Linear stability of a gas boundary layer flowing past a thin liquid film over a flat plate

Abstract

The flow of a gas stream past a flat plate under the influence of rainfall is investigated. As raindrops sediment on the flat plate, they coalesce to form a water film that flows under the action of shear from the surrounding gas stream. In the limit of (a) large Reynolds number, Re, in the gas phase, (b) small rainfall rate, r˙, compared to the free-stream velocity, U_∞, and (c) small film thickness compared to the thickness of the boundary layer that surrounds it, a similarity solution is obtained that predicts growth of the liquid film like x^3/4; x denotes dimensionless distance from the leading edge. The flow in the gas stream closely resembles the Blasius solution, whereas viscous dissipation dominates inside the film. Local linear stability analysis is performed, assuming nearly parallel base flow in the two streams, and operating in the triple-deck regime. Two distinct families of eigenvalues are identified, one corresponding to the well-known Tollmien–Schlichting (TS) waves that originate in the gas stream, and the other corresponding to an interfacial instability. It is shown that, for the air–water system, the TS waves are convectively unstable whereas the interfacial waves exhibit a pocket of absolute instability, at the streamwise location of the applied disturbance. Moreover, it is found that as the inverse Weber number (We⁻¹) increases, indicating the increasing effect of surface tension compared to inertia, the pocket of absolute instability is translated towards larger distances from the leading edge and the growth rate of unstable waves decreases, until a critical value is reached, We⁻¹ ≈ We⁻¹_c, beyond which the family of interfacial waves becomes convectively unstable. Increasing the inverse Froude number (Fr⁻¹), indicating the increasing effect of gravity compared to inertia, results in the pocket of absolute instability shrinking until a critical value is reached, Fr⁻¹ ≈ Fr⁻¹_c, beyond which the family of interfacial waves becomes convectively unstable. As We⁻¹ and Fr⁻¹ are further increased, interfacial waves are eventually stabilized, as expected. In this context, increasing the rainfall rate or the free-stream velocity results in extending the region of absolute instability over most of the airfoil surface. Owing to this behaviour it is conjectured that a global mode that interacts with the boundary layer may arise at the interface and, eventually, lead to three-dimensional waves (rivulets), or, under extreme conditions, even premature separation.