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 x3/4; 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−1c,
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−1c, 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.
Prediction and Control of Laminar-turbulent Transition in High-speed Boundary-Layer Flows