Keywords: Q-schemes, coupled conservation laws, source terms, 1D shallow water equations, two-layer flows, hyperbolic systems, maximal two-layer exchange flows, Strait of Gibraltar, internal tides.

This paper deals with the numerical simulation of flows of stratified fluids through channels with irregular geometry. Channel cross-sections are supposed to be symmetric but not necessarily rectangular. The fluid is supposed to be composed of two shallow layers of immiscible fluids of constant densities, and the flow is assumed to be one-dimensional. Therefore, the equations to be solved are a coupled system composed of two Shallow Water models with source terms involving depth and breadth functions. Extensions of the Q-schemes of van Leer and Roe are proposed where a suitable treatment of the coupling and source terms is performed by adapting the techniques developed in [6], [5] and [2]. We also present a numerical scheme that deals with the numerical difficulties related to the appearance of the so-called wet/dry fronts. Wet/dry fronts appears in two-layer fluids when the thickness of at least one of the layers vanishes. In this case, a front develop separating two-layer and one layer flows or two layer flows and dry bed. These fronts appear also in practical applications in coastal simulations or in lock-exchange experiments, in which the two layers are initially separated by a dam that is suddenly broken. An enhanced consistency condition, the so-called C-property, introduced in [1] is extended and a general result providing sufficient conditions to ensure this property is shown. Then, some numerical tests to validate the resulting schemes are presented. Finally we apply the numerical scheme to the simulation of the flow through the Strait of Gibraltar. Real bathymetric and coast-line data are considered to include in the model the main features of the abrupt geometry of this natural strait connecting the Atlantic Ocean and the Mediterranean Sea. A steady state solution is obtained from lock-exchange initial conditions. This solution is then used as initial condition to simulate the main semidiurnal and diurnal tidal waves in the Strait of Gibraltar through the imposition of suitable boundary conditions obtained from observed tidal data. Comparisons between numerical results and observed data are also presented.

1

A. Bermúdez, M.E. Vázquez, "Upwind methods for hyperbolic conservation laws with source terms". Computers and Fluids, 23(8):1049-1071, 1994. doi:10.1016/0045-7930(94)90004-3

2

M.J. Castro, J. Macías, C. Parés, "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. doi:10.1051/m2an:2001108

3

M.J. Castro, J.Macías, C. Parés, J.A. Rubal, M.E. Vázquez-Cendón, "A two-layer numerical model for solving exchange flows through channels with irregular geometry", Proceedings of "ECCOMAS 2001", Swansea. 2001.

4

M.J. Castro et. all "Numerical simulation of two-layer Shallow Water flows through channels with irregular geometry", Accepted in J. Comp. Physics. 2003. doi:10.1016/j.jcp.2003.08.035

5

P. García-Navarro, M.E. Vázquez-Cendón. "On numerical treatment of the source terms in the shallow water equations". Computers and Fluids 29(8):17-45, 2000. doi:10.1016/S0045-7930(99)00038-9

6

M.E. Vázquez-Cendón. "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. doi:10.1006/jcph.1998.6127

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