A numerical reduced model for thin liquid films sheared by a gas flow

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 liquid–gas 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.