Simultaneous measurement of temperature and velocity fields in convective air flows

Based on the review papers of Smith et al [13] and Dabiri [14], some of the fundamentals of TLCs, especially some of the colour-dependent parameters, are described in the following. It is well-known that the colour of a TLC droplet depends not only on its temperature but also on, e.g., the angle between the incident illumination and line of view, the background light and the age of the droplet. Due to the angular dependence, a local calibration provides far better accuracy than a global calibration. Another important parameter is the size of the droplets. Hence, a major requirement is that the produced particles have a sharp size distribution. In this respect the minimal/maximal possible size of the TLCs, which depends on two factors, must be detected. On one hand the TLCs must maintain their temperature-dependent reflection of different wavelengths (the play of colours), i.e. they must be as large as possible, and on the other hand, they need the best possible following characteristics, i.e. they must be as small as possible.

From previous studies by Czapp [18] and investigations of the colour play of the crystal as a function of the particle generator settings, we know that the diameter of the TLCs should be of the order of 10 μm. An estimate of the reaction time of the TLCs to temperature changes, the following behaviour and the sinking velocity is given in Czapp [18] and Raffel et al [12]. There, the reaction time based on the heat transport from the fluid to crystal via convection (τ_α) is estimated as $\alpha = Nu\cdot \lambda _{t,\mathrm{air}}\cdot d_\mathrm{p}^{-1}$ using Nu = 2 for very creeping flows. Here, α is the heat transmission coefficient. The other possible heat transfer mechanism from the fluid to the crystal is heat radiation (τ_σ), which can be approximated using the Stefan–Boltzmann law and the emissivity = 1. Once heat is transferred from the fluid to the crystal, or vice versa, the surrounding is colder than the crystal. Thus, the crystal itself must transport the heat to its interior by heat conduction (τ_λ), which can be estimated based on the heat conductivity of the crystal λ_p. Finally, the helical structure of the crystal, i.e. the colour of the crystal, must change according to the temperature of the crystal. Since this relaxation time τ_relax is hard to estimate we refer to the value 100 ms for cholesteryl ester and 1–10 ms for chiral nematic crystals published in [22–25]. Thus, based on the above discussed assumptions, one obtains the following estimates for the relevant reaction times for the colour change of a TLC particle with a diameter d_p = 10 μm:

$$\begin{eqnarray*} &&\tau _\alpha \approx \frac{C_p\cdot d_\mathrm{p}}{6\cdot \alpha }\approx \frac{C_p\cdot d_\mathrm{p}^2}{12\cdot \lambda _{t,\mathrm{air}}}=1.0 \,\mathrm{ms},\\ &&\tau _\sigma \approx \frac{C_p\cdot d_\mathrm{p}}{24\cdot \sigma \cdot T^3}=0.8 \,\mathrm{s},\\ &&\tau _\lambda \approx \frac{C_p\cdot d_\mathrm{p}^2}{4\cdot \lambda _\mathrm{p}}=0.3 \ldots 2.5 \,\mathrm{ms},\\ &&\tau _\mathrm{relax}=1\ldots 10\,\mathrm{ms}\ \mbox{(100 ms for cholesteryl}\nonumber\\ &&\mbox{ ester crystals})\ [22\hbox{--}25], \end{eqnarray*}$$

with the heat capacity per volume of C_p, the Stefan–Boltzmann constant σ and the ambient temperature T. The above shown three orders of magnitude higher reaction times of the radiative heat transport to the particle (τ_σ) are negligible compared to the heat transport via convection (τ_α). Contrarily, the time constant for the internal heat conduction within the particle (τ_λ) is like that of τ_α of the order of milliseconds as well. Another important time constant is the time that the TLC needs to change its colour after having changed its temperature τ_relax. For thin layers of chiral nematic crystals, which is the crystal type we use in our experiments, values in the range 1–10 ms are reported for τ_relax in the literature [22–25]. Although the lateral dimensions of the TLCs might affect τ_relax, the values for small droplets of chiral nematic crystals are similar. Dabiri [14] reports response times for TLC particles from Hallcrest with a diameter of 10 μm of 1–4 ms. Tropea et al [15] give a response time of 3–10 ms for TLC tracer particles with diameters of 20–50 μm. However, much larger values of up to 100 milliseconds are reported for another crystal type, i.e. cholesteryl ester (cholesteric) crystals [22–24]. According to the discussion above we expect the overall thermal response time τ_t to be below 10 ms for our chiral nematic crystals (Hallcrest, R20C6W) with a mean diameter of approx 10 μm. It is a function of the processes discussed above, some of which proceed in parallel while others take place consecutively. However, our analysis revealed the relaxation time as the limiting time scale, while direct measurements of τ_relax of dispersed TLC particles in air have not been reported so far and constitute an open issue. However, in spite of the fast response time compared to conventional temperature sensors, precise temperature measurements are possible only if the TLC particles are able to adapt their temperature on time scales comparable to that of the local convective velocity and the spatial resolution of the measurement. To quantify this, we estimate the maximal travelled distance δ_min = τ_t · u_max during the reaction time τ_t, which ought to be smaller than the minimum of the interrogation window size and the size of the smallest coherent structures, i.e. approximately 10 times the Kolmogorov length η_K [26]:

$$\begin{eqnarray*} \delta _\mathrm{min} & < d_\mathrm{int. window}\\ \delta _\mathrm{min} & < 10\cdot \eta _\mathrm{K}. \end{eqnarray*}$$

In our measurements, the maximal fluid velocity is u_max ≈ 0.1 m s⁻¹, thus δ_min ≈ 10⁻³ m, while 10 · η_K ≈ 10⁻² m (for Ra ≈ 10⁸ and Pr ≈ 0.7) [26] and the interrogation window size amounts to d_{int. window} ≈ 5 × 10⁻³ m. Hence, the thermal reaction time is short enough for the here considered flows.

An overview of the parameter range in which the technique is applicable according to the above discussed limitations is depicted in figure 1. The dark grey shaded triangle in figure 1, in which the measurement technique can be applied, reflects that the spatial resolution not only decreases with increasing velocity but also crucially depends on the reaction time, which lies between 1 and 10 ms. The region of resolvable spatial fluctuations depending on the maximal fluid velocity u is highlighted in light and dark grey for τ_t = 1 ms and τ_t = 10 ms, respectively. For the interrogation size of the measurements presented below, which is marked as the black dot in figure 1, the response time of τ_t = 10 ms is small enough. However, it must be noted that if this is not the case, backtracing of the particle positions using the measured velocity field with sophisticated tracing algorithms allows enlargement of the parameter range in figure 1 in which the measurement technique can be applied.

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Other important parameters, which might limit the accuracy, are the mechanical reaction time τ_m and the sinking velocity due to gravity v_g. They can be estimated (following Raffel et al [12]) assuming Stokes friction:

$$\begin{eqnarray*} v_\mathrm{g}&=&d_\mathrm{p}^2\cdot \frac{\rho _\mathrm{p}-\rho }{18\cdot \eta }\cdot g \approx 3\, \mathrm{mm}\, \mathrm{s}^{-1}\\ \tau _\mathrm{m}&=&d_\mathrm{p}^2\cdot \frac{\rho _\mathrm{p}-\rho }{18\cdot \eta }\approx 0.3\, \mathrm{ms} \end{eqnarray*}$$

using the particle density ρ_p ≈ 1.01 × 10³ kg m⁻³ (ρ_TLC = 1.00–1.02 × 10³ kg m⁻³ given by Hallcrest for the unencapsulated TLCs used by us), the gravitational acceleration gas well as the density ρ ≈ 1.2 kg m⁻³ and the dynamic viscosity η ≈ 1.8 × 10⁻⁵ of air at 20 °C. Although this sinking velocity is lowered by the Brownian motion, it is still a limiting factor for the total measurement time without supplying new particles and the accuracy of the velocity measurement. Again, based on the assumption of Stokes, Raffel et al [12] describe the following behaviour of the smallest particles while travelling through a shock wave:

$$\begin{eqnarray*} &&v_\mathrm{p}(t) = v_\infty \left[1-\mathrm{exp}\left(-\frac{t}{\tau _\mathrm{m}}\right) \right], \end{eqnarray*}$$

with the velocity of the particle v_p(t) and the velocity of the surrounding fluid v_∞. These estimates result for a d_p = 10 μm TLC particle in an air flow in a time of t ⩽ 3 ms that the particle needs to reach 99% of the fluid velocity. Thus, TLC particles with d_p ⩽ 10 μm follow rather slow fluid flows almost instantaneously.

To calculate the temperature from the colour of the TLC particles, a transformation of the colour space must be applied. For the TLC particles used with this specific diameter, a calibration of the temperature-dependent hue value [14] has to be performed, so that a temperature accuracy better than ±0.3 K is achieved. Thereby the hue is, besides saturation and value or saturation and lightness, one of the three parameters of the hue/saturation/value (HSV) or hue/lightness/saturation (HLS) colour space, respectively. For graphical representations of the colour spaces, see figure 2. The parameter value is the only non-obvious parameter, and can be understood as the brightness (or darkness) value of the colour with V = 0 being black and V = 1 being the pure colour. The hue saturation intensity colour space is another possible colour space which can be used for the calibration of temperatures to colours. Its difference from the HLS colour space is only the third parameter: lightness is the relative light intensity and intensity is the absolute one. Using either one of these does not matter much. For completeness it should be noted that the hue saturation brightness (HSB) colour space, which some of the literature refers to, is exactly the same as the HSV colour space, only naming the parameter, more obviously, brightness instead of value. Nevertheless, in the following, we will use the name HSV instead of HSB because it is the more common expression.

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The hue value h can be calculated from the RGB values (red: r, green: g, blue: b) [27] by:

$$\begin{equation*} h(r,g,b)=\left\lbrace \begin{array}{@{}ll@{}} \frac{g-b}{{\rm max}(r,g,b)-{\rm min}(r,g,b)}, & \mbox{if }r={\rm max}(r,g,b)\\ \frac{b-r}{{\rm max}(r,g,b)-{\rm min}(r,g,b)}+2, & \mbox{if }g={\rm max}(r,g,b)\\ \frac{r-g}{{\rm max}(r,g,b)-{\rm min}(r,g,b)}+4, & \mbox{if }b={\rm max}(r,g,b).\end{array}\right. \end{equation*}$$

The parameters saturation and lightness, intensity or value can be used to correct artefacts like black or white pixels of the image and for the filtering process which is needed for the separation of the particles from the background. A possible realization of this process is described in section 2.4. All colour information needed to calculate the temperature is expressed with the single parameter hue, hence this parameter is used to calculate the temperature from the colour image.

For the production of tiny unencapsulated TLC particles, a generator is needed that fulfils the following requirements. Firstly, to investigate systems with air exchange, continuous production of the particles must be realized. Secondly, the particles have to exhibit a sharp size distribution in order to allow for well-defined colour play and finally, the mean particle size, however, should be adjustable by the operational parameters of the generator in order to allow matching of the size-dependent particle colour play to the actual measurement. The final particles must, on one hand, maintain their temperature-dependent reflection of different wavelengths (the play of colours), i.e. they must be as large as possible. On the other hand, they need the best possible following characteristics, i.e. they must be as small as possible.

While in previous studies, particle generation based on the Rayleigh instability [28], which potentially provides some of the above mentioned requirements, was employed [19], two different generators have been tested. One is based on an air-atomizing nozzle (0.25 mm nozzle diameter, liquid nozzle: PF1050, air nozzle: PA64, ©Spraying Systems) and the other one is an airbrush system (0.3 mm nozzle diameter). An inverted grey scale image of the particle jet produced by the airbrush system is presented in figure 3. Both generators provide a particle production rate that is sufficient for measurements on the square meter scale. To generate the tracer particles, the TLC mixture is first dissolved in isopropyl alcohol. In the second step, the mixture is atomized by the generator. After evaporation of the isopropyl alcohol from the atomized droplets, atomized TLC tracers remain. However, the resulting particles do not have a monodisperse size distribution, see figure 4(a) and (b). Due to the above mentioned aspects, it is almost impossible to measure the TLC particle size directly. Therefore we used water instead of the TLC mixture and measured the particle size distribution produced with the generators by applying phase Doppler anemometry (PDA). The diameter statistics of the generated water droplets for both generators at their working points are presented in figure 4. All droplet sizes were measured with a PDA system assuming spherical droplets. The particle generators used provide a sufficient number of particles with the desired density and size distribution and are therefore well suited for simultaneous PIT/PIV investigations. A further analysis of the particle diameter distribution of the generators revealed that the airbrush system provides droplets whose diameters follow a log-normal distribution, see the Gaussian curve on a logarithmic x-axis (inlay of figure 4(a)). For the air-atomizing nozzle, no such simple particle distribution was obtained. The semi-logarithmic plot (inlay of figure 4(b)) reveals that there is a rather high proportion of large particles. These particle statistics underline that the airbrush system provides a sharper size distribution and, therefore, appears to be more suitable for our purposes. Nevertheless, more development work can be performed with respect to the generation of TLC particles, to further optimize, e.g. the dispersity of the generated particles.

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In our experiments, the mass mixing ratio between the TLC mixture and isopropyl alcohol was set to 1: 2 or 1: 1, eventually resulting in a TLC particle size smaller than the produced droplets by a factor of approximately 1.4 or 1.3 in diameter, respectively. Of course, variation of the mixing ratio opens up another possibility to vary the TLC tracer particle size.

For the results presented in this paper, we used the airbrush system and a mixing ratio of 1: 1 for the particle generation. The size of most of the produced TLC-isopropyl alcohol droplets turned out to be approximately 10 μm, see figure 4(a). Thus, the TLC particle diameter should be in the range of d_p ≈ 7 μm. The particles produced with this combination of generator and mixing ratio, led to the so far most convincing results in simultaneous PIV and PIT. Please note that according to the assumptions given by Raffel et al [12], discussed above, the mechanical reaction time until the particles reach 99% of the fluid velocity is in the order of τ_m ≈ 1.5 ms. For the rather slow flows considered here, the particles are assumed to follow the surrounding fluid instantaneously.

One of the main components in a simultaneous PIT/PIV setup is the white light sheet. For high quality PIV measurements, it should provide an intensity that is high enough and a collimation that is good enough to allow sharp imaging of single tracer particles. This implies also that the pulses of the light source must be short enough to not smear the particle images. Furthermore, the flashing frequency should be as high as possible, to achieve the best possible time resolution as well.

The white light sheet employed in our measurements is based on light emitting diodes (LEDs) and was specially developed for this purpose. The LEDs have the advantages of low cost, easy handling and good luminous efficiency. However, their drawbacks are the low coherence and high divergence of the emitted light and their finite size, i.e. they cannot be considered as point sources. A successful implementation of LEDs for PIV investigations was already reported in Willert et al [29]. Due to the need for high light intensity, white colour and good focusability at the same time in our application, we developed a light sheet source based on 60 ©OSRAM Platinum Dragon LEDs, LW W5SN-KYLY-JKQL, which are arranged in a one-dimensional array on a printed circuit board. Each single LED is equipped with a clip-on lens (©SHOWIN Clip-On Lens for Osram GD+ LED) in order to reduce the angle of radiation from approximately 120° to approximately 15°. Another main advantage of the applied clip-on optics is their small spatial extension, which is just a bit larger than the LEDs, allowing the achievement of a spatial density of 11.6 LEDs per 10 cm. The one-dimensional array of LEDs is used in combination with a slit aperture and a cylindrical lens. By adjusting the distances between the different elements, one can manipulate the brightness and divergence of the produced light sheet. A sketch of the light sheet optics is shown in figure 5.

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Each of the white light emitting LEDs has a luminous flux of 191 lm at a forward current of 1 A [30]. PIV investigations require a pulsed light source, this implies that the LEDs can be operated at an increased forward current of 3.3 A. For reasonable PIV results in thermal convection at Pr ≈ 0.7 and moderate Ra numbers, a pulse length equal to or shorter than approximately 1 ms is sufficiently short. The duration of the light pulses and the time delay between the first and the second pulse for PIV can be controlled with an external trigger unit using TTL signals. The time response of the light source to a double pulse TTL trigger signal was measured using a photodiode, placed in the light sheet, and a oscilloscope. For the characterization of the light source only, the delay between the two pulses was set to τ = 5 ms and the duration of each pulse to 1 ms. While the pulse duration is kept at 1 ms during our actual experimental investigations, the delay time is adjusted to τ = 16 ms in order to get a maximal pixel displacement in the order of Δx ≈ 10 px. Figure 6 shows the time traces of the light sheet output and the trigger pulse. Thereby, figure 6(a) depicts both pulses necessary for PIV investigations, and (b) sketches a magnification of just one pulse, to show the reaction of the light output on a trigger signal. The rise and fall times of the light output are 44 μs and 63 μs, respectively.

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The recorded pulse duration of 1 ms is sufficiently short for investigation of rather slow flows, e.g. indoor climatization, but may not be applicable for high flow velocities (also depending on the used camera and lens system). At high flow velocities, such a long pulse duration will lead to elliptically stretched rather than sharp circular images of the tracer particles, and therefore will lead to decreasing accuracy of the velocity evaluation. We also tested a shorter light pulse length of 0.5 ms, which makes it possible to measure faster flows as well. Due to a better signal to noise ratio and no drawbacks in the quality of the particle images, we used a pulse length of 1 ms for the study presented here. It should be noted that τ = 5 ms was fixed for the characterization of the light source, only. During the measurements, τ was adjusted to get pixel displacement values which are suitable for the PIV interrogation, see section 3.2.

Figure 7(a) presents intensity profiles of the lightsheet, which were measured using a photodiode mounted onto a traverse stage. For all presented distances (100 mm ⩽ x ⩽ 500 mm) from the front lens of the light sheet optics, a Gaussian profile was found. To further characterize the light sheet, the maximal values of the photo voltage as well as the full width of the profiles at half maximum (fwhm) are presented in figure 7(b) for distances from 50 to 1000 mm. One finds that for x ⩽ 500 mm the maximal photo voltage (i.e. the luminous intensity of the light sheet) drops only by 25%, while the thickness in terms of fwhm is always less than 10 mm. For larger distances, the maximal photo voltage drops rather quickly, to approximately 30% of its maximum, and the light sheet thickness increases up to 18 mm at x = 1000 mm.

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Thus, summarizing, we can state that the developed white light sheet optics produces a light sheet with a thickness (fwhm) of approximately 9 mm, very low divergence and rather homogeneous illumination over a distance of 500 mm.

The repetition rate is, at present, set to 4 Hz and in our investigations limited by the cameras used rather than by the light sheet optics. It is much higher than the repetition rate of approximately 1/5 Hz that Czapp [31] was able to realize. Furthermore, the new frequency of 4 Hz allows for well time-resolved studies of rather slow processes, e.g. indoor climatization or convection driven flows at low and moderate Rayleigh numbers in which we are interested.

A characteristic difficulty of PIT of air flows arises from the large difference of index of refraction between fluid and particle. While this is advantageous for parallel PIV measurements, the high amount of Mie-scattered white light tends to overcast the colour information of the inelastically scattered light. Precise PIT measurements thus require special attention during image processing in order to separate the temperature-dependent colour information from the background noise. In order to keep the amount of scattered white light to a minimum, the PIT camera was positioned at an angle of approximately 45° in the backward direction, which already reduces the background significantly. Such a camera setup for PIV leads to perspective errors associated with the out of plane velocity component. While the backward position of the colour camera is essential to receive sufficient image quality for PIT, the viewing angle of the PIV camera can be chosen independently. In the current study the PIV camera has been installed viewing slightly into backward scattering for practical reasons. However, in future measurements the PIV camera can be operated perpendicular to the field of view as well, while the colour camera is maintained in the backward direction, as long as the optical accessibility of the setup allows it.

The images for both PIV and PIT are recorded simultaneously with two Pixelfly (©PCO) cameras, one grey level double shutter and one colour camera. The double shutter camera is used for the well-known PIV evaluation (see e.g. [12]). For the temperature measurements the colour camera is used. This camera records red/green/blue (RGB) images with a Bayer filter (step 1 in figure 8). Both images are deskewed and overlapped using calibration grids and are projecting tools similar to those used for stereoscopic PIV.

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Since no neutral sunlight is used for illumination, a specific white balance is fundamental for acquiring realistic and reproducible colours. Therefore, we use a neutral greycard which has a remission of light that is independent of the wavelength of the incident light. Such a card is well-known in photography. It allows the determination of a reference value for each setup, in particular for a specific arrangement of the illumination, and thus to choose the right white balance which guarantees a neutral colour and reproducibility.

For filtering and temperature analysis, these images are converted into the HSV or the HLS colour space (step 2 in figure 8). Due to the finite width of the particle size distribution, the recorded images reveal a rather high amount of background noise. In order to take only the colourful particles into account, a threshold for the value (HSV) is used in order to filter the hue values (step 3 and 4). Different filter algorithms could also be based on the saturation (HSV or HLS) or the lightness (HLS). Finally, the local temperatures can be calculated using hue–temperature calibration functions (step 5).

The main challenge is to identify those pixels which are neither background noise, nor small particles scattering white light. This can be done in different colour spaces. For a realization in the HLS colour space, only those pixels should be taken into account that have a high saturation and a lightness in the range of < L < 1 − with > 0. In the HSV colour space, both parameters saturation and value must be above a certain threshold. However, we could get rather good results by using a local filter which is just based on the value parameter of the HSV colour space. Therefore, we will focus on the HSV colour space and in the following all given parameters, hue, saturation and value, will correspond to this colour space.

As an example, a synthetic image with different coloured particles and background noise is generated. It is shown in figure 9(a). The background noise is set to 0.1 ∈ [0, 1], the particles with a size of 3 × 3 px are placed on top of the noise and have linearly decreasing intensity from left to right of 0.9 to 0. Thus, close to the right side of the image the particles vanish within the noise. Such a configuration is chosen to see how robust the filtering algorithm is with respect to the signal to noise ratio. The colours of the particles are red at the top, then passing to yellow, green, blue and purple, until they become red at the bottom again. Therefore, the hue increases from top to bottom from H = 0 to H = 1.

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Figure 9(b) presents the pixelwise calculated hue map of the test image. One finds a random distribution of all hue values, clearly showing that the hue cannot directly be used for the calculation of the colour and therefore the temperature distribution.

The next possible step is to apply a median filter. The result for a 16 × 16 px median filter is shown in figure 9(c). The tendency of increasing hue from top to bottom is visible now, nevertheless all mean hue values are in the vicinity of H = 0.5. This is caused by the impact of the background noise with random hue values and clearly shows that an image which has more background than particles cannot be simply evaluated by calculating the mean hue within interrogation windows.

The idea is to use interrogation windows again, but to take only those pixels into the calculation of the mean hue value for this interrogation window that are not background noise. Due to slightly inhomogeneous illumination in almost all experimental setups, we applied such a filter with a local rather than a global threshold for the identification of particles. Therefore, we chose the V value, i.e. the brightness (see figure 9(d)), of the colour image and localize those pixels where the value of V has increased from the background level. Within each interrogation window, we take only those pixels into account for the calculation of the mean hue for the whole interrogation window, where the highest V-values are located. To clarify this procedure, figure 9(e) shows a scatter plot of hue versus V-value for a 16 × 16 px interrogation window. For low values of V (V ≲ 0.1), one finds all H ∈ [0, 1], but for high values of V, only a rather sharp hue distribution appears. We identify those pixels as the particles and the rest as the background noise. For the calculation of the mean hue, we take into account only the pixels with the highest value of V, which are marked with a box; in this example the threshold was set to four. Both the size of the interrogation windows as well as the threshold for how many pixels should be taken into account must be adjusted to the particle density of the image.

Applying this procedure to the synthetic image results in figure 9(f). One can see three things: firstly, the hue gradient from almost H = 0 at the top to almost H = 1 at the bottom can be reproduced. Secondly, even at a very low signal to noise ratio, on the right side of the image, the algorithm reproduces the hue values correctly. Thirdly, in red regions where H ≈ 0 or H ≈ 1, some errors are produced due to fact the hue is periodic and that a pure red colour is either H = 0 or H = 1. However, for the calibration of the liquid crystals this fact should not be bothersome, because the liquid crystals turn from red to yellow and green to blue until they become colourless again, meaning that H ≈ 1 will not appear using liquid crystals.

Although filtering can be realized in a similar way in the HLS colour space, filtering with the V-value in the HSV colour space yields good results for almost all investigated particle images so far, so we concentrate on this technique for now.

To obtain the absolute temperature information from the hue values, a hue–temperature calibration must be conducted. The calibration function found is only valid for the specific setup, i.e. only for the specific arrangement of cameras and light-sheet optics as well as for the crystal type and the particle generator used. Typically, during periods of stable layering of the temperatures, the different hue values and the corresponding temperatures are recorded. Due to the difficulty of generating these stable temperatures within our geometry and due to the high time effort of this calibration technique, we decided to apply, as a fast approach, a dynamic calibration to correlate the recorded hue values to the absolute temperatures. The main advantage is the rapidity of the calibration process. Therefore, a very quickly reacting, precisely calibrated glassbed NLC thermistor with a diameter of only 400 μm is placed within the measurement plane. Using this thermistor, which is calibrated with an accuracy of σ_{T, therm.} = 0.05 K, the temperatures are recorded at a high frequency (compared to the recording frequency of the PIT/PIV images) of approximately 13 Hz. For the hue–temperature calibration, the hue values are evaluated as described in the previous section and those hue values recorded next to the position of the thermistor are correlated to the temperatures. This results in time series for hue and temperature, as shown in figure 10(a). By eye, one can already see that the two quantities correlate with each other. It must be noted that the response time of the crystals is much lower than that of the 400 μm thermistor, whose thermal inertia acts as a low pass filter and therefore averages the high frequent temperature fluctuation caused by turbulence.

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Plotting the temperature as a function of the hue value confirms this, see figure 10(b). A linear fit to the measured data results in T [°C] = m · H + b = 14.8( ± 0.7) · H + 18.5( ± 0.1). The fluctuations of the signal incorporate different contributions. However, the main part is ascribed to the different reaction times of the thermistor and the TLCs. The latter are able to react to the temperature changes in the turbulent flow, i.e. the TLCs register more temperature fluctuations than the thermistor. However, the resulting uncertainty of the calibration parameters can be easily reduced by extending the calibration measurements, because the uncertainty of the mean hue value in a certain hue interval σ_〈H〉 scales like $\sigma _{\langle H\rangle }=\sigma _H/\sqrt{n}$, where σ_H is the standard deviation of the hue values in the interval and n is the number of recorded values. Using the fit errors of the linear regression, the uncertainty of the thermistor and the Gaussian error propagation for our actual calibration measurement, a hue dependent calibration uncertainty of $\sigma _{T_\mathrm{cal,min}}=\sqrt{(\sigma _m\cdot H_\mathrm{min})^2+\sigma _b^2+\sigma _\mathrm{T,therm}^2}=0.15\,\mathrm{K}$ at H_min = 0.14 is found, which increases to $\sigma _{T_\mathrm{cal,max}}=\sqrt{(\sigma _m\cdot H_\mathrm{max})^2+\sigma _b^2+\sigma _\mathrm{T,therm}^2}=0.18\,\mathrm{K}$ at H_max = 0.21. The corresponding error interval is indicated in figure 10(b) by the dashed lines.

However, these uncertainties are valid for absolute temperature measurements only. The error of relative temperature measurements due to the uncertainty of the calibration, following the Gaussian error propagation, is much smaller and given by $\sigma _{\Delta T_\mathrm{cal,max}}=\sqrt{(\sigma _m\cdot\Delta H_\mathrm{max})^2}=0.05\,\mathrm{K}$. For the actual measurement, however, other noise sources, which can be expected to average out during calibration, come into play and have to be considered. Most noteworthy for PIT is the pixel-hue noise, which is the variation of the hue value in a single interrogation window at an imaginary constant temperature. This error contribution can be caused by colour-noise of the PIT-camera, colour-noise of the light source and the size distribution of the TLC particles, whose colour play is known to depend on the actual particle size. Obviously, this noise can be decreased at the expense of spatial resolution by enlarging the interrogation window size. For an interrogation window size of 12 × 12 px in combination with a threshold of 64, i.e. the settings used in the following, this error has been estimated to amount to σ_hue ≈ 0.0016, as described in the following. As already stated in section 2.1, the smallest coherent structures to be expected in our flow measure around 10 mm, while the dimensions of our interrogation windows for PIT are as small as 5 × 5 mm². Accordingly, the hue fluctuations of a single interrogation window reflect measurement noise only. In order to determine this value, we computed the mean spatial standard deviation of the measured hue values in quadratic conglomerates of adjacent interrogation windows, which we averaged over the whole field of view and the whole image series. The results are depicted in figure 11. In order to determine the standard deviation of the hue value for a single interrogation window, the measured data were fitted with a power law and extrapolated to the value for n_int.win = 1, which is then interpreted as the hue noise error of a single interrogation window.

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Using again Gaussian error propagation, we obtain $\sigma _{T_\mathrm{max}}=\sqrt{(\sigma _m\cdot H_\mathrm{max})^2+\sigma _b^2+\sigma _\mathrm{T,therm}^2+(\sigma _\mathrm{hue}\cdot m)^2}=0.19\,\mathrm{K}$ and $\sigma _{\Delta T_\mathrm{max}}=\sqrt{(\sigma _m\cdot \Delta H_\mathrm{max})^2+2\cdot (\sigma _\mathrm{hue}\cdot m)^2}=0.06\,\mathrm{K}$ as maximal values for the absolute and the relative error, respectively. This results in a dynamic range for the actual measurement of (H_max − H_min)/σ_hue ≈ 43.

At this point it should be noted though, that the calibration described above has to be improved in the next steps by incorporating more measurement data, considering the impact of different viewing angles in the same field of view and systematically investigating the use of the evaluation parameters, which influence and can thus be used to optimize bandwidth and the dynamic range.

Other filtering and calibration procedures, as proposed in the literature, e.g. Rao and Zang [32] proposing a 5 × 5 median filter, Grewal et al [33] introducing neural networks and Roesgen and Totaro [34] suggesting a statistical technique using a proper orthogonal decomposition, are promising approaches to further improve the accuracy of the technique.

The investigated mixed convection cell has a square cross section of 500 mm × 500 mm and an aspect ratio between length and height of Γ_xz = 5 (see figure 12(a)). The air inflow is supplied through a slot below the ceiling, while exhaust is provided by another slot above the floor. Both vents extend over the whole length of the cell. The bottom and ceiling of the cell can be maintained at different temperatures. This setup allows generating mixed convection under well-defined conditions, and serves as a generalized model system for geometrically more complex technical configurations. The flow field and the heat transport within this container have already been presented in some publications: Westhoff et al [35] investigated the low-frequency oscillations of the large-scale circulation, Kühn et al [36] applied a newly developed tomographic PIV technique to large volumes, and Schmeling et al [37] studied the large-scale flow structures and the heat transport using PIV and temperature probes. Again Schmeling et al [38] investigated the large-scale circulations and their oscillations by means of many temperature probes within the cell combined with smoke visualizations.

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For the first experiments with the combined PIT/PIV technique, no external pressure gradient is applied between the inlet and the outlet of the enclosure. Thus, the measurements are conducted rather in pure thermal convection than in mixed convection. Nevertheless, due to the still existing openings of both the inflow and outflow channels, a slow fluid exchange with the surrounding takes place. The fields of view with extensions of approximately 300 × 150 mm² for PIT and 225 × 150 mm² for PIV are located close above the heated bottom plate and close to the front side of the enclosure, see figure 12(b) and the coordinate systems in figure 16, where y = 0 and z = 0 is the lower front edge of the enclosure. The difference of the sizes of the fields of view is caused by the different angles under which the cameras are operated. Remapping and superposing the images is realized based on a well-known technique for stereoscopic PIV (e.g. [12]) using recordings of a calibration grid.

The TLC particles are generated with the airbrush system, which is supposed to inject the particles into the settling chamber in front of the inlet for mixed convection cases. During the feasibility tests, as already mentioned, the external volume flow is switched off in order to provide a longer retention period in the cell, therefore, the particles have to be sprayed directly into the enclosure. For illumination of the particles, the specially developed white light sheet based on white LEDs is used. The particle images are recorded with a double shutter grey level and a single shutter colour CCD camera, so that temperatures could be determined from the local hue values and velocities from the local particle image displacements between subsequent recordings. The use of two cameras provides better results than using just a single double shutter colour CCD camera. This is because the less sensitive colour camera can be operated in async. shutter mode, recording only the first of the two light pulses, while the highly sensitive grey level camera is operated in the double shutter mode for the PIV evaluation. Furthermore, the signal to noise ratio of the grey level camera is lower than that of the colour camera, which leads to a higher quality of the PIV results.

A raw particle colour image is presented in figure 13(a), to depict the quality and the contrast of the colour images used for the temperature evaluation. A plume of warmer air can already be seen in this picture: the blue part in the centre indicates higher hue values than those of the green and red surroundings. Both frames of the B/W image after high and low pass filtering are shown in figure 13(b). Further, information on the applicability of PIV is given in figure 14, which presents (a) the correlation plane (left) for one interrogation window (24 × 24 px), including the corresponding windows of the two frames (middle and right). The size of the particle images, which is approximately 2 × 2 to 3 × 3 px, the suitable number density of particle images per interrogation window and the clear peak in the correlation plane indicate the high quality of the PIV measurements. Additionally, figure 14(b) reflects the 2D and (c) the 1D pixel displacement distributions; these depict the good quality of the PIV results in terms of the absence of peak locking.

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After rectifying, overlapping and evaluating the data, simultaneously recorded instantaneous temperature and velocity fields as depicted in figure 15 can be calculated. The formation of a warm plume which rises from the lower thermal boundary layer can be seen. The top hat of the plume as well as both side vorticities and the root with quickly rising warm air are clearly visible. Such a detailed recorded picture of temperature and velocity of a rising thermal plume has, as far as we know, not been previously recorded experimentally.

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For the velocity calculation, we use a pulse delay of τ = 16 ms at a light pulse length of 1 ms, which leads to a pixel displacement up to 10 px and therefore a maximal particle shift during one light pulse of less than Δx ≈ 0.7 px. Furthermore, we use an interrogation window size of 24 × 24 px, an overlap of 50%, a three point Gaussian fit for the peak detection, and a seven step multigrid evaluation starting at a window size of 256 × 256 px and disabled sub-pixel image shifting. This results in a vector spacing of approximately 2.5 mm. For the sake of clarity, only every second vector in each direction is shown in figures 15 and 16. The hue values, i.e. the temperature values as well, are calculated using a filter size of 12 × 12 pixel and a threshold of 64, see section 2.4.

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To conclude this subsection, a short discussion of possible errors and limitations resulting from the use of TLC tracer particles, LED based white light illumination, as well as the camera systems will be given. First to be mentioned is the spatial homogeneity of the particle distribution, as in any PIV evaluation. This is already rather good, see e.g. the particle images in figure 13(b). The homogeneity of the illumination is also no problem for this field of view, but might be a more important issue with larger measurement planes due to the divergence of the LEDs. The light sheet thickness, which is approximately 10 mm, is in reasonably good agreement with the interrogation window size used for PIV (≈5 × 5 mm²), but imposes a restriction on the spatial resolution for non-purely 2D flows. These points may cause some errors and limitations, but do not influence the results in our studies negatively so far. Secondly, there are restrictions regarding the maximal flow velocity that can be studied. These are the following behaviour of the tracer particles, which is discussed in detail in section 2.2. The duration of the light pulse limits, in combination with the imaging optics used, the maximal flow velocity as well, because particle images would be smeared out or stretched to stripes for too long pulses in combination with fast flows. In our system, a pulse duration of 1 ms still leads to sharp particle images, see e.g. the particle images in figure 13(b) and 14(a). Thirdly, the accuracy of the temperature and the sensitivity of the hue measurements have been analysed and discussed in section 2.5. Currently, the achievable accuracy in the temperature measurement is limited by the accuracy of the hue temperature calibration, while the hue sensitivity would offer more headroom. Nevertheless, relative temperatures can already be measured with an accuracy of $\sigma _{\Delta T_\mathrm{max}}=0.06\,\mathrm{K}$, which is reflected in the colour bar of the contour plots of our results accordingly.

The convective air flow investigated here is induced by a temperature gradient between the bottom and the top plates of ΔT = 7.2 K, while the external pressure gradient is set to zero. This results in the following characteristic numbers: Rayleigh number Ra ≈ 9.0 × 10⁷, Reynolds number based on the inflow velocity Re = 0 and Prandtl number Pr = 0.71. Thus, except for the unsuppressed fluid exchange through the openings of the inlet and the outlet, pure thermal convection is investigated.

The results of simultaneous PIT and PIV measurements in our rectangular convection cell are presented in figure 16, which shows the temperature as contour and velocity, calculated from image pairs by means of well-established PIV algorithms [12], as a vector field. Thereby, the maximal observed velocity is approximately 0.12 m s⁻¹. The images are recorded at a recording frequency of 4 Hz. Figure 16 shows twelve consecutive images, i.e. the time difference between two images is 0.25 s and the series spans a duration of 3 s. A supplemental movie created out of 100 single images at a recording frequency of 4 Hz, i.e. spanning a time period of approximately 100 s is available at stacks.iop.org/MST/25/035302/mmedia.

The parameters used for the PIV and PIT evaluation are the same as described in the previous section, besides the fact that now only every second vector is shown in each direction for the sake of clarity in figure 16 and every third in the movie. Finally it should be noted that no outlier detection is used during the PIV interrogation, but for the visualization, especially in the supplemental movie, a value blanking using a threshold for the velocity is applied to remove distracting vectors.

Figure 16(a) is again the warm plume that was already depicted and described in the section above (figure 15). The time series of images here (a) to (l) shows the breakdown of the typical plume structure towards a thin region of warm rising air at a rather high time resolution of 4 Hz. Thereby, the vortex on the left side of the plume is almost at rest during these 3 s, while the right one convects out of the field of view, leading to the breakdown of the typical mushroom structure. Due to the fact that the velocity and temperature fields are recorded as 2D fields in a plane, it must be taken into account that there will be an out-of-plane component which is not recorded here.

Nevertheless, from the fluid dynamical point of view, the instantaneous temperature and velocity fields recorded at a high frequency open new possibilities in, e.g., analysing the correlation between these two quantities and studying their time evolution. Furthermore, simultaneous measurements will help to further investigate the dynamical processes which lead to, e.g., reorientations and oscillations within convective flows observed, amongst others, in our convection cell [38].