There are many new developments in scientific computing, in its application to fluid flows and wave phenomena, which warrant their consolidation in a single source, covering some of the key developments. I have been convinced by many students and peers that there is a definitive need for a single source book which deals with topics covered here. I would like to acknowledge their inspiration. My main motivation in writing this manuscript is to communicate something new and powerful as opposed to conventional derivatives of products churned out by existing schools of thought.
However, this book also provides general introduction to computational fluid dynamics (CFD), using well tested classical methods of solving partial differential equations (PDEs) for the sake of completeness. These are to be found in Chaps. 1 to9 and 13, but re-interpreted using the spectral analysis method introduced in Chaps. 4,8 and 10. This provides an unity of approach in understanding numerical methods for parabolic, elliptic and hyperbolic PDEs. The spectral analysis tool has been refined in recent years by the author's group, with which disparate methods can be easily compared.
This spectral analysis enables one, as shown in Chap. 8, that celebrated von Neumann analysis is actually flawed, in which it is assumed that for a linear dynamical system the error and signal follow the same dynamics. While this appeals to anyone as logical in an intuitive framework, the correct error analysis shows that this is indeed a false assumption when viewed for the numerical solution of the one-dimensional convection equation, which is an ideal model equation for non-dissipative, non-dispersive system. A correct error analysis for this equation shows that numerical schemes must be neutrally stable and dispersion error-free to provide high accuracy solution. The error dynamics is noted to be distinctly different from the signal for this simple model equation. One of the singular achievement of this analysis is identification of the correct numerical dispersion relation, which in turn has enabled the creation of a general spectral theory of analysis of numerical methods over the full domain for any discrete computations. Such analysis can also be done for linear shallow water equation, as an example for dispersive dynamic system and that also shows different error dynamics from signal behaviour.

Foreword xiii

Preface xvii

Chapter 1

Basic Ideas of Scientific Computing 1

      1.1 Overview on Scientific Computing 1

      1.2 Major Milestones in Electronic Computing 2

      1.3 Supercomputing and High Performance Computing 3

      1.3.1 Parallel and cluster computing 5

      1.3.2 Algorithmic issues of HPC 5

      1.4 Computational Fluid Mechanics 5

      1.5 Role of Computational Fluid Mechanics 6

Chapter 2

Governing Equations in Fluid Mechanics 8

      2.1 Introduction 8

      2.2 Basic Equations of Fluid Mechanics 8

      2.2.1 Finite control volume 9

      2.2.2 Infinitesimal fluid element 9

      2.2.3 Substantive derivative 10

      2.3 Equation of Continuity 11

      2.4 Momentum Conservation Equation 11

      2.5 Energy Conservation Equation 14

      2.6 Alternate Forms of Energy Equation 16

      2.7 The Energy Equation in Conservation Form 17

      2.8 Notes on Governing Equations 17

      2.9 Strong Conservation and Weak Conservation Forms 18

      2.10 Boundary and Initial Conditions (Auxiliary Conditions) 19

      2.11 Equations of Motion in Non-Inertial Frame 19

      2.12 Equations of Motion in Terms of Derived Variables 23

      2.13 Vorticity-Vector Potential Formulation 25

      2.14 Pressure Poisson Equation 26

      2.15 Comparison of Different Formulations 28

      2.16 Other Forms of Navier-Stokes Equation 28

Chapter 3

Classification of Quasi-Linear Partial Differential Equations 31

      3.1 Introduction 31

      3.2 Classification of Partial Differential Equations 31

      3.3 Relationship of Numerical Solution Procedure and Equation Type 35

      3.4 Nature of Well-Posed Problems 36

      3.5 Non-Dimensional Form of Equations 37

Chapter 4

Waves and Space-Time Dependence in Computing 38

      4.1 Introduction 38

      4.2 The Wave Equation 39

      4.2.1 Plane waves 42

      4.2.2 Three-dimensional axisymmetric wave 43

      4.2.3 Doppler shift 43

      4.2.4 Surface gravity waves 43

      4.3 Deep and Shallow Water Waves 47

      4.4 Group Velocity and Energy Flux 47

      4.4.1 Physical and computational implications of group velocity 49

      4.4.2 Wave-packets and their propagation 50

      4.4.3 Waves over layer of constant depth 50

      4.4.4 Waves over layer of variable depth H(x) 52

      4.4.5 Wave refraction in shallow waters 52

      4.4.6 Finite amplitude waves of unchanging form in dispersive medium 53

      4.5 Internal Waves at Fluid Interface: Rayleigh-Taylor Problem 54

      4.5.1 Internal and surface waves in finite over an infinite deep layer of fluid 55

      4.5.2 Barotropic or surface mode 57

      4.5.3 Baroclinic or internal mode 57

      4.5.4 Rotating shallow water equation and wave dynamics 57

      4.6 Shallow Water Equation (SWE) 59

      4.6.1 Various frequency regimes of SWE 61

      4.7 Additional Issues of Computing: Space-Time Resolution of Flows 63

      4.7.1 Spatial scales in turbulent flows 63

      4.8 Two- and Three-Dimensional DNS 65

      4.9 Temporal Scales in Turbulent Flows 67

      4.10 Computing Time-Averaged and Unsteady Flows 68

Chapter 5

Spatial and Temporal Discretizations of Partial Differential Equations 71

      5.1 Introduction 71

      5.2 Discretization of Differential Operators 72

      5.2.1 Functional representation by the laylor series 72

      5.2.2 Polynomial representation of function 73

      5.3 Discretization in Non-Uniform Grids 74

      5.4 Higher Order Representation of Derivatives Using Operators 75

      5.5 Higher Order Upwind Differences 77

      5.5.1 Symmetric stencil for higher derivatives 78

      5.6 Numerical Errors 79

      5.7 lime Integration 79

      5.7.1Single-step methods 80

      5.7.2 Single-step multi-stage methods 82

      5.7.3 Runge-Kutta methods 83

      5.7.4 Multi-step time integration schemes 90

Chapter 6

Solution Methods for Parabolic Partial Differential Equations 92

      6.1 Introduction 92

      6.2 Theoretical Analysis of the Heat Equation 92

      6.3 A Classical Algorithm for Solution of the Heat Equation 94

      6.4 Spectral Analysis of Numerical Methods 94

      6.4.1 A higher order method or Milne's method 95

      6.5 Treating Derivative Boundary Condition 95

      6.6 Stability; Accuracy and Consistency of Numerical Methods 96

      6.6.1 Richardsonzs method 97

      6.6.2 Du Fort-Frankel method 99

      6.7 Implicit Methods 101

      6.8 Spectral Stability Analysis of Implicit Methods 102

      Appendix-I 104

Chapter 7

Solution Methods for Elliptic Partial Differential Equations 106

      7.1 Introduction 106

      7.2 Jacobi or Richardson Iteration 108

      7.3 Interpretation of Classical Iterations 109

      7.4 Different Point and Line Iterative Methods 111

      7.4.1 Gauss-Seidel point iterative method 111

      7.4.2 Line Jacobi method 112

      7.4.3 Explanation of line iteration methods 112

      7.5 Analysis of Iterative Methods 113

      7.6 Convergence Theorem for Stationary Linear Iteration 114

      7.7 Relaxation Methods 115

      7.8 Efficiency of Iterative Methods and Rate of Convergence 116

      7.8.1 Method of acceleration due to Lyustemik 117

      7.9 Alternate Direction Implicit (ADI) Method 117

      7.9.1 Analysis of ADI method 119

      7.9.2 Choice of acceleration parameters 121

      7.9.3 Estimates of maximum and minimum eigenvedues 122

      7.9.4 Explanatory notes on ADI and other variant methods 122

      7.10 Method of Fractional Steps 123

      7.11 Multi-Grid Methods 124

      7.11.1 Iwo-Gria method 126

      7.11.2 Multi-Grid method 128

      7.11.3 Other classifications of multi-grid method 129

Chapter 8

Solution of Hyperbolic PDEs: Signal and Error Propagation 130

      8.1 Introduction 130

      8.2 Classical Methods of Solving Hyperbolic Equations 130

      8.2.1 Explicit methods 131

      8.3 Implicit Methods 133

      8.4 General Characteristics of Various Methods for Linear Problems 134

      8.5 Non-linear Hyperbolic Problems 134

      8.6 Error Dynamics: Beyond von Neumann Analysis 135

      8.6.1 Dispersion error and its quantification 138

      8.7 Role of Group Velocity and Focussing 142

      8.7.1 Focussing phenomenon 144

Chapter 9

Curvilinear Coordinate and Grid Generation 150

      9.1 Introduction 150

      9.2 Generalized Curvilinear Scheme 151

      9.3 Reciprocal or Dual Base Vectors 152

      9.4 Geometric Interpretation of Metrics 152

      9.5 Orthogonal Grid System 153

      9.6 Generalized Coordinate Transformation 154

      9.7 Equations for the Metrics 154

      9.8 Navier-Stokes Equation in the Transformed Plane 156

      9.9 Linearization of Fluxes 159

      9.10 Thin Layer Navier-Stokes Equation 161

      9.11 Grid Generation 162

      9.12 Types of Grid 162

      9.13 Grid Generation Methods 163

      9.14 Algebraic Grid Generation Method 164

      9.14.1 One-dimensional stretching functions 164

      9.15 Grid Generation by Solving Partial Differential Equations 165

      9.16 Elliptic Grid Generators 165

      9.17 Hyperbolic Grid Generation Method 166

      9.18 Orthogonal Grid Generation for Navier-Stokes Computations 166

      9.19 Coordinate Transformations and Governing Equations in Orthogonal System 169

      9.19.1 Gradient operator 170

      9.19.2 Divergence operator 170

      9.19.3 The Laplacian operator 171

      9.19.4 The curl operator 171

      9.19.5 The line integral 172

      9.19.6 The surface integral 172

      9.19.7 The volume integral 172

      9.20 The Gradient and Laplacian of Scalar Function 172

      9.21 Vector Operators of a Vector Function 173

      9.22 Plane Polar Coordinates 173

      9.23 Navier-Stokes Equation in Orthogonal Formulation 174

      9.24 Improved Orthogonal Grid Generation Method for Cambered Airfoils 176

      9.24.1 Orthogonal grid generation for GA(W)-1 airfoil 176

      9.24.2 Orthogonal grid generation for an airfoil with roughness element 181

      9.24.3 Solutions of Navier-Stokes equation for flow past SHM-1 airfoil 182

      9.24.4 Compressible flow past NACA 0012 airfoil 18

      9.25 Governing Euler Equation, Auxiliary Conditions, Numerical Methods and Results 184

      9.26 Flow Field Calculation Using Overset or Chimera Grid Technique 187

Chapter 10

Spectral Analysis of Numerical Schemes and Aliasing Error 196

      10.1 Introduction 196

      10.2 Spatial Discretization of First Derivatives 198

      10.2.1 Second order central differencing (CD2) scheme 198

      10.3 Discrete Computing and Nyquist Criterion 198

      10.4 Spectral Accuracy of Differentiation 198

      10.5 Spectral Analysis of Fourth Order Central Difference Scheme 200

      10.6 Role of Upwinding 200

      10.6.1 First order upwind scheme (UD₁) 200

      10.6.2 Third order upwind scheme (UD₃) 202

      10.7 Numerical Stability and Concept of Feedback 203

      10.8 Spectral Stability Analysis 204

      10.9 High Accuracy Schemes for Spatial Derivatives 204

      10.10 Temporal Discretization Schemes 207

      10.10.1 Euler time integration scheme 207

      10.10.2 Four-stage Runge-Kutta (RK4) method 212

      10.11 Multi-Ume Level Discretization Schemes 212

      10.11.1 Mid-point leapfrog scheme 214

      10.11.2 Second order Adams-Bashforth scheme 223

      10.12 Aliasing Error 232

      10.12.1 Why aliasing error is important? 233

      10.12.2 Estimation of aliased component 240

      10.13 Numerical Estimates of Aliasing Error 243

      10.14 Controlling Aliasing Error 248

      10.14.1 Aliasing removal by zero padding 253

      10.14.2 Aliasing removal by phase shifts and grid-staggering 254

Chapter 11

Higher Accuracy Methods 256

      11.1 Introduction 256

      11.2 The General Compact Schemes 257

      11.2.1 Approximating first derivatives by central scheme 257

      11.3 Method for Solving Periodic Tridiagonal Matrix Equation 258

      11.4 An Example of a Sixth Order Scheme 260

      11.5 Order of Approximation versus Resolution 262

      11.6 Optimization Problem Associated with Discrete Evaluation of First Derivatives 267

      11.7 An Optimized Compact Scheme For First Derivative by Grid Search Method 270

      11.8 Upwind Compact Schemes 271

      11.9 Compact Schemes with Improved Numerical Properties 274

      11.9.1 OUCS1 scheme 274

      11.9.2 OUCS2 scheme 274

      11.9.3 OUCS3 scheme 275

      11.9.4 OUCS4 scheme 277

      11.10 Approximating Second Derivatives 278

      11.11 Optimization Problem for Evaluation of the Second Derivatives 280

      11.12 Solution of One-Dimensional Convection Equation 281

      11.13 Symmetrized Compact Difference Schemes 286

      11.13.1 High accuracy symmetrized compact scheme 291

      11.13.2 Solving bidirectional wave equation 295

      11.13.3 Transitional channel flow 300

      11.13.3.1 Establishment of equilibrium flow 302

      11.13.3.2 Receptivity of channel flow to convecting single viscous vortex 303

      11.13.4 Transitional channel flow created by vortex street 305

      11.14 Combined Compact Difference (CCD) Schemes 308

      11.14.1 A new combined compact difference (NCCD) scheme 314

      11.14.2 Solving the Stommel Ocean Model problem 319

      11.14.3 Operational aspects of the CCD schemes 32

      11.14.4Calibrating NCCD method to solve Navier-Stokes equation for 2D lid-driven cavity problem 322

      11.15 Diffusion Discretization and Dealiasing Properties of Compact Schemes 326

      11.15.1 Dynamics and aliasing in square LDC problem 331

      11.15.2 Receptivity calculation of an adverse pressure gradient flow 334

Chapter 12

Introduction to Finite Volume and Finite Element Methods 341

      12.1 Introduction 341

      12.2 Finite Volume Method 342

      12.3 Finite Volume Discretization for Two-Dimensional Flows 343

      12.4 Geometric Constraints of FVM 343

      12.5 FVM for Three-Dimensional Flows 343

      12.6 Evaluating Viscous Terms 345

      12.7 High Resolution Finite Volume Upwind Schemes 347

      12.8 Properties of Reconstruction Schemes 349

      12.9 Spectral Resolution of Flux-Vector Splitting (FVS) Scheme 350

      12.10 Dispersion Relation Preservation Property 357

      12.11 Solution of ID Convection Equation 357

      12.12 Explaining the Gibbs' Phenomenon for FVS-FVM 358

      12.13 Derivation of the FV2S Scheme 361

      12.13.1 Spectral properties of the FV2S scheme 362

      12.13.2 Test cases for the FV2S scheme 368

      12.14 Introduction to Finite Element Methods 374

      12.15 Weighted Residual Methods 375

      12.15.1 The collocation method 376

      12.15.2 The subdomain method 376

      12.15.3 The least square method 376

      12.15.4 The Bubnov-Galerkin method 376

      12.15.5 General weighted residual method or the Petrov-Galerkin method 377

      12.16 Finite Element Approximation and Discretization 378

      12.16.1 FEM basis functions 378

      12.17 Dispersion Properties of the Galerkin FEM 383

      12.18 The Petrov-Galerkin Finite Elements Method 386

      12.18.1 Further notes on the SUPG method 388

      12.19 Higher Order Basis Function for the Bubnov-Galerkin FEM 389

      12.19.1 Solution of ID convection equation by the quadratic basis function Galerkin method (G2FEM) 390

      12.20 A Comparative Study of FEM, FVM and FDM 399

Chapter 13

Solution of Navier-Stokes Equation 405

      13.1 Introduction 405

      13.2 Stream Function-Vorticity Formulation for 2D Flows 406

      13.3 Start-up and Initial Condition 408

      13.4 Solution of Stream Function Equation (SFE) 410

      13.5 Wall-Vorticity Estimation 412

      13.6 Solution of Vorticity Transport Equation (VTE) 413

      13.6.1 Explicit upwind methods 413

      13.7 Implicit Upwind Methods 414

      13.8 Solution of Navier-Stokes Equation using Pressure-Velocity Formulation 414

      13.8.1 The MAC method of Harlow and Welch 414

      13.8.2 Operator splitting projection methods or fractional step methods 415

      13.9 Solution of Navier-Stokes Equation: Comparative Study of Different Formulations 416

      13.1 0Vorticity-Velocity Formulation: Detailed Study 419

      13.11 Solution of Flow in Lid Driven Cavity by (V, ω)-Formulation 427

      13.12 Role of Grid Staggering for LDC Flow using (V, ω)-Formulation 431

      13.13 Receptivity of Flow Past a Flat Plate using (V, ω)-Formulation 433

      13.14 Effects of Initial Acceleration: Solution of Navier-Stokes Equation 436

Chapter 14

Recent Developments in Discrete Finite Difference Computing 442

      14.1 Introduction 442

      14.2 One-Dimensional Filters for DES, LES and DNS 443

      14.3 Design, Order and Transfer Function of Explicit Filters 445

      14.4 Transfer Functions of Filters for Non-Periodic Problems 449

      14.5 Numerical Amplification and Dispersion Properties of Filters 451

      14.6 Upwind One-Dimensional Filter: A New Approach 452

      14.7 Application of One-Dimensional Filters 455

      14.7.1 Accelerated flow past a NACA 0015 aerofoil 456

      14.7.2 Flow past a cylinder executing rotary oscillation 458

      14.8 Two-Dimensional Higher Order Filters 465

      14.8.1 Implementation procedure for two-dimensional filters 469

      14.8.2 Performance comparison between ID and 2D filters 470

      14.9 Solutions of Navier-Stokes Equation 471

      14.9.1 Flow past a cylinder executing rotary oscillation 471

      14.9.2 Accelerated flow past a NACA 0015 aerofoil 475

      14.9.3 Equilibrium flow past SHM-1 aerofoil 476

      14.9.4 Algorithmic cost estimate for 2D filters 478

      14.10 Optimal Time Advancing DRP Schemes 479

      14.10.1 Formulating an optimization problem for time integration 482

      14.10.2 Optimized RK2 scheme coupled to CD₂,CD_(4) and CD_(6) schemes 486

      14.10.3 Optimized RK3 scheme coupled with CD₂,CD_(4) and CD_(6) schemes 487

      14.10.4 Optimized RK4 scheme coupled with CD₂,CD_(4) and CD_(6) schemes 488

      14.11 Numerical Properties of Coupled Temporal and Spatial Schemes 190

      14.11.1 Solving 1D convection problem 492

      14.11.2 Solving 2D lid-driven cavity (LDC) problem 198

      14.11.3 Benchmark Problem for DRP schemes for application in computational aeroacoustics (CAA) 503

      14.12 Optimized Explicit Runge-Kutta DRP Schemes for Compact Spatial Discretization Schemes 506

      14.13 Anisotropy of Numerical Wave Solutions by Finite Difference Methods 512

      14.14 Analysis of Numerical Methods for One-Dimensional Skewed Wave Propagation 513

      14.15 Finite Difference Spatial Semi-Discretization in 2D 514

      14.15.1 Fourier analysis of finite difference semi-discretization 515

      14.16 Analysis of Full Discretization in 2D 517

      14.16.1 An example of Skewed wave propagation at θ= 30° 524

      14.17 Analysis of Numerical Methods for Linearized Rotating Shallow Water Wave Equation 526

Exercises 535

References 546

Index 563

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