Numerical study of thermal convection induced by centrifugal buoyancy in a rotating cylindrical annulus

Thermal convection in a cylindrical annulus with solid-body rotation and subject to an inward radial heating is investigated by the linear stability analysis and direct numerical simulation using the periodic boundary conditions. The Archimedean buoyancy is neglected in order to investigate the properties of thermal convection induced solely by the centrifugal buoyancy. The Coriolis buoyancy, which is usually neglected in similar studies, is included in the flow equations. The critical state occurs in the form of columnar vortices drifting in the retrograde direction. The critical parameters of these convective columnar vortices are determined for different values of the radius ratio and the Prandtl number (Pr). Instantaneous flow and temperature fields are computed for $\mathrm{Pr}=1$ in order to investigate higher instability modes and chaotic states. The kinetic energy and the coefficient of heat transfer by columnar vortices are evaluated for a large range of values of the Rayleigh number.

Figure 1

Flow geometry: A cylindrical annulus in solid-body rotation about the $z$ axis with inward heating $\left(T_o>T_i\right)$.

Figure 2

Variation of the critical Rayleigh number $\left(\mathrm{R}{\mathrm{a}}_c\right)$ with the radius ratio (η).

Figure 3

Variations of (a) the critical wave number $q_c$ and (b) the critical frequency multiplied by $\mathrm{Pr}/\tau {\gamma }_a$ with the radius ratio (η) for $\mathrm{Pr}=10$ and 100.

Figure 4

Grid system for $\eta =0.5$ (every other grid point is plotted in each direction for clarity); (a) $r$-φ plane, (b) $r\text{−}z$ plane. Here, $x=r-\eta /(1-\eta )$.

Figure 5

Instantaneous flow and temperature fields on the $r$-φ plane for $\mathrm{Ra}=1811\left(\epsilon =0.019\right)$, $\mathrm{Pr}=1,{\gamma }_a={10}^{-2}$, and $\eta =0.5$; (a) contour of axial vorticity $\left({\omega }_z\right)$, (b) contour of temperature, (c) vector plot of the velocity field and contour of pressure. Here, $\epsilon =(\mathrm{Ra}-\mathrm{R}{\mathrm{a}}_c)/\mathrm{R}{\mathrm{a}}_c$ is the criticality.

Figure 6

Instantaneous flow and temperature fields on the $r$-φ plane for $\mathrm{Ra}=1755\left(\epsilon =0.021\right)$, $\mathrm{Pr}=1,{\gamma }_a={10}^{-2}$, and $\eta =0.8$; (a) contour of axial vorticity $\left({\omega }_z\right)$, (b) contour of temperature, (c) contour of pressure.

Figure 7

Time variations of velocity and temperature for $\mathrm{Ra}=1811\left(\epsilon =0.019\right)$, $\mathrm{Pr}=1,{\gamma }_a={10}^{-2}$, and $\eta =0.5$; (a) space-time diagram of radial velocity component along the axial direction at $\left(x=0.5,\phi =\pi ,z\right)$, (b) space-time diagram of temperature along the azimuthal direction at $\left(x=0.5,\phi ,z=\mathrm{\Gamma }/2\right)$, (c) time histories of temperature (red line) and radial velocity component (blue line) at $\left(x=0.5,\phi =\pi ,z=\mathrm{\Gamma }/2\right)$, (d) corresponding time frequency spectrum of the temperature where $f_d=6.18\times 10^{-2}$, (e) phase portrait of $\left(u_r,u_{\phi }\right)$ at $\left(x=0.5,\phi =\pi ,z=\mathrm{\Gamma }/2\right)$.

Figure 8

Perturbation mode for $\mathrm{Pr}=1,{\gamma }_a={10}^{-2}$, and $\eta =0.5$; (a) growth and saturation phases of the perturbation amplitude for $\mathrm{Ra}=1811$: there is a strong decrease near the time origin because of the random noise; (b) growth rate as function of Ra.

Figure 9

Steady wave state for $\mathrm{Ra}=3.375\times {10}^4$; (a) cross section of the axial vorticity, (b) cross section of the temperature field, (c) cross section of the pressure. Temporal characteristics of the steady waves (SW); (d) time signals of radial (red line) and azimuthal (blue line) velocity components at $\left(x=0.5,\phi =\pi ,z=\mathrm{\Gamma }/2\right)$, (e) time variations of the temperature (red line) at $\left(x=0.5,\phi =\pi ,z=\mathrm{\Gamma }/2\right)$ and the Nusselt number (blue line) on the inner cylinder, (f) corresponding frequency spectrum of the temperature, (g) phase portrait of $\left(u_r,u_{\phi }\right)$ at $\left(x=0.5,\phi =\pi ,z=\mathrm{\Gamma }/2\right)$.

Figure 10

Radial profile of (a) the mean azimuthal velocity $\left(u_{\phi }\right)$, (b) the mean temperature for different values of Ra, for $\eta =0.5,\mathrm{Pr}=1$, and ${\gamma }_a={10}^{-2}$.

Figure 11

Quasiperiodic state for $\mathrm{Ra}=1.35\times {10}^5$; (a) cross section of the axial vorticity, (b) cross section of the temperature, (c) cross section of the pressure. Temporal characteristics of the amplitude vacillation (AV); (d) time signal of radial (red line) and azimuthal (blue line) velocity components at $\left(x=0.5,\phi =\pi ,z=\mathrm{\Gamma }/2\right)$, (e) time signals of the temperature (red line) at $\left(x=0.5,\phi =\pi ,z=\mathrm{\Gamma }/2\right)$ and the Nusselt number (blue line) on the inner cylinder, (f) corresponding frequency spectrum of the temperature, (g) corresponding frequency spectrum of the Nusselt number, (h) phase portrait of $\left(u_r,u_{\phi }\right)$ at $\left(x=0.5,\phi =\pi ,z=\mathrm{\Gamma }/2\right)$.

Figure 12

Modulated amplitude vacillation for $\mathrm{Ra}=3.75\times {10}^5$; (a) cross section of the axial vorticity, (b) cross section of the temperature, (c) cross section of the pressure. Temporal characteristics of the modulated amplitude vacillation (MAV); (d) time signals of radial (red line) and azimuthal (blue line) velocity components at $\left(x=0.5,\phi =\pi ,z=\mathrm{\Gamma }/2\right)$, (e) time signal of the temperature (red line) at $\left(x=0.5,\phi =\pi ,z=\mathrm{\Gamma }/2\right)$ and the Nusselt number (blue line) on the inner cylinder, (f) corresponding time frequency spectrum of the temperature, (g) corresponding frequency spectrum of the Nusselt number, (h) phase portrait of $\left(u_r,u_{\phi }\right)$ at $\left(x=0.5,\phi =\pi ,z=\mathrm{\Gamma }/2\right)$.

Figure 13

Chaotic flow for $\mathrm{Ra}=3.375\times {10}^6$; (a) cross section of the axial vorticity, (b) cross section of the temperature, (c) cross section of the pressure. Temporal characteristics of the chaotic flow regime; (d) time signals of radial (red line) and azimuthal (blue line) velocity components at $\left(x=0.5,\phi =\pi ,z=\mathrm{\Gamma }/2\right)$, (e) time signals of the temperature (red line) at $\left(x=0.5,\phi =\pi ,z=\mathrm{\Gamma }/2\right)$, and the Nusselt number (blue line) on the inner cylinder, (f) corresponding frequency spectrum of the temperature, (g) corresponding frequency spectrum of the Nusselt number, (h) phase portrait of $\left(u_r,u_{\phi }\right)$ at $\left(x=0.5,\phi =\pi ,z=\mathrm{\Gamma }/2\right)$.

Figure 14

Flow regime diagrams for $\mathrm{Pr}=1,{\gamma }_a={10}^{-2}$, and $\eta =0.5$; (a) variation of the number of pairs of columnar vortices vs Ra; the solid line is the marginal stability curve; (b) variation of the frequency vs Ra.

Figure 15

Radial profiles of different terms of the kinetic energy balance equation for $\eta =0.5,\mathrm{Pr}=1$, and ${\gamma }_a={10}^{-2}$.

Figure 16

Variation of the mean kinetic energy with Ra for $\eta =0.5,\mathrm{Pr}=1$, and ${\gamma }_a={10}^{-2}$.

Figure 17

Heat transfer for $\eta =0.5,\mathrm{Pr}=1$, and ${\gamma }_a={10}^{-2}$; (a) profiles of averaged radial heat current across the cylindrical surface of radius $r$ for different values of Ra $(x$: see caption of Fig. 4), (b) variation of the Nusselt number with Ra $(+$: comparison with King et al. [32]); inset gives the linear growth near the threshold.

Figure 18

Cross sections of the temperature field and pressure for $\mathrm{Ra}=2.4\times {10}^5$ (a), (b) without and (c), (d) with Coriolis buoyancy terms.

Figure 19

Isosurfaces of 3D vortical structure for $\mathrm{Pr}=1,{\gamma }_a={10}^{-2}$, and $\eta =0.5$; (a) $\mathrm{Ra}=1811\left(Q={10}^{-4}\right)$, (b) $\mathrm{Ra}=1.35\times {10}^5\left(Q=0.004\right)$, (c) $\mathrm{Ra}=2.4\times {10}^5\left(Q=0.004\right)$, (d) $\mathrm{Ra}=3.75\times {10}^5\left(Q=0.004\right)$, (e) $\mathrm{Ra}=3.375\times {10}^6\left(Q=0.01\right)$, (f) $\mathrm{Ra}=9.375\times {10}^6\left(Q=0.01\right)$.

Figure 20

Instantaneous contours of axial vorticity $\left({\omega }_z\right)$ on $r$-φ plane for $\mathrm{Pr}=1,{\gamma }_a={10}^{-2}$, and $\eta =0.5$.

Figure 21

Instantaneous contours of temperature on $r$-φ plane for $\mathrm{Pr}=1,{\gamma }_a={10}^{-2}$, and $\eta =0.5$.

Figure 22

Instantaneous vector plots and contours of pressure (every other vector is plotted in each direction for clarity) on the $r$-φ plane for $\mathrm{Pr}=1,{\gamma }_a={10}^{-2}$, and $\eta =0.5$.