Instability and secondary motion in a rotating channel flow

Abstract

An experiment with the pressure-driven flow down a long rotating channel is described. For zero rotation the flow is quasi-parabolic, laminar, and one-dimensional up the channel. With slight rotation Ω there is a weak double-vortex secondary circulation aligned with the channel. At intermediate Ω there exists an instability in the form of logitudinal rolls of non-dimensional wavenumber 5. The instability disappears at high rotation rates. The general stability problem for a rotating zonal flow $\overline{U}(y)$ is considered theoretically. For perturbations independent of the co-ordinate in the direction of the flow, the problem is exactly analogous to the stability problem of a temperature-stratified fluid where the stratification $\overline{T}_z(z)$ corresponds to the quantity \[ (\partial\overline{U}/\partial y)(1/2\Omega)-1. \] This analogy extends to much more general mean fields (e.g. non-linear or time dependent) than does the oft-quoted analogy between thermal convection and cylindrical Couette flow. The instability theory is in qualitative agreement with the experiment.