Free-Surface Flow
Computational Methods
- 1st Edition - November 7, 2018
- Author: Nikolaos D. Katopodes
- Language: English
- Paperback ISBN:9 7 8 - 0 - 1 2 - 8 1 5 4 8 5 - 4
- eBook ISBN:9 7 8 - 0 - 1 2 - 8 1 5 4 8 6 - 1
Free-Surface Flow: Computational Methods presents a detailed analysis of numerical schemes for shallow-water waves. It includes practical applications for the numerical simulatio… Read more
Free-Surface Flow: Computational Methods presents a detailed analysis of numerical schemes for shallow-water waves. It includes practical applications for the numerical simulation of flow and transport in rivers and estuaries, the dam-break problem and overland flow. Closure models for turbulence, such as Reynolds-Averaged Navier-Stokes and Large Eddy Simulation are presented, coupling the aforementioned surface tracking techniques with environmental fluid dynamics. While many computer programs can solve the partial differential equations describing the dynamics of fluids, many are not capable of including free surfaces in their simulations.
- Provides numerical solutions of the turbulent Navier-Stokes equations in three space dimensions
- Includes closure models for turbulence, such as Reynolds-Averaged Navier-Stokes, and Large Eddy Simulation
- Practical applications are presented for the numerical simulation of flow and transport in rivers and estuaries, the dam-break problem and overland flow
Civil and Environmental Engineering, Coastal Engineering, and Ocean Engineering
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
91.3 Finite-Difference Approximations
141.3.1 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
211.4.1 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
391.5.1 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
491.7 Algorithmic Dissipation
511.7.0.1 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
711.9.1 Positivity and Monotonicity 71
1.10 Least-Squares Approximation
74Problems
76References
792. Finite-Difference Methods for Diffusion
2.1 Introduction
822.2 Explicit Scheme for Diffusion (FTCS)
842.2.1 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
962.3.1 Spurious Waves 97
2.3.2 Stability of FTCS Scheme 98
2.4 Leapfrog Scheme
1022.4.1 Stability Analysis of Leapfrog Scheme 103
2.5 du Fort-Frankel Scheme
1052.6 Implicit Scheme for Diffusion
1072.6.1 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
1112.7.1 Stability of Crank-Nicolson Scheme 112
2.7.2 Weighted Average Explicit-Implicit Scheme 112
Problems
115References
1173. Finite-Difference Methods for Advection
3.1 Introduction
1203.2 The Numerical Method of Characteristics
1223.2.1 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)
1303.3.1 Accuracy of Upwind Scheme 131
3.4 The Courant-Friedrichs-Lewy (CFL) Condition
1373.4.1 Stability of Explicit Upwind Scheme 138
3.5 Centered Explicit Scheme (FTCS)
1403.6 Implicit Upwind Scheme (BTBS)
1423.6.1 Stability of the BTBS Scheme 143
3.7 Lax-Friedrichs Scheme
1463.7.1 Stability Analysis 147
3.8 Leapfrog Scheme
1503.8.1 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
1613.9.1 Fourier Analysis of Lax-Wendroff Scheme 163
3.9.2 Two-Step Lax-Wendroff-Richtmyer Scheme 164
3.10 Beam and Warming Scheme
1663.10.1 Stability Analysis 167
3.11 Parasitic Waves, Dissipation, and Dispersion
1693.11.1 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
1793.12.1 Steady State Solution 181
3.12.2 Generalized Upwind Method 184
3.13 Transient Advection-Diffusion Schemes
1883.13.1 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
195References
1974. Finite-Element and Finite-Volume Methods for Scalar
Transport
4.1 Introduction
2004.1.1 Variational Principles 200
4.1.1.1 Functional for Steady State Diffusion 201
4.2 The Finite-Element Method (FEM)
2034.2.1 Basis Functions 204
4.2.2 FEM Approximation of the Functional 205
4.3 Method of Weighted Residuals
2074.3.1 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
2134.5 Finite-Element Model for Transient Diffusion
2174.5.1 Time Domain Discretization 218
4.6 Finite-Element Model for Advection
2214.6.1 Semi-Discrete Form 222
4.6.2 Advection of a Sharp Concentration Front 223
4.7 Petrov-Galerkin Modification
2264.7.1 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
2394.9 Finite Volume Method for Advection
2414.9.1 Conservative Fluxes 242
4.9.2 Upwind Finite Volume Scheme 244
4.9.3 QUICK Scheme for Advection 244
4.10 Total Variation Diminishing
2474.11 Superbee Limiter for Advection
2484.11.1 Comparison With the Petrov-Galerkin Finite-Element
Model 249
4.12 Discontinuous Galerkin Method
2524.12.1 Linear Advection Equation 254
4.12.2 Stability Analysis 255
Problems
257References
2585. Finite-Difference Methods for Equilibrium Problems
5.1 Introduction
2625.2 Domain Discretization
2635.2.1 Choice of Computational Nodes 267
5.3 Equilibrium Problems
2695.3.1 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
2785.4.1 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
2925.5.1 Conjugate Gradient Method 293
5.6 Matrix Preconditioning
2965.6.1 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
3105.7.1 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
3325.8.1 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
3385.9.1 Dirichlet Boundaries 338
5.9.2 Neumann Boundaries 341
Problems
345References
3486. Methods for Two-Dimensional Scalar Transport
6.1 Introduction
3526.2 Finite-Difference Models for Diffusion
3536.2.1 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
3606.3.1 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
3776.4.1 Stability of Crank-Nicolson Scheme 377
6.4.2 Cross-Wind Diffusion 380
6.5 Finite-Element Analysis
3836.5.1 Two-Dimensional Shape Functions 385
6.6 Galerkin Formulation
3886.6.1 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
400References
4027. Methods for Open-Channel Flow
7.1 The Method of Characteristics
4067.1.1 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
4207.2.1 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
4417.3.1 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
4667.4.1 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
4847.5.1 Stability Analysis 486
7.5.2 Computational Results 487
7.5.3 Serre Equations 490
7.5.4 Finite-Volume Methods 491
Problems
493References
4958. Methods for Two-Dimensional Shallow-Water Flow
8.1 Introduction
5028.2 The Numerical Method of Bicharacteristics
5048.2.1 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
5268.3.1 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
5378.4.1 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
5468.5.1 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
564References
5659. Methods for Incompressible Viscous Flow
9.1 Introduction
5709.2 Projection Method
5759.2.1 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
5919.3.1 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
6009.4.1 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
612References
61310. Deforming Grid Methods
10.1 Introduction
61610.2 Finite-Difference Projection Method
61910.2.1 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
62810.3.1 Finite-Element Solution 630
10.3.1.1 Backwater Subdomain 631
10.3.1.2 Tailwater Subdomain 632
10.4 FEM for Viscous Flow
63810.4.1 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
67410.5.1 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
69110.6.1 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
708References
70911. Marker and Cell Method
11.1 Introduction
71411.2 Particle-In-Cell Method
71611.2.1 Computational Results 718
11.3 Marker-And-Cell Method
71911.3.1 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
73911.4.1 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
74811.5.0.1 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
75511.6.0.1 Particle Movement 756
11.6.0.2 Computational Results 757
11.7 Recent Developments
760Problems
763References
76412. Volume of Fluid Method
12.1 Introduction
76812.2 Simple Line Interface Calculation
77012.3 Fractional Volume of Fluid
77212.3.1 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
78512.4.0.1 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
801References
80213. Level Set Method
13.1 Introduction
80613.2 Implicit Surfaces
80713.3 Level Set Method
80813.3.1 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
81613.5 Computational Results
81913.5.0.1 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
826References
82814. Smoothed Particle Hydrodynamics
14.1 Introduction
83214.2 Integral Representation of Fluid Properties
83414.2.1 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
83914.3.1 Summation Representation of Derivatives 840
14.4 SPH for Viscous Flow
84314.4.1 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
84814.5.1 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
85014.6.0.1 Stability Conditions 851
14.6.1 Enhanced SPH Methods 852
14.7 Practical Implementation
85514.8 Computational Results
85714.8.1 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
865References
866Epilogue 867
Note
869Bibliography 871
Index 875
- Edition: 1
- Published: November 7, 2018
- Language: English
NK
Nikolaos D. Katopodes
Nikolaos D. Katopodes, University Michigan Ann Arbor, Department of Civil & Environmental Engineering, Ann Arbor, United States. Dr. Katopodes has chaired or co-chaired 28 PhD student theses. His research has resulted in over 200 publications, and several software packages that are used worldwide for the analysis and control of free-surface flows.
Affiliations and expertise
Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, USARead Free-Surface Flow on ScienceDirect