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The numerical treatment of wet/dry fronts in shallow flows: application to one-layer and two-layer systems. (English) Zbl 1121.76008
Summary: This paper deals with the numerical difficulties related to the appearance of the so-called wet/dry fronts that may occur during the simulation of free-surface waves in shallow fluids and internal waves in stratified fluids composed by two shallow layers of immiscible liquids. In the one-layer case, the fluid is supposed to be governed by the shallow water equations. In the case of two layers, the system to solve is formulated under the form of two-coupled system of shallow water equations. In both cases, the discretization of the equations are performed by means of the \(Q\)-schemes of Roe upwinding the source term developed in [M. E. Vázquez-Cendón, J. Comput. Phys. 148, No. 2, 497–526 (1999; Zbl 0931.76055), P. Garcia-Navarro and M. E. Vázquez-Cendón, Comput. Fluids 29, No. 8, 951–979 (2000; Zbl 0986.76051)] for the one-layer system, and M. Castro, J. Macías, and C. Parés, M2AN, Math. Model. Numer. Anal. 35, No. 1, 107–127 (2001; Zbl 1094.76046)] for the two-layer system. These schemes satisfy the so-called \(\mathcal C\)-property: they solve exactly steady solutions corresponding to water at rest.
In order to handle properly with wet/dry fronts, a numerical scheme has to verify also an extended \(\mathcal C\)-property: it has to solve exactly steady solutions corresponding to water at rest including wet/dry transitions. In [P. Brufau, “Simulacion bidimensional de flujos hidrodinamicos transitorios en geometrias irregulares”, Ph.D. thesis, Univ. Zaragoza, Zaragoza (2000); P. Brufau, M. E. Vazquez-Cendon and P. García-Navarro, Int. J. Numer. Methods Fluids 39, No. 3, 247–275 (2002; Zbl 1094.76538); A. Ferreiro, “Resolucion y validacion experimental del modelo de aguas poco profundas unidimensional incluyendo areas secas”, Trabajo de DEA, Univ. Santiago, Compostela (2002)] some numerical schemes satisfying this property has been obtained. In this paper, we propose an improvement of these schemes and its extension to two-layer systems.
We present some numerical results: for one-layer fluids, we compare the numerical results with some measurements corresponding to a physical model. For two-layer systems, we use the numerical scheme to perform a lock-exchange experiment, and we verify its validity by comparing the steady state reached with an approximate solution obtained by L. Armi and D. M. Farmer in [ J. Fluid Mech. 163, 27–58 (1986), J. Fluid Mech. 164, 27–51 (1986; Zbl 0587.76168)] by using a simplified model.

76B07 Free-surface potential flows for incompressible inviscid fluids
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B55 Internal waves for incompressible inviscid fluids
76M20 Finite difference methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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[1] Vázquez-Cendón, M.E., Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry, J. comp. physics, 148, 497-526, (1999) · Zbl 0931.76055
[2] García-Navarro, P.; Vázquez-Cendón, M.E., On numerical treatment of the source terms in the shallow water equations, Computers and fluids, 29, 8, 17-45, (2000) · Zbl 0986.76051
[3] Castro, M.J.; Macías, J.; Parés, C., A Q-scheme for a class of systems of coupled conservation laws with source term. application to a two-layer 1-D shallow water system, Math. model. and numer. an., 35, 1, 107-127, (2001) · Zbl 1094.76046
[4] Brufau, P., Simulación bidimensional de flujos hidrodinámicos transitorios en geometrĺas irregulares, Thesis doctoral, univ. de Zaragoza, (2000)
[5] Brufau, P.; Vázquez-Cendón, M.E.; García-Navarro, A numerical model for the flooding and drying of irregular domain, J. numer. meth. fluids, 39, 247-275, (2002) · Zbl 1094.76538
[6] Ferreiro, A., Resolución y validación experimental del modelo de aguas poco profundas unidimensional incluyendo áreas secas, ()
[7] Armi, L., The hydraulics of two flowing layers with different densities, J. fluid mech., 163, 27-58, (1986)
[8] Armi, L.; Farmer, D., Maximal two-layer exchange through a contraction with barotropic net flow, J. fluid mech., 164, 27-51, (1986) · Zbl 0587.76168
[9] Castro, M.J.; García-Rodríguez, J.A.; Macías, J.; Parés, C.; Vázquez-Cendón, M.E., Two-layer numerical model for solving exchange flows in channels with arbitrary section, J. comput. physics., 195, 202-235, (2004) · Zbl 1087.76077
[10] González-Vida, J.M., Desarrollo de esquemas numéricos para el tratamiento de frentes seco-mojado en sistemas de aguas someras, ()
[11] Bermúdez, A.; Vázquez, M.E., Upwind methods for hyperbolic conservation laws with source terms, Computers and fluids, 23, 8, 1049-1071, (1994) · Zbl 0816.76052
[12] Vázquez-Cendón, M.E., Estudio de esquemas descentrados para su aplicación a las leyes de conservación hiperbólicas con Términos fuente, ()
[13] Roe, P.L., Approximate Riemann solvers, parameter vectors and difference schemes, J. comput. phys., 43, 357-371, (1981) · Zbl 0474.65066
[14] Harten, A., On a class of high resolution total-variation-stable finite-difference schemes, SIAM J. of numer. anal., 21, 1, 1-23, (1984) · Zbl 0547.65062
[15] Farmer, D.; Armi, L., Maximal two-layer exchange over a sill and through a combination of a sill and contraction with barotropic flow, J. fluid mech., 164, 53-76, (1986) · Zbl 0587.76169
[16] Castro, M.J.; Macías, J.; Parés, C.; Rubal, J.A.; Vázquez Cendón, M.E., Two-layer numerical model for solving exchange flows through channels with irregular geometry, () · Zbl 1177.76280
[17] García Lafuente, J.; Almazán, J.L.; Castillejo, F.; Khribeche, A.; Hakimi, A., Sea level in the strait of gibraltar: tides, International hydrographic review, LXVII, 1, 111-130, (1990)
[18] Roe, P.L., Upwinding differenced schemes for hyperbolic conservation laws with source terms, (), 41-51
[19] Toro, E.F., Riemann solvers and numerical methods for fluid dynamics. A practical introduction, (1997), Springer · Zbl 0888.76001
[20] Toro, E.F.; Vázquez-Cendón, M.E., Model hyperbolic systems with source terms: exact and numerical solutions, () · Zbl 0989.65095
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