A finite volume shock‐capturing solver of the fully coupled shallow water‐sediment equations

Summary This paper describes a numerical solver of well-balanced, 2D depth-averaged shallow water-sediment equations. The equations permit variable variable horizontal fluid density and are designed to model water-sediment flow over a mobile bed. A Godunov-type, HLLC finite volume scheme is used to solve the fully coupled system of hyperbolic conservation laws which describe flow hydrodynamics, suspended sediment transport, bedload transport and bed morphological change. Dependent variables are specially selected to handle the presence of the variable density property in the mathematical formulation. The model is verified against analytical and semi-analytical solutions for bedload transport and suspended sediment transport, respectively. The well-balanced property of the equations is verified for a variable-density dam break flow over discontinuous bathymetry. Simulations of an idealised dam-break flow over an erodible bed are in excellent agreement with previously published results ([1]), validating the ability of the model to capture the complex interaction between rapidly varying flow and an erodible bed and validating the eigenstructure of the system of variable-density governing equations. Flow hydrodynamics and final bed topography of a laboratory-based 2D partial dam breach over a mobile bed are satisfactorily reproduced by the numerical model. Comparison of the final bed topographies, computed for two distinct sediment transport methods, highlights the sensitivity of shallow water-sediment models to the choice of closure relationships. This article is protected by copyright. All rights reserved.

[1]  Fayssal Benkhaldoun,et al.  Well-balanced finite volume schemes for pollutant transport by shallow water equations on unstructured meshes , 2007, J. Comput. Phys..

[2]  Jian Guo Zhou,et al.  Lattice Boltzmann morphodynamic model , 2014, J. Comput. Phys..

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

[4]  Enrique D. Fernández-Nieto,et al.  Sediment transport models in Shallow Water equations and numerical approach by high order finite volume methods , 2008 .

[5]  Rui M. L. Ferreira,et al.  Dam-break flows over mobile beds: experiments and benchmark tests for numerical models , 2012 .

[6]  Ali Vakil,et al.  Cyclic steps and roll waves in shallow water flow over an erodible bed , 2012, Journal of Fluid Mechanics.

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

[8]  Kuo Yan,et al.  Coastal flood inundation and morphodynamic change , 2010 .

[9]  Gareth Pender,et al.  Reconciled bedload sediment transport rates in ephemeral and perennial rivers , 2010 .

[10]  A. Borthwick,et al.  1‐D numerical modelling of shallow flows with variable horizontal density , 2009 .

[11]  Hervé Capart,et al.  Intense bed-load due to a sudden dam-break , 2013, Journal of Fluid Mechanics.

[12]  Norihiro Izumi,et al.  Purely erosional cyclic and solitary steps created by flow over a cohesive bed , 2000, Journal of Fluid Mechanics.

[13]  Guy Simpson,et al.  Coupled model of surface water flow, sediment transport and morphological evolution , 2006, Comput. Geosci..

[14]  János Józsa,et al.  Variable density bore interaction with block obstacles , 2011 .

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

[16]  Guus S. Stelling,et al.  A staggered conservative scheme for every Froude number in rapidly varied shallow water flows , 2003 .

[17]  Justin Hudson,et al.  Formulations for Numerically Approximating Hyperbolic Systems Governing Sediment Transport , 2003, J. Sci. Comput..

[18]  Hervé Capart,et al.  Formation of a jump by the dam-break wave over a granular bed , 1998, Journal of Fluid Mechanics.

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

[20]  Gareth Pender,et al.  Computational Dam-Break Hydraulics over Erodible Sediment Bed , 2004 .

[21]  Javier Murillo,et al.  A Riemann solver for unsteady computation of 2D shallow flows with variable density , 2012, J. Comput. Phys..

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

[23]  Wei Li,et al.  Morphological modeling using a fully coupled, total variation diminishing upwind‐biased centered scheme , 2013 .

[24]  Guus S. Stelling,et al.  A finite volume analogue of the P1NC-P1 finite element: With accurate flooding and drying , 2010 .

[25]  Junke Guo,et al.  Logarithmic matching and its applications in computational hydraulics and sediment transport , 2002 .

[26]  Qiuhua Liang,et al.  Adaptive quadtree simulation of shallow flows with wet-dry fronts over complex topography , 2009 .

[27]  Javier Murillo,et al.  Energy balance numerical schemes for shallow water equations with discontinuous topography , 2013, J. Comput. Phys..

[28]  Javier Murillo,et al.  A 2D weakly-coupled and efficient numerical model for transient shallow flow and movable bed , 2014 .

[29]  Roger Alexander Falconer,et al.  Comparison between TVD-MacCormack and ADI-type solvers of the shallow water equations , 2006 .

[30]  Zhixian Cao,et al.  EQUILIBRIUM NEAR-BED CONCENTRATION OF SUSPENDED SEDIMENT , 1999 .

[31]  Fayssal Benkhaldoun,et al.  An unstructured finite‐volume method for coupled models of suspended sediment and bed load transport in shallow‐water flows , 2013 .

[32]  Javier Murillo,et al.  One-Dimensional Riemann Solver Involving Variable Horizontal Density to Compute Unsteady Sediment Transport , 2016 .

[33]  Tao Sun,et al.  Transportational cyclic steps created by flow over an erodible bed. Part 2. Theory and numerical simulation , 2005 .

[34]  Z. Cao,et al.  Mathematical modelling of alluvial rivers: reality and myth. Part 1: General review , 2002 .

[35]  Alistair G.L. Borthwick,et al.  One-dimensional modelling of fluvial bed morphodynamics , 2008 .

[36]  Javier Murillo,et al.  An Exner-based coupled model for two-dimensional transient flow over erodible bed , 2010, J. Comput. Phys..

[37]  Rui M. L. Ferreira,et al.  Two-dimensional depth-averaged modelling of dam-break flows over mobile beds , 2013 .

[38]  Octavio E. Sequeiros,et al.  Cyclic steps: A phenomenon of supercritical shallow flow from the high mountains to the bottom of the ocean , 2010 .

[39]  Ilektra-Georgia Apostolidou.  Variable density shallow flow model for flood simulation , 2011 .

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

[41]  Yves Zech,et al.  Two‐dimensional two‐layer shallow water model for dam break flows with significant bed load transport , 2013 .