Instability of flow around a rotating, semi-infinite cylinder

Abstract

Stability of flow around a rotating, semi-infinite cylinder placed in an axial stream is investigated. Assuming large Reynolds number, the basic flow is computed numerically as described by Derebail Muralidhar et al. [Proc. R. Soc. London, Ser. A 472, 20150850 (2016)], while numerical solution of the local stability equations allows calculation of the modal growth rates and hence determination of flow stability or instability. The problem has three nondimensional parameters: the Reynolds number Re, the rotation rate S, and the axial location Z. Small amounts of rotation are found to strongly affect flow stability. This is the result of a nearly neutral mode of the nonrotating cylinder which controls stability at small S. Even small rotation can produce a sufficient perturbation that the mode goes from decaying to growing, with obvious consequences for stability. Without rotation, the flow is stable below a Reynolds number of about 1060 and also beyond a threshold Z. With rotation, no matter how small, instability is no longer constrained by a minimum Re nor a maximum Z. In particular, the critical Reynolds number goes to zero as Z → ∞, so the flow is always unstable at large enough axial distances from the nose. As Z is increased, the flow goes from stability at small Z to instability at large Z. If the critical Reynolds number is a monotonic decreasing function of Z, as it is for S between about 0.0045 and 5, there is a single boundary in Z, which separates the stable from the unstable part of the flow. On the other hand, when the critical Reynolds number is nonmonotonic, there can, depending on the choice of Re, be several such boundaries and flow stability switches more than once as Z is increased. Detailed results showing the critical Reynolds number as a function of Z for different rotation rates are given. We also obtain an asymptotic expansion of the critical Reynolds number at large Z and use perturbation theory to further quantify the behavior at small S.