Nonlinear free-surface Rayleigh-Taylor instability

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Abstract
The nonlinear evolution of a Rayleigh-Taylor (RT) unstable free surface is studied by three independent approaches. (i) The method of least-squares approximation (LSA) is critically examined and applied to the general RT initial-value problem. It extends previous results of perturbation theories to higher orders and describes the appearance of bubbles and spikes for both single- and multiple-wavelength surface perturbations. Computational limitations, however, are found for the steady-state bubble regime where the number of harmonics becomes exceedingly large. (ii) A mathematically consistent sinusoidal flow model is developed valid for certain nonuniform gravitational accelerations. Its general properties are discussed, including a spike singularity and a unique steady-state bubble shape. As a special case, Layzer’s model is obtained and compared with the LSA calculations. (iii) Steady-state bubbles are described more generally in terms of source potentials. A one-parameter family of possible bubble shapes and corresponding gravitational potentials can be derived. It includes the steady-state sinusoidal flow model and yields improved analytic expressions for the constant-acceleration bubble parameters. From this model it is also concluded that flow singularities can limit the general applicability of Fourier analyses to free-boundary flows.

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