Numerical Computation of Internal and External Flows: The Fundamentals of Computational Fluid Dynamics (eBook)

Cover 1
Copyright Page 5
Contents 7
Preface to the Second Edition 16
Nomenclature 19
Introduction: An Initial Guide to CFD and to this Volume 22
I.1 The position of CFD in the world of virtual prototyping 22
I.1.1 The Definition Phase 23
I.1.2 The Simulation and Analysis Phase 24
I.1.3 The Manufacturing Cycle Phase 26
I.2 The components of a CFD simulation system 32
I.2.1 Step 1: Defining the Mathematical Model 32
I.2.2 Step 2: Defining the Discretization Process 34
I.2.3 Step 3: Performing the Analysis Phase 36
I.2.4 Step 4: Defining the Resolution Phase 37
I.3 The structure of this volume 39
References 41
Part I: The Mathematical Models for Fluid Flow Simulations at Various Levels of Approximation 42
Chapter 1 The Basic Equations of Fluid Dynamics 48
Objectives and guidelines 48
1.1 General form of a conservation law 50
1.2 The mass conservation equation 61
1.3 The momentum conservation law or equation of motion 64
1.4 The energy conservation equation 68
A1.5 Rotating frame of reference 75
A1.6 Advanced applications of control volume formulations 78
Summary of the basic flow equations 81
Conclusions and main topics to remember 84
References 84
Problems 84
Chapter 2 The Dynamical Levels of Approximation 86
Objectives and guidelines 86
2.1 The Navier–Stokes equations 91
2.2 Approximations of turbulent flows 107
2.3 Thin shear layer approximation (TSL) 115
2.4 Parabolized Navier–Stokes equations 115
2.5 Boundary layer approximation 116
2.6 The distributed loss model 117
2.7 Inviscid flow model: Euler equations 118
2.8 Potential flow model 119
2.9 Summary 122
References 122
Problems 124
Chapter 3 The Mathematical Nature of the Flow Equations and Their Boundary Conditions 126
Objectives and guidelines 126
3.1 Simplified models of a convection–diffusion equation 129
3.2 Definition of the mathematical properties of a system of PDEs 132
3.3 Hyperbolic and parabolic equations: characteristic surfaces and domain of dependence 138
3.4 Time-dependent and conservation form of the PDEs 143
3.5 Initial and boundary conditions 151
A.3.6 Alternative definition: compatibility relations 153
Conclusions and main topics to remember 157
References 158
Problems 158
Part II: Basic Discretization Techniques 162
Chapter 4 The Finite Difference Method for Structured Grids 166
Objectives and guidelines 166
4.1 The basics of finite difference methods 168
4.2 Multidimensional finite difference formulas 181
4.3 Finite difference formulas on non-uniform grids 190
A4.4 General method for finite difference formulas 201
A4.5 Implicit finite difference formulas 210
Conclusions and main topics to remember 216
References 217
Problems 218
Chapter 5 Finite Volume Method and Conservative Discretization with an Introduction to Finite Element Method 224
Objectives and guidelines 224
5.1 The conservative discretization 225
5.2 The basis of the finite volume method 230
5.3 Practical implementation of finite volume method 237
A5.4 The finite element method 246
Conclusions and main topics to remember 262
References 263
Problems 264
Chapter 6 Structured and Unstructured Grid Properties 270
Objectives and guidelines 270
6.1 Structured Grids 271
6.2 Unstructured grids 282
6.3 Surface and volume estimations 288
6.4 Grid quality and best practice guidelines 295
Conclusions and main topics to remember 297
References 298
Part III: The Analysis of Numerical Schemes 300
Chapter 7 Consistency, Stability and Error Analysis of Numerical Schemes 304
Objectives and guidelines 304
7.1 Basic concepts and definitions 306
7.2 The Von Neumann method for stability analysis 313
7.3 New schemes for the linear convection equation 324
7.4 The spectral analysis of numerical errors 334
Conclusions and main topics to remember 353
References 353
Problems 354
Chapter 8 General Properties and High-Resolution Numerical Schemes 358
Objectives and guidelines 358
8.1 General formulation of numerical schemes 360
8.2 The generation of new schemes with prescribed order of accuracy 375
8.3 Monotonicity of numerical schemes 386
8.4 Finite volume formulation of schemes and limiters 410
Conclusions and main topics to remember 421
References 424
Problems 427
Part IV: The Resolution of Numerical Schemes 432
Chapter 9 Time Integration Methods for Space-discretized Equations 434
Objectives and guidelines 434
9.1 Analysis of the space-discretized systems 435
9.2 Analysis of time integration schemes 450
9.3 A selection of time integration methods 462
A9.4 Implicit schemes for multidimensional problems: approximate factorization methods 496
Conclusions and main topics to remember 503
References 504
Problems 506
Chapter 10 Iterative Methods for the Resolution of Algebraic Systems 512
Objectives and guidelines 512
10.1 Basic iterative methods 514
10.2 Overrelaxation methods 526
10.3 Preconditioning techniques 533
10.4 Nonlinear problems 539
10.5 The multigrid method 541
Conclusions and main topics to remember 554
References 554
Problems 556
Appendix A: Thomas Algorithm for Tridiagonal Systems 557
Part V: Applications to Inviscid and Viscous Flows 562
Chapter 11 Numerical Simulation of Inviscid Flows 566
Objectives and guidelines 566
11.1 The inviscid Euler equations 569
11.2 The potential flow model 577
11.3 Numerical solutions for the potential equation 579
11.4 Finite volume discretization of the Euler equations 595
11.5 Numerical solutions for the Euler equations 604
Conclusions and main topics to remember 617
References 618
Chapter 12 Numerical Solutions of Viscous Laminar Flows 620
Objectives and guidelines 620
12.1 Navier–Stokes equations for laminar flows 622
12.2 Density-based methods for viscous flows 625
12.3 Numerical solutions with the density-based method 631
12.4 Pressure correction method 646
12.5 Numerical solutions with the pressure correction method 659
12.6 Best practice advice 661
Conclusions and main topics to remember 665
References 666
Index 668
A 668
B 668
C 668
D 669
E 670
F 670
G 671
H 672
I 672
J 672
K 673
L 673
M 673
N 674
O 674
P 674
Q 675
R 675
S 675
T 676
U 677
V 677
W 677
Z 677
Colour Plates 678