Rayleigh-Taylor Instability and WENO Schemes

John Costello, University of Arizona

The problem of the Rayleigh-Taylor instability is a problem in high-Reynolds number fluid mechanics. We’re interested in the interaction between two near-inviscid fluids of markedly different densities which start evolving from an "inverted" stratification – the denser fluid is layered on top of the more rarefied one. The situation is difficult to model experimentally because often the "noise" involved in the experimental setup makes the experiments irrepeatable, and it’s difficult to model computationally because of the presence of a very sharp boundary layer at the beginning of the time evolution. Current approaches to the problem being worked on at Argonne National Labs involve relaxing the condition of having a sharp boundary between the two fluids, and replacing it with a smoothly varying density distribution; this results in a problem which is readily modeled via spectral methods.

Finite difference methods, when used to model this kind of problem, often show oscillatory instability at the boundary layer (this is most easily seen in a naive approach to Burgers’ equation with an initial condition which leads to a shock – as the shock develops, finite difference schemes develop spurious high-frequency oscillations in the area of the shock). However, a recent development in finite difference and finite volume schemes shows some promise in being able to handle these oscillations. Weighted Essentially Non-Oscillatory (WENO) schemes use an adaptive stencil to eliminate oscillations around a shock while maintaining optimally high order in the areas where the solution is smooth.

We present in this an overview of the WENO strategy as well as preliminary applications to a two-dimensional version of the Rayleigh-Taylor problem.

Abstract Author(s): John Costello