2.1 Experimental configuration

The experimental convection cell is a 41. 5 cm × 10. 5 cm × 41. 5 cm rectangular cell, with aspect ratios $\unicode[STIX]{x1D6E4}_{xz}=1$ and $\unicode[STIX]{x1D6E4}_{yz}=0.253$. The details of this cell have been published in our previous works (Salort et al. Reference Salort, Liot, Rusaouen, Seychelles, Tisserand, Creyssels, Castaing and Chillà2014; Liot et al. Reference Liot, Ehlinger, Rusaouën, Coudarchet, Salort and Chillà2017). The only difference is that the walls are now made of glass, instead of PMMA, to avoid spurious optical index gradients in the walls. Because the heat conductivity of glass is larger than PMMA, this means that we have to work with a bulk temperature close to the temperature of the room, to avoid heat flux across the walls. Because the cell cannot be fully insulated to preserve optical access, only a sufficiently high heating power can be used for the heat losses to remain negligible. The experimental conditions are summarized in table 1.

We recall the main characteristics. The bottom plate is rough, made of aluminium alloy (5083) anodized in black. The roughness consists in periodic parallelepiped roughness elements $2~\text{mm}\times 5~\text{mm}$, machined directly into the plate. The top plate is a 4 cm thick smooth copper plate coated with a thin layer of nickel. Plate temperatures are measured with PT-100 temperature sensors. The bottom plate is heated with Kapton heaters glued at the bottom of the plate, and powered with a constant voltage. The top plate is cooled with the circulation of a mixture of water and ethylene glycol, flowing into a tight meander machined at the top of the copper plate, and temperature regulated by a Lauda RP845 chiller.

2.2 Numerical configuration

The numerical configuration is detailed in Belkadi (Reference Belkadi2019). We consider a physical configuration close to the experimental one: a 40 cm high Rayleigh–Bénard cell of square vertical cross-section, filled with water at the mean temperature ($T_{mean}$), with a rough bottom plate and a smooth top plate. The roughness consists in regularly spaced parallelepiped roughness elements. The main characteristics of the physical problem are given in table 1. In this paper, the Rayleigh number is set to $Ra=2\times 10^{9}$.

The fluid flow is governed by the Boussinesq equations, where the gravity vector $\boldsymbol{g}$ is aligned with the $-\boldsymbol{e}_{z}$ direction. A no-slip condition for the velocity field is applied on the walls. The temperature difference $\unicode[STIX]{x0394}T$ is imposed on the top and bottom walls including the roughness element surfaces by applying isothermal boundary conditions ($T_{top}=T_{mean}-\unicode[STIX]{x0394}T/2$ and $T_{bot}=T_{mean}+\unicode[STIX]{x0394}T/2$), while adiabaticity $\unicode[STIX]{x2202}T/\unicode[STIX]{x2202}\boldsymbol{n}=0$ is satisfied on sidewalls.

The Boussinesq solver, called SUNFLUIDH, uses a finite volume discretization on staggered grids. The time marching is treated by a semi-implicit scheme combined with a prediction–correction scheme for the velocity–pressure coupling, with a global second-order accuracy. The pressure problem is solved by a multi-grid method. Roughness elements are modelled through a loop truncation technique. A domain decomposition method using MPI is implemented for parallel computation.

The simulation has been performed on a non-uniform grid with 768 × 384 × 768 cells in the ($\boldsymbol{e}_{x},\boldsymbol{e}_{y},\boldsymbol{e}_{z}$) directions respectively. The mesh has been particularly refined between roughness elements, and the cell size is everywhere finer than the Kolmogorov scale. The time step, equal to 0.011925 s, meets the Courant–Friedrichs–Lewy criterion. The spatial resolution has been checked by evaluating the numerical convergence of time-averaged Nusselt numbers obtained by different methods (space averages on the horizontal planes or volume averages based on the dissipation rates) as proposed by Shishkina et al. (Reference Shishkina, Stevens, Grossmann and Lohse2010). Statistics have been determined over more than one hour and 20 min (physical time) to obtain these values. They converge with a deviation of less than 1 % of the average Nusselt number, as specified in table 1.

3.1 Experimental set-up

There are several possible shadowgraph set-ups (Settles Reference Settles2001). The classical, and most used set-up requires parallel incident light. This is obtained with either a concave mirror or a lens. However, for fields of view larger than 15 cm, this is impractical, or would require the use of a parabolic telescope mirror. The alternative, that we used in this work, is to implement direct shadowgraphy in diverging light with a small, or punctual, bright source of light. This is the method originally used by Marat (Reference Marat1780).

Figure 1.

Optical set-up for the shadowgraph experiment. (a) Quartz iodine lamp, heat absorbing filter and diaphragm; (b) Rayleigh–Bénard convection cell; (c) ground glass diffuser; (d) PCO 1600 camera.

In this work, the convection cell is illuminated by a punctual source of white light, as shown in figure 1. The cell depth is four times smaller than the width and height, so that the flow is quasi-bidimensional and the integration along the depth does not significantly blur the convection patterns.

The shadowgraph pattern is captured on a projection screen, and recorded with a PCO 1600 digital camera. The resolution is $1600~\text{pixels}\times 1200~\text{pixels}$ and the frame rate is 10 f.p.s., corresponding to 800 images per turnover, where the turnover time $T_{turnover}$ is

(3.1)$$\begin{eqnarray}T_{turnover}=\frac{4h}{v_{wind}},\end{eqnarray}$$

and $v_{wind}\approx 2~\text{cm}~\text{s}^{-1}$. Twenty series of 2400 images were recorded, amounting to 80 min, which is 60 turnover times. The recorded image, $I(x,z)$, is determined by the second derivative of the refractive index,

(3.2)$$\begin{eqnarray}\frac{I(x,z)}{I_{0}}=\frac{\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D702}-\unicode[STIX]{x1D6FE})}{\unicode[STIX]{x1D702}}\int _{0}^{D}\left(\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}x^{2}}+\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}z^{2}}\right)(\ln n)\,\text{d}y,\end{eqnarray}$$

where $I_{0}$ is the intensity of the incident light, $\unicode[STIX]{x1D6FE}=15~\text{cm}$ is the distance between the cell and the projection screen and $\unicode[STIX]{x1D702}=265~\text{cm}$ is the distance between the light source and the projection screen. Since in this set-up, $\unicode[STIX]{x1D702}/\unicode[STIX]{x1D6FE}\gg 1$, the sensitivity function $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D702}-\unicode[STIX]{x1D6FE})/\unicode[STIX]{x1D702}$ is not very large, and nearly proportional to $\unicode[STIX]{x1D6FE}$, and the magnification factor $m$

(3.3)$$\begin{eqnarray}m=\frac{\unicode[STIX]{x1D702}}{\unicode[STIX]{x1D702}-\unicode[STIX]{x1D6FE}},\end{eqnarray}$$

is close to unity. This is close to the case of parallel light, and is sufficient to image plumes in Rayleigh–Bénard convection.

In practice, in the case of thermal convection, the variation of optical index is caused by variations of temperature, which are small in the bulk. Equation (3.2) can thus be written in a simpler form,

(3.4)$$\begin{eqnarray}\frac{I(x,z)}{I_{0}}\propto \unicode[STIX]{x1D6FE}\int _{0}^{D}\left(\frac{\unicode[STIX]{x2202}^{2}T}{\unicode[STIX]{x2202}x^{2}}+\frac{\unicode[STIX]{x2202}^{2}T}{\unicode[STIX]{x2202}z^{2}}\right)\,\text{d}y.\end{eqnarray}$$

Figure 2.

(a) Experimental full field shadowgraph image at $Ra=5.6\times 10^{10}$. The colour bar indicates the relative light intensity, $I(x,z)/I_{0}(x,z)-1.0$, where $I_{0}(x,z)$ is the time-averaged image of the cell (see text). (b) Synthetic full field shadowgraph image from the numerical simulation at $Ra=2\times 10^{9}$, computed from the full three-dimensional temperature field using (3.4). (c) Snapshot of the mid-plane ($y=10~\text{cm}$) temperature of the numerical simulation at the same time as the shadowgraph. Only the fluid trapped between the roughness elements is visible at this particular position (see the dashed line in (d)). (d) Snapshot of the transverse ($x=29~\text{cm}$) temperature. The blue dashed lines in (b,c,d) show the intersection of the planes.

A snapshot of the experimental shadowgraph is shown in figure 2(a). The raw image is numerically enhanced by dividing the field by the reference image $I_{0}(x,z)$. This reference image can be obtained with the fluid at rest, or by averaging the recording in time. Indeed, the plume pattern is lighter and darker, and averages to zero. We found no practical difference between the two estimates of $I_{0}(x,z)$, and found it more practical to use the time average. As shown in the figure, the variations of light intensity induced by the thermal plumes is $\pm$20 %.

3.2 Synthetic shadowgraph in numerical simulations

A numerical shadowgraph can be derived from the temperature field using (3.4), at regular time steps 0.5366 s, and produces a sequence of synthetic images, similar to those obtained in the experiments (see figure 2b). To that end, 5524 integrated two-dimensional (2-D) Laplacian fields were saved, spanning over 49 min in dimensional time, with a numerical frame rate of 1.86 f.p.s. The mean wind is $0.4~\text{cm}~\text{s}^{-1}$, so the numerical frame rate corresponds to 745 images per turnover, and a total recording time of 7.4 turnovers. The time resolution is therefore of a similar order of magnitude as the experimental recording frame rate, but with a shorter duration in terms of turnover times. However, the Rayleigh number is also smaller, so fewer turnover times are sufficient to reach statistical convergence. This allows us to use the same type of image, and similar processing tools, both for the experiment and the numerics.

The comparison with the experimental shadowgraph, shown in figure 2(a), evidences more plumes, and smaller plumes in the experiment. This is due to the difference in Rayleigh numbers. The numerical shadowgraph of the present work can also be compared to experimental shadowgraphs obtained in water at lower Rayleigh numbers (Wang et al. Reference Wang, Lai, Song and Tong2018; Tummers & Steunebrink Reference Tummers and Steunebrink2019), and higher Rayleigh numbers (Xie & Xia Reference Xie and Xia2017).

In this paper, we focus on the analysis of the shadowgraph itself, and use the numerical simulations to validate the interpretation of the field. With that aim, it is not necessary to have experiments and numerical simulations in the same range of Rayleigh numbers. The role of the Rayleigh number, and the various regimes in rough cells, is beyond the scope of the present work, but would benefit from the tools described here, in particular in situations where shadowgraphs may be the only available investigation tool.

A snapshot of the mid-plane temperature field is shown in figure 2(c). The mean temperature value is larger than $40\,^{\circ }\text{C}$, because of the enhanced heat transport of the rough surface at the bottom. Only a few plumes can be identified on the temperature snapshot, both on the top and bottom surfaces, while many more plumes can be seen on the numerical shadowgraph. This is due to the integration along $y$ in the latter, which adds up contributions of plumes at several $y$ positions, as well as the contribution of their 3-D structure. The far lower number of visible plumes on the mid-plane temperature, versus integrated fields such as the shadowgraph, can also be observed experimentally, e.g. with laser-induced fluorescence (Sakakibara & Adrian Reference Sakakibara and Adrian2004), or thermochromic liquid crystals (Du & Tong Reference Du and Tong2001), but there are few direct comparisons.

Figure 3.

Snapshot of the numerical fields at $Ra=2\times 10^{9}$. (a) Numerical shadowgraph computed with (3.4). (b) Local variance of the shadowgraph image. (c) Local 2-D thermal dissipation rate, averaged over $y$, $\langle \unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703},2D}\rangle _{y}=\langle \unicode[STIX]{x1D705}((\unicode[STIX]{x2202}_{x}T)^{2}+(\unicode[STIX]{x2202}_{z}T)^{2})\rangle _{y}$, normalized by the global thermal dissipation rate, $\langle \unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}\rangle _{x,y,z,t}=\unicode[STIX]{x1D705}(\unicode[STIX]{x0394}T^{2}/h^{2})Nu$. (d) Time-averaged cross-correlation between the local variance of the shadowgraph image and the local 2-D thermal dissipation rate averaged over $y$ (average value is 63.8 %).

Figure 3 compares the synthetic shadowgraph image, computed from (3.4), i.e. the $y$-averaged 2-D Laplacian of temperature, and the $y$-averaged 2-D thermal dissipation rate, $\langle \unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703},2D}\rangle _{y}$, which is given by

(3.5)$$\begin{eqnarray}\langle \unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703},2D}\rangle _{y}(x,z)=\unicode[STIX]{x1D705}\left\langle \left(\frac{\unicode[STIX]{x2202}T}{\unicode[STIX]{x2202}x}\right)^{2}+\left(\frac{\unicode[STIX]{x2202}T}{\unicode[STIX]{x2202}z}\right)^{2}\right\rangle _{y}.\end{eqnarray}$$

There is no straightforward mathematical relation between $I(x,z)$ and $\langle \unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703},2D}\rangle _{y}$ in the general case. However, figure 4 sketches the case of a warm fluid parcel over a mostly isothermal background. It is represented by a Gaussian profile for simplicity, but a similar result would be obtained for other bell-shaped curves. The dissipation rate,

(3.6)$$\begin{eqnarray}\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703}}\sim (\unicode[STIX]{x2202}T/\unicode[STIX]{x2202}x)^{2},\end{eqnarray}$$

and the Laplacian,

(3.7)$$\begin{eqnarray}\unicode[STIX]{x2202}^{2}T/\unicode[STIX]{x2202}x^{2},\end{eqnarray}$$

have similar shapes, with two maxima located roughly at the warm parcel mid-height.

Figure 4.

Simple sketch of patch of hot fluid. (a) Temperature profile. (b) Thermal dissipation profile. (c) Shadowgraph profile.

In the case of the Laplacian, the maxima are positive, and the minimum is negative. This can be observed in both the experimental and numerical shadowgraph images, where a plume pattern consists of darker and lighter boundaries (see figures 2 and 3). The thermal dissipation rate is always positive, and more closely resembles the shadowgraph local variance (discussed in the next section). The qualitative conclusion is that the structures evidenced in the shadowgraph images correspond to both areas of high thermal dissipation rate marking out the plume, which could also be used as a criterion to detect thermal plumes (Belkadi Reference Belkadi2019).

To estimate the similarity of the local variance of the shadowgraph image (figure 3b) and the $y$-averaged thermal dissipation, $\langle \unicode[STIX]{x1D716}_{\unicode[STIX]{x1D703},2D}\rangle _{y}$ (figure 3c), one can compute the time-averaged cross-correlation, $C(x,z)$,

(3.8)$$\begin{eqnarray}C(x,z)=\frac{\langle S(x,z,t)D(x,z,t)\rangle _{t}-\langle S(x,z,t)\rangle \langle D(x,z,t)\rangle _{t}}{(\langle S(x,z,t)^{2}\rangle _{t}-\langle S(x,z,t)\rangle _{t}^{2})^{1/2}\times (\langle D(x,z,t)^{2}\rangle _{t}-\langle D(x,z,t)\rangle _{t}^{2})^{1/2}},\end{eqnarray}$$

where $S(x,z,t)$ is the local variance of the shadowgraph, and $D(x,z,t)$ is the $y$-averaged thermal dissipation. As shown in figure 3d, the cross-correlation is higher than 60 % almost everywhere in the cell, and it averages to 63.8 %. This shows that the two fields are indeed correlated.

4.1 Plume detection technique

Figure 5.

Spatio-temporal diagram of the shadowgraph intensity $I(x,z_{0},t)$. Experimental diagrams at $Ra=5.6\times 10^{10}$: (a) $z_{0}=5~\text{cm}$ (bottom), (b) $z_{0}=20.75~\text{cm}$, (c) $z_{0}=36.5~\text{cm}$ (top). Numerical diagrams at $Ra=2\times 10^{9}$: (d) $z_{0}=4.8~\text{cm}$ (bottom), (e) $z_{0}=20~\text{cm}$, (f) $z_{0}=35~\text{cm}$ (top). The $t$ axis of the numerical shadowgraph is reversed to visually match the experimental diagrams where the roll direction is opposite.

One common way to find advected structures in a sequence of images is to build a spatio-temporal diagram, $I(x,z_{0},t)$, by choosing a vertical position $z_{0}$, or $I(x_{0},z,t)$, by choosing a horizontal position $x_{0}$ and plotting the image intensity as a function of $x$ and $t$, respectively $z$ and $t$. As shown in figure 5, advected coherent structures yield inclined lines. Similarly to other such analyses in Rayleigh–Bénard convection (Chillà et al. Reference Chillà, Ciliberto, Innocenti and Pampaloni1993; Taberlet et al. Reference Taberlet, Plihon, Auzémery, Sautel, Panel and Gibaud2018), these lines are a clear evidence of the mean wind, in opposite directions near each plate, with no clear advection at the centre.

By measuring the slope of those segments, one can get estimates for the average horizontal velocity (respectively the vertical velocity), and a density of plumes. This method is very common in the field of geophysical fluid dynamics, where it is used to find propagating wave fronts, and is sometimes referred to as a Hovmöller diagram (Lovegrove, Read & Richards Reference Lovegrove, Read and Richards2000; Flór, Scolan & Gula Reference Flór, Scolan and Gula2011).

In the remainder of the paper, we use the terms plume velocity to refer to velocities obtained from the analysis of plume advection (shadowgraph spatio-temporal diagrams, or ‘time-of-flight’ correlation time from thermometers), and fluid velocity to refer to either the numerical simulation velocity field, or experimental PIV measurements.

To obtain a spatio-temporal diagram that can be further successfully post-processed, the contrast of the image must first be enhanced, so that plumes are more visible. The ultimate contrast would consist of texture segmentation, where pixels that belong to a plume are separated from pixels that belong to the bulk. One possible estimate for this type of task is based on the local variance and the local regularity (Pascal et al. Reference Pascal, Pustelnik, Abry, Serres and Vidal2018).

A local variance image is obtained from a standard deviation filtering operation, which consists of computing a new image where each pixel is the standard deviation of the neighbourhood around the corresponding pixel in the original image. High values in this new image correspond to areas where there are large changes in the original image intensity. Low values correspond to areas where the original image intensity is more homogeneous. Hence, boundaries of plumes where there are sharp temperature gradients will produce high local variance values.

Figure 6(a,c) shows the local variance of the shadowgraph image, computed in neighbourhoods of $3\times 3~\text{pixels}$, both for the experimental shadowgraph and for the numerical shadowgraph. The contrast is indeed well enhanced.

Figure 6.

Local spatial variance of the shadowgraph images shown in figure 2, computed in a neighbourhood of $3\times 3~\text{pixels}$, for the experimental image at $Ra=5.6\times 10^{10}$ (a), and for the numerical shadowgraph at $Ra=2\times 10^{9}$ (c). Texture segmentation obtained from thresholding the local variance, (b) for the experimental image with a threshold value 0.025 (in units of relative light intensity), and (d) for the numerical shadowgraph with a threshold value $3.1~\text{K}~\text{cm}^{-1}$ (in units of integrated temperature Laplacian).

In the case of the numerical shadowgraph, the resolution is enough to observe the spatial oscillation pattern of each plume. Larger values of neighbourhood would not remove these oscillations, but would lead to blurred images. As previously discussed, they are the result of the double derivative of the spatial temperature profile of the plume (see sketch in figure 4), and should also be visible in the experimental shadowgraph. However, the plumes in the experiments are too small, compared to the resolution of the camera, to fully resolve the structure within each plume, and the oscillation patterns are no longer visible on the local variance image.

A threshold applied on the local variance yields a segmented picture, where pixels belonging to a plume are separated from those belonging to the bulk. However, in the case of Rayleigh–Bénard convection, the plume density is too high for such a binary decomposition, as can be seen in figure 6(b,d). Therefore, we use the local variance only to enhance the contrast. The spatio-temporal diagrams computed from the local variance images are shown in figure 7 for the case of a horizontal line near the top of the cell, and in figure 8 for the case of a vertical line at $x_{0}=8~\text{cm}$.

Figure 7.

Spatio-temporal diagrams built from the horizontal line at $z_{0}$ of the local spatial variance shown in figure 6. (a) Experimental diagram at $Ra=5.6\times 10^{10}$ for $z_{0}=38.5~\text{cm}$ (top). (b) Reconstructed line segments from the experimental diagram, using OpenCV Line Segment Detector. (c) Numerical diagram at $Ra=2\times 10^{9}$ for $z_{0}=36~\text{cm}$ (top). The $t$ axis is reversed to visually match the experimental diagram because the mean wind goes in the opposite direction. (d) Reconstructed line segments from the numerical diagram, using OpenCV Line Segment Detector.

Figure 8.

Spatio-temporal diagram built from the vertical line at $x_{0}=8~\text{cm}$ of the local spatial variance shown in figure 6. (a) Experimental diagram at $Ra=5.6\times 10^{10}$. (b) Reconstructed lines from the experimental diagram, using OpenCV Line Segment Detector. (c) Numerical diagram at $Ra=2\times 10^{9}$. (d) Reconstructed lines from the numerical diagram, using OpenCV Line Segment Detector.

The experimental and numerical spatio-temporal diagrams are very similar, and also closely resemble the diagrams obtained by Chillà et al. (Reference Chillà, Ciliberto, Innocenti and Pampaloni1993) from the temperature gradient with a sweeping laser method at a lower Rayleigh number ($Ra=10^{8}$). All feature inclined lines, which are a signature of the advection of plumes.

In the present work, the contrast of the inclined lines is good enough to proceed to the detection of line segments. We use the algorithm of Grompone von Gioi et al. (Reference Grompone von Gioi, Jakubowicz, Morel and Randall2012), implemented in the free OpenCV library (Bradski Reference Bradski2000). The detected segments in the case of a spatio-temporal diagram built from a horizontal line at $z=z_{0}$, are shown in figure 7(b,d), restricted to segments at the centre of the cell, $\unicode[STIX]{x1D709}<x<\ell -\unicode[STIX]{x1D709}$, and with durations $\unicode[STIX]{x1D70F}$ of several time steps. For both the experimental and numerical line segments, we choose $\unicode[STIX]{x1D709}=\ell /4$, to keep only the contribution of the nearly horizontal wind, and remove the corner flows and the jets. The duration is $\unicode[STIX]{x1D70F}=0.4~\text{s}$ for the experimental data, and $\unicode[STIX]{x1D70F}=2.3~\text{s}$ for the numerical data. The difference comes from the typical velocity that is much lower in the numerical set-up, due to the lower Rayleigh number.

4.2 Plume density

Each detected segment corresponds to the advection of a thermal plume. However, because some segments are not detected, and because one segment could possibly originate from a group of plumes, the number of detected lines is not directly equal to the number of plumes. Nevertheless, the slope of the detected segments yields an estimate of the plume horizontal velocity, $U_{x}(x,z_{0},t)$ for diagrams built from horizontal lines at $z=z_{0}$, or the plume vertical velocity $U_{z}(x_{0},z,t)$ for diagrams built from vertical lines at $x=x_{0}$, and the number of detected lines, $N_{pl,h}$ and $N_{pl,v}$, are estimates of the plume density in both cases.

The detected segments in the case of a spatio-temporal diagram built from a vertical line are shown in figure 8(b,d). In this convention where the origin is at the bottom, we get a positive vertical velocity for $z\rightarrow 0$, and a negative vertical velocity for $z\rightarrow h$. This is a signature of rising plumes at the bottom, and falling plumes at the top. In the case of asymmetrical cells, where only the bottom plate is rough, this analysis allows us to directly compare plume statistics of the smooth and rough plates, with no difference in the optical and image processing parameters.

Figure 9.

Number of detected segments in the spatio-temporal diagram, per unit time. (a,b) Spatio-temporal diagram from horizontal lines, detecting plumes moving horizontally, as a function of the plate distance $z$, restricted to the centre area, $\unicode[STIX]{x1D709}<x<\ell -\unicode[STIX]{x1D709}$. Blue circles: upper smooth plate. Red squares: lower rough plate. (c,d) Spatio-temporal diagram from vertical lines, detecting plumes moving vertically, as a function of the distance from the wall $x$, restricted to the centre area, $\unicode[STIX]{x1D709}<z<h-\unicode[STIX]{x1D709}$. Blue circles: falling plumes. Red squares: rising plumes. (e,f) Total number of plumes in the cell, at a given altitude $z$. (a,c,e) Experiment at $Ra=5.6\times 10^{10}$. (b,d,f) Numerical simulation at $Ra=2\times 10^{9}$.

The sets of line segments can be restricted to those at the centre of the cell, $\unicode[STIX]{x1D709}<x<\ell -\unicode[STIX]{x1D709}$, and $\unicode[STIX]{x1D709}<z<h-\unicode[STIX]{x1D709}$, to obtain profiles similar to the kind of profiles found with a probe moving along the axis. We choose $\unicode[STIX]{x1D709}/h=1/4$ which is a compromise to have enough events to statistically converge the profile, and is sufficiently away from the walls to be considered nearly at the centre. The resulting number of plumes (per unit time) is shown in figure 9(a–d), for horizontally advected plumes $N_{pl,h}^{centre}(z_{0})$, and vertically advected plumes $N_{pl,v}^{centre}(x_{0})$.

It is also possible to count the total number of plumes, as a function of plate distance, $N_{pl}^{tot}(z)$. Indeed, for a vertical line at $x=x_{0}$, which yields vertically advected plumes at $x=x_{0}$, each detected line segment in the diagram has spatio-temporal coordinates $(z_{1},t_{1},z_{2},t_{2})$. One can build the histogram of the space coordinates of their centre positions, $N_{pl,v}(x_{0},z)$. The total number of plumes $N_{pl}^{tot}$, shown in figure 9(e,f), is

(4.1)$$\begin{eqnarray}N_{pl}^{tot}(z)=N_{pl,h}(z)+\mathop{\sum }_{x_{0}}N_{pl,v}(x_{0},z),\end{eqnarray}$$

where $N_{pl,h}(z)$ is the number of advected plumes on the full horizontal line at $z$.

This measurement of the plume density can be compared with predictions and measurements in the literature. In their analysis of the role of plumes in thermal convection, Grossmann & Lohse (Reference Grossmann and Lohse2004) predicted that the number of plumes, $N_{pl}^{tot}$ scales like

(4.2)$$\begin{eqnarray}N_{pl}\sim Nu.\end{eqnarray}$$

In our case, this means that the number of plumes in the experimental case ($Nu=298.2$) should be three times larger than the number of plumes in the simulation ($Nu=107$). As can be seen in figure 9(e,f), for $z/h\sim 0.2$, we find indeed approximately 5 plumes per second in the numerical simulation, and 15 plumes per second in the experiment.

One expected effect of plate roughness is to change the emission of thermal plumes. Du & Tong (Reference Du and Tong2000) found that the roughness elements on the plate enhance the detachment of the thermal boundary layer. Although the details of the cell geometry are different in our case, we also previously evidenced bursts of plume emissions near the rough plate, typically between $40\unicode[STIX]{x1D6FF}_{th}$ and $120\unicode[STIX]{x1D6FF}_{th}$ away from the plate, where

(4.3)$$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{th}=\frac{H}{2Nu}\end{eqnarray}$$

is the thermal boundary layer thickness (Salort et al. Reference Salort, Liot, Rusaouen, Seychelles, Tisserand, Creyssels, Castaing and Chillà2014).

In the experiment the thermal boundary layer is typically $\unicode[STIX]{x1D6FF}_{th}\simeq 700~\unicode[STIX]{x03BC}\text{m}$, more than twice smaller than the height of the roughness elements. In this regime, the plate is hydrodynamically rough, and the heat transfer is enhanced.

However, as can be seen in figure 9(a), the average numbers of detected plumes in the present experimental case on the upper and lower halves of the cell only slightly differ. There are more plumes in the range $50\unicode[STIX]{x1D6FF}_{th}<z<150\unicode[STIX]{x1D6FF}_{th}$, which is where a burst of plumes was previously observed. There are no visible differences far from the plates, when the number of detected plumes vanishes. There seems to be a difference very close to the plate in figure 9(a), with less detected plumes in the rough case. We do not know, at this stage, if our analysis holds so close to the plate, where spurious reflections are possible. We intend to operate a similar experiment on a smooth symmetrical cell to compare. However, further away from the plate, no spurious optical effect is expected. A similar result is obtained from the number of rising and falling plumes in the jets (figure 9c); the number of detected rising plumes is slightly higher than the number of detected falling plumes.

In the numerical simulation, the Rayleigh number is $Ra=2\times 10^{9}$, and the thermal boundary layer thickness is typically $\unicode[STIX]{x1D6FF}_{th}\simeq 1.868~\text{mm}$, almost ten times smaller than the height of the roughness element. Compared to the experimental case, figure 9(b) shows that the plume statistics more strongly differ between the top and the bottom; there are more plumes near the bottom plate. Figure 9(d) shows also that there are more rising plumes than falling plumes.

The stronger plume enhancement observed in the numerical simulation may be a signature of Regime III. Indeed, the experiment lies in Regime II at a relatively high Rayleigh number, and the numerical simulation lies in Regime III, at lower Rayleigh number. In these conditions, the flow is more plume dominated in the numerical simulations than in the experiment, which is more turbulence dominated. It is, however, a general feature that the rough plate yields more total plumes in both cases.

4.3 Velocity statistics

Figure 10.

(a,c) Experimental horizontal (respectively vertical) velocity profile, obtained from the slope of the shadowgraph spatio-temporal diagram segments at $Ra=5.6\times 10^{10}$ (blue circles), or from PIV at $Ra=6.2\times 10^{10}$ (solid dark green line). (b,d) Numerical horizontal (respectively vertical) velocity profile, obtained from the slope of the numerical shadowgraph spatio-temporal diagram segments at $Ra=2\times 10^{9}$ (blue circles), from the mid-plane slice of the horizontal velocity average (solid dark green line), from the $y$-integrated horizontal velocity average (dashed light green line). The $\langle U_{x}\rangle$ and $\langle U_{z}\rangle$ axes of the simulation are reversed to visually match the experimental profile, where the roll direction is opposite. The average is both temporal and spatial for $\unicode[STIX]{x1D709}<x<\ell -\unicode[STIX]{x1D709}$ (respectively $\unicode[STIX]{x1D709}<z<h-\unicode[STIX]{x1D709}$), and $\unicode[STIX]{x1D709}=\ell /4$ (respectively $\unicode[STIX]{x1D709}=h/4$).

The analysis of the segments in the spatio-temporal diagram built from horizontal lines at $z=z_{0}$ yields sets of plume horizontal velocities, $U_{x}(x,z_{0},t)$. From these sets, one can infer the average horizontal velocity, $\langle U_{x}\rangle _{t,x}(z_{0})$, as a function of the plate distance, see figure 10(a,b). The average is both temporal and spatial, but only lines near the centre $\unicode[STIX]{x1D709}<x<\ell -\unicode[STIX]{x1D709}$ are considered. In the experiment, these profiles can be compared to velocity profiles previously obtained from PIV in similar conditions, and an identical geometry. In the numerical simulation, we use the average velocities of the mid-plane, at $y=10~\text{cm}$, as a reference.

These velocity profiles can be compared to those found by Belmonte, Tilgner & Libchaber (Reference Belmonte, Tilgner and Libchaber1994) and Vasiliev et al. (Reference Vasiliev, Sukhanovskii, Frick, Budnikov, Fomichev, Bolshukhin and Romanov2016) in a smooth Rayleigh–Bénard cell filled with water, where the velocity was measured with a pH dye technique. They found a maximum for the velocity at $z/h\simeq 0.05$, i.e. at $z>\unicode[STIX]{x1D6FF}_{th}$. Although our full cell images, either shadowgraph or PIV, do not have the resolution to accurately measure at the scale of boundary layer, similar maxima can be observed in our profiles, and at similar $z$.

Similarly, the analysis of the segments of the spatio-temporal diagram built from the vertical lines at $x=x_{0}$ yields sets of plume vertical velocities, $U_{z}(x_{0},z,t)$. From these sets, one can infer the average vertical plume velocity $\langle U_{z}\rangle _{t,z}(x_{0})$, as a function of the distance from the wall, see figure 10(c,d).

The plume velocity profiles, obtained from spatio-temporal diagrams or from fluid velocity measurement (PIV in experiment, mid-plane velocity in numerical simulation), have a similar order of magnitude. It is of the order of $2~\text{cm}~\text{s}^{-1}$ in the experiment, and $0.4~\text{cm}~\text{s}^{-1}$ in the numerical simulation. But the profiles do not completely collapse.

We use the numerical simulation to determine if this observed discrepancy is the consequence of the shadowgraph method, rather than an actual difference between plume velocities and fluid velocity. In principle, a difference between average plume velocity and average fluid velocity is not unphysical. Indeed, there is a selection bias; while the advection velocity of a plume should indeed be the same as the local velocity of the fluid, statistics based on plume advection measurements do not account for velocities of fluid parcels between plumes, or in areas where there is no plume. Additionally, the velocity field is divergence free, so a fluid parcel moving up (respectively down) must induce that there is fluid moving down (respectively up). This down-moving fluid parcel is not necessarily a falling plume, as it is moving down because of incompressibility and not because it is colder. The fluid velocity average accounts for all fluid movements, while the plume velocity statistics only account for the contribution of plumes.

Yet, the integration along the $y$ axis could lower the average, as fluid parcels slowed by the drag on the vertical walls will contribute to the integral. Integration may also smooth larger velocity events. Therefore, to check this hypothesis, we compute a profile of the $y$-averaged horizontal fluid velocity. This is similar to what we would get if PIV were integrated in the same manner as the shadowgraph is. This profile is shown in light green dashed line in figure 10(b). As expected, it is slightly lower than the profile obtained without integration in the $y$ direction. However, it does not differ from the non-integrated fluid velocity as much as the plume velocity does. This indicates that the integration along $y$ is not responsible for the observed difference between the plume velocity average and the fluid velocity average.

It is very broadly assumed in the literature that the velocity of advected plumes can be a satisfactory estimate for the velocity of the ‘wind’, which is then used to infer a Reynolds number. In practice, this is used mostly in cases where direct velocity measurement is not possible, using the correlation time between thermometers in the rising (or falling) jets (Wu Reference Wu1991; Chavanne et al. Reference Chavanne, Chillà, Chabaud, Castaing and Hébral2001; Sun & Xia Reference Sun and Xia2005; He et al. Reference He, van Gils, Bodenschatz and Ahlers2015; Musilová et al. Reference Musilová, Králik, Mantia, Macek, Urban and Skrbek2017), or correlation of shadowgraph images (Funfschilling & Ahlers Reference Funfschilling and Ahlers2004; Brown, Funfschilling & Ahlers Reference Brown, Funfschilling and Ahlers2007). While we find that the average plume velocity yields indeed the same order of magnitude as that of the mean fluid velocity, this analysis shows that care must be taken when comparing the Reynolds numbers computed in this fashion between different cells. He & Tong (Reference He and Tong2011) have analysed the temperature correlation in homogeneous convection. Their result is in agreement with predictions from He & Zhang (Reference He and Zhang2006), and showed that estimates of velocity can be inferred. However, to interpret this velocity estimate properly, there is an underlying assumption that the plume velocity statistics and local velocity statistics are identical. Besides the present work, we are not aware of another direct comparison in the literature between the statistics of plume advection velocity versus local fluid velocity.

In particular, the velocity estimate from rising (or falling) plumes in the jets very close to the walls is close to the maximum fluid velocity, yet still lower. Further away from the wall, average plume velocities are found to be higher than the average fluid velocity. Assuming the velocity profiles shown in figure 10(c) bear some universality, let us discuss the consequence for those results in the literature. They are all performed in cylindrical cells. Let us note by $\unicode[STIX]{x1D6F7}$ their diameter, and $r$ the distance from the wall. Wu (Reference Wu1991) and Musilová et al. (Reference Musilová, Králik, Mantia, Macek, Urban and Skrbek2017) have thermistors at a lateral position $r/\unicode[STIX]{x1D6F7}=0.05$, He et al. (Reference He, van Gils, Bodenschatz and Ahlers2015) have thermistors at a lateral position $r/\unicode[STIX]{x1D6F7}\sim 0.013$. For those, the plume velocity may be 40 % smaller than the fluid velocity, at least in this range of Rayleigh numbers. Chavanne et al. (Reference Chavanne, Chillà, Chabaud, Castaing and Hébral2001) have thermistors at a lateral position $r/\unicode[STIX]{x1D6F7}=0.25$, where plume velocity is slightly larger than average fluid velocity, but the fluid velocity is more than twice smaller than the velocity closer to the walls.

Figure 11.

Velocity PDF computed from the shadowgraph spatio-temporal diagram lines (full circles), from PIV measurements for the experimental data or the horizontal velocity mid-plane slice for the numerical simulation (open squares). Experimental data at $Ra=5.6\times 10^{10}$: (a) 5 cm above the bottom plate. (b) Mid-height. (c) 5 cm below the top plate. Numerical simulations at $Ra=2\times 10^{9}$: (d) 5 cm above the bottom plate. (e) Mid-height. (f) 5 cm below the top plate.

Finally, the velocity probability density functions (PDFs) are shown in figure 11. The PDFs of the experimental and numerical data are mostly similar.

The velocity PDF, obtained from the fluid velocity (PIV measurements in the experiment, mid-plane velocity in the numerical simulation), and those obtained from the shadowgraph spatio-temporal diagram, are similar at $z=5~\text{cm}$ and $z=35~\text{cm}$. However, they seem to evidence exponential tails which may seem to be in contradiction with the data from Qiu et al. (Reference Qiu, Shang, Tong and Xia2004) in a cylindrical cell, or Belmonte et al. (Reference Belmonte, Tilgner and Libchaber1994) in a cubic cell, where Gaussian velocity PDFs are measured at the centre of the convection cell. But Zhou, Sun & Xia (Reference Zhou, Sun and Xia2007b) report an exponential velocity PDF, away from the sidewalls in a rectangular cell, with a geometry much closer to ours. The difference between cylindrical and cubic versus rectangular may arise from the contribution of azimuthal motion which is reported to have a Gaussian amplitude PDF (Xi, Zhou & Xia Reference Xi, Zhou and Xia2006).

In the experiment, small differences in the operating conditions (see table 1) could account for the slight differences. Some velocity PDFs obtained from PIV exhibit secondary peaks (clearly visible for example in figure 11(c), at $z\approx -2~\text{cm}~\text{s}^{-1}$ and $z\approx -3~\text{cm}~\text{s}^{-1}$). These velocities are close to integer pixel displacements and are due to peak locking, which are known artefacts of the sub-pixel estimator used by the PIV algorithm Raffel et al. (Reference Raffel, Willert, Wereley and Kompenhans1998).

The PDFs are, however, very different at the centre of the cell, where the fluid velocity exhibits a symmetric histogram centred on zero, while the plume velocity PDFs evidence two peaks. The reason is that our method can only capture advected plumes, in one direction or another, but cannot resolve vanishing velocities, where no plume is advected.

The plume velocity histograms always evidence events in both directions ($U_{x}>0$ and $U_{x}<0$), but the most probable plume velocity value matches the most probable fluid velocity. The plume velocity distributions at $z=5~\text{cm}$ and $z=35~\text{cm}$ are slightly skewed, which may be a signature of the two kinds of plumes that are captured; plumes emitted from the top plate and those emitted from the bottom plate, which each have their PDF. The final PDF is the superposition of the PDFs of these two kinds of plumes.