The bifurcation structure of viscous steady axisymmetric vortex breakdown with open lateral boundaries

Abstract

The effect of small viscosity on the behavior of the incompressible axisymmetric flow with open lateral and outlet boundaries near the critical swirling number has been studied by numerical simulations and asymptotic analysis. This work extends the theoretical studies of Wang and Rusak and numerical results of Beran and Culik to the case of flow with open lateral and outlet boundaries. In the inviscid limit the columnar flow state constitutes a solution that is known to become unstable at a particular swirl parameter. An asymptotic expansion shows that for small perturbations about this inviscid state an exchange of stability gives rise to a double saddle node bifurcation. The solution of the Euler equations breaks into two branches of the Navier–Stokes equations with a gap between the branches in which no near-columnar flow can exist. Around this region, two steady-state solutions exist for the same boundary conditions, one close to the columnar state and the other corresponding to either an accelerated or a decelerated state. This bifurcation structure is verified by numerical simulations, where the Navier–Stokes solutions are computed using branch continuation techniques based on the recursive projection method. For relatively small Reynolds numbers the numerically computed bifurcation curve does not exhibit any characteristic fold, and thus no hysteresis behavior. In this case, only a single equilibrium solution is found to exist, which changes monotonically from the quasicolumnar state to the breakdown solution. For large Reynolds numbers, however, the numerically determined bifurcation diagram confirms the fold structure characterized by the disappearance of the nearly columnar state via a saddle node bifurcation. Using the minimum axial velocity on the axis as a measure of the flow state we show that the agreement between theory and numerics is asymptotically good.