On the flow past a quarter infinite plate using Oseen's equations

Abstract

This paper is concerned with the determination, on the basis of Oseen's equations, of the flow past a quarter-plane with its leading edge normal to, and its side edge parallel to, a uniform incident stream. The solution is completed, except for a region in the vicinity of the corner, correct to orderv^1/2for small kinematic viscosity ν.Away from the vicinity of the side edge the flow will approximate to the two-dimensional flow past a semi-infinite plate. This two-dimensional flow can be built up successively, if we like to think in terms of boundary conditions at the plate rather than at the edge of the boundary layer, from the potential flow associated with the uniform stream, a shear layer introduced to remove the tangential slip and a potential flow to remove the normal velocity at the plate associated with the shear layer. In the vicinity of the plate the three together give the usual boundary-layer solution.We start our solution from this same basis, namely, the potential flow associated with the uniform stream and the shear layer to restore the no-slip condition. As a first approximation, neglecting the effects of the edges, this will be the same as for the two-dimensional problem. The normal velocity introduced by this shear layer has to be compensated by a potential flow (see §4). This potential flow in turn (and here our problem diverges significantly from the two-dimensional problem) introduces tangential velocities with components parallel to both leading and side edges which require the introduction of a further shear layer. Over the main body of the plate this secondary shear layer is of a conventional form (§5) but requires special examination near the edges. In §6 it is shown how Carrier & Lewis's (1949) solution can be modified to give the flow near the leading edge away from the tip and in §7, the core of the paper, the flow near the side edge is determined.In the vicinity of the side edge the extra potential flow has no component in the direction of that edge and so the solution given by Howarth (1950) for the corresponding unsteady problem is applicable. What emerges from the present calculations, however, is that Howarth's application of Rayleigh's analogy to give the excess skin friction is seriously incomplete. For, whilst this argument gives correctly the local increase of order ν in skin friction in the immediate vicinity of the side edge, it omits the widespread effects of the secondary shear layer. These are found to be of the same order in ν as the local effects.The cross-flow in the side edge region has features of special interest. Its determination depends on a knowledge of the potential flow associated with the primary shear layer and so it depends, for instance, on the shape of the leading edge and is not, as appears to have been assumed up to now, determined completely by local conditions. This is further exemplified by the fact that it cannot be expressed in terms of what would be regarded as the natural boundary-layer variables but involves quite separately the distance from the leading edge.