 293
7.5 Applications: Nonlinear Problem  ......Page 294
7.5.1 FTCS Explicit  ......Page 296
7.5.3 DuFort-Frankel Explicit  ......Page 299
7.5.4 MacCormack Explicit  ......Page 300
7.5.5 MacCormack Implicit  ......Page 301
7.5.7 BTBCS Implicit  ......Page 303
7.5.8 Modified Runge-Kutta  ......Page 304
7.5.9 Second-Order TVD Schemes  ......Page 306
7.6 Summary Objectives  ......Page 308
7.7 Problems  ......Page 309
8.1 Introductory Remarks  ......Page 316
8.2 Incompressible Navier-Stokes Equations  ......Page 317
8.2.1 Primitive Variable Formulations  ......Page 318
8.2.2 Vorticity-Stream Function Formulations  ......Page 321
8.2.3 Comments on Formulations  ......Page 323
8.3 Poisson Equation for Pressure: Primitive Variables  ......Page 324
8.4 Poisson Equation for Pressure: Vorticity-Stream Function Formulation  ......Page 325
8.5 Numerical Algorithms: Primitive Variables  ......Page 328
8.5.1.1 Artificial Compressibility  ......Page 329
8.5.1.2 Solution on a Regular Grid  ......Page 330
8.5.1.3 Crank-Nicolson Implicit  ......Page 335
8.6 Boundary Conditions  ......Page 336
8.6.1 Body Surface  ......Page 337
8.6.3 Symmetry  ......Page 339
8.6.6 An Example  ......Page 340
8.7 Staggered Grid  ......Page 342
8.7.1 Marker and Cell Method  ......Page 344
8.7.2 Implementation of the Boundary Conditions  ......Page 346
8.7.3 DuFort-Frankel Scheme  ......Page 347
8.7.4 Use of the Poisson Equation for Pressure  ......Page 348
8.7.5 Unsteady Incompressible Navier-Stokes Equations  ......Page 349
8.8 Numerical Algorithms: Vorticity-Stream Function Formulation  ......Page 351
8.8.1 Vorticity Transport Equation  ......Page 352
8.9 Boundary Conditions  ......Page 357
8.9.1 Body Surface  ......Page 358
8.9.3 Symmetry  ......Page 360
8.9.4 Inflow  ......Page 361
8.10 Application  ......Page 362
8.11 Temperature Field  ......Page 365
8.11.1 The Energy Equation  ......Page 366
8.11.2 Numerical Schemes  ......Page 368
8.11.3 Boundary Conditions  ......Page 369
8.12 Problems  ......Page 371
9.1 Introductory Remarks  ......Page 372
9.2 Transformation of the Governing Partial Differential Equations  ......Page 376
9.3 Metrics and the Jacobian of Transformation  ......Page 377
9.4 Grid Generation Techniques  ......Page 378
9.5 Algebraic Grid Generation Techniques  ......Page 379
9.7 Elliptic Grid Generators  ......Page 397
9.7.1 Simply-Connected Domain  ......Page 399
9.7.2 Doubly-Connected Domain  ......Page 409
9.7.3 Multiply-Connected Domain  ......Page 415
9.8.1 Grid Point Clustering  ......Page 418
9.8.2 Orthogonality at the Surface  ......Page 421
9.9 Hyperbolic Grid Generation Techniques  ......Page 425
9.10 Parabolic Grid Generators  ......Page 432
9.11 Problems  ......Page 434
Appendix A: An Introduction to Theory of Characteristics: Wave Propagation  ......Page 440
Appendix B: Tridiagonal System of Equations  ......Page 452
Appendix C: Derivation of Partial Derivatives for the Modified Equations  ......Page 457
Appendix D: Basic Equations of Fluid Mechanics  ......Page 459
Appendix E: Block-Tridiagonal System of Equations  ......Page 478
Appendix F: Derivatives in the Computational Domain  ......Page 487
References  ......Page 492
Index  ......Page 495