On the bifurcation structure of axisymmetric vortex breakdown in a constricted pipe

The bifurcation structure is presented for an axisymmetric swirling flow in a constricted pipe, using the pipe geometry of Beran and Culick [J. Fluid Mech. 242, 491 (1992)]. The flow considered has been restricted to a two-dimensional parameter space comprising the Reynolds number Re and the relative swirl V ₀ of the incoming swirling flow. The bifurcation diagram is constructed by solving the time-dependent axisymmetric Navier-Stokes equations. The stability of the steady results presented by Beran and Culick, obtained from a steady axisymmetric Navier-Stokes code, has been confirmed. Further, the steady solution branch has also been extended to much larger V₀ values. At larger V₀, a stable unsteady solution branch has been identified. This unsteady branch coexists with the previously found stable steady solution branch and originates via a turning point bifurcation. The bifurcation diagram is of the type described by Benjamin [Proc. R. Soc. London Ser. A 359, 1 (1978)] as the canonical unfolding of a pitchfork bifurcation. This type of bifurcation structure in the two-dimensional parameter space (Re, V₀), suggests the possibility of hysteresis behavior over some part of parameter space, and this is observed in the present study. The implications of this on the theoretical description of vortex breakdown and the search for a criterion for its onset are discussed.