A structure for laminar flow past a bluff body at high Reynolds number

Abstract

The steady planar symmetric motion of an incompressible fluid, past a symmetric bluff body fixed in an otherwise uniform stream, is considered for large Reynolds numbers Re. A laminar-flow structure is proposed which consists primarily of (a) the large-scale flow and (b) the smaller, body-scale, flow. Here (a) involves a pair of massive, effectively inviscid, recirculating eddies set up behind the body and bounded by viscous shear layers. Each eddy has small constant vorticity and its length and width both increase linearly with Re, so that the large-scale potential flow outside the eddies is significantly disturbed from the oncoming stream. This reduces the effective free stream acting on (b). The latter has the Kirchhoff property of a parabolic growth in the eddy width downstream; but its eddy vorticity is non-uniform and substantial, contrary to the Kirchhoff and Prandtl–Batchelor models, and secondary separation is possible. The non-uniform vorticity is provoked by the thick return jet, which is forced back along the centreline in (a) from downstream. Buffer zones, e.g. of length ∝ Re^½, are required to join (b) fully to (a). The resulting drag coefficient c_D is believed to be O(1) generally, and is controlled, along with the eddy length and vorticity, by a combination of the viscous–inviscid flow problems posed in both (a) and (b). A special case of small c_D is also covered. The structure seems self-consistent so far, and tends to compare reasonably well with recent numerical solutions of the Navier–Stokes equations at increased Re.

An Appendix describes the inviscid parts of (a) for relatively thin eddies.