The Kelvin-Helmholtz instability of the gas-liquid interface of a sonic gas jet submerged in a liquid

Abstract

It is well known that the small perturbation equation governing steady or mildly unsteady potential flow in a sonic gas jet is nonlinear. However, for a sonic gas jet submerged in a liquid with a disturbance on the gas-liquid interface, it is shown that the transient motion of the gas dominates, and the nonlinear term due to accumulation of disturbances in the basic flow becomes negligible; the condition necessary for the applicability of the linearized governing equation is obtained. It is demonstrated that most gas-jet/liquid systems of physical interest satisfy this condition and that the margin with which this condition is satisfied improves as the wave velocity of the disturbance or, more particularly, as the stagnation pressure or density of the gas for a given gas-liquid system increases. The Kelvin-Helmholtz instability of the gas-liquid interface of a sonic gas jet submerged in a liquid is predominantly governed by the transfer of energy from the gas phase to the liquid layer, both through wave-drag and ‘lift’ components of the pressure perturbation; at and above the cut-off wavenumber, which only exists for very low viscosity liquids owing to the stabilizing effect of surface tension, the pressure perturbation becomes in phase with the wave amplitude. It is shown that for low viscosity liquids the phase angle between the pressure perturbation exerted by the gas phase on the liquid a t the gas-liquid interface and the wave amplitude, which is the measure of the relative effectiveness of the ‘lift’ and wave-drag components of the pressure perturbation, is a function of the density ratio (ratio of gas density a t throat conditions to liquid density). At low density ratios both of these components are operative; however, at high density ratios the wave-drag component becomes dominant. The analysis further shows that the cut-off wave- number and the wavenumber a t maximum instability decrease with increasing density ratio. For highly viscous liquids and liquids having finite viscosity the pressure perturbation is always out of phase with the wave amplitude, and no cut-off wavenumber exists, i.e. the gas-liquid interface is always unstable in spite of the stabilizing effect of viscosity and surface tension.