Chaotic Rayleigh-Bénard convection with finite sidewalls

We explore the role of finite sidewalls on chaotic Rayleigh-Bénard convection. We use large-scale parallel spectral-element numerical simulations for the precise conditions of experiment for cylindrical convection domains. We solve the Boussinesq equations for thermal convection and the conjugate heat transfer problem for the energy transfer at the solid sidewalls of the cylindrical domain. The solid sidewall of the convection domain has finite values of thickness, thermal conductivity, and thermal diffusivity. We compute the Lyapunov vectors and exponents for the entire fluid-solid coupled problem. We quantify the chaotic dynamics of convection over a range of thermal sidewall boundary conditions. We find that the dynamics become less chaotic as the thermal conductivity of the sidewalls increases as measured by the value of the fractal dimension of the dynamics. The thermal conductivity of the sidewall is a stabilizing influence; the heat transfer between the fluid and solid regions is always in the direction to reduce the fluid motion near the sidewalls. Although the heat interaction for strongly conducting sidewalls is only about 1% of the heat transfer through the fluid layer, it is sufficient to reduce the fractal dimension of the dynamics by approximately 25% in our computations.

Figure 1

A schematic of the cylindrical convection domain used in the numerical simulations. (a) A vertical cross-section through the domain showing the $r^{*}-z^{*}$ plane. This projection clearly shows the depth $d$ of the layer, the radius of the region containing the fluid $r_1$, and the outer radius of the solid sidewall $r_2$. The solid sidewall is gray and the fluid filled region is white. The bottom surface is hot at temperature $T_h$ and the top surface is cold at temperature $T_c$. (b) A projection of the domain in the $x^{*}-y^{*}$ or $r^{*}-\theta$ plane. Panels (a) and (b) are drawn to scale. In our notation, a superscript $*$ indicates a coordinate in dimensional form.

Figure 2

Typical flow field patterns of chaotic Rayleigh-Bénard convection for (a) perfectly insulating sidewalls ${\beta }_k=0$ and (b) perfectly conducting sidewalls ${\beta }_k=\infty$. The solid black line marks the location of the sidewall boundary. The color contours are of the temperature field at the horizontal midplane ($z=1/2$), where red is hot rising fluid and blue is cold falling fluid. For perfectly insulating or perfectly conducting sidewalls, the sidewall is represented as a boundary condition and we do not need to solve the conjugate heat transfer problem.

Figure 3

The time variation of the heat transfer through the fluid layer for perfectly insulating sidewalls ${\beta }_k=0$. (a) The heat transfer through the bottom plate $Q_b\left(t\right)$ (red, solid) and the heat transfer through the top plate $|Q_t\left(t\right)|$ (blue, dashed). The axis on the right shows these heat transfers as measured by the Nusselt numbers ${\text{Nu}}_b\left(t\right)$ and ${\text{Nu}}_t\left(t\right)$, respectively. The long-time average of the Nusselt number is $〈\text{Nu}〉\approx 1.93$. (b) The time variation of $Q_{bt}\left(t\right)$. For an insulating sidewall $Q_{bt}=d\mathcal{E}/dt$ from Eq. (31). The long-time average from our numerical simulation is $〈d\mathcal{E}/dt〉=5.2\times {10}^{-3}\approx 0$ and the standard deviation of the fluctuations is $\delta (d\mathcal{E}/dt)=0.25$.

Figure 4

The time variation of $d\mathcal{E}/dt$ as a function of the thermal conductivity ratio ${\beta }_k$ for chaotic convection with finite sidewalls. Curves are shown for three representative cases: ${\beta }_k=0.01$ (red, solid), ${\beta }_k=1$ (blue, dashed), and ${\beta }_k=100$ (green, dash-dotted). $d\mathcal{E}/dt$ is computed using Eq. (31) by numerically evaluating $Q_b$, $Q_t$, and $Q_w$. For all values of ${\beta }_k$ the values of $d\mathcal{E}/dt$ are similar to the results shown here where the long-time average $d\mathcal{E}/dt\approx 0$ and the standard deviation of the fluctuations is $\delta (d\mathcal{E}/dt)\approx 0.25$.

Figure 5

The time variation of the heat transfer through the bottom, top and sidewalls for chaotic convection as a function of the thermal conductivity ratio ${\beta }_k$. The solid (blue) line is $Q_{bt}\left(t\right)$ and the dashed (red) line is $Q_w\left(t\right)$. (a) ${\beta }_k=0.01$, (b) ${\beta }_k=1$, and (c) ${\beta }_k=100$.

Figure 6

Typical flow field patterns of chaotic Rayleigh-Bénard convection for three domains with different values of the thermal conductivity ratio ${\beta }_k$ where (a) ${\beta }_k=0.1$, (b) ${\beta }_k=1.0$, (c) ${\beta }_k=10$. From top to bottom the sidewalls go from insulating to conducting relative to the fluid. The solid black line marks the location of fluid-solid boundary. For the fluid region, the color contours of the temperature field at the horizontal mid-plane ($z=1/2$) are shown where red is hot rising fluid and blue is cold falling fluid. For the solid region, red is a hot region and blue is a cold region.

Figure 7

Typical flow fields for ${\beta }_k=\{0.1,1,10\}$. The color contours are of the temperature field at a slice through the $r\text{−}z$ plane where the images are a close-up view near the sidewall region. The black solid line represents the fluid-solid boundary. The left part is the solid sidewall region and the right part is the fluid region. For the fluid region, the color contours of the temperature field are red for hot rising fluid and blue for cold falling fluid. For the solid region red is hot and blue is cold. (a) ${\beta }_k=0.1$, (b) ${\beta }_k=1.0$, (c) ${\beta }_k=10$.

Figure 8

A three-dimensional view of the relationship between the heat flux through the sidewalls and the flow field pattern. The color contours are of the temperature field at a midplane slice ($z=1/2$). In the fluid region red is hot rising fluid and blue is cold falling fluid and in the solid sidewall region the color represents the temperature of the solid material. A vertical slice is shown at the interface between the solid and the fluid. The heat flux through the sidewall is shown using a gray scale, where light indicates regions on the sidewall where the heat flux is from the solid region into the fluid filled region and dark regions indicate where the heat flux is from the fluid filled region into the solid sidewall. Overall, the domain is tilted at an angle of ${30}^{\circ }$ with respect to the vertical to clearly illustrate the heat flux on the sidewall interface. The color scale ($-0.45$ for blue, 0.55 for red) and gray scale ($-0.5$ for black, 0.5 for white) are the same for both panels. (a) ${\beta }_k=0.1$, (b) ${\beta }_k=10$.

Figure 9

(a) The variation of the leading order Lyapunov exponent ${\lambda }_1$ with ${\beta }_k$. (b) The variation of the fractal dimension $D_{\lambda }$ with ${\beta }_k$. In both panels, the upper dashed line is the perfectly insulating result (${\beta }_k=0$) and the lower dashed line is the perfectly conducting result (${\beta }_k=\infty$).

Figure 10

The spatial variation of the leading order Lyapunov vector for (a) ${\beta }_k=0.01$ and (b) ${\beta }_k=100$. The color contours are of the temperature perturbation field at the horizontal mid-plane $\delta T^{\left(1\right)}(x,y,z=1/2,t_0)$ at some instant of time $t_0$. Red represents positive magnitude and blue represent negative magnitude. The black contour lines indicate the temperature contour at $T_f(x,y,z=1/2,t_0)=T_s(x,y,z=1/2,t_0)=1/2$. In the fluid, this indicates the location of the convection rolls and in the solid it indicates how the temperature variation is related to the convection rolls. The black circle indicates the boundary between the fluid and solid regions and is drawn for reference.

Figure 11

A close-up view of the spatial variation of the leading order Lyapunov vector near the wall for the dynamics shown in Fig. 10 where (a) ${\beta }_k=0.01$ and (b) ${\beta }_k=100$. In panel (a) it is clear that the magnitude of the Lyapunov vector is large for the entire region near the sidewall. In panel (b), the Lyapunov vector nearly vanishes near the sidewall for a strongly conducting sidewall. This illustrates the stabilizing effect of conducting sidewalls.