Late-time quadratic growth in single-mode Rayleigh-Taylor instability

Abstract

The growth of the two-dimensional single-mode Rayleigh-Taylor instability (RTI) at low Atwood number (A=0.04) is investigated using Direct Numerical Simulations. The main result of the paper is that, at long times and sufficiently high Reynolds numbers, the bubble acceleration becomes stationary, indicating mean quadratic growth. This is contrary to the general belief that single-mode Rayleigh-Taylor instability reaches a constant bubble velocity at long times. At unity Schmidt number, the development of the instability is strongly influenced by the perturbation Reynolds number, defined as Rep≡λsqrt[Agλ/(1+A)]/ν. Thus, the instability undergoes different growth stages at low and high Rep. A new stage, chaotic development, was found at sufficiently high Rep values, after the reacceleration stage. During the chaotic stage, the instability experiences seemingly random acceleration and deceleration phases, as a result of complex vortical motions, with strong dependence on the initial perturbation shape (i.e., wavelength, amplitude, and diffusion thickness). Nevertheless, our results show that the mean acceleration of the bubble front becomes constant at late times, with little influence from the initial shape of the interface. As Rep is lowered to small values, the later instability stages, chaotic development, reacceleration, potential flow growth, and even the exponential growth described by linear stability theory, are subsequently no longer reached. Therefore, the results suggest a minimum Reynolds number and a minimum development time necessary to achieve all stages of single-mode RTI development, requirements which were not satisfied in the previous studies of single-mode RTI.