The effect of slight viscosity on a near-critical swirling flow in a pipe

Abstract

The effect of slight viscosity on a near-critical axisymmetric incompressible swirling flow in a straight pipe is studied. We demonstrate the singular behavior of a regular-expansion solution in terms of the slight viscosity around the critical swirl. This singularity infers that large-amplitude disturbances may be induced by the small viscosity when the incoming flow to the pipe has a swirl level around the critical swirl. It also provides a theoretical understanding of Hall’s boundary layer separation analogy to the vortex breakdown phenomenon. In order to understand the nature of flows in this swirl range, we develop a small-disturbance analysis. It shows that a small but finite viscosity breaks the transcritical bifurcation of solutions of the Euler equations at the critical swirl into two branches of solutions of the Navier–Stokes equations. These branches fold at limit points near the critical swirl with a finite gap between the two branches. This means that no near-columnar equilibrium state can exist for an incoming flow with swirl close to the critical level and the flow must develop large disturbances in this swirl range. Beyond this range, two equilibrium states may exist under the same inlet/outlet conditions. When the flow Reynolds number is decreased this special behavior uniformly changes into a branch of a single equilibrium state for each incoming swirl. We also derive a weakly nonlinear approach to study the effect of slight viscosity on standing waves in a long pipe. This special behavior of viscous solutions shows good agreement with the numerical simulations of the axisymmetric Navier–Stokes equations by Beran and Culick and provides a theoretical understanding of these computations. The relevance of the results to the axisymmetric vortex breakdown in a pipe is also discussed.