Influence of counter-rotating von Kármán flow on cylindrical Rayleigh-Bénard convection

The axisymmetric flow in an aspect-ratio-one cylinder whose upper and lower bounding disks are maintained at different temperatures and rotate at equal and opposite velocities is investigated. In this combined Rayleigh-Bénard/von Kármán problem, the imposed temperature gradient is measured by the Rayleigh number Ra and the angular velocity by the Reynolds number Re. Although fluid motion is present as soon as $\text{Re}\ne 0$, a symmetry-breaking transition analogous to the onset of convection takes place at a finite Rayleigh number higher than that for $\text{Re}=0$. For $\text{Re}<95$, the transition is a pitchfork bifurcation to a pair of steady states, while for $\text{Re}>95$, it is a Hopf bifurcation to a limit cycle. The steady states and limit cycle are connected via a pair of saddle-node infinite-period bifurcations except very near the Takens-Bogdanov codimension-two point, where the scenario includes global bifurcations. Detailed phase portraits and bifurcation diagrams are presented, as well as the evolution of the leading part of the spectrum, over the parameter ranges $0\le \text{Re}\le 120$ and $0\le \text{Ra}\le 30000$.

Figure 1

Cylinder with height and radius equal to one. The bottom disk is heated and rotated in the clockwise direction, while the top disk is cooled and rotated in the counterclockwise direction.

Figure 2

(Color online) Convection thresholds as a function of Reynolds number. Solid blue curve: transition to steady convection. Dashed red curve: transition to oscillatory convection. Dotted black curve: fit to even fourth-order polynomial [Eq. (9)]. Points: thresholds obtained from time-dependent code. Letters indicate parameter values for flows presented in Figs. 3, 4 (B, basic state), Figs. 5, 6 (S, steady convection), and Figs. 7, 8 (O, oscillatory convection).

Figure 3

(Color online) Basic flow at $\text{Re}=40$, $\text{Ra}=2000$. From left to right: $U_{\theta }$, $\Psi$, and t $T$. Ranges for $U_r$ and $U_z$ are $\left[-0.05,0.05\right]$ and $\left[-0.04,0.04\right]$, respectively.

Figure 4

(Color online) Basic flow at $\text{Re}=90$, $\text{Ra}=8000$. From left to right: $U_{\theta }$, $\Psi$ and $T$. Ranges for $U_r$ and $U_z$ are $\left[-0.11,0.11\right]$ and $\left[-0.07,0.07\right]$, respectively.

Figure 5

(Color online) Convective state at $\text{Re}=40$, $\text{Ra}=4000$. From left to right: $U_{\theta }$, $\Psi$, and $T$. Ranges for $U_r$ and $U_z$ are $\left[-0.11,0.19\right]$ and $\left[-0.29,0.11\right]$, respectively.

Figure 6

(Color online) Convective state at $\text{Re}=90$, $\text{Ra}=12500$. From left to right: $U_{\theta }$, $\Psi$, and $T$. Ranges for $U_r$ and $U_z$ are $\left[-0.15,0.18\right]$ and $\left[-0.24,0.10\right]$, respectively.

Figure 7

Near-sinusoidal limit cycle at $\text{Ra}=20000$, $\text{Re}=110$. $\Psi$ is shown at times near 0, $\tau /8$, $\tau /4$, $\tau /2$, $3\tau /4$, $\tau$ indicated on time series in Fig. 9. Times are given in units of $1/\Omega$. The overall limit cycle has reflection symmetry: the states in the second half of the cycle are related by reflection symmetry to those in the first half of the cycle. The vortex in the upper right corner is smaller during the first half of the cycle and larger during the second half. The streamfunction contours are not equally spaced, but instead chosen to illustrate topological features of the flow.

Figure 8

Near-heteroclinic limit cycle at $\text{Ra}=20000$, $\text{Re}=63$. $\Psi$ is shown at times indicated on time series in Fig. 9: $t=22.6$; four instants in $68\le t\le 73$, during which the flow changes a great deal; and $t=86.5$, approximately a half-period $\tau /2\approx 57.5$ after $t=22.6$. The vortex in the lower right corner grows and the vortex in the upper left corner folds and then divides into three small vortices. Two of these disappear, leaving a small vortex in the upper right corner. As in Fig. 7, the states in the second half of the cycle are related by reflection symmetry to those in the first half of the cycle and the streamfunction contours are chosen to illustrate topological features of the flow.

Figure 9

Time series (left) and phase portraits (right) corresponding to limit cycles at $\text{Ra}=20000$. Dots in time series refer to visualizations in Figs. 7, 8. Dots in phase portraits are plotted at equally spaced times so that their density reflects the rate at which the limit cycle is traversed. Top: at $\text{Re}=110$, the oscillations are regular and near-sinusoidal. Bottom: at $\text{Re}=63$, the oscillations have a much longer period and consist mainly of two long plateaus with abrupt gradients between them.

Figure 10

Variation in square frequency with Re. Left: $1/{\tau }^2$ for $\text{Ra}=20000$ (hollow dots) and for $\text{Ra}=18000$ (solid dots). Right: ${\left(\text{Re}/\tau \right)}^2$ for $\text{Re}=20000$ (hollow dots) and for $\text{Ra}=18000$ (solid dots).

Figure 11

(Color online) Curves of bifurcation points in the (Re,Ra) plane showing first pitchfork (blue, solid, ${\text{PF}}_1$), second pitchfork (green, short-dashed, ${\text{PF}}_2$), Hopf (red, long-dashed, $\text{H}$), and saddle-node (black, dotted, SN) bifurcations. Dots indicate codimension-two points, where curve ${\text{PF}}_2$ meets curves SN $\left(90,14300\right)$ and $\text{H}$ $\left(95,11750\right)$. An enlargement of the region inside the square is shown in Fig. 12. Phase portraits shown as insets. (a) and (b) Below ${\text{PF}}_1$ and $\text{H}$, the only solution is the stable basic flow, which is (a) a node or (b) a spiral focus. (c) Between ${\text{PF}}_1$ and ${\text{PF}}_2$, the basic flow is unstable and there exist two stable asymmetric convective states. (d) and (e) Between ${\text{PF}}_2$ and SN, there exist five states: the unstable basic flow and two stable and two unstable convective states. (f) and (g) Between SN and $\text{H}$, the unstable basic flow is surrounded by a stable limit cycle, which is near-heteroclinic near SN (f) and near-sinusoidal near $\text{H}$ (g).

Figure 12

(Color online) Left: enlargement of square in Fig. 11. The additional curve (violet, dash-dotted) indicates the secondary Hopf bifurcation SH which exists between codimension-two points $\left(95,11750\right)$ (where it meets $\text{H}$) and $\left(74,17100\right)$ (where it meets SN). Labels correspond to those of Fig. 11. Behavior in the square is described by TB normal form (12). Right: behavior of TB normal form (12). Shown are the pitchfork bifurcation P, Hopf bifurcation H, secondary Hopf bifurcation SH, as well as the saddle-node of periodic orbits $\mathrm{SNP}$ and the gluing bifurcation G which are contained in the unfolding of the TB point. Phase portraits (b), (c), (g) as in Fig. 11. (h) Between P and SH a stable limit cycle surrounds the unstable basic and convective flows. (i) Between SH and G, small unstable limit cycles surround each stable convective state. (j) Between G and $\mathrm{SNP}$, two large limit cycles, one stable and one unstable, surround the three steady states. Hatched region indicates bistability between limit cycle and steady states.

Figure 13

(Color online) Bifurcation diagrams along lines in central (Re,Ra) diagram. $T\left(1/2,1/2\right)$ is plotted as a function of Ra for fixed Re in the diagrams along the bottom, and as a function of Re for fixed Ra in the diagrams in the column on the right. Diagrams contain steady branches with zero (blue, solid), one (green, long-dashed) and two (red, short-dashed) unstable directions. Black dots indicate limit cycles.

Figure 14

(Color online) Real part $\sigma$ of leading eigenvalues as a function of Ra for six values of Re. For $\text{Re}=0$, the eigenvalues are real and cross transversely. Zero crossings at $\text{Ra}=2260$ and $\text{Ra}=6640$ correspond to pitchfork bifurcations. Purely thermal and azimuthal eigenvalues at $\sigma =-9.87$ and $\sigma =-24.6$, respectively, are independent of Ra. When $\text{Re}>0$, the transverse crossings become complex-conjugate pairs, over Ra intervals which widen with increasing Re. By $\text{Re}=96$, the bifurcating eigenvalue is a complex-conjugate pair, leading to a Hopf bifurcation at $\text{Ra}=11856$.

Figure 15

(Color online) Leading eigenvectors at $\text{Ra}=10000$. Left column: $\text{Re}=0$. (a) and (b) eigenvectors responsible for transition to convection with one and two toroidal rolls, respectively, with $\sigma =38.5$ and $\sigma =15.9$. (c) and (d) thermal and azimuthal velocity eigenvectors, respectively, with $\sigma =-9.87\approx -{\pi }^2$ and $\sigma =-24.6\approx -{\pi }^2-j_{11}^2$. (e) Eigenvector with two vertically stacked rolls, with $\sigma =-21.2$. Right column: $\text{Re}=96$. (f) and (g) first complex-conjugate pair, with $\sigma \pm i\omega =-2.24\pm i3.45$. (h) and (i) second complex-conjugate pair, with $\sigma \pm i\omega =-19.2\pm i4.35$.