A reversed-flow singularity in interacting boundary layers

Abstract

It is proposed that a singularity can occur in interacting boundary layers when reversed flow is present. The theoretical breakdown/singularity takes place in general at a critical point between separation and reattachment and it involves viscous-inviscid interaction between the sublayers that develop on either side of the critical point, because of a streamfunction behaviour like normal distance to the power K near the wall and possibly in mid-eddy. Moreover, the breakdown is for a value of the controlling parameter not much beyond the value that first produces separation. A study of the acceptable K range tends to suggest that the singularity arises either for the value K = 3/2, corresponding to the local pressure gradient and skin friction being singular like streamwise distance to the powers -3/5 and -1/5 respectively, subject to logarithmic modifications. or, more likely, for the limit value K = 1, which is associated with a discontinuity appearing in the pressure and velocity profile. Comparisons with interactive-boundary-layer and Navier-Stokes computations seem fairly supportive. The occurrence of the singularity offers an explanation for the appearance of a pronounced second minimum in numerous computed skin-friction plots, the numerical difficulties encountered in all reversed-flow computations, and the start of airfoil stall. The theory of the breakdown (which is modified for unsteady motions in an allied paper) also has widespread application, being relevant to all the interactive separating flows known to date, including internal and external two-dimensional or three-dimensional boundary layers and wakes.