Pré-Publication, Document De Travail Année : 2025

A comprehensive list of stationary solutions of shallow water models

Résumé

The shallow water equation of Barré de Saint-Venant describes the evolution of the water height and of the horizontal velocity of a column of water. It is valid when the horizontal length scale of the flow is much greater than its vertical length scale. This model is widely employed to model rivers and lake. It can be enriched to take into account a shear velocity along the vertical. In this work we classify all the stationary solutions of the shallow water equation with a shear velocity on a bounded domain. Boundary conditions are fixed at the inlet and the outlet, and their number depends on whether the regime is supercritical (Froude number larger than 1) or subcritical. The stationary solutions consist of a succession of regular parts where the momentum and hydraulic head are constant, separated by discontinuity (also called hydraulic jumps) through which the hydraulic head decreases but the momentum and momentum flux are constant. We exhibit the form of the stationary solution and its conditions of existence for any set of boundary conditions on monotonic, bumped shaped and hollow shaped topographies. We find that in some cases and for supercritical flow at the inlet, two different stationary solutions can fulfilled the same set of boundary conditions. The solutions with shocks on increasing topography are sparsely documented and seem to be instable. Eventually we consider a topography made of a succession of N decreasing bumps and prove in the subcritical case that up to 2^{N-1} solutions with discontinuities on decreasing topography only may coexist.

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hal-05025901 , version 1 (08-04-2025)

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  • HAL Id : hal-05025901 , version 1

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Nina Aguillon, Emmanuel Audusse, Nelly Boulos, Martin Parisot. A comprehensive list of stationary solutions of shallow water models. 2025. ⟨hal-05025901⟩
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