Boundary layer flow of air past solid surfaces in the presence of rainfall

The steady two-dimensional laminar flow of an air stream, flowing past a solid surface at high Reynolds number, is examined in the presence of rainfall. As raindrops sediment on the surface they coalesce and form a continuous water film that flows due to shear, pressure drop and gravity, in general. In the limit as the boundary layer and film thickness remain smaller than the radius of curvature of the surface a simplified lubrication-type formulation describes the flow field in the film, whereas the usual boundary layer formulation is applied in the gas phase. In the case of a flat plate and close to the leading edge, x → 0, a piecewise-self-similar solution is obtained, according to which creeping flow conditions prevail in the film and its thickness grows like x3/4, whereas the Blasius solution is recovered in the air stream. Numerical solution of the governing equations in the two phases and for the entire range of distances from the leading edge, x = O(1), shows that the film thickness increases as the rainfall rate, r˙, increases or as the free-stream velocity, U∞, decreases and that the region of validity of the asymptotic result covers a wide range of the relevant problem parameters. In the case of flow past a NACA-0008 airfoil at zero angle of attack a Goldstein singularity may appear far downstream on the airfoil surface due to adverse pressure gradients, indicating flow reversal and eddy formation inside the liquid film, and, possibly, flow separation. However, when the effect of gravity becomes evident in the film flow, as the Froude number decreases, and provided gravity acts in such a way as to negate the effect of the adverse pressure gradient, the location of the singularity is displaced towards the trailing edge of the airfoil and the flow pattern resembles that for flow past a flat plate. The opposite happens when gravity is aligned with the adverse pressure gradient. In addition it was found that there exists a critical water film thickness beyond which the film has a lubricating effect delaying the appearance of the singularity. Below this threshold the presence of the liquid film actually enhances the formation of the singularity.