The Rayleigh–Taylor instability for inviscid and viscous fluids

Abstract

The classical Rayleigh–Taylor instability occurs when two inviscid fluids, with a sharp interface separating them, lie in two horizontal layers with the heavier fluid above the lighter one. A small sinusoidal disturbance on the interface grows rapidly in time in this unstable situation, as the heavier upper fluid begins to move downwards through the lighter lower fluid. This paper presents a novel numerical method for computing the growth of the interface. The technique is based on a spectral representation of the solution. The results are accurate right up to the time when a curvature singularity forms at the interface and the inviscid model loses its validity. A spectral method is then presented to study the same instability in a viscous Boussinesq fluid. The results are shown to agree closely with the inviscid calculations for small to moderate times. However, the high interface curvatures that develop in the inviscid model are prevented from occurring in viscous fluid by the growth of regions of high vorticity at precisely these singular points. This leads to over-turning of the interface, to form mushroom-shaped profiles. It is shown that different initial interface configurations can lead to very different geometrical outcomes, as a result of the flow instability. These can include situations when detached bubbles form in the fluid.