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The non-linear dynamics of thin liquid films sheared by a laminar gas flow in a channel is investigated. Such a two-layer flow is driven by pressure gradient and possibly by the gravity force. We describe the liquid phase with a long-wave integral model, with the aim to save computational cost with respect to the full Direct Numerical Simulation (DNS) of the Navier-Stokes equations. We derive this long-wave model by the integration of the Navier-Stokes equations over the film thickness, and by an asymptotic expansion up to the first order in terms of a long-wave parameter. These depth-integrated (or shallow water) equations are discretized by means of an augmented system, which holds an evolution equation for the surface tension in order to avoid numerical instabilities of classical upwind and centered schemes. On the other side, we study the gas phase with compressible Navier-Stokes equations, and we discretize them by means of a low-Mach scheme, accounting also for moving meshes (ALE). In order to analyze liquidgas interactions, we introduce then a coupling methodology between depth-integrated equations and Navier-Stokes equations. This approach represents a compromise between the two existing methods: the full DNS, and the full long-wave model applied to both phases. In order to validate this approach, we present comparisons with DNS, showing a good agreement of spatio-temporal evolutions of the film thickness and the stress field. Furthermore, interfacial shear stress and pressure gradient evolutions are shown to be in accordance with those provided by two-layer second-order low-dimensional models.

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In this article we present a 2D conservative remapping method which relies on exact polygonal mesh intersection, high accurate polynomial reconstruction (up to degree 5) and a posteriori stabilization based on MOOD paradigm [19, 29, 30, 70]. This paradigm does not compute any sort of a priori limiter for the polynomial reconstructions. Instead it rather observes if the candidate solution after remapping does not fulfill user-given criteria, and, in this case, locally to some so-called problematic cells, the method decrements the polynomial degree for the reconstructions. These problematic cells are then recomputed. Next they are again tested against the validity criteria and, possibly, some decrementation is applied. Such iterative procedure always ends either with a valid remapped solution obtained from polynomial degrees greater than 0, or, in the worst case scenario, with a cell remapped without polynomial reconstruction (formally with a first order accurate remap scheme). Numerical results assess the behavior of such remapping method on pure advection equations and on some emulation of system of equations.