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.

The Evolution of Interfaces for Underwater Supersonic Gas Jets

“…For high-speed gas jets, previous scholars proposed that the K-H instability is the dominant mechanism [21]; however, after the gas is injected into the water, the high speed will rapidly decline after being hindered by the aqueous medium. The dominant mechanism of instability varies with time and space.…”

“…Due to the large density gradient and velocity gradient between the gas and the surrounding fluid water, the jets will be governed by Rayleigh-Taylor (R-T) instability and Kelvin-Helmholtz (K-H) instability. Chawla [20,21] studied the K-H instability of the sonic gas jet underwater. He found that pressure disturbance, liquid viscosity, and surface tension affect the stability of the interface.…”

Submerged Gas Jet into a Liquid Bath: A Review

“…Chawla 44 investigated the stability of the gas/liquid interface of a sonic gas jet submerged in an infinite mass of liquid under the action of a pressure perturbation, liquid viscosity, and surface tension (see Figure 18). The nonlinear wave equation for the gas jet was subjected to kinematic boundary conditions at the gas/liquid interface.…”

“…Furthermore, the nonlinear equations were literalized by addition of a perturbation equation and the stability criteria that were applicable for different flows (such as subsonic, sonic and supersonic) were determined. Figure 19 shows the gas jet issuing from a jet orifice of radius r. The flow at the 44 ). orifice was assumed to be uniform, with the gas jet having a constant mean radius equal to the orifice radius.…”

Formation and Development of Submerged Air Jets

“…Formation of jet as well as jet length in submerged condition depends on a number of factors such as gas velocity, nozzle geometry, liquid properties, size and depth of the liquid column, etc. 2,3 The theoretical studies dealing with submerged gas jets focused mostly on stability analysis of gas jets in terms of the disturbance generated on the jet surface 4,5 . Koria 6 calculated the non-buoyant jet length of gas jets in a metal bath for both subsonic and supersonic regions.…”