Numerical Computation of Internal and External Flows, Volume 1: The Fundamentals of Computational Fluid Dynamics

Numerical Computation of Internal and External Flows, Second Edition......Page 4
Copyright Page......Page 5
Contents......Page 7
Preface to the Second Edition......Page 16
Nomenclature......Page 19
I.1 The position of CFD in the world of virtual prototyping......Page 22
I.1.1 The Definition Phase......Page 23
I.1.2 The Simulation and Analysis Phase......Page 24
I.1.3 The Manufacturing Cycle Phase......Page 26
I.2.1 Step 1: Defining the Mathematical Model......Page 32
I.2.2 Step 2: Defining the Discretization Process......Page 34
I.2.3 Step 3: Performing the Analysis Phase......Page 36
I.2.4 Step 4: Defining the Resolution Phase......Page 37
I.3 The structure of this volume......Page 39
References......Page 41
Part I: The Mathematical Models for Fluid Flow Simulations at Various Levels of Approximation......Page 42
Objectives and guidelines......Page 48
1.1 General form of a conservation law......Page 50
1.2 The mass conservation equation......Page 61
1.3 The momentum conservation law or equation of motion......Page 64
1.4 The energy conservation equation......Page 68
A1.5 Rotating frame of reference......Page 75
A1.6 Advanced applications of control volume formulations......Page 78
Summary of the basic flow equations......Page 81
Problems......Page 84
Objectives and guidelines......Page 86
2.1 The Navier–Stokes equations......Page 91
2.2 Approximations of turbulent flows......Page 107
2.4 Parabolized Navier–Stokes equations......Page 115
2.5 Boundary layer approximation......Page 116
2.6 The distributed loss model......Page 117
2.7 Inviscid flow model: Euler equations......Page 118
2.8 Potential flow model......Page 119
References......Page 122
Problems......Page 124
Objectives and guidelines......Page 126
3.1 Simplified models of a convection–diffusion equation......Page 129
3.2 Definition of the mathematical properties of a system of PDEs......Page 132
3.3 Hyperbolic and parabolic equations: characteristic surfaces and domain of dependence......Page 138
3.4 Time-dependent and conservation form of the PDEs......Page 143
3.5 Initial and boundary conditions......Page 151
A.3.6 Alternative definition: compatibility relations......Page 153
Conclusions and main topics to remember......Page 157
Problems......Page 158
Part II: Basic Discretization Techniques......Page 162
Objectives and guidelines......Page 166
4.1 The basics of finite difference methods......Page 168
4.2 Multidimensional finite difference formulas......Page 181
4.3 Finite difference formulas on non-uniform grids......Page 190
A4.4 General method for finite difference formulas......Page 201
A4.5 Implicit finite difference formulas......Page 210
Conclusions and main topics to remember......Page 216
References......Page 217
Problems......Page 218
Objectives and guidelines......Page 224
5.1 The conservative discretization......Page 225
5.2 The basis of the finite volume method......Page 230
5.3 Practical implementation of finite volume method......Page 237
A5.4 The finite element method......Page 246
Conclusions and main topics to remember......Page 262
References......Page 263
Problems......Page 264
Objectives and guidelines......Page 270
6.1 Structured Grids......Page 271
6.2 Unstructured grids......Page 282
6.3 Surface and volume estimations......Page 288
6.4 Grid quality and best practice guidelines......Page 295
Conclusions and main topics to remember......Page 297
References......Page 298
Part III: The Analysis of Numerical Schemes......Page 300
Objectives and guidelines......Page 304
7.1 Basic concepts and definitions......Page 306
7.2 The Von Neumann method for stability analysis......Page 313
7.3 New schemes for the linear convection equation......Page 324
7.4 The spectral analysis of numerical errors......Page 334
References......Page 353
Problems......Page 354
Objectives and guidelines......Page 358
8.1 General formulation of numerical schemes......Page 360
8.2 The generation of new schemes with prescribed order of accuracy......Page 375
8.3 Monotonicity of numerical schemes......Page 386
8.4 Finite volume formulation of schemes and limiters......Page 410
Conclusions and main topics to remember......Page 421
References......Page 424
Problems......Page 427
Part IV: The Resolution of Numerical Schemes......Page 432
Objectives and guidelines......Page 434
9.1 Analysis of the space-discretized systems......Page 435
9.2 Analysis of time integration schemes......Page 450
9.3 A selection of time integration methods......Page 462
A9.4 Implicit schemes for multidimensional problems: approximate factorization methods......Page 496
Conclusions and main topics to remember......Page 503
References......Page 504
Problems......Page 506
Objectives and guidelines......Page 512
10.1 Basic iterative methods......Page 514
10.2 Overrelaxation methods......Page 526
10.3 Preconditioning techniques......Page 533
10.4 Nonlinear problems......Page 539
10.5 The multigrid method......Page 541
References......Page 554
Problems......Page 556
Appendix A: Thomas Algorithm for Tridiagonal Systems......Page 557
Part V: Applications to Inviscid and Viscous Flows......Page 562
Objectives and guidelines......Page 566
11.1 The inviscid Euler equations......Page 569
11.2 The potential flow model......Page 577
11.3 Numerical solutions for the potential equation......Page 579
11.4 Finite volume discretization of the Euler equations......Page 595
11.5 Numerical solutions for the Euler equations......Page 604
Conclusions and main topics to remember......Page 617
References......Page 618
Objectives and guidelines......Page 620
12.1 Navier–Stokes equations for laminar flows......Page 622
12.2 Density-based methods for viscous flows......Page 625
12.3 Numerical solutions with the density-based method......Page 631
12.4 Pressure correction method......Page 646
12.5 Numerical solutions with the pressure correction method......Page 659
12.6 Best practice advice......Page 661
Conclusions and main topics to remember......Page 665
References......Page 666
C......Page 668
D......Page 669
F......Page 670
G......Page 671
J......Page 672
M......Page 673
P......Page 674
S......Page 675
T......Page 676
Z......Page 677
Colour Plates......Page 678