An assessment of theories modeling vortex breakdown as a transition between cylindrical flow states

This study assesses the validity of two theories proposed to explain vortex breakdown occurring in swirling flows in pipes [Benjamin, J. Fluid Mech. 14, 593-629 (1962) and Wang and Rusak, J. Fluid Mech. 340, 177-223 (1997)]. Both model vortex breakdown as a steady, inviscid, streamwise transition between axisymmetric, cylindrical (streamwise-invariant) flow states, with the downstream conjugate state predicted differently by each based on the upstream inflow state. In this study, these conjugate solutions are computed for three distinct swirling inflow profiles by solving the Bragg-Hawthorne equation based on the inflow conditions. It is first shown that the adjacent conjugate solution proposed by Benjamin exhibits stronger flow reversal when the inflow swirl strength is decreased. This is in direct contradiction to trends observed in experiments, indicating that this aspect of the theory is invalid. Following this, the global minimizer conjugate solution proposed by Wang and Rusak is examined. In addition to numerical computations, an analytic expression for this conjugate solution is derived for the case of a Rankine vortex as the inflow. For various inflow profiles, it is shown that these conjugate solutions exhibit many trends similar to those observed in experiments. However, the results also indicate that the stagnation zone associated with these solutions expands radially in an unbounded fashion in the absence of confinement effects, implying that viscous effects might play a crucial role in limiting the radial expansion of the flow. Finally, based on these results and the inverse relationship between the swirl parameter and Mach number, it is argued that modeling vortex breakdown as directly analogous to a gasdynamic normal shock wave is erroneous.