A moving-wall boundary layer with reverse flow

Abstract

The problem of uniform flow past a flat plate whose surface has a constant velocity λU opposite in direction to that of the mainstream is considered for large values of the Reynolds number R. In a previous communication (Klemp & Acrivos 1972) it was shown that, if the region of reverse flow which is established next to the plate as a consequence of its motion is O(R^−1/2) in thickness, the appropriate laminar boundary-layer equations have a solution provided λ ≤ 0·3541. Here the analysis is extended to the range λ > 0·3541, which cannot be treated using a conventional boundary-layer approach. Specifically, it is found that for λ > 0·3541 the flow consists of three overlapping domains: (a) the external uniform flow; (b) a conventional boundary layer with reverse flow for x_s < x < 1, where x_s, refers to the point of detachment of the ψ = 0 streamline and x = 1 is the trailing edge of the plate; and (c) an inviscid collision region in the neighbourhood of x_s, having dimensions O(R^−1/2) in both the streamwise and the normal direction, within which the reverse moving stream collides with the uniform flow, turns around and then proceeds downstream. It is established furthermore that x_s = 0 for 0 ≤ λ ≤ 1 and that x_s < 0 for λ > 1.

Also, detailed streamline patterns were obtained numerically for various λ's in the range of 0 ≤ λ ≤ 2 using a novel computational scheme which was found to be more efficient than that previously reported. Interestingly enough, the drag first decreased with λ, reached a minimum at λ = 0.3541, and then increased monotonically until, at λ = 2, it was found to have attained essentially the value predicted from the asymptotic λ → ∞ similarity solution available in the literature. Thus it is felt that the present numerical results plus the two similarity solutions for λ = 0 and for λ → ∞ fully describe the high-R steady flow for all non-negative values of λ.