On oscillatory Kelvin-Helmholtz instability of liquids with comparable viscosities

Abstract

We study the behavior of a two-dimensional system of immiscible incompressible viscous liquids in a rectangular cavity under horizontal harmonic high-frequency vibrations. In this system, influence of pulsating motions of the liquids causes the development of Kelvin-Helmholtz instability leading to the formation of a quasi-stationary relief on the interfacial surface in the form of a frozen wave. A direct numerical modeling of the relief formation was carried out with the use of AN-SYS Fluent software; the influence of the viscosity ratio on the parameters of the frozen wave and the flow generated by vibrations near the interface was investigated. The dependencies of the characteristics of the wave relief on supercriticality and vibration frequency are plotted for different ratios of liquid viscosities. In the cases of vibrations with sufficiently high frequency, the obtained numerical results are consistent with analytical theory developed for ideal liquids. For lower frequencies, the observed relief wavelength turns out to be noticeably longer than the theoretical predictions. For high vibration frequencies, with increasing supercriticality the wavelength increases monotonically, while for low frequencies it decreases monotonically. The relief height increases as the intensity of vibration influence increases; with significant supercriticality, it exceeds the theoretical values obtained in the ideal fluid approximation. The generation of reverse averaged flow by vibrations near the interface is found when viscosity of the upper fluid is low. In this case, the flows in the upper and lower liquids are directed opposite to each other and a noticeable decrease in the average flow generation intensity is observed.