Flow along streamwise corners revisited

Abstract

In this paper we investigate the three-dimensional laminar incompressible steady flow along a corner formed by joining two similar quarter-infinite unswept wedges along a side-edge. We show that a four-region construction of the potential flow arises naturally for this flow problem, the formulation being generally valid for a corner of an arbitrary angle (π−2α), including the limiting cases of semi- and quarter-infinite flat-plate configurations. This construction leads to five distinct three-dimensional boundary-layer regions, whereby both the spanwise length and velocity scales of the blending intermediate layers are O(δ), with Re−1/2 [Lt ] δ [Lt ] 1, Re being a reference Reynolds number supposed to be large. This reveals crucial differences between concave and convex corner flows. For the latter flow regime, the corner-layer motion is shown to be mainly controlled by the secondary flow which effectively reduces to that past sharp wedges with solutions being unique and existing only for favourable streamwise pressure gradients. In this regime, the corner-layer thickness is shown to be O(Re−0.5+α/π/δ2α/π), −½π [les ] α [les ] 0, which is much smaller than O(Re−1/2) for concave corner flows.Crucially, our numerical results show conclusively that, for α ≠ 0, closed streamwise symmetrically disposed vortices are generated inside the intermediate layers, confirming thus the prediction made by Moore (1956) for a rectangular corner, which has so far remained unconfirmed in the literature.For almost planar corners, three-dimensional corner boundary-layer features are shown, as in (Smith 1975), to arise when α ∼ O(1/ln Re). On the other hand, we show that the flow past a quarter-infinite flat plate would be attained when both values of the streamwise pressure gradient and external corner angle (π+2α) become O(1/ln Re) or smaller.Numerical results for all these flow regimes are presented and discussed.