Rayleigh-Bénard Convection in Open and Closed Rotating Cavities

Buoyancy effects can be significant in the rotating annular cavities found between compressor discs in gas-turbine engines, where Rayleigh numbers above $1012$ are common. In some engines, the cavity is “closed” so that the air is confined between four rotating surfaces: two discs and inner and outer cylinders. In most engines, however, the cavity is “open” and there is an axial throughflow of cooling air at the center. For open rotating cavities, a review of the published evidence suggests a Rayleigh–Bénard type of flow structure, in which, at the larger radii, there are pairs of cyclonic and anti-cyclonic vortices. The toroidal circulation created by the axial throughflow is usually restricted to the smaller radii in the cavity. For a closed rotating annulus, solution of the unsteady Navier–Stokes equations, for Rayleigh numbers up to $109$⁠, show Rayleigh–Bénard convection similar to that found in stationary enclosures. The computed streamlines in the $r$-$θ$ plane show pairs of cyclonic and anti-cyclonic vortices; but, at the larger Rayleigh numbers, the computed isotherms suggest that the flow in the annulus is thermally mixed. At the higher Rayleigh numbers, the computed instantaneous Nusselt numbers are unsteady and tend to oscillate with time. The computed time-averaged Nusselt numbers are in good agreement with the correlations for Rayleigh–Bénard convection in a stationary enclosure, but they are significantly higher than the published empirical correlations for a closed rotating annulus.