Potential flow models of Rayleigh-Taylor and Richtmyer-Meshkov bubble fronts

A potential flow model of Rayleigh-Taylor and Richtmyer-Meshkov bubbles on an interface between an incompressible fluid and a constant supporting pressure (Atwood number A=1) is presented. In the model, which extends the work of Layzer [Astrophys. J. 122, 1 (1955)], ordinary differential equations for the bubble heights and curvatures are obtained by considering the potential flow equations near the bubble tips. The model is applied to two-dimensional single-mode evolution as well as two-bubble competition, for both the Rayleigh-Taylor (RT) and the Richtmyer-Meshkov (RM) instabilities, the latter treated in an impulse approximation. The model predicts that the asymptotic velocity of a single-mode RM bubble of wavelength λ decays as λt^-1, in contrast with the constant asymptotic velocity attained in the RT case. Bubble competition, which is believed to determine the multimode front evolution, is demonstrated for both the RT and RM instabilities. The capability of the model to predict bubble growth in a finite-thickness fluid layer is shown. Finally, the model is applied to the evolution of three-dimensional modes with an initial rectangular geometry. The model yields the aspect ratio dependence of the early nonlinear stages, in agreement with third-order perturbation theory. However, in the late nonlinear stage, the model predicts that the bubbles forget the initial geometry and attain the fastest growing shape, with a circular tip. The model results are in good agreement with full hydrodynamic simulations and analytic solutions, where available.