TWO DIMENSIONAL FINITE VOLUME MODEL FOR SIMULATING UNSTEADY TURBULENT FLOW AND SEDIMENT TRANSPORT by CHUNSHUI YU A Dissertation Submitted to the Faculty of the DEPARTMENT OF HYDROLOGY AND WATER RESOURCES In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOPHY WITH A MAJOR IN HYDROLOGY In the Graduate College THE UNIVERSITY OF ARIZONA 2013 2 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE As members of the Dissertation Committee, we certify that we have read the dissertation prepared by Chunshui Yu entitled “Two dimensional finite volume model for simulating unsteady turbulent flow and sediment transport” and recommend that it be accepted as fulfilling the dissertation for the Degree of Doctor of Philosophy. Date: July 15, 2013 Dr. Jennifer Duan Date: July 15, 2013 Dr. Jim Yeh Date: July 15, 2013 Dr. Juan Valdes Date: July 15, 2013 Dr. Kevin Lansey Final approval and acceptance of this dissertation is contingent upon the candidate’s submission of the final copies of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement. Date: July 15, 2013 Dissertation Director: Dr. Jennifer Duan 3 STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at the University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. Signed: s 4 ACKNOWLEDGMENTS “HAPPY is the boy who discovers the bent of his life-work during childhood. That, indeed, was my good fortune.” – Sven Hedin, 1925, “MY LIFE AS AN EXPLORER” I express my deepest sense of gratitude to my advisor, Dr. Jennifer Duan, for her guidance, encouragement and support throughout the completion of this study. I am equally grateful to my committee members, Dr. Jim Yeh, Dr. Juan Valdes and Dr. Kevin Lansey, for their encouragements, insights and suggestions to my study. Also I am very grateful to other faculty and staff members in the Department of Hydrology and Water Resources and Department of Civil Engineering and Engineering Mechanics. I am thankful to my colleagues Yang Bai, Jaeho Shim, Khalid A Abdalrazaak al Asadi, Ari Posner, Anu Acharya, Lei Liu and Shiyan Zhang for all the help they offered in my study. I am also heartily thankful to my friends I met here in the desert: Deqiang Mao, Zhe Li, Liang Xue, Hui Xiong, Zhufeng Fang, Jianwei Xiang and Junfeng Zhu. The present work was supported by NSF CAREER Award EAR-0846523 to the University of Arizona. 5 DEDICATION To my family 6 TABLE OF CONTENTS LIST OF FIGURES ..................................................................................................................... 8 ABSTRACT .............................................................................................................................. 10 CHAPTER 1 INTRODUCTION .............................................................................................. 11 1.1 Statement of Purpose ....................................................................................................... 11 1.2 Methodology.................................................................................................................... 12 1.3 Outline of Dissertation .................................................................................................... 13 CHAPTER 2 GOVERNING EQUATIONS ............................................................................. 14 2.1 Shallow Water Equations ................................................................................................ 14 2.2 Eigenstructure of Shallow Water Equations.................................................................... 17 2.3 The Riemann Problem ..................................................................................................... 19 2.4 Shocks, Rarefactions and Contact Discontinuities .......................................................... 21 2.5 Exact Riemann Solution .................................................................................................. 27 2.6 Approximations of Shallow Water Equations ................................................................. 31 CHAPTER 3 GODUNOV-TYPE FINITE VOLUME METHOD ........................................... 33 3.1 Godunov-Type Finite Volume Method ........................................................................... 33 3.2 TVD Scheme ................................................................................................................... 35 3.3 MUSCL Scheme .............................................................................................................. 38 3.4 WENO Scheme ............................................................................................................... 40 3.5 Approximate Riemann Solver ......................................................................................... 42 3.6 Discretization of Time ..................................................................................................... 46 CHAPTER 4 NUMERICAL MODELING OF KINEMATIC WAVE EQUATION .............. 49 4.1 Introduction ..................................................................................................................... 49 4.2 Numerical Model ............................................................................................................. 52 4.3 Shock Wave ..................................................................................................................... 54 4.4 Rarefaction Wave ............................................................................................................ 57 4.5 Wave Steepening ............................................................................................................. 59 4.6 Uniform Rainfall-Runoff Overland Flow ........................................................................ 61 4.7 Steady Non-uniform Rainfall-Runoff Overland Flow .................................................... 66 4.8 Rainfall-Runoff over Non-uniform Overland Slope ....................................................... 69 4.9 Summary.......................................................................................................................... 71 7 TABLE OF CONTENTS - continued CHAPTER 5 A SURFACE FLOW ROUTING METHOD (SWE-KWA) .............................. 72 5.1 Introduction ..................................................................................................................... 72 5.2 Numerical Modeling ........................................................................................................ 77 5.3 One-Dimensional Rainfall-Runoff Case ......................................................................... 81 5.4 Two-Dimensional Rainfall-Runoff over a V-Shaped Catchment ................................... 87 5.5 Goodwin Creek Experimental Watershed ....................................................................... 92 5.6 Summary.......................................................................................................................... 95 CHAPTER 6 DEPTH-AVERAGED TURBULENCE MODELING ....................................... 97 6.1 Introduction ..................................................................................................................... 97 6.2 Numerical Modeling ...................................................................................................... 100 6.3 Dam-Break Flow over a Triangular Sill ........................................................................ 104 6.4 Dam-Break Flow through an Isolated Obstacle ............................................................ 108 6.5 Malpasset Dam-Break ................................................................................................... 115 6.6 Summary........................................................................................................................ 118 CHAPTER 7 NUMERICAL MODELING OF DAM-BREAK FLOW OVER MOVABLE BEDS ....................................................................................................................................... 120 7.1 Introduction ................................................................................................................... 120 7.2 Numerical Modeling ...................................................................................................... 122 7.3 1D Dam-Break Flow over Mobile Bed ......................................................................... 124 7.4 2D Dam-Break Flow over Mobile Bed ......................................................................... 127 7.5 Summary........................................................................................................................ 132 CHAPTER 8 CONCLUSIONS AND FUTURE DIRECTIONS ............................................ 133 8.1 Conclusions ................................................................................................................... 134 8.2 Future Directions ........................................................................................................... 139 REFERENCES ........................................................................................................................ 143 8 LIST OF FIGURES Fig. 2.1. Symbol definitions of shallow water equations .............................................................. 15 Fig. 2.2. Initial conditions of a dam-break problem ..................................................................... 19 Fig. 2.3. Time evolution of a dam-break problem ........................................................................ 20 Fig. 2.4. A rarefaction wave.......................................................................................................... 21 Fig. 2.5. A right rarefaction wave ................................................................................................. 22 Fig. 2.6. A left rarefaction wave ................................................................................................... 23 Fig. 2.7. A contact discontinuity ................................................................................................... 24 Fig. 2.8. A shock wave.................................................................................................................. 25 Fig. 2.9. A left shock wave ........................................................................................................... 25 Fig. 2.10. A right shock wave ....................................................................................................... 26 Fig. 2.11. Possible wave patterns of the Riemann problem .......................................................... 28 Fig. 2.12. Wave structure of a general Riemann problem solution .............................................. 28 Fig. 3.1. A two-dimensional Cartesian control volume ................................................................ 33 Fig. 3.2. Reconstruct variables at interfaces ................................................................................. 34 Fig. 3.3. Evolve the fluxes across interfaces ................................................................................. 35 Fig. 3.4. Average cells variables ................................................................................................... 35 Fig. 3.5. TVD schemes and non-TVD scheme ............................................................................. 36 Fig. 3.6. Generation of oscillation by the Lax-Wendroff scheme ................................................ 37 Fig. 3.7. Reconstruction using minmod limiter and superbee limiter ........................................... 39 Fig. 3.8. A demonstration of WENO scheme ............................................................................... 41 Fig. 4.1. Results of shock wave test case ...................................................................................... 56 Fig. 4.2. Calculated total variations for the shock wave test case ................................................ 56 Fig. 4.3. Results of rarefaction wave test case .............................................................................. 58 Fig. 4.4. Calculated total variation for the rarefaction wave test case .......................................... 59 Fig. 4.5. Initial condition of long time evolution case .................................................................. 60 Fig. 4.6. Oscillations of the MacCormack scheme ....................................................................... 60 Fig. 4.7. Results of wave steepening test case .............................................................................. 62 Fig. 4.8. Hydrograph of uniform rainfall-runoff case ................................................................... 65 Fig. 4.9. Non-uniform rainfall distribution ................................................................................... 66 Fig. 4.10. Results of non-uniform rainfall-runoff test case........................................................... 67 Fig. 4.11. Hydrograph of non-uniform rainfall-runoff test case ................................................... 68 Fig. 4.12. Hydrographs of the non-uniform overland slope test case ........................................... 70 9 LIST OF FIGURES - continued Fig. 5.1. Concept view of surface flow, including channel flow and overland flow .................... 72 Fig. 5.2. Hypothetical values of friction term ............................................................................... 79 Fig. 5.3. The flow chart of processes within each time step ......................................................... 82 Fig. 5.4. Schematic description of one-dimensional rainfall-runoff case ..................................... 83 Fig. 5.5. Discharge hydrograph of one-dimensional rainfall-runoff case ..................................... 83 Fig. 5.6. Flow depth (top) and velocity (bottom) of 1D rainfall-runoff case (1600s) .................. 85 Fig. 5.7. Flow depth (top) and velocity (bottom) of 1D rainfall-runoff case (2600s) .................. 86 Fig. 5.8. Schematic description of V-shaped catchment case ....................................................... 87 Fig. 5.9. Discharge hydrograph of two-dimensional V-shaped catchment case ........................... 89 Fig. 5.10. Discharge hydrograph of overland flow and channel flow .......................................... 89 Fig. 5.11. Flow depth (top) and velocity (bottom) along channel (80 min) ................................. 90 Fig. 5.12. Flow depth (top) and velocity (bottom) along hill slope cross-sections....................... 91 Fig. 5.13. DEM map of Goodwin Creek Experimental Watershed .............................................. 93 Fig. 5.14. Discharge hydrograph of Goodwin Creek case ............................................................ 94 Fig. 6.1. Flume set-up, initial conditions and gauge locations for test case 1 (in meters) .......... 104 Fig. 6.2. Comparisons between measured data and calculate results for test case 1 .................. 106 Fig. 6.3. Comparisons of simulated flow depth with and without turbulence model ................. 107 Fig. 6.4. Flume set-up, initial conditions and gauge locations for test case 2 (in meters) .......... 109 Fig. 6.5. Comparisons between observations and calculated results for test case 2 ................... 111 Fig. 6.6. Comparisons of calculated flow depth with and without the turbulence model .......... 114 Fig. 6.7. Demonstration of Cartesian cut-cell grid for test case 3 .............................................. 116 Fig. 6.8. Inundation map at t = 1200 s for Malpasset case ......................................................... 116 Fig. 6.9. Comparisons between surveyed data and calculated results ........................................ 117 Fig. 6.10. Comparisons between measured data of physical model and calculated results ........ 118 Fig. 7.1. Concept view of sediment-Laden flow ......................................................................... 120 Fig. 7.2. Schematic diagram of 1D dam-break flow over mobile bed ........................................ 124 Fig. 7.3. Computational results on 1D dam-break flow over mobile bed ................................... 125 Fig. 7.4. Concentration of total load sediment at time = 0.75 s .................................................. 126 Fig. 7.5. Plane view of 2D UCL dam-break experiment ............................................................ 127 Fig. 7.6. Observed and calculated water surface for 2D test case (1) ......................................... 128 Fig. 7.7. Observed and calculated bed profiles for 2D test case (1) ........................................... 129 Fig. 7.8. Observed and calculated water surfaces for 2D test case (2) ....................................... 130 Fig. 7.9. Observed and calculated bed profiles for 2D test case (2) ........................................... 131 10 ABSTRACT The two-dimensional depth-averaged shallow water equations have attracted considerable attentions as a practical way to solve flows with free surface. Compared to three-dimensional Navier-Stokes equations, the shallow water equations give essentially the same results at much lower cost. Solving the shallow water equations by the Godunov-type finite volume method is a newly emerging area. The Godunov-type finite volume method is good at capturing the discontinuous fronts in numerical solutions. This makes the method suitable for solving the system of shallow water equations. In this dissertation, both the shallow water equations and the Godunov-type finite volume method are described in detail. A new surface flow routing method is proposed in the dissertation. The method does not limit the shallow water equations to open channels but extends the shallow water equations to the whole domain. Results show that the new routing method is a promising method for prediction of watershed runoff. The method is also applied to turbulence modeling of free surface flow. The k turbulence model is incorporated into the system of shallow water equations. The outcomes prove that the turbulence modeling is necessary for calculation of free surface flow. At last part of the dissertation, a total load sediment transport model is described and the model is tested against 1D and 2D laboratory experiments. In summary, the proposed numerical method shows good potential in solving free surface flow problems. And future development will be focusing on river meandering simulation, non-equilibrium sediment transport and surface flow – subsurface flow interaction. 11 CHAPTER 1 INTRODUCTION 1.1 Statement of Purpose Open channel flow in natural rivers can be characterized as shallow water flow [Vreugdenhil, 1994]. The general characteristic of such flow is that the vertical dimension is much smaller than the typical horizontal scales. Shallow water flows are nearly horizontal, which allows a considerable simplification in the mathematical formulation and numerical solution by assuming the pressure distribution to be hydrostatic. However, they are not exactly two-dimensional. The flow exhibits a threedimensional structure due to bottom friction. Yet, in many shallow water flows these threedimension effects are not essential and it is sufficient to consider the depth-averaged form, which is two-dimensional in the horizontal plane. The restricted form, commonly indicated by the term “shallow water equations”, is the topic of this research. For many practical applications, the twodimensional depth-averaged shallow water equations give essentially the same information as the three-dimensional model at much lower cost. The main goal of this research is to simulate the unsteady turbulent shallow water flow and sediment transport using the Godunov-type finite volume method. The Godunov-type shock capturing methods have been successfully applied to many scientific disciplines. Their application to shallow water flows is more recent but the experience so far is encouraging. These 12 methods are indeed very attractive for their ability to resolve strong wave interaction and flows including steep gradients such as bores, shear waves and contact discontinuities. Godunov-type methods have the ability to sharply resolve these waves, along with their correct propagation speeds and free from unphysical oscillations in their vicinity. In the French scientific community, the shallow water equations are commonly referred as the Saint-Venant equations, although it appears that Saint-Venant derived only the onedimensional version [Hervouet, 2007]. In the dissertation, we will use the term “shallow water equations” throughout. 1.2 Methodology For the shallow water flows, one-dimensional and two-dimensional numerical models are developed from scratch. The governing equations for unsteady turbulent flow simulation are the depth-averaged Navier-Stokes equations with k-e model for turbulence closure. While for sediment transport simulation, the shallow water equations are solved together with the total sediment load transport equation and bed elevation change equation. The Godunov-type finite volume method is used to solve those differential equations. The domain is discretized by the Cartesian mesh. Different approximate Riemann solvers are tested in the study, including Roe solver, HLL solver, HLLC solver and LLF solver. For the states reconstruction, the piecewise constant method (first-order accuracy), the MUSCL scheme (second-order accuracy) and the WENO scheme (fifth-Order accuracy) are tested in the study. Different slope limiters, including MINMOD limiter, SUPERBEE limiter, van Leer Limiter and MC limiter are tested. The numerical models are written in FORTRAN programming language. The code developing environment is based on the Microsoft Visual Studio and the Intel FORTRAN 13 complier. Input files are generated by the Matlab scripts. And the post-processing tool is the Tecplot 360 software. The OpenMP API is used in the codes to execute the codes in parallel mode. All the codes are running on Microsoft Windows 7 64-bits system. 1.3 Outline of Dissertation In Chapter 2, the shallow water equations are introduced. The eigenstructure of shallow water equations are described. To derive the exact Riemann solution, the simple waves, including shocks, rarefactions and contact discontinuities are discussed. Besides, the approximations of shallow water equations, including kinematic wave equation and diffusive wave equation, are presented. In Chapter 3, the Godunov-type finite volume method is elaborated. The TVD schemes, slope limiters and reconstruction methods are discussed in details. Chapter 4 discusses the numerical methods for solving the kinematic wave equation. The MacCormack scheme, as a non-TVD scheme, generates oscillations near the discontinuous front, while the MUSCL scheme and the WENO scheme show smooth solutions. In Chapter 5, a surface flow routing method is described. The method is based on the shallow water equations and the kinematic wave approximation. The method is stable for simulating rainfall-runoff process over arbitrarily sloped land surface. Since the shallow water equations are used in the overland flow routing method, the real roughness coefficients are used in the calculation. Chapter 6 describes a numerical model based on two-dimensional depth-averaged Reynolds Averaged Navier Stokes equations. The coupled system is solved simultaneously by the HLLC Riemann solver. Chapter 7 presents a coupled unsteady flow and sediment transport model that simultaneously solves the shallow water equations and the total load sediment transport equation. 14 CHAPTER 2 GOVERNING EQUATIONS Equation Section 2 Shallow water flow refers to a nearly horizontal free-surface flow. Typical shallow water flows include: atmospheric flow, coastal flow, river/open channel flow, tsunami, and dam-break flow, among others. The key assumption made in the derivation of shallow water equations is the hydrostatic pressure distribution. This results from assuming that the acceleration of the water particles in vertical direction has a negligible effect on the distribution of pressure. The resulting shallow water equations are a 2D unsteady system of hyperbolic partial differential equations. 2.1 Shallow Water Equations The governing equations of shallow water flow are the shallow water equations (SWE). The two-dimensional shallow water equations are a system of three equations which represent the conservations of mass and momentum. The definitions of symbol for shallow water equations are depicted in Fig. 2.1 The primitive variables U used in the shallow water equations are: h U u v (2.1) where h is flow depth; u and v are depth-averaged flow velocities in x and y directions, respectively. And the conservative variables Q are defined as: 15 q1 h Q q2 hu q hv 3 (2.2) where q1 is mass flux; q2 and q3 are momentum fluxes in x and y directions. Surface h Bottom b z Datum y x Fig. 2.1. Symbol definitions of shallow water equations The shallow water equations can be expressed in primitive variables or conservative variables. Both forms are equivalent in mathematic formulation, but this is no longer true in numerical formulation [Toro, 2001; Vreugdenhil, 1994]. In conservative form, the two-dimensional shallow water equations looks like: Q F(Q) G(Q) S0 Sf t x y (2.3) 16 where Q is the conservative variables; F(Q) and G(Q) are the advective fluxes in x and y directions, respectively; S 0 is the bed slope source term; S f is the bed friction source term. The advective fluxes F(Q) and G(Q) are given as: hu 2 2 F(Q) hu gh / 2 , huv hv G(Q) hvu hv 2 gh 2 / 2 (2.4) where g is the gravity acceleration. The bed slope source term S 0 equals: 0 S0 ghS0 x ghS 0y where S0 x (2.5) b b and S0 y are the bed slopes in x and y directions, receptively; b is the x y bed elevation. And bed friction source term S f equals: 0 S f C f u u C u v f (2.6) where C f gn 2h 1/3 is the drag coefficient; n is Manning’s roughness coefficient; u u2 v2 . 17 2.2 Eigenstructure of Shallow Water Equations The homogeneous part of shallow water equations can be written in quasi-linear form as: Q Q Q A(Q) B(Q) 0 t x y (2.7) where A(Q) and B(Q) are the Jacobian matrices corresponding to the advective fluxes F(Q) and G(Q) . The Jacobian matrices A(Q) and B(Q) are given by: 0 A(Q) c 2 u 2 uv 1 0 2u 0 v u (2.8) 0 B(Q) uv c2 v 2 0 v 1 u 0 2v (2.9) and where c gh is the celerity. The eigenvalues of A(Q) and B(Q) are given by: 1 u c λ 2 u for A(Q) u c 3 and (2.10) 18 1 v c λ 2 v for B(Q) v c 3 (2.11) It is easy to prove that the linear combination of the Jacobian matrices A(Q) and B(Q) has three real eigenvalues and the shallow water equations are hyperbolic system. When the whole domain is wet, the eigenvalues are all distinct and the system is strictly hyperbolic. The right eigenvectors of A(Q) and B(Q) are given by: R r1 r2 0 1 1 r3 u c 0 u c for A(Q) v 1 v (2.12) R r1 r2 0 1 1 r3 u 1 u for B(Q) v c 0 v c (2.13) and The left eigenvectors of A(Q) and B(Q) are given by: 1 0 l1 u c L l 2 v 0 1 for A(Q) l (u c ) 1 0 3 (2.14) 0 1 l1 v c L l 2 u 1 0 for B(Q) l ( v c ) 0 1 3 (2.15) and 19 The right and left eigenvectors of the Jacobian matrices A(Q) and B(Q) are bi-orthonormal [Toro, 2001; Vreugdenhil, 1994]. 2.3 The Riemann Problem The Riemann problem for shallow water equations is defined as an initial-value problem (IVP): Q F(Q) 0 t x Q L , if x 0 Q( x,0) Q R , if x 0 (2.16) An interesting case of Riemann problem is the dam-break problem (Fig. 2.2). Assume that a rectangular channel has two uniform water levels, both at rest, separated by a wall at x 0 . h QL QR 0 x Fig. 2.2. Initial conditions of a dam-break problem The initial conditions for this one-dimensional dam-break problem are given as: h h QL L , QR R 0 0 (2.17) 20 h QL QR 0 (a) x u QL QR 0 (b) Rarefaction t x Shock QL (c) QR 0 Fig. 2.3. Time evolution of a dam-break problem (a) flow depth; (b) flow velocity; (c) wave structure on the phase plane x 21 If the wall vanishes instantaneously at time t 0 , then the time evolution is described by Fig. 2.3. Two waves emerge from the dam-break process. A right-facing wave travels into the left portion of domain, raising the flow depth abruptly. This is called a shock/bore wave, a discontinuous wave. The left-facing wave moves to the left region and has the effect of reducing the height of water surface. This is called a rarefaction/depression wave, a smooth wave. In the Riemann problem, the generalization of dam-break problem, the velocities are allowed to be distinct from zero. There are three wave families, which are associated with three distinct eigenvalues, in the two-dimensional shallow water equations. 2.4 Shocks, Rarefactions and Contact Discontinuities In the solution of Riemann problem for the shallow water equations, the left and right waves are shocks or rarefactions, while the middle wave is always a contact discontinuity. Rarefaction wave (Fig. 2.4) is a smooth wave and all flow variables vary continuously across the wave. However, across the bounding characteristics corresponding to the head and tail, the derivative in x direction is discontinuous. λ(UL) t λ(UR) UL UR 0 Fig. 2.4. A rarefaction wave x 22 For the shallow water equations, the rarefactions are associated with the characteristic field i 1 or i 3 . The characteristic field is divergence: (U L ) (U R ) (2.18) h* hR U* u* , U R uR v v * R (2.19) Assume the primitive variables: on the left side and right side of a right rarefaction wave associated with the characteristic field 3 u c (Fig. 2.5). t λ3(U*) λ3(UR) U* UR 0 x Fig. 2.5. A right rarefaction wave Across the right rarefaction wave, we have the following relationships between the left side and the right side of the wave: u* 2c* uR 2cR v* vR (2.20) 23 Assume the primitive variables: hL h* U L uL , U* u* v v L * (2.21) on the left side and right side of a left rarefaction wave associated with the characteristic field 1 u c (Fig. 2.6). t λ1(UL) λ1(U*) UL U* 0 x Fig. 2.6. A left rarefaction wave Across the left rarefaction wave, we have the following relationships between the left side and the right side of the wave: u* 2c* uL 2cL v* vL (2.22) Contact discontinuity is discontinuous wave across which the tangential velocity jumps discontinuously. The states on either side are connected through a discontinuity, which associated with the characteristic field 2 u (Fig. 2.7). The characteristic field is parallel: 24 (U*L ) (U*R ) t (2.23) λ2(U*L)=λ2(U*R) U*L U*R 0 x Fig. 2.7. A contact discontinuity Suppose that the states on the left and right sides of the wave are: h*L h*R U*L u*L , U*R u*R v v *L *R (2.24) then we have the following relationships: h* L h* R u* L u* R v v *L *R (2.25) Shock wave (Fig. 2.8) is a compressive wave and the states on both sides are connected through a single jump discontinuity. The shocks are associated with the characteristic field i 1 or i 3 . The characteristic field is convergent: (U L ) S (U R ) (2.26) 25 where S is the shock wave speed. t S λ(UL) λ(UR) UL UR 0 x Fig. 2.8. A shock wave The left shock wave is associated the characteristic field 1 u c (Fig. 2.9). t SL λ1(UL) λ1(U*) UL U* 0 x Fig. 2.9. A left shock wave Assume the states on the left and right side of a left shock wave are: hL h* U L uL , U* u* v v L * (2.27) 26 Based on the given initial condition U L , the state behind the shock is determined by: 1 1 8( Fr Fr ) 2 L S h* hL 2 u* uL cL (1 hL / h* )( FrL FrS ) v* vL (2.28) The Froude numbers FrL and FrS are defined as: FrL uL S , FrS L cL cL (2.29) where S L is the left shock speed and FrS is called shock Froude number. The right shock wave is associated with the characteristic field 3 u c (Fig. 2.10). t SR λ3(U*) λ3(UR) U* UR 0 Fig. 2.10. A right shock wave Denote the states on both sides as: x 27 h* hR U* u* , U R uR v v * R (2.30) Then the states behind the wave are: 1 1 8( Fr Fr ) 2 S R h* hR 2 u* uR cR (1 hR / h* )( FrS FrR ) v* vR (2.31) The Froude numbers are given as: FrR uR S , FrS R cR cR (2.32) where S R is the right shock speed. All possible situations involving a single wave arising as solution of the Riemann problem for the shallow water equations are listed here. Then this knowledge will be applied to the exact solution of the Riemann problem. 2.5 Exact Riemann Solution There are four possible wave patterns that may occur in the solution of the Riemann problem (Fig. 2.11). There are three waves in each solution. In general the left and right are shocks or rarefactions depending on the initial conditions. The middle wave is always a contact discontinuity, across which the tangential velocity component changes discontinuously. 28 t t 0 x t 0 x 0 x t 0 x Fig. 2.11. Possible wave patterns of the Riemann problem Star Region t λ1 λ2 U*L λ3 U*R UL UR 0 Fig. 2.12. Wave structure of a general Riemann problem solution x 29 The structure of the solution of the Riemann problem is depicted in Fig. 2.12. There are three waves, which are associated with the eigenvalues 1 , 2 and 3 . The three waves separate four constant states by U L , U*L , U*R , and U R . The region between the left and right waves is call the star region and is subdivided into two sub-regions. According to the single wave solutions, the four constant states are: hL h* h* hR U L uL , U*L u* , U*R u* , U R uR v v v v L L R R (2.33) where h* and u* are the constant states in the star region. The solution for h* is given by the root of the algebraic equation: f (h) f L (h, hL ) f R (h, hR ) u (2.34) where u uR uL ; the depth function f L and f R are: 2( gh ghL ), if h hL (rarefaction) fL 1 h hL g ( h hL ) , if h hL (shock) 2 hhL (2.35) 2( gh ghR ), fR 1 h hR g ( h hR ) 2 hhR if h hR (rarefaction) , if h hR (shock) and the u* equals: 1 1 u* (uL uR ) ( f R (h* , hR ) f L (h* , hL )) 2 2 (2.36) 30 There is no closed-form solution of Eq. (2.34) for h* and the equation has to be solved by a nonlinear solver. The rest of the solution is to complete the wave structure by using single wave relations. If h* hL , then the left wave is a shock with shock speed: S L u L c L qL 1 ( h* hL )h* qL 2 hL2 (2.37) If h* hR , then the right wave is a shock with shock speed: S R u R c R qR 1 ( h* hR )h* qR 2 hR2 (2.38) If h* hL , then the left wave is a rarefaction and the head and tail speeds are: S HL uL cL STL u* c* (2.39) If h* hR , then the right wave is a rarefaction and the head and tail speeds are: S HR uR cR STR u* c* The solution of the tangential velocity is: (2.40) 31 vL , if v ( x, t ) v , if R x u* t x u* t (2.41) 2.6 Approximations of Shallow Water Equations The dimensionless one-dimensional shallow water equations can be written as [Toro, 2001]: h* * * t * x* ( h u ) 0 * * * * 2 u u* u 1 h k (1 (u ) ) t * x* ( Fr* )2 x* h* (2.42) where k is the kinematic wave number and defined as: k S0 L0 H 0 Fr02 (2.43) Kinematic wave approximation: for k 50 , the flow is dominated by the slope term and friction term: k (1 u2 )0 h (2.44) The kinematic wave equation is obtained: h 3/2 (h ) 0 t x Diffusion wave approximation: when Fr reduces to: (2.45) 1 and k is large, the momentum equation 32 h u2 Fr 2 k (1 ) x h (2.46) Then the diffusion wave equation is derived: h 3/2 h (h (1 ( Fr 2 k ) 1 )1/2 ) 0 t x x (2.47) 33 CHAPTER 3 GODUNOV-TYPE FINITE VOLUME METHOD Equation Section (Next) 3.1 Godunov-Type Finite Volume Method The integral form of the shallow water equations can be written as: Qd F(Q)d G(Q)d Sd t (3.1) where represents the domain; is the boundary of the domain. In the presence of discontinuities, including shocks, the integral form still holds. y Δx j+1/2 Δy j j-1/2 i-1/2 i i+1/2 Fig. 3.1. A two-dimensional Cartesian control volume x 34 Consider a control volume as depicted in Fig. 3.1. The integral form (3.1) of shallow water equations becomes: Qin,j1 Qin, j t t (Fi 1/2, j Fi 1/2, j ) (Gi , j 1/2 Gi , j 1/2 ) tSi , j x y (3.2) This is the basis of finite volume method, which is a conservative numerical method. The Godunov-type finite volume method is based on the Reconstruct-Evolve-Average (REA) process. 1. Reconstruct (Fig. 3.2): to build the flow variables at the interfaces between cells, from the cell averages. In the simplest case, this is a piecewise constant function that takes the value at time step n. h x Fig. 3.2. Reconstruct variables at interfaces 2. Evolve (Fig. 3.3): to solve the local Riemann problems by an exact Riemann solver or an approximate Riemann solver. The calculations are based on the reconstructed values at both sides of interfaces. 35 t x Fig. 3.3. Evolve the fluxes across interfaces 3. Average (Fig. 3.4): to update the flow variables at each cell. h x Fig. 3.4. Average cells variables 3.2 TVD Scheme The total variation (TV) of a numerical scheme is given by: TV ui 1 ui (3.3) i A numerical scheme is said to be total variation diminishing (TVD) is: TV (u( n 1) ) TV (u( n ) ) (3.4) 36 (a) (b) Fig. 3.5. TVD schemes and non-TVD scheme 37 Harten and Ami [1983] proved that a monotone scheme is TVD and a TVD scheme is monotonicity preserving. TVD schemes can capture sharper shocks on coarse grid and there are no spurious oscillations in TVD solutions. Here is an example of TVD schemes and non-TVD scheme. In Fig. 3.5, the kinematic wave equation is solved by MacCormack scheme (non-TVD), MUSCL scheme (TVD) and WENO scheme (TVD). Near the shock front, the solution of MacCormack scheme, which is a non-TVD scheme, is polluted by oscillations. As the time proceeding, the total variation of MacCormack scheme keeps increasing. This is consistent with the oscillation in the solution. On the other hand, the TVD schemes can capture sharper shock front without any oscillations and keep the total variations constant. Another example is shown in Fig. 3.6. In the example, the variables are reconstructed by the Lax-Wendroff scheme. The values at the interfaces are overshoot and this brings oscillations to the solution in the next time step. This suggests that near a discontinuity we may want to limit the slope, using a value that is small in magnitude in order to avoid oscillations. h h x Fig. 3.6. Generation of oscillation by the Lax-Wendroff scheme x 38 3.3 MUSCL Scheme The MUSCL (Monotone Upstream-centered Schemes for Conservation Laws) is the first high-order (2nd order), total variation diminishing (TVD) scheme [van Leer, 1979]. The idea of MUSCL scheme is to replace the piecewise constant approximation of Godunov’s scheme by slope limited, reconstructed left and right extrapolated states at the cell interfaces. Since the evolving and averaging steps in the Godunov-type method cannot possibly increase the total variation, it is only the reconstruction that will bring oscillations to the solution. A wide variety of slope limiters have been proposed in the literature. Here are the slope limiters used in this dissertation. Minmod limiter: Q Qi 1 Qi 1 Qi slope minmod i , x x (3.5) where the minmod function is defined as: a, if a b and ab 0 minmod(a, b) b, if b a and ab 0 0, if ab 0 (3.6) slope minmod( slope(1) , slope(2) ) (3.7) Superbee limiter: where 39 Q Qi Qi Qi 1 slope(1) minmod i 1 ,2 x x Q Qi Qi Qi 1 slope(2) minmod 2 i 1 , x x (3.8) MC limiter: Q Qi 1 Qi Qi 1 Qi 1 Qi slope minmod i 1 ,2 ,2 x x x (3.9) van Leer limiter: slope sr sl sl sr sl sr (3.10) where Qi 1 Qi s r x s Qi Qi 1 l x (3.11) The behaviors of the minmod limiter and supperbee limiter are depicted in Fig. 3.7. h h x Fig. 3.7. Reconstruction using minmod limiter and superbee limiter x 40 3.4 WENO Scheme WENO stands for “Weighted Essentially Non-Oscillatory” and the first WENO scheme was provided by Liu et al. [1994]. Jiang and Shu [1996] presented a general framework to construct arbitrary high order WENO schemes. Many of details about WENO schemes are discussed in Shu [1999; 2009]. Instead of using the piecewise linear reconstruction procedure in the MUSCL scheme, the WENO scheme uses the WENO reconstruction procedure to obtain an approximation to the cell interface. The WENO reconstruction procedure consists of following steps: Step 1. Calculate the smoothness indicators: 13 1 2 2 1 12 ( hi 2 2hi 1 hi ) 4 ( hi 2 4hi 1 3hi ) 13 1 2 2 2 ( hi 1 2hi hi 1 ) ( hi 1 hi 1 ) 12 4 13 1 2 2 3 12 ( hi 2hi 1 hi 2 ) 4 (3hi 4hi 1 hi 2 ) (3.12) Step 2. Calculate the third-order approximations at cell interfaces: 1 7 11 (1) h h h hi i 1/2 i 2 i 1 3 6 6 1 5 1 (2) hi 1/2 hi 1 hi hi 1 6 6 3 1 5 1 (3) hi 1/2 3 hi 6 hi 1 6 hi 2 (3.13) Step 3. Calculate the nonlinear weights: wj j wj , with w j , j 1,2,3 ( j )2 w1 w2 w3 (3.14) 41 where 106 for actual calculations; j is the linear weights and is given by: 1 1 5 5 , 2 , 3 16 8 16 (3.15) Step 4. Calculate the fifth-order approximation as a convex combination of the three third-order approximations: (2) (3) hi 1/2 w1hi(1) 1/2 w2 hi 1/2 w3hi 1/2 (3.16) A demonstration is sketched in Fig. 3.8. h(1) h(2) h h(3) i-2 i-1 i i+1 i+2 i+3 Fig. 3.8. A demonstration of WENO scheme x 42 3.5 Approximate Riemann Solver The process of solving the Riemann problems by an exact Riemann solver includes iterative procedures. On a modest 100 100 two-dimensional grid, for example, one must solve roughly 20,000 Riemann problems in each time step. As the iterative solutions are too costly, some approximations have to be made. Roe solver [Roe, 1981]: Roe’s solver is the first successful approximate Riemann solver. Consider approximate solutions which are exact solutions to an approximate problem: Q Q A(Q) 0 t x (3.17) where the constant matrix A should satisfy the following properties: (1) It constitutes a linear mapping from the vector space Q to F. (2) As QL QR Q , A(QL , QL ) A(Q) , where A F / Q . (3) For any Q L and Q R , A(QL , QR ) (QL QR ) FL FR . (4) The eigenvectors of A are linearly independent. The Roe flux for the shallow water equations is given as: 1 1 FRoe (F(QL ) F(QR )) R Λ LQ 2 2 where the Roe averages are defined as: (3.18) 43 h hL hR uL hL uR hR u hL hR v h vR hR v L L hL hR c 1 (c 2 c 2 ) L R 2 (3.19) 0 1 1 R u c 0 u c v 1 v (3.20) 1 h ( h u ) c 2 LQ h v 1 ( h h u ) c 2 (3.21) uc 0 0 (3.22) The average right eigenvectors are: The wave strengths are: The average eigenvalues are: 0 u 0 0 0 u c To avoid unphysical solution, an entropy fix method is needed. When i i / 2, i =4(i , R i , L ), i 1 or 3 (3.23) 44 the eigenvalue should be replaced with: i i2 i i 4 (3.24) HLL solver [Harten, 1984]: HLL stands for Harten-Lax-van Leer. The HLL flux across the cell interface is calculated by: F HLL F(Q L ), if sL 0 * (Q L , Q R ) F (Q L , Q R ), if sL 0 sR F(Q ), if sR 0 R (3.25) where the flux at the star region F* (QL , QR ) is determined by: F* (QL , QR ) sR F(QL ) sL F(QR ) sL sR (QR QL ) sR sL (3.26) The left wave speed sL and right wave speed sR are estimated by: sL min(u ghL , us cs ) sR max(u ghR , us cs ) (3.27) us (uL uR ) / 2 ( ghL ghR ) cs (uL uR ) / 4 ( ghL ghR ) / 2 (3.28) where us and cs are defined as: For dry bed situations, the estimated wave speeds are replaced by the exact dry front speed: 45 sL uL ghL right dry bed: sR uL 2 ghL left dry bed: sL uR 2 ghR sR uR ghR (3.29) HLLC solver: the HLLC (Harten-Lax-van Leer-Contact) solver was introduced by Toro et al. [1994]. It is a modification of the HLL solver whereby the missing contact and shear waves in the shallow water equations are restored. The HLLC flux equals: FL , if 0 sL F , if s 0 s L * *L F*R , if s* 0 sR FR , if 0 sR (3.30) F*L FL sL (Q*L QL ) F*R FR sR (Q*R QR ) (3.31) F HLLC where The intermediate states are given by: Q*K s uK hK K sK s* 1 s* , K L or R v K (3.32) The wave speed sL and sR can be estimated by Eq. (3.27). And the middle wave speed s* can be estimated by: s* sL hR (uR sR ) sR hL (uL sL ) hR (uR sR ) hL (uL sL ) (3.33) 46 LLF solver: the LLF (Local Lax-Friedrichs) solver, also called Rusanov solver, is a kind of centered scheme that does not require explicit local information on wave propagation. This makes it suitable for some very complex system of conservation laws. The LLF flux is: 1 1 FLLF (FL FR ) smax (QR QL ) 2 2 (3.34) smax max( uL cL , uR cR ) (3.35) where If set smax x t (3.36) the LLF solver degenerates to LxF (Lax-Friedrichs) scheme. For a scalar conservation law, the LLF solver has the same form as the Roe solver. 3.6 Discretization of Time MUSCL-Hancock scheme: this is a second-order, two steps, predictor-corrector scheme [van Leer, 1984]. The predictor step uses a non-conservative approach, that defines an intermediate value over a time interval t : 2 Q( n 1/2) Q( n ) t F(Q(Kn ) ) 2 k (3.37) where Qk is the reconstructed value at the cell interface. The corrector step is a fully conservative step. The intermediate solution from the predictor step is used to define a set of left 47 and right states for a series of Riemann problems. The solution of these Riemann problems are used to update the flow solution over time interval t : Q( n 1) Q( n ) t F(Q(Ln,k1/2) , Q(Rn,k 1/2) ) (3.38) k where F(QnL,k1/2 , QnR,k1/2 ) is the flux obtained by solving a local Riemann problem. TVD Runge-Kutta scheme: the scheme was introduced by Gottlieb and Shu [1998]. The scheme can keep the total variation diminishing along the time axis. The second-order TVD Runge-Kutta scheme is an explicit two-step scheme: Q(*) Q( n ) t L(Q( n ) ) ( n 1) 1 ( n ) 1 (*) 1 Q Q t L(Q(*) ) Q 2 2 2 (3.39) where L(Q) is an operator including Riemann fluxes and source terms. The second-order scheme is suitable for the MUSCL scheme. Third-order TVD Runge-Kutta scheme includes three steps: Q(1) Q( n ) t L(Q( n ) ) Q(2) 3 Q( n ) 1 Q(1) 1 t L(Q(1) ) 4 4 4 1 2 2 Q( n 1) Q( n ) Q(2) t L(Q(2) ) 3 3 3 The third-order scheme is often used with the WENO scheme. (3.40) 48 Our experiences show that the TVD Runge-Kutta scheme is a better choice for the shallow water equations than the MUSCL-Hancock scheme, especially for the wetting and drying processes. 49 CHAPTER 4 NUMERICAL MODELING OF KINEMATIC WAVE EQUATION Equation Section (Next) 4.1 Introduction The kinematic wave equation was first developed by Lighthill and Whitham [1955]. The equation is based on the assumptions that the acceleration term and the pressure gradient term in the momentum equation are negligible, so that the energy slope is equal to the bottom slope. The kinematic wave model is widely used to simulate the overland flow [Ponce, 1991; Singh, 2001]. Henderson [1966] showed that natural flood waves behave nearly the same as kinematic waves in steep slope ( S0 0.002 ). Vieira [1983] concluded that the kinematic wave equation can be used on natural slopes with the kinematic wave number, k 50 . Ponce [1991] compared the kinematic wave equation with the unit hydrograph as a practical method of overland flow routing. Singh [2001] concluded that the kinematic wave equation is applicable to surface water hydrology, vadose zone hydrology, river and costal hydrology, erosion and sediment transport, etc. The focus of the kinematic wave equation has been on hydrographs at outlet of watershed. This research will change the focus and concentrate on the details of flow field, including flow depth and flow velocity. The kinematic wave equation is a first-order hyperbolic partial differential equation (PDE). For a hyperbolic equation, the disturbance will travel along the characteristics of the equation in a finite propagation speed. This feature distinguishes the hyperbolic equations from 50 elliptic and parabolic equations. On the other hand, the kinematic wave equation also belongs to a kind of equations called conservation laws [LeVeque, 2002; Toro, 2009]. Since the flux term is a nonlinear function of conservative variables, the solution does not propagate uniformly but deforms as it evolves. Even the initial conditions be continuous and smooth, the hyperbolic conservation laws can develop discontinuities in the solution, for example, shock waves. Both shock wave and rarefaction wave are the intrinsic features of hyperbolic equations. Lighthill and Whitham [1955] discussed the formations of shock wave and rarefaction wave. Kibler and Woolhiser [1970] investigated the structure and general properties of shock waves and developed a numerical procedure for shock fitting. Eagleson [1970] found that using nonuniform flow depth as initial condition, non-uniform rainfall in the source term, or increasing inflows as the boundary condition may cause the formation of kinematic shock wave. Borah and Prasad [1980] presented the propagating shock-fitting scheme (PSF) to simulate overland flow with shock waves. Singh [2001] found three factors that affect the shock wave formation: 1) initial and boundary conditions; 2) lateral inflow and outflow, and 3) watershed geometric characteristics. Due to the complex geometry, non-uniform roughness and non-uniform rainfall pattern, it is impossible to derive a general analytical solution for the kinematic wave equation. Singh [2001] summarized three numerical techniques for solving the kinematic wave equation: (1) method of characteristic, (2) Lax-wendroff finite difference method, and (3) finite element method. Numerical diffusion and numerical dispersion were observed when using the finite difference schemes. Kazezyılmaz-Alhan et al. [2005] evaluated several finite difference schemes for solving kinematic wave equation: the linear explicit scheme, the four-point Pressimann implicit scheme, and the MacCormack scheme. The study found the MacCormack scheme is better than the four-point implicit finite difference scheme for shock capture. However, 51 Kazezyılmaz-Alhan et al. [2005] didn’t explicitly examine the dispersion occurred at the shock and rarefaction waves from non-uniformly distributed rainfall. The stability of the classical MacCormack scheme at the presence of shock and rarefaction wave is also unknown. Recently, the Godunov-type finite volume method has been widely used in solving shallow water equations [LeVeque, 2002; Toro, 2009] because of its wide applicability, strong stability, and high accuracy. One of the most popular Godunov-type methods is a second-order, TVD (Total Variation Diminishing) scheme, namely the MUSCL (Monotone Upstream-centered Schemes for Conservation Laws) scheme [van Leer, 1979]. The MUSCL scheme is a highresolution scheme because (1) the spatial accuracy of the scheme is equal to or higher than second order; (2) the scheme is free from numerical oscillations or wiggles; (3) high-resolution is produced around discontinuities. In general, the high-resolution schemes are considered as tradeoffs between computational cost and desired accuracy [Harten, 1983; Toro, 2009]. Another popular but relatively new method is the high-order WENO (Weighted Essentially NonOscillatory) finite volume scheme [C W Shu, 1999]. High-order means the order of accuracy is equal to or higher than the third-order. According to Shu [2009], the WENO scheme is suitable for the complicated problems, such as flow having both shocks and complicated smooth structures (e.g., small perturbation). Although the computational cost of high-order WENO scheme can be three to ten times than a second-order high-resolution scheme, it is still preferable because of its high-order accuracy in both time and space. The applications of those two high resolution schemes to solve the kinematic wave equation have not been studied. Whether or not these finite volume schemes have advantages over the commonly used finite difference schemes are examined in the research. 52 4.2 Numerical Model The one dimensional kinematic wave equation for flows over a slope is given by [Eagleson, 1970; Lighthill and Whitham, 1955]: h q i0 t x (4.1) where h is the depth of flow; q is the discharge per unit width; i0 i f is the rain excess; i is the intensity of rainfall; f is the infiltration rate; t is the time; x is the downslope distance. For the overland flow, the discharge q is defined as: q hm (4.2) where m is the exponential, and α is the coefficient. For fully turbulent flow, the coefficients are given by Ponce [1989b]: 1 5 S0 , m n 3 (4.3) where n is the Manning’s roughness coefficient; S0 is the bottom slope. Since the kinematic wave equation is a nonlinear hyperbolic partial differential equation, different numerical schemes exhibit different amounts of numerical diffusion and dispersion depending on the nature of schemes. Numerical diffusion often presents itself as the attenuation of the kinematic wave, while numerical dispersion is responsible for oscillations or negative outflows near large surface gradients. To provide comparisons with classical finite difference schemes, a study of two high-resolution Godunov-type finite volume schemes, the MUSCL finite 53 volume scheme and the WENO finite volume scheme, is presented in this paper. All the selected schemes are explicit, but differ in order of accuracy. The MUSCL scheme is a second-order scheme while the WENO scheme is fifth-order. The MacCormack scheme [MacCormack, 2003] is a commonly used finite difference scheme to solve hyperbolic PDEs. This two-step scheme is second-order accurate in both time and space. Compared to the first-order scheme, the MacCormack scheme does not introduce numerical diffusion in the solution. However, the numerical dispersion can be introduced in the region of large surface gradients. The MacCormack scheme is also a variation of the two-step Lax-Wendroff scheme. It includes two steps: a predictor step followed by a corrector step. The predictor step uses forward difference approximations while the corrector step uses backward difference approximations for spatial derivatives. The order of differencing can be reversed from time step to time step (i.e., forward/backward differencing followed by backward/forward differencing). The time step used in the predictor step is t in contrast to t / 2 used in the corrector step. Predictor step: hin 1 hin t n ( qi 1 qin ) i0 t x (4.4) Corrector step: 1 1 t hin 1 (hin hin 1 ) ( qin 1 qin11 ) i0 t 2 2 x (4.5) where subscript i is the spatial index; superscript n , n 1 , and n 1 are the temporal indices; t is time step; x is the space step. 54 The MUSCL scheme is described in section (3.3) and the WENO scheme is described in section (3.4). The following sections evaluate the selected schemes in a variety of flow conditions because a given scheme does well for a test case does not guarantee it will do well for another test case. In order to evaluate the performances of these schemes, it is necessary to choose test cases with exact solutions. For this reason, the one-dimensional propagations of a shock wave and a rarefaction wave are selected. Then the behaviors of these schemes in a long time period are tested by a wave steepening example. Finally, two synthetic rainfall-runoff cases and one overland flow over a varied slope are selected to demonstrate the actual performance of those schemes. 4.3 Shock Wave The shock wave and rarefaction wave are the intrinsic features of the hyperbolic equations, and thus the kinematic wave equation. The celerity of kinematic wave is defined as [Lighthill and Whitham, 1955]: c mhm1 mv (4.6) where v is flow velocity. Consider the following initial-value problem for the kinematic wave equation: hL , if x x0 h( x,0) hR , if x x0 (4.7) If we assume hL hR , we will have cL cR .This means that a shock wave will arise. The exact shock wave solution for the kinematic wave equation is: 55 hL , for ( x x0 ) / t S h ( x, t ) hR , for ( x x0 ) / t S (4.8) where S is the shock wave speed. For the kinematic wave equation, according to the RankineHugoniot jump condition [Toro, 2009], S is equal to: S hRm hLm hR hL (4.9) In the study, the following parameters are used: the channel length L 10.0 m ; the bed slope S0 0.0016 ; the Manning’s coefficient n 0.025 s/m1/3 ; the simulation time t 3.5 s ; the time step t 0.01 s ; the grid spacing x 0.1 m . The initial condition is: hL 1.0 m and hR 0.5 m . The boundary conditions are zero-depth gradient at both the entrance and the exit of the channel. The flow depth and velocity profiles calculated by the numerical schemes and the exact solution are plotted in Fig. 4.1. The results show that the numerical diffusion for both flow depth and velocity near the front of shock wave is small for the MUSCL and WENO schemes. But the results of flow depth and velocity from the MacCormack scheme are oscillatory. The wiggled surface oscillations are largest at the discontinuous wave front, and the solutions eventually diverge. To avoid oscillatory solutions, the total variation needs to decrease or remain constant with time. The total variations of flow depth from those schemes are shown in Fig. 4.2, which indicate that the total variation increases with time using the MacCormack scheme, while it remains constant for both MUSCL and WENO schemes. This implies that the MacCormack scheme will introduce oscillations at the presence of shock waves, but MUSCL and WENO 56 schemes can preserve the functional properties of the shock wave. Therefore, the solutions using the MacCormack scheme are dispersive and inaccurate for capturing the shock wave. 1.5 Flow Depth (m) 1 0.5 EXACT MacCormack MUSCL WENO 0 0 2 4 6 Channel Distance (m) 8 10 Fig. 4.1. Results of shock wave test case 2.5 MacCormack MUSCL WENO Total Variation (m) 2 1.5 1 0.5 0 0 0.5 1 1.5 2 Time (s) 2.5 3 3.5 Fig. 4.2. Calculated total variations for the shock wave test case 57 4.4 Rarefaction Wave Reconsider the initial-value problem described in Eq. (20), and assume hL hR , we have cL cR . This time, instead of generating a shock wave, a rarefaction wave is generated near the discontinuity since the celerity at the head of the discontinuity is greater than that at the tail and, consequently, the discontinuity continually expands as it propagates. For the kinematic wave equation, the exact solution of rarefaction wave is: x x0 hL , for t cL x x0 h hL x x0 h hL ( cL ) R , for cL cR t cR cL t x x0 cR hR , for t (4.10) The parameters used in the rarefaction test case are summarized as: the channel length L 10.0 m ; the bed slope S0 0.0016 ; the Manning’s coefficient n 0.025 s/m1/3 ; the simulation time t 3.0 s ; the time step t 0.01 s ; and the grid spacing x 0.1 m . The initial condition is: hL 0.5 m and hR 1.0 m . The boundary conditions are zero-depth gradient at both ends of the channel. The calculated flow depth and velocity profiles are plotted in Fig. 4.3. The results are similar to the shock wave case: the MacCormack scheme generated notable oscillations at the tail of the rarefaction wave. Taylor et al. [1972] tested the MacCormack scheme for solving the Burgers’ equation and found that the MacCormack scheme is unstable for rarefaction wave under some conditions. This study finds that the same phenomenon occurred to the kinematic wave equation. Since the oscillations occur at the tail of the rarefaction wave, where flow depth 58 is smaller, unrealistic negative flow depths can be induced at the tails of rarefaction waves. This can also be proved by calculating the total variation of flow depth as shown in Fig. 4.4. The total variations from both MUSCL and WENO schemes are constant. But the values from the MacCormack scheme increase suddenly to a peak value and then gradually decrease to a constant value (about 0.9) greater than the initial total variation (=0.5). Although the total variation shows a decreasing trend, the scheme cannot keep the total variation not to increase with time for the entire simulation time. Therefore, oscillations occur at the discontinuous front, the tail of rarefaction wave. Therefore, for the rarefaction wave, the MacCormack scheme is not as stable as the MUSCL or the WENO scheme. Both the shock and rarefaction wave test cases suggest that the MacCormack scheme is not a total variation diminishing (TVD) scheme, and the oscillations will be generated near the discontinuous wave fronts. To simulate unsteady flows having shock or rarefaction waves, the MUSCL scheme and WENO schemes should be used. 1.5 Flow Depth (m) 1 0.5 EXACT MacCormack MUSCL WENO 0 0 2 4 6 Channel Distance (m) 8 Fig. 4.3. Results of rarefaction wave test case 10 59 1.5 Total Variation (m) MacCormack MUSCL WENO 1 0.5 0 0 0.5 1 1.5 Time (s) 2 2.5 3 Fig. 4.4. Calculated total variation for the rarefaction wave test case 4.5 Wave Steepening One of the prominent features of the hyperbolic equation is that discontinuities will be generated even the initial water surface is smooth. So it is important for a scheme to preserve the sharpness of discontinuous fronts in a long-time simulation. This test case is to test the long time behavior of these schemes, especially the ability of anti-diffusion. Here are the parameters in the test case: the channel length L 10.0 m ; the bed slope S0 0.0016 ; the Manning’s coefficient n 0.025 s/m1/3 ; the simulation time t 100.0 s ; the time step t 0.01 s ; and the grid spacing x 0.1 m . The initial condition is: h max 0.5, 1.0 ( x 1.0)2 (4.11) This initial condition will create a parabolic perturbation in the channel as shown in Fig. 4.5. The boundary conditions are periodic flow depths at both ends of the channel. 60 1.5 Flow Depth (m) 1 0.5 0 0 2 4 6 Channel Distance (m) 8 10 Fig. 4.5. Initial condition of long time evolution case 1.5 MacCormack Flow Depth (m) 1 0.5 0 0 2 4 6 Channel Distance (m) 8 Fig. 4.6. Oscillations of the MacCormack scheme 10 61 The parabolic perturbation will produce a “wave steepening” effect in the domain [Henderson, 1966; Toro, 2009]. The reason is that the celerity of the perturbation is faster than the ambient fluid. The head of the perturbation is a compressive region while the tail of the perturbation is an expansive region. This perturbation is a combination of a shock wave (head) and a rarefaction wave (tail). This scenario is detrimental to the MacCormack scheme because the oscillations occurred behind the shock wave will drown out the parabolic perturbation immediately (Fig. 4.6). Consequently, the MacCormack scheme cannot converge to a stable solution, but amplified oscillations. This further proves that the MacCormack scheme is an unstable scheme for flows of discontinuous waves. The results of MUSCL scheme and WENO scheme are plotted in Fig. 4.7. After 100 seconds, the fronts of the perturbation are still sharp. The numerical diffusion of the MUSCL scheme is a little larger than that of the WENO scheme because the MUSCL scheme is a second-order scheme while the WENO scheme is third-order accurate near the discontinuity. 4.6 Uniform Rainfall-Runoff Overland Flow The uniform rainfall-runoff overland flow can be solved by the method of characteristics [Eagleson, 1970; Kazezyılmaz-Alhan et al., 2005]. At a given rainfall excess in a specified duration, the outflow hydrograph ( q qL , x L ) can be solved analytically. Assuming a constant rainfall excess ( i0 ) and an initial zero flow depth, h 0 (t 0 and 0 x L) (4.12) and the boundary condition at the channel entrance, h 0 (t 0 and x 0) (4.13) 62 The solutions of two possible outflow hydrographs are summarized below: 0.6 MUSCL WENO 0.59 0.58 Flow Depth (m) 0.57 0.56 0.55 0.54 0.53 0.52 0.51 0.5 0 2 4 6 Channel Distance (m) 8 10 MUSCL WENO 0.57 0.565 Flow Depth (m) 0.56 0.555 0.55 0.545 0.54 0.535 0.53 0.525 3 3.5 4 Channel Distance (m) Fig. 4.7. Results of wave steepening test case 4.5 63 Case 1 ( tr tc ): hL i0t , for t tc qL h , to solve hL use: hL i0tc , for tc t tr m 1 1 L hL [hLi0 m(t tr )], for t tr (4.14) h i t , for t t L 0 r m qL hL , to solve hL use: hL i0tr , for tr t t p m 1 1 L hL [hLi0 m(t tr )], for t t p (4.15) m L Case 2 ( tr tc ): where tr is the duration of rainfall, and tc is the time of concentration: 1/ m Li1 m tc 0 (4.16) and t p is defined as: tc* tr * L t p tr , tc m1 , hLr i0tr m hLr (4.17) In Kazezyılmaz-Alhan et al. [2005]’s hypothetical experiment, the duration of rainfall is longer than the time of concentration ( tr tc ). The experimental parameters are summarized here: the channel length L 182.88 m ; the bed slope S0 0.0016 ; the Manning’s coefficient n 0.025 s/m1/3 ; the duration of rainfall tr 0.5 h ; the rainfall excess i0 50.8 mm/h ; simulation time t 1 h ; the time step t 1.0 s ; and the grid spacing x 1.83 m . The initial 64 condition is h 0.0 m . The boundary conditions are zero-depth gradient at the outlet and zero depth at the inlet. Fig. 4.8 plots the outflow hydrographs calculated by the numerical schemes and the exact solution calculated by Eq. (28). Since there isn’t any discontinuity/perturbation in the domain, the results obtained by the three numerical schemes are very close to the exact solution. The comparisons of the peak of the hydrographs show that the WENO scheme yields the closest results to the exact solution without any oscillation. When the outflow discharge reaches the peak, flow in the channel reaches steady state. To check if the mass is conserved in the simulation, the differences between the total outflow volume and the total rainfall volume are calculated, and then non-dimensionalized by the total rainfall volume. The percentage flow differences relative to the total rainfall are 0.65 % for the MacCormack scheme, 0.35% for the MUSCL scheme, and 0.30 % for the WENO scheme, respectively. All three schemes preserve mass conservation very well at the absence of discontinuous waves. The MUSCL scheme and WENO scheme are slightly more conservative than MacCormack scheme. 65 0.18 EXACT MacCormack MUSCL WENO 0.16 Flow Discharge (m2/min) 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0 500 1000 1500 2000 Time (s) 2500 3000 3500 0.159 EXACT MacCormack MUSCL WENO 0.158 Flow Discharge (m2/min) 0.157 0.156 0.155 0.154 0.153 0.152 0.151 0.15 0.149 1400 1500 1600 1700 1800 1900 Time (s) Fig. 4.8. Hydrograph of uniform rainfall-runoff case 66 4.7 Steady Non-uniform Rainfall-Runoff Overland Flow To compare the stability of different schemes, a steady non-uniform rainfall-runoff case is tested. The test conditions are the same as the uniform rainfall-runoff case except for the rainfall excess. The following steady non-uniform rainfall excess is used in this case (Fig. 4.9): 2.0 50.8 0.0328 50.8 ( x 91.44)2 , 73.15m x 109.72m i0 (mm / h) 50.8, elsewhere (4.18) The duration of rainfall is infinite. The simulated results are showed in Fig. 4.10 and Fig. 4.11. The depth of the runoff generated by the non-uniform rainfall is not smooth and a shock wave is generated. Oscillations occurred in the solution by using the MacCormack scheme. Solutions by both the MUSCL and WENO schemes are stable free of oscillations. 200 180 Rainfall Intensity (mm/hr) 160 140 120 100 80 60 40 20 0 0 20 40 60 80 100 120 Channel Distance (m) 140 160 Fig. 4.9. Non-uniform rainfall distribution 180 67 0.02 MacCormack MUSCL WENO 0.018 0.016 Flow Depth (m) 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0 0 0.017 20 40 60 80 100 120 Channel Distance (m) 140 160 180 MacCormack MUSCL WENO Flow Depth (m) 0.016 0.015 0.014 0.013 0.012 140 145 150 155 160 165 Channel Distance (m) 170 175 180 Fig. 4.10. Results of non-uniform rainfall-runoff test case The computation is run with a DELL XPS notebook (Intel i5 M450 2.4GHz CPU with 4GB memory). The programs are developed using Matlab 2011b running on Microsoft Windows 7.0 OS. For this simulation, the CPU times using the MacCormack, MUSCL, and WENO 68 schemes are 5.8, 13.5, and 38.7s, respectively. As expected, the MacCormack scheme runs the fastest, while the WENO scheme is three times slower than the MUSCL scheme. 0.2 0.18 Flow Discharge (m2/min) 0.16 0.14 0.12 0.1 0.08 0.06 0.04 MacCormack MUSCL WENO 0.02 0 0 500 1000 1500 2000 Time (s) 2500 3000 3500 0.13 Flow Discharge (m2/min) 0.12 0.11 0.1 0.09 0.08 0.07 MacCormack MUSCL WENO 0.06 0.05 700 800 900 1000 1100 1200 Time (s) Fig. 4.11. Hydrograph of non-uniform rainfall-runoff test case 69 4.8 Rainfall-Runoff over Non-uniform Overland Slope Since overland flow always travels in channels of varying slopes, this requires the numerical scheme for solving the kinematic wave equation to be valid for changing bed slopes. Prior test cases have shown that the MacCormack scheme is unstable for flows with discontinuous waves (e.g., shock, rarefaction). Therefore, the MacCormack scheme is not applicable to non-uniform rainfall on a uniform slope. To verify its applicability on a changing slope, this study hypothesizes an experiment of rainfall and overland flow in a watershed having the same experimental conditions as in Kazezyılmaz-Alhan et al. [2005] except the upstream half of the channel has a slope different from the downstream half ( S0 0.0016 ). This study adopted two different slopes, 0.0032 and 0.016, for the upstream half of the channel. The simulated water surfaces are shown in Fig. 4.12. As the slope in the upstream half channel increases to 0.016, ten times of the downstream slope, the MacCormack scheme generated severe oscillations. At the slope, two times of the downstream slope, the MacCormack scheme yields relative smooth solutions similar to those from the MUSCL and WENO schemes. This test verifies that the MacCormack scheme is not only unstable for shock and rarefaction waves but also for flows on rapidly varying slopes. In summary, the testing cases listed in Section 4.1-4.6 pointed out the limitations of the MacCormack scheme for solving the kinematic wave equation and proved the validity of high resolution schemes, the MUSCL and WENO scheme. Although the MacCormack scheme has been widely used to solve the kinematic wave equation, it should be avoided at the presence of shock/rarefaction waves, or flow over rapidly varying sloped land surfaces. Instead, the MUSCL and the WENO scheme should be used in spite of the slight increase of computation cost. 70 0.2 MacCormack MUSCL WENO 0.18 Flow Discharge (m2/min) 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0 500 1000 1500 2000 Time (s) 2500 3000 3500 0.2 MacCormack MUSCL WENO 0.18 Flow Discharge (m2/min) 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0 500 1000 1500 2000 Time (s) 2500 3000 3500 Fig. 4.12. Hydrographs of the non-uniform overland slope test case 71 4.9 Summary In this study, the applicability of three numerical schemes (MacCormack scheme, MUSCL scheme and WENO scheme) to solve the kinematic wave equation is investigated. The schemes are examined by using six test cases: shock wave, rarefaction wave, wave steepening, overland flow from uniform or non-uniform rainfall, and overland flow over varying bed slopes. The results show that oscillations appeared in the solutions using the MacCormack scheme when the shock and rarefaction waves are present, or bed slopes changing rapidly. Since the MacCormack scheme is not a TVD scheme, the oscillations seriously affect the stability of the solution, and make the solution diverge. On the other hand, the MUSCL and WENO schemes consistently perform better than the MacCormack scheme. There is no oscillation in the solutions using the MUSCL scheme or the WENO scheme. The high-order WENO scheme shows the best resolution power in all test cases. Although the computational costs are higher than that of the MacCormack scheme, to ensure numerical stability, the MUSCL and WENO schemes are the stable and accurate techniques for solving the kinematic wave equation. 72 CHAPTER 5 A SURFACE FLOW ROUTING METHOD (SWE-KWA) Equation Section (Next) 5.1 Introduction Surface flow routing, including overland and channel flow routing, refers to the process that a precipitation generated surface runoff moves over land surface from source areas to an outlet (Fig. 5.1). Fig. 5.1. Concept view of surface flow, including channel flow and overland flow It’s an important component in the hydrologic and hydraulic researches including but not limited to rainfall-runoff modeling [Freeze and Harlan, 1969], flooding routing [Hunter et al., 2007; Sturm, 2001] and soil erosion [Johnson et al., 2000; Woolhiser et al., 1990]. Up to date, a 73 considerable amount of algorithms has been developed for modeling surface flow routing [Beven, 2011]. Most of them are based on the two-dimensional shallow water equations (SWE), also known as Saint-Venant equations or dynamic wave equations, or its simplifications, e.g., the diffusion wave equation (DWE) and the kinematic wave equation (KWE) [Vieira, 1983]. Overland flow is the flow that runs across the land surface before it enters a channel. [Woolhiser and Liggett, 1967] proved that the solution of overland flow produced by the kinematic wave equation is practically similar to the solution of the shallow water equations. In engineering applications [Woolhiser et al., 1990], [Julien et al., 1995] and [Downer and Ogden, 2004], the kinematic wave equation is a good approximation for most overland flows of large Froude number. For overland flow over mild slopes, several models based on the diffusion wave equation have been published [Abbott et al., 1986; Gottardi and Venutelli, 1993; Panday and Huyakorn, 2004; Qu and Duffy, 2007] which are, in theoretical perspective, more accurate than kinematic wave equation. In contrast to the overland flow, it is popular to simulate the channel flow by the diffusion wave equation. [Ponce, 1991; Ponce et al., 1978] found that a model based on diffusion wave equation would satisfy the requirements of most flood wave propagation cases in channels. [Downer and Ogden, 2004] solve the diffusion wave equation for channel flow routing where backwater is present. Other channel flow models based on the diffusion wave equation include [Panday and Huyakorn, 2004; Qu and Duffy, 2007], among others. A common practice in the surface flow routing is to calculate the overland flow and the channel flow separately using two models [Abbott et al., 1986; Downer and Ogden, 2004; Ivanov et al., 2004; Julien et al., 1995; Panday and Huyakorn, 2004; Qu and Duffy, 2007; Woolhiser et 74 al., 1990]. This practice makes it necessary to define a channel network with all the information including cross sections and connection links. Then the channel network needs to be coupled with the overland flow grid. These increase the difficulties of data pre-processing and decrease the running efficiency of program. Nowadays, a promising trend in this area is to solve the surface flow problem by the fully two-dimensional shallow water equations [Cea and Vazquez-Cendon, 2012; Cea et al., 2010; J Kim et al., 2012; Li and Duffy, 2011]. [Cea et al., 2010] compared the shallow water model with the diffusion wave model for flood inundation in urban areas and they concluded that the inertia terms becomes very significant in the rapidly varying flow. The numerical schemes used in these models are all based on the explicit Godunov-type finite volume method [LeVeque, 2002; Toro, 2009], which makes the models suitable for solving complex unsteady flow problems. Some typical problems include dam-break flow, hydraulic jump, flow convergence and divergence, and so on. Those problems are hard to solve by the traditional hydrologic models [J Kim et al., 2012]. Moreover, as pointed out by [Cea et al., 2010], hydrodynamic models can provide more accurate and more detailed information about the flow field, including flow depth, velocity, and shear stress. These flow characteristics are essential for predicting soil erosion, sediment transport, contaminant propagation and flash-flood forecasts. The solution of shallow water equations by the Godunov-type finite volume method originates in hydraulic research of dam-break flow [Toro, 2001]. Toro and Garcia-Navarro [Toro and Garcia-Navarro, 2007] gave a thorough review about this topic. A short list of engineering applications in the past ten years includes: Valiani et al. [Valiani et al., 2002] demonstrated the applicability of the high-resolution Godunov-type numerical model through a comparison between calculated data and field measurements for the Malpasset dam-break event; Begnudelli 75 and Sanders [L. Begnudelli and Sanders, 2007] calculated the ST. Francis dam-break flood with the MUSCL-Hancock scheme [Bradford and Sanders, 2002] and the Volume/Free-surface Relationships (VFRs) [L. Begnudelli and Sanders, 2006]; George [George, 2011] simulated the Malpasset dam-break flood by using the an augmented Riemann solver [George, 2008] and an adaptive refined mesh. Problems arise, however, when attempts are made to apply the method to the thin overland flow routing. The problem is stability of a shallow water equations solver. Audusse and Bristeau [Audusse and Bristeau, 2005] pointed out that most numerical schemes for shallow water equation based on the Godunov-type finite volume method do not preserve the positivity of the water depth. Xing et al. [Xing et al., 2010] reported that there may be an incompatibility between the well-balanced scheme and the positivity-preserving scheme. The reason for this problem comes from the discretization of the source terms in the shallow water equations, including the gravity term and the friction term [Murillo and Garcia-Navarro, 2010]. According to our experience, this problem might become far more serious when the slope of the terrain is steep and the flow depth is very small, i.e., in the overland flow situation. Although a lot of solutions to this problem have been proposed [Audusse and Bristeau, 2005; Cea and Vazquez-Cendon, 2012; Murillo and Garcia-Navarro, 2010; 2012; Xing et al., 2010], no real world application exists which overcomes this problem. Here the “real world application” means the flow problem in which the minimum flow depth has the same order as the precipitation rate (10-5 m/s – 10-8 m/s). By thoroughly analyzing the stability problem of the shallow water equation, we found that the kinematic wave approximation (KWA) can help solve this problem. The concept, kinematic wave approximation, has long been discussed in the research of surface flow routing 76 [Morris and Woolhiser, 1980; Ponce et al., 1978; Richardson and Julien, 1994; Vieira, 1983; Woolhiser and Liggett, 1967]. Woolhiser and Liggett [Woolhiser and Liggett, 1967] proposed a dimensionless number, i.e. the kinematic wave number, to evaluate when the kinematic wave equation is an accurate approximation of shallow water equations. The kinematic wave number is a function of bed slope and flow depth and represents the relative importance of different terms in the momentum equations. Vieira [Vieira, 1983] divided the Froude number vs. the kinematic wave number diagram to several zones, which represent suitable approximations to the shallow water equations. Based on these analyses, it is reasonable to calculate the surface flow by different approximation according to the local flow situations. And more importantly, the kinematic wave approximation is far more stable than a shallow water equation solver when the terrain is rough and the flow depth is thin, i.e. overland flow. The objective of this paper is to present an innovative algorithm for surface flow routing. The algorithm is based on the shallow water equation and the kinematic wave approximation. During the running time, the solver will divide the computational domain into the shallow water zone and the kinematic wave zone. An approximate solution based on the kinematic wave equation is applied to the kinematic wave zone. This approximation is equivalent to a simplified implicit Euler and helps to keep the balance between the gravity term and the friction term. The SWE-KWA integrated solution of shallow water equations and the kinematic wave equation provides a practical way to solve the surface flow routing problem over a large scale watershed while keeping the solution as accurate as close as the shallow water equations. 77 5.2 Numerical Modeling The two-dimensional shallow water equations are described in Chapter 2. The Godunovtype finite volume method is described in Chapter 3. The stability of numerical method is constrained by the Courant–Friedrichs–Lewy condition (CFL condition). The CFL number Cmax is defined as [LeVeque, 2002; Toro, 2001]: Cmax smax t x (5.1) where smax u gh is the maximum wave speed at a time step. For the advection dominated flow, the CFL number should be less than 1. However, the CFL condition is only a necessary condition and is not sufficient for the stability of the shallow water equations. More and more research [L. Begnudelli and Sanders, 2007; Fraccarollo and Toro, 1995; Wu et al., 2012; Yu and Duan, 2012] have proved this. The flow depth of a typical channel flow is often bigger than 10-3 m, whereas the flow depth of a typical overland flow is often less than 10-3 m. Our previous study [Yu and Duan, 2012] suggested that the CFL condition is adequate for the channel flow but inadequate for the overland flow. Under the overland flow condition, flow depth is small and the kinematic wave number is large. The advection terms in the shallow water equations are negligible, and flow is dominated by the slope source term and the friction source term. This means that the CFL condition loses its constraints and new stability criteria, therefore, has to be explored. 78 Consider a one-dimensional problem and assume that the flow is dominated by the slope and friction source terms. Then the advection terms are cancelled out and the discretized form for the x-direction momentum equation is written as: (hu) ( n 1) (hu) (n) t ( gh S0 gn (n) 2 u( n ) u( n ) (h ( n ) )1/3 ) (5.2) Eq.(5.2) can be rewritten as: u( n 1) where Fr h( n ) u( n ) g t ( S0 sgn(u ( n ) ) gn 2 ( Fr ( n ) )2 (h ( n ) ) 1/3 ) h( n 1) (5.3) u is the Froude number; sgn(u( n ) ) represents the sign of u ( n ) . In Eq.(5.3), the gh slope term is independent of flow variables, and the solution remains stable while the flow depth is not very small because the friction term is inversely proportional to flow depth. When flow depth decreases to a tiny value, this relationship brings a disastrous result to the numerical solution. This situation is shown in Fig. 5.2, where a series of hypothetical values of n and Fr are used to calculate the friction term and assume h(n ) 1 . The results show that the friction h ( n 1) term can be several orders larger than the slope term ( S0 0.01 in general) which in turn results in an unrealistic large velocity. Therefore, the solution of shallow water equations is unstable for overland flow, typically with very shallow flow depth. To solve the stability problem, we incorporate a kinematic wave approximation to the numerical solution of the shallow water equations. It is well known that flow field is dominated by the gravity and friction terms when water depth is small and kinematic wave number is large. 79 Under this situation, the kinematic wave equation is an accurate approximation to the shallow water equation [Vieira, 1983; Woolhiser and Liggett, 1967]. Instead of updating the velocity by Eq.(5.3), we can calculate the velocity approximately by: u ( n 1) 1 S0 (h ( n 1) )2/3 n (5.4) This approximate solution is based on the kinematic wave assumption, so we call it the Kinematic Wave Approximation (KWA) method. 10 10 Friction Term (-) 10 10 10 10 10 10 4 Fr = Fr = Fr = Fr = Fr = Fr = 3 2 1 0.5, 1.0, 1.5, 0.5, 1.0, 1.5, n= n= n= n= n= n= 0.01 0.01 0.01 0.1 0.1 0.1 0 -1 -2 -3 -4 10 -10 10 10 -8 10 -6 10 -4 10 -2 10 0 h (m) Fig. 5.2. Hypothetical values of friction term The kinematic wave assumption automatically preserves the exact balance between the slope term and the friction term. Flow velocity becomes nearly zero as flow depth decrease to a very small value. In practice, we have tested the flow depth as small as 1010 m . We found the method still keeps its stability and generates reasonable solutions. 80 The entire calculation processes within each time step are summarized in Fig. 5.3. In the program, two different switch criteria are adapted to determine whether the shallow water equations or the kinematic wave approximation should be used. One criterion is the kinematic wave number as in [Vieira, 1983; Woolhiser and Liggett, 1967], and the other is the local flow depth hKWA . The implementations of other flow routing components, including precipitation, interception and infiltration, are similar to CASC2D [Julien et al., 1995; Sánchez, 2002]. The precipitation intensity is interpolated by the inverse distance squared approximations: NRG im ( jrg , krg ) d2 i ( j, k ) m 1 NRG m 1 2 m 1 d m (5.5) where i( j, k ) is the rainfall intensity in cell ( j, k ) ; im ( jrg , krg ) is the measured rainfall intensity by the m-th rainfall gauge at ( jrg , krg ) ; d m is the distance from cell (i, j ) to m-th rainfall gauge at ( jrg , krg ) ; NRG is the total number of rainfall gauges. The interception loss is modeled by the concept of interception depth. The intercepted water does not reach the land surface and the rainfall excess rate is set equal to zero in each time step until the interception depth has been satisfied. The Green-Ampt model is used for the calculation of soil infiltration. By neglecting the ponding on the surface, the Green-Ampt model can be written as: H f Md f K s 1 F (5.6) 81 where f is the infiltration rate; K s is the saturated hydraulic conductivity; H f is the capillary pressure head at the wetting front; M d is the soil moisture deficit; F is the total infiltrated depth. At each time step, after the calculation of the surface flow part, the infiltrated water is subtracted from the surface flow. 5.3 One-Dimensional Rainfall-Runoff Case The first test case is a typical one-dimensional, rainfall-runoff flow routing over a constant hill slope. Similar cases can be found in [He et al., 2008; Kazezyilmaz-Alhan and Medina, 2007; J Kim et al., 2012], among others. The sketch of the case is shown in Fig. 5.4. The analytical solution of flow field based on the kinematic wave equation has been derived in [Ponce, 1989a]. The differences between the solutions of the kinematic wave equation and the shallow water equations will be discussed below. The parameters used in the test are summarized here: the plane length L 182.88 m ; the Manning’s coefficient n 0.025 s/m1/3 ; the bed slope S0 0.0016 ; the rainfall excess rate i0 50.8 mm/h ; the rainfall duration tr 0.5 h ; the total simulation time tmax 1.0 h . The boundary conditions are no-flow condition at the inlet and zero-gradient condition at the outlet. In the calculation, the domain is divided into 100 cells and the grid spacing is x 1.83 m . The constant time step ( t 1.0 s ) is used in the calculation. 82 Fig. 5.3. The flow chart of processes within each time step 83 Fig. 5.4. Schematic description of one-dimensional rainfall-runoff case 0.2 Exact KW KWA SWE+KWA 0.18 0.16 Discharge (m2/s) 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0 500 1000 1500 2000 Time (m) 2500 3000 3500 Fig. 5.5. Discharge hydrograph of one-dimensional rainfall-runoff case The calculated discharge hydrograph is shown in Fig. 5.5. In the figure, the ‘Exact KW’ is the analytical solution based the kinematic wave equation. For the ‘KWA’ solution, the minimum flow depth for using the kinematic wave 84 approximation, hKWA , is set to 0.1 m. Since the maximum flow depth is less than 0.1 m, the entire domain is calculated by the kinematic wave approximation method. As to the ‘SWE-KWA’ solution, hKWA equals 0.001 m and it represents the solution of the algorithm proposed in this paper. In general, the numerical solutions using the KWA of the combination of SWE and KWA match well with the analytical solution, which proves that the kinematic wave equation is an accurate approximation to the shallow water equations when the flow depth is thin. However, there are some differences near the peak discharge and the recession limb. The solutions by KWA and SWE-KWA methods showed some diffusion at the peak flow due to the 1st order scheme used in the paper. To investigate the reasons of these differences, flow field at time 1600 s and 2600 s, are plotted in Fig. 5.6 and Fig. 5.7, respectively. Different flow depth criteria hKWA are used in the solutions: hKWA 0.1 m for solution 1 hKWA 0.01 m for solution 2 h KWA 0.001 m for solution 3 (5.7) In Fig. 5.6, it is obvious to notice the behavior of the algorithm. Solution 2 clearly shows the feature of the algorithm: Solution 2 jumps immediately from the solution 1 to solution 3 near the switch criteria hKWA 0.01 m . This jump represents the difference between the shallow water equations and the kinematic wave equation. And the jump brings some numerical wiggle to the solution. The numerical wiggle is also obviously shown in Fig. 5.7. So to avoid this numerical wiggle, a smaller hKWA is recommended to use in the test cases. 85 0.025 Flow Depth (m) 0.02 0.015 0.01 0.005 0 0 Solutoin 1 Solution 2 Solution 3 20 40 60 80 100 Channel Length (m) 120 140 160 180 Flow Velocity (m/s) 0.15 0.1 0.05 0 0 Solutoin 1 Solution 2 Solution 3 20 40 60 80 100 Channel Length (m) 120 140 160 180 Fig. 5.6. Flow depth (top) and velocity (bottom) of 1D rainfall-runoff case (1600s) 86 0.015 Flow Depth (m) 0.01 0.005 0 0 Solutoin 1 Solution 2 Solution 3 20 40 60 80 100 Channel Length (m) 120 140 160 180 0.1 0.09 0.08 Flow Velocity (m/s) 0.07 0.06 0.05 0.04 0.03 0.02 Solutoin 1 Solution 2 Solution 3 0.01 0 0 20 40 60 80 100 Channel Length (m) 120 140 160 180 Fig. 5.7. Flow depth (top) and velocity (bottom) of 1D rainfall-runoff case (2600s) 87 5.4 Two-Dimensional Rainfall-Runoff over a V-Shaped Catchment This hypothetical experiment is the two-dimensional surface flow over a tilted V-shaped catchment generated by a constant rainfall event. The case is extensively used in recent surface flow routing researches [Di Giammarco et al., 1996; He et al., 2008; J Kim et al., 2012; Lai, 2009; Panday and Huyakorn, 2004; Sulis et al., 2010] to test the validity of different algorithms: kinematic wave equation, diffusive wave equation and coupled overland flow/channel network. The schematic description of the case is shown in Fig. 5.8. The catchment is a symmetrical Vshaped valley including two parts: one main channel and two hill slopes on both sides. Fig. 5.8. Schematic description of V-shaped catchment case 88 The parameters are listed here: the length of hill slope is 1000 m; the width of hill slope is 800m; the length of channel is 1000 m; the width of channel is 20 m. The hill slopes have two slopes: 0.05 in x direction and 0.02 in y direction; the channel has a single slope of 0.02 in y direction. The Manning’s roughness coefficients are 0.015 on the hill slopes and 0.15 in the main channel. The rainfall excess rate is 10.8 mm/h and rainfall duration is 90 min. The domain is discretized into 162 by 100 cells. The total simulation time is 180 min and the time step t 1.0 s . The zero-gradient boundary condition is applied at the outlet while the other three sides have zero-flow boundary conditions. The simulated flow hydrograph is shown in Fig. 5.9. For the KWA solution, the whole domain is calculated by the solution of kinematic wave equation. The SWE-KWA solution is calculated by the proposed algorithm. As seen in Fig. 5.9, both solutions reach the same peak discharges, but the KWA solution is slightly faster than the SWE-KWA solution. In Fig. 5.10, the total discharge is divided into the channel flow discharge and the overland flow discharge. For the KWA solution, more flow is routed through the hill slopes than the SWE-KWA solution. And the SWE-KWA solution routes more flow through channel than the KWA solution. Besides, for the V-shaped catchment, more flow is routed through the hill slopes than the channel. More details are revealed by examining the flow field. At 80 min, flow reaches a steady state. Flow depth and velocity at each cross section along the channel and hill slopes are shown in Fig. 5.11 and Fig. 5.12 respectively. For the SWE+KWA solution, both flow depth and velocity are larger than the KWA solution at each cross-section. However, for the cross sections the hill slope, both solutions have similar flow depths and KWA solutions yield larger flow velocity than the SWE+KWA solution. The SWE+KWA solution presents a different discharge 89 pattern from the KWA solution. Does this mean the SWE+KWA solution is better or worse than the KWA solution? More works are needed before we make a decision. 5 KWA SWE+KWA 4.5 4 Discharge (m3/s) 3.5 3 2.5 2 1.5 1 0.5 0 0 20 40 60 80 100 Time (min) 120 140 160 180 Fig. 5.9. Discharge hydrograph of two-dimensional V-shaped catchment case 5 KWA SWE+KWA 4.5 4 Overland flow Discharge (m3/s) 3.5 3 2.5 2 Channel flow 1.5 1 0.5 0 0 20 40 60 80 100 Time (min) 120 140 160 180 Fig. 5.10. Discharge hydrograph of overland flow and channel flow 90 0.3 KWA SWE+KWA 0.25 Flow Depth (m) 0.2 0.15 0.1 0.05 0 0 100 200 300 400 500 600 Channel Length (m) 700 800 900 1000 0.5 KWA SWE+KWA 0.45 Flow Velocity Magnitude (m/s) 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 100 200 300 400 500 600 Channel Length (m) 700 800 900 Fig. 5.11. Flow depth (top) and velocity (bottom) along channel (80 min) 1000 91 -3 6 x 10 KWA SWE+KWA 5 x = 600 m Flow Depth (m) 4 3 x = 200 m 2 1 0 0 100 200 300 400 500 600 Channel Length (m) 700 800 900 1000 0.6 KWA SWE+KWA Flow Velocity Magnitude (m/s) 0.5 x = 600 m 0.4 0.3 x = 200 m 0.2 0.1 0 0 100 200 300 400 500 600 Channel Length (m) 700 800 900 1000 Fig. 5.12. Flow depth (top) and velocity (bottom) along hill slope cross-sections 92 5.5 Goodwin Creek Experimental Watershed The Goodwin Creek Experimental Watershed is chosen to test the developed model. The watershed is in the Panola County, Mississippi. As a tributary of Long Creek, it flows into the Yocona River, Yazoo River Basin. The area of the watershed is 21.3 km2. The watershed elevation ranges from 71 m to 128 m above the sea level. The Digital Elevation Model (DEM) of the Goodwin Creek watershed is shows in Fig. 5.13. The DEM resolution used in the test is 30 m by 30 m. The rainfall event of Oct. 17, 1981 has been simulated by [Lai, 2009; Sánchez, 2002]. The soil type, land use, precipitation and DEM data are described in detail by [Blackmarr, 1995; Sánchez, 2002]. We use the same calculation methods, including precipitation, interception and infiltration, as [Julien et al., 1995]. Six stream gauges are used in the study of [Sánchez, 2002]. The duration of the rainfall is 4.8 hours. Rainfall data are available for 16 rain gauges and the inverse distance method is used to interpolate the precipitation data over the entire watershed. Seven soil types are used in the simulation. The model uses the Green-Ampt approximation for soil infiltration. The land use is divided into four types: forest, pasture, water and cultivated. The grid size is 30 m by 30 m. And the time step for the simulation is 1.0 s. Instead of using unstructured grid [Lai, 2009; Sánchez, 2002], our model directly uses the DEM mesh only. Because our surface flow routing algorithm integrates the solutions of SWE and KWA, the coupling between channel network and overland flow is not necessary in our model. The channel flow and the overland flow are calculated by either the SWE or KWA depending on the local flow depth. This increases the computing efficiency and decreases its 93 complexity. And the most important is that it improves the accuracy of simulation by adaptively applying the shallow water equation and kinematic wave approximation to the whole domain. 6000 120 5000 115 110 105 100 3000 95 90 2000 Elevation (m) y (m) 4000 85 80 1000 75 0 0 70 1000 2000 3000 4000 5000 x (m) 6000 7000 8000 9000 Fig. 5.13. DEM map of Goodwin Creek Experimental Watershed The measured and simulated flow hydrographs at the six gauges are compared with the results of CASC2D in Fig. 5.14. In CASC2D results, the following roughness coefficients are used: forest (n = 0.25), water (n = 0.01), cultivated (n = 0.15) and pasture (n = 0.2). These roughness coefficients are the calibrated parameters. Since our SWE-KWA method is based on shallow water equations, we changed these roughness coefficients to: forest (n = 0.05), water (n = 0.01), cultivated (n = 0.03) and pasture (n = 0.04). Overall, results from our model match the peak times and the peak discharges very well. At the internal gauges, the results from our model are slightly better than CASC2D results. However, the results from our model are deviated from 94 the measurements at the outlet. A possible explanation is that the parameters are not calibrated for the shallow water equations based algorithms, like our SWE-KWA algorithm. 8 12 Measured Data CASC2D SWE-KWA 7 Measured Data CASC2D SWE-KWA 10 Flow Runoff (mm/hr) Flow Runoff (mm/hr) 6 5 4 3 8 6 4 2 2 1 0 0 (a) 200 400 600 800 Time (min) 1000 1200 1400 0 0 (b) 14 Flow Runoff (mm/hr) Flow Runoff (mm/hr) 8 6 1400 Measured Data CASC2D SWE-KWA 6 4 2 2 200 400 600 800 Time (min) 1000 1200 1400 0 0 (d) 200 400 600 800 Time (min) 1000 1200 1400 18 Measured Data CASC2D SWE-KWA 16 14 14 12 12 10 8 6 10 8 6 4 4 2 2 200 400 600 800 Time (min) 1000 1200 1400 Measured Data CASC2D SWE-KWA 16 Flow Runoff (mm/hr) Flow Runoff (mm/hr) 1200 8 4 0 0 1000 10 18 (e) 600 800 Time (min) 12 10 (c) 400 14 Measured Data CASC2D SWE-KWA 12 0 0 200 (f) 0 0 200 400 600 800 Time (min) 1000 1200 1400 Fig. 5.14. Discharge hydrograph of Goodwin Creek case . (a) Gauge 1; (b) Gauge 4); (c) Gauge 6; (d) Gauge 7; (e) Gauge 8; (f) Gauge 14 95 5.6 Summary The challenge of routing flow over hill slopes and channels has been addressed with the SWE-KWA algorithm which is based on the shallow water equations and the kinematic wave approximation. The stability analysis reveals that the friction source term loses its stability as the flow depth decreases, while the slope source term keeps stable. This breaks the balance between the friction source term and the slope source term. The concept of kinematic wave approximation is introduced and leads to a simplified implicit Euler scheme which forces the balance between the friction term and the slope term. The SWE-KWA method is based on the shallow water equations. This makes it suitable to solve flow problems like backwater effect, flow choking and hydraulic jump which are hardly addressed by traditional routing methods. The method produces detailed flow information, including flow depth and velocity. The information is helpful to flood prediction and hazard analysis. The test results show that the proposed SWE-KWA method improves the stability of the program. In particular, the minimum allowable flow depth equals 10-10 m. Considering the common rainfall excess rate (10-5 m/s – 10-8 m/s), the SWE-KWA method is suitable for the rainfall-runoff routing applications. The results also show the differences between the kinematic wave equation and the shallow water equation. The kinematic wave equation predicts higher velocity for the overland flow. Whereas the shallow water equations higher flow depth and velocity for the channel flow. These differences affect the parameters’ calibration of rainfall-runoff models. Further efforts are needed to investigate this impact to model calibration. 96 Being different with traditional surface flow routing methods, the SWE-KWA method is based on the DEM data only and it does not need a channel network. On the one hand, this decreases the complexity of program and improves the efficiency of program. On the other hand, this makes the proposed method extremely suitable for the flow routing problems in ungauged watersheds. 97 CHAPTER 6 DEPTH-AVERAGED TURBULENCE MODELING Equation Section (Next) 6.1 Introduction Many free surface flows are considered as unsteady turbulent shallow water flows, such as open channel flow, dam-break flow, flash flood and tsunami, etc. Because the horizontal scale of the shallow water flow is far larger than the vertical scale, two dimensional depthaveraged models are often used for simulating the hydrodynamics of unsteady turbulent flow [Valiani et al. 2002, Duan 2004, Begnudelli et al. 2006, Murillo et al. 2009, Jim et al. 2010, Wang et al. 2011]. The governing equations, i.e. depth-averaged RANS equations, are a set of non-linear conservation equations. The solution of this system of governing equations may become discontinuous, even if the system starts from smooth initial conditions, representing physical processes, such as surges, hydraulic jumps and bore interactions. Among many numerical schemes to solve the depth-averaged RANS equations, Godunov-type finite volume method (FVM) with an approximate Riemann solver is an accurate, effective and robust method [LeVeque, 2002; Toro, 2009]. The high order Godunov-type method can precisely capture the flow with shock waves without spurious oscillation and with minimal numerical diffusion. However, the simulation of unsteady turbulent flow using the Godunov-type method has been based on the solutions of shallow water equations in which the eddy viscosity terms are neglected in the momentum equations [Valiani et al. 2002, Begnudelli et al. 2006, Begnudelli et al. 2008, Wang et al. 2011]. The importance of eddy viscosity terms to the solutions of 98 Godunov-type finite volume method has not been studied systematically in literature, with reference to this kind of numerical scheme. In general, the depth-averaged eddy viscosity terms in momentum equations are calculated by a depth-averaged turbulence model, such as the parabolic eddy viscosity model [Duan, 2004; Rodi and Research, 1993], the standard k model [Rodi and Research, 1993; Wu, 2004], the non-linear k model [Wu, 2004], algebraic stress model [Cea et al., 2007]. Babarutsi et al. [1996] reported that depth-integrated turbulence closure models are insensitive to the eddy viscosity for the friction-dominated zone. Based on this conclusion, Begnudelli et al. [2010] suggested a simple algebraic turbulence model to balance the model complexity and performance. Since the k model is non-linear and more complicated than the algebraic model, k model is less stable than the algebraic model. To ensure the stability of the solution, the calculated values (e.g., velocity, depth) are often bounded by the limiters [Shettar and Murthy, 1996]. Murillo et al. [2009] solved a system of discretized shallow water equations coupled with the k equations for rapidly varying unsteady flow, and proved that the uncoupled solver failed to predict correct results. For unsteady turbulent flow simulation, the selected numerical method faces two major challenges. The first challenge is the ability to capture the wet/dry fronts. Two methods, front tracking and front capture, are often used to deal with the wetting and drying processes. Since the front capturing method is easier to implement than the front tracking, the front capturing method is often adopted in this study. To distinguish wet and dry cells, Cea et al. [2007] used a wet-dry tolerance parameter. To precisely reconstruct the free surface in partially wetted cells, Begnudelli et al. [2006] proposed the Volume/Free-surface Relationships algorithm. Brufau et al. [2004] presented a wetting-drying condition algorithm that generates zero numerical error in mass 99 conservation. Lai [2010] developed a wetting-drying algorithm including cell-coloring, edgesearching and rewetting operations. All those methods can result in water over-drafted cells when using the multi-time step scheme [L. Begnudelli and Sanders, 2006; Brufau et al., 2004]. Two second order explicit schemes are often used in solving the shallow-water equations: the predictor-corrector scheme from MUSCL-Hancock method [L. Begnudelli and Sanders, 2007; Causon et al., 2000; Murillo et al., 2009] and the TVD Runge-Kutta scheme [Liang and Marche, 2009; Song et al., 2011]. The advantage of the TVD Runge-Kutta scheme over the predictorcorrector scheme is that the TVD Runge-Kutta scheme can prevent the appearance of oscillations by guaranteeing the TVD property in every intermediate step [Gottlieb and Shu, 1998]. This advantage becomes significant when calculating the dry-wet fronts because the scheme helps to guarantee a positive water depth as time proceeds. Therefore, this study will use the TVD Runge-Kutta scheme instead of the predictor-corrector scheme in the MUSCL-Hancock method. The second challenge is the bed slope source term. Zhou et al. [2001] proposed the Surface Gradient Method (SGM) to treat the bed slope source term. The SGM is a second-order accurate scheme with well-balanced property. But for the cells near the wet-dry fronts, the method is easy to lose stability. To solve this problem, Brufau et al. [2002] modified the bed slope at the wet-dry front. Bradford and Sanders (2005) suggested the Double Minmod limiter to resolve discontinuity when applying the 2nd order Roe’s solver. Cea et al. [2007] applied a kinetic reflection condition at the cell face of wet-dry front. Liang et al. [2009] derived prebalanced shallow-water equations with consideration of pressure balancing. Wang et al. [2011] developed a simplified local bed modification method without including the extra balancing term. This study employs the 2nd order scheme at the internal cells and 1st order scheme for the 100 boundary cells. An extra flux term due to bed slope source term is derived to balance the momentum equations. To accommodate irregular boundaries, the body-fitted unstructured mesh [L. Begnudelli and Sanders, 2006; Murillo et al., 2009; Zhao et al., 1994] is often used. One of the alternative approaches is the Cartesian cut-cell method [Causon et al., 2000; H J Kim et al., 2010; Zhou et al., 2004]. The accuracy of Cartesian cut-cell method has been assessed by Coirier et al. [1995]. Compared to the unstructured mesh, the Cartesian cut-cell mesh is easy to generate and efficient to implement. This study adopted the Cartesian cut-cell approach to accommodate the irregular boundaries in natural rivers. The method cuts the boundaries of a domain out of a background uniform rectangular mesh. The cells of the domain are divided into uniform rectangular internal cells and irregular boundary cells. It is straight forward to calculate the gradient and flux of the internal cells. The memory usage of internal cells is lower than that of irregular cells. The boundary cells are irregular with three, four or five edges. If the area of a boundary cell is less than a minimum value, the cell should be merged with one of its neighbor cell. To cope with more complex geometries, the Cartesian cut-cell method can be easily extended to non-uniform Cartesian mesh and incorporated into the adaptive mesh refinement [Causon et al., 2000]. 6.2 Numerical Modeling The depth-averaged RANS can be written in differential form as: Q Fx Fy Dx Dy S0 S f S k t x y x y (6.1) 101 where Q is the conservative variables; Fx and Fy are the advective fluxes in x and y direction; Dx and D y are the diffusive fluxes in x and y direction; S0 is the bed slope source term; S f is the bed friction source term; Sk is the source term for k model. The conservative variables Q and primitive variables U are defined as: h h hu u Q hv ,U v hk k h (6.2) where h is flow depth; u and v are the depth-averaged velocities in x and y direction, respectively; k is the turbulent kinetic energy; is the energy dissipation rate. The advective fluxes Fx and Fy are defined as: hu hv 2 huv huu gh / 2 , Fy hvv gh 2 / 2 Fx hvu hku hkv h u h v where g is the gravity acceleration. The diffusive fluxes Dx and D y are defined as: (6.3) 102 0 0 h( t )( u v ) 2h( t ) u y x x u v v h ( )( ) 2h( t ) t y x , D y y Dx h t k h t k k x k y h t h t x y (6.4) where is the kinematic viscosity of water; t is the eddy viscosity and is calculated as t C k2 ; C , k and are the constants for k model and are listed in Eq. (9). The bed slope source term S0 is defined as: 0 gh b x b S0 gh y 0 0 (6.5) where b is the bed elevation. The bed friction source term is defined as: C f u S f C v f u2 v2 u2 v2 0 0 0 (6.6) 103 where C f gnm2 / h1/3 is the drag coefficient, and nm is Manning’s roughness coefficient. Sk , the source term for k model, is defined as: 0 0 0 Sk Ph Pkb h 2 C 1 Ph P b C 2h k k (6.7) and Ph , Pkb and P b are defined as: Ph h t (2( u 2 v u v ) 2( ) 2 ( ) 2 ) x y y x Pkb C f 1/2U *3 (6.8) P b C C 2C1/2C f 3/4U *4 / h where U* C f (u 2 v 2 ) is bed friction velocity. The coefficients are listed below [Rodi and Research, 1993; Wu, 2004]: C 0.09, C 1 1.44, C 2 1.92, k 1.0, 1.3, C 1.8 3.6 (6.9) The governing equations are solved by the Godunov-type finite volume method and the local Riemann problems are solved by the HLLC approximate Riemann solver. Second order MUSCL scheme and TVD Runge-Kutta scheme are used in the problem. The developed model is tested using two experimental cases and one field case. The following are detailed descriptions of those test cases and simulation results. The computing platform is a DELL XPS Studio 15 with Intel i5 CPU and 4GB memory. 104 6.3 Dam-Break Flow over a Triangular Sill The experiment is designed by the European IMPACT project [Soares-Frazão, 2007]. The focus is to test the performances of numerical model with and without the turbulence model to predict flood propagation over an initially dry bed, using 1D dam-break flow. The experiment was conducted in a 5.6 m long and 0.5 m wide straight rectangular channel with glass walls. The experimental set-up is shown in Fig. 6.1. The upstream reservoir is 2.39 m long and initially filled with 0.111 m of water at rest. From the gate to the bump, the channel is dry. The triangular bump is 0.065 m high and has a slope of 0.14 at both sides. Downstream from the bump, a pool contains 0.02 m of water at rest, and a wall closes the end of the channel. The Manning friction coefficient of the flume is 0.011 s/m-1/3 . Three resistive gauges are used to record stages at three distinct locations (Fig. 6.1), and those measurements are verified by digital image measurements. Fig. 6.1. Flume set-up, initial conditions and gauge locations for test case 1 (in meters) The computational domain is discretized by a Cartesian mesh of 56 10 cells. Three different grid sizes, 5 cm (112 10 cells), 6.7 cm (84 8 cells), and 10 cm (56 5 cells), are used. Slip boundary conditions are used at the walls. The CFL number is set to 0.5. Time step is adjusted at each time step to satisfy the constant CFL number. The time steps are 0.032 – 0.062 sec (grid size = 10 cm), 0.021 – 0.040 sec (grid size = 6.7 cm), and 0.015 – 0.030 sec (grid size = 105 5.0 cm). The CPU times are 1.34 sec, 3.92 sec, and 8.22 sec for grid sizes, 10 cm, 6.7 cm, and 5.0 cm, respectively. If no turbulence model is used, the CPU time for 5.0 cm grid is 6.83 sec. Fig. 6.2 shows the calculated water depth changes with time at three different gauge locations, compared with the gauge measurements. As the grid size reduces, simulated water depths gradually converge to the measurements at Gauge 1 and 2, especially during the first 10 seconds. For Gauge 3, the simulated results by using three different grids are nearly identical. The propagation of flood wave is captured perfectly at the left side of the bump. Both magnitude error and phase error are negligible at Gauge 3. Additionally, after about 20 seconds, results by using three different grids are nearly the same at Gauge 1 and 2. This indicates that results from all three grids are accurate when flow approaches steady state after the initial dam break. When the grid size is less or equal to 5 cm, the solutions are largely converged. Since initially, the water depth at the right side of the bump is 2 cm, the left of the bump is dry until the dam-break flow arrives. The bump is partially submerged at the right side. Therefore, there is a stationary steady flow with wet/dry fronts over uneven bottom at the beginning of simulation. Fig. 6.2 shows this steady flow at Gauge 1 and Gauge 2 is predicted precisely by our numerical model. The dry bed at Gauge 3 before the dam-break flow arrives is also predicted accurately. Those results verify the well-balanced property of the model. When the dam-break flow advances in the downstream direction, a surge is formed at t = 1.92 sec and a jet flow is observed at t = 2.6 sec. Both these flow features are predicted by the model. After t = 5 sec, the reflected bore wave comes back to the bump, mixes with the dambreak flow resulting in a significant increase of flow depth. The calculated profiles with x 5cm agree well with the gauge records. 106 h (m) 0.1 0.05 Observation 10.0 cm 6.7 cm 5.0 cm 0 0 5 10 15 20 (a) 25 30 35 40 45 t (s) h (m) 0.1 0.05 Observation 10.0 cm 6.7 cm 5.0 cm 0 0 5 10 15 20 (b) 25 30 35 40 45 t (s) h (m) 0.1 0.05 Observation 10.0 cm 6.7 cm 5.0 cm 0 (c) 0 5 10 15 20 25 30 35 40 45 t (s) Fig. 6.2. Comparisons between measured data and calculate results for test case 1 (a) gauge 1, (b) gauge 2 and (c) gauge 3 107 h (m) 0.1 0.05 Observation Without turbulence model With turbulence model 0 (a) 0 5 10 15 20 25 30 35 40 45 t (s) h (m) 0.1 0.05 Observation Without turbulence model With turbulence model 0 (b) 0 5 10 15 20 25 30 35 40 45 t (s) h (m) 0.1 0.05 Observation Without turbulence model With turbulence model 0 (c) 0 5 10 15 20 25 30 35 40 45 t (s) Fig. 6.3. Comparisons of simulated flow depth with and without turbulence model (a) gauge 1, (b) gauge 2 and (c) gauge 3 Fig. 6.3 shows the differences of simulated water depth at three gauges with and without the turbulence model by using 5 cm sized grid. At Gauge 3, there is very minor difference between results with and without the turbulence model. At Gauge 1 and 2, results with the turbulence model is a little worse than those without the turbulence model for the first 10 108 seconds, then the results become nearly the same regardless of turbulence model. This attributes to the inaccuracy of the standard model for approximating turbulence in unsteady flow because all the coefficients are empirically determined from measurements of steady turbulent flow. Direct measurements of unsteady turbulent flow are needed to better define those coefficients. The small differences between results with and without turbulence model indicate the impacts of turbulence model on 1D model are insignificant. Perhaps, this explains that several depth-averaged flow model predicted reasonable flow field without turbulence closure models [Valiani et al. 2002, Begnudelli et al. 2006, Wang et al. 2011]. 6.4 Dam-Break Flow through an Isolated Obstacle The experiment is also a part of the IMPACT project [Soares-Frazão and Zech, 2007]. The objective is to investigate the effects of a single obstacle on a dam-break flow. The set-up of the experiments is sketched in Fig. 6.4. The experiments were carried out in a 36 m long and 3.6 m wide open channel with a composite cross section, which is rectangular at the top and trapezoidal near the bed. An impermeable wall with a 1 m wide gate was installed at 6.9 m from the flume entrance to serve as the reservoir. The channel bed is horizontal. The bottom roughness coefficient is 0.010 s/m1/3 . An isolated obstacle representing a building is placed at the downstream channel at 64 degree to the side of the flume. At the beginning, the reservoir is filled up with water to h = 0.40 m and the channel is filled with a thin layer of water (h = 0.02 m). As soon as the gate opens, flow will inundate the building. Water levels and velocities were measured respectively using resistive gauges and Acoustic Doppler Velocimeters (ADV) at five gauges surrounding the building and one gauge inside the reservoir. Detailed description of the experiments can be found at [Soares-Frazão and Zech, 2007]. 109 Fig. 6.4. Flume set-up, initial conditions and gauge locations for test case 2 (in meters) The computational domain is discretized by a Cartesian mesh. Transmissive boundary condition is used at the outlet (east side) of the flume. All other wall boundaries are designated as slip boundary conditions. The CFL number is equal to 0.5. The time steps are 0.015 – 0.024 sec (grid size = 10 cm with 360 36 cells), 0.01 – 0.016 sec (grid size = 6.7 cm with 538 54 cells) and 0.007 – 0.012 sec (grid size = 5.0 cm with 720 72 cells). The CPU times are 21.36 sec for 10 cm sized grid, 76.19 sec for 6.7 cm, and 197.59 sec for 5.0 cm. If no turbulence model is used, the CPU time for 5.0 cm grid is 159.31 sec. Fig. 6.5 shows the comparisons of the simulated water depth and corresponding measurements at six gauges with three different sized grids. At Gauge 2, only the 5 cm grid is able to capture the wave front. Finer grids yield slightly better results, but cannot match the instantaneous fluctuations of flow depth at the gauges around the obstacle. This means the simulation error is no longer depending on grid size, but the turbulence model and the dispersion terms resulting from the discrepancies between the actual velocity and depth-averaged velocity. As mentioned earlier, none of the turbulence closure model has been verified by measurements of unsteady turbulent flow. And there is no mathematical formula to approximate the vertical velocity profiles for flow field around an obstacle. 110 0.2 Observation 10.0 cm 6.7 cm 5.0 cm h (m) 0.15 0.1 0.05 0 (a) 0 5 10 15 t (s) 20 25 30 0.2 Observation 10.0 cm 6.7 cm 5.0 cm h (m) 0.15 0.1 0.05 0 (b) 0 5 10 15 t (s) 20 25 30 0.2 Observation 10.0 cm 6.7 cm 5.0 cm h (m) 0.15 0.1 0.05 0 (c) 0 5 10 15 t (s) 20 25 30 111 0.2 Observation 10.0 cm 6.7 cm 5.0 cm h (m) 0.15 0.1 0.05 0 (d) 0 5 10 15 t (s) 20 25 30 0.2 Observation 10.0 cm 6.7 cm 5.0 cm h (m) 0.15 0.1 0.05 0 (e) 0 5 10 15 t (s) 20 25 30 0.6 Observation 10.0 cm 6.7 cm 5.0 cm 0.5 h (m) 0.4 0.3 0.2 0.1 0 (f) 0 5 10 15 t (s) 20 25 30 Fig. 6.5. Comparisons between observations and calculated results for test case 2 (a) gauge 1, (b) gauge 2, (c) gauge 3, (d) gauge 4, (e) gauge 5 and (f) gauge 6 112 The dispersion terms in the momentum equations cannot be quantified. At present, the results by using 5 cm grid are the best if using the standard model without the dispersion terms. For the gauges around the obstacle, there is a delay of arriving time of dam-break flow in the calculated results. The delays are nearly constant, about 1/4 seconds, for all five gauges. Noël et al. (2003) found that it is a measurement error in system level. The hydraulic jump formed from the wave reflection against the obstacle is well captured by Gauge 2. The model predicted this hydraulic jump correctly at t = 20 sec. At Gauge 6 inside the reservoir, a pattern of wave superposition was observed. These waves are resulted from the reflection waves as the back water waves reach the different sides of the reservoir. Although discrepancies between simulated and measured water depths exist, in general, this experimental dam-break flow passing through the obstacle is generally well predicted by this model. But due to inaccuracy of turbulence model and neglecting of dispersion terms, the model did not predict the water surface fluctuations very well, especially when the flow field is highly three-dimensional around the obstacle. This is clearly the limitation of the depth-averaged flow model. Fig. 6.6 shows the results with and without turbulence model at six gauges. At Gauge 1, the results with turbulence model over-predict flow depth. Gauge 2 locates at where the dambreak wave reflects against the obstacle and results in an oblique hydraulic jump. The results with a turbulence closure model agree better with the measurements. Gauge 3, 4, and 5, are in zones where several oblique hydraulic jumps are superimposed by reflection waves from the obstacle and side walls. The results are nearly the same regardless of turbulence model, although only the mean values of flow depths are predicted. Those results demonstrate that the turbulence model is important in simulating flow field around an obstacle. 113 0.2 Observation Without turbulence model With turbulence model h (m) 0.15 0.1 0.05 0 (a) 0 5 10 15 t (s) 20 25 30 0.2 Observation Without turbulence model With turbulence model h (m) 0.15 0.1 0.05 0 (b) 0 5 10 15 t (s) 20 25 30 0.2 Observation Without turbulence model With turbulence model h (m) 0.15 0.1 0.05 0 (c) 0 5 10 15 t (s) 20 25 30 114 0.2 Observation Without turbulence model With turbulence model h (m) 0.15 0.1 0.05 0 (d) 0 5 10 15 t (s) 20 25 30 0.2 Observation Without turbulence model With turbulence model h (m) 0.15 0.1 0.05 0 (e) 0 5 10 15 t (s) 20 25 30 0.6 Observation Without turbulence model With turbulence model 0.5 h (m) 0.4 0.3 0.2 0.1 0 (f) 0 5 10 15 t (s) 20 25 30 Fig. 6.6. Comparisons of calculated flow depth with and without the turbulence model (a) gauge 1, (b) gauge 2, (c) gauge 3, (d) gauge 4, (e) gauge 5 and (f) gauge 6 115 However, the accuracy of simulated results is limited by the adapted turbulence model and inability of depth-averaged model. The further improvements of modeling results rely on more accurate turbulence model for unsteady turbulence flow and dispersion terms, rather than numerical schemes. Besides, the results also show that the developed numerical scheme is a well-balanced robust scheme for simulating unsteady turbulent flow. 6.5 Malpasset Dam-Break The Malpasset dam-break case is performed by the CADAM project. The aim is to validate numerical simulations of flooding waves due to dam failure. It is a strict test of the reliability of the numerical model using natural river data. The domain size is about 18000 10000 m. The water elevation is 100 m inside the reservoir and the floodplain is assumed initially dry. The Manning’s roughness coefficient is set to n = 0.033 s/m1/3 . This case is selected to test the model’s applicability to an irregular domain. The initial background Cartesian grid size is 1000 500 , and then is cut by the irregular boundary. The cell size is 18 20 m. After removing the invalid cells outside the boundary, the domain has 163,882 Cartesian cells including internal cells and cut cells. The simulation domain with 163,882 cells cannot be shown in a figure because each cell would be too small to differentiate. To illustrate the Cartesian cut-cell domain, Fig. 6.7 shows a demonstrative Cartesian cut cell grid with a background grid of 200 100 . All the boundaries of cut cells are set to wall boundary conditions. The CFL number used in the calculation equals to 0.5. 116 Fig. 6.7. Demonstration of Cartesian cut-cell grid for test case 3 Fig. 6.8. Inundation map at t = 1200 s for Malpasset case 117 100 Measured Calculated (m) 80 60 40 20 P1 P3 P5 P7 P9 P12 P13 P15 P17 Gauge (a) 100 Measured Calculated (m) 80 60 40 20 P2 (b) P4 P6 P8 P10 P11 P14 P16 Gauge Fig. 6.9. Comparisons between surveyed data and calculated results (a) gauges of police (right side), (b) gauges of police (left side) 118 100 Measured Calculated (m) 80 60 40 20 S6 S7 S8 S9 S10 S11 S12 S13 S14 Gauge Fig. 6.10. Comparisons between measured data of physical model and calculated results Fig. 6.8 shows the flow depth and flood extent at t = 1200 sec when the flood nearly reaches the sea. The available measurement data include 17 police surveyed locations (start with P) and 9 physical model simulated points (start with S). Fig. 6.9 compares the police surveyed data with our model simulated results. Fig. 6.10 is the comparison between the physical model and numerical model. The calculated results agree well with the measurements from both the field survey and the physical model. The results also verify the ability of the Cartesian cut-cell method to treat irregular boundaries. 6.6 Summary In this research, a second order finite volume model for depth-averaged RANS equations coupled with the k turbulence model is developed. A Godunov-type cell-centered finite volume method is adopted in which an augmented HLLC approximate Riemann solver is used to calculate the inter-cell fluxes. The second order MUSCL type linear reconstruction using the 119 gradients of primitive variables is implemented to restore flow states at the interface of cells. At the wet/dry fronts, the numerical scheme is reduced to first order approximation to avoid unrealistic flux from the MUSCL scheme. This technique requires the addition of an extra flux to keep the well-balanced property for the solution at the wet/dry fronts. The developed model is tested by two experimental and one field dam-break flow cases. For 1D dam-break flow, the simulated results agree well with the measurements regardless the addition of a turbulence model. For the two-dimensional case, without a turbulence model, the simulation results cannot predict the observed stages. By adding a turbulence model, simulation results, while capturing the main features of the experiment, still miss some of the details. This indicates that a turbulence model is necessary for simulating unsteady turbulent flows near obstacles due to the complex turbulence activities. However, the observed water surface oscillations cannot be simulated by the current model due to the lack of an accurate turbulent closure model for unsteady turbulent flow and the neglecting of dispersion terms. Obviously, direct solutions of Navier-Stokes equations, such as a 3D DNS model, should be applied to fully capture the water surface changes. In case that 2D model is necessary, a well verified turbulence closure model for unsteady turbulent flow, and vertical velocity profiles for flow around an obstacle and near a wall are needed to improve the model’s accuracy. Furthermore, the model adapted the Cartesian cut-cell technique to accommodate the irregular boundaries. This method requires high resolution meshes to precisely represent the boundary irregularity, which may require high computing powers. To reduce computing cost, parallel computing and adaptive mesh techniques will be developed in the near future. 120 CHAPTER 7 NUMERICAL MODELING OF DAM-BREAK FLOW OVER MOVABLE BEDS Equation Section (Next) 7.1 Introduction Non-cohesive bed materials start motion as soon as the shear stress applied on the bed surface is greater than the critical shear stress. Generally, coarse particles roll and slide in a thin layer near the bed, and finer particles enter suspension (Fig. 7.1). The total sediment load can be divided into bedload and suspended load. Numerous sediment transport equations have been developed in the past and significant progresses have been made. Suspended Load Bed Load Rolling Saltating Sliding Stream Bed Fig. 7.1. Concept view of sediment-Laden flow 121 While field observations are rare due to obvious reasons, laboratory experiments are largely constrained by the comparatively small spatial scales and may not be able to fully reveal the mechanism of sediment transport. Over the recent decades, numerical modeling techniques for the prediction of dam-break flows have been developed. Those efforts are aimed to enhance the understanding of dam failure induced floods. Cao et al. [2004] presented one of the first numerical model hydraulics of dam-break flows and resultant sediment transport and morphological evolution. The numerical model is based on the one-dimensional shallow water equations and the total load transport equation. The system of equations is solved by TVD version of the second-order weighted average flux method in conjunction with the HLLC approximate Riemann solver and SUPERBEE slope limiter. Wu and Wang [2007] established a one-dimensional numerical model to simulate the fluvial processes under dam-break flow over movable beds. The model is based on the generalized shallow water equations. The sediment model computes the non-equilibrium transport of bed load and suspended load. The effects of sediment concentration on sediment settling and entrainment are considered in determining the sediment settling velocity and transport capacity. The system of equations is solved by the first-order explicit Euler scheme and upwind scheme. Zhang et al. [2013] developed a one-dimensional finite volume model for simulating non-equilibrium sediment transport in unsteady flow. The governing equations are mass and momentum conservation equations for sediment-laden flow and the sediment continuity equation for both bed load and suspended load transport. The Rouse profile is modified to consider the 122 non-equilibrium transport of suspended sediment. The concept of adaption length is used to model the spatial lag between the instantaneous flow properties and the rate of bed load transport. In this research, the Cao et al. [2004] model is extended to two-dimensional case. The numerical model is tested against 1D and 2D dam-break experiments. 7.2 Numerical Modeling The mass and momentum conservation equations for the sediment-laden flow and the mass conservation equations for total sediment load and bed change are: ( h ) ( hu ) ( hv ) b t x y b t 1 ( huv ) ( hu ) 2 2 gh( S0 x S fx ) t x ( hu 2 gh ) y 1 ( hv ) ( hvu ) ( hv 2 gh 2 ) gh( S0 y S fy ) x y 2 t ( hCt ) ( huCt ) ( hvCt ) ED t x y b D E 1 t (7.1) where t is time; x and y are spatial coordinates; h is flow depth; u and v are flow velocities in x and y coordinates; Ct is volumetric concentration of total load sediment; b is bed elevation; is porosity of bed sediment; (1 Ct ) w Ct s is density of sediment-laden flow; b w (1 ) s is density of bed; w and s are the density of water and sand, respectively; D is sediment deposition rate; E is sediment entrainment rate; S0 x and S0 y are gravity source terms in x and y directions; S fx and S fy are friction source terms in x and y directions. 123 Eq. (7.1) can be reformulated by eliminating flow density on the left hand sides. The new equations read: h ( hu ) ( hv ) E D t x y 1 1 2 ( huv ) ( hu ) 2 t x ( hu 2 gh ) y gh( S0 x S fx ) Scx Smx 1 ( hv ) ( hvu ) ( hv 2 gh 2 ) gh( S0 y S fy ) Scy Smy x y 2 t ( hCt ) ( huCt ) ( hvCt ) ED t x y b D E 1 t (7.2) The gravity source terms are defined as: S0 x b b and S0 y x y (7.3) The friction source terms are defined as: n2u u2 v w n2v u2 v w S fx and S fy h 4/3 h 4/3 (7.4) The concentration source terms are: Scx ( s w ) gh 2 Ct ( w ) gh 2 Ct and Scy s 2 x 2 y (7.5) The momentum exchange source terms are: Smx ( b ) E D u (1 ) and Smy ( b ) E D v (1 ) (7.6) 124 The sediment deposition rate equals: D Ct0 (1 Ct )m (7.7) and the sediment entrainment rate equals: ( ) u 2 v 2 h 1d 0.2 , if c c E 0, else (7.8) where min(2,(1 ) / Ct ) ; 0 is settling velocity; m 2 ; c is critical Shields parameter; is Shields parameter; d is sediment particle diameter; is empirical constant. This system of equations is solved by the first-order Godunov-type finite volume method. The HLL approximate Riemann solver is used to solve local Riemann problems. 7.3 1D Dam-Break Flow over Mobile Bed The first test case is one-dimensional laboratory experiment of dam-break waves propagating over non-cohesive sediment bed [Spinewine and Zech, 2007]. The flume is equipped with a fast downward-moving gate. The flood wave generates intense geomorphic changes as bed sediments are entrained and deposited due to flow generated shear stress. 0.35 m Reservoir 0.10 m Sand Bed 3.0 m Gate Sand Bed 3.0 m Fig. 7.2. Schematic diagram of 1D dam-break flow over mobile bed 125 Calculated surface Calculated bed Observed surface Observed bed Time = 0.25 s 0.4 0.4 0.3 Elevation (m) Elevation (m) 0.3 0.2 0.1 0 0 -2 -1 0 X (m) 1 2 -0.1 -3 3 Calculated surface Calculated bed Observed surface Observed bed Time = 0.75 s 0.4 Elevation (m) Elevation (m) 0 X (m) 1 2 3 Calculated surface Calculated bed Observed surface Observed bed 0.3 0.2 0.1 0.1 0 0 -2 -1 0 X (m) 1 2 -0.1 -3 3 Calculated surface Calculated bed Observed surface Observed bed Time = 1.25 s 0.4 -2 -1 0 X (m) 1 2 3 Calculated surface Calculated bed Observed surface Observed bed Time = 1.50 s 0.4 0.3 Elevation (m) 0.3 0.2 0.2 0.1 0.1 0 0 -0.1 -3 -1 0.4 0.2 -0.1 -3 -2 Time = 1.00 s 0.3 Elevation (m) 0.2 0.1 -0.1 -3 Calculated surface Calculated bed Observed surface Observed bed Time = 0.50 s -2 -1 0 X (m) 1 2 3 -0.1 -3 -2 -1 0 X (m) 1 Fig. 7.3. Computational results on 1D dam-break flow over mobile bed 2 3 126 The schematic plot is shown is Fig. 7.2. The flume is 6 m long, 0.25 m wide and 0.7 m deep. A central gate is used to simulate an idealized dam. The flow is imaged through the sidewalls with fast digital cameras. The following parameters are used in the simulation: sediment particle density s 2683kg / m3 ; median size d50 1.82mm ; settling velocity 0 0.16m / s ; bed porosity 0.47 ; Manning coefficient n 0.0165 . 1 0.9 Concentration of Total Load 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -3 -2 -1 0 X (m) 1 2 3 Fig. 7.4. Concentration of total load sediment at time = 0.75 s The comparison of the calculated results and the measured results are plotted in Fig. 7.3. As shown in the figure, the calculated results show good agreement with measured results. A hydraulic jump is formed near the initial dam position. The occurrence of the hydraulic jump is well predicted by the numerical model. The result does not show significant initial scour hole. This result agrees well with the observations. In Fig. 7.4, high sediment concentration is accumulated at the front of dam-break flow and the peak value predicted by the model is 0.3. 127 This high sediment concentration is not surprising and this means that the flow picks up the sediments at the front and transports them to the downstream. 7.4 2D Dam-Break Flow over Mobile Bed This test case is a benchmark experiment (NSF-PIRE project “Modelling of Flood Hazards and Geomorphic Impacts of Levee Breach and Dam Failure”) carried out at UCLBelgium to investigate the 2D morphological evolution of a mobile bed under dam-break flow [Soares-Frazao et al., 2012]. The objective is to provide a test case to validate numerical models for the simulation of dam-break flow over a mobile bed. The plan view of the flume is shown in Fig. 7.5. The flume is 3.6 m wide and about 36 m long. The partial dam-break is represented by rapidly lifting the 1 m wide gate between two impervious blocks. The experiment lasted 20 s. Then, the gate was closed and the flow was stopped. There are 8 gauges in the flume. Their exact positions are stated in [Soares-Frazao et al., 2012]. 1.00 m 9.00 m y 1.30 m Reservoir 9.20 m 1.00 m Gate 1.30 m 9.83 m 4, 4' 3, 3' 2, 2' 1, 1' 8 8' 7 7' 6 6' 5 5' x 1.00 m Gauges for test case (1) Sand Bed 15.00 m Gauges for test case (2) 1.76 m Fig. 7.5. Plane view of 2D UCL dam-break experiment Water Level (m) 5 10 Time (s) 0.3 Gauge 1 15 20 Calculated Observed 0.2 0.1 0 Water Level (m) 0 0 5 10 Time (s) 0.3 Gauge 3 15 20 Calculated Observed 0.2 0.1 0 0 5 10 Time (s) 0.3 Gauge 5 15 20 Calculated Observed 0.2 0.1 0 0 5 10 Time (s) Gauge 7 15 20 Water Level (m) 0.1 0.3 Calculated Observed 0.2 0.1 0 Water Level (m) 0.2 0 Water Level (m) Calculated Observed 0 5 10 Time (s) 0.3 Gauge 2 15 20 Calculated Observed 0.2 0.1 0 Water Level (m) 0.3 0 5 10 Time (s) 0.3 Gauge 4 15 20 Calculated Observed 0.2 0.1 0 Water Level (m) Water Level (m) 128 0 5 10 Time (s) 0.3 Gauge 6 15 20 Calculated Observed 0.2 0.1 0 0 5 10 Time (s) Gauge 8 15 20 Fig. 7.6. Observed and calculated water surface for 2D test case (1) 129 0.2 Bed Elevation (m) Y = 0.20 m Calculated Observed 0.15 0.1 0.05 0 1 2 3 4 5 6 7 8 X (m) 0.2 Bed Elevation (m) Y = 0.70 m Calculated Observed 0.15 0.1 0.05 0 1 2 3 4 5 6 7 8 X (m) 0.2 Bed Elevation (m) Y = 1.45 m Calculated Observed 0.15 0.1 0.05 0 1 2 3 4 5 6 7 X (m) Fig. 7.7. Observed and calculated bed profiles for 2D test case (1) 8 Water Level (m) 0 Water Level (m) 0 5 10 Time (s) Gauge 1 15 20 0.3 0.2 0.1 0 Calculated Observed 0 5 10 Time (s) Gauge 3 15 20 0.3 0.2 0.1 0 Water Level (m) Calculated Observed Calculated Observed 0 5 10 Time (s) Gauge 5 15 20 0.3 0.2 Calculated Observed 0.1 0 0 5 10 Time (s) Gauge 7 15 20 Water Level (m) 0.1 0.3 0.2 0.1 0 Water Level (m) 0.2 Calculated Observed 0 5 10 Time (s) Gauge 2 15 20 0.3 0.2 0.1 0 Water Level (m) 0.3 Calculated Observed 0 5 10 Time (s) Gauge 4 15 20 0.3 0.2 0.1 0 Water Level (m) Water Level (m) 130 Calculated Observed 0 5 10 Time (s) Gauge 6 15 20 0.3 0.2 Calculated Observed 0.1 0 0 5 10 Time (s) Gauge 8 15 20 Fig. 7.8. Observed and calculated water surfaces for 2D test case (2) 131 0.2 Bed Elevation (m) Y = 0.20 m Calculated Observed 0.15 0.1 0.05 0 1 2 3 4 5 6 7 8 X (m) 0.2 Bed Elevation (m) Y = 0.70 m Calculated Observed 0.15 0.1 0.05 0 1 2 3 4 5 6 7 8 X (m) 0.2 Bed Elevation (m) Y = 1.45 m Calculated Observed 0.15 0.1 0.05 0 1 2 3 4 5 6 7 X (m) Fig. 7.9. Observed and calculated bed profiles for 2D test case (2) 8 132 The experiment consists of two cases: case (1) is a wet bed case and case (2) is a dry bed case. The initial conditions and boundary condition for both cases are described in [SoaresFrazao et al., 2012]. The following parameters are used in the simulation: sediment particle density s 2630kg / m3 ; median size d50 1.61mm ; bed porosity 0.42 ; Manning coefficient n 0.0165 . Fig. 7.6 and Fig. 7.8 show the calculated and observed time series of water surface elevations. The calculated water levels at gauges 1 and 4 are lower than the observed ones. This may be due to 3D features of the flow near the corners. The bed elevations are shown in Fig. 7.7 and Fig. 7.9. The figures show that reasonable agreements between the measured results and the calculated results can be obtained. Both calculation and observation show that significant erosion led to a scour hole downstream of the dam. The deposition process is controlled by the shock waves generated by the partial dam-break process. The shock waves spread to the sidewalls and then reflected from the sidewalls. The calculated scour pit is smoother than the observed one. Considering the complex process of the test case, the model reproduces generally well the erosion and deposition patterns observed in the experiment. 7.5 Summary A coupled two-dimensional shallow water hydrodynamic model is presented for the dambreak flow over mobile bed. The model is applied to the 1D and 2D test cases. The model can satisfactorily reproduce the measured hydrographs and bed profiles for both dry bed and wet bed cases. There would be more improvements by introducing the non-equilibrium sediment transport concept in the future development of the work. 133 CHAPTER 8 CONCLUSIONS AND FUTURE DIRECTIONS The dissertation is focusing on solving the shallow water equations by the Godunov-type finite volume method. In Chapter 2, the fundamental theory of shallow water equations are presented. This part of information is helpful in understanding the behavior of the shallow water equations and makes the dissertation more self-dependent. The wave structures of the equations are described here. Based on these wave structures, the exact solutions of the Riemann problem are derived. The Godunov-type finite volume method is presented in Chapter 3. Different numerical schemes, including first-order Godunov scheme, second-order MUSCL scheme, and fifth-order WENO scheme are described in the chapter. The core of the Godunov-type finite volume method is to solve local Riemann problems by an approximate Riemann solver. Roe solver, HLL solver, HLLC solver, and LLF solver are described. Then in Chapter 4, different numerical methods for the kinematic wave equation are discussed. The numerical tests show that the classical McCormack scheme generated excessive oscillations at the discontinuous front, including both the shock wave and rarefaction wave. On the contrary, the Godunov-type finite volume schemes, including MUSCL scheme and WENO scheme, can prevent the solution from oscillations. 134 A new surface flow routing method is proposed in Chapter 5. The stability of shallow water equations is analyzed in the chapter. The analysis shows that the shallow water equations will bring unrealistic values to the solution when the depth is thin. The new method is based on the shallow water equations coupled with the kinematic wave approximation. The numerical test about the Goodwin Creek watershed proved that the method is promising surface flow routing, especially for ungauged watershed. In Chapter 6, the numerical model is extended to turbulence modeling. The k turbulence model is incorporated into the shallow water equations. The HLLC approximate Riemann solver is used in the numerical method to solve the coupled system of equations simultaneously. The results show that it is necessary to apply a turbulence model to simulating the complicated flow fields. The applicability of shallow water equations to simulate sediment transport is explored in Chapter 7. In the chapter, a total load sediment transport model is developed. The model is verified using 1D and 2D experimental case of dam break flow, and showed good agreements with measurements. 8.1 Conclusions In this dissertation, the Godunov-type finite volume method has been used for solving shallow water equations and its various extensions (depth-average RANS equations and total load sediment transport equations) and simplification (kinematic wave equation). All the applications, including surface flow routing, turbulence modeling and sediment transport, show that the shallow water equations, which is based on the concept of shallow water flow, is a powerful tool for the modeling of free surface flows. A good example to understand 135 this is the dam-break problem. The dam-break flow is a strong unsteady flow with vertical motions. The test cases in chapter 6 prove that the numerical model can capture long wave trends of the dam-break flow and it is still a challenge to capture short wave perturbations of the flow. The essential difference between the finite difference method and the finite volume method is that the finite volume method is based on integral form of governing equations, while the finite difference is directly based on differential form of governing equations. This makes the finite volume method suitable for capturing discontinuous fronts in the solution. These discontinuous fronts represent the nature of hyperbolic equations. First-order Godunov scheme, second-order MUSCL scheme and fifth-order WENO scheme are used in the research. Numerical tests prove that the Godunov scheme is a stable scheme with obvious numerical diffusions. On the contrary, the WENO scheme generates least numerical diffusions and consumes most significant CPU times. The MUSCL scheme represents the balance between accuracy and expense. Time discretization schemes used in the research include first-order explicit Euler scheme, second-order MUSCL-Hancock scheme, second-order TVD Runge-Kutta scheme and thirdorder TVD Runge-Kutta scheme. When stability is a major concern, explicit Euler scheme is always the first choice. In the first step of MUSCL-Hancock scheme, the fluxes across interfaces are calculated by flux function. This may bring intense instability to the system when the interfaces are near wet/dry fronts. Wet/dry fronts are big challenges for numerical models. Near wet/dry fronts, the flow state is discontinuous and the flow depth is thin and shallow. Common errors in numerical models, like negative flow depth and unrealistic flow velocity, can always be found here. 136 Besides, it is difficult to define the wetting-drying process for a multistep time discretization scheme. So, near the wet/dry fronts, the best choice is to downgrade from higher-order numerical scheme to first-order scheme. A similar case is to generate body-fitted mesh by the Cartesian cut-cell method. When the flow around cut boundaries are simple and smooth, the cut-cell method will generate accurate results. On the other hand, regular cells are more suitable for discontinuous and complicated boundaries. The well balanced property is important and easily overlooked. The surface gradient method is the common method that holds the well balanced property of numerical model. The essence of the surface gradient method is a second-order approximation. And this method may generate a positive flow depth for a dry cell. The Roe solver is the first approximate Riemann solver with wide applications. The Roe solver depends on decomposition of Jacobian matrix of hyperbolic equations. For some situations (i.e. sediment transport model with Exner equation), it is difficult to express the Jacobian matrix or to decompose the Jacobian matrix in analytic forms. And these may limit the application of Roe solver. Although numerical diffusions are significant, the LLF solver is the easiest approximate Riemann solver, especially for extended shallow water systems. The LLF solver has the lowest explicit level of upwinding. The HLL and the HLLC solvers are the most popular Riemann solvers. It is easy to extend the solvers to multi-equation systems. And the HLLC solver is more accurate than the 137 HLL solver for two-dimensional systems. But unfortunately, it is difficult to estimate wave speeds used in the solvers. It is crucial to solve a strongly nonlinear system by a coupled way. A typical example is the St. Venant-Exner equations. If the system is solved by two separated steps, the eigenstructures of the system are changed. And this will lead to a radical error when the flow is supercritical flow. For situations like this, the coupled way is the only choice. On the other hand, the fractional step method is an under-estimated method. Although it is a two-step method, the fractional step method can still keep the eigen-structure of the coupled system. As to source terms of the shallow water equations, the friction term is the common source of instability. To restore the flow velocity, the term has to be divided by the flow depth. This will bring unrealistic value to the system when the flow depth is thin and the term will dominate the momentum equations. On the contrary, the gravity term is a stable term. This term can only dominate the momentum equations when slopes are higher than normal values. And the scale effect of DEM mesh may bring higher slope to the system. For a stable numerical model, one of the key features is to keep the balance between the gravity term and the friction term. The kinematic wave equation is a simplification of the shallow water equations and it is based on the assumption that the gravity term is balanced by the friction term. So it is a natural choice to calculate the flow velocity by the kinematic wave approximation when the flow depth is thin or the slope is high. Since this approximation is an implicit scheme, it is an unconditional stable method. Besides, it can force the balance between the gravity term 138 and the friction term. These features make the SWE-KWA method a promising scheme for surface flow routing. One of the drawbacks of kinematic wave approximation is over-estimation of flow velocity. This is the nature of the kinematic wave equation. Choosing a smaller threshold value may diminish the over-estimations. Another drawback of KWA is violation of well-balance property. Since the flow velocity is calculated from bed slope, the well-balance property does not hold for KWA method. Our test results show that turbulence model is necessary for two-dimensional shallow water flow model. The turbulent flow model based on the shallow water equations can capture low-frequency, long wave trends and cannot capture high-frequency, short wave perturbations. The explanation for this phenomenon is that the shallow water flow represents interactions between the gravity term and the friction term. These interactions result in the long wave trends while the short wave perturbations are caused by pressure term which is cancelled out in shallow water equations. The effect of turbulence is represented by the eddy viscosity term which is a second-order diffusion term. In our current model, this term is calculated by explicit scheme. Sometimes this will violate the stability condition for diffusion term. Another issue about turbulence modeling is the turbulence source term. The accuracy of our model decreases in the region near solid boundaries. The sediment transport model of our research is based on total load concept. The total load transport equation has the same form as the solute transport equation. This makes the equation easier to solve. A common strategy is to extend the HLL solver or the HLLC solver to 139 solve the system of equations in a coupled way. The definition of wave speed just follows the definition in the shallow water equations. There are two more source terms in the momentum equations of total load sediment transport equations. One source term represents the effect of variable concentration along flow directions and the other one indicates the momentum exchange between water column and mobile bed. The effects of both terms are significant for dam-break flows and should not be omitted from numerical models. The Godunov-type finite volume method is an explicit scheme and the method belongs to the easy parallel problem. The OpenMP parallel API is used in our program to convert the program to a parallel running program. 8.2 Future Directions Based on the progresses made in the dissertation, our methods have shown good potentials in these areas: surface flow routing, turbulent flow modeling, and total load sediment transport. Our future directions will be focused on river meandering simulation, surface flow routing over large scale raw DEM data and surface flow – subsurface flow interaction. River meandering is one of the most common phenomena in fluvial rivers. For many years, the river meandering has attracted the interest of researches in geomorphology, river hydraulics, riparian ecology and petroleum engineering. From a fluid mechanics point of view, meandering rivers are a dynamic system far from equilibrium. The evolution of river is driven by fluid dynamic and morphodynamic processes. These processes cause lateral bank erosion, continuous migration of rivers, and cutoffs that prevent self-intersections of rivers and reduce length and sinuosity of rivers. 140 Our numerical model of meandering river system will be the combination of turbulent flow model and total load sediment transport model. The whole system includes the shallow water equations (three equations), the k turbulence model (two equations) and the total load transport equation. The HLLC solver will be extended to six-equation system. A purely geometric approach will be implemented to account for lateral bank erosions. Because of development of secondary flow near meandering channel, integration of product of the discrepancy between the depth-averaged and the actual velocity can no long be neglected. So the dispersion terms will be added to momentum equations and this will improve the accuracy of the numerical model. Another potential option is to solve the whole system of equations by an implicit scheme. Since several diffusion terms are included in the system, an implicit scheme can provide an unconditional stable solution to the model. The difficult part about this idea lies in the solution of Riemann problems since all of approximate Riemann solvers discussed in the research are not differentiable. Because the Jacobian matrix is not available, a Jacobian-free Newton-Krylov solver becomes an obvious choice. The development of a fully implicit solver will not only benefit the calculation of turbulence term, but also the calculation of dispersion term. The stability problem that caused by thin flow depth has been solved in this research. The next step is to extend the method to large scale raw DEM data. Here the raw DEM data means two things. Firstly, the channel networks may be automatically extracted from the DEM data. The extraction of channel networks is based on fluid dynamic algorithm. And the flow directions are calculated according to runtime flow states. If the resolution of DEM data is too coarse for the 141 channel flow routing, a sub-scale channel may be superimposed to the watershed model. Subscale means the width of channel is smaller than the size of cell. Secondly, depressions in the DEM data, like lakes and reservoirs, do not need to be smoothed out of raw DEM data. A well-balanced scheme would satisfy the requirement and restore the right flow directions. Adaptive mesh refinement is a must for a large scale routing method. In general the overland flows are nearly uniform, steady flows, while the channel flows are non-uniform, unsteady flows. For the overland flows, a larger mesh size is acceptable. On the other hand, a smaller mesh size is suitable for channel flows. A quadtree mesh is a common choice for the adaptive mesh. The quadtree mesh has a simple structure and is easy to implement the quadtree structure. The local timestep refinement can boost the runtime efficiency along the time axis. For a highly unsteady flow, a smaller time step is an appropriate choice. The method relaxes the global CFL condition to local CFL conditions. Both adaptive mesh refinement and local timestep refinement are applicable to parallel computer environments. The system may be separated by the coarsest mesh size and timestep. As to the parallel API, the OpenMP API will be used on local computers and the MPI API will be used on cluster of computers. Since the Godunov-type method is an explicit scheme, it is simple to assign the calculation tasks to different threads, different CPUs and different computers without any pains. The last direction discussed here is the surface-subsurface interaction. In our current model, the Green-Ampt model is used. This model is easy to implement and has higher runtime 142 efficiency. But this model is not suitable for three-dimensional groundwater flows. So in the future direction, the Green-Ampt model will be replaced by the variably saturated groundwater flow equation. The variable saturated groundwater flow equation is a non-linear diffusion equation. Implicit schemes are often used in solving the equation. But implicit schemes are not compatible with explicit schemes used in solving surface flows. To overcome this incompatible problem, the variable saturated groundwater flow equation has to be solved by an explicit scheme. Solving a diffusion equation by an explicit scheme will bring stability problem to the system. So the system needs to be carefully tested and examined. It is true that a full three-dimensional free surface model will definitely give more accurate and detail results of flow field. However, a two-dimensional depth-averaged model could be more preferable and practical because of being computational-efficient. 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