Congested shallow water model: roof modelling in free surface flow

Edwige Godlewski (1, 2) , Martin Parisot (2) , Jacques Sainte-Marie (1, 2, 3) , Fabien Wahl (1, 2)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 ANGE - Numerical Analysis, Geophysics and Ecology
3 Cerema Direction Centre-Est - Centre d'Etudes et d'Expertise sur les Risques, l'Environnement, la Mobilité et l'Aménagement - Direction Centre-Est

We are interested in the modeling and the numerical approximation of flows in the presence of a roof, for example flows in sewers or under an ice floe. A shallow water model with a supplementary congestion constraint describing the roof is derived from the Navier-Stokes equations. The congestion constraint is a challenging problem for the numerical resolution of hyperbolic equations. To overcome this difficulty, we follow a pseudo-compressibility relaxation approach. Eventually, a numerical scheme based on a Finite Volume method is proposed. The well-balanced property and the dissipation of the mechanical energy, acting as a mathematical entropy, are ensured under a non-restrictive condition on the time step in spite of the large celerity of the potential waves in the congested areas. Simulations in one dimension for transcritical steady flow are carried out and numerical solutions are compared to several analytical (stationary and non-stationary) solutions for validation.

Keywords: Congested hyperbolic model; Unilateral constraint; Well-balanced scheme; Entropic scheme; Shallow water equations

Domains: Numerical Analysis [math.NA]; Modeling and Simulation; Fluid mechanics [physics.class-ph]; Analysis of PDEs [math.AP]; Geophysics [physics.geo-ph]

Edwige Godlewski, Martin Parisot, Jacques Sainte-Marie, Fabien Wahl. Congested shallow water model: roof modelling in free surface flow. ESAIM: Mathematical Modelling and Numerical Analysis, 2018, 52 (5), pp.1679 - 1707. ⟨10.1051/m2an/2018032⟩. ⟨hal-01368075v3⟩