Pure single-mode Rayleigh-Taylor instability for arbitrary Atwood numbers

The validity of theoretical investigation on Rayleigh-Taylor instability (RTI) with nonlinearity is quite important, especially for the simplest and the commonest case of a pure single-mode RTI, while its previous explicit solution in weakly nonlinear scheme is found to have several defections. In this paper, this RTI is strictly solved by the method of the potential functions up to the third order at the weakly nonlinear stage for arbitrary Atwood numbers. It is found that the potential solution includes terms of both the stimulating and inhibiting RTI, while the terms of the decreasing RTI are omitted in the classical solution of the weakly nonlinear scheme, resulting in a big difference between these two results. For the pure single-mode cosine perturbation, comparisons among the classical result, the present potential result and numerical simulations, in which the two dimensional Euler equations are used, are carefully performed. Our result is in a better agreement with the numerical simulations than the classical one before the saturation time. To avoid the tedious expressions and improve a larger valid range of the solution, the method of the Taylor expansion is employed and the velocities of the bubble and spike are, respectively, obtained. Comparisons between the improved and the simulation results show that the improved theory can better predict the evolution of the interface from the linear to weakly nonlinear, even to later of the nonlinear stages.