Rayleigh-Taylor instability at large times

The potential flow of an inviscid incompressible heavy fluid lying above a light one is investigated. The asymptotic stage is described by an unsteady discontinuity, which approximates the flow in the neighborhood of the tongue, and by a steady flow outside this narrow region. Consequences of the conservation laws which make it possible to check the accuracy of the solution of the steady-state problem are obtained. A steady-state solution is constructed for Froude numbers 0<Fr<0.37. The position of the discontinuity, the corresponding values of the complex potential and the pressure, the average acceleration of the tongue, and the dependence of the parameters of the limiting regime on the values of the conservation integrals (in particular, the Froude numbers) and the level of the initial energy perturbation are determined from the conservation laws. At the apex of the bubble the derivative of the free-surface curvature has a discontinuity when Fr≠0.23, which makes it possible to attribute the choice of solution obtained in a number of studies to the presence of an artificial surface tension.