On boundary-layer flow past two-dimensional obstacles

Abstract

A complete description is sought for the two-dimensional laminar flow response of an incompressible boundary layer encountering a hump on an otherwise smooth boundary. Given that the typical Reynolds number Re (based on the development length L* of the boundary layer) is large, the flow characteristics depend on only two parameters, the non-dimensional length and height scales l, h of the obstacle. For short humps of length less than the familiar O(Re^−⅜) triple-deck size the critical height scale, which produces a nonlinear interaction and hence the prospect of separation, is of order $Re^{-\frac{1}{2}}l^{\frac{1}{3}}$. For long humps whose length is greater than the triple-deck size the corresponding critical height scale is much bigger, of order $l^{\frac{5}{3}}$. Height scales below critical produce only a weak flow response while height scales above critical force relatively large-scale separated motions to occur. In the paper the flow structures and typical solutions produced by two representative cases, a short obstacle of length comparable with the oncoming boundary-layer thickness and a long obstacle of height comparable with the boundary-layer thickness, are mainly considered. The former case is controlled by the unknown pressure force induced locally in the flow near the hump and by two length scales, that of the hump itself and that of the longer triple deck. The latter case is governed mainly by the inviscid externally produced pressure force. Alternatively, however, all the dominant flow properties in both cases can be obtained as special or limiting solutions of the triple-deck problem. Comparisons between the cases studied are also presented.