Viscous instabilities in trailing vortices at large swirl numbers

Abstract

This paper deals with the temporal stability of the $q$-vortex trailing line vortex model. We describe a family of viscous instabilities existing in a range of parameters which is usually assumed to be stable, namely large swirl parameters ($q\,{>}\,1.5$) and large Reynolds numbers. These instabilities affect negative azimuthal wavenumbers ($m\,{<}\,0$) and take the form of centre-modes (i.e. with a structure concentrated along the vortex centreline). They are related to a family of viscous modes described by Stewartson, Ng & Brown (1988) in swirling Poiseuille flow, and are the temporal counterparts of weakly amplified spatial modes recently computed by Olendraru & Sellier (2002). These instabilities are studied numerically using an original and highly accurate Chebyshev collocation method, which allows a mapping of the unstable regions up to $\hbox{\it Re}\,{\approx}\,10^6$ and $q\,{\approx}\,7$. Our results indicate that in the limit of very large Reynolds numbers, trailing vortices are affected by this kind of instability whatever the value of the swirl number.