Solution properties and approximate Riemann solvers for two-layer shallow flow models Benoit Spinewine a,⇑ , Vincent Guinot b,1 , Sandra Soares-Frazão a,2 , Yves Zech c,3 a Fonds de la Recherche Scientifique – FNRS, and Université catholique de Louvain, Institute of Mechanics, Materials and Civil Engineering, Place du Levant, 1, B-1348 Louvain-la-Neuve, Belgium b Université Montpellier 2, Maison des Sciences de l’Eau, 34095 Montpellier Cedex 5, France c Université catholique de Louvain, Institute of Mechanics, Materials and Civil Engineering, Place du Levant, 1, B-1348 Louvain-la-Neuve, Belgium article info Article history: Received 9 March 2009 Received in revised form 16 November 2010 Accepted 2 January 2011 Available online 18 January 2011 Keywords: Two-layer shallow flow Riemann solvers Hyperbolic systems of conservation laws Source term discretization Hyperbolicity abstract The 4  4 system of governing equations for two-layer shallow flow models is known to exhibit particular behaviours such as loss of hyperbolicity under certain flow configurations. An eigenvalue analysis of the conservation part of the equations shows that the loss of hyperbolicity is due only to the reaction exerted by each fluid onto the other at the interface between the fluids. Three Riemann solvers derived from the HLL formalism are presented. In the first solver, the pressure-induced terms are accounted for by the source term; in the second solver, they are incorporated into the fluxes; the third solver uses the same formulation as the first, except that the mass and momentum balance for the bottom layer are replaced with the balance equations for the system formed by the two layers as a whole. Numerical results using the three solvers are presented for (1) static conditions such as two fluids of identical densities at rest above each other, (2) dam-break flows involving the collapse of a body of light fluid over a uniform layer of a denser fluid, and (3) Liska and Wendroff’s ill-posed test cases [24] involving two-layer flows over a topographic bump. The three solvers produce quasi-undistinguishable results for the dam-break flows, and produce sharp solutions over the full range of density ratio, from 0 to 1. However, only the third sol- ver allows a strict preservation of static configurations. Moreover, a method is proposed to assess the con- vergence of the numerical solutions in the configurations for which no analytical solution can be obtained. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction Many types of flows of geophysical interest can be approxi- mated as layered systems made of two shallow fluids of distinct densities. The two fluids can further differ in terms of rheology, compressibility, viscosity, and potential for mixing. The density contrast between the two layers (that is, the ratio of the density of the ‘‘light’’ fluid in the upper layer to that of the ‘‘dense’’ fluid in the lower layer) may be very low, as is the case e.g. for two bodies of water with a distinct salinity or two layers of air with a temperature contrast, or extremely large, as is the case, e.g. for water running under air. The present study focuses on a two-layer shallow-water system of incompressible, immiscible and inviscid fluids with a free surface, with a density contrast ranging between 0 and 1. The two-layer shallow-water equations with a free surface rep- resent a 4  4 system of conservation equations [1]. Under the hypothesis that the two fluids are immiscible and inviscid, the lay- ers do not undergo friction or mixing. They thus interact mainly through pressure terms caused by a sloping free surface or a slop- ing internal interface separating the two fluids. A known peculiar aspect of the corresponding system of equations is that it may ex- hibit a loss of hyperbolicity. The eigenvalues of the system become complex under certain conditions [2,3,22,23]. The threshold for the appearance of such a behaviour is related to the density and veloc- ity contrast between the layers, and to the so-called stability Froude number [31]. The possible loss of hyperbolicity has been well-known for dec- ades [26]. Physically, the loss of hyperbolicity could be linked to the emergence of shear instabilities [25]. From a mathematical point of view, it results in ill-posedness of the problems to be solved, as illustrated by a number of publications [24,28,33]. Numerically, traditional schemes that apply a symmetrical 0045-7930/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compfluid.2011.01.001 ⇑ Corresponding author. Tel.: +32 (0)10 47 20 61; fax: +32 (0)10 47 21 79. E-mail addresses: benoit.spinewine@uclouvain.be (B. Spinewine), guinot@ msem.univ-montp2.fr (V. Guinot), sandra.soares-frazao@uclouvain.be (S. Soares- Frazão), yves.zech@uclouvain.be (Y. Zech). 1 Tel.: +33 (0)4 67 14 90 45; fax: +33 (0)4 67 14 47 74. 2 Tel.: +32 (0)10 47 21 20; fax: +32 (0)10 47 21 79. 3 Tel.: +32 (0)10 47 21 21; fax: +32 (0)10 47 21 79. Computers & Fluids 44 (2011) 202–220 Contents lists available at ScienceDirect Computers & Fluids journal homepage: www.elsevier.com/locate/compfluid