Optimal growth of the Batchelor vortex viscous modes

Abstract

We present calculations of optimal linear growth in the Batchelor (or q) vortex. The level of transient growth is used to quantify the effect of the viscous centre modes found at large Reynolds number and large swirl. The viscous modes compete with inviscid-type transients, which are seen to provide faster growth at short times. Following a smooth transition, the viscous modes emerge as dominant in a different regime at later times. A comparison is drawn with two-dimensional shear flows, such as boundary layers, in which weak instability modes (Tollmien–Schlichting waves) also compete with inviscid transients (streamwise streaks). We find the competition to be more evenly balanced in the Batchelor vortex, because the inviscid transients are damped faster in a swirling jet than a two-dimensional shear flow, so that despite their weak growth rates the viscous modes may be relevant in some situations.