Temporal evolution of bubble tip velocity in classical Rayleigh-Taylor instability at arbitrary Atwood numbers

Abstract

In this research, the temporal evolution of the bubble tip velocity in Rayleigh-Taylor instability (RTI) at arbitrary Atwood numbers and different initial perturbation velocities with a discontinuous profile in irrotational, incompressible, and inviscid fluids (i.e., classical RTI) is investigated. Potential models from Layzer [Astrophys. J. 122, 1 (1955)] and perturbation velocity potentials from Goncharov [Phys. Rev. Lett. 88, 134502 (2002)] are introduced. It is found that the temporal evolution of bubble tip velocity [u(t)] depends essentially on the initial perturbation velocity [u(0)]. First, when the u(0)<C(1)uasp, the bubble tip velocity increases smoothly up to the asymptotic velocity (uasp) or terminal velocity. Second, when C(1)uasp≤u(0)<C(2)uasp, the bubble tip velocity increases quickly, reaching a maximum velocity and then drops slowly to the uasp. Third, when C(2)uasp≤u(0)<C(3)uasp, the bubble tip velocity decays rapidly to a minimum velocity and then increases gradually toward the uasp. Finally, when u(0)≥C(3)uasp, the bubble tip velocity decays monotonically to the uasp. Here, the critical coefficients C(1),C(2), and C(3), which depend sensitively on the Atwood number (A) and the initial perturbation amplitude of the bubble tip [h(0)], are determined by a numerical approach. The model proposed here agrees with hydrodynamic simulations. Thus, it should be included in applications where the bubble tip velocity plays an important role, such as the design of the ignition target of inertial confinement fusion where the Richtmyer-Meshkov instability (RMI) can create the seed of RTI with u(0)∼uasp, and stellar formation and evolution in astrophysics where the deflagration wave front propagating outwardly from the star is subject to the combined RMI and RTI.