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.

Numerical computation of the Rayleigh-Taylor instability for a viscous fluid with regularized interface properties.

In this article, the computation of the linear growth rates and eigenfunctions of the viscous version of the Rayleigh-Taylor instability by numerically solving the corresponding eigenvalue problem in the case of one-dimensional (1D) and two-dimensional (2D) geometries is studied. The 1D version is first validated in the particular inviscid case to be compared to the previous literature. The most unstable mode, also known as the first mode, which has the maximal linear growth rate has been extensively studied in previous literature. Higher modes have smaller eigenvalues, but the corresponding eigenfunctions present a more complex structure that contains multipeak shapes. In the extension to the 2D geometry, the length of the domain limits the wave number of the eigenvectors computed. In the extension to the 2D geometry the length of the domain limits the wave number of the eigenvectors computed. The importance of extending the results to the two-dimensional case is twofold. First, it opens up the possibility of generalizing the computation to more complex geometries that could contain fixed or floating objects and, second, allows the computation of flow instabilities in nonzero basic flows that could come from the steady Navier-Stokes solutions.