Abstract

The two-layer problem is defined as the coexistence of two immiscible fluids, separated by an interface surface. Under the shallow-flow hypothesis, 1D models are based on a four equations system accounting for the mass and momentum conservation in each fluid layer. Mathematically, the system of conservation laws modelling 1D two-layer flows has the important drawback of loss of hyperbolicity, causing that numerical schemes based on the eigenvalues of the Jacobian become unstable. In this work, well-balanced FORCE scheme is proposed for 1D two-layer shallow flows. The FORCE scheme combines the first-order Lax–Friedrichs flux and the second-order Lax–Wendroff flux. The scheme is supplemented with a hydrostatic reconstruction procedure in order to ensure the well-balanced behaviour of the model for steady flows even under wet–dry conditions. Additionally, a method to obtain high-accuracy numerical solutions for two-layer steady flows including friction dissipation is proposed to design reference benchmark tests for model validation. The enhanced FORCE scheme is faced to lake-at-rest benchmarking tests and steady flow cases including friction, demonstrating its well-balanced character. Furthermore, the numerical results obtained for highly unsteady two-layer dambreaks are used to analyse the robustness and accuracy of the model under a wide range of flow conditions.

HIGHLIGHTS

  • In this work, a new 1D well-balanced FORCE scheme is proposed able to deal with steady and highly unsteady two-layer shallow flows, including wet–dry fronts.

  • The enhanced FORCE scheme is supplemented with a new hydrostatic reconstruction procedure, based on the definition of a virtual free surface and a virtual pressure surface for each layer, in order to ensure the well-balanced behaviour at the intercell edges for steady flows.

  • The proper treatment of the wet–dry fronts in both the upper and the lower layer is ensured by the correct balance between the convective fluxes and the integrated source terms at the wet–dry intercell edges.

  • Additionally, a method to obtain high-accuracy numerical solutions for two-layer steady flows including friction dissipation is also proposed in order to design reference benchmark tests for two-layer models validation.

  • The numerical results obtained with the well-balanced FORCE scheme for steady and highly unsteady two-layer benchmark tests are used to analyse the robustness and accuracy of the proposed model under a wide range of flow conditions.

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