Numerical Modeling in Open Channel Hydraulics (eBook)

Preface 6
Contents 9
1 Open Channel Flow Equations 14
1.1 Basic Definitions 14
1.2 General Equations for Incompressible Liquid Flow 23
1.3 Derivation of 1D Dynamic Equation 25
1.4 Derivation of 1D Continuity Equation 33
1.5 System of Equations for Unsteady Gradually Varied Flow in Open Channel 34
1.6 Steady Gradually Varied Flow in Open Channel 37
1.6.1 Derivation of Governing Equation from the Energy Equation 38
1.6.2 Derivation of Governing Equation from the System of Saint-Venant Equations 40
1.7 Storage Equation 43
1.8 Equation of Mass Transport 45
1.8.1 Mass Transport in Flowing Water 46
1.8.2 Derivation of the Mass Transport Equation 48
1.9 Thermal Energy Transport Equation 56
1.10 Types of Equations Applied in Open Channel Hydraulics 62
References 63
2 Methods for Solving Algebraic Equations and Their Systems 65
2.1 Solution of Non-linear Algebraic Equations 65
2.1.1 Introduction 65
2.1.2 Bisection Method 66
2.1.3 False Position Method 68
2.1.4 Newton Method 70
2.1.5 Simple Fixed-Point Iteration 74
2.1.6 Hybrid Methods 78
2.2 Solution of Systems of the Linear Algebraic Equations 80
2.2.1 Introduction 81
2.2.2 Gauss Elimination Method 84
2.2.3 LU Decomposition Method 88
2.3 Solution of Non-linear System of Equations 91
2.3.1 Introduction 91
2.3.2 Newton Method 92
2.3.3 Picard Method 94
References 96
3 Numerical Solution of Ordinary Differential Equations 97
3.1 Initial-Value Problem 97
3.1.1 Introduction 97
3.1.2 Simple Integration Schemes 99
3.1.3 Runge--Kutta Methods 105
3.1.4 Accuracy and Stability 111
3.2 Initial Value Problem for a System of Ordinary Differential Equations 115
3.3 Boundary Value Problem 119
References 122
4 Steady Gradually Varied Flow in Open Channels 123
4.1 Introduction 123
4.1.1 Governing Equations 123
4.1.2 Determination of the Water Surface Profiles for Prismatic and Natural Channel 124
4.1.3 Formulation of the Initial and Boundary Value Problems for Steady Flow Equations 128
4.2 Numerical Solution of the Initial Value Problem for Steady Gradually Varied Flow Equation in a Single Channel 129
4.2.1 Numerical Integration of the Ordinary Differential Equations 130
4.2.2 Solution of the Non-linear Algebraic Equation Furnished by the Method of Integration 132
4.2.3 Examples of Numerical Solutions of the Initial Value problem 137
4.2.4 Flow Profile in a Channel with Sudden Change of Cross-Section 140
4.2.5 Flow Profile in Ice-Covered Channel 143
4.3 Solution of the Boundary Problem for Steady Gradually Varied Flow Equation in Single Channel 145
4.3.1 Introduction to the Problem 146
4.3.2 Direct Solution Using the Newton Method 147
4.3.3 Direct Solution Using the Newton Method with Quasi--Variable Discharge 151
4.3.4 Direct Solution Using the Improved Picard Method 153
4.3.5 Solution of the Boundary Problem Using the Shooting Method 156
4.4 Steady Gradually Varied Flow in Open Channel Networks 159
4.4.1 Formulation of the Problem 159
4.4.2 Numerical Solution of Steady Gradually Varied Flow Equations in Channel Network 161
References 169
5 Partial Differential Equations of Hyperbolic and Parabolic Type 170
5.1 Types of Partial Differential Equations and Their Properties 170
5.1.1 Classification of the Partial Differential Equations of 2nd Order with Two Independent Variables 170
5.1.2 Classification of the Partial Differential Equations via Characteristics 172
5.1.3 Classification of the Saint Venant System and Its Characteristics 176
5.1.4 Well Posed Problem of Solution of the Hyperbolic and Parabolic Equations 180
5.1.5 Properties of the Hyperbolic and Parabolic Equations 185
5.1.6 Properties of the Advection-Diffusion Transport Equation 189
5.2 Introduction to the Finite Difference Method 194
5.2.1 Basic Information 194
5.2.2 Approximation of the Derivatives 196
5.2.3 Example of Solution: Advection Equation 205
5.3 Introduction to the Finite Element Method 208
5.3.1 General Concept of the Finite Element Method 208
5.3.2 Example of Solution: Diffusion Equation 214
5.4 Properties of the Numerical Methods for Partial Differential Equations 220
5.4.1 Convergence 220
5.4.2 Consistency 222
5.4.3 Stability 223
References 228
6 Numerical Solution of the Advection Equation 229
6.1 Solution by the Finite Difference Method 229
6.1.1 Approximation with the Finite Difference Box Scheme 229
6.1.2 Stability Analysis of the Box Scheme 232
6.2 Amplitude and Phase Errors 235
6.3 Accuracy Analysis Using the Modified Equation Approach 241
6.4 Solution of the Advection Equation with the Finite Element Method 249
6.4.1 Standard Finite Element Approach 249
6.4.2 Donea Approach 254
6.4.3 Modified Finite Element Approach 256
6.4.3.1 The Concept of the Modified Finite Element Method 256
6.4.3.2 Solution of the Advection Equation Using the Modified Finite Element Method 258
6.4.3.3 Stability Analysis of the Modified Finite Element Method 260
6.4.3.4 Accuracy Analysis Using the Modified Equation Approach 261
6.5 Numerical Solution of the Advection Equation with the Method of Characteristics 262
6.5.1 Problem Presentation 263
6.5.2 Linear Interpolation 264
6.5.3 Quadratic Interpolation 265
6.5.4 Holly--Preissmann Method of Interpolation 266
6.5.5 Interpolation with Spline Function of 3rd Degree 268
References 271
7 Numerical Solution of the Advection-Diffusion Equation 272
7.1 Introduction to the Problem 272
7.2 Solution by the Finite Difference Method 273
7.2.1 Solution Using General Two Level Scheme with Up-Winding Effect 274
7.2.2 The Difference Crank-Nicolson Scheme 278
7.2.3 Numerical Diffusion Versus Physical Diffusion 281
7.2.4 The QUICKEST Scheme 286
7.3 Solution Using the Modified Finite Element Method 288
7.4 Solution of the Advection-Diffusion Equation with the Splitting Technique 292
7.5 Solution of the Advection-Diffusion Equation Using the Splitting Technique and the Convolution Integral 298
7.5.1 Governing Equation and Splitting Technique 298
7.5.2 Solution of the Advective-Diffusive Equation by Convolution Approach 299
7.5.3 Solution of the Advective-Diffusive Equation with Variable Parameters and Without Source Term 302
7.5.4 Solution of the Advective-Diffusive Equation with Source Term 304
7.5.5 Solution of the Advective-Diffusive Equation in an Open Channel Network 306
References 309
8 Numerical Integration of the System of Saint Venant Equations 310
8.1 Introduction 310
8.2 Solution of the Saint Venant Equations Using the Box Scheme 311
8.2.1 Approximation of Equations 311
8.2.2 Accuracy Analysis Using the Modified Equation Approach 317
8.3 Solution of the Saint Venant Equations Using the Modified Finite Element Method 322
8.3.1 Spatial and Temporal Discretization of the Saint Venant Equations 322
8.3.2 Stability Analysis of the Modified Finite Element Method 329
8.3.3 Numerical Errors Generated by the Modified Finite Element Method 334
8.4 Some Aspects of Practical Application of the Saint Venant Equations 338
8.4.1 Formal Requirements and Actual Possibilities 339
8.4.2 Representation of the Channel Cross-Section 339
8.4.3 Initial and Boundary Conditions 342
8.4.4 Unsteady Flow in Open Channel Network 346
8.5 Solution of the Saint Venant Equations with Movable Channel Bed 351
8.5.1 Full System of Equations for the Sediment Transport 352
8.5.2 Initial and Boundary Conditions for the Sediment Transport Equations 356
8.5.3 Numerical Solution of the Sediment Transport Equations 358
8.6 Application of the Saint Venant Equations for Steep Waves 360
8.6.1 Problem Presentation 360
8.6.2 Conservative Form of the Saint Venant Equations 362
8.6.3 Solution of the Saint Venant Equations with Shock Wave 365
References 373
9 Simplified Equations of the Unsteady Flow in Open Channel 375
9.1 Simplified Forms of the Saint Venant Equations 375
9.2 Simplified Flood Routing Models in the Form of Transport Equations 380
9.2.1 Kinematic Wave Equation 380
9.2.2 Diffusive Wave Equation 381
9.2.3 Linear and Non-linear Forms of the Kinematic and Diffusive Wave Equations 385
9.3 Mass and Momentum Conservation in the Simplified Flood Routing Models in the Form of Transport Equations 386
9.3.1 The Mass and Momentum Balance Errors 388
9.3.2 Conservative and Non-conservative Forms of the Non-linear Advection-Diffusion Equation 391
9.3.3 Possible Forms of the Non-linear Kinematic Wave Equation 392
9.3.4 Possible Forms of the Non-linear Diffusive Wave Equation 396
9.4 Lumped Flood Routing Models 398
9.4.1 Standard Derivation of the Muskingum Equation 398
9.4.2 Numerical Solution of the Muskingum Equation 400
9.4.3 The Muskingum--Cunge Model 402
9.4.4 Relation Between the Lumped and Simplified Distributed Models 406
9.5 Convolution Integral in Open Channel Hydraulics 409
9.5.1 Open Channel Reach as a Dynamic System 409
9.5.2 IUH for Hydrological Models 415
9.5.3 An Alternative IUH for Hydrological Lumped Models 419
References 423
Index 425