Interplay of Kelvin–Helmholtz instability with acoustics in a viscous potential flow

Abstract

Among the hydrodynamic instabilities influencing the evolution, stabilization, and control of flows, the Kelvin–Helmholtz (KH) instability mode is a profound trigger to induce unsteadiness and turbulence—either within a single fluid, by means of a velocity shear, or along the interface of multiple fluids. This mechanism has been analytically studied by Funada and Joseph [“Viscous potential flow analysis of Kelvin–Helmholtz instability in a channel,” J. Fluid Mech. 445, 263 (2001)], for the surface separating two fluids within the approximation of inviscid and viscous potential flows. The present investigation extends the Funada–Joseph formulation to incorporate the effect of imposed acoustic waves on the system under consideration. Specifically, the KH–acoustic interaction is studied by employing a modification of the Bychkov approach [V. Bychkov, “Analytical scalings for flame interaction with sound waves,” Phys. Fluids 11, 3168 (1999)], which has been originally derived for the acoustic coupling to the combustion instability. The analytic formulae for the dispersion relations, growth rates, and neutral curves describing the perturbed interface of the KH instability/acoustic region are derived. Specifically, the limits for stable/unstable regimes as a function of hydrodynamic and acoustic parameters are identified. Two interacting modes are of particular interest: resonant and parametric modes, characterized by acoustic fields having the same frequency (resonant) and twice the frequency (parametric) of the instability oscillations. It is shown that while relatively weak acoustics provide a promising contribution to stabilize the KH instability, those of higher strength can excite the parametric instability. Overall, a comprehensive parametric study of the KH–acoustic coupling and stability limits shows that a global stability region may exist between that of the resonant and parametrically unstable regimes.