Boussinesq approximation for Rayleigh-Taylor and Richtmyer-Meshkov instabilities

Abstract

We apply numerical and analytic techniques to study the Boussinesq approximation in Rayleigh-Taylor and Richtmyer-Meshkov instabilities. In this approximation, one sets the Atwood number A equal to zero except where it multiplies the acceleration g or velocity-jump v. While this approximation is generally applied to low-A systems, we show that it can be applied to high-A systems also in certain regimes and to the "bubble" part of the instability, i.e., the penetration depth of the lighter fluid into the heavier fluid. It cannot be applied to the spike. We extend the Boussinesq approximation for incompressible fluids and show that it always overestimates the penetration depth but the error is never more than about 41%. The effect of compressibility is studied by analytic techniques in the linear regime which indicate that compressibility has the opposite effect and the Boussinesq approximation underestimates bubbles by about 14%. We also present direct numerical simulations of two compressible systems which have approximately the same A v: a low-A air/CO 2 system shocked at M s = 1.57, and a high-A air/SF 6 system shocked at M s = 1.24. While the bubbles are approximately equal, the lower-A system has a shorter (less penetrating) spike; however, because its mushrooms are more tightly wound, the low-A system has the larger interface area. C 2014 Author(s)

Simulating coronal condensation dynamics in 3D

“…determines the morphology and the evolution of the plasma undergoing nonlinear Rayleigh-Taylor instability, since for A close to 0, Rayleigh-Taylor instability shows a symmetric mixing for both finger-like plasma structures for the heavy down falling fluid and bubble-like ones for the light fluid that rises; while for A close to 1, falling spikes have larger growth rates and penetrate deeper in the opposite region than rising bubbles (Mikaelian, 2014). The vast majority of the blobs appears to have Atwood numbers close to unity, suggesting a blob density of about 20 times higher than the plasma elements underneath the blob.…”

Exact, approximate, and hybrid treatments of viscous Rayleigh-Taylor and Richtmyer-Meshkov instabilities

“…[38,39,40]. We pointed out that inviscid linear results remain the same under this approximation, but nonlinear results do not [39]. Here let us point out that viscous linear results also remain the same in our approximate treatment, but the exact and hybrid results do not remain the same, as the reader can easily verify from the expressions given in this paper.…”

“…Recent considerations of the Boussinesq approximation in the context of RT and RM instabilities can be found in Refs. [38,39,40]. We pointed out that inviscid linear results remain the same under this approximation, but nonlinear results do not [39].…”

Effects of Error on the Onset and Evolution of Rayleigh–Taylor Instability

“…Using same fluid alleviates the problem of tracking the interface. Large discontinuous jump in temperature (ΔT = 70K ) will not allow one to apply Boussinesq approximation [6]. Most of RTI simulations [2,3,5] are reported using incompressible formulation.…”

“…Present use of compressible NSE [1,4] helps remove this major discrepancy. It is noted [6] that the issue of compressibility is highly challenging and direct numerical simulations are the best way to assess the validity of Boussinesq approximation. Thus the time evolution of the RTI is traced here by high resolution dispersion relation preserving (DRP) compact scheme for DNS.…”