A well-balanced unsplit finite volume model with geometric flexibility

A two-dimensional finite volume model is developed for the unsteady, and shallow water equations on arbitrary topography. The equations are discretized on quadrilateral control volumes in an unstructured arrangement. The HLLC Riemann approximate solver is used to compute the interface fluxes and the MUSCL-Hancock scheme with the surface gradient method is employed for second-order accuracy. This study presents a new method for translation of discretization technique from a structured grid description based on the traditional (i, j) duplet to an unstructured grid arrangement based on a single index, and efficiency of proposed technique for unsplit finite volume method. In addition, a simple but robust well-balanced technique between fluxes and source terms is suggested. The model is validated by comparing the predictions with analytical solutions, experimental data and field data including the following cases: steady transcritical flow over a bump, dam-break flow in an adverse slope channel and the Malpasset dam-break in France.

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

[2]  Roger Alexander Falconer,et al.  A boundary-fitted numerical model for flood routing with shock capturing , 2007 .

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

[4]  Paolo Mignosa,et al.  Numerical simulation and experimental verification of Dam-Break flows with shocks , 2000 .

[5]  Alireza Zia,et al.  Simple efficient algorithm (SEA) for shallow flows with shock wave on dry and irregular beds , 2008 .

[6]  Ming-Hseng Tseng.  Improved treatment of source terms in TVD scheme for shallow water equations , 2004 .

[7]  E. Toro.  Shock-Capturing Methods for Free-Surface Shallow Flows , 2001 .

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

[9]  Valerio Caleffi,et al.  Well-Balanced Bottom Discontinuities Treatment for High-Order Shallow Water Equations WENO Scheme , 2009 .

[10]  Jihn-Sung Lai,et al.  Finite‐volume multi‐stage schemes for shallow‐water flow simulations , 2008 .

[11]  Pilar García-Navarro,et al.  Zero mass error using unsteady wetting–drying conditions in shallow flows over dry irregular topography , 2004 .

[12]  K. S. Erduran,et al.  Performance of finite volume solutions to the shallow water equations with shock‐capturing schemes , 2002 .

[13]  Benedict D. Rogers,et al.  Mathematical balancing of flux gradient and source terms prior to using Roe's approximate Riemann solver , 2003 .

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

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

[16]  Emmanuel Audusse,et al.  A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows , 2004, SIAM J. Sci. Comput..

[17]  N. Cheng,et al.  Source term treatment of SWEs using surface gradient upwind method , 2012 .

[18]  Brett F. Sanders,et al.  Mesh type tradeoffs in 2D hydrodynamic modeling of flooding with a Godunov-based flow solver , 2014 .

[19]  E. Toro,et al.  Restoration of the contact surface in the HLL-Riemann solver , 1994 .

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

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

[22]  Qiuhua Liang,et al.  Simulation of dam‐ and dyke‐break hydrodynamics on dynamically adaptive quadtree grids , 2004 .

[23]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[24]  Derek M. Causon,et al.  Numerical solutions of the shallow water equations with discontinuous bed topography , 2002 .

[25]  Eleuterio F. Toro,et al.  Experimental and numerical assessment of the shallow water model for two-dimensional dam-break type problems , 1995 .

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

[27]  M. Aliparast,et al.  Two-dimensional finite volume method for dam-break flow simulation , 2009 .

[28]  P. Lax,et al.  On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws , 1983 .

[29]  Sandra Soares Frazao,et al.  An eigenvector‐based linear reconstruction scheme for the shallow‐water equations on two‐dimensional unstructured meshes , 2007 .

[30]  Solution of the 2‐D shallow water equations with source terms in surface elevation splitting form , 2007 .

[31]  Qiuhua Liang,et al.  Flood Simulation Using a Well-Balanced Shallow Flow Model , 2010 .

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