On the nonlinear development of two-dimensional and three-dimensional perturbations in the presence of Rayleigh-Taylor instability

Abstract

The nonlinear evolution of two-dimensional and three-dimensional perturbations of finite amplitude in the presence of Rayleigh-Taylor instability is investigated. It is assumed that the problem is one of potential flow. The solution is constructed by the Fourier method [1]. In the two-dimensional case the conformal mapping method [2, 3] is employed, which makes it possible to consider the strongly nonlinear stages of development of the perturbations, including the formation of surfaces with multiply valued dependence of the variables in Cartesian coordinates. The construction of the mappings reduces to the solution of the Hilbert problem, which is given in the form of Schwartz integrals [4]. Explicit expressions for these integrals [5], obtained with the aid of Fourier series, are employed. Effective computational algorithms are developed and a series of numerical investigations is carried out. Inter alia, a destabilizing effect of the short-wave components is detected, the regularizing action of the surface tension is demonstrated, and the characteristic times of nonlinear development of the perturbations and the characteristic spectral distributions are found. The role of three-dimensional effects, characterized by a decrease in the rate of development of perturbations, is investigated.