Improved FVM for two-layer shallow-water models: Application to the Strait of Gibraltar q Manuel J. Castro a , Jose ´ A. Garcı ´a-Rodrı ´guez a , Jose ´ M. Gonza ´lez-Vida b , Jorge Macı ´as a, * , Carlos Pare ´s a a Departamento de Ana ´ lisis Matema ´ tico, Facultad de Ciencias, Universidad de Ma ´ laga, Campus de Teatinos, s/n 29080 Ma ´ laga, Spain b Departamento de Matema ´ tica Aplicada, ETSIT, Universidad de Ma ´ laga, Campus de Teatinos, s/n 29080 Ma ´ laga, Spain Available online 7 November 2006 Abstract 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 cou- pled system composed of two Shallow Water models with source terms involving depth and breadth functions. Extensions of Roe’s Q- scheme are proposed where a suitable treatment of the coupling and source terms is performed by adapting the techniques developed in [Va ´zquez-Cendo ´ n ME. Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. J Comp Phys 1999;148:497–526; Garcı ´a-Navarro P, Va ´zquez-Cendo ´n ME. On numerical treatment of the source terms in the shallow water equations. Comput Fluids 2000;29(8):17–45; Castro MJ, Macı ´as J, Pare ´s 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 Numer An 2001;35(1):107–27]. 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 and some tests on friction sen- sitivity are also presented. Ó 2006 Elsevier Ltd. and Civil-Comp Ltd. All rights reserved. Keywords: Q-schemes; Coupled conservation laws; Source terms; 1D shallow water equations; Two-layer exchange flows; Hyperbolic systems; Strait of Gibraltar; Internal tides; Fortnightly and monthly signal; Friction effects 1. Introduction Two layer flows, or flows which can be idealized as such, occur naturally in estuary flows, ocean currents, and atmo- spheric flows. Two layer flows also occur as a result of man’s interaction with natural flows by the addition of dif- ferent density pollutants. This kind of flows are interesting from a theoretical and practical point of view. In this work, our interest focused on the computation of maximal and tidally induced two-layer exchange flows through the Strait of Gibraltar. In this narrows, connecting the Atlantic Ocean with the Mediterranean Sea, two layers of different waters can be distinguished: the colder and less saline Atlantic water flowing at surface and penetrating into the Mediterranean, and the deeper, denser Mediterranean water flowing into the Atlantic. The observation of this simplified picture shows that the simpler model to be used to simulate the flow in this region must unavoidably con- sider this two-layer structure. 0965-9978/$ - see front matter Ó 2006 Elsevier Ltd. and Civil-Comp Ltd. All rights reserved. doi:10.1016/j.advengsoft.2006.09.012 q This research has been partially supported by the CICYT (project BFM2003-07530-C02-02). * Corresponding author. Tel.: +34 952131898; fax: +34 952131894. E-mail address: macias@anamat.cie.uma.es (J. Macı ´as). www.elsevier.com/locate/advengsoft Advances in Engineering Software 38 (2007) 386–398