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.
A nonlinear theory is developed to describe the cylindrical Richtmyer-Meshkov instability (RMI) of an impulsively accelerated interface between incompressible fluids, which is based on both a technique of Padé approximation and an approach of perturbation expansion directly on the perturbed interface rather than the unperturbed interface. When cylindrical effect vanishes (i.e., in the large initial radius of the interface), our explicit results reproduce those [Q. S.-I. Sohn, Phys. Fluids 9, 1106 (1996)] related to the planar RMI. The present prediction in agreement with previous simulations [C. Matsuoka and K. Nishihara, Phys. Rev. E 73, 055304(R) (2006)] leads us to better understand the cylindrical RMI at arbitrary Atwood numbers for the whole nonlinear regime. The asymptotic growth rate of the cylindrical interface finger (bubble or spike) tends to its initial value or zero, depending upon mode number of the initial cylindrical interface and Atwood number. The explicit conditions, directly affecting asymptotic behavior of the cylindrical interface finger, are investigated in this paper. This theory allows a straightforward extension to other nonlinear problems related closely to an instable interface. V C 2014 AIP Publishing LLC. [http://dx
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