Transition in the axisymmetric jet

Abstract

The unsteady laminar flow from a point source of momentum is considered. Dimensional considerations lead to a formulation of the problem which is self-similar in time. Three limiting cases are examined. In the limit t → ∞ the solution corresponds to the classic steady solution first discovered by Landau (1944). The limit t → 0 was examined recently by Sozou & Pickering (1977) and was shown to correspond to the flow from an unsteady dipole of linearly increasing strength. More recently Sozou (1979) determined an analytic solution for the creeping flow limit Re → 0. In the present work, unsteady particle trajectories for each of these cases are examined by reducing the particle path equations to an autonomous system with the Reynolds number as a parameter. Transition of the jet is examined as a bifurcation of this system. In the case of the creeping-flow solution, the particle-path pattern exhibits a structure which is not easily discerned in any of the other variables which govern the flow. For sufficiently small Reynolds number the particle paths converge to a single stable node which lies on the axis of the jet. At a Reynolds number of 6·7806 the pattern bifurcates to a saddle lying on the axis of the jet plus two stable nodes lying symmetrically to either side of the axis. At a Reynolds number of 10·09089 the pattern bifurcates a second time to form a saddle and two stable foci.