Nonlinear saturation amplitude in the Rayleigh-Taylor instability at arbitrary Atwood numbers with continuous profiles

We present an approximate method to derive an analytic formula for the Nonlinear Saturation Amplitude (NSA, ηs), in the two-dimensional Rayleigh-Taylor Instability (RTI) for incompressible fluids, with sharp density profiles, to analyze the Atwood number (AT) effects. Our analytic formula indicates that the Atwood number effects remarkably affect the NSA of the RTI. The density gradient effects (i.e., the effects of interface width) on the NSA are investigated by the Direct Numerical Simulation (DNS). In our DNS, it is found that the NSA is fundamentally dependent on kL, where L=min(ρ/|dρ/dx|) is the minimum density gradient scale length and k is the perturbation wave number. For fixed AT, kηs increases with kL, and for fixed L, kηs increases with kL. The results of DNS show that our analytic formula , is recovered, when kL→0. In our DNS, for large kL, the NSA can approach and even exceed its wavelength, which cannot be predicted by the classical estimation, 0.1λ (Jacobs J. W. and Catton I., J. Fluid Mech., 187 (1988) 329).