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Numerical simulation of two-layer shallow water flows through channels with irregular geometry. (English) Zbl 1087.76077
Summary: 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 [M. E. Vázquez-Cendón, J. Comput. Phys. 148, No. 2, 497–526 (1999; Zbl 0931.76055); Comput. Fluids 29, No. 8, 17 ff (2000); M. Castro et al., M2AN, Math. Model. Numer. Anal. 35, No. 1, 107–127 (2001; Zbl 1094.76046)]. An enhanced consistency condition, the so-called C-property, introduced in [Comput. Fluids 23, No. 8, 1049–1071 (1994; Zbl 0816.76052)] is extended to this case ,and a general result providing sufficient conditions to ensure this property is shown. Then, some numerical tests to validate the resulting schemes are presented. First, we verify that, in practice, the numerical schemes satisfy the \(C\)-property, even for extremely irregular channels. Then, in order to validate the schemes, we compare some approximate steady solutions obtained with the generalized \(Q\)-scheme of Van Leer with those obtained by using the asymptotic techniques developed by L. Armi and D. Farmer [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 455, No. 1989, 3221–3258 (1999; Zbl 0951.76021)] for channels with simplified geometries. 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.

MSC:
76M20 Finite difference methods applied to problems in fluid mechanics
76B70 Stratification effects in inviscid fluids
76U05 General theory of rotating fluids
86A05 Hydrology, hydrography, oceanography
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