Extreme multiplicity in cylindrical Rayleigh-Bénard convection. I. Time dependence and oscillations

Rayleigh-Bénard convection in a cylindrical container can take on many different spatial forms. Motivated by the results of Hof et al. [Phys. Fluids 11, 2815 (1999)], who observed coexistence of several stable states at a single set of parameter values, we have carried out simulations at the same Prandtl number, that of water, and a radius-to-height aspect ratio of two. We have used two kinds of thermal boundary conditions: perfectly insulating sidewalls and perfectly conducting sidewalls. In both cases we obtain a wide variety of coexisting steady and time-dependent flows.

Figure 1

Patterns observed in the experiment of Hof et al. at $\text{Ra}=14200$. (a) three rolls, (b) two rolls, (c) inverted two rolls, (d) four rolls, (e) inverted four rolls, (f) Mercedes, (g) inverted Mercedes, and (h) axisymmetric pattern. Dark areas correspond to hot (rising) and bright to cold (descending) fluid.

Figure 2

Geometry and coordinate system.

Figure 3

(Color online) Schematic diagram of stability ranges and transitions between convective patterns as a function of Rayleigh number for $\Gamma =2$, $\text{Pr}=6.7$ and insulating sidewalls. Stars denote solutions obtained from a sudden start to the Rayleigh number indicated, using as an initial condition a slightly perturbed conductive state. (a) Initial perturbation. (b)–(e) Convective patterns obtained from a sudden start to Rayleigh numbers indicated. Visualization of the temperature in the cylinder's midplane, dark, hot (ascending), bright, cold (descending) fluid.

Figure 4

(Color online) Evolution from three-roll pattern. (a)–(c) Evolution at $\text{Ra}=2000$ leading towards dipole. (d)–(g) Three-roll patterns at different Rayleigh numbers. (h)–(i) Asymmetric three-roll pattern with central roll shifted to the right, obtained for Ra between $20000$ and $25000$.

Figure 5

Dependence of the three-roll patterns on Rayleigh number: (a) temperature at $r=0.3$, $z=0.25$, and $\theta =0$; (b) energy.

Figure 6

(Color online) (a)–(c) Four-roll patterns for different Rayleigh numbers. (d)–(h) Evolution of the four-roll pattern towards (probably transitional) cross pattern at $\mathrm{Ra}=25000$

Figure 7

(Color online) Evolution from pizza pattern. (a)–(d) To four-roll pattern at $\mathrm{Ra}=5000$. (e)–(i) To three-roll pattern at $\mathrm{Ra}=10000$. (j)–(l) Through eight to final torus pattern at $\mathrm{Ra}=14200$. (m)–(s) Through a series of intermediate states towards final two-roll pattern at $\mathrm{Ra}=16000$. (t)–(z) Through a series of intermediate states towards final two-roll pattern at $\mathrm{Ra}=29000$.

Figure 8

(Color online) (a)–(b) Two-roll patterns. (c)–(f) Axisymmetric patterns. (g)–(l) Mercedes patterns. Rayleigh numbers as indicated below each visualization.

Figure 9

(Color online) Evolution from dipole pattern to three-roll pattern at $\mathrm{Ra}=5000$. (a)–(f) Visualizations at time indicated. Evolution as a function of time of (g) energy and (h) vertical velocity at one point. A long-lasting transient state is visible between $t=3$ and $t=12$

Figure 10

(Color online) Evolution from dipole pattern. (a)–(c) Through dipole smile into final CO pattern at $\mathrm{Ra}=10000$. (d)–(e) Transitional patterns observed during evolution from dipole into three-roll pattern. (f) Initial dipole and (g)–(j) final patterns at Rayleigh numbers indicated.

Figure 11

Squares, energy of the flow evolved from dipole state as a function of Rayleigh number; crosses, energy of the flow evolved from three-roll state (for comparison). At $\mathrm{Ra}=10000$ the dipole pattern evolved into a CO pattern.

Figure 12

(Color online) Rotating S pattern at $\text{Ra}=12500$. (a)–(f) Visualizations at six different times. (g) Evolution in time of vertical velocity at two points with the same r and z and different θ.

Figure 13

(a) Frequency and (b) energy of the rotating S pattern as a function of Rayleigh number.

Figure 14

(Color online) Evolution from dipole-shaped perturbation into a stable smile pattern at $\text{Ra}=16000$.

Figure 15

Dependence on Rayleigh number of (a) energy and (b) scaled temperature deviation $\overline{H}/\text{Ra}$ for all stable convective patterns obtained for insulating sidewalls.

Figure 16

(Color online) Schematic diagram of stability ranges and transitions between convective patterns as a function of Rayleigh number for $\Gamma =2$, $\text{Pr}=6.7$ and conducting sidewalls. Stars denote solutions obtained from a slight perturbation of the conductive state, at given Ra. (a) Arbitrary perturbation used for initializing simulations and (b)–(j) final patterns at various Rayleigh numbers.

Figure 17

(Color online) Roll patterns at different Rayleigh numbers. (a)-(c) Three-roll patterns. (d)-(f) Four-roll patterns.

Figure 18

(Color online) Oscillatory X/X pattern at $\mathrm{Ra}=35000$.

Figure 19

(Color online) (a)–(d) Convective patterns evolved from star flow. (e) Evolution of the system initialized with star pattern at $\text{Ra}=2000$: energy as a function of time. Between $t=5$ and $35$ there is a long-lasting transient axisymmetric pattern.

Figure 20

(Color online) Cross pattern evolved at $\mathrm{Ra}=10000$ from da Vinci flow.

Figure 21

Dependence of Rayleigh number of (a) energy and (b) scaled temperature deviation $\overline{H}/\text{Ra}$ for all stable convective patterns obtained for conducting sidewalls.