Linear and nonlinear instabilities in rotating cylindrical Rayleigh-Bénard convection

Linear and nonlinear convection in a rotating annular cylinder, under experimental boundary conditions, heated from below and rotating about a vertical axis are investigated. In addition to the usual physical parameters such as the Rayleigh and Taylor number, an important geometric parameter, the ratio of the inner to outer radius, enters into the problem. For intermediate ratios, linear stability analysis reveals that there exist two countertraveling convective waves which are nonlinearly significant: a retrograde wave located near the outer sidewall and a prograde wave adjacent to the inner sidewall. Several interesting phenomena of nonlinear convection are found: (i) tempospatially modulated countertraveling waves caused by an instability of the Eckhaus-Benjamin-Feir type, (ii) destructive countertraveling waves in which the existence or disappearance of the prograde wave is determined by its relative phase to the retrograde wave, and (iii) a saddle-node-type bifurcation in which the prograde wave takes an infinite amount of time to pass over the retrograde wave.

Figure 1

Geometry of a rotating annular cylinder of depth $d$ with cross section defined by $\left(r_{i}d\right)⩽s⩽\left(r_{o}d\right)$, $s$ being the distance from the rotation axis.

Figure 2

The Rayleigh number at the onset of convective instabilities as a function of the inner radius $r_i$ with a fixed $r_o=2.0$: ${\mathrm{Ra}}_o$ (solid line) represents the Rayleigh number required to excite the retrogradely propagating wave $({\omega }_o>0)$ while ${\mathrm{Ra}}_i$ (dashed line) is for the progradely propagating wave $({\omega }_i<0)$. In all linear solutions, the Taylor number is fixed to be $\mathrm{Ta}={10}^6$ with (a) for $\mathrm{Pr}=7.0$ and (b) for $\mathrm{Pr}=0.7$.

Figure 3

Contours of $u_z$ (a) in the middle plane $z=1∕2$ and (b) in a vertical plane for the retrograde mode concentrating near the outer sidewall with ${\mathrm{Ra}}_o=34406$, $m_o=8$, and ${\omega }_o=4.104$ for $\chi =0.75$, $\mathrm{Ta}={10}^6$, and $\mathrm{Pr}=7.0$.

Figure 4

Contours of $u_z$ (a) in the middle plane $z=1∕2$ and (b) in a vertical plane for the prograde mode concentrating near the inner sidewall with ${\mathrm{Ra}}_i=35667$ with $m_i=7$, ${\omega }_i=-4.053$ for $\chi =0.75$, $\mathrm{Ta}={10}^6$, and $\mathrm{Pr}=7.0$.

Figure 5

A sketch of several grid points in cylindrical coordinates.

Figure 6

Two nonlinear solutions in the form of a steadily retrogradely traveling wave obtained for $r_i=1.5$ and $r_o=2.0$: contours of $u_z$ at the $z=1∕2$ horizonal plane for (a) $\mathrm{Ra}=34600$ and (b) $\mathrm{Ra}=34800$.

Figure 7

A time-dependent nonlinear solution for $r_i=1.5$ and $r_o=2.0$ at $\mathrm{Ra}=38000$: (a) $u_z(s=r_i+0.1)$ and (b) $u_z(s=r_o-0.1)$ at $z=1∕2$ as function of $\varphi$ plotted at five equally spaced instants $t=t_0+(j-1)0.2$, $j=1,2,\dots ,5$.

Figure 8

Patterns of long wavelength modulated convection for $r_i=1.5$ and $r_o=2.0$ at $\mathrm{Ra}=38000$: contours of $u_z$ at the $z=1∕2$ horizonal plane for two different instants.

Figure 9

Two nonlinear solutions for $r_i=1.5$ and $r_o=2.0$: (a) kinetic energies $E_{\mathrm{kin}}$ and (b) $(\mathrm{Nu}-1)$ as a function of time for two different Rayleigh numbers $\mathrm{Ra}=40000$ and 50 000 (from bottom up).

Figure 10

The pattern of nonlinear convection for $r_i=1.5$ and $r_o=2.0$ obtained for $\mathrm{Ra}=40000$ at two different instants: (a) at an instant when $E_{\mathrm{kin}}$ and Nu reach a minimum while (b) shows an instant when $E_{\mathrm{kin}}$ and Nu reach a maximum.

Figure 11

Nonlinear convection in the form of the two-layered steadily retrogradely traveling wave for $r_i=1.5$ and $r_o=2.0$: contours of $u_z$ in the $z=1∕2$ horizonal plane for (a) $\mathrm{Ra}=67500$ and (b) $\mathrm{Ra}=70000$ in a drifting frame.

Figure 12

Three nonlinear solutions for $r_i=1.5$ and $r_o=2.0$: (a) kinetic energies $E_{\mathrm{kin}}$ and (b) $(\mathrm{Nu}-1)$ as a function of time for three different Rayleigh numbers with $\mathrm{Ra}=60000,65000,66000$ (from bottom up).

Figure 13

The time-dependent patterns of nonlinear convection for $r_i=1.5$ and $r_o=2.0$ at $\mathrm{Ra}=65000$, showing contours of $u_z$ at the $z=1∕2$ horizonal plane at six equally spaced instants during a period of the oscillation.

Figure 14

Three nonlinear solutions for $\chi =0.75$ and $r_o=2.0$ at $\mathrm{Pr}=7.0$ (a) kinetic energies $E_{\mathrm{kin}}$ and (b) $(\mathrm{Nu}-1)$ as a function of time for three different Rayleigh numbers: $\mathrm{Ra}=50000,60000,65000$ (from bottom up).