Two-dimensional Kelvin–Helmholtz instabilities of multi-component fluids

Abstract

The Kelvin–Helmholtz instability of multi-component (more than two) incompressible and immiscible fluids is studied numerically using a phase-field model. The instability is governed by the modified Navier–Stokes equations and the multi-component convective Cahn–Hilliard equations. A finite difference method is used to discretize the governing system. To solve the equations efficiently and accurately, we employ the Chorin’s projection method for the modified Navier–Stokes equations and the recently developed practically unconditionally stable method for the multi-component Cahn–Hilliard equations. Through our model and numerical solution, we investigate the effects of surface tension, density ratio, magnitude of velocity difference, and forcing on the Kelvin–Helmholtz instability of multi-component fluids. It is shown that increasing the surface tension or the density ratio reduces the growth of the Kelvin–Helmholtz instability. And it is also observed that as the initial horizontal velocity difference gets larger, the interface rolls up more. We also found that the billow height reaches its maximum more slowly as the initial wave amplitude gets smaller. And, for the linear growth rate for the Kelvin–Helmholtz instability of two-component fluids, the simulation results agree well with the analytical results. From comparison between the numerical growth rate of two- and three-component fluids, we observe that the inclusion of extra layers can alter the growth rate for the Kelvin–Helmholtz instability. Finally, we simulate the billowing cloud formation which is a classic example of the Kelvin–Helmholtz instability and cannot be seen in binary fluids. With our multi-component method, the details of the real flow (e.g., the asymmetry in the roll-up and the self-interaction of the shear layer) are well captured.