The primary flow transition in a differentially heated rotating channel of fluid with O(2) symmetry

Abstract

The primary flow transition in a periodic differentially heated rotating channel of fluid with O(2) symmetry is studied. This transition occurs when a time-independent flow that is uniform along the channel bifurcates to a stationary wave flow. The fluid is modelled using the Navier–Stokes equations in the Boussinesq approximation and the flow transition is found using linear stability analysis. The computation of the flow transition curve is performed efficiently by replacing the relevant eigenvalue problem with an equivalent bordered linear system, and by implementing a pseudoarclength continuation strategy that is appropriate for large-scale systems. The dynamics of the fluid near the transition are deduced by applying centre manifold reduction and normal form theory. The reduction produces analytical expressions for the normal form coefficients in terms of functions that must be computed numerically.The results indicate that the transition to stationary wave flow occurs via a supercritical pitchfork bifurcation to a group orbit, sometimes referred to as a pitchfork of revolution. Furthermore, at several points along the transition curve two such pitchfork bifurcations occur simultaneously, which physically corresponds to the interaction of two stationary wave modes. An analysis of the normal form equations that are associated with the steady-state mode interactions shows the possibility of bistability and hysteresis of the stationary waves. Many of the results obtained in the channel model show a remarkable quantitative similarity to those of theoretical and experimental studies of analogous experiments using a cylindrical annulus, even though the difference in the symmetries of the systems ensure certain qualitative differences. This suggests that the dynamics of the fluid are dominated by the differential heating and the rotation, and not by the curvature of the system.