Szymkiewicz Numerical Modeling in Open Channel Hydraulics

1. Auflage 2010
ISBN: 978-90-481-3674-2
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 370 Seiten

Open channel hydraulics has always been a very interesting domain of scienti c and engineering activity because of the great importance of water for human l- ing. The free surface ow, which takes place in the oceans, seas and rivers, can be still regarded as one of the most complex physical processes in the environment. The rst source of dif culties is the proper recognition of physical ow processes and their mathematical description. The second one is related to the solution of the derived equations. The equations arising in hydrodynamics are rather comp- cated and, except some much idealized cases, their solution requires application of the numerical methods. For this reason the great progress in open channel ow modeling that took place during last 40 years paralleled the progress in computer technique, informatics and numerical methods. It is well known that even ty- cal hydraulic engineering problems need applications of computer codes. Thus, we witness a rapid development of ready-made packages, which are widely d- seminated and offered for engineers. However, it seems necessary for their users to be familiar with some fundamentals of numerical methods and computational techniques applied for solving the problems of interest. This is helpful for many r- sons. The ready-made packages can be effectively and safely applied on condition that the users know their possibilities and limitations. For instance, such knowledge is indispensable to distinguish in the obtained solutions the effects coming from the considered physical processes and those caused by numerical artifacts.

Autoren/Hrsg.

Szymkiewicz, Romuald

Inhaltsverzeichnis

1;Preface;6
2;Contents;9
3;1 Open Channel Flow Equations;14
3.1;1.1 Basic Definitions;14
3.2;1.2 General Equations for Incompressible Liquid Flow;23
3.3;1.3 Derivation of 1D Dynamic Equation;25
3.4;1.4 Derivation of 1D Continuity Equation;33
3.5;1.5 System of Equations for Unsteady Gradually Varied Flow in Open Channel;34
3.6;1.6 Steady Gradually Varied Flow in Open Channel;37
3.6.1;1.6.1 Derivation of Governing Equation from the Energy Equation;38
3.6.2;1.6.2 Derivation of Governing Equation from the System of Saint-Venant Equations;40
3.7;1.7 Storage Equation;43
3.8;1.8 Equation of Mass Transport;45
3.8.1;1.8.1 Mass Transport in Flowing Water;46
3.8.2;1.8.2 Derivation of the Mass Transport Equation;48
3.9;1.9 Thermal Energy Transport Equation;56
3.10;1.10 Types of Equations Applied in Open Channel Hydraulics;62
3.11;References;63
4;2 Methods for Solving Algebraic Equations and Their Systems;65
4.1;2.1 Solution of Non-linear Algebraic Equations;65
4.1.1;2.1.1 Introduction;65
4.1.2;2.1.2 Bisection Method;66
4.1.3;2.1.3 False Position Method;68
4.1.4;2.1.4 Newton Method;70
4.1.5;2.1.5 Simple Fixed-Point Iteration;74
4.1.6;2.1.6 Hybrid Methods;78
4.2;2.2 Solution of Systems of the Linear Algebraic Equations;80
4.2.1;2.2.1 Introduction;81
4.2.2;2.2.2 Gauss Elimination Method;84
4.2.3;2.2.3 LU Decomposition Method;88
4.3;2.3 Solution of Non-linear System of Equations;91
4.3.1;2.3.1 Introduction;91
4.3.2;2.3.2 Newton Method;92
4.3.3;2.3.3 Picard Method;94
4.4;References;96
5;3 Numerical Solution of Ordinary Differential Equations;97
5.1;3.1 Initial-Value Problem;97
5.1.1;3.1.1 Introduction;97
5.1.2;3.1.2 Simple Integration Schemes;99
5.1.3;3.1.3 Runge--Kutta Methods;105
5.1.4;3.1.4 Accuracy and Stability;111
5.2;3.2 Initial Value Problem for a System of Ordinary Differential Equations;115
5.3;3.3 Boundary Value Problem;119
5.4;References;122
6;4 Steady Gradually Varied Flow in Open Channels;123
6.1;4.1 Introduction;123
6.1.1;4.1.1 Governing Equations;123
6.1.2;4.1.2 Determination of the Water Surface Profiles for Prismatic and Natural Channel;124
6.1.3;4.1.3 Formulation of the Initial and Boundary Value Problems for Steady Flow Equations;128
6.2;4.2 Numerical Solution of the Initial Value Problem for Steady Gradually Varied Flow Equation in a Single Channel;129
6.2.1;4.2.1 Numerical Integration of the Ordinary Differential Equations;130
6.2.2;4.2.2 Solution of the Non-linear Algebraic Equation Furnished by the Method of Integration;132
6.2.3;4.2.3 Examples of Numerical Solutions of the Initial Value problem;137
6.2.4;4.2.4 Flow Profile in a Channel with Sudden Change of Cross-Section;140
6.2.5;4.2.5 Flow Profile in Ice-Covered Channel;143
6.3;4.3 Solution of the Boundary Problem for Steady Gradually Varied Flow Equation in Single Channel;145
6.3.1;4.3.1 Introduction to the Problem;146
6.3.2;4.3.2 Direct Solution Using the Newton Method;147
6.3.3;4.3.3 Direct Solution Using the Newton Method with Quasi--Variable Discharge;151
6.3.4;4.3.4 Direct Solution Using the Improved Picard Method;153
6.3.5;4.3.5 Solution of the Boundary Problem Using the Shooting Method;156
6.4;4.4 Steady Gradually Varied Flow in Open Channel Networks;159
6.4.1;4.4.1 Formulation of the Problem;159
6.4.2;4.4.2 Numerical Solution of Steady Gradually Varied Flow Equations in Channel Network;161
6.5;References;169
7;5 Partial Differential Equations of Hyperbolic and Parabolic Type;170
7.1;5.1 Types of Partial Differential Equations and Their Properties;170
7.1.1;5.1.1 Classification of the Partial Differential Equations of 2nd Order with Two Independent Variables;170
7.1.2;5.1.2 Classification of the Partial Differential Equations via Characteristics;172
7.1.3;5.1.3 Classification of the Saint Venant System and Its Characteristics;176
7.1.4;5.1.4 Well Posed Problem of Solution of the Hyperbolic and Parabolic Equations;180
7.1.5;5.1.5 Properties of the Hyperbolic and Parabolic Equations;185
7.1.6;5.1.6 Properties of the Advection-Diffusion Transport Equation;189
7.2;5.2 Introduction to the Finite Difference Method;194
7.2.1;5.2.1 Basic Information;194
7.2.2;5.2.2 Approximation of the Derivatives;196
7.2.3;5.2.3 Example of Solution: Advection Equation;205
7.3;5.3 Introduction to the Finite Element Method;208
7.3.1;5.3.1 General Concept of the Finite Element Method;208
7.3.2;5.3.2 Example of Solution: Diffusion Equation;214
7.4;5.4 Properties of the Numerical Methods for Partial Differential Equations;220
7.4.1;5.4.1 Convergence;220
7.4.2;5.4.2 Consistency;222
7.4.3;5.4.3 Stability;223
7.5;References;228
8;6 Numerical Solution of the Advection Equation;229
8.1;6.1 Solution by the Finite Difference Method;229
8.1.1;6.1.1 Approximation with the Finite Difference Box Scheme;229
8.1.2;6.1.2 Stability Analysis of the Box Scheme;232
8.2;6.2 Amplitude and Phase Errors;235
8.3;6.3 Accuracy Analysis Using the Modified Equation Approach;241
8.4;6.4 Solution of the Advection Equation with the Finite Element Method;249
8.4.1;6.4.1 Standard Finite Element Approach;249
8.4.2;6.4.2 Donea Approach;254
8.4.3;6.4.3 Modified Finite Element Approach;256
8.4.3.1;6.4.3.1 The Concept of the Modified Finite Element Method;256
8.4.3.2;6.4.3.2 Solution of the Advection Equation Using the Modified Finite Element Method;258
8.4.3.3;6.4.3.3 Stability Analysis of the Modified Finite Element Method;260
8.4.3.4;6.4.3.4 Accuracy Analysis Using the Modified Equation Approach;261
8.5;6.5 Numerical Solution of the Advection Equation with the Method of Characteristics;262
8.5.1;6.5.1 Problem Presentation;263
8.5.2;6.5.2 Linear Interpolation;264
8.5.3;6.5.3 Quadratic Interpolation;265
8.5.4;6.5.4 Holly--Preissmann Method of Interpolation;266
8.5.5;6.5.5 Interpolation with Spline Function of 3rd Degree;268
8.6;References;271
9;7 Numerical Solution of the Advection-Diffusion Equation;272
9.1;7.1 Introduction to the Problem;272
9.2;7.2 Solution by the Finite Difference Method;273
9.2.1;7.2.1 Solution Using General Two Level Scheme with Up-Winding Effect;274
9.2.2;7.2.2 The Difference Crank-Nicolson Scheme;278
9.2.3;7.2.3 Numerical Diffusion Versus Physical Diffusion;281
9.2.4;7.2.4 The QUICKEST Scheme;286
9.3;7.3 Solution Using the Modified Finite Element Method;288
9.4;7.4 Solution of the Advection-Diffusion Equation with the Splitting Technique;292
9.5;7.5 Solution of the Advection-Diffusion Equation Using the Splitting Technique and the Convolution Integral;298
9.5.1;7.5.1 Governing Equation and Splitting Technique;298
9.5.2;7.5.2 Solution of the Advective-Diffusive Equation by Convolution Approach;299
9.5.3;7.5.3 Solution of the Advective-Diffusive Equation with Variable Parameters and Without Source Term;302
9.5.4;7.5.4 Solution of the Advective-Diffusive Equation with Source Term;304
9.5.5;7.5.5 Solution of the Advective-Diffusive Equation in an Open Channel Network;306
9.6;References;309
10;8 Numerical Integration of the System of Saint Venant Equations;310
10.1;8.1 Introduction;310
10.2;8.2 Solution of the Saint Venant Equations Using the Box Scheme;311
10.2.1;8.2.1 Approximation of Equations;311
10.2.2;8.2.2 Accuracy Analysis Using the Modified Equation Approach;317
10.3;8.3 Solution of the Saint Venant Equations Using the Modified Finite Element Method;322
10.3.1;8.3.1 Spatial and Temporal Discretization of the Saint Venant Equations;322
10.3.2;8.3.2 Stability Analysis of the Modified Finite Element Method;329
10.3.3;8.3.3 Numerical Errors Generated by the Modified Finite Element Method;334
10.4;8.4 Some Aspects of Practical Application of the Saint Venant Equations;338
10.4.1;8.4.1 Formal Requirements and Actual Possibilities;339
10.4.2;8.4.2 Representation of the Channel Cross-Section;339
10.4.3;8.4.3 Initial and Boundary Conditions;342
10.4.4;8.4.4 Unsteady Flow in Open Channel Network;346
10.5;8.5 Solution of the Saint Venant Equations with Movable Channel Bed;351
10.5.1;8.5.1 Full System of Equations for the Sediment Transport;352
10.5.2;8.5.2 Initial and Boundary Conditions for the Sediment Transport Equations;356
10.5.3;8.5.3 Numerical Solution of the Sediment Transport Equations;358
10.6;8.6 Application of the Saint Venant Equations for Steep Waves;360
10.6.1;8.6.1 Problem Presentation;360
10.6.2;8.6.2 Conservative Form of the Saint Venant Equations;362
10.6.3;8.6.3 Solution of the Saint Venant Equations with Shock Wave;365
10.7;References;373
11;9 Simplified Equations of the Unsteady Flow in Open Channel;375
11.1;9.1 Simplified Forms of the Saint Venant Equations;375
11.2;9.2 Simplified Flood Routing Models in the Form of Transport Equations;380
11.2.1;9.2.1 Kinematic Wave Equation;380
11.2.2;9.2.2 Diffusive Wave Equation;381
11.2.3;9.2.3 Linear and Non-linear Forms of the Kinematic and Diffusive Wave Equations;385
11.3;9.3 Mass and Momentum Conservation in the Simplified Flood Routing Models in the Form of Transport Equations;386
11.3.1;9.3.1 The Mass and Momentum Balance Errors;388
11.3.2;9.3.2 Conservative and Non-conservative Forms of the Non-linear Advection-Diffusion Equation;391
11.3.3;9.3.3 Possible Forms of the Non-linear Kinematic Wave Equation;392
11.3.4;9.3.4 Possible Forms of the Non-linear Diffusive Wave Equation;396
11.4;9.4 Lumped Flood Routing Models;398
11.4.1;9.4.1 Standard Derivation of the Muskingum Equation;398
11.4.2;9.4.2 Numerical Solution of the Muskingum Equation;400
11.4.3;9.4.3 The Muskingum--Cunge Model;402
11.4.4;9.4.4 Relation Between the Lumped and Simplified Distributed Models;406
11.5;9.5 Convolution Integral in Open Channel Hydraulics;409
11.5.1;9.5.1 Open Channel Reach as a Dynamic System;409
11.5.2;9.5.2 IUH for Hydrological Models;415
11.5.3;9.5.3 An Alternative IUH for Hydrological Lumped Models;419
11.6;References;423
12;Index;425