falling and sheared films

This work is devoted to the analysis of a counter-current gas-liquid film flow in an inclined channel. Our purpose is to consider how travelling waves generated at the free surface of the film by the classical Kapitza instability mechanism are affected by the gaseous turbulent flow. We aim at reproducing the experimental results obtained by Sophie mergui and Nicolas Kofman at FAST laboratory. Travelling wave solutions, i.e. waves which remain stationary in their moving frame, have been found numerically. The approach is one-sided as the interfacial stress is a function of the interface position only (we use Camassa's model to compute the shear exerted by the gas flow onto the liquid). A pseudo-spectral approach is followed where a projection of the unknowns onto Chebyshev polynomial functions is performed and the primitive equations are evaluated at the Gauss-Lobatto points, which results in the elimination of the normal coordinate. By invoking a penalization method to account for the continuity of the stresses at the free surface, an autonomous dynamical system of large dimension is obtained. Travelling wave solutions are then obtained as Hopf bifurcations of the Nusselt flat-film solution by means of a predictor-corrector continuation method (ATO07p software. A good agreement is found at a relatively low superficial gas speed. Though only qualitative, the proposed one-sided modelling is able to retrieve qualitatively the experimental observations: enhancement of wave amplitude, lowering of phase velocity and reduction of the number of capillary ripples.