Instability of a free swirling jet driven by a half–line vortex

Abstract

By studying similarity solutions of the Navier–Stokes equations, which represent swirling jets of a viscous incompressible fluid, we develop a new stability approach, and elucidate the nature of perturbations that cause hysteresis and break axisymmetry. As an example, we consider a jet in an infinite fluid driven by a half–line vortex: a model of a tornado and of a leading–edge vortex above the delta wing of an aircraft. The approach reduces the problem of spatial stability of these strongly non–parallel flows to an ordinary differential system and thereby eases the analysis. We show how non–uniqueness of the solutions appears through cusp and fold catastrophes as the swirl Reynolds number, Res , increases, and find that the fold instability is due to disturbances at the outer boundary of the flow. Also, we study the breaking of axisymmetry due to steady three–dimensional disturbances, and reveal that a helical instability occurs due to disturbances at the inner boundary of the flow. Both the fold and helical instabilities occur for moderate values of Res . Finally, we deduce an amplitude equation, similar to the Ginzburg–Landau equation, to describe the weakly nonlinear spatio–temporal growth of disturbances when Res is slightly above its critical value for linear stability. Thus, our results reveal new features of axisymmetric and helical vortex breakdown in jets.