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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 equal to zero except where it multiplies the acceleration or velocity-jump Δv. While this approximation is generally applied to low- systems, we show that it can be applied to high- 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 Δv: a low-A air/CO system shocked at M = 1.57, and a high-A air/SF system shocked at M = 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.


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