Rayleigh–Taylor instability for adiabatically stratified fluids

Abstract

A linear analysis of the effects of compressibility on the stability of two superposed isentropic fluids is presented. The results of the analysis, which differ from those available in the literature for other unperturbed stratifications, are illustrated with several numerical examples. It is found that, in the present conditions, compressibility has a stabilizing effect at small wavelengths and a destabilizing effect at long wavelengths. The magnitude of these effects is, however, small in most circumstances. A physical basis for the interpretation of the results is also described in qualitative terms. This discussion sheds some light on the nature of the differences between the present results and those relative to the case of isothermal stratification.

Compressibility effects in Rayleigh–Taylor instability-induced flows

“…The SS and TS cases (Lezzi & Prosperetti 1989;Livescu 2004) may be recast in a similar form. An attempt at solving the three-layer configuration (Hoshoudy 2007) has to be confirmed (Livescu 2008).…”

“…a relationship between the growth rate s and the wavenumber k. A twodimensional solution may be sought, as the linear stability problem depends only on the norm of the wavenumber of the perturbation, k = k 2 x + k 2 y . One may also use the hypothesis of irrotational velocity vector field (Landau & Lifshitz 1959;Plesset & Hsieh 1964;Lezzi & Prosperetti 1989). The first step is to solve the linear equation for the perturbation in each fluid before imposing a matching condition at the interface, which gives the dispersion relation.…”

“…Particular perfect fluid RTI Compressibility effects are studied only through the EOS's parameters as the stratification is not recognized as a parameter (Lezzi & Prosperetti 1989). The TT case (Blake 1972;Mathews & Blumenthal 1977;Baker 1983;Bernstein & Book 1983;Newcomb 1983;Book 1986;Ribeyre et al 2004) is obtained for Re Pr = 0 and we have the following inequality for the linear growth rate:…”

“…Bernstein & Book (1983) also investigated the isothermal case and found that the growth rate decreases with increasing adiabatic index g. The variation is expected to be 'a few tens of percent' for materials of interest in inertial confinement fusion. Lezzi & Prosperetti (1989) investigated the behaviour of isentropic perturbations from an isentropic hydrostatic equilibrium state. They provided a careful parameter study the main result of which is that increasing the compressibility of the heavy fluid (decreasing g) stabilizes the instability whereas increasing compressibility of the light fluid destabilizes the flow.…”

Bell-Plesset effects for an accelerating interface with contiguous density gradients

“…A related and coupled issue is to what degree, if any, will Bell-Plesset (BP) geometric growth [15] be affected by contiguous density profiles. Previous work by Bernstein and Book [16] and Lezzo and Prosperetti [17] has considered the RT question, though only in a slab geometry. Physically, the effect of contiguous density profiles on a classical (discontinuous) interface (At≠0) should be manifested most at wavelengths on the order of the density-gradient scalelength or greater, not less as in the converse case of a finite density gradient on the interface [4][5][6][7][8].…”