Thermal convection in a co-rotating cylindrical annulus

We investigate thermal convection in a fluid of thermal expansion coefficient $\alpha $, kinematic viscosity $\nu $, thermal diffusivity $\kappa $ in a cylindrical annulus of inner radius $a$ and outer radius $b $with a solid body rotation of angular frequency $\Omega $ and an inward heating with a temperature difference $\Delta $T. The control parameters are $\eta =$ a/b, Pr $= \quad \nu $/$\kappa $ and the Rayleigh number Ra $= \quad \alpha \Delta $T gd$^{\mathrm{3}}$/$\nu \kappa $ where the centrifugal gravity g$_{\mathrm{c}} \quad = \quad \Omega^{\mathrm{2}}$(a$+$b)/2. We adopt the generalized Boussinesq approximation. Linear stability analysis shows that for infinite annulus, the threshold Ra$_{\mathrm{c}}$ decreases with $\eta $ and tends to the value Ra$_{\mathrm{c}}=$1708 when $\eta \to $1 and that critical modes are columnar vortices. Direct numerical simulations using periodic boundary conditions in the axial direction, show that the columnar vortices appear via a supercritical bifurcation. Higher modes of columnar vortices have been determined using the frequency spectra and the Nusselt number for Pr$=$1 and $\eta \quad =$ 0.5: drifting vortices, vacillation modes and chaotic modes have been identified from Ra$=$1700 to Ra$=$10$^{\mathrm{7}}$ The contribution of the centrifugal buoyancy to the variation of the kinetic energy in the flow is analysed.