Inviscid instability of an incompressible flow between rotating porous cylinders to three-dimensional perturbations

TL;DR

This paper investigates the stability of inviscid flow between rotating porous cylinders, revealing conditions under which three-dimensional perturbations cause instability, and demonstrates that viscous effects have a limited impact on the instability at high Reynolds numbers.

Contribution

It proves the stability of purely radial flow and axisymmetric perturbations, and analyzes the onset of three-dimensional instability in flows with azimuthal components, including viscous corrections.

Findings

Purely radial flow is stable.

Flows with azimuthal components are unstable to 3D perturbations when azimuthal velocity is large.

Viscous corrections are small at high Reynolds numbers, indicating real flow instability.

Abstract

We study the stability of two-dimensional inviscid flows in an annulus between two porous cylinders with respect to three-dimensional perturbations. The basic flow is irrotational, and both radial and azimuthal components of the velocity are non-zero. The direction of the radial flow can be from the inner cylinder to the outer one (the diverging flow) or from the outer cylinder to the inner one (the converging flow). It had been shown earlier in Ref. \cite{IM2013a} that, independent of the direction of the radial flow, the basic flow can be unstable to small two-dimensional perturbations. In the present paper, we prove first that purely radial flow is stable and that flows with both radial and azimuthal components are always stable to axisymmetric perturbations. Then we show that both the diverging and converging flows are unstable with respect to non-axisymmetric three-dimensional perturbations provided that the ratio of the azimuthal component of the velocity to the radial one is sufficiently large. Neutral curves in the space of parameters of the problem are computed and it is demonstrated that for any ratio of the radii of the cylinders, the most unstable modes (corresponding to the smallest ratio of the azimuthal velocity to the radial one) are the two-dimensional ones. We also consider the corresponding viscous stability problem and construct an asymptotic expansion of its solutions for large radial Reynolds numbers. We compute the first-order viscous correction to inviscid eigenvalues and show that the asymptotic results give a good approximation to the viscous eigenvalues even for moderate values of radial Reynolds number, which indicates that the instability may be observed in real flows.