Simultaneous tomographic particle image velocimetry and thermometry of turbulent Rayleigh–Bénard convection

The workflow of the combined Tomo-PIV–PIT is illustrated in figure 3. Four b/w cameras monitored the convection sample from different viewing angles. They provided four image pairs of the TLC particles at two time instants, t and t + Δt. The mapping functions from 3D ($\vec{X}$) to 2D ($\vec{x}$) space were obtained performing a volume self-calibration [32] with a calibration plate. This yielded the fitted 38 parameters for a Soloff polynomial function F [33] describing the translation $F(\vec{X}) = \vec{x}$ for each camera involved. Using an in-house Tomo-PIV code, which is described in detail in [10, 34], the simultaneous algebraic reconstruction technique (SMART) [35] was employed to determine the intensity distribution in 3D space, which was afterwards subdivided into interrogation volumes. Subsequently, a 3D cross-correlation between the two time steps is conducted in order to calculate the average displacement within each volume. By combining the time separation Δt with the displacement, the velocity field is obtained.

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In addition, a color CCD camera recorded the color information of the TLC particles used to determine the lightness and the hue maps, as explained in more detail in section 3.3. To relate the two-dimensional images recorded by the color camera to the desired 3D fields in the measurement volume, the volume self-calibration with the four b/w cameras was repeated after adding the color camera to obtain the required translation function for it. The lightness of the pixels on the color images serves for the particle detection, while the hue maps are used for the color-temperature calibration resulting in the temperature maps.

The calibration was performed by setting the temperatures of the heating and cooling plate to the same temperature value $T_\mathrm{h}\,=\,T_\mathrm{c}\,=\,T_\mathrm{cal}$. After an average settling time of eight hours, 10 000 color images of the TLC particles were recorded. Then, the hue values H were determined at all points in the measurement volume. Subsequently, $T_\mathrm{cal}$ was changed to determine the corresponding hue values. Since H is affected by the viewing angle of the camera, it was necessary to divide the color image into finite subregions. The calibration curve translating H into T was obtained for each subregion, see section 3.3.

The positions of the TLC particles were determined based on the 3D intensity maps provided by Tomo-PIV, see figure 3 (top right), which considers only the four b/w cameras. Afterwards, the temperature values obtained by PIT were assigned to the 3D positions defining 3D PIT, see figure 3 (center). For this assignment the translation function $F(\vec{X})$ of the color camera was used. Ambiguities can occur when a particle on the camera is assigned to multiple 3D positions resulting from the same line of sight. Assuming that the particle closest to the camera covers the more distant ones, the ambiguity was resolved by assigning the temperature value only to the 3D position closest to the camera.

The 3D particle positions were obtained from the intensity maps calculated in the SMART reconstruction during the Tomo-PIV evaluation. Due to a blurring applied during the reconstruction the particles were represented as a Gaussian distribution. Thus, a Gaussian curve fitting was used for the position recovery of the particles. This fitting employs the ROOT Toolkit [36] to recover the 3D position and to save computational time.

In brief, the workflow of the particle position reconstruction process can be summarized as follows: The positions were reconstructed by taking the highest intensity value found in an intensity map into account first. At the location of this value an initial estimation of the particle position uses one 1D Gaussian curve fit for each spatial direction. These results were then used as starting values for a subsequent 3D Gaussian fit improving the accuracy. Moreover, this procedure proved to be more stable than a direct 3D Gaussian curve fit to the data. The determined central value of the fit represents the position of the particle in 3D space. Afterwards, all intensities considered for the fit were subtracted from the intensity map. The procedure was reiterated to reconstruct as many particles as possible until the noise threshold of 75 intensity units was reached where no further particles could be identified.

Due to the high computational effort of the process—nine hours on average per map using a single core—a parallel implementation was employed. The curve fitting was performed using 20 threads in parallel on an Intel Xeon E5-2680V2 with Hyper-Threading enabled reducing the real time to less than two hours per map. Exclusive subvolumes were assigned to the individual processes in order to avoid ambiguities which might be introduced by removing already used intensities.

A color camera recorded the scattered light of the TLC particles and thus the contained temperature information. This data is typically acquired in the common RGB color space, yet it is known that other representations are more suitable for the required color-temperature calibration [16]. The two most common color spaces are hue-saturation-lightness (HSL) and hue-saturation-value (HSV). In the present study, a transformation of the color space from RBG to HSL was applied to calculate the temperature from the color information of the TLC particles. Since H represents the raw color, it is only sensitive to the color composition and therefore used for the color-temperature calibration. It is commonly calculated as follows:

$$\begin{equation} H : = \left\{\begin{array}{ll} \displaystyle 0, & \mathrm{for}\; \mathrm{max}(r,g,b) = \mathrm{min}(r,g,b) \\ \displaystyle 60^\circ\cdot\left( 0 + \frac {g - b} {\mathrm{max}(r,g,b) - \mathrm{min}(r,g,b)} \right), & \mathrm{for}\; \mathrm{max}(r,g,b) = r~\mathrm{and}~g \geqslant b\\ \displaystyle 60^\circ\cdot\left( 2 + \frac {b - r} {\mathrm{max}(r,g,b) - \mathrm{min}(r,g,b)} \right), & \mathrm{for}\; \mathrm{max}(r,g,b) = g \\ \displaystyle 60^\circ\cdot\left( 4 + \frac {r - g} {\mathrm{max}(r,g,b) - \mathrm{min}(r,g,b)} \right), & \mathrm{for}\; \mathrm{max}(r,g,b) = b \\ \displaystyle 60^\circ\cdot\left( 6 + \frac {g - b} {\mathrm{max}(r,g,b) - \mathrm{min}(r,g,b)} \right), & \mathrm{for}\; \mathrm{max}(r,g,b) = r~\mathrm{and}~g\lt b\end{array}.\right. \end{equation} \tag{ 1 }$$

The saturation S indicates the colorfulness, i.e. S = 0 describes grayscale and S = 1 the pure color. The lightness, L, is usually considered as a combined brightness of the color given by

$$\begin{equation*} L \,: = \, \frac{\mathrm{max}(r,g,b) + \mathrm{min}(r,g,b)}{2}. \end{equation*}$$

For completeness, the parameter V represents the raw brightness given by $V \, = \,\mathrm{max}(r, g, b)$. Both values, L and V, have in common that 0 means black and 1 represents the bright color. The HSL color space was chosen for the present study and L was used for particle detection since slightly more particles were detected as compared to V.

A sketch of the employed processing chain for the color-temperature calibration is presented in figure 4. Five major steps are performed to determine the necessary dependence of the color on the temperature:

• Setting the calibration temperature $T_\mathrm{cal}$.
• Transforming the color space from red-green-blue (RGB) to HSL.
• Determining the central hue values in interrogation windows using a Gaussian curve fit.
• Reiterating for multiple calibration temperatures.
• Employing the H(T)-value pairs in a curve fit to estimate the color-temperature calibration curves.

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To avoid scaling issues, H and L are given in units of 2¹² = 4096 in accordance with the maximum bit depth of the color camera. Using the information delivered by the calibration images, only the TLC particles with a certain quality i.e. brightness $L \,\gt \, L_\mathrm{min}$ were taken into account and the 2D hue maps were determined based on the 2D color images. Here, only the pixel coordinate defines the viewing angle which is presented as a 2D sketch of the imaging process in figure 5. The light source illuminates the particles from the left-hand side. Two lines of sight each crossing two particles are indicated. On one line of sight (red) a constant angle α with respect to the incident light is present, although the two indicated particles are positioned at different depth. Similarly, for the other line of sight (purple) a constant angle β is present which is different from the former one and again two particles at different depth are shown. Thus, it is not necessary to consider the depth-wise (and the parallel directions) in the volume separately and it is sufficient to employ small interrogation windows with a size of 100 × 100 pixels on the 2D hue maps to compensate for the viewing angle-dependent color reflection of the TLC particles.

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Within each of these windows, a 1D Gaussian curve fit was performed for the reduced hue-value distribution. The deduced center value, $H_\mathrm{c}$, was stored together with the corresponding calibration temperature for the desired temperature range to reach a sufficient number of values pairs $T(H_\mathrm{c})$. This allowed to determine the functional dependence T(H) using a curve fit with the following function to estimate the parameters p_i :

$$\begin{align} \begin{array}{lccc} T(H) = & \displaystyle {-\frac{p_0}{H-p_1}}&+{p_2}&+{p_3\cdot(H-p_1)}.\\ &\Uparrow&\Uparrow&\Uparrow\\ &\mathrm{~divergence~}&\mathrm{~offset~}&\mathrm{~constant~slope~} \end{array} \end{align} \tag{ 2 }$$

While parameter p₀ is a scaling parameter, p₁ shifts the hue values and causes a divergence in the analytical function at the position of the blue stop of the TLC tracer particles. p₂ levels the temperature curve in accordance with the red start of the particles and parameter p₃ reflects the linear dependence of temperature and hue value. The obtained functions were used to translate the hue values into temperatures. For the presented measurement, this calibration routine results in 14 × 11 = 154 calibration curves.

For the Tomo-PIV a particle image density of 0.035 particles per pixel (ppp) on the four camera images was realized. From the latter, 1000 instantaneous velocity fields were determined using the Tomo-PIV processing path of the evaluation workflow, see figure 3. The number of fields was chosen as a compromise between suppression of turbulent fluctuations and high statistics. The particles on the camera images were identified using an image pre-processing approach consisting of a minimum image subtraction, image normalization with 101 × 101 pixel kernel, high-pass filter with 31 × 31 pixel kernel, thresholding at 600 intensity units and a three-point Gaussian fit with 3 × 3 pixel kernel. The obtained particle distributions were then used for the intensity reconstruction, see figure 3 (top right), employing 1000 × 1000 × 500 voxels for 500 × 500 × 250 mm³ in physical space. Afterwards, a cross-correlation algorithm with grid refinement was applied to determine the velocity fields. The interrogation windows started with a size of $(144\mathrm{\,voxels})^3$ and were refined to $(50\mathrm{\,voxels})^3$ with an additional overlap of 50%. For the resulting velocity fields, a basic outlier detection was performed flagging only 0.9 percent of the vectors as invalid. Those were replaced using the surrounding 26 vectors, and the first reiteration of the outlier detection did not reveal any remaining invalid vectors.

In order to determine the dominant behavior of the investigated flow, the resulting velocity fields were statistically evaluated by means of temporal averaging $\lt.\gt_t$. Figure 6 shows the time-averaged velocity field using three-component vectors. Color-coding and vector scaling reflect the velocity magnitude, $\lt\overline{\vec{V}}\gt_t$. Increased velocities between 4 and 6 mm s⁻¹ along a circular path within the X-Y planes can be seen. There is an upstream region at the left side of the sample (X ≈ 50 mm) and a downstream region at the right side (X ≈ 450 mm). In RBC, this structure is typically termed large-scale circulation (LSC). Its normal vector points in Z-direction but is not exactly aligned. An important characteristic of the LSC is its highest velocity magnitude of the time-averaged velocity field:

$$\begin{equation*} {V}_\mathrm{mag}^{\mathrm{max}} \, = \, 5.82 \, \mathrm{mm}\kern1.5pt \mathrm{s}^{-1}. \end{equation*}$$

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Moreover, ${V}_\mathrm{mag}^{\mathrm{max}}$ can be used to determine statistical convergence by expressing the measurement duration $T_\mathrm{meas}$ by means of a simplified circulation period of the LSC, $T_\mathrm{LSC}$. In order to do so, a circular flow structure covering the sample is assumed and with a length along the circular path amounting to $l_\mathrm{circ}$. Here, it is compared to the distance s₁₀₀₀ which the flow covers within the 1000 evaluated timesteps:

$$\begin{equation*} \frac{T_\mathrm{meas}}{T_\mathrm{LSC}}\, = \,\frac{s_{1000}}{l_\mathrm{circ}}\, = \,\frac{\Delta t\,\cdot\, 1000\,\cdot\, {V}_\mathrm{mag}^{\mathrm{max}}}{2\pi \cdot \frac{L_\mathrm{H}}{2}}\,=\,0.5. \end{equation*}$$

Thus, the low-frequency oscillations of the LSC remain present, whereas the high-frequency turbulent fluctuations are suppressed.

The fundamental flow properties are studied by means of a histogram plot of the velocity magnitude obtained from the 1000 instantaneous fields presented in figure 7. The absolute number of values per bin is shown for a bin width of $\Delta_{v_\mathrm{mag}} \, = \, 0.0035 \,$mm s⁻¹. The complete distribution over the entire velocity range shown in blue in figure 7 reflects a maximum value of 18 mm s⁻¹. In the upper right corner, a close-up view of the distribution between 13.5 and 17.5 mm s⁻¹ is plotted in green. Since absolute velocity values larger than $v_\mathrm{max} \, = \, 16.1 \,$mm s⁻¹, indicated by the red line in figure 7, are rather rare, the latter velocity was considered as the maximum velocity in the instantaneous fields and was used to estimate the dynamic range of the measurements. From the smoothness of the distribution it is concluded that no peak locking occurs indicates the high quality of the Tomo-PIV.

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Figure 8 shows an example of an instantaneous color image recorded by the PCO Pixelfly color camera reflecting a particle image density of 0.025 ppp. It must be noted that the color scale provided by TLC particles is counter-intuitive as red reflects the lowest and blue the highest temperature values. The color play of the TLC particles is illustrated by green crystals on the left-hand side and red particles on the right-hand side—which can generally be attributed to the dependence on the viewing angle [16] and illustrates the necessity for the smaller calibration windows to compensate for this reflection characteristic of the TLC particles, see section 3.3.

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In order to prove the applicability of the TLC particles to the temperature measurements, the dependence of the hue values on the temperature was investigated. First, the TLC particles had to be identified on the color-camera images. After the color-space transformation, a noise level of up to L ≈ 300 was reported. Here, the TLC particles were separated from the noise using a threshold at L = 800. Afterwards, the hue value distribution of the identified particles was studied by means of histogram plots presented in blue in figure 9 for four calibration temperatures. Ten thousand color images were taken into account to determine the distribution for each calibration point. In addition, the approximate regions for red, blue and green to violet are shown in the background. In general, an increase in temperature resulted in a shift of the distribution from red (18 ^∘C, cold) towards blue (20 ^∘C, warm). Further, it can be seen that there are no values for the violet area. This was to be expected based on with the typical behavior of the TLC particles [16] and shows that the implemented Bayer filter and the hue value calculation work as assumed. The distributions at 20 ^∘C and 22 ^∘C were quite similar revealing that a change in the temperature between 20 ^∘C and 22 ^∘C does not affect H. Thus, it can be concluded that the particles' temperature resolution decreases as soon as the temperature exceeds 20 ^∘C. Further, the 19 ^∘C calibration point showed a large concentration of values in the green region and an additional, however, about 3.5 times smaller, agglomeration in the blue region. This proved the necessity for the individual calibration windows and the associated calibration functions to account for this non-uniform behavior.

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Consequently, the calibration approach using 100 × 100 pixels interrogation windows was employed. An exemplary result at T = 20 ^∘C for an interrogation window taken from the middle of the camera image is visualized in figure 10(a). The H-distribution is shown as a blue histogram. Following the procedure described in section 3.3, the corresponding 1D Gaussian curve was determined and superimposed as a black line. Based on the Gaussian curve, the center value $H_\mathrm{c}\,=\,2488$ was determined as highlighted in red in figure 10(a). By repeating this procedure for all interrogation windows the calibration points $T(H_\mathrm{c})$ were obtained.

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Based on these calibration points for all interrogation windows, calibration curves were determined by fitting the Function (2). As an example, figure 10(b) shows the calibration curve obtained for the interrogation window considered in figure 10(a). The black data points represent the five calibration temperatures $T(H_\mathrm{c})$ with $H_\mathrm{c}$ representing the central hue value as discussed above. Additionally, the fitted curve is shown in red and the parameters determined from the fitting procedure are indicated in green at the bottom. Since the resulting calibration curve describes the data with negligible uncertainty, it can be used to transform hue values into temperatures. It must be noted that an ensemble of 14 × 11 = 154 curves was used for the measurements presented below.

An impression of the 2D temperature distribution is provided in figure 11, where the color coding reflects the temperature in ^∘C. The image is colored in black for regions without temperature information obtained from the TLC particles. The field of view corresponds to the raw color image of figure 8. Compared to the calibration images, a less restrictive brightness threshold of L = 600 was applied for particle detection for the measurement series. The associated H was used to determine the temperature. The main part of the temperature map revealed temperatures around 18 ^∘C. Nevertheless, in the lower left corner, an increased temperature of 19 ^∘C to 20 ^∘C was determined. This will be investigated in more detail in the following section 4.3.

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To give an example of the estimation of the 3D particle position with the Gaussian curve fitting procedure described in section 3.2, the reconstructed position of a rather challenging asymmetric intensity distribution at location Y ≈ 3 mm, and Z ≈ 248 mm along X is plotted in dependency of X in figure 12. The data points are marked with blue crosses and the fitted Gaussian curve is shown in dashed green with its central value indicated in red. The intensities on the left side of the Gaussian distribution are missing, while the right side shows the expected Gaussian shape. The reason for this could be a minor misalignment of the cameras or defective/noisy pixels. When comparing the distribution with the possible position measures—mean value (447.4 mm), highest value (445.8 mm) and center of Gaussian curve fit (446.3 mm)—the results were similar. Yet, the position of the highest value suffered from a general binning and the mean value was easily offset by missing values. Only the Gaussian curve fit recovered a trustworthy position of $X_\mathrm{fit, 1d} \, = \, 446.3$ mm. Analogously, the coordinates $Y_\mathrm{fit, 1d}$ and $Z_\mathrm{fit, 1d}$ are determined. As described before, these three results were used as starting values for a complete 3D Gaussian curve fit which resulted in a central value of $\vec{X}_\mathrm{fit, 3d}$.

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The temperature value of the position which corresponds to the above-determined 3D position was then assigned to the latter using the translation functions from the volume self-calibration. The resulting 3D velocity and temperature fields will be discussed in the following section.

In order to unite the measured PIV and PIT data, the reconstructed 3D particle positions were combined with the 2D temperature maps from PIT using the mapping functions obtained during the volume self-calibration. On average 11 000 temperature values in 3D are obtained, while the effect of mapping multiple 3D positions with the same temperature value occurs for less than 0.1%. For such a low effect this ambiguity is solved reliably by assigning the temperature value only to the 3D position closest to the camera. A resulting instantaneous 3D velocity and temperature field is shown in figure 13. The spheres indicate the temperatures at the corresponding 3D positions in the volume using color coding. The direction of the velocity is indicated by 3C vectors, while the scaling reflects the velocity magnitude. The velocity field was determined for the entire volume, whereas the temperature measurement was limited to the upper half of the convection sample. For the sake of clarity, only every 100th vector and temperature value are shown. The structure of the LSC in this instantaneous field is similar to the structure in the time-averaged velocity field discussed in figure 6. Still, as expected for turbulent RB convection, smaller turbulent flow structures occurred throughout the measurement volume, which can also be resolved by the experimental set-up. Figure 13 shows the high upward velocities occurring within the LSC for low X values on the left and the high downward velocities for high X values on the right. The temperature values are dominantly distributed close to 18 ^∘C with some increased temperature values reported for X ≈ 100 mm.

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In contrast to the velocities, which were determined on a regular Eulerian grid, the temperature values were measured at the positions of the TLC particles. Thus, to be able to analyze both quantities at the same position, a 3D linear (also called tri-linear) interpolation of the temperature values measured at the particle position on the structured Eulerian grid was performed. Figure 14 reflects a resulting instantaneous velocity and temperature field in terms of scaled 3C velocity vectors which are color-coded by the temperature values at the corresponding positions. A reference vector with a velocity magnitude of 5 mm s⁻¹ is shown at the top left. Velocity vectors at positions where no temperature value is available are visualized in gray. Outlier temperatures resulting from sporadic color fluctuations on the CCD sensor of the color camera have been removed using a simple threshold at $T_\mathrm{max}\, = \,25\,^\circ$C and the corresponding vectors are also colored in gray. Looking at the direction of the velocity vectors, the LSC can be easily identified rotating around its center located in the middle of the measurement volume. Most of the interpolated temperatures in figure 14 show values close to the average sample temperature of 18 ^∘C. For the considered fully turbulent RBC it is well-known that the temperature gradients are relatively high close to the heated/cooled plates and rather low in the bulk [18, 37]. In this respect, the measured temperatures in the bulk region are close to the mean temperature $\overline{T}\, = \, 18\,^\circ$C. Moreover, figure 14 also reveals the existence of isolated structures by showing increased temperatures at low X values and high Y values followed by a colder structure in the upstream direction. In addition, a temperature increase to 19 ^∘C with an upward flow direction on the left side of the sample (X ≈ 100 mm, Y ≈ 350 mm) is resolved. This reflects a part of the LSC which transports warm fluid from the heating plate to the cooling plate.

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In order to quantify the dependence of upward velocity and temperature, a 2D histogram of the local temperature T and the upward velocity v_y was determined. Figure 15 shows the histogram of the instantaneous velocity and temperature field at first time step. The absolute number of pairs per bin is used as color key and the trendline is indicated in green. The velocity was divided into bins with a size of $\Delta_{v_y} \, = \, \frac{1}{3}\,$mm s⁻¹ and the temperature bins had a size of $\Delta_T \, = \, \frac{1}{14}\,^\circ$C. The dependence of T(v_y ) was quantified by means of a linear curve fitting using an orthogonal distance regression in the following form:

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$$\begin{equation} T(v_y)\, = \,m\cdot v_y\,+\,T_0. \end{equation} \tag{ 3 }$$

Figure 15 reveals a distribution of temperature values mainly between 18 ^∘C and 19 ^∘C, while the velocity values range from −3 mm s⁻¹ to 5 mm s⁻¹. The fitted trendline parameters were $m \, = \,0.0165\, \pm \,0.0048 \,\frac{\mathrm{s} \cdot \,^\circ\mathrm{C}} {\mathrm{mm}}$ with a corresponding ordinate section $T_0 \, = \, 18.56 \, \pm \, 0.01 \,^\circ \mathrm{C}$. This reflects two important characteristics of the flow. First, a temperature distribution around $\overline{T}$ was to be expected for RBC. In the present study, with the limited investigated subvolume the average temperature T₀ was slightly above $\overline{T}$. Secondly, due to buoyancy, small warm structures drive the flow upwards, whereas colder ones are associated with downwards velocities [4]. For the investigated subvolume this was confirmed by a positive value for m reflecting that increased values for T yield a higher upward velocity v_y .

In the following analysis, all evaluated time steps were taken into account to make statements about the average behavior of the convective flow in the RBC sample. For this purpose, the upward velocity, the temperature and their correlation were examined based on the histograms shown in figure 16: the green histogram at the top reflects the frequency distribution of the vertical flow velocity with a bin width of $\Delta_{v_y} \, = \, 0.03$ mm s⁻¹. In blue, on the right, the corresponding temperature distribution is shown with a bin width of $\Delta_T\,=\,0.01\,^\circ$C. The correlation of the two quantities was again determined with the help of a 2D histogram, analogous to figure 15. The bin widths were chosen identically to those of the 1D histograms. The green velocity histogram shows a smooth distribution shifted towards the positive range. The shift results from the fact that the field of view of the temperature measurement was limited and thus the same limitation applies to the considered velocities. In figure 16, the blue histogram of the temperature distribution also shows a smooth curve, ranging from slightly less than 18 ^∘C to approximately 20 ^∘C. It is also evident that the most common value is around 18.5 ^∘C, which is slightly higher than the average sample temperature of $\overline {T}\,=\,18\,^\circ$C. This observation is in good agreement with the flow field, as the temperature was actually measured in the region where the warm fluid flow rises from the heating plate. The 2D histogram illustrates the dominant dependence of the fluid temperature on the upward velocity. The presented distribution reflects an inclined ellipse indicating that higher temperatures were observed for higher vertical velocities. Thus, it can be concluded that the previously described positive correlation also persists over time indicating a stable LSC alignment.

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As discussed in a previous study with the same in-house Tomo-PIV implementation [10, 34], the uncertainty of the velocity field is based on the reconstruction precision in the voxel space and the subsequent cross-correlation yielding $\sigma_\mathrm{tomo}\, = \,0.2$ voxels per time step. For cubical voxels—as used in the present study—the uncertainty in terms of the velocity measurement can be determined by the following equation:

$$\begin{eqnarray} &&\sigma_v = \frac{s_\mathrm{vox}\cdot\sigma_\mathrm{tomo}}{\Delta t} \end{eqnarray} \tag{ 4 }$$

with the voxel size along one direction $s_\mathrm{vox}$ given in [mm voxel⁻¹] and the used time separation Δt given in seconds. The uncertainty regarding the velocity measurement was estimated in accordance with equation (4). A total of 1000 × 1000 × 500 voxels were used for 500 × 500 × 250 mm³ in physical space resulting in a voxel size of $s_\mathrm{vox}\, = \,0.5$ mm voxel⁻¹. Taking the realized time separation of Δt = 0.150 s into account leads to the upper limit in terms of the uncertainty of the velocity measurement of $\sigma_\mathrm{V}\,=\,0.66\,$mm s⁻¹. As shown in figure 7, the instantaneous velocity distribution reached values up to $v_\textrm{max}\,=\,16\,$mm s⁻¹ resulting in a dynamic range of more than 24 levels for the velocity measurement.

With a density similar to the density of water ($\rho_\mathrm{P}\,=\,1050$ kg m⁻³), the following behavior of the TLC seeding particles was investigated by employing the Stokes number (St) [38]

$$\begin{eqnarray} &&\mathrm{St}\,=\,\frac{\rho_\mathrm{P} D_\mathrm{P}^{2}|\vec{u}|}{18 \mu L_\mathrm{H}} \end{eqnarray} \tag{ 5 }$$

with µ being the dynamic viscosity, a characteristic velocity $|\vec{u}|$ and the particle diameter $D_\mathrm{P}$. The criterion $\mathrm{St}\,\lt\lt\,1$ determines the limit for feasible following behavior of the particles within the flow. To investigate the highest St possible within the system, $|\vec{u}|$ is chosen as $v_\textrm{max}\, = \,16 \,$mm s⁻¹ resulting in St  =  0.007 thereby fulfilling the criterion. Thus, it is apparent that the following behavior of the TLC particles does no affect the velocity measurements.

The uncertainty of the measured temperature fields depends on the accuracy of the calibration temperature $\sigma_{T, \mathrm{sensors}}$, of the determination of the hue values in the interrogation windows $\sigma_{T, \mathrm {Hue}}$ and of the analytical calibration curves $\sigma_ {T, \mathrm{calib}}$.

The temperature calibration points were determined using various temperature sensors to measure the average fluid temperature. The fluid temperature during the calibration process was determined by means of a calibrated semiconductor temperature sensor in the fluid with an accuracy of 0.01 K at 20 ^∘C. This measurement was supported by 32 Pt₁₀₀₀ temperature sensors with class AA accuracy, i.e. every single sensor yields an accuracy of approximately 0.1 K, or combined 0.018 K. This provided a combined measurement accuracy slightly better than $\sigma_{T, \mathrm{sensors}} \,\lt\,0.01$ K.

Determining the hue values in an interrogation window implies an associated uncertainty $\sigma_{T, \mathrm {Hue}}$, which was defined by the width of the Gaussian distribution while estimating the central hue values $H_\mathrm{c}$. The uncertainty was estimated using the 2σ interval of this distribution, which means that 95 % of the data is described well, to derive the variation of the temperature response for all calibration curves. The largest deviation of all curves was assumed to be the upper limit for $\sigma_{T, \mathrm {Hue}}$. The dominant uncertainty resulted from determining the central hue values H_c for the interrogation windows: $\sigma_{T, \mathrm {Hue}} \, \lt\, 0.094$ K.

The subsequent determination of the analytical calibration curves was also subject to an uncertainty and amounts to $\sigma_ {T, \mathrm{calib}}$ and turned out to be negligibly small: $\sigma_ {T, \mathrm{calib}} \, \lt\, 0.002$ K.

Following the Gaussian error propagation, which assumes uncorrelated uncertainties, the combined uncertainty amounted to $\sigma_ {T} \, \lt\, 0.095$ K. This corresponds to a high dynamic range of 21 levels for the studied temperature measurement range of 2 K. Temperature regions of specific interest can be addressed directly by using specifically tailored TLC particles with a corresponding temperature range—which, however, may result in a reduced dynamic range or an altered absolute accuracy for temperature measurements.

For the current particles per pixel densities of 0.035 and 0.025 on the b/w cameras and color camera, respectively, an ambiguity during the temperature assignment occurred only for 0.1% of the temperature values and was resolved by using only the 3D coordinate of the particle closest to the camera. Thus, there was no significant effect on the temperature measurement. Yet, for higher particles per pixel densities the ambiguities will increase and this straightforward strategy could fail.

When it comes to estimating the temperature uncertainties, it should be noted that aging effects of the TLC particles could not be taken into account. However, in order to minimize unquantifiable uncertainties related to aging of particles, the measurements were carried out directly after adding the TLC particles to the water-glycol mixture. Subsequently, the T(H)-calibration was conducted based on five data points, which were chosen as a trade-off: The aim was to acquire as many T(H)-calibration points with a small deviation from the temperature equilibrium as possible while at the same time keeping the calibration process as short as possible to avoid aging of the TLC particles. It took one day for the sample to reach the thermal equilibrium, thus generating one temperature calibration point per day. The acquired calibration data allowed for three points within the linear T(H)-regime and two within the tail. Like this, the applicable temperature range could be maximized. While the main part of the measurement data is described within the linear regime, it must be noted that the calibration points within the tail region correspond to a temperature range where the TLC particles show only minor color changes in case of temperature changes.