Numerical Modelling of Rapidly Varied River Flow
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The University of Nottingham School of Civil Engineering NUMERICAL MODELLING OF RAPIDLY VARIED RIVER FLOW SANG-HEON LEE, BSc. GEORGE GREEN LIBRARY OF SCIENCE AND ENGINEERING-( Thesis submitted to the University of Nottingham for the degree of Doctor of Philosophy NOVEMBER 2010 Abstract A new approach to solve shallow water flow problems over highly irregular geometry both correctly' and efficiently is presented in this thesis. Godunov-type schemes which are widely used with the fmite volume technique cannot solve the shallow water equations correctly unless the source terms related to the bed slope and channel width variation are discretized properly, because Godunov-type schemes were developed on the basis of homogeneous governing equations which is not compatible with an inhomogeneous system. The main concept of the new approach is to avoid a fractional step method and transform the shallow water equations into homogeneous form equations. New definitions for the source terms which can be incorporated into the homogeneous form equations are also proposed in this thesis. The modification to the homogeneous form equations combines the source terms with the flux term and solves them by the same solution structure of the numerical scheme. As a result the source terms are automatically discretized to achieve perfect balance with the flux terms without any special treatment and the method does not introduce numerical errors. Another point considered to achieve well-balanced numerical schemes is that the channel geometry should be reconstructed in order to be compatible with the numerical flux term which is computed with piecewise constant initial data. In this thesis, the channel geometry has been changed to have constant state inside each cell and, consequrently, each cell interface is considered as a discontinuity. The definition of the new flux related to the source terms has been obtained on the basis of the modified channel geometry. A simple and accurate algorithm to solve the moving boundary problem in two- dimensional modelling case has also been presented in this thesis. To solve the moving boundary condition, the locations of all the cell interfaces between the wet and dry cells have been detected first and the integrated numerical fluxes through the interfaces have been controlled according to the water surface level of the wet cells. The proposed techniques were applied to several well-known conservative schemes including Riemann solver based and verified against benchmark tests and natural river flow problems in the one and two dimensions. The numerical results shows good agreement with the analytical solutions, if available, and recorded data from other literature. The proposed approach features several advantages: 1) it can solve steady problems as well as highly unsteady ones over irregular channel geometry, 2) the numerical discretization of the source terms is always performed as the same way that the flux term is treated, 3) as a result, it shows stong applicability to various conservative numerical schemes, 4) it can solve the moving (wetting/drying) boundary problem correctly. The authour believes that this new method can be a good option to simulate natural river flows over highly irregular geometries. Publication A part of the research conducted for this thesis has resulted in the following publication: Lee S and Wright N : Simple and efficient solution of the shallow water equations with source terms. International Journal for Numerical Methods in Fluids, volume 63, Issue 3: 313-340, 30/5/2010 Acknowledgements I would like to thank my supervisor Dr. Nigel Wright for his support and assistance throughout this Ph.D.. I would also like to thank Dr. Ignacio Villanueva for his advice about numerical methods and Riemann solver based techniques. I am grateful for the fmancial support received from the Korean Government including the Ministry of Land, Transport and Maritime Affairs which made the study possible. Finally, a special thank should go to my wife Woon-Jung and two lovely daughters Kyoung-Eun and Young-Eun who supported and encouraged me during my study. Contents Chapter 1 Introduction •.••••.....••.•.•......••••.••.•.•••.••..•••.•••••••••·················1 Chapter 2 Hydraulic River Modelling •·······••···••·••••••••·••••••····•·•····..··5 2.1 Steady Flow Model 5 2.1.1 Uniforn Flow Model 5 2.1.2 Non-uniform. Flow Model ···············································6 2.2 Hydrodynamic Model 8 2.2.1 Shallow Water Model 10 2.2.2 Further Simplification of the Shallow Water Model ..... 19 Chapter 3 Literature Review ....••.•..••...••..•......•.......•...•....•.••....•..······21 3.1 Governing Equations ~ 21 3.2 Hyperbolic Conservation Law 22 3.3 Numerical Discretization 24 3.3.1 Finite Difference Method 24 3.3.2 Finite Volume Method 29 3.4 Roe's Approximate Solution 35 3.4.1 The Shallow Water Equations and Roe's Riemann Solver ................................................................................................ 36 3.4.2 Second-order Schemes and High Resolution Method ... 42 3.4.3 Discretization of the Source Term.s ·53 3.4.4 Two-dimensional Modelling 64 3.4.5 Wetting/Drying Boundary Condition 73 -j- 3.5 Application of Riemann Solver Based Scheme to Natural River Flows ·········78 3.6 Conclusion 83 Chapter 4 Homogeneous Form of the Shallow Water Equations ··85 4.1 Introduction 85 4.2 Homogeneous Form of the Shallow Water Equations .......... ·87 4.3 Conservative Schemes 93 4.3.1 Roe's Approximate Riemann Solver ·························..··93 4.3.2 HLL Approximate Riemann Solver 97 4.3.3 Lax-Friedrichs Scheme 100 4.3.4 Lax-WendroffScheme 103 4.3.5 MacCormack Scheme 107 4.4 Numerical Tests and Results 109 4.4.1 Test Problems 111 4.4.2 Results and Discussion 116 4.5 Conclusion 120 Chapter 5 Extension of the Homogeneous Approach to Two-dimensions and Wetting/Drying Problem ·········140 5.1 The Two-dimensinal Shallow Water Equations 140 5.2 Homogeneous Form of the Shallow Water Equations ........ ·141 5.2.1 Numerical Discretization 142 5.2.2 Defmition of Source Flux Vector 144 -jj- 5.3 Conservative Schemes 146 5.4 The WettinglDryingProblem ······················..··········..······...··151 5.5 Numerical Tests and Results ················································155 5.6 Conclusion 172 Chapter 6 Application: flood modelling over natural geometry ·174 6.1 One-dimensional Modelling of the Im-jin River Flood· ..·..·.. 174 6.1.1 Introduction 174 6.1.2 Overview of the Study Site and Input Data 176 6.1.3 Numerical Results 178 6.2 Two-dimensional Modelling of Floodplain Inundation ....... 187 6.2.1 Introduction 187 6.2.2 Numerical Results 189 6.3 Numerical Modelling of Urban Flooding 192 6.3.1 Introduction 192 6.3.2 Overview of the Study Site and Input Data 192 6.3.3 Numerical Results 194 6.3 Summary 200 Chapter 7 Conclusions ..•......•...................•........................···············201 7.1 Summary ························201 7.2 Conclusions 202 7.3 Recommended Future Work 205 References ••••••..••.••....•........••••••.•••••••••••.•••.••.•.·····································207 - iii- List of Figures Figure 2.1 Definition sketch for the derivation of unsteady flow equations: (a)Control volume, section view, (b)cross seciton and (c) plan view [28] 12 Figure 2.2 Flow in a channel with free surface under gravity [63] 16 Figure 2.3 Discretization elements in the quasi two-dimensional model [4] .......... ·20 Figure 3.1 Numerical domain of the finite volume method at time nflt ·30 Figure 3.2 Initial data of the Riemann problem ·31 Figure 3.3 Piecewise constant distribution of initial data 32 Figure 3.4 The solution structureof a Riemann problem 33 Figure 3.5 (a) The Sweby digram of several flux limiters and (b) Second order TVD region 44 Figure 3.6 The concept of the upwind approach [80] ·..·56 Figure 3.7 Details of the two-dimensional cells [11] 66 Figure 3.8 1D-2D hybrid discretization of channel junction [71] 81 Figure 4.1 Reconstruction of geometry: (a) section view and (b) plan view 89 Figure 4.2 Ideal dam-break problem with Roe's solver ·97 Figure 4.3 The structure ofHLL Riemann solver 98 - iv- Figure 4.4 Ideal dam-break problem with HLL solver 99 Figure 4.5 Ideal dam-break problem with Lax-Friedrichs (LF) scheme 103 Figure 4.6 Ideal dam-break problem with Lax- Wendroff (LW) scheme 106 Figure 4.7 Ideal dam-break problem with TVD Lax-Wendroff (TVD-LW) scheme ............................................................................................................ ··············106 Figure 4.8 Ideal dam-break problem with MacCormack and TVD MacCormack scheme ........................................................................................................................... ·..·.. 108 Figure 4.9 Cross section used for test problems 110 Figure 4.10 Bed elevation and width variation for Problems 1 and 2 ····...... ·········114 Figure 4.11 Water surface and bed elevation for Problem 3-1 ..·· ·..·· ·····114 Figure 4.12 Water surface and bed elevation for Problem 3-2 ·...... ··.... ············..···115 Figure 4.13 Bed level and widthvariation for Problem 4 115 Figure 4.14 Water surface level and discharge profile at t = 10000s in Problem 1: (a) Roe, (b) HLL, (c) LF, (d) TVD-LW and (e) TVD-MC ..·····...... ··...... ·..··...... 123 Figure 4.15 Water surface level and discharge profile at t = 10,800s in Problem 2 with rectangular channel: (a) Roe, (b) HLL, (c) LF, (d) TVD-LW and (e) TVD- MC 125 -v- Figure 4.16 Water surface level and discharge profile at t = 10,800s in Problem 2 with trapezoidal