Closure of "Finite Volume Model for Shallow Water Equations with Improved Treatment of Source Terms"

A simple yet precise relation between the flux gradient and the bed slope source term is presented, which produces a net force within the cell with an inclined water surface, but ensures still water condition when there is no flow across the boundaries. The proposed method consists of calculating the pressure term based on the water depths at the cell vertices, which may be computed by a higher order scheme and the bed slope source term by a centered discretization technique. The methodology is demonstrated with a Godunov-type upwind finite volume formulation. The inviscid fluxes are calculated using Roe’s approximate Riemann solver and a second-order spatial accuracy is obtained by implementing multidimensional gradient reconstruction and slope limiting techniques. The accuracy and applicability of the numerical model is verified with a couple of test problems and a real flow example of tidal water movement in a stretch of River Hooghly in India.

[1]  M. Hanif Chaudhry,et al.  Open-Channel Flow , 2007 .

[2]  Alessandro Valiani,et al.  Divergence Form for Bed Slope Source Term in Shallow Water Equations , 2006 .

[3]  Pilar García-Navarro,et al.  A HIGH-RESOLUTION GODUNOV-TYPE SCHEME IN FINITE VOLUMES FOR THE 2D SHALLOW-WATER EQUATIONS , 1993 .

[4]  An improved, robust implicit solution for the two dimensional shallow water equations on unstructured grids , 2004 .

[5]  N. Katopodes,et al.  Computing Two-Dimensional Dam-Break Flood Waves , 1978 .

[6]  J. Cunge,et al.  Practical aspects of computational river hydraulics , 1980 .

[7]  D. Zhao,et al.  Approximate Riemann solvers in FVM for 2D hydraulic shock wave modeling , 1996 .

[8]  Alfredo Bermúdez,et al.  Upwind methods for hyperbolic conservation laws with source terms , 1994 .

[9]  Tullio Tucciarelli,et al.  Finite-element modeling of floodplain flow , 2000 .

[10]  V. T. Chow.  Open-channel hydraulics , 1959 .

[11]  M. Berzins,et al.  An unstructured finite-volume algorithm for predicting flow in rivers and estuaries , 1998 .

[12]  Robert J. Connell,et al.  Two-Dimensional Flood Plain Flow. I: Model Description , 2001 .

[13]  Scott F. Bradford,et al.  Finite-Volume Model for Shallow-Water Flooding of Arbitrary Topography , 2002 .

[14]  Brett F. Sanders M. Iahr.  High-resolution and non-oscillatory solution of the St. Venant equations in non-rectangular and non-prismatic channels , 2001 .

[15]  K. Anastasiou,et al.  SOLUTION OF THE 2D SHALLOW WATER EQUATIONS USING THE FINITE VOLUME METHOD ON UNSTRUCTURED TRIANGULAR MESHES , 1997 .

[16]  Randall J. LeVeque,et al.  Balancing Source Terms and Flux Gradients in High-Resolution Godunov Methods , 1998 .

[17]  R L Stockstill,et al.  Finite-Element Model for High-Velocity Channels , 1995 .

[18]  I W Dand,et al.  SHALLOW WATER HYDRODYNAMICS , 1971 .

[19]  D. Zhao,et al.  Finite‐Volume Two‐Dimensional Unsteady‐Flow Model for River Basins , 1994 .

[20]  C. Hirsch,et al.  Numerical Computation of Internal and External Flows. By C. HIRSCH. Wiley. Vol. 1, Fundamentals of Numerical Discretization. 1988. 515 pp. £60. Vol. 2, Computational Methods for Inviscid and Viscous Flows. 1990, 691 pp. £65. , 1991, Journal of Fluid Mechanics.

[21]  Matthew E. Hubbard,et al.  Regular Article: Multidimensional Slope Limiters for MUSCL-Type Finite Volume Schemes on Unstructured Grids , 1999 .

[22]  D. Causon,et al.  The surface gradient method for the treatment of source terms in the shallow-water equations , 2001 .

[23]  Jean-Antoine Désidéri,et al.  Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes , 1998 .

[24]  C. P. Skeels,et al.  Evaluation of some approximate Riemann solvers for transient open channel flows , 2000 .

[25]  P. García-Navarro,et al.  On numerical treatment of the source terms in the shallow water equations , 2000 .

[26]  Valerio Caleffi,et al.  Case Study: Malpasset Dam-Break Simulation using a Two-Dimensional Finite Volume Method , 2002 .

[27]  Ruxun Liu,et al.  A comparative study of finite volume methods on unstructured meshes for simulation of 2D shallow water wave problems , 2000 .

[28]  Pilar García-Navarro,et al.  Flux difference splitting and the balancing of source terms and flux gradients , 2000 .

[29]  Jonathan Richard Shewchuk,et al.  Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator , 1996, WACG.

[30]  S. Roberts,et al.  Robust and efficient solution of the 2D shallow water equation with domains containing dry beds , 2000 .

[31]  Pilar García-Navarro,et al.  Efficient construction of high‐resolution TVD conservative schemes for equations with source terms: application to shallow water flows , 2001 .

[32]  Abioala A. Akanbi,et al.  Model for Flood Propagation on Initially Dry Land , 1988 .

[33]  Paul Glaister.  Approximate Riemann solutions of the shallow water equations , 1988 .

[34]  M. Vázquez-Cendón.  Improved Treatment of Source Terms in Upwind Schemes for the Shallow Water Equations in Channels with Irregular Geometry , 1999 .

[35]  Marinko Nujić,et al.  Efficient implementation of non-oscillatory schemes for the computation of free-surface flows , 1995 .

[36]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[37]  Pilar García-Navarro,et al.  Flux difference splitting for 1D open channel flow equations , 1992 .

[38]  Pilar García-Navarro,et al.  A numerical model for the flooding and drying of irregular domains , 2002 .

[39]  Tae Hoon Yoon,et al.  Finite volume model for two-dimensional shallow water flows on unstructured grids , 2004 .

[40]  M. Hanif Chaudhry,et al.  Explicit Methods for 2‐D Transient Free Surface Flows , 1990 .

[41]  H. Kamath,et al.  A high-resolution procedure for Euler and Navier-Stokes computations on unstructured grids , 2000 .

[42]  Davood Farshi,et al.  Discussion of “Finite Volume Model for Two-Dimensional Shallow Water Flows on Unstructured Grids” by Tae Hoon Yoon and Seok-Koo Kang , 2005 .

[43]  Pilar García-Navarro,et al.  Unsteady free surface flow simulation over complex topography with a multidimensional upwind technique , 2003 .

[44]  D. Holmes,et al.  Solution of the 2D Navier-Stokes equations on unstructured adaptive grids , 1989 .

[45]  M. Hanif Chaudhry,et al.  Depth-averaged open-channel flow model , 1995 .