The Nonlinear Asymptotic Stage of the Rayleigh-Taylor Instability with Wide Bubbles and Narrowing Spikes

TL;DR

This paper analyzes the nonlinear asymptotic behavior of the Rayleigh-Taylor instability, focusing on wide bubbles and narrowing spikes, using matched asymptotic expansions to describe flow dynamics and solution stability.

Contribution

It introduces an asymptotic solution framework for the Rayleigh-Taylor instability that accounts for initial energy and surface tension effects, extending previous Froude number analyses.

Findings

Existence of solutions within the same Froude number range as prior studies.

Discontinuity in free-surface curvature derivative at a specific Froude number.

Identification of artificial surface tension effects in numerical solutions.

Abstract

The potential flow of an incompressible inviscid heavy fluid over a light one is considered. The integral version of the method of matched asymptotic expansion is applied to the construction of the solution over long intervals of time. The asymptotic solution describes the flow in which a bubble rises with constant speed and the "tongue" is in free fall. The outer expansion is stationary, but the inner one depends on time. It is shown that the solution exists within the same range of Froude number obtained previously by Vanden-Broeck (1984a,b). The Froude number and the solution depend on the initial energy of the disturbance. At the top of the bubble, the derivative of the free-surface curvature has a discontinuity when the Froude number is not equal to 0.23. This makes it possible to identify the choice of the solution obtained in a number of studies with the presence of an artificial numerical surface tension. The first correction term in the neighborhood of the tongue is obtained when large surface tension is included.