Combined effect of the density and velocity gradients in the combination of Kelvin–Helmholtz and Rayleigh–Taylor instabilities

Abstract

We have derived explicit analytic formulas for the linear growth rate and the frequency in the combination of Kelvin–Helmholtz (KH) and Rayleigh–Taylor (RT) instabilities in fluids with continuous density and velocity profiles. It is found that the density gradient effect (i.e., the density transition layer) decreases the linear growth rate in the RT instability (RTI), especially for the short perturbation wavelength. The linear growth rate for the KH instability (KHI) is increased by the density gradient effect but decreased by the velocity gradient effect (i.e., the velocity transition layer). The frequency in the KHI is reduced by both the density gradient effect and the velocity gradient effect. In most cases, both the linear growth rate and the frequency are decreased by the combination of density and velocity transition layers, i.e., the combined effect of density and velocity gradients stabilizes the KHI. The density gradient effect has an opposite influence on the linear growth rates of the RTI and KHI. Therefore, in real system, there is a competition between the growths of the RTI and KHI which plays an important role in the material transport or mixture. If the widths of density and velocity transition layers have the same dimensionless values, the combined linear growth rate in the combination of KHI and RTI increases with the acceleration but decreases with the width of density (velocity) transition layer.