Dynamic heterogeneity and conditional statistics of non-Gaussian temperature fluctuations in turbulent thermal convection

Non-Gaussian fluctuations with an exponential tail in their probability density function (PDF) are often observed in nonequilibrium steady states (NESSs) and one does not understand why they appear so often. Turbulent Rayleigh-Bénard convection (RBC) is an example of such a NESS, in which the measured PDF $P\left(\delta T\right)$ of temperature fluctuations $\delta T$ in the central region of the flow has a long exponential tail. Here we show that because of the dynamic heterogeneity in RBC, the exponential PDF is generated by a convolution of a set of dynamics modes conditioned on a constant local thermal dissipation rate $\epsilon$. The conditional PDF $G\left(\delta T\right|\epsilon )$ of $\delta T$ under a constant $\epsilon$ is found to be of Gaussian form and its variance ${\sigma }_T^2$ for different values of $\epsilon$ follows an exponential distribution. The convolution of the two distribution functions gives rise to the exponential PDF $P\left(\delta T\right)$. This work thus provides a physical mechanism of the observed exponential distribution of $\delta T$ in RBC and also sheds light on the origin of non-Gaussian fluctuations in other NESSs.

Figure 1

Measured unconditional PDF $P\left(\delta T\right)$ of temperature fluctuations $\delta T$ at the cell center for three different values of $\text{Ra}$: $1.3\times {10}^9$ (blue squares), $3.8\times {10}^9$ (black triangles), and $8.3\times {10}^9$ (red circles). In the plot, $\delta T$ is normalized by its rms value ${\sigma }_0$, and the error bars show the standard deviation of the red circles. The solid line shows a plot of Eq. (4).

Figure 2

(a) Measured conditional PDF $G\left(\delta T\right|\epsilon )$ for five different values of $\epsilon$ in units of their rms value ${\sigma }_{\epsilon }$ (from bottom to top): $\epsilon /{\sigma }_{\epsilon }=0.2$ (black circles), 0.5 (red circles), 2 (blue circles), 5 (green circles), and 10 (purple circles). The measurements were made at the cell center with $\text{Ra}=8.3\times {10}^9$. In the plot, $\delta T$ is conditioned at a fixed value of $\epsilon$ and is normalized by the corresponding rms value ${\sigma }_T\left(\epsilon \right)$. For clarity, the vertical scale of the four PDFs with $\epsilon /{\sigma }_{\epsilon }\ge 0.5$ is multiplied by a factor 10, ${10}^2,{10}^3$, and ${10}^4$, respectively. The solid lines represent Gaussian functions expressed by Eq. (2) with different variance ${\sigma }_T^2\left(\epsilon \right)$. (b) Similar plot of $G\left(\delta T\right|\epsilon )$ obtained at the cell center with $\text{Ra}=1.8\times {10}^9$. The data sets are obtained at five different values of $\epsilon$ in units of their rms value ${\sigma }_{\epsilon }$ (from bottom to top): $\epsilon /{\sigma }_{\epsilon }=2$ (black), 4 (red), 6 (blue), 8 (green), and 10 (purple). For clarity, the vertical scale of the four PDFs with $\epsilon /{\sigma }_{\epsilon }\ge 4$ is multiplied by a factor 4, $4^2,4^3$, and $4^4$, respectively. The solid lines show the Gaussian fits of Eq. (2) to the data points with different values of ${\sigma }_T\left(\epsilon \right)$.

Figure 3

Obtained values of ${\sigma }_T^2\left(\epsilon \right)/{\sigma }_0^2$ as a function of $\epsilon /{\sigma }_{\epsilon }$ (red circles). The measurements were made at the cell center with $\text{Ra}=8.3\times {10}^9$. The error bars indicate the bin size $\delta \epsilon$ used to digitize the measured $\epsilon$. The solid line shows a power-law fit of Eq. (5) with $a=2.853$ and $\alpha =0.363$ to the data points for $\epsilon /{\sigma }_{\epsilon }>0.043$.

Figure 4

Measured unconditional PDF $P\left(\delta T\right)$ of temperature fluctuations $\delta T$ in the boundary-layer region with $z/\delta =0.14$ (black squares) and $z/\delta =0.46$ (red circles). The measurements were made in the $\mathrm{\Gamma }=1$ upright cylinder with fixed values of $\text{Ra}=6.8\times {10}^9$ and $\text{Pr}=4.4$ (water). In the plot, $\delta T$ is normalized by its rms value ${\sigma }_0$. The solid line shows a plot of the standard Gaussian function.

Figure 5

Measured PDF $f\left(\epsilon \right)$ of the local thermal dissipation rate $\epsilon$ at the cell center for three different values of $\text{Ra}$: $1.3\times {10}^9$ (green squares), $3.8\times {10}^9$ (blue triangles), and $8.3\times {10}^9$ (red circles). In the plot, $\epsilon$ is normalized by its rms value ${\sigma }_{\epsilon }$, and the error bars show the standard deviation of the circles. The solid line is a plot of Eq. (6) with $a=2.853$ and $\alpha =0.363$.

Figure 6

Numerically obtained PDF $F\left({\sigma }_T^2\right)$ of temperature variance ${\sigma }_T^2$ for $\text{Ra}=8.3\times {10}^9$ at the cell center. The error bars show the standard deviation of the data points. The solid line is a plot of Eq. (3).