# American Institute of Mathematical Sciences

June  2011, 15(4): 917-934. doi: 10.3934/dcdsb.2011.15.917

## Multilayer Saint-Venant equations over movable beds

 1 LAGA, Université Paris 13, 99 Av J.B. Clement, 93430 Villetaneuse, France, France 2 Laboratoire d’Hydraulique Saint-Venant, 6 Quai Watier, BP 49, 78401 Chatou, France 3 School of Engineering and Computing Sciences, University of Durham, South Road, Durham DH1 3LE, United Kingdom

Received  April 2010 Revised  September 2010 Published  March 2011

We introduce a multilayer model to solve three-dimensional sediment transport by wind-driven shallow water flows. The proposed multilayer model avoids the expensive Navier-Stokes equations and captures stratified horizontal flow velocities. Forcing terms are included in the system to model momentum exchanges between the considered layers. The topography frictions are included in the bottom layer and the wind shear stresses are acting on the top layer. To model the bedload transport we consider an Exner equation for morphological evolution accounting for the velocity field on the bottom layer. The coupled equations form a system of conservation laws with source terms. As a numerical solver, we apply a kinetic scheme using the finite volume discretization. Preliminary numerical results are presented to demonstrate the performance of the proposed multilayer model and to confirm its capability to provide efficient simulations for sediment transport by wind-driven shallow water flows. Comparison between results obtained using the multilayer model and those obtained using the single-layer model are also presented.
Citation: Emmanuel Audusse, Fayssal Benkhaldoun, Jacques Sainte-Marie, Mohammed Seaid. Multilayer Saint-Venant equations over movable beds. Discrete & Continuous Dynamical Systems - B, 2011, 15 (4) : 917-934. doi: 10.3934/dcdsb.2011.15.917
##### References:
 [1] E. Audusse and M. O. Bristeau, A well-balanced positivity preserving second order scheme for shallow water flows on unstructured meshes,, JCP, 206 (2005), 311. doi: doi:10.1016/j.jcp.2004.12.016. Google Scholar [2] E. Audusse, F. Bouchut, M. O. Bristeau, R. Klein and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows,, SIAM J. Sci. Comp., 25 (2004), 2050. doi: doi:10.1137/S1064827503431090. Google Scholar [3] E. Audusse, M. O. Bristeau, B. Perthame and J. Sainte-Marie, "A Multilayer Saint-Venant Model With Mass Exchange: Derivation and Numerical Validation,", M2AN, (2010). Google Scholar [4] F. Benkhaldoun, S. Sahmim and M. Seaid, A two-dimensional finite volume morphodynamic model on unstructured triangular grids,, Int. J. Num. Meth. Fluids, 63 (2010), 1296. Google Scholar [5] F. Bouchut and T. Morales de Luna, An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment,, M2AN, 42 (2008), 683. Google Scholar [6] F. Benkhaldoun, S. Sahmim and M. Seaid, Solution of the sediment transport equations using a finite volume method based on sign matrix,, SIAM J. Sci. Comp., 31 (2009), 2866. doi: doi:10.1137/080727634. Google Scholar [7] A. Bermúdez, C. Rodríguez and M. A. Vilar, Solving shallow water equations by a mixed implicit finite element method,, IMA J Numer Anal., 11 (1991), 79. doi: doi:10.1093/imanum/11.1.79. Google Scholar [8] M. J. Castro, J. Macías and C. Parés, A Q-scheme for a class of coupled conservation laws with source term. Application to a two-layer 1d shallow water system,, M2AN, 35 (2001), 107. Google Scholar [9] A. Crotogino and K. P. Holz, Numerical movable-bed models for practical engineering,, Applied Mathematical Modelling, 8 (1984), 45. doi: doi:10.1016/0307-904X(84)90176-8. Google Scholar [10] J. A. Dutton, "The Ceaseless Wind: An Introduction to the Theory of Atmospheric Motion,", Dover Publications Inc, (1987). Google Scholar [11] A. J. Grass, "Sediment Transport by Waves and Currents,", SERC London Cent. Mar. Technol. Report No: FL29, (1981). Google Scholar [12] J. Hudson and P. K. Sweby, Formations for numerically approximating hyperbolic systems governing sediment transport,, J. Scientific Computing, 19 (2003), 225. doi: doi:10.1023/A:1025304008907. Google Scholar [13] E. Meyer-Peter and R. Müller, Formulas for bed-load transport,, In: Report on 2nd meeting on international association on hydraulic structures research, 8 (1948), 39. Google Scholar [14] E. Miglio, A. Quarteroni and F. Saleri, Finite element approximation of quasi-3D shallow water equations,, Computer Methods in Applied Mechanics and Engineering, 174 (1999), 355. doi: doi:10.1016/S0045-7825(98)00304-1. Google Scholar [15] B. Perthame, "Kinetic Formulation of Conservation Laws,", Oxford University Press, (2004). Google Scholar [16] D. Pritchard and A. J. Hogg, On sediment transport under dam-break flow,, J. Fluid Mech., 473 (2002), 265. doi: doi:10.1017/S0022112002002550. Google Scholar [17] G. Rosatti and L. Fraccarollo, A well-balanced approach for flows over mobile-bed with high sediment-transport,, J. Comput. Physics, 220 (2006), 312. doi: doi:10.1016/j.jcp.2006.05.012. Google Scholar [18] G. K. Vallis, "Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation,", Cambridge University Press, (2006). doi: doi:10.1017/CBO9780511790447. Google Scholar

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##### References:
 [1] E. Audusse and M. O. Bristeau, A well-balanced positivity preserving second order scheme for shallow water flows on unstructured meshes,, JCP, 206 (2005), 311. doi: doi:10.1016/j.jcp.2004.12.016. Google Scholar [2] E. Audusse, F. Bouchut, M. O. Bristeau, R. Klein and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows,, SIAM J. Sci. Comp., 25 (2004), 2050. doi: doi:10.1137/S1064827503431090. Google Scholar [3] E. Audusse, M. O. Bristeau, B. Perthame and J. Sainte-Marie, "A Multilayer Saint-Venant Model With Mass Exchange: Derivation and Numerical Validation,", M2AN, (2010). Google Scholar [4] F. Benkhaldoun, S. Sahmim and M. Seaid, A two-dimensional finite volume morphodynamic model on unstructured triangular grids,, Int. J. Num. Meth. Fluids, 63 (2010), 1296. Google Scholar [5] F. Bouchut and T. Morales de Luna, An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment,, M2AN, 42 (2008), 683. Google Scholar [6] F. Benkhaldoun, S. Sahmim and M. Seaid, Solution of the sediment transport equations using a finite volume method based on sign matrix,, SIAM J. Sci. Comp., 31 (2009), 2866. doi: doi:10.1137/080727634. Google Scholar [7] A. Bermúdez, C. Rodríguez and M. A. Vilar, Solving shallow water equations by a mixed implicit finite element method,, IMA J Numer Anal., 11 (1991), 79. doi: doi:10.1093/imanum/11.1.79. Google Scholar [8] M. J. Castro, J. Macías and C. Parés, A Q-scheme for a class of coupled conservation laws with source term. Application to a two-layer 1d shallow water system,, M2AN, 35 (2001), 107. Google Scholar [9] A. Crotogino and K. P. Holz, Numerical movable-bed models for practical engineering,, Applied Mathematical Modelling, 8 (1984), 45. doi: doi:10.1016/0307-904X(84)90176-8. Google Scholar [10] J. A. Dutton, "The Ceaseless Wind: An Introduction to the Theory of Atmospheric Motion,", Dover Publications Inc, (1987). Google Scholar [11] A. J. Grass, "Sediment Transport by Waves and Currents,", SERC London Cent. Mar. Technol. Report No: FL29, (1981). Google Scholar [12] J. Hudson and P. K. Sweby, Formations for numerically approximating hyperbolic systems governing sediment transport,, J. Scientific Computing, 19 (2003), 225. doi: doi:10.1023/A:1025304008907. Google Scholar [13] E. Meyer-Peter and R. Müller, Formulas for bed-load transport,, In: Report on 2nd meeting on international association on hydraulic structures research, 8 (1948), 39. Google Scholar [14] E. Miglio, A. Quarteroni and F. Saleri, Finite element approximation of quasi-3D shallow water equations,, Computer Methods in Applied Mechanics and Engineering, 174 (1999), 355. doi: doi:10.1016/S0045-7825(98)00304-1. Google Scholar [15] B. Perthame, "Kinetic Formulation of Conservation Laws,", Oxford University Press, (2004). Google Scholar [16] D. Pritchard and A. J. Hogg, On sediment transport under dam-break flow,, J. Fluid Mech., 473 (2002), 265. doi: doi:10.1017/S0022112002002550. Google Scholar [17] G. Rosatti and L. Fraccarollo, A well-balanced approach for flows over mobile-bed with high sediment-transport,, J. Comput. Physics, 220 (2006), 312. doi: doi:10.1016/j.jcp.2006.05.012. Google Scholar [18] G. K. Vallis, "Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation,", Cambridge University Press, (2006). doi: doi:10.1017/CBO9780511790447. Google Scholar
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