A study of bubble growth in the compressible Rayleigh–Taylor and Richtmyer–Meshkov instabilities

Within the framework of modified Layzer-type potential flow theory [V. N. Goncharov, “Analytical model of nonlinear, single-mode, classical Rayleigh-Taylor instability at arbitrary Atwood numbers,” Phys. Rev. Lett. 88, 134502 (2002)], we study bubble growth in compressible Rayleigh–Taylor (RT) and Richtmyer–Meshkov (RM) instabilities. It is known from adiabatic equations that the density ρ and adiabatic index γ are compressibility-related factors for a given static pressure p. Here, we introduce a dynamically varying stagnation point pressure P̃=p±12ρ̃η̇02, which relates time-varying quantities, such as fluid density ρ̃, pressure P̃, and bubble tip velocity η̇0, and then, we analytically derive the governing equations for time evolution of bubbles in the RT and RM instabilities of compressible fluids. For the RT instability, the upper fluid adiabatic index γu and density ρu increase the bubble amplitude and velocity, but they decrease the bubble curvature radius at the early stage, while the lower fluid adiabatic index γl and density ρl have opposite effects on those of γu and ρu, which is consistent with recent results. For the RM instability, γu and ρu decrease the bubble amplitude and velocity, but they increase the bubble curvature radius at the early stage; however, γl and ρl have opposite effects on those of γu and ρu. Moreover, we find a good agreement between our three-dimensional results of the RM bubble amplitude and recent numerical simulations.