Rayleigh-Bénard convection in a vertical annular container near the convection threshold

The instabilities and transitions of flow in an annular container with a heated bottom, a cooled top, and insulated sidewalls are studied numerically. The instabilities of the static diffusive state and of axisymmetric flows are investigated by linear stability analysis. The onset of convection is independent of the Prandtl number but determined by the geometry of the annulus, i.e., the aspect ratio $\Gamma$ (outer radius to height) and radius ratio $\delta$ (inner radius to outer radius). The stability curves for onset of convection are presented for $0.001\le \delta \le 0.8$ at six fixed aspect ratios: $\Gamma =1$, 1.2, 1.6, 1.75, 2.5, and 3.2. The instability of convective flow (secondary instability), which depends on both the annular geometry and the Prandtl number, is studied for axisymmetric convection. Two pairs of geometric control parameters are chosen to perform the secondary instability analysis—$\Gamma =1.2$, $\delta =0.08$ and $\Gamma =1.6$, $\delta =0.2$—and the Prandtl number ranges from 0.02 to 6.7. The secondary instability exhibits some similarities to that for convection in a cylinder. A hysteresis stability loop is found for $\Gamma =1.2$, $\delta =0.08$ and frequent changes of critical mode with Prandtl number are found for $\Gamma =1.6$, $\delta =0.2$. The three-dimensional flows beyond the axisymmetry-breaking bifurcations are obtained by direct numerical simulation for $\Gamma =1.2$, $\delta =0.08$.

Figure 1

Configuration of the annular system.

Figure 2

Variation of critical Rayleigh number for the primary thresholds with radius ratio at six aspect ratios: $\Gamma =1$, 1.2, 1.6, 1.75, 2.5, and 3.2. The results for radius ratio in the range $0.001\le \delta \le 0.4$ are shown in (a), (c), (e), and in the range $0.4\le \delta \le 0.8$ are shown in (b), (d), (f). The instability modes are all steady. Upward triangles indicate $m=1$ modes at the transition; circles, $m=2$ modes; squares, $m=3$ modes; diamonds, $m=4$ modes; downward triangles, $m=5$ modes; circles with crosses, $m=6$ modes; circles with slashes, $m=7$ modes; rightward triangles, $m=8$ modes; leftward triangles, $m=9$ modes; thick solid lines, $m=0$ modes.

Figure 3

Variation of renormalized critical azimuthal wave number ($m/\Gamma$; the $m=0$ mode is not included) for primary thresholds with radius ratio at six aspect ratios: $\Gamma =1$, 1.2, 1.6, 1.75, 2.5, and 3.2.

Figure 4

Axisymmetric base flow at $\Gamma =1.2$, $\delta =0.08$, $\mathrm{Pr}=0.02$, and $\mathrm{Ra}=2400$. Left: Stream function contours. Right: Isotherms.

Figure 5

Stability curves for three-dimensional instability of axisymmetric primary flow. (a) $\Gamma =1.2$. Dashed line, $\delta =0$; solid line, $\delta =0.08$. (b) $\Gamma =1.6$, $\delta =0.2$. The range for the Prandtl number is $0.02\le \mathrm{Pr}\le 6.7$. Curves with open symbols represent steady transitions; curves with filled symbols, oscillatory transitions. Stars indicate $m=1$ modes at the transition; open circles, $m=2$ modes; open squares, $m=3$ modes; circles with slashes, $m=7$ modes; open rightward triangles, $m=8$ modes; open leftward triangles, $m=9$ modes; open diamonds, $m=10$ modes; open downward triangles, $m=11$ modes; circles with $\mathsf{x}$'s, $m=12$ modes; circles with crosses, $m=13$ modes; filled circles, $m=2$ modes; filled squares, $m=3$ modes; filled diamonds, $m=4$ modes; filled downward triangles, $m=5$ modes; and filled upward triangles, $m=6$ modes.

Figure 6

Variation of critical angular frequency ${\omega }_{\mathrm{cr}}$ with Prandtl number Pr. Filled squares represent $m=3$ modes; filled diamonds, $m=4$ modes; filled downward triangles, $m=5$ modes; filled upward triangles, $m=6$ modes.

Figure 7

Contours of vertical velocity on the $z=0.5$ plane at $\Gamma =1.2$, $\delta =0.08$, $\mathrm{Pr}=0.02$. (a) $\mathrm{Ra}=2400$; (b) $\mathrm{Ra}=5000$. Ten levels are drawn between the values of $-0.35$ and 0.25 (a) and between the values of $-5$ and 5 (b). Dark area (solid lines) represents ascending fluid, and light area (dashed lines) represents descending fluid.

Figure 8

Contours of vertical velocity on the $z=0.5$ plane in a period at $\Gamma =1.2$, $\delta =0.08$, $\mathrm{Pr}=0.02$, $\mathrm{Ra}=6400$. The oscillation period $T=2.14$, and patterns at $t=0$, 0.15, 0.3, 0.45, 0.64, and 0.81 T are plotted. Ten levels are drawn between the values of $-4.5$ and 5.8 in each figure. Dark area (solid lines) represents ascending fluid and light area (dashed lines) represents descending fluid.

Figure 9

Contours of vertical velocity on the $z=0.5$ plane in a period at $\Gamma =1.2$, $\delta =0.08$, $\mathrm{Pr}=0.2$. (a) $\mathrm{Ra}=23000$, oscillation period $T=0.318$, and patterns at $t=0$, 0.15, 0.3, 0.45, 0.64, and 0.81 T are plotted. (b) $\mathrm{Ra}=46000$, oscillation period $T=0.092$, and patterns at $t=0$, 0.16, 0.32, 0.5, 0.68, and 0.84 T are plotted. Ten levels are drawn between the values of $-28$ and 25 (a) and between the values of $-50$ and 50 (b). Dark area (solid lines) represents ascending fluid and light area (dashed lines) represents descending fluid.

Figure 10

Contours of vertical velocity on the $z=0.5$ plane at $\Gamma =1.2$, $\delta =0.08$, $\mathrm{Pr}=0.7$. (a) $\mathrm{Ra}=25000$; (b) $\mathrm{Ra}=50000$. Ten levels are drawn between the values of $-50$ and 25 (a) and between the values of $-70$ and 70 (b). Dark area (solid lines) represents ascending fluid and light area (dashed lines) represents descending fluid.

Figure 11

Contours of vertical velocity on the $z=0.5$ plane in a period at $\Gamma =1.2$, $\delta =0.08$, $\mathrm{Pr}=0.02$, $\mathrm{Ra}=6400$. Oscillation period $T=0.0794$, and patterns at $t=0$, 0.15, 0.3, 0.45, 0.64, and 0.81 T are plotted. Ten levels are drawn between the values of $-75$ and 75 in each figure. Dark area (solid lines) represents ascending fluid and light area (dashed lines) represents descending fluid.