2 Experiment

The experimental apparatus and procedures used in this experiment are similar to those described previously (He & Tong Reference He and Tong2009; Wang, He & Tong Reference Wang, He and Tong2016; Wang et al. Reference Wang, Xu, He, Yik, Wang, Schumacher and Tong2018b ), and here we only mention some key points. The experiment is conducted in two convection cells with different geometries. One of them is a vertical thin disk with its circular cross-section aligned parallel to gravity, as shown in figure 1(a). The cell has a diameter $D=188~\text{mm}$ and thickness $L=20~\text{mm}$ , and thus the corresponding aspect ratio is $\unicode[STIX]{x1D6E4}\equiv L/D\simeq 0.1$ . The top and bottom $1/3$ of the circular side wall are made of copper of 8 mm in thickness. The surface of the copper plates in contact with the convecting fluid is electroplated with a thin layer of nickel. The remaining $1/3$ of the side wall on both sides is made of transparent Plexiglas of 18 mm in thickness. The two flat end walls of the cell are also made of the same type of Plexiglas. Two silicon rubber film heaters connected in parallel are sandwiched on the back side of the bottom conducting plate to provide constant and uniform heating. The top copper plate is in contact with a cooling chamber consisting of two water channels. The temperature of the top plate is maintained by a temperature-controlled circulator (NESLAB, RTE740), which circulates cooling water with a temperature stability of 10 mK. The temperature of the top and bottom plates is measured at a rate of 2 Hz by calibrated thermistors with an accuracy of 5 mK. They are embedded in each plate at a distance of 1 mm away from the surface of the conducting plate. This cell was used in the previous convection experiments (Song, Villermaux & Tong Reference Song, Villermaux and Tong2011; Wang et al. Reference Wang, He and Tong2016, Reference Wang, Lai, Song and Tong2018a ,Reference Wang, Xu, He, Yik, Wang, Schumacher and Tong b ).

Figure 1. Sketch of the experimental set-up for the measurement of local temperature fluctuations at different locations along the central axis of (a) the vertical thin disk and (b) the $\unicode[STIX]{x1D6E4}=1$ upright cylinder. The black arrows indicate the direction of the large-scale circulation.

The second cell is an upright cylinder, as shown in figure 1(b). The inner diameter of the cell is $D_{up}=19.0~\text{cm}$ and its height is $H=19.0~\text{cm}$ . The corresponding aspect ratio of the cell is $\unicode[STIX]{x1D6E4}\equiv D_{up}/H=1$ . The side wall of the cell is made of a transparent Plexiglas ring with a wall thickness of 6 mm. The top and bottom plates of the cell are made of brass and their surfaces are electroplated with a thin layer of gold. The Plexiglas ring side wall is sandwiched between the two plates and is sealed to the top and bottom plates via two rubber O-rings. Except for the difference in shape, all other aspects of the cell, such as the heating (cooling) of the bottom (top) plate and temperature measurement of the conducting plates, remain the same as those for the vertical thin disk. The temperature standard deviation of the top and bottom plates is found to be within $1\,\%$ of $\unicode[STIX]{x0394}T$ used in the experiment. This cell was also used in previous convection experiments (He & Tong Reference He and Tong2009; He, Ching & Tong Reference He, Ching and Tong2011; Wang et al. Reference Wang, Xu, He, Yik, Wang, Schumacher and Tong2018b ).

In the experiment, the entire convection cell is placed inside a square thermostat box, whose temperature matches the mean temperature of the bulk fluid (maintained at $40\pm 0.1\,^{\circ }\text{C}$ ), in order to prevent heat exchange between the convecting fluid and the surroundings. The Rayleigh number $Ra$ for the thin disk is defined as $Ra\equiv \unicode[STIX]{x1D713}g\unicode[STIX]{x0394}TD^{3}/(\unicode[STIX]{x1D708}\unicode[STIX]{x1D705})$ , where $g$ is the gravitational acceleration, $\unicode[STIX]{x0394}T$ is the vertical temperature difference across the cell and $\unicode[STIX]{x1D713}$ , $\unicode[STIX]{x1D708}$ and $\unicode[STIX]{x1D705}$ are, respectively, the thermal expansion coefficient, viscous and thermal diffusivity of the convecting fluid. For the upright cylinder, the cell height $H$ is used to replace $D$ . The Prandtl number $Pr$ is defined as $Pr\equiv \unicode[STIX]{x1D708}/\unicode[STIX]{x1D705}$ . Two working fluids are used in the experiment. For the thin disk, a 20 wt. % aqueous solution of glycerin with $Pr=7.6$ is used. For the upright cylinder, both distilled water with $Pr=4.4$ and the 20 wt. % aqueous solution of glycerin with $Pr=7.6$ are used. In the experiment, the temperature $T_{b}$ of the bottom heating plate is varied in the range $42.4\lesssim T_{b}\lesssim 60.2\,^{\circ }\text{C}$ , and the temperature $T_{t}$ of the top cooling plate is varied in the range $13.8\lesssim T_{t}\lesssim 37.8\,^{\circ }\text{C}$ . Correspondingly, $Ra$ is varied in the range $9\times 10^{8}\lesssim Ra\lesssim 1.2\times 10^{10}$ and $Pr$ is fixed for a given fluid. The black arrows in figure 1 show the direction of the large-scale circulation (LSC). To pin down the azimuthal motion of the LSC in the upright cylinder, the cylindrical cell is tilted slightly at a small angle ( ${<}1^{\circ }$ ) by adjusting the levelling plate of the cell. It has been shown (Ahlers, Brown & Nikolaenko Reference Ahlers, Brown and Nikolaenko2006a ) that such a small tilt to the $\unicode[STIX]{x1D6E4}=1$ upright cylinder does not affect the convective flow very much, as the single-roll structure of the LSC is quite stable in the $\unicode[STIX]{x1D6E4}=1$ cell (Qiu & Tong Reference Qiu and Tong2001). The LSC structure in the $\unicode[STIX]{x1D6E4}=0.5$ upright cylinder, however, is more sensitive to the cell tilt and so are the temperature fluctuations (Chillá et al. Reference Chillá, Rastello, Chaumat and Castaing2004).

The local fluid temperature is measured using a glass-encapsulated thermistor of diameter 0.17 mm with an accuracy of 5 mK (AB6E3-B05KA202R). Details about the temperature calibration and measurements have been reported elsewhere (He & Tong Reference He and Tong2009). To guide the two thermistors into the thin disk, we install a horizontal and a vertical stainless steel tube in the cell, as shown in figure 1(a). The horizontal tube is fixed to measure the mean temperature $T_{0}$ at the cell centre. The vertical tube is used to measure the temperature time series data $T(t)$ with varying distance $z$ away from the centre of the bottom plate. The vertical tube is mounted on a translational stage, which is controlled by a stepping motor with a position resolution of 50  $\unicode[STIX]{x03BC}$ m. The measurements of $T_{0}$ and $T(t)$ are made, respectively, at the sampling rates of 2 Hz and 15 Hz. Typically, we take 1 h long time series data ( $5.4\times 10^{4}$ data points) at each location of $z$ for the measurement of $T(t)$ . The measurement of temperature fluctuations in the upright cylinder is made in a similar way, except that a vertical tube is installed to measure $T_{0}$ , as shown in figure 1(b).

3 Experimental results

3.1 Turbulent temperature fluctuations in the thin disk

Figure 2(a) shows the evolution of the measured local temperature variations $T(t)$ for four different values of distance $z$ away from the centre of the bottom plate, and at a fixed value of $Ra=5.3\times 10^{9}$ and $Pr=7.6$ in the thin disk. Figure 2(b) shows the corresponding PDFs of the measured $T(t)$ in figure 2(a). In a recent experiment and direct numerical simulation, we have shown that there is a well-developed thermal BL near the bottom (top) conducting plate of the thin disk cell. A typical value of the BL thickness $\unicode[STIX]{x1D6FF}$ at this $Ra$ is $\unicode[STIX]{x1D6FF}\simeq 0.55~\text{mm}$ . Heat transport inside the BL is accomplished mainly by thermal diffusion, which establishes a large mean temperature gradient across the BL. In the region outside the BL, however, heat transport is dominated by turbulent convection and the mean temperature along the central axis of the bulk region is approximately constant. At $z=0.2~\text{mm}$ which is in the inner region of the BL, the measured $T(t)$ (black curve in figure 2 a) reveals large symmetric fluctuations, and thus their PDF (black squares in figure 2 b) has a symmetric shape with a broad width. In addition, the measured mean temperature $\langle T\rangle$ at this location is much larger than that at the other three positions.

Figure 2. (a) Evolution of the measured local temperature variations $T(t)$ for 4 different values of distance $z$ away from the centre of the bottom plate (bottom to top): $z=0.2~\text{mm}$ (black curve), $z=4.9~\text{mm}$ (red curve), $z=61.6~\text{mm}$ (green curve) and $z=94.0~\text{mm}$ (blue curve). The measurements are made in the thin disk at $Ra=5.3\times 10^{9}$ and $Pr=7.6$ (20 wt. % aqueous solution of glycerin). (b) Corresponding PDFs of the measured $T(t)$ in (a) at $z=0.2~\text{mm}$ (black squares), $z=4.9~\text{mm}$ (red circles), $z=61.6~\text{mm}$ (green up triangles) and $z=94.0~\text{mm}$ (blue down triangles).

At $z=4.9~\text{mm}$ which is outside the BL and in the mixing zone (Wang et al. Reference Wang, He and Tong2016), the measured $T(t)$ (red curve in figure 2 a) reveals large temperature spikes in one direction, resulting from warm thermal plumes passing through the measuring position. These warm plumes are intermittently emitted from the bottom BL. In addition to the hot spikes, the measured $T(t)$ also shows small continuous fluctuations resulting from the turbulent background. As shown in figure 2(b), the intermittent temperature spikes in the time series $T(t)$ give rise to an exponential tail in the corresponding temperature PDF $P(T)$ (red circles with $T\gtrsim 41\,^{\circ }\text{C}$ ). The small background fluctuations, on the other hand, produce a symmetric peak in the measured $P(T)$ .

By increasing $z$ further into the turbulent bulk region, the warm plumes are mixed and dissipated by the flow field. As a result, the intermittent spikes in $T(t)$ and the asymmetric tail in the resulting $P(T)$ gradually diminish. The green curve in figure 2(a) and green up triangles in figure 2(b) show an example at $z=61.8~\text{mm}$ . In addition, the standard deviation of temperature fluctuations also decreases with increasing $z$ . Finally, at the cell centre ( $z=94.0~\text{mm}$ ), the standard deviation of temperature fluctuations becomes the smallest (blue curve in figure 2 a), and the resulting $P(T)$ shows the smallest width (blue down triangles in figure 2 b). Because of the reflection symmetry at the cell centre, the measured $P(T)$ has a symmetric shape as expected. An interesting feature shown in figure 2(b) is that once outside the thermal BL, both the peak position and left side of the peak of all the measured $P(T)$ remain unchanged with $z$ . This result suggests that temperature fluctuations in the bulk region outside the BL have a fairly uniform turbulent background (Zhou & Xia Reference Zhou and Xia2013).

Figure 3. Sketch of the four flow regions in the thin disk cell, namely, boundary layer (BL, $z\lesssim 2\unicode[STIX]{x1D6FF}$ ), mixing zone ( $0.01\lesssim z/D\lesssim 0.1$ ), turbulent bulk ( $0.1\lesssim z/D\leqslant 0.5$ ) and cell centre ( $z/D=0.5$ ). The vertical dashed line shows the central $z$ -axis of the cell. The BL can be further divided into an inner region ( $z\lesssim 0.6\unicode[STIX]{x1D6FF}$ ) and an outer region ( $0.6\unicode[STIX]{x1D6FF}\lesssim z\lesssim 2\unicode[STIX]{x1D6FF}$ ), in which thermal plumes start to form. The black arrow indicates the direction of the large-scale circulation.

Figure 3 shows a sketch of the four flow regions in the thin disk cell, as discussed above. In particular, the BL can be further divided into two sub-regions: inner and outer regions. In the inner region of the BL ( $z\lesssim 0.6\unicode[STIX]{x1D6FF}$ ), the mean value of the local temperature decreases linearly with distance $z$ and heat is transported primarily by diffusion. In the outer region of the BL ( $0.6\unicode[STIX]{x1D6FF}\lesssim z\lesssim 2\unicode[STIX]{x1D6FF}$ ), the mean value of the local temperature gradually approaches a constant value and thermal plumes start to form. These thermal plumes are emitted intermittently into the mixing zone ( $0.01\lesssim z/D\lesssim 0.1$ ) outside the BL and then are carried away by the LSC (as marked by the black arrow in figure 3). Because of turbulent mixing and thermal dissipation, the thermal plumes in the turbulent bulk region ( $0.1\lesssim z/D\lesssim 0.5$ ) become less energetic and fewer in number compared with those in the mixing zone.

Figure 4. Measured PDF $P(\unicode[STIX]{x1D6FF}T)$ of the normalized temperature fluctuation $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}$ for different values of $Ra$ with a fixed value $Pr=7.6$ (20 wt. % aqueous solution of glycerin). The measurements are made in the thin disk and in four different regions of distance $z$ away from the bottom plate: (a) $z/\unicode[STIX]{x1D6FF}\lesssim 0.62$ (inner region of the BL), (b) $z/D\simeq 0.026$ (mixing zone), (c) $z/D\simeq 0.33$ and (d) $z/D=0.5$ (cell centre). The dashed lines in (a–d) show the Gaussian function given in (3.1). The solid line in (b) shows a fit of (4.8) to the data points with $\unicode[STIX]{x1D6FC}_{1}=0.25\pm 0.05$ and $\unicode[STIX]{x1D6FD}=0.1\pm 0.03$ . The solid line in (c) shows a fit (4.6) to the data points with $\unicode[STIX]{x1D6FC}_{1}=1.1\pm 0.2$ . The solid line in (d) is a fit of (4.11) to the data points with $\unicode[STIX]{x1D6FC}=1.7\pm 0.3$ .

We now discuss the scaling properties of the measured PDF $P(\unicode[STIX]{x1D6FF}T)$ of temperature fluctuations $\unicode[STIX]{x1D6FF}T$ from the mean value with varying values of $z$ and $Ra$ . Here $\unicode[STIX]{x1D6FF}T=T(t)-\langle T\rangle$ with $\langle T\rangle$ being the local mean temperature. It is found that the temperature PDFs obtained at a fixed value of $z$ and different values of $Ra$ can all be brought into coincidence, by normalizing $\unicode[STIX]{x1D6FF}T$ by its root mean square (r.m.s.) value $\unicode[STIX]{x1D70E}_{T}$ . Plots of $P(\unicode[STIX]{x1D6FF}T)$ versus $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}$ remain unchanged in the $Ra$ range studied and only $\unicode[STIX]{x1D70E}_{T}$ changes with $Ra$ . Figure 4 shows the measured $P(\unicode[STIX]{x1D6FF}T)$ as a function of $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}$ for different values of $Ra$ in four regions of $z$ . It is seen that the measured $P(\unicode[STIX]{x1D6FF}T)$ in each region of $z$ has a unique functional form independent of $Ra$ .

Figure 4(a) shows the measured $P(\unicode[STIX]{x1D6FF}T)$ as a function of normalized temperature fluctuations $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}$ when $z$ is in the inner region of the BL with $z/\unicode[STIX]{x1D6FF}\lesssim 0.62$ , where the mean temperature gradient is almost constant. It is seen that all of the measured PDFs $P(\unicode[STIX]{x1D6FF}T)$ can be well described by a Gaussian function (dashed line)

(3.1) $$\begin{eqnarray}\displaystyle P(\unicode[STIX]{x1D6FF}T)=\frac{1}{\sqrt{2\unicode[STIX]{x03C0}}\unicode[STIX]{x1D70E}_{T}}e^{-\unicode[STIX]{x1D6FF}T^{2}/(2\unicode[STIX]{x1D70E}_{T}^{2})}, & & \displaystyle\end{eqnarray}$$

and only $\unicode[STIX]{x1D70E}_{T}$ changes with $z$ . For RBC in an rectangular cell, the measured $P(\unicode[STIX]{x1D6FF}T)$ inside the BL was also found to have a similar Gaussian form (Zhou & Xia Reference Zhou and Xia2013). In the outer region of the BL with $0.62\lesssim z/\unicode[STIX]{x1D6FF}\lesssim 2$ , while plots of $P(\unicode[STIX]{x1D6FF}T)$ versus $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}$ remain unchanged with $Ra$ at a fixed value of $z/\unicode[STIX]{x1D6FF}$ , the shapes of the measured $P(\unicode[STIX]{x1D6FF}T)$ for different values of $z$ start to show systematic deviations from the Gaussian function. These deviations cause the measured $P(\unicode[STIX]{x1D6FF}T)$ to become asymmetric and skewed toward the positive side of $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}$ with the skewness increasing with $z$ .

When $z$ moves beyond the BL region ( $z/\unicode[STIX]{x1D6FF}\gtrsim 2$ or equivalently $z/D\gtrsim 0.01$ ), the BL thickness $\unicode[STIX]{x1D6FF}$ is no longer the scaling length. Instead, the normalized variance profile $\unicode[STIX]{x1D70E}_{T}^{2}(z)/(\unicode[STIX]{x1D70E}_{T}^{2})_{0}$ , with $(\unicode[STIX]{x1D70E}_{T}^{2})_{0}$ being the maximal value of $\unicode[STIX]{x1D70E}_{T}^{2}$ , was found to scale with $z/D$ in the range $0.01\lesssim z/D\lesssim 0.1$ (mixing zone) (Wang et al. Reference Wang, He and Tong2016, Reference Wang, Xu, He, Yik, Wang, Schumacher and Tong2018b ). Figure 4(b) shows an example of the measured $P(\unicode[STIX]{x1D6FF}T)$ in the mixing zone for different values of $Ra$ . The measurements are made at approximately the same position with $z/D\simeq 0.026$ . All of the PDFs collapse onto a single curve, which has a Gaussian-like main portion (dashed line) for small temperature fluctuations ( $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}\lesssim 1$ ) and a long exponential tail for large temperature fluctuations ( $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}\gtrsim 2$ ). As shown in figure 2(a) (red curve), the long exponential tail is caused by the warm plumes, which are intermittently emitted from the bottom BL and pass through the measuring position. For other values of $z$ in the mixing zone, the measured PDFs $P(\unicode[STIX]{x1D6FF}T)$ retain the same characteristic shape as that shown in figure 4(b), but they vary slightly with $z$ so that plots of $P(\unicode[STIX]{x1D6FF}T)$ versus $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}$ do not overlap with each other.

As $z$ is further moved to the bulk region, with the values of $z$ being larger than the cell thickness ( $0.1\lesssim z/D\lesssim 0.4$ ), the shape of the measured PDFs $P(\unicode[STIX]{x1D6FF}T)$ changes to another form due to the strong mixing in this region. Figure 4(c) shows an example of the measured $P(\unicode[STIX]{x1D6FF}T)$ in this region for different values of $Ra$ . The measurements are made at approximately the same position with $z/D\simeq 0.33$ . All of the PDFs collapse onto a single curve, which has a Gaussian-like main portion (dashed line) for small temperature fluctuations ( $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}\lesssim 1.5$ ) and an asymmetric tail which smoothly evolves to an exponential shape for large temperature fluctuations ( $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}\gtrsim 2.5$ ).

As $z$ moves toward the central region of the cell ( $0.4\lesssim z/D\lesssim 0.5$ ), temperature fluctuations become more and more symmetric and so does the measured $P(\unicode[STIX]{x1D6FF}T)$ . Figure 4(d) shows the measured PDFs $P(\unicode[STIX]{x1D6FF}T)$ at the cell centre with different values of $Ra$ . Again, all of the measured PDFs collapse onto a single curve, which is symmetric and has a Gaussian cap (dashed line) for small temperature fluctuations $-2.5\lesssim \unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}\lesssim 2.5$ and an exponential-like tail which falls off slower than a Gaussian.

3.2 Turbulent temperature fluctuations in the $\unicode[STIX]{x1D6E4}=1$ upright cylinder

We now examine the scaling properties of the measured $P(\unicode[STIX]{x1D6FF}T)$ in the $\unicode[STIX]{x1D6E4}=1$ upright cylinder. Similar to the situation in the thin disk, once $\unicode[STIX]{x1D6FF}T$ is normalized by its r.m.s. value $\unicode[STIX]{x1D70E}_{T}$ , the temperature PDFs obtained at a fixed value of $z$ and different values of $Ra$ all fall onto a single master curve. Figure 5 shows the measured $P(\unicode[STIX]{x1D6FF}T)$ as a function of $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}$ for different values of $Ra$ with fixed $Pr=4.4$ (water). The measurements are made in four representative regions of distance $z$ away from the bottom plate. In each region of $z$ , the measured $P(\unicode[STIX]{x1D6FF}T)$ has a unique functional form, which remains unchanged with $Ra$ in the $Ra$ range studied. In the inner region of the BL with $z/\unicode[STIX]{x1D6FF}\lesssim 0.46$ , the measured $P(\unicode[STIX]{x1D6FF}T)$ , as shown in figure 5(a), can also be well described by the Gaussian function given in (3.1) (dashed line).

In a recent experiment, Wang et al. (Reference Wang, Xu, He, Yik, Wang, Schumacher and Tong2018b ) reported that a mixing zone exists outside the BL in the $\unicode[STIX]{x1D6E4}=1$ upright cylinder in the range of $0.04\lesssim z/H\lesssim 0.3$ for $Pr=4.4$ and $0.06\lesssim z/H\lesssim 0.3$ for $Pr=7.6$ . Two characteristic forms of $P(\unicode[STIX]{x1D6FF}T)$ are observed in the mixing zone. Figure 5(b) shows a characteristic form of $P(\unicode[STIX]{x1D6FF}T)$ in the inner region of the mixing zone ( $0.04\lesssim z/H\lesssim 0.07$ ) for different values of $Ra$ and $Pr\simeq 4.4$ . The shape of the measured $P(\unicode[STIX]{x1D6FF}T)$ in this region is similar to that in the thin disk, namely, all the PDFs have a Gaussian-like main portion (dashed line) for small temperature fluctuations ( $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}\lesssim 1$ ) and a long exponential tail for large temperature fluctuations ( $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}\gtrsim 2$ ). In the outer region of the mixing zone ( $0.1\lesssim z/H\lesssim 0.3$ ), the measured $P(\unicode[STIX]{x1D6FF}T)$ has a new characteristic form, which is not observed in the thin disk. Figure 5(c) shows an example of the measured $P(\unicode[STIX]{x1D6FF}T)$ in this region. All of the PDFs for different values of $Ra$ collapse onto a single curve, which has a Gaussian cap (dashed line) for small temperature fluctuations ( $-1\lesssim \unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}\lesssim 1$ ) and two asymmetric exponential tails for larger temperature fluctuations. For the tail with $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}\lesssim -1$ , it smoothly connects to the Gaussian cap and we believe that it results from the cold plumes passing through the measuring position. There is an abrupt change in the local slope of the measured $P(\unicode[STIX]{x1D6FF}T)$ between the Gaussian cap and the exponential tail with $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}\gtrsim 2$ . This result indicates that the properties of the cold and warm plumes in the outer region of the mixing zone are different.

When $z$ moves into the bulk region ( $0.3\lesssim z/H\lesssim 0.5$ ), the measured $P(\unicode[STIX]{x1D6FF}T)$ has a small Gaussian cap with two smooth but asymmetric exponential tails (not shown here). As $z$ moves toward the cell centre ( $z/H=0.5$ ), the asymmetry between the two exponential tails reduces and finally vanishes at the cell centre. Figure 5(d) shows the measured PDFs $P(\unicode[STIX]{x1D6FF}T)$ at the cell centre for different values of $Ra$ . Except for a small round-off near the origin, all of the PDFs are symmetric and have a long exponential tail (solid line) on each side of the distribution with its amplitude varied by more than 4 decades, which falls off much slower than a Gaussian (Kadanoff Reference Kadanoff2001). The exponential PDF $P(\unicode[STIX]{x1D6FF}T)$ was also observed in the previous convection experiments with $Ra$ varied in a wide range from $4\times 10^{7}$ up to $10^{15}$ and in various convecting fluids (Castaing et al. Reference Castaing, Gunaratne, Heslot, Kadanoff, Libchaber, Thomae, Wu, Zaleski and Zanetti1988; Sano et al. Reference Sano, Wu and Libchber1989; Gollub et al. Reference Gollub, Clarke, Gharib, Lane and Mesquita1991; Solomon & Gollub Reference Solomon and Gollub1991; Belmonte et al. Reference Belmonte, Tilgner and Libchaber1994; Niemela et al. Reference Niemela, Skrbek, Sreenivasan and Donnelly2000; Du & Tong Reference Du and Tong2001; Zhou & Xia Reference Zhou and Xia2002; He et al. Reference He, Tong and Xia2007; Wei & Ahlers Reference Wei and Ahlers2016).

Figure 5. Measured PDF $P(\unicode[STIX]{x1D6FF}T)$ of the normalized temperature fluctuation $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}$ for different values of $Ra$ with a fixed value $Pr=4.4$ (water). The measurements are made in the $\unicode[STIX]{x1D6E4}=1$ upright cylinder and in four representative regions of distance $z$ away from the bottom plate centre: (a) $z/\unicode[STIX]{x1D6FF}\lesssim 0.46$ (inner region of the BL), (b) $z/H=0.05$ (inner mixing zone), (c) $z/H=0.16$ (outer mixing zone) and (d)  $z/H=0.5$ (cell centre). The dashed lines in (a–c) show a Gaussian function given in (3.1). The solid line in (b) shows a fit of (4.8) to the data points with $\unicode[STIX]{x1D6FC}_{1}=0.31\pm 0.06$ and $\unicode[STIX]{x1D6FD}=0.18\pm 0.05$ . The solid line in (c) shows a fit of (4.13) to the data points with $\unicode[STIX]{x1D6FC}_{1}=0.33\pm 0.06$ , $\unicode[STIX]{x1D6FC}_{2}=0.92\pm 0.15$ and $\unicode[STIX]{x1D6FD}=0.18\pm 0.05$ . The solid line in (d) is a fit of (4.11) to the data points with $\unicode[STIX]{x1D6FC}=0.30\pm 0.06$ .

Figure 6 shows $P(\unicode[STIX]{x1D6FF}T)$ as a function of $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}$ for different values of $Ra$ with fixed $Pr=7.6$ (20 wt. % aqueous solution of glycerin). It is seen that the measured PDFs $P(\unicode[STIX]{x1D6FF}T)$ in the four representative regions of $z$ for the 20 wt. % glycerin solution have similar behaviours to those as shown in figure 5 for water. At a fixed location $z$ , the measured PDFs $P(\unicode[STIX]{x1D6FF}T)$ with different values of $Ra$ all fall onto a single master curve, which remains unchanged with $Ra$ in the $Ra$ range studied. The shape of $P(\unicode[STIX]{x1D6FF}T)$ continuously evolves with $z$ , from a Gaussian form in the inner region of the BL to an exponential form at the cell centre. For the two working fluids used in the experiment, previous studies (Wu & Libchaber Reference Wu and Libchaber1991; Zhang, Childress & Libchaber Reference Zhang, Childress and Libchaber1997; Chillá et al. Reference Chillá, Rastello, Chaumat and Castaing2004; Ahlers et al. Reference Ahlers, Brown, Araujo, Funfschilling, Grossmann and Lohse2006b ) have shown that the non-Oberbeck–Boussinesq (NOB) effect of distilled water is very weak. The 20 wt. % aqueous solution of glycerin, on the other hand, has a stronger NOB effect, as its viscosity depends sensitively on the fluid temperature. It is seen from figures 5 and 6 that the four distinct forms of the measured $P(\unicode[STIX]{x1D6FF}T)$ in the four flow regions remain approximately the same in the two working fluids, suggesting that the characteristic behaviours of the measured $P(\unicode[STIX]{x1D6FF}T)$ are not influenced very much by the NOB effect.

Figure 6. Measured PDF $P(\unicode[STIX]{x1D6FF}T)$ as a function of $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}$ for different values of $Ra$ with a fixed value $Pr=7.6$ (20 wt. % aqueous solution of glycerin). The measurements are made in the $\unicode[STIX]{x1D6E4}=1$ upright cylinder and in four representative regions of distance $z$ away from the bottom plate: (a) $z/\unicode[STIX]{x1D6FF}\lesssim 0.52$ (inner region of the BL), (b)  $z/H=0.062$ (inner mixing zone), (c) $z/H=0.18$ (outer mixing zone) and (d) $z/H=0.5$ (cell centre). The dashed lines in (a–c) show a Gaussian function given in (3.1). The solid line in (b) shows a fit of (4.8) to the data points with $\unicode[STIX]{x1D6FC}_{1}=0.24\pm 0.04$ and $\unicode[STIX]{x1D6FD}=0.10\pm 0.03$ . The solid line in (c) shows a fit of (4.13) to the data points with $\unicode[STIX]{x1D6FC}_{1}=0.30\pm 0.05$ , $\unicode[STIX]{x1D6FC}_{2}=1.05\pm 0.2$ and $\unicode[STIX]{x1D6FD}=0.10\pm 0.03$ . The solid line in (d) is a fit of (4.11) to the data points with $\unicode[STIX]{x1D6FC}=0.20\pm 0.04$ .

4 Further theoretical analysis

Based on the above experimental results, we now derive expressions of the temperature PDF $P(\unicode[STIX]{x1D6FF}T)$ in different regions. We assume that temperature fluctuations in the convecting fluid are made from two independent sources, as shown in (1.3). The turbulent background is assumed to be homogeneous, so that $\unicode[STIX]{x1D6FF}T_{B}(t)$ is homogeneous both in space and time and has a Gaussian distribution

(4.1) $$\begin{eqnarray}\displaystyle P_{B}(\unicode[STIX]{x1D6FF}T_{B})=\frac{1}{\sqrt{2\unicode[STIX]{x03C0}}\unicode[STIX]{x1D70E}_{B}}e^{-\unicode[STIX]{x1D6FF}T_{B}^{2}/(2\unicode[STIX]{x1D70E}_{B}^{2})},\quad \unicode[STIX]{x1D6FF}T_{B}\in (-\infty ,+\infty ), & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}_{B}$ is the r.m.s. value of $\unicode[STIX]{x1D6FF}T_{B}(t)$ . Because thermal plumes have a non-uniform spatial distribution in the convection cell, $\unicode[STIX]{x1D6FF}T_{P}(t)$ can be further divided into $\unicode[STIX]{x1D6FF}T_{P}^{h}(t)$ for warm plumes near the bottom conducting plate and $\unicode[STIX]{x1D6FF}T_{P}^{c}(t)$ for cold plumes near the top conducting plate. As mentioned in Introduction, the conditional PDF for temperature fluctuations resulting from thermal plumes alone has a simple exponential form (He et al. Reference He, Wang and Tong2018). Therefore, the PDFs for $\unicode[STIX]{x1D6FF}T_{P}^{h}(t)$ and $\unicode[STIX]{x1D6FF}T_{P}^{c}(t)$ can be expressed, respectively, as

(4.2) $$\begin{eqnarray}\displaystyle P_{h}(\unicode[STIX]{x1D6FF}T_{P}^{h})=\frac{1}{\unicode[STIX]{x1D70E}_{h}}e^{-\unicode[STIX]{x1D6FF}T_{P}^{h}/\unicode[STIX]{x1D70E}_{h}},\quad \unicode[STIX]{x1D6FF}T_{P}^{h}\in [0,+\infty ) & & \displaystyle\end{eqnarray}$$

and

(4.3) $$\begin{eqnarray}\displaystyle P_{c}(\unicode[STIX]{x1D6FF}T_{P}^{c})=\frac{1}{\unicode[STIX]{x1D70E}_{c}}e^{\unicode[STIX]{x1D6FF}T_{P}^{c}/\unicode[STIX]{x1D70E}_{c}},\quad \unicode[STIX]{x1D6FF}T_{P}^{c}\in (-\infty ,0], & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}_{h}$ and $\unicode[STIX]{x1D70E}_{c}$ are, respectively, the r.m.s. value of $\unicode[STIX]{x1D6FF}T_{P}^{h}(t)$ and $\unicode[STIX]{x1D6FF}T_{P}^{c}(t)$ . The condition $\unicode[STIX]{x1D6FF}T_{P}^{h}\geqslant 0$ applies for warm plumes and $\unicode[STIX]{x1D6FF}T_{P}^{c}\leqslant 0$ applies for cold plumes, which state that the thermal plumes are fluid particles with their temperature warmer (or colder) than the surroundings.

In the lower half of the bulk region, one finds more warm plumes than cold plumes. In this case, the local temperature fluctuation can be expressed as

(4.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}T(t)\simeq \unicode[STIX]{x1D6FF}T_{B}(t)+\unicode[STIX]{x1D6FF}T_{P}^{h}(t), & & \displaystyle\end{eqnarray}$$

and its PDF is a convolution of (4.1) and (4.2) with

(4.5) $$\begin{eqnarray}\displaystyle P(\unicode[STIX]{x1D6FF}T) & = & \displaystyle \int _{0}^{+\infty }P_{B}(\unicode[STIX]{x1D6FF}T-\unicode[STIX]{x1D6FF}T_{P}^{h})P_{h}(\unicode[STIX]{x1D6FF}T_{P}^{h})\,\text{d}(\unicode[STIX]{x1D6FF}T_{P}^{h})\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{2\unicode[STIX]{x1D70E}_{h}}e^{[(1/2)(\unicode[STIX]{x1D70E}_{B}/\unicode[STIX]{x1D70E}_{h})^{2}-\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{h}-1]}\text{erfc}\left(\frac{\unicode[STIX]{x1D70E}_{B}^{2}/\unicode[STIX]{x1D70E}_{h}-\unicode[STIX]{x1D6FF}T-\unicode[STIX]{x1D70E}_{h}}{\sqrt{2}\unicode[STIX]{x1D70E}_{B}}\right),\end{eqnarray}$$

where $\text{erfc}(x)$ is the complementary error function. Similar convolution was also used in the study of scaler mixing (Le Borgne et al. Reference Le Borgne, Huck, Dentz and Villermaux2017). In (4.5), we have dropped off the mean value of $\unicode[STIX]{x1D6FF}T$ so that $P(\unicode[STIX]{x1D6FF}T)$ has a zero mean. The mean value of $\unicode[STIX]{x1D6FF}T$ represents the contribution of thermal plumes to the mean temperature field, which is small but spatially heterogeneous. As the focus of this paper is on local temperature fluctuations, we will not consider the local mean value of $\unicode[STIX]{x1D6FF}T$ in the rest of our analysis. From (4.5), one can show that $P(\unicode[STIX]{x1D6FF}T)$ has a r.m.s. value of $\unicode[STIX]{x1D70E}_{T}=(\unicode[STIX]{x1D70E}_{B}^{2}+\unicode[STIX]{x1D70E}_{h}^{2})^{1/2}$ . With the normalized variable $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}$ , equation (4.5) can be rewritten as

(4.6) $$\begin{eqnarray}\displaystyle P(\unicode[STIX]{x1D709};\unicode[STIX]{x1D6FC}_{1})=\unicode[STIX]{x1D70E}_{T}P(\unicode[STIX]{x1D70E}_{T}\unicode[STIX]{x1D709})=\frac{\sqrt{1+\unicode[STIX]{x1D6FC}_{1}^{2}}}{2}e^{[\unicode[STIX]{x1D6FC}_{1}^{2}/2-\unicode[STIX]{x1D709}\sqrt{1+\unicode[STIX]{x1D6FC}_{1}^{2}}-1]}\text{erfc}\left(\frac{\unicode[STIX]{x1D6FC}_{1}^{2}-\unicode[STIX]{x1D709}\sqrt{1+\unicode[STIX]{x1D6FC}_{1}^{2}}-1}{\sqrt{2}\unicode[STIX]{x1D6FC}_{1}}\right), & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}_{1}=\unicode[STIX]{x1D70E}_{B}/\unicode[STIX]{x1D70E}_{h}$ . The solid line in figure 4(c) shows a fit of (4.6) to the data points with $\unicode[STIX]{x1D6FC}_{1}=1.1\pm 0.2$ as a fitting parameter. Excellent agreement is found between the theoretical prediction and experimental data. The obtained value of $\unicode[STIX]{x1D6FC}_{1}=1.1$ suggests that the two fluctuation sources have comparable r.m.s. values in the bulk region of the thin disk.

Figure 7. Comparison of the measured power spectra $P_{T}(f)$ of local temperature fluctuations $\unicode[STIX]{x1D6FF}T(t)$ in three representative regions of distance $z$ away from the bottom plate: $z/D=0.026$ (mixing zone, black curve), $z/D=0.33$ (bulk region, red curve) and $z/D=0.5$ (cell centre, blue curve). The measurements are conducted at $Ra=5.3\times 10^{9}$ and $Pr=7.6$ in the thin disk. For clarity, the measured $P_{T}(f)$ is shifted vertically.

In the above, the two signals $\unicode[STIX]{x1D6FF}T_{B}(t)$ and $\unicode[STIX]{x1D6FF}T_{P}^{h}(t)$ are assumed to be continuous in time. However, this is not always the case. For example, it was reported (Song et al. Reference Song, Villermaux and Tong2011) that warm plumes emit periodically from the lower BL to the mixing zone, so that $\unicode[STIX]{x1D6FF}T_{P}^{h}(t)$ in this region is not continuous in time. Figure 7 shows a comparison of the measured power spectra $P_{T}(f)$ of $\unicode[STIX]{x1D6FF}T(t)$ in three representative regions of the thin disk cell. It is seen that in the mixing zone ( $z/D=0.026$ ), the measured $P_{T}(f)$ (black curve) reveals a peak at $f\simeq 0.02~\text{Hz}$ , indicating a periodic emission of warm plumes from the lower BL (Song et al. Reference Song, Villermaux and Tong2011). The frequency peak in $P_{T}(f)$ gradually diminishes with increasing $z$ and eventually vanishes in the bulk region (red curve) and at the cell centre (blue curve), as a result of turbulent mixing and plume–plume interactions, which smear out the coherence of plume emission. This finding suggests that at certain times $t$ , $\unicode[STIX]{x1D6FF}T_{P}^{h}(t)\simeq 0$ and $\unicode[STIX]{x1D6FF}T(t)\simeq \unicode[STIX]{x1D6FF}T_{B}(t)$ . To characterize this temporal intermittency in the mixing zone, we introduce two duty cycle parameters with $\unicode[STIX]{x1D6FD}$ being used to describe the probability for $\unicode[STIX]{x1D6FF}T(t)=\unicode[STIX]{x1D6FF}T_{B}(t)+\unicode[STIX]{x1D6FF}T_{P}^{h}(t)$ and $1-\unicode[STIX]{x1D6FD}$ for the probability that $\unicode[STIX]{x1D6FF}T(t)=\unicode[STIX]{x1D6FF}T_{B}(t)$ . In this case, the PDF of $\unicode[STIX]{x1D6FF}T(t)$ can be written as

(4.7) $$\begin{eqnarray}\displaystyle P(\unicode[STIX]{x1D6FF}T) & = & \displaystyle (1-\unicode[STIX]{x1D6FD})P_{B}(\unicode[STIX]{x1D6FF}T)+\unicode[STIX]{x1D6FD}\int _{0}^{+\infty }P_{B}(\unicode[STIX]{x1D6FF}T-\unicode[STIX]{x1D6FF}T_{P}^{h})P_{h}(\unicode[STIX]{x1D6FF}T_{P}^{h})\,\text{d}(\unicode[STIX]{x1D6FF}T_{P}^{h})\nonumber\\ \displaystyle & = & \displaystyle \frac{1-\unicode[STIX]{x1D6FD}}{\sqrt{2\unicode[STIX]{x03C0}}\unicode[STIX]{x1D70E}_{B}}e^{[-(\unicode[STIX]{x1D6FF}T+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D70E}_{h})^{2}/(2\unicode[STIX]{x1D70E}_{B}^{2})]}+\frac{\unicode[STIX]{x1D6FD}}{2\unicode[STIX]{x1D70E}_{h}}e^{[(1/2)(\unicode[STIX]{x1D70E}_{B}/\unicode[STIX]{x1D70E}_{h})^{2}-\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{h}-\unicode[STIX]{x1D6FD}]}\text{erfc}\left(\frac{\unicode[STIX]{x1D70E}_{B}^{2}/\unicode[STIX]{x1D70E}_{h}-\unicode[STIX]{x1D6FF}T-\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D70E}_{h}}{\sqrt{2}\unicode[STIX]{x1D70E}_{B}}\right).\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

In (4.7) $P(\unicode[STIX]{x1D6FF}T)$ has a r.m.s. value $\unicode[STIX]{x1D70E}_{T}=(\unicode[STIX]{x1D70E}_{B}^{2}+(2\unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6FD}^{2})\unicode[STIX]{x1D70E}_{h}^{2})^{1/2}$ . With the normalized variable $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}$ , equation (4.7) can be rewritten as

(4.8) $$\begin{eqnarray}\displaystyle P(\unicode[STIX]{x1D709};\unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D6FD})=\frac{(1-\unicode[STIX]{x1D6FD})\unicode[STIX]{x1D6FE}}{\sqrt{2\unicode[STIX]{x03C0}}\unicode[STIX]{x1D6FC}_{1}}e^{[-(\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D709})^{2}/(2\unicode[STIX]{x1D6FC}_{1}^{2})]}+\frac{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}}{2}e^{[\unicode[STIX]{x1D6FC}_{1}^{2}/2-\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D709}-\unicode[STIX]{x1D6FD}]}\text{erfc}\left(\frac{\unicode[STIX]{x1D6FC}_{1}^{2}-\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D709}-\unicode[STIX]{x1D6FD}}{\sqrt{2}\unicode[STIX]{x1D6FC}_{1}}\right), & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}=(\unicode[STIX]{x1D6FC}_{1}^{2}+2\unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6FD}^{2})^{1/2}$ .

It is found that the theoretical prediction given in (4.8) can well describe the experimental results in the mixing zone of both the thin disk cell and upright cylinder. The solid line in figure 4(b) shows a fit of (4.8) to the data points with $\unicode[STIX]{x1D6FC}_{1}=0.25\pm 0.05$ and $\unicode[STIX]{x1D6FD}=0.1\pm 0.03$ as two fitting parameters. The solid line in figure 5(b) shows a fit of (4.8) to the data points in the inner mixing zone of the $\unicode[STIX]{x1D6E4}=1$ upright cylinder filled with water ( $Pr=4.4$ ) using $\unicode[STIX]{x1D6FC}_{1}=0.31\pm 0.06$ and $\unicode[STIX]{x1D6FD}=0.18\pm 0.05$ as two fitting parameters. Similarly, the solid line in figure 6(b) shows a fit of (4.8) to the data points in the inner mixing zone of the $\unicode[STIX]{x1D6E4}=1$ upright cylinder filled with the 20 wt. % aqueous solution of glycerin ( $Pr=7.6$ ) using $\unicode[STIX]{x1D6FC}_{1}=0.24\pm 0.04$ and $\unicode[STIX]{x1D6FD}=0.1\pm 0.03$ .

When the measuring position $z$ moves to the cell centre, both warm and cold plumes contribute to temperature fluctuations. In this case, the local temperature fluctuation can be expressed as

(4.9) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}T(t)=\unicode[STIX]{x1D6FF}T_{B}(t)+\unicode[STIX]{x1D6FF}T_{P}^{h}(t)+\unicode[STIX]{x1D6FF}T_{P}^{c}(t), & & \displaystyle\end{eqnarray}$$

and its PDF can be written as

(4.10) $$\begin{eqnarray}\displaystyle P(\unicode[STIX]{x1D6FF}T) & = & \displaystyle \int _{-\infty }^{0}\int _{0}^{+\infty }P_{B}(T-\unicode[STIX]{x1D6FF}T_{P}^{h}-\unicode[STIX]{x1D6FF}T_{P}^{c})P_{h}(\unicode[STIX]{x1D6FF}T_{P}^{h})P_{c}(\unicode[STIX]{x1D6FF}T_{P}^{c})\text{d}(\unicode[STIX]{x1D6FF}T_{P}^{h})\,\text{d}(\unicode[STIX]{x1D6FF}T_{P}^{c})\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{4\unicode[STIX]{x1D70E}_{P}}e^{(\unicode[STIX]{x1D70E}_{B}/\unicode[STIX]{x1D70E}_{P})^{2}/2}\left[e^{\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{P}}\text{erfc}\left(\frac{\unicode[STIX]{x1D70E}_{B}^{2}+\unicode[STIX]{x1D6FF}T\unicode[STIX]{x1D70E}_{P}}{\sqrt{2}\unicode[STIX]{x1D70E}_{P}\unicode[STIX]{x1D70E}_{B}}\right)+e^{-\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{P}}\text{erfc}\left(\frac{\unicode[STIX]{x1D70E}_{B}^{2}-\unicode[STIX]{x1D6FF}T\unicode[STIX]{x1D70E}_{P}}{\sqrt{2}\unicode[STIX]{x1D70E}_{P}\unicode[STIX]{x1D70E}_{B}}\right)\right],\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}_{P}\equiv \unicode[STIX]{x1D70E}_{h}=\unicode[STIX]{x1D70E}_{c}$ , because the system is symmetric at the cell centre. In (4.10), $P(\unicode[STIX]{x1D6FF}T)$ has a r.m.s. value $\unicode[STIX]{x1D70E}_{T}=(\unicode[STIX]{x1D70E}_{B}^{2}+2\unicode[STIX]{x1D70E}_{P}^{2})^{1/2}$ . With the normalized variable $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}$ , equation (4.10) can be rewritten as

(4.11) $$\begin{eqnarray}\displaystyle P(\unicode[STIX]{x1D709};\unicode[STIX]{x1D6FC}) & = & \displaystyle \frac{\sqrt{\unicode[STIX]{x1D6FC}^{2}+2}}{4}e^{\unicode[STIX]{x1D6FC}^{2}/2}\left[e^{\unicode[STIX]{x1D709}\sqrt{\unicode[STIX]{x1D6FC}^{2}+2}}\text{erfc}\left(\frac{\unicode[STIX]{x1D6FC}^{2}+\unicode[STIX]{x1D709}\sqrt{\unicode[STIX]{x1D6FC}^{2}+2}}{\sqrt{2}\unicode[STIX]{x1D6FC}}\right)\right.\nonumber\\ \displaystyle & & \displaystyle \left.+e^{-\unicode[STIX]{x1D709}\sqrt{\unicode[STIX]{x1D6FC}^{2}+2}}\text{erfc}\left(\frac{\unicode[STIX]{x1D6FC}^{2}-\unicode[STIX]{x1D709}\sqrt{\unicode[STIX]{x1D6FC}^{2}+2}}{\sqrt{2}\unicode[STIX]{x1D6FC}}\right)\right],\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D70E}_{B}/\unicode[STIX]{x1D70E}_{P}$ . Equation (4.11) is a symmetric function of $\unicode[STIX]{x1D709}$ with the exponential distribution ( $\unicode[STIX]{x1D6FC}=0$ ) and Gaussian distribution ( $\unicode[STIX]{x1D6FC}=+\infty$ ) being its two asymptotic forms.

It is found that the theoretical prediction given in (4.11) can well describe the experimental results at the centre of both convection cells. The solid line in figure 4(d) shows a fit of (4.11) to the data points in the thin disk with $\unicode[STIX]{x1D6FC}=1.7\pm 0.3$ as a fitting parameter. The solid line in figure 5(d) shows a fit of (4.11) to the data points at the centre of the $\unicode[STIX]{x1D6E4}=1$ upright cylinder filled with water ( $Pr=4.4$ ) using $\unicode[STIX]{x1D6FC}=0.30\pm 0.06$ . Similarly, the solid line in figure 6(d) shows a fit of (4.11) to the data points at the centre of the $\unicode[STIX]{x1D6E4}=1$ upright cylinder filled with the 20 wt. % aqueous solution of glycerin ( $Pr=7.6$ ) using $\unicode[STIX]{x1D6FC}=0.20\pm 0.04$ .

Figure 8. Measured temperature PDF $P(\unicode[STIX]{x1D6FF}T)$ as a function of $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}$ for different values of $Ra$ and at a fixed value of $Pr=4.4$ (a) and $Pr=7.6$ (b). The measurements are conducted at the centre of the $\unicode[STIX]{x1D6E4}=1$ upright cylinder, and the data in (a) and (b) are the same as those shown in figure 5(d) and figure 6(d), respectively. For clarity, the measured PDFs are plotted on a linear scale showing an expanded view of the round-off cap region near $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}\simeq 0$ . The solid lines are the fits of (4.11) to the data points with $\unicode[STIX]{x1D6FC}=0.30\pm 0.06$ (a) and $\unicode[STIX]{x1D6FC}=0.20\pm 0.04$ (b). The horizontal dashed line shows the expected peak value $1/\sqrt{2\unicode[STIX]{x03C0}}$ for a normalized Gaussian distribution, and the horizontal dotted line shows the expected peak value $1/\sqrt{2}$ for a normalized exponential distribution.

While the effect of background fluctuations at the cell centre is relatively small, it nevertheless provides a measurable mixing effect to smear out the distribution of thermal plumes. Mathematically, a pure exponential distribution has a normalized form, $P(\unicode[STIX]{x1D709})=(1/\sqrt{2})\exp (-\sqrt{2}|\unicode[STIX]{x1D709}|)$ , so that its peak value at $\unicode[STIX]{x1D709}=0$ is $P_{max}=1/\sqrt{2}\simeq 0.707$ . On the other hand, the peak value of a normalized Gaussian distribution is $P_{max}=1/\sqrt{2\unicode[STIX]{x03C0}}\simeq 0.399$ . Figure 8 shows a magnified view of the measured $P(\unicode[STIX]{x1D6FF}T)$ near the origin. It is seen that all the measured values of $P_{max}$ for different values of $Ra$ and $Pr$ are in between $1/\sqrt{2}$ (dotted lines) and $1/\sqrt{2\unicode[STIX]{x03C0}}$ (dashed lines), indicating the existence of a Gaussian component. In a physical system, this Gaussian component at the origin precludes the non-analytic sharp peak of the exponential distribution (Wei & Ahlers Reference Wei and Ahlers2016).

Equation (4.11) can well describe the temperature PDFs obtained not only from the above experiments but also from previous experiments (Wu & Libchaber Reference Wu and Libchaber1992; Du & Tong Reference Du and Tong2001). Figure 9 shows the measured $P(\unicode[STIX]{x1D6FF}T)$ as a function of $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}$ for a low-temperature (5 K) helium gas ( $Pr=0.7$ ) over a range of $Ra$ . The measurements were conducted at the centre of a $\unicode[STIX]{x1D6E4}=0.5$ upright cylinder (Wu & Libchaber Reference Wu and Libchaber1992). The solid line shows a fit of (4.11) to the data points with $\unicode[STIX]{x1D6FC}=1.2\pm 0.2$ as a fitting parameter. Similar measurements at the centre of a $\unicode[STIX]{x1D6E4}=6.7$ upright cylinder with $Pr=0.7$ (Wu & Libchaber Reference Wu and Libchaber1992) are also found to be well described by (4.11) with $\unicode[STIX]{x1D6FC}=1.5\pm 0.3$ (not shown here). These results further confirm that our model is quite general and can be used to describe temperature PDFs in a closed convection cell with different aspect ratios and filled with a working fluid with different values of $Ra$ and $Pr$ .

Finally, we consider a general case described in (4.9), in which $\unicode[STIX]{x1D6FF}T_{P}^{h}(t)$ is intermittent with its duty ratio being $\unicode[STIX]{x1D6FD}$ . The corresponding PDF $P(\unicode[STIX]{x1D6FF}T)$ is given as

(4.12) $$\begin{eqnarray}\displaystyle P(\unicode[STIX]{x1D6FF}T) & = & \displaystyle (1-\unicode[STIX]{x1D6FD})\int _{-\infty }^{\unicode[STIX]{x1D70E}_{c}}P_{B}(\unicode[STIX]{x1D6FF}T-\unicode[STIX]{x1D6FF}T_{P}^{c})P_{c}(\unicode[STIX]{x1D6FF}T_{P}^{c})\,\text{d}(\unicode[STIX]{x1D6FF}T_{P}^{c})\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D6FD}\int _{-\infty }^{\unicode[STIX]{x1D70E}_{c}}\int _{-\unicode[STIX]{x1D70E}_{h}}^{+\infty }P_{B}(\unicode[STIX]{x1D6FF}T-\unicode[STIX]{x1D6FF}T_{P}^{h}-\unicode[STIX]{x1D6FF}T_{P}^{c})P_{h}(\unicode[STIX]{x1D6FF}T_{P}^{h})P_{c}(\unicode[STIX]{x1D6FF}T_{P}^{c})\,\text{d}(\unicode[STIX]{x1D6FF}T_{P}^{h})\,\text{d}(\unicode[STIX]{x1D6FF}T_{P}^{c})\nonumber\\ \displaystyle & = & \displaystyle \frac{1-\unicode[STIX]{x1D6FD}}{2\unicode[STIX]{x1D70E}_{c}}e^{[(\unicode[STIX]{x1D70E}_{B}/\unicode[STIX]{x1D70E}_{c})^{2}/2+(\unicode[STIX]{x1D6FF}T+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D70E}_{h})/\unicode[STIX]{x1D70E}_{c}-1]}\text{erfc}\left(\frac{\unicode[STIX]{x1D70E}_{B}^{2}+(\unicode[STIX]{x1D6FF}T+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D70E}_{h}-\unicode[STIX]{x1D70E}_{c})\unicode[STIX]{x1D70E}_{c}}{\sqrt{2}\unicode[STIX]{x1D70E}_{B}\unicode[STIX]{x1D70E}_{c}}\right)\nonumber\\ \displaystyle & & \displaystyle +\,\frac{\unicode[STIX]{x1D6FD}}{2(\unicode[STIX]{x1D70E}_{h}+\unicode[STIX]{x1D70E}_{c})}e^{[(\unicode[STIX]{x1D70E}_{B}/\unicode[STIX]{x1D70E}_{h})^{2}/2-(\unicode[STIX]{x1D6FF}T-\unicode[STIX]{x1D70E}_{c})/\unicode[STIX]{x1D70E}_{h}-\unicode[STIX]{x1D6FD}]}\text{erfc}\left(\frac{\unicode[STIX]{x1D70E}_{B}^{2}-(\unicode[STIX]{x1D6FF}T+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D70E}_{h}-\unicode[STIX]{x1D70E}_{c})\unicode[STIX]{x1D70E}_{h}}{\sqrt{2}\unicode[STIX]{x1D70E}_{B}\unicode[STIX]{x1D70E}_{h}}\right)\nonumber\\ \displaystyle & & \displaystyle +\,\frac{\unicode[STIX]{x1D6FD}}{2(\unicode[STIX]{x1D70E}_{h}+\unicode[STIX]{x1D70E}_{c})}e^{[(\unicode[STIX]{x1D70E}_{B}/\unicode[STIX]{x1D70E}_{c})^{2}/2+(\unicode[STIX]{x1D6FF}T+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D70E}_{h})/\unicode[STIX]{x1D70E}_{c}-1]}\text{erfc}\left(\frac{\unicode[STIX]{x1D70E}_{B}^{2}+(\unicode[STIX]{x1D6FF}T+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D70E}_{h}-\unicode[STIX]{x1D70E}_{c})\unicode[STIX]{x1D70E}_{c}}{\sqrt{2}\unicode[STIX]{x1D70E}_{B}\unicode[STIX]{x1D70E}_{c}}\right).\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

In (4.12), $P(\unicode[STIX]{x1D6FF}T)$ has a r.m.s. value $\unicode[STIX]{x1D70E}_{T}=(\unicode[STIX]{x1D70E}_{B}^{2}+\unicode[STIX]{x1D70E}_{c}^{2}+(2\unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6FD}^{2})\unicode[STIX]{x1D70E}_{h}^{2})^{1/2}$ . With the normalized variable $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}$ , equation (4.12) can be rewritten as

(4.13) $$\begin{eqnarray}\displaystyle P(\unicode[STIX]{x1D709};\unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D6FC}_{2},\unicode[STIX]{x1D6FD}) & = & \displaystyle \frac{(1-\unicode[STIX]{x1D6FD})\unicode[STIX]{x1D6FE}_{2}}{2}e^{[\unicode[STIX]{x1D6FC}_{2}^{2}/2+\unicode[STIX]{x1D6FE}_{2}\unicode[STIX]{x1D709}+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FC}_{2}/\unicode[STIX]{x1D6FC}_{1}-1]}\text{erfc}\left(\frac{\unicode[STIX]{x1D6FC}_{2}^{2}+\unicode[STIX]{x1D6FE}_{2}\unicode[STIX]{x1D709}+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FC}_{2}/\unicode[STIX]{x1D6FC}_{1}-1}{\sqrt{2}\unicode[STIX]{x1D6FC}_{2}}\right)\nonumber\\ \displaystyle & & \displaystyle +\,\frac{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D6FE}_{2}}{2(\unicode[STIX]{x1D6FC}_{1}+\unicode[STIX]{x1D6FC}_{2})}e^{[\unicode[STIX]{x1D6FC}_{1}^{2}/2-\unicode[STIX]{x1D6FE}_{1}\unicode[STIX]{x1D709}+\unicode[STIX]{x1D6FC}_{1}/\unicode[STIX]{x1D6FC}_{2}-\unicode[STIX]{x1D6FD}]}\text{erfc}\left(\frac{\unicode[STIX]{x1D6FC}_{1}^{2}-\unicode[STIX]{x1D6FE}_{1}\unicode[STIX]{x1D709}+\unicode[STIX]{x1D6FC}_{1}/\unicode[STIX]{x1D6FC}_{2}-\unicode[STIX]{x1D6FD}}{\sqrt{2}\unicode[STIX]{x1D6FC}_{1}}\right)\nonumber\\ \displaystyle & & \displaystyle +\,\frac{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D6FE}_{2}}{2(\unicode[STIX]{x1D6FC}_{1}+\unicode[STIX]{x1D6FC}_{2})}e^{[\unicode[STIX]{x1D6FC}_{2}^{2}/2+\unicode[STIX]{x1D6FE}_{2}\unicode[STIX]{x1D709}+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FC}_{2}/\unicode[STIX]{x1D6FC}_{1}-1]}\text{erfc}\left(\frac{\unicode[STIX]{x1D6FC}_{2}^{2}+\unicode[STIX]{x1D6FE}_{2}\unicode[STIX]{x1D709}+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FC}_{2}/\unicode[STIX]{x1D6FC}_{1}-1}{\sqrt{2}\unicode[STIX]{x1D6FC}_{2}}\right),\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

with $\unicode[STIX]{x1D6FC}_{1}=\unicode[STIX]{x1D70E}_{B}/\unicode[STIX]{x1D70E}_{h}$ , $\unicode[STIX]{x1D6FC}_{2}=\unicode[STIX]{x1D70E}_{B}/\unicode[STIX]{x1D70E}_{c}$ , $\unicode[STIX]{x1D6FE}_{1}=(\unicode[STIX]{x1D6FC}_{1}^{2}+\unicode[STIX]{x1D6FC}_{1}^{2}/\unicode[STIX]{x1D6FC}_{2}^{2}+2\unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6FD}^{2})^{1/2}$ and $\unicode[STIX]{x1D6FE}_{2}=(\unicode[STIX]{x1D6FC}_{2}^{2}+(2\unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6FD}^{2})\unicode[STIX]{x1D6FC}_{2}^{2}/\unicode[STIX]{x1D6FC}_{1}^{2}+1)^{1/2}$ .

Equation (4.13) is the most general form of $P(\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T})$ and includes the asymptotic cases given by (4.6), (4.8) and (4.11). In the case $\unicode[STIX]{x1D6FC}_{2}=+\infty$ and $\unicode[STIX]{x1D6FD}=1$ , equation (4.13) reduces to (4.6). In the case $\unicode[STIX]{x1D6FC}_{2}=+\infty$ , equation (4.13) reduces to (4.8). In the case $\unicode[STIX]{x1D6FC}_{1}=\unicode[STIX]{x1D6FC}_{2}=\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}=1$ , equation (4.13) reduces to (4.11). It is found that (4.13) can well describe the measured $P(\unicode[STIX]{x1D6FF}T)$ in the outer mixing zone of the $\unicode[STIX]{x1D6E4}=1$ upright cylinder, in which cold plumes are continuous (far away from the upper cooling plate and thus lose their intermittency) and warm plumes are highly intermittent (close to the lower heating plate). The solid line in figure 5(c) shows a fit of (4.13) to the data points in the outer mixing zone of the $\unicode[STIX]{x1D6E4}=1$ upright cylinder filled with water ( $Pr=4.4$ ) using $\unicode[STIX]{x1D6FC}_{1}=0.33\pm 0.06$ , $\unicode[STIX]{x1D6FC}_{2}=0.92\pm 0.15$ and $\unicode[STIX]{x1D6FD}=0.18\pm 0.05$ . Similarly, the solid line in figure 6(c) shows a fit of (4.13) to the data points in the outer mixing zone of the $\unicode[STIX]{x1D6E4}=1$ upright cylinder filled with the 20 wt. % aqueous solution of glycerin ( $Pr=7.6$ ) using $\unicode[STIX]{x1D6FC}_{1}=0.30\pm 0.05$ , $\unicode[STIX]{x1D6FC}_{2}=1.05\pm 0.2$ and $\unicode[STIX]{x1D6FD}=0.10\pm 0.03$ .

Figure 9. Measured temperature PDF $P(\unicode[STIX]{x1D6FF}T)$ as a function of $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}$ at the centre of a $\unicode[STIX]{x1D6E4}=0.5$ upright cylinder (red line). The measurements were conducted at a fixed value of $Pr=0.7$ (a low-temperature helium gas) in the $Ra$ range from $4\times 10^{9}$ to $6\times 10^{14}$ . The data are taken from figure 1(b) in Wu & Libchaber (Reference Wu and Libchaber1992). The solid line shows a fit of (4.11) to the data points with $\unicode[STIX]{x1D6FC}=1.2\pm 0.2$ . The dashed and dotted lines show, respectively, a simple exponential distribution function and a Gaussian distribution function for comparison.

Table 1. Obtained values of the parameters $\unicode[STIX]{x1D6FC}_{1}$ , $\unicode[STIX]{x1D6FC}_{2}$ and $\unicode[STIX]{x1D6FD}$ in (4.13) for three different flow regions in the two convection cells. The numbers with an error bar indicate the fitted values with a fitting uncertainty and those without an error bar are the expected values in theory.

Table 1 gives a summary of the final fitting results obtained in the two convection cells. It is seen that the fitted values of $\unicode[STIX]{x1D6FC}_{1}=\unicode[STIX]{x1D70E}_{B}/\unicode[STIX]{x1D70E}_{h}$ in the thin disk cell increase monotonically, as the measuring position $z$ moves from the mixing zone to the cell centre (see figure 3). This result suggests that the contribution of warm plumes to the local temperature fluctuations decreases with increasing $z$ , as expected. However, the fitted values of $\unicode[STIX]{x1D6FC}_{1}$ in the upright cylinder do not change much in the three different flow regions. The fitted values of $\unicode[STIX]{x1D6FC}_{1}$ in the turbulent bulk region and at the centre of the upright cylinder are much smaller than those obtained in the thin disk. These findings indicate that the effect of thermal plumes in the upright cylinder is much larger than that in the thin disk. This is because the quasi-two-dimensional LSC in the thin disk is more stable than the LSC in the upright cylinder, which gives rise to a more organized spatial distribution of thermal plumes, reducing their probabilities to go through the bulk region. Such an aspect-ratio-dependent $P(\unicode[STIX]{x1D6FF}T)$ at the cell centre was also reported previously by Solomon & Gollub (Reference Solomon and Gollub1991).

It is also seen that the fitted value of $\unicode[STIX]{x1D6FD}$ at $Pr=7.6$ is $\unicode[STIX]{x1D6FD}=0.1$ , which indicates that the warm plumes are active for only 10 % of the measuring time, so that they are highly intermittent in the mixing zone. The fact that the obtained value of $\unicode[STIX]{x1D6FD}$ in the thin disk is the same as that in the upright cylinder suggests that the fitted value of $\unicode[STIX]{x1D6FD}=0.1$ is not sensitive to the changes of cell geometry and $Ra$ . Table 1 also reveals that the fitted values of $\unicode[STIX]{x1D6FC}_{1}$ and $\unicode[STIX]{x1D6FD}$ in the upright cylinder decrease slightly with increasing $Pr$ . This is because the thermal plumes in the mixing zone become more coherent with increasing $Pr$ . Here ‘coherent’ has two meanings. First, because the warm plumes at larger $Pr$ have a longer lifetime, they are relatively hotter than the warm plumes at lower $Pr$ (at the same position $z$ ). As a result, the plume-induced temperature fluctuations have a larger amplitude, leading to a smaller value of $\unicode[STIX]{x1D6FC}_{1}$ at larger $Pr$ . Second, because the thermal plumes at larger $Pr$ are more localized in space (less thermal diffusion) and move faster (more buoyant), they have a shorter time passing through the temperature probe and thus a smaller value of $\unicode[STIX]{x1D6FD}$ .

5 Summary

We have carried out a systematic study of spatial variations of the PDF $P(\unicode[STIX]{x1D6FF}T)$ for temperature fluctuations $\unicode[STIX]{x1D6FF}T$ in turbulent Rayleigh–Bénard convection. The local temperature measurements were made along the central axis of two closed convection cells: one is a thin circular disk cell and the other is an upright cylinder of aspect ratio unity. In the experiment, the Rayleigh number $Ra$ was varied in the range between $9\times 10^{8}$ and $1.2\times 10^{10}$ with two fixed Prandtl numbers $Pr=4.4$ and $Pr=7.6$ . The measured $P(\unicode[STIX]{x1D6FF}T)$ reveals distinct changes in four different flow regions in the lower half of the cell, namely, the thermal BL near the bottom conducting place, mixing zone outside the BL, lower turbulent bulk region and cell centre. Because of the symmetry of the cell shape, one expects that similar regions exist in the upper half of the cell. These distinct changes in the functional form of $P(\unicode[STIX]{x1D6FF}T)$ remain unchanged for different values of $Ra$ and $Pr$ , once the normalized temperature fluctuation $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}$ is used as the scaling variable. Here $\unicode[STIX]{x1D70E}_{T}$ is the r.m.s. value of $\unicode[STIX]{x1D6FF}T$ .

In the inner region of the BL, where the mean temperature gradient remains approximately constant, the measured $P(\unicode[STIX]{x1D6FF}T)$ has a Gaussian form in both convection cells. In the mixing zone outside the BL, the measured $P(\unicode[STIX]{x1D6FF}T)$ in the thin disk cell has a Gaussian shape in the main body but with an exponential tail for large positive values of $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}$ . Such an exponential tail is caused by the periodic emission of warm plumes from the bottom BL, which also gives rise to an abrupt change in the local slope of the measured exponential tail of $P(\unicode[STIX]{x1D6FF}T)$ . A similar $P(\unicode[STIX]{x1D6FF}T)$ is also found in the upright cylinder. In the outer region of the mixing zone, the measured $P(\unicode[STIX]{x1D6FF}T)$ in the upright cylinder evolves to a new shape containing two asymmetric exponential tails for large absolute values of $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}$ . This new shape of $P(\unicode[STIX]{x1D6FF}T)$ is not observed in the thin disk cell. In the lower turbulent bulk region of both convection cells, the abrupt change in the local slope of the measured exponential tail of $P(\unicode[STIX]{x1D6FF}T)$ disappears, suggesting that the effect of the periodic emission of thermal plumes is diminished. In this case, the measured $P(\unicode[STIX]{x1D6FF}T)$ in the thin disk has a Gaussian shape in the main body and a smooth exponential tail for large positive values of $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}$ . The measured $P(\unicode[STIX]{x1D6FF}T)$ in the upright cylinder has a Gaussian cap and two asymmetric but smooth exponential tails for large absolute values of $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}$ . At the centre of both convection cells, because of the symmetry of the convection cell, the measured $P(\unicode[STIX]{x1D6FF}T)$ has a symmetric shape with a Gaussian cap for small absolute values of $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}$ and two long exponential tails for large absolute values of $\unicode[STIX]{x1D6FF}T/\unicode[STIX]{x1D70E}_{T}$ .

Based on the above experimental findings, we proposed a theoretical model which assumes that temperature fluctuations in different regions of the convection cell are all made from two independent sources, namely, a homogeneous background which obeys Gaussian statistics and non-uniform thermal plumes with an exponential distribution. As a result, the temperature PDF $P(\unicode[STIX]{x1D6FF}T)$ is uniquely described by a convolution of the two modes of fluctuations with a relative strength, which varies among the four different flow regions. The predicted analytic expressions of $P(\unicode[STIX]{x1D6FF}T)$ , as shown in (4.6), (4.8), (4.11) and (4.13), agree well with the experimental results obtained in the present experiment and also in previous experiments. Our model thus provides a unique theoretical framework with a common set of parameters to quantitatively describe the effect of turbulent background, thermal plumes and their spatio-temporal intermittency on the temperature PDF $P(\unicode[STIX]{x1D6FF}T)$ . Such an understanding of turbulent fluctuations with decomposition of fluctuation modes based on the coherent structures in the flow is very useful not only for the study of temperature fluctuations in turbulent thermal convection but also for the understanding of turbulent mixing problems of practical interest.