The late-time dynamics of the single-mode Rayleigh-Taylor instability

Abstract

We report on numerical simulations of the detailed evolution of the single mode Rayleigh-Taylor [Lord Rayleigh, Scientific Papers II (Cambridge University Press, Cambridge, 1900), p. 200; G. I. Taylor, “The instability of liquid surfaces when accelerated in a direction perpendicular to their plane,” Proc. R. Soc. London, Ser. A 201, 192 (1950)10.1098/rspa.1950.0052; S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Oxford University Press, Oxford, 1961)] instability to late times and high aspect ratios. In contrast to established potential flow models that predict a terminal velocity and a constant Froude number at low Atwood numbers, we observe a complex sequence of events that can be summarized in four stages: I. Exponential growth of imposed perturbations, II. Saturation to terminal velocity, III. Reacceleration to a higher Froude number, and IV. Chaotic mixing. The observed reacceleration away from the Froude number predicted by potential flow theory is attributed to the appearance of secondary Kelvin–Helmholtz structures, and described with a modification to the potential flow model proposed by Betti and Sanz [R. Betti and J. Sanz, “Bubble acceleration in the ablative Rayleigh-Taylor instability,” Phys. Rev. Lett. 97, 205002 (2006)10.1103/PhysRevLett.97.205002]. The secondary KH instability is in turn sensitive to several parameters, and can be suppressed at large Atwood numbers, as well as viscosity (physical or numerical), with the bubble/spike velocity in each case reverting to the potential flow value. Our simulations delineate the change in dynamics of the primary and secondary instabilities due to changes in these flow parameters. When the flow is allowed to evolve to late times, further instability is observed, resulting in chaotic mixing which is quantified here. The increased atomic mixing due to small-scale structures results in a dramatic drop in the late-time Froude number. Spike behavior resembles bubbles at low A, while for large A, spikes approach free-fall – thus, the notion of a terminal velocity appears not to be applicable to spikes at any density difference. We expect the results to be relevant to turbulent mix models that are based on bubble growth and interaction.