A sensitivity study of vortex breakdown onset to upstream boundary conditions

Abstract

This paper theoretically investigates the influence of the upstream boundary conditions on the bifurcation structure leading to vortex breakdown. The axisymmetric flow of an inviscid fluid in a pipe of constant cross-section and finite axial length is considered. Solutions bifurcating from the columnar solution at criticality are analysed via a weakly nonlinear expansion and computed in the fully nonlinear regime using numerical continuation, until a centreline recirculation is found at the pipe outlet. Bifurcation diagrams are determined for a parametric family of inflows describing a wide range of axial and azimuthal profiles, the third inlet condition being chosen either as a fixed azimuthal vorticity or as a vanishing radial velocity. Including the traditional picture given by Wang & Rusak (J. Fluid Mech., vol. 340, 1997a, p. 177), six different diagrams are found to be possible. In particular, a scenario of smooth transition to breakdown may exist as the swirl is increased, with no loss of stability and no hysteresis, breakdown appearing for swirl levels larger than the critical swirl in a pipe. This transition involves a new type of flow akin to a pre-breakdown flow. Our results, furthermore, suggest that rigidly rotating Poiseuille flow could correspond to the limit for which breakdown is impossible because it is predicted at infinitely large swirl numbers. We finally find that flows with a large rotational core are particularly sensitive to an accurate modelling of the upstream boundary conditions, weakly confined vortices being much more robust.