Non-uniqueness and bifurcation in annular and planar channel flows

Abstract

High-Reynolds-number steady flow in an annular pipe which encounters a shallow axisymmetric expansion or indentation in the walls is studied using interactive boundary-layer theory. The flow upstream of the indentation (x < 0) is fully developed; the ratio of the shear rate on the outer wall to that on the inner wall is denoted by ρ (0 < ρ < 1): similarity solutions are found for the case where the wall perturbations are proportional to . The solution is unique in a constriction, when the pressure gradient (represented by a parameter b) is favourable (b < 0). In an expansion, however, with an adverse pressure gradient, three different solutions are found if b exceeds a critical value bc. When ρ ≠ 1, one of these solutions, representing a flow that is attached on the inner wall and separated (i.e. has negative wall shear) on the outer, is a continuation of the unique doubly attached flow at small b. The other two, one separated on the inner and not the outer wall and the other separated on both walls, arise from a saddle-node bifurcation at b = bc. The doubly separated flow is never stable, as observed in diffusers. In the case of a planar channel (ρ = 1) symmetry is restored, and the non-uniqueness arises through a supercritical pitchfork bifurcation. This agrees with previous computations on channel flow, but not with Jeffery-Hamel flow, for which the bifurcation is subcritical.