The Nonlinear Stability of Vortex Flows and Vortex Breakdown

Abstract

On the condition of zero gradient in z direction$(\partial \,/\,\partial \,z = 0)$, a new model is proposed for the study on the stability of vortex motion. One beneficial character of the model is that two of the three vorticity transport equations, in fact, become linear, but the stretching of vorticity is still an essential mechanism. In this way the complexity of the stability of vortex motion is much simplified so that the nonlinear instability of vortex motion can be easily studied both analytically and numerically. We find that the deceleration of the axial velocity, which is a characteristic of the approach flow for vortex breakdown, is caused by the nonlinear instability which leads to the production of a negative azimuthal component of vorticity. The results show that the nonlinear term:

$(\frac{{V'_r }}{r} + \frac{1}{r}\frac{{\partial V'_\theta }}{{\partial \theta }})\Omega '_\theta$

appearing in the governing equations of perturbation plays a crucial role in the stability of vortex motion. It is conjectured that the nonlinear term is also crucial for fully three-dimensional vortex motion. In the light of the nonlinear instability, it seems that the vortex breakdown is a kind of unsteady separation phenomenon in three-dimensional flows, where the separation criterion can be extended from the MRS criterion in two dimensions.