Free-Surface Flow

1. Basic Concepts

1.1 Introduction 4

1.1.1 “Newton’s Rules” for Computational Modeling 5

1.1.2 Computational Models 6

1.2 The Taylor Series

9

1.3 Finite-Difference Approximations

14

1.3.1 Forward Differences 15

1.3.2 Backward Differences 16

1.3.3 Central Differences 16

1.3.4 Second-Order, One-Sided Differences 17

1.3.5 Identity and Shift Operators 18

1.3.6 Linear Difference Equations 18

1.4 Initial-Value Problems for ODE’s

21

1.4.1 Basic Numerical Models 22

1.4.2 Truncation Error and Order of Accuracy 23

1.4.3 Stability, Consistency, and Convergence 24

1.4.4 Absolute Stability 26

1.4.5 Runge-Kutta Methods 29

1.4.6 Linear Multi-Step Methods 34

1.4.7 Backward-Difference Methods 36

1.5 Boundary-Value Problems

39

1.5.1 Steady-State Diffusion 39

1.5.2 Solution of a Tri-Diagonal System 40

1.5.3 The Thomas Algorithm 44

1.5.4 Natural Boundary Conditions 45

1.5.5 Variable Grid Computations 46

1.6 Error Norms

49

1.7 Algorithmic Dissipation

51

1.7.0.1 Backward Difference Model 52

1.7.1 Damping Effect of 2nd Derivative Operator 53

1.7.2 Order of Dissipation 54

1.7.3 Algorithmic Dispersion 54

1.8 von Neumann Stability Analysis 57

1.8.1 Representation of Oscillatory Data – Wave Aliasing 58

1.8.2 Discrete Fourier Series Representation 60

1.8.3 The Fourier Symbol 61

1.8.4 Temporal Evolution 62

1.8.5 Propagation Factor 64

1.8.6 Algorithmic Dissipation – Condition for Stability 65

1.8.7 Algorithmic Celerity – Dispersion 66

1.8.8 Algorithmic Portrait 66

1.8.9 Construction of Phase and Amplitude Graphs 67

1.8.10 PDE’s With Variable Coefficients 69

1.9 Stability, Consistency, and Convergence

71

1.9.1 Positivity and Monotonicity 71

1.10 Least-Squares Approximation

74

Problems

76

References

79

2. Finite-Difference Methods for Diffusion

2.1 Introduction

82

2.2 Explicit Scheme for Diffusion (FTCS)

84

2.2.1 Results and Error Estimates 86

2.2.2 Stability 88

2.2.3 Propagation of Information 88

2.2.4 Discretization of Discontinuous Initial Data 90

2.2.5 Boundary Effects 92

2.2.6 Natural Boundary Conditions 92

2.2.7 Simulation of a Point Source 93

2.2.8 Accuracy of FTCS Scheme 94

2.3 Oscillatory Initial Data and Spurious Signals

96

2.3.1 Spurious Waves 97

2.3.2 Stability of FTCS Scheme 98

2.4 Leapfrog Scheme

102

2.4.1 Stability Analysis of Leapfrog Scheme 103

2.5 du Fort-Frankel Scheme

105

2.6 Implicit Scheme for Diffusion

107

2.6.1 Natural Boundary Conditions 108

2.6.2 Accuracy of BTCS Scheme 109

2.6.3 Stability of BTCS Scheme 109

2.7 Crank-Nicolson Implicit Scheme

111

2.7.1 Stability of Crank-Nicolson Scheme 112

2.7.2 Weighted Average Explicit-Implicit Scheme 112

Problems

115

References

117

3. Finite-Difference Methods for Advection

3.1 Introduction

120

3.2 The Numerical Method of Characteristics

122

3.2.1 Curvilinear Characteristic Network 123

3.2.2 Characteristic Scheme on a Cartesian Grid 126

3.2.3 The Effect of Interpolation 128

3.3 Explicit Upwind Scheme (FTBS)

130

3.3.1 Accuracy of Upwind Scheme 131

3.4 The Courant-Friedrichs-Lewy (CFL) Condition

137

3.4.1 Stability of Explicit Upwind Scheme 138

3.5 Centered Explicit Scheme (FTCS)

140

3.6 Implicit Upwind Scheme (BTBS)

142

3.6.1 Stability of the BTBS Scheme 143

3.7 Lax-Friedrichs Scheme

146

3.7.1 Stability Analysis 147

3.8 Leapfrog Scheme

150

3.8.1 Propagation Properties 151

3.8.2 Stability Analysis 153

3.8.3 Dispersion Control 156

3.8.3.1 Leapfrog-Trapezoidal Scheme 157

3.8.3.2 Leapfrog-RAW Scheme 157

3.9 The Lax-Wendroff Scheme

161

3.9.1 Fourier Analysis of Lax-Wendroff Scheme 163

3.9.2 Two-Step Lax-Wendroff-Richtmyer Scheme 164

3.10 Beam and Warming Scheme

166

3.10.1 Stability Analysis 167

3.11 Parasitic Waves, Dissipation, and Dispersion

169

3.11.1 Leapfrog Scheme 170

3.11.2 Lax-Wendroff Scheme 171

3.11.3 Frequency Analysis 172

3.11.4 Group Velocity 175

3.12 Advection Coupled With Diffusion

179

3.12.1 Steady State Solution 181

3.12.2 Generalized Upwind Method 184

3.13 Transient Advection-Diffusion Schemes

188

3.13.1 Centered Explicit Scheme 188

3.13.2 Crank-Nicolson Scheme 191

3.13.3 Stability of Crank-Nicolson Scheme 192

3.13.4 Boundary Conditions 192

Problems

195

References

197

4. Finite-Element and Finite-Volume Methods for Scalar

Transport

4.1 Introduction

200

4.1.1 Variational Principles 200

4.1.1.1 Functional for Steady State Diffusion 201

4.2 The Finite-Element Method (FEM)

203

4.2.1 Basis Functions 204

4.2.2 FEM Approximation of the Functional 205

4.3 Method of Weighted Residuals

207

4.3.1 Optimal Least-Squares Distance 207

4.3.2 Inner Product Space 208

4.3.3 Minimization of the Finite-Element Residual 209

4.3.4 Linear Finite Elements 210

4.3.5 Local Coordinates 211

4.4 Diffusion Matrix and Load Vector

213

4.5 Finite-Element Model for Transient Diffusion

217

4.5.1 Time Domain Discretization 218

4.6 Finite-Element Model for Advection

221

4.6.1 Semi-Discrete Form 222

4.6.2 Advection of a Sharp Concentration Front 223

4.7 Petrov-Galerkin Modification

226

4.7.1 Dissipative Galerkin Model 228

4.7.2 Fourier Stability Analysis 229

4.7.3 Phase and Amplitude Portraits 230

4.7.4 Anti-Dissipative Behavior 231

4.7.5 Preserving Monotonicity 233

4.7.6 Selective Dissipation and Shock Capturing 235

4.7.7 Fully Discrete Monotone DG Model 237

4.8 Finite-Volume Method for Diffusion

239

4.9 Finite Volume Method for Advection

241

4.9.1 Conservative Fluxes 242

4.9.2 Upwind Finite Volume Scheme 244

4.9.3 QUICK Scheme for Advection 244

4.10 Total Variation Diminishing

247

4.11 Superbee Limiter for Advection

248

4.11.1 Comparison With the Petrov-Galerkin Finite-Element

Model 249

4.12 Discontinuous Galerkin Method

252

4.12.1 Linear Advection Equation 254

4.12.2 Stability Analysis 255

Problems

257

References

258

5. Finite-Difference Methods for Equilibrium Problems

5.1 Introduction

262

5.2 Domain Discretization

263

5.2.1 Choice of Computational Nodes 267

5.3 Equilibrium Problems

269

5.3.1 Finite-Difference Solution of Laplace’s Equation 270

5.3.2 Sources and Anisotropic Media 271

5.3.3 Natural Node Ordering 272

5.3.4 The Right Hand Side Vector 273

5.3.5 The Coefficient Matrix of the Discrete Laplacian 274

5.3.6 Fast Poisson Solvers 275

5.3.7 The Residual Equation 276

5.4 Iterative Solution of Sparse Systems

278

5.4.1 Relaxation Methods 278

5.4.2 Over Relaxation 282

5.4.3 Application of SOR to a Square Domain 283

5.4.4 Convergence of the Iterations 284

5.4.5 The Spectral Radius 286

5.4.6 Optimum Relaxation Factor 287

5.4.7 Comparison of Relaxation Methods 289

5.4.8 Impact of Problem Size 290

5.5 Optimization Methods for Solving Sparse Systems of Linear

Equations

292

5.5.1 Conjugate Gradient Method 293

5.6 Matrix Preconditioning

296

5.6.1 Preconditioned Conjugate Gradient Method 296

5.6.1.1 Incomplete Factorization 296

5.6.1.2 LDU Factorization 299

5.6.2 Incomplete Factorization 300

5.6.3 Incomplete Cholesky Factorization Algorithm 301

5.6.4 Preconditioned Conjugate Gradient Method 302

5.6.5 Modified Incomplete Cholesky Factorization 304

5.6.6 Convergence Tests 308

5.7 Multigrid Methods

310

5.7.1 Diffusion of Iteration Error 310

5.7.2 Eigenvalues of the Iteration Matrix 313

5.7.2.1 Higher Dimensions 316

5.7.3 Modes of the Jacobi Iteration 319

5.7.4 Behavior on Coarse Grid 322

5.7.5 Elements of Multigrid Method 323

5.7.6 Inter-Grid Operations 324

5.7.6.1 Prolongation 324

5.7.7 Restriction 326

5.7.8 Cycling Schemes 327

5.7.9 Multigrid Solution of Laplace Equation 330

5.8 Multi-Domain Methods

332

5.8.1 Schwarz Alternating Method 332

5.8.1.1 General Boundary Conditions 333

5.8.2 Steklov-Poincaré Method 334

5.8.3 Schur Complement and Iterative Substructuring 336

5.9 Irregular Boundaries

338

5.9.1 Dirichlet Boundaries 338

5.9.2 Neumann Boundaries 341

Problems

345

References

348

6. Methods for Two-Dimensional Scalar Transport

6.1 Introduction

352

6.2 Finite-Difference Models for Diffusion

353

6.2.1 Explicit Method (FTCS) for Diffusion 353

6.2.2 Stability of 2D-FTCS 355

6.2.2.1 The Relaxation Analogy 356

6.2.3 Alternating Direction Implicit (ADI) Scheme 356

6.2.4 Stability of ADI Scheme 359

6.3 Finite-Difference Models for Advection

360

6.3.1 The Method of Characteristics for 2D Advection 360

6.3.2 Stability of 2D Method of Characteristics 363

6.3.3 Upwind Method (FTBS) for Advection 365

6.3.4 Stability of 2D-Upwind Scheme for Advection 366

6.3.5 Modified Equation of the Upwind Scheme 369

6.3.6 2D Lax-Friedrichs Scheme 371

6.3.7 Stability Analysis of Lax-Friedrichs Scheme 372

6.3.8 2D Lax-Wendroff Scheme 372

6.3.9 Stability Analysis of 2D Lax-Wendroff Scheme 374

6.4 Advection Coupled With Diffusion

377

6.4.1 Stability of Crank-Nicolson Scheme 377

6.4.2 Cross-Wind Diffusion 380

6.5 Finite-Element Analysis

383

6.5.1 Two-Dimensional Shape Functions 385

6.6 Galerkin Formulation

388

6.6.1 Transformation of Shape Function Derivatives 389

6.6.2 Transformation of Integrals to Local Coordinates 390

6.6.3 Finite Element Equations 390

6.6.4 Gaussian Quadrature 391

6.6.4.1 Transient Advection-Diffusion Problems 392

6.6.5 Petrov-Galerkin Approximation 393

6.6.6 Large-Scale Applications 395

Problems

400

References

402

7. Methods for Open-Channel Flow

7.1 The Method of Characteristics

406

7.1.1 Kinematic Waves 406

7.1.2 Kinematic Shock Model 408

7.1.3 Dynamic Waves 409

7.1.4 Massau’s Method 412

7.1.5 Moving Boundaries 415

7.1.6 Hartree’s Method 416

7.1.6.1 Moving Boundaries 418

7.1.6.2 Shock Fitting 419

7.2 Finite-Difference Methods

420

7.2.1 Naive FTCS Scheme 420

7.2.1.1 Boundary Conditions 421

7.2.1.2 Stability Analysis 423

7.2.2 Lax-Friedrichs Scheme 425

7.2.3 Lax-Wendroff Scheme 426

7.2.3.1 Two Step Version of LW Scheme 427

7.2.3.2 Boundary Conditions 428

7.2.3.3 Stability Analysis 429

7.2.4 The Preissmann Implicit Scheme 431

7.2.4.1 Double Sweep Method 434

7.2.4.2 Stability Analysis 436

7.2.5 Implicit ENO Method 437

7.2.5.1 Computational Results 439

7.3 FEM for Open-Channel Flow

441

7.3.1 Bubnov-Galerkin Method (BG) 443

7.3.1.1 Computational Results 445

7.3.1.2 Stability Analysis 446

7.3.2 Taylor-Galerkin Method 449

7.3.2.1 Stability Analysis 452

7.3.3 Petrov-Galerkin Method 453

7.3.4 Dissipative Galerkin Scheme (DG) 456

7.3.4.1 Stability Analysis 457

7.3.5 Characteristic Galerkin Scheme (CG) 460

7.3.5.1 Stability Analysis 461

7.3.6 Comparative Analysis of Petrov-Galerkin Schemes 462

7.4 Finite-Volume Methods for Open-Channel Flow

466

7.4.1 The Riemann Problem 467

7.4.2 Numerical Flux Functions 468

7.4.3 Transcritical Depression Waves 471

7.4.4 Source Term Discretization 472

7.4.5 Extension to Second Order Accuracy 474

7.4.6 Flux Limiting 476

7.4.7 Stability Analysis 477

7.4.8 Computational Results 478

7.4.9 Zero-Inertia Deforming-Cell Model 479

7.4.9.1 Inflow Boundary 482

7.4.9.2 Surge Front 482

7.5 Dispersive Waves

484

7.5.1 Stability Analysis 486

7.5.2 Computational Results 487

7.5.3 Serre Equations 490

7.5.4 Finite-Volume Methods 491

Problems

493

References

495

8. Methods for Two-Dimensional Shallow-Water Flow

8.1 Introduction

502

8.2 The Numerical Method of Bicharacteristics

504

8.2.1 Parametric Form of Characteristic Relations 504

8.2.2 Direct Tetrahedral Network 505

8.2.3 Inverse Tetrahedral Network 506

8.2.4 Inverse Pentahedral Network 508

8.2.4.1 Discrete Compatibility Equations 511

8.2.4.2 Predictor Step 512

8.2.4.3 Corrector Step 513

8.2.4.4 Bicharacteristic Tangency Condition 515

8.2.4.5 Bivariate Interpolation of Initial Data 516

8.2.4.6 Stability Analysis 518

8.2.4.7 Moving Grid Algorithm 521

8.2.4.8 Boundary Conditions 523

8.2.4.9 Computational Results 524

8.3 Finite-Difference Models

526

8.3.1 Leendertse Scheme 526

8.3.1.1 Stability Analysis 529

8.3.2 Computational Results 531

8.3.3 MacCormack Scheme 532

8.3.3.1 Boundary Conditions 533

8.3.3.2 Stability Analysis 535

8.3.3.3 Computational Results 535

8.4 Finite-Element Models

537

8.4.1 Deforming Element Formulation 538

8.4.2 The Dissipative Interface 540

8.4.3 Deforming Flow Domain 543

8.4.4 Computational Results 544

8.5 Finite-Volume Models

546

8.5.1 Structured Grid Model 547

8.5.2 The MUSCL Scheme for Two-Dimensional Flow 550

8.5.3 Boundary Conditions 553

8.5.4 Source Term Discretization 554

8.5.4.1 Hydrostatic Imbalance 555

8.5.5 Critical Flow Sections 556

8.5.6 Stability Analysis 556

8.5.7 Wave Propagation on Dry Terrain 557

8.5.7.1 Steep Slopes With Low Runoff 559

8.5.8 Computational Results 560

Problems

564

References

565

9. Methods for Incompressible Viscous Flow

9.1 Introduction

570

9.2 Projection Method

575

9.2.1 2D Staggered Grid Discretization 577

9.2.2 Time Integration 578

9.2.2.1 Stability Condition 579

9.2.2.2 Semi-Implicit Formulation 580

9.2.3 Spatial Discretization 580

9.2.3.1 Averaging Errors 581

9.2.4 Upwinding of Advective Terms 582

9.2.5 Boundary Conditions 583

9.2.6 Computational Results 584

9.2.7 Higher-Order Projection methods 585

9.2.7.1 Block LU Factorization 587

9.2.7.2 Strong-Stability-Preserving Methods 589

9.3 Finite-Element Methods

591

9.3.1 Mixed Element Formulation 592

9.3.2 Lagrange Multiplier Approach 595

9.3.3 Penalty Methods 596

9.3.4 Artificial Compressibility 598

9.4 Finite-Volume Methods

600

9.4.1 Semi-Implicit Method for Pressure-Linked Equations

(SIMPLE) 600

9.4.1.1 SIMPLE Algorithm 602

9.4.2 FVM on Collocated Grids 605

9.4.3 Pressure-Implicit With Splitting of Operator (PISO) 607

9.4.3.1 PISO Algorithm 608

9.4.3.2 Stability Analysis 609

Problems

612

References

613

10. Deforming Grid Methods

10.1 Introduction

616

10.2 Finite-Difference Projection Method

619

10.2.1 Flow With Small Density Gradients 619

10.2.2 Staggered Spatial Discretization 620

10.2.3 Computational Results 624

10.3 FEM for Ideal Fluid Flow

628

10.3.1 Finite-Element Solution 630

10.3.1.1 Backwater Subdomain 631

10.3.1.2 Tailwater Subdomain 632

10.4 FEM for Viscous Flow

638

10.4.1 Boundary Conditions 639

10.4.2 Steady, Two-Dimensional Flow 641

10.4.2.1 Domain Discretization 641

10.4.2.2 Method of Weighted Residuals 642

10.4.2.3 Local Coordinates 643

10.4.2.4 Formulation of Global Matrices 644

10.4.2.5 Computation of Free-Surface 646

10.4.2.6 Computational Results 650

10.4.3 Unsteady Viscous Flow 653

10.4.3.1 Formulation of Residuals 653

10.4.3.2 Time Integration Scheme 655

10.4.3.3 Unsteady Flow Simulations 656

10.4.4 Extended Finite Element Method 660

10.4.5 Three-Dimensional Deforming FEM 662

10.4.5.1 Upstream Weighting 665

10.4.5.2 Deforming Element Formulation 667

10.4.5.3 Evaluation of Element Matrices 667

10.4.5.4 Nonlinear System Solver 669

10.4.5.5 Computational Results 670

10.4.6 ALE FEM in Three Dimensions 671

10.5 Structured Finite-Volume Method

674

10.5.1 Conservation Form of Equations 674

10.5.2 Velocity of Nodal Motion 675

10.5.3 Finite Volume Equations 676

10.5.4 Time Integration 678

10.5.4.1 Free Surface Elevation 679

10.5.4.2 The Dynamic Pressure Solver 681

10.5.5 Scalar Transport 683

10.5.6 Spatial Discretization 684

10.5.7 Computational Results 685

10.6 Unstructured Large-Scale Models

691

10.6.1 Vertical Coordinates 691

10.6.2 Governing Equations 693

10.6.3 z-Level Unstructured Grid 694

10.6.4 Numerical Algorithm 697

10.6.4.1 Drag Boundary Conditions 698

10.6.5 Discrete Continuity Equation 699

10.6.6 Advection of Momentum 699

10.6.6.1 Horizontal Diffusion of Momentum 702

10.6.6.2 Non-Hydrostatic Pressure 703

10.6.6.3 Discretized Transport Equations 704

10.6.6.4 Stability Conditions 705

10.6.7 Computational Results 706

Problems

708

References

709

11. Marker and Cell Method

11.1 Introduction

714

11.2 Particle-In-Cell Method

716

11.2.1 Computational Results 718

11.3 Marker-And-Cell Method

719

11.3.1 2D MAC Method 719

11.3.2 Initial and Boundary Conditions 723

11.3.2.1 Inflow Boundary 724

11.3.2.2 Outflow Boundary 725

11.3.2.3 Free-Slip Wall Boundary 725

11.3.2.4 No-Slip Wall Boundary 725

11.3.2.5 Permeable Wall Boundary 726

11.3.2.6 Corner Boundary 726

11.3.2.7 Free-Surface Boundary 727

11.3.3 Modified Free-Surface Condition 731

11.3.4 Particle Movement 733

11.3.5 The Overall Algorithm 734

11.3.6 Stability Conditions 735

11.3.7 Laminar Flow Applications 736

11.4 Turbulent Flow Simulation

739

11.4.1 The Donor Cell Upwind Scheme 742

11.4.1.1 Boundary Conditions for Turbulent Flow 744

11.4.2 Turbulent Flow Applications 745

11.5 Semi-Implicit MAC Method

748

11.5.0.1 Streamwise Momentum Equation 748

11.5.0.2 Vertical Momentum Equation 751

11.5.1 Enforcement of Incompressibility 754

11.6 Extension to Inclined Channels

755

11.6.0.1 Particle Movement 756

11.6.0.2 Computational Results 757

11.7 Recent Developments

760

Problems

763

References

764

12. Volume of Fluid Method

12.1 Introduction

768

12.2 Simple Line Interface Calculation

770

12.3 Fractional Volume of Fluid

772

12.3.1 Pressure Definition in a Surface Cell 773

12.3.2 Advection of Fractional Volume of Fluid 774

12.3.3 Subgrid Computations 778

12.3.3.1 Computational Results 778

12.3.4 Piece-Wise Linear Interface Calculation 779

12.3.4.1 The Interface Normal 781

12.3.5 Intersection With Cell Edges 782

12.4 Analytical Reconstruction Methods

785

12.4.0.1 Interface Position 786

12.4.1 Lagrangian Advection of the Interface 789

12.4.2 Extension to Three Dimensions 790

12.4.3 Computational Results 793

12.4.4 Eulerian Advection of the Interface 793

12.4.4.1 Sudden Closing of Sluice Gate 793

12.4.4.2 Fluid-Structure Interaction 795

12.4.4.3 Two-Phase Flow: Breaking Waves 796

12.4.4.4 Two-Phase Flow: Bubble Formation 796

Problems

801

References

802

13. Level Set Method

13.1 Introduction

806

13.2 Implicit Surfaces

807

13.3 Level Set Method

808

13.3.1 The Level Set Function 808

13.3.2 Evolution of the Level Set Function 810

13.3.3 Free-Surface Thickness 810

13.3.4 The Signed Distance Function 811

13.3.5 Re-Initialization of the Level Set Function 813

13.3.5.1 Smoothing the Signed Distance Function 815

13.4 WENO Scheme for Interface Advection

816

13.5 Computational Results

819

13.5.0.1 Multi-Marker, Level Set Method 819

13.5.0.2 Iso-Geometric Analysis Model 820

13.5.0.3 Immersed Boundary – Level Set Method 821

13.5.1 Comparison of Volume of Fluid and Level Set Methods 824

Problems

826

References

828

14. Smoothed Particle Hydrodynamics

14.1 Introduction

832

14.2 Integral Representation of Fluid Properties

834

14.2.1 Selection of SPH Kernel 834

14.2.2 Approximate Kernel Functions 835

14.2.3 Accuracy of SPH Approximation 837

14.2.4 Evaluation of Derivatives 838

14.3 Summation Representation of Fluid Properties

839

14.3.1 Summation Representation of Derivatives 840

14.4 SPH for Viscous Flow

843

14.4.1 Conservation of Mass 843

14.4.2 Conservation of Momentum 844

14.4.2.1 Viscosity Models 845

14.4.2.2 Artificial Viscosity 845

14.4.2.3 Equation of State 846

14.4.3 Adaptive Smoothing Length 847

14.5 Boundary Conditions

848

14.5.1 No-Slip Wall Boundary 848

14.5.2 Free-Slip Wall Boundary 849

14.5.3 Free Surface Boundary 849

14.6 Propagation of Particles

850

14.6.0.1 Stability Conditions 851

14.6.1 Enhanced SPH Methods 852

14.7 Practical Implementation

855

14.8 Computational Results

857

14.8.1 Two-Dimensional Dam-Break Wave 857

14.8.2 Impact and Ricochet of Plunging Jet 857

14.8.3 Ice-Shelf Dynamics 859

14.8.4 Three-Dimensional Dam-Break Model 860

14.8.5 Simulation of Spillway Flow 863

14.8.6 Combined SPH and Level Set Method 863

Problems

865

References

866

Epilogue 867

Note

869

Bibliography 871

Index 875