Effect of gravity and surface tension on compressible Kelvin-Helmholtz instability

For an ideal incompressible fluid, an interface of tangential velocity discontinuity necessarily undergoes the Kelvin-Helmholtz instability (KHI), with growth rate proportional to the magnitude of discontinuity in tangential velocity. Compressibility acts to weaken KHI, and even suppresses KHI when the Mach number of velocity discontinuity is greater than √8 (Landau 1944). In this investigation, by adding density jump across the interface of velocity discontinuity, we explore the effect of gravity force and surface tension on the compressible KHI. In case a heavy fluid lies on a light fluid, the problem becomes interaction of KHI with the Rayleigh-Taylor instability of a compressible fluid. In case a light fluid lies on a heavy fluid, the gravity force as well as the surface tension acts as restoring forces. We are concerned with the influence of these restoring forces on the compressible KHI. The results should follow Krein's theory of Hamiltonian spectra. By extending the wave-energy formula of incompressible flows to compressible flows, we identify negative-energy modes and, from this viewpoint, clarify the roles of the gravity and the surface tension in stabilization / destabilization of KHI.