Extended computation of the viscous Rayleigh-Taylor instability in a horizontally confined flow

In this article, the classical Rayleigh-Taylor instability is extended to situations where the fluid is completely confined, in both the vertical and horizontal directions. This article starts with the two-dimensional (2D) viscous periodic case with finite height where the effect of adding surface tension to the interface is analyzed. This problem is simulated from a dual perspective: first, the linear stability analysis obtained when the Navier-Stokes equations are linearized and regularized in terms of density and viscosity; and second, looking at the weakly compressible version of a multiphase smoothed particle hydrodynamics (WCSPH) method. The evolution and growth rates of the different fluid variables during the linear regime of the SPH simulation are compared to the computation of the eigenvalues and eigenfunctions of the viscous version of the Rayleigh-Taylor stability (VRTI) analysis with and without surface tension. The most unstable mode, which has the maximal linear growth rate obtained with both approaches, as well as other less unstable modes with more complex structures are reported. The classical horizontally periodic (VRTI) case is now adapted to the case where two additional left and right walls are included in the problem, representing the cases where a two-phase flow is confined in a accelerated tank. This 2D case where no periodic assumptions are allowed is also solved using both techniques with tanks of different sizes and a wide range of Atwood numbers. The agreement with the linear stability analysis obtained by a Lagrangian method such as multiphase WCSPH is remarkable.