Quantifying spatiotemporal chaos in Rayleigh-Bénard convection

Using large-scale parallel numerical simulations we explore spatiotemporal chaos in Rayleigh-Bénard convection in a cylindrical domain with experimentally relevant boundary conditions. We use the variation of the spectrum of Lyapunov exponents and the leading-order Lyapunov vector with system parameters to quantify states of high-dimensional chaos in fluid convection. We explore the relationship between the time dynamics of the spectrum of Lyapunov exponents and the pattern dynamics. For chaotic dynamics we find that all of the Lyapunov exponents are positively correlated with the leading-order Lyapunov exponent, and we quantify the details of their response to the dynamics of defects. The leading-order Lyapunov vector is used to identify topological features of the fluid patterns that contribute significantly to the chaotic dynamics. Our results show a transition from boundary-dominated dynamics to bulk-dominated dynamics as the system size is increased. The spectrum of Lyapunov exponents is used to compute the variation of the fractal dimension with system parameters to quantify how the underlying high-dimensional strange attractor accommodates a range of different chaotic dynamics.

Figure 1

A spatiotemporally chaotic flow field for $\epsilon =4.27$, $\sigma =1$, and $\Gamma =10$. Contours are shown of the temperature field at a midplane slice where $z=1/2$. Light regions are hot rising fluid, and dark regions are cool falling fluid. This flow-field image is at time $t=610.5$.

Figure 2

The convergence of the fractal dimension $D_{\lambda }$ in time. Results are shown for four different values of the reduced Rayleigh number $\epsilon$ where $\Gamma =10$ and $\sigma =1$. The time scale has been normalized by the horizontal diffusion time for heat, ${\tau }_H={\Gamma }^2$. The convergence is quite slow and remains noisy over the entire range shown.

Figure 3

The flow field and the variation of the Nusselt number $N$ with time for periodic dynamics. The simulation parameters are $\Gamma =5$, $\sigma =1$, and $\epsilon =1.93$. (a) The flow field at $t=579$. (b) The flow field at $t=581$. (c) The variation $N\left(t\right)$ for one period of the dynamics. The vertical lines represent the instances of time of the two flow-field images.

Figure 4

The time variation of the first three instantaneous Lyapunov exponents $\tilde{\lambda }$ for time-periodic dynamics. The simulation parameters are $\Gamma =5$, $\sigma =1$, and $\epsilon =1.93$. The Lyapunov exponents have been normalized by the maximum value of ${\lambda }_1$ for ease of comparison. The values for ${\tilde{\lambda }}_1$, ${\tilde{\lambda }}_2$, and ${\tilde{\lambda }}_3$ are given by the solid, dashed, and dash-dot lines, respectively.

Figure 5

(a) The spectrum of Lyapunov exponents ${\lambda }_k$. Also shown are the error bars that are computed from the standard deviation of ${\lambda }_k$ about its mean value at long times. (b) The instantaneous cross-correlation between the leading order Lyapunov exponent and the remaining exponents in the spectra ${\lambda }_j$ for $j=2,...,30$. The simulation parameters for both panels are $\sigma =1$, $\Gamma =10$, and $\epsilon =2.51$.

Figure 6

Overlay of a grayscale contours of the midplane temperature perturbation field with solid black lines representing the convection roll boundaries for different aspect ratios: (a) $\Gamma =5$, (b) $\Gamma =10$, (c) $\Gamma =15$, and (d) $\Gamma =30$. For $\Gamma =30$ the image is plotted at half scale to be able to fit on this figure. The parameters are $\epsilon =2.51$ and $\sigma =1$.

Figure 7

The spatial variation of the time-averaged magnitude of the thermal perturbation field ${\langle \delta T(x,y)\rangle }_t$ evaluated at the horizontal midplane. (a) $\Gamma =5$, (b) $\Gamma =10$, (c) $\Gamma =15$, and (d) $\Gamma =30$. The simulation parameters are $\epsilon =2.51$ and $\sigma =1$. The image for $\Gamma =30$ is plotted at half scale to fit on this figure. In the color contour, red regions (located primarily near the boundary for small $\Gamma$ and at the bulk of the domain for large $\Gamma$) correspond to the large magnitude of the perturbation, and blue regions (located mainly at the bulk of the domain for small $\Gamma$ and near the boundary for large $\Gamma$) are associated with the small magnitude of the perturbation.

Figure 8

The radial variation of the azimuthal and time-averaged thermal perturbation field ${\langle \delta T\left(\overline{r}\right)\rangle }_{t,\theta }$. The aspect ratios are $5\le \Gamma \le 30$.

Figure 9

(a) The variation of the fractal dimension with Rayleigh number for $\Gamma =10$ and $\sigma =1$. The circles are data points from the simulations, and the solid line is the curve fit $D_{\lambda }=0.095{\epsilon }^4+19.4$. (b) The variation of the fractal dimension with Prandtl number for $\Gamma =10$ and $\epsilon =2.51$. The circles are data points from the simulations, and the solid line is a curve fit as $D_{\lambda }=44.95{\sigma }^{-1.06}-20.91$ for $1⩽\sigma <2$ and $D_{\lambda }=0$ for $\sigma ⩾2$.

Figure 10

The variation of the natural chaotic length scale (${\xi }_{\delta }$), the wavelength of the pattern (${\xi }_L$), and the ratio of ${\xi }_L/{\xi }_{\delta }$ with $\epsilon$ for $\Gamma =10$ and $\sigma =1$. The open squares show ${\xi }_{\delta }$, the open circles show ${\xi }_L$, the open triangles demonstrate ${\xi }_L/{\xi }_{\delta }$, and the solid lines illustrate curve fits for ${\xi }_{\delta }$ and ${\xi }_L$ as ${\xi }_{\delta }=2.32-0.03{\epsilon }^{2.4}$ and ${\xi }_L=5.07-4.75{\epsilon }^{-0.99}$ and the linear fit for the ratio as ${\xi }_L/{\xi }_{\delta }=0.73\epsilon -0.36$.

Figure 11

The flow-field patterns for different Prandtl numbers. In each case $\epsilon =2.51$ and $\Gamma =10$. Panel (a) $\sigma =1$, (b) $\sigma =3$, (c) $\sigma =5$, (d) $\sigma =7$.