ISBN: 354042895X
TITLE: Computational Methods in Environmental Fluid Mechanics
AUTHOR: Olaf Kolditz
TOC:

Part I - Continuum Mechanics 1
1 Balance Equations of Fluid Mechanics 3
1.1 General Conservation Law 3
1.1.1 Basic Equations of Fluid Dynamics 3
1.1.2 Conservation Quantities 5
1.1.3 Lagrangian Description of Motion 5
1.1.4 Eulerian Description of Motion 7
1 1.5 Reynolds Transport Theorem 9
1.1.6 Fluxes 9
1.1.7 General Balance Equation 11
1.2 Mass Conservation 12
1.3 Momentum Conservation 13
1.3.1 General Momentum Equation 13
1.3.2 Stress Tensor - sigma 14
1.3.3 Kinematics of a Fluid Element 17
1.3.4 Viscous Stress Tensor - tau 19
1.3.5 Euler Equations 21
1.3.6 Navier-Stokes Equations 21
1.3.7 Stokes Equations 21
1.3.8 Darcy Equations 22
1.4 Energy Conservation 22
1.4.1 Energy Balance 22
1.4.2 Energy Fluxes 23
1.4.3 Energy Sources 23
1.4.4 Integral Energy Balance Equation 23
1.4.5 Differential Energy Balance Equation 23
1.4.6 Balance Equation of Kinetic Energy 25
1.4.7 Balance Equation of Internal Energy 25
1.5 Problems 27
2 Turbulence 29
2.1 Physics of Turbulence 29
2.2 Reynolds Equations 34
2.3 Turbulence Models 36
2.3.1 Mixing Length Model 39
2.3.2 k-epsilon Model 40
3 Porous Media 45
3.1 Multiphase Media 45
3.2 Macroscopic Equations 47
3.3 Isothermal Consolidation of Porous Media 50
3.3.1 Mass Conservation 50
3.3.2 Momentum conservation 53
3.4 Mass Transport in Porous Media 58
3.5 Heat Transport in Porous Media 60
4 Problem Classification 63
4.1 Mathematical Classification 63
4.2 Physical Classification 64
4.3 Elliptic Equations 66
4.4 Parabolic Equations 67
4.5 Hyperbolic Equations 68
4.6 Equation Types 70
4.7 Boundary Conditions 70
4.7.1 General Remarks 70
4.7.2 Mass Fluxes 71
4.7.3 Momentum Fluxes 72
4.7.4 Energy Transfer 75
4.8 Problems 76
Bibliography 77
Part II - Numerical Methods 79
5 Numerical Methods 81
5.1 Solution Procedure 81
5.2 Theory of Discrete Approximation 83
5.2.1 Terminology 83
5.2.2 Errors and Accuracy 84
5.2.3 Convergence 85
5.2.4 Consistency 86
5.2.5 Stability 86
5.3 Solution Process 86
5.3.1 Linear Solver 87
5.3.2 Non-Linear Solver 91
5.4 Problems 96
6 Finite Difference Method 97
6.1 Approximation of Derivatives 97
6.1.1 Taylor Series Expansion (TSE) 97
6.1.2 First-Order Derivatives 98
6.1.3 Second-Order Derivatives 99
6.2 Diffusion Equation 100
6.2.1 Explicit and Implicit Schemes 100
6.2.2 Explicit FTCS Scheme 101
6.2.3 Fully Implicit Scheme 105
6.2.4 Crank-Nicolson Scheme (CNS) 107
6.2.5 Generalized Scheme 108
6.2.6 Initial and Boundary Conditions 108
6.3 Advection Equation 110
6.3.1 Explicit FTCS Scheme 110
6.3.2 Upwind Difference Representation 111
6.3.3 Leapfrog and Lax-Wendroff Schemes 113
6.3.4 Crank-Nicolson Scheme (CNS) 114
6.3.5 Euler-Taylor Scheme 115
6.3.6 Numerical Dispersion and Dissipation 115
6.4 Transport Equation 116
6.4.1 Steady Transport Equation 117
6.4.2 Explicit Schemes for the Linear Transport Equation 121
6.4.3 Implicit Schemes for the Linear Transport Equation 121
6.5 Burgers Equation 122
6.5.1 Explicit Schemes 124
6.5.2 Implicit Schemes 125
6.6 Problems 126
7 Finite Element Method 129
7.1 Domain Discretization 130
7.2 Equation Discretization 132
7.2.1 Variational Principles 133
7.2.2 Method of Weighted Residuals 134
7.2.3 Galerkin Method 135
7.3 Interpolation and Shape Functions 137
7.3.1 General Method for Shape Function Derivation 137
7.3.2 Shape Function Conditions 138
7.3.3 Isoparametric Mapping 139
7.4 Element Shape Functions 140
7.4.1 1-D Linear Bar Elements 140
7.4.2 1-D Quadratic Elements 144
7.4.3 2-D Triangular Elements 145
7.4.4 2-D Bilinear Quadrilateral Elements 148
7.4.5 3-D Tetrahedral Elements 150
7.4.6 3-D Triangular Prismatic Elements 152
7.4.7 3-D Hexahedrai Element 154
7.5 Network Simulation 157
7.5.1 Pipe Networks - 1-D Elements in R^3 157
7.5.2 Fracture Networks - 2-D Triangular Elements in R^3 159
7.6 Diffusion Equation 161
7.6.1 Steady Diffusion Equation 161
7.6.2 Unsteady Diffusion Equation 163
7.7 Transport Equation 166
7.7.1 Unsteady Transport Equation 166
7.8 Problems 170
8 Finite Volume Method 173
8.1 Discretization Process 174
8.2 Evaluation of Time Derivatives 176
8.3 Evaluation of Cell Area/Volume 176
8.4 Evaluation of Fluxes 178
8.4.1 Two-Dimensional Schemes 178
8.4.2 Central Scheme on a Cartesian Mesh 179
8.4.3 Upwind Scheme on a Cartesian Mesh 181
8.4.4 Integration Formulas 182
8.5 Diffusion Equation 183
8.6 Advection Equation 185
8.7 Problems 187
Bibliography 189
Part III - Software-Engineering 191
9 Object-Oriented Methods for Hydrosystem Modeling 193
9.1 Introduction 193
9.2 Investigation Objects 194
9.2.1 Geometric modeling 194
9.2.2 Processes 196
9.3 Mathematical Objects 196
9.4 Numeric Objects 199
9.4.1 Discretization Process 199
9.4.2 Solution Process 200
9.4.3 Common Methods 200
9.5 Informatic Objects 201
9.5.1 Data Structures 202
9.6 Software Objects 203
10 Object-Oriented Programming Techniques 207
10.1 Object Orientation 207
10.2 Object Design 208
10.2.1 Mathematical Objects 208
10.2.2 Numerical Objects 209
10.2.3 Algebraic Equations - Solver 211
10.2.4 Model-Kernel Concept 213
10.3 Object Implementation 214
10.3.1 Object 214
10.3.2 Nodes and Elements 215
10.3.3 Node Related Objects 217
10.3.4 Element Related Objects 219
10.3.5 Equation Systems and Solver 222
10.3.6 Models 224
10.3.7 FEM - Application 227
10.4 Graphical User Interface 228
10.4.1 Shell - Objects 228
10.4.2 Application Embedding 229
10.4.3 Object - GUI - Interface 230
11 Element Implement at ion 233
11.1 2-D Linear Triangular Elements 233
11.1.1 Data Input 233
11.1.2 Element Geometry 234
11.1.3 Element Matrices 235
11.1.4 Equation System 235
11.1.5 Element Resultants 235
11.2 Problems 236
Bibliography 237
Part IV - Selected Topics 239
12 Non-Linear Flow in Fractured Media 241
12.1 Introduction and Background 241
12.1.1 Theoretical Issues 241
12.1.2 Experimental Issues 244
12.2 Governing Equations 249
12.3 Numerical Scheme 251
12.3.1 Galerkin Method 251
12.3.2 Finite Element Approach 253
12.3.3 Evaluation of Element, Matrices in Local Coordinates 253
12.3.4 Resolution of the Non-Linearities 254
12.4 Non-Linear Flow in Single Fractures 255
12.4.1 Numerical Properties 256
12.4.2 Effects of Non-Linearity 259
12.4.3 Fracture Roughness 260
12.5 Non-linear Flow in Fracture Systems 261
12.6 Concluding Remarks 266
12.7 Appendix 266
Bibliography 269
13 Heat Transport in Fractured-Porous Media 271
13.1 Introduction 271
13.2 Governing Equations 275
13.3 Numerical Procedure 277
13.4 Example - Rosemanowes Hot Dry Rock Site 277
13.4.1 Data 277
13.4.2 Simulation Results 280
13.4.3 Discussion 289
13.5 Case Study: Soultz-sous-Forts 293
Bibliography 296
14 Density Dependent Flow in Porous Media 301
14.1 Introduction 302
14.1.1 Background 302
14.1.2 Prior work concerning variable-density flow and transport 303
14.2 Governing Equations 305
14.2.1 Macroscopic balance and constitutive equations 306
14.2.2 Boussinesq approximation of the mass balance of the fluid 309
14.2.3 Different formulations of the balance equation of solute mass 309
14.3 Finite Element Formulations 311
14.4 Examples 314
14.4.1 The Henry problem 315
14.4.2 The Elder problem 318
14.4.3 The salt dome problem 323
Bibliography 329
15 Multiphase Flow in Deformable Porous Media 333
15.1 Macroscopic Balance Equations 333
15.1.1 Mass Conservation in Static Porous Media 333
15.1.2 Mass Conservation in Deformable Porous Media 335
15.1.3 Momentum Conservation 336
15.2 Constitutive Relationships 337
15.2.1 Saturation 337
15.2.2 Density 338
15.2.3 Capillary Pressure 339
15.2.4 Relative Permeability 343
15.2.5 Total and Effective Stress 345
15.3 Governing Equations 347
15.3.1 Static Porous Medium - Two Phase Flow 348
15.3.2 Deformable Porous Medium - Richards Approximation 349
15.3.3 Deformable Porous Medium - Two Phase Flow 350
15.4 Finite Element Formulations 352
15.4.1 Mass Balance of Porous Medium 352
15.4.2 Equilibrium Problem 354
15.5 Implementation 356
15.5.1 Model 358
15.5.2 Loop 360
15.5.3 Element Matrices and Resultants 361
15.5.4 Equation System 361
15.5.5 Solver 361
15.5.6 Materials 363
15.5.7 Data Input 364
15.6 Examples 364
15.6.1 Benchmark Problem - Liakopoulos Experiment 364
15.6.2 Moisture Swelling of Bentonites 367
Bibliography 371
Index 373
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