An arbitrary Lagrangian-Eulerian σ (ALES) model with non-hydrostatic pressure for shallow water flows

Abstract An arbitrary Lagrangian-Eulerian model in the σ coordinate system (ALES) is developed for shallow water flows, based on the unsteady Reynolds-averaged Navier–Stokes equations. Unlike the conventional σ coordinate system, non-hydrostatic pressure is incorporated and the effect of a moving free-surface included giving a general scheme. The standard k − ϵ turbulence model is used to calculate the eddy viscosity. Wave flows over bars are appropriate test cases for which experimental data are available and comparisons are favourable. The model is also applied to the important practical problem of wave and wave/current flows over a trench, for which experimental data are available only for the steady current alone case. The inclusion of vertical grid velocity has negligible effect on computational efficiency. The solution of Poisson's equation for non-hydrostatic pressure at each time step is solved efficiently by the conjugate gradient method but represents the main component of computer time. The model may readily be extended to three dimensions.