Weakly nonlinear incompressible Rayleigh–Taylor–Kelvin–Helmholtz instability in plane geometry

Abstract

A weakly nonlinear theoretical model is established for the two-dimensional incompressible Rayleigh–Taylor–Kelvin–Helmholtz instability (RT–KHI). The evolution of the perturbation interface is analytically studied by the third-order solution of the planar RT–KHI induced by a single-mode surface perturbation. The difference between the weakly nonlinear growth for Rayleigh–Taylor instability (RTI), Kelvin–Helmholtz instability (KHI), and RT–KHI in plane geometry is discussed. The trend of bubble and spike amplitudes with the Atwood number and the Richardson number is discussed in detail. The bubble and spike amplitudes of RT–KHI change from the KHI case to the RTI case as the Richardson number increases. The deflecting distance of bubble and spike vertices becomes smaller compared to the KHI case as the Richardson number increases. The dependence of the nonlinear saturation amplitude of RT–KHI on the Atwood number, the Richardson number, and the initial perturbation is obtained. The Richardson number is as vital to the nonlinear saturation amplitude as the Atwood number. It is found that the variation of the nonlinear saturation amplitude with the Atwood number at different Richardson numbers is divided into three parts, namely, “RTI-like part,” “transition part,” and “KHI-like part.” In the transition part, the trend of the nonlinear saturation amplitude increasing with the Atwood number is completely opposite to the RTI and KHI cases. Finally, the theory is compared to the numerical simulation under identical initial conditions and displays good correspondence in the linear and weakly nonlinear stages.