Application of 4D two-colour LIF to explore the temperature field of laterally confined turbulent Rayleigh–Bénard convection

Recent studies show the significant effect of the third dimension and flow unsteadiness of laterally confined Rayleigh–Bénard convection (RBC). However, there are limited studies investigating 4D flow properties. The application of 4D two-colour laser-induced fluorescence to explore a laterally confined RBC at high Rayleigh number of Ra=9.9×107\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ra=9.9\times {10}^{7}$$\end{document} and Prandtl number of Pr=6.1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Pr=6.1$$\end{document} is investigated using a scanning laser system. A two-colour, two dyes approach was employed to resolve the laser sheet intensity variations due to refractive index variations caused mainly by the generation of thermal plumes. Two temperature-sensitive fluorescent dyes with opposite sensitivities were used to enhance the overall temperature sensitivity to 7.3%/°C. Details of the experimental procedure and optical system employed to reach such a high sensitivity are demonstrated. From the whole field temperature distribution, the dimensionless heat transfer coefficient, the Nusselt number, and its evolution were calculated for both hot and cold boundaries. Temperature field and Nusselt number obtained from 3D and 2D fields are reported to compare the results for these two scenarios. Thermal plumes were found to have a conical shape in the laterally confined RBC compared to the conventional mushroom shape. From visualization of the time-resolved 3D temperature field and 2D distribution of the Nusselt number, it was also found that only by volumetric measurement, temporal and spatial variations of the temperature and heat transfer can be evaluated. This shows that to evaluate the classic and ultimate theories, volumetric measurement is required for a coherent understanding of the physics.


Introduction
Buoyancy-driven flow in an enclosure heated from below and cooled from above is known as Rayleigh-Bénard convection (RBC) (Ahlers et al. 2009). Controlling parameters of RBC are defined as Rayleigh number, Ra ≡ gΔTh 3 ∕ , Prandtl number, Pr ≡ ∕ , and aspect ratio of the enclosure, Γ ≡ w∕h , where is the thermal expansion coefficient, w and h are the width and height of the convection cell respectively, g is gravitational acceleration, is kinematic viscosity, is thermal diffusivity of the fluid, and ΔT = T H − T C is the temperature difference between the hot, T H , and cold, T C , horizontal boundaries (Shishkina 2021). By exceeding the Rayleigh number from a critical value, convection initiates by the rise and fall of thermal plumes (Xi et al. 2004). After this, development of the turbulent flow leads to the formation of unsteady circulation structures (Krishnamurti and Howard 1981). It is known that these circulating structures are driven by the formation of the thermal plumes (Ahlers et al. 2009).
RBC has been a focus of interest due to its many applications in engineering, studying the physics of the atmosphere, and in studying bifurcation and chaos (Ahlers et al. 2009). Cubical and cylindrical enclosures are the most conventional shapes that have been used for studying RBC (Shishkina 2021). Among these two enclosures, both unit aspect ratio and high aspect ratio cells have been studied extensively (Shishkina 2021). However, recent investigations have found that the physics of heat and flow is quite different in lateral confined enclosures (low aspect ratio/ vertically slender) (Zwirner et al. 2020). Most of these studies are however, numerical works and are based on relevant experimental works that are limited to measurements of flow properties such as velocity and temperature at a limited number of points (Ecke and Shishkina 2022). Although these experimental studies were insightful, whole field time-resolved measurements of flow properties are required to provide supporting evidence for the numerical studies.
The heat flux which is transferred by convection is defined as Q = h c A T w − T ∞ in which h c is the convection coefficient, A is the area of the solid boundary which is in contact with the fluid flow, T w and T ∞ are the temperature of the solid boundary and fluid flow, respectively (Adrian 2013). To measure the heat transfer and calculate the dimensionless heat transfer coefficient, Nusselt number, which is defined as Nu = h c l∕ , where l is the characteristics length, measuring the temperature is necessary. Generally, in experimental works this is done by measuring the temporal variation of the temperature of the certain points of the domain, then the spatial averaged heat transfer can be calculated based on the number and distribution of the temperature sensors (thermocouple or thermistor) (Adrian 2013). In RBC, similar procedure is usually taken to calculate the Nusselt number both for hot and cold boundaries (Ecke and Shishkina 2022).
To characterize and quantify the heat transport of RBC, two different theories have been proposed and investigated since 1954 (Lohse and Xia 2010). The classical theory claims that the dimensionless heat transfer coefficient, the Nusselt number, Nu , of this system is only a function of Rayleigh number as Nu ∼ Ra 1∕3 (Malkus 1954;Priestley 1954;Doering 2020). However, the ultimate theory declares that heat transport of this system is also a function of the Prandtl number as Nu ∼ Pr 1∕2 Ra 1∕2 (Kraichnan 1962;Grossmann and Lohse 2011;Doering 2020). Many investigations have been untaken to unravel the physics of the heat transfer of an RBC and to explode the potential fit of either theory (Doering 2019).
In experimental works on RBC, it can be seen that the measurement of flow properties such as temperature is usually limited to certain points in the domain (Ecke and Shishkina 2022). Although whole field measurement of the temperature field was always a focus of interest (Sakakibara and Adrian 1999), recently it can be seen that there are more attempts to apply whole field measurement techniques to measure the flow properties of the RBC (Moller et al. 2021;Schiepel et al. 2021). Theoretically, to calculate the Nusselt number solving the governing equations of the flow field is required (Ecke and Shishkina 2022). In RBC by using the Oberbeck-Boussinesq approximation the Navier-Stokes equation is described as: in which u is the velocity of the fluid flow, P is the pressure, and ê y is the unit vector in vertical direction. The energy and continuity equations also can be defined as: As a result, in numerical studies of RBC, the calculation of the Nusselt number can be determined since the solution of above governing equations results in the calculation of the required velocity and temperature fields (Ecke and Shishkina 2022). In experimental studies, however, calculation of average Nusselt number is more usual due to the limited number of certain points on the boundary of the fluid domain due to the difficulties in the measurement of the velocity and temperature of the full spatial and temporal domain (Ecke and Shishkina 2022).
Other than heat transfer, knowledge of the temperature field and its evolution and dynamics of the evolving thermal plumes has also been a focus (Lohse and Xia 2010). An example of (1) t u + u ⋅ ∇u + ∇P = ∇ 2 u + gTê y this is the application of RBC in the design of a new DNA polymerase chain reaction (PCR) which operates based on the circulation of flow in an enclosure with different temperature zones (Khodakov et al. 2021). It is known that in confined RBC (confined only in one direction), 2D measurement and simulation can be used to capture the true physics (Chong and Xia 2016). However, the knowledge of the whole field in lateral confined RBC is necessary since generally the variation in the 3 rd dimension is significant (Zwirner et al. 2020). These points highlight the necessity to understand the time-resolve 3D data of the temperature of the whole domain of RBC to have a coherent understanding of the physics of this thermofluid system.
There are different techniques to measure the fluid temperature within an RBC such as thermocouples, liquid crystal (Fujisawa et al. 2005), phosphorescent (Hu and Koochesfahani 2006), and fluorescent thermography (Sakakibara et al. 1993). Planar laser-induced fluorescence (PLIF) is a non-intrusive fluorescent thermography technique that can be applied for temperature measurement of liquid or gaseous flow fields (Sakakibara and Adrian 2004). To increase the temperature sensitivity and reduce the effect of uncertainty sources such as the laser power fluctuation and local density gradient, twocolour PLIF has been developed (Coppeta and Rogers 1998;Sakakibara and Adrian 1999). Rhodamine B and Rhodamine 110 are a common pair of fluorescent dyes for two-colour PLIF in aqueous flow (Kim and Kihm 2001;Song and Nobes 2011). However, due to the low temperature sensitivity of this pair of fluorescent dyes, there are several studies attempted to enhance the temperature sensitivity either by finding a new pair of fluorescent dyes or by changing the wavelength of the excitant source (Sutton et al. 2008;Shafii et al. 2010;Estrada-Pérez et al. 2011).
The experimental methodology for applying scanning 4D two-colour LIF on a laterally confined RBC is investigated in this paper. The challenges and the solutions for achieving a relatively high temperature sensitivity is discussed. The evolution of the 3D temperature field from the onset of convection, t = 0 s to t = 200 s is captured using this method. From the temperature distribution near the horizontal boundaries, the whole field Nusselt number and its evolution is calculated for both hot and cold boundaries of RBC during the whole 200 s of the experiment. Results from the whole field measurement is also compared to the planar results to specify the role of 3 rd dimension in characterizing the physics of temperature distribution and heat transfer of the present RBC.

Two-colour PLIF
PLIF thermometry works based on the fluorescence light of some fluorescent materials (Walker 1987). Electrons of fluorescent molecules can be excited and move to a higher energy state by absorbing photons from a light sourse (Walker 1987). When this electron returns to the ground state, the fluorescence process occurs with the emission of a photon of light at a specific wavelength (Walker 1987). Usually the emitted signal of a fluorescent material has a longer wavelength than the excitant light (Walker 1987).
The fluorescent intensity can be expressed as: where K is a parameter representing the detection collection efficiency, depends mainly on optical apparatuses specification, I 0 is the incident laser intensity, C is the concentration of the fluorescent dye, is the absorption coefficient, L is the sampling length along the incident beam/sheet, and Φ is the photoluminescence quantum efficiency. For some molecules, the quantum efficiency Φ is temperature dependent. Based on that, by collecting the emitted signal of the fluorescent dye the temperature field can be quantified in a liquid/gas flow. For one-colour PLIF, the emission from a single fluorescent dye, over a single spectral band can be used to measure the temperature field (Nakajima et al. 1991;Sakakibara et al. 1993). In this scenario, spatial and temporal variations of the laser intensity can affect the emitted signal of the fluorescent dye (Sakakibara and Adrian 1999). A good example of this is the formation of thermal plumes which is one of the characteristics of RBC (Sakakibara and Adrian 2004). Formation of thermal plumes in RBC leads to formation of different regions with variable temperature and concentration which influences the refractive index distribution of the whole field (Sakakibara and Adrian 2004). As an example, Fig. 1 shows the calibrated fluorescence signal (one-colour) during the formation of hot plumes from a hot surface, excited by a laser sheet. As can be seen in this figure, the region at the (4) I f = I 0 K LCΦ Fig. 1 Signal of a temperature sensitive fluorescent dye (one-colour) showing the rise of the thermal plumes in an enclosure heated from below. Colour bar indicates the temperature range after applying the temperature calibration top of the thermal plumes is quite smooth. But looking to the region filled with thermal plumes, some horizontal streaks can be observed which are due to beam steering as a results of refractive index variation in the flow field caused by the variation of the density of fluid by thermal plumes.
A common method to eliminate the effect of laser intensity variations is to use a ratiometric approach (Coolen et al. 1999). Applying a two-colour two dye approach is the most common method to apply ratiometric PLIF. Using this approach, the temperature field will be calculated based on: where D1 and D2 are representing the fluorescent signals of the first and second dyes. Assuming that same laser light with the same optical apparatus are used to collect the signal of both fluorescent dyes. Hence the temperature can be obtained considering fixed concentrations of a mixture of two fluorescent dyes, C(D1) and C(D2) the fluorescence intensity ratio R of the two dyes where R = I f (D1)∕I f (D2) can be defined as R ≡ AF(T) (Shafii et al. 2010). In this definition, A is a constant representing the optical properties of the experimental setup, and F(T) represents the overall temperature response function that is connected to the temperature dependence of the absorption coefficients, (D1) and (D2) and the quantum efficiencies of the two dyes, Φ(D1) and Φ(D2).

Fluorescent dyes and optical filters
Rhodamine B (RhB) is a popular fluorescent dye that has been used for thermometry by one-colour PLIF (Kim and Kihm 2001). The maximum temperature sensitivity that has been reported for this dye is around − 2%/°C (Sakakibara et al. 1993). To apply two-colour two dye PLIF, Rhodamine 110 is the fluorescent dye that is commonly used with RhB (Kim and Kihm 2001). Since this fluorescent dye is non-sensitive to temperature the overall temperature sensitivity of (5) this pair of fluorescent dye stays the same as the one-colour RhB (Sakakibara and Adrian 1999). Table 1 shows studies that used this pair of fluorescent dyes to apply two-colour PLIF for thermometry. As can be seen, considering the different range of temperature and the wavelength of the laser, the maximum temperature sensitivity that has been obtained is similar to the one-colour PLIF reported by (Sakakibara and Adrian 1999).
There have been several attempts to enhance the temperature sensitivity by finding new temperature sensitive fluorescent dyes, pairing different dyes, and changing the experimental conditions to apply two-colour PLIF (Crimaldi 2008). The highest temperature sensitivity that has been found was reported by pairing Fluorescein, with positive temperature sensitivity, with Kiton red (Kr) and RhB, both with negative temperature sensitivity (see Table 1). Comparing the RhB with Kr, based on what has been reported by Sutton et al. (2008), Kr-Fl 27 has a higher temperature sensitivity than RhB-Fl 27. It is worth noting that there are two different types of Fluorescein dye have been applied for thermometry, Sodium Fluorescein (Fl) and Fluorescein 27 (Fl 27) with slightly different properties and temperature sensitivities (Sutton et al. 2008).
A list of experiments specifications of thermometry using Fl, Fl 27, and Kr is shown in Table 2. As can be seen, Fl emission signal is a function of laser wave length. The emitted signal of this dye has a positive temperature sensitivity when a 514 nm or 532 nm laser is used. However, it is negative and drastically lower when it is excited by a 488 nm laser light. For Kr, the temperature sensitivity is reported to be ~ − 1.5%/°C using 532 and 514 nm laser light.
Based on the literature, using Fl or Fl 27 with Kr can be used as two dyes for applying two-colour PLIF to achieve a high temperature sensitivity. However, since there is some lack of information regarding these three fluorescent dyes, such as the temperature sensitivity of Fl 27 when it is excited by a 488 nm laser, and it is different for each experimental setup, the temperature sensitivity of them was tested separately. This was carried out by calibrating the temperature  Sutton et al. (2008) versus the emitted light of each fluorescent mixture in the temperature range shown in Fig. 2 using a dye calibration cell (LaVision GmbH) with a procedure similar to (Sakakibara and Adrian 1999). The properties of these three fluorescent dyes can be found in Table 3. As shown in Fig. 2a, Fl and Fl 27 have a sensitivity of + 2.4%/°C and + 2.2%/°C, respectively when they are excited by a 532 nm laser. When they are excited by a 488 nm source, the sensitivity of both of them was found to be as low as − 0.3%/°C. The emission spectrum of Kr at different temperatures given in Estrada-Pérez et al. (2011) highlights that Kr has more sensitivity at the emission peak at 514 nm. Therefore, the temperature sensitivity of Kr signal when it is excited by a 532 nm laser is examined, once with a long pass filter with cut off wave length of 600 nm and once with a band pass filter capturing the signal of the peak of the emitted signal in the range of 589-625 nm. As shown in Fig. 2b, temperature sensitivity for the band pass and long pass filters is found to be − 2.1%/°C and − 1.7%/°C respectively, showing the advantage of the band pass filter.
To apply the two-colour PLIF in this study, an aqueous mixture of Sodium Fluorescein (Fl) and Kiton red (Kr) were used. The variation of the absorption and emission spectra of the dyes with the wavelength is illustrated in Fig. 3, in addition to the location of the illumination laser at = 532 nm. A diagram depicting the fluorescent signals and optics is illustrated in Fig. 4. As shown in Fig. 3, the laser light and absorption spectrum of Fl has a very small overlap which leads to a significant decrease in the emission signal of Fl (Coppeta and Rogers 1998) which leads to a decrease in  signal-to-noise ratio, S/N of the Fl emitted light. To increase the intensity of the signal and prevent reduction in S/N ratio, the pH of this aqueous solution was increased to 11 by adding Sodium Hydroxide (NaOH) since the emission intensity of Fl has a direct relation with the pH of the aqueous solution (Estrada-Pérez et al. 2011). As shown in Fig. 3, the absorption spectrum of the Kr and emission spectrum of Fl have a considerable overlap. To prevent the absorption of the signal which is emitted by Fl, the concentration of both dyes was set to 10 −7 mol/l or 5.8 × 10 −5 g/l for Kr and 3.8 × 10 −5 g/l for Fl by accurately measurement the mass fraction of the dye powder in the mixture using a precision mass balance. The Fl signal, which is marked in light green in Figs. 3 and 4, was collected using a band pass filter (#86-992, Edmund Optics Inc.) with a central wavelength (CWL) of 525 nm and full width at half maximum (FWHM) of 50 nm. The 532 nm laser light is also used to excite the Kr. To capture the Kr signal with maximum temperature sensitivity and to avoid receiving the Fl emitted signal, a band pass filter (#84-118, Edmund Optics Inc.) with CWL of 607 nm and FWHM of 36 nm was utilized, as shown in orange in Figs. 3 and 4. As shown in Fig. 4, the emitted signal of the fluid domain can be divided into the three main signals of the Fl, Kr, and noise sources such as the laser light. A dichroic mirror with a cut-on wavelength of 567 nm (DMLP567L, ThorLabs Inc.) was used to divide the signals from the flow field to each camera. Using the dichroic mirror instead of a beam splitter was advantageous to maintain the whole energy of Fl emitted light. To avoid receiving the signal of the laser, shown in Fig. 4, a notch filter (#86-130, Edmund Optics Inc.) with a central wavelength (CWL) of 532 nm and FWHM of 17 nm was utilized as shown in Fig. 3. Figure 5 shows the result of the intensity versus temperature calibration of the Fl-Kr mixture. The intensity-temperature calibration of the fluorescent dyes which leads to Fig. 5a has been carried out using a dye calibration cell (LaVision GmbH) with the same conditions of the experiments, ranging from 5 °C < T < 47 °C. Fl has a positive temperature sensitivity of ~ +1.3%/°C, and Kr has a temperature sensitivity of ~ − 1.9%/°C. As can be seen, the temperature sensitivity of the signal of each dye is less comparing to their sensitivity when they were measured individually (as shown in Fig. 2). One reason for this is increasing the pH of mixture for increasing the signal of Fl (Coolen et al. 1999). Figure 3 also highlight a notable overlap between the emission spectra of Kr and emission spectra of Fl which can lead to a decrease in temperature sensitivity of Kr. The ratiometric intensity-temperature graph, Fig. 5b, indicates the temperature measurement with the overall temperature sensitivity of  Table 1. Data points of the calibration graph of the two-colour method (Fig. 5b) were obtained by dividing individual data points of the linear calibration graphs of Kr and Fl (Fig. 5a). Dividing the linear, single-colour calibration data for the two dyes leads to the nonlinear ratiometric response shown in Fig. 5b of the two-colour sensitivity with temperature. The general overall temperature sensitivity of ~ 7.3%/°C is calculated based on the mean linear slope of the graph in Fig. 5b following the approach used by Sutton et al. (2008) and Shafii et al. (2010).

Measurement system
Previous investigations (Estrada-Pérez et al. 2011) showed that a 532nm laser can provide the highest temperature sensitivity in comparison to the lower excitation wavelengths such as 526, 515, 510 and 488 nm when Fl 27 is used for Fluorescence thermometry. The results of temperature sensitivity of Fl27 indicated in Fig. 2a agree with Pérez et al. results, and from the same figure it can be seen that Fl has the same temperature sensitivity variation as Fl 27 using 532 nm and 488 nm excitation wavelengths. As a result, it can be inferred that using 532 nm laser lead to maximum temperature sensitivity using Fl. For the experiments of this work, a diode pump laser with a maximum power of 2 W and wavelength of 532 nm was used to excite the fluorescent dyes.
A schematic of the optical measurement system implemented to apply 4D two-colour LIF is illustrated in Fig. 6. Two scanning mirrors (GVS002, ThorLabs Inc.) were used first to make the laser sheet (scanning mirror-y) and then to scan the channel depth, z-direction (scanning mirror-z) with the laser sheet itself. As depicted in Fig. 6, two double convex lenses were set in front of the laser beam to minimize the beam and, respectively, sheet thickness which led to FWHM of 120 μm. A double convex lens with a focal length of 80 mm also positioned in front of the scanning mirrors, collimates the laser sheet.
To collect signals of the fluorescent dyes, two 8-bit highspeed CMOS cameras (Flare 12M125, IO Industries Inc.) with a maximum frame rate of 220 fps and resolution of 2048 pixels × 2048 pixels were utilized. A two-channel function generator controlled the frequency and amplitude of the scanning mirrors. A signal generator was also utilized to synchronize the cameras and the two scanning mirrors. A timing plot of the signals generated to control the scanning mirrors and synchronize them with the cameras is illustrated in Fig. 7. In this figure, Signal A is the waveform that scans the laser sheet in the depth of the test section. The waveform's initial voltage defines the initial position of the laser sheet in the z-direction obtained by the voltage-space calibration.
To apply 4D LIF, 9 slices with a 0.5 mm spacing and 0.5 mm offset from the two vertical boundaries were used to scan the depth of the channel, z-direction. Signal B in Fig. 7   Schematic of the optical measurement system used to apply 4D two-colour LIF is set to trigger the laser beam as it was scanned through the z-direction as is shown in Fig. 6. The periodic-pulse, Signal C is a TTL signal generated to control the framerate of the cameras and synchronize them with the scanning mirrors. For each plane a single TTL signal was generated to capture a frame per plane. To freeze the motion of the thermal plumes and temperature variations, the frequency of the scan is set to be much higher than the time scale of the flow, free falling time defined as t f ≡ h∕ √ gΔTh . As a result, having a 9-slice planar temperature field stack allowed the generation of the 3D time-resolved temperature field.
For the PLIF application, a scanning laser beam to generate the laser sheet is advantageous comparing to generation of a static laser sheet by bulk optics such as cylindrical and Powell lenses (Crimaldi 2008). Based on a numerical simulation (Crimaldi 2008), it was shown that to capture the flow features such as the thermal plumes and local temperature variations using a scanning system leads to a sharper image of those features. Using a scanning system also allows a relatively thin laser sheet to be used which avoids capturing the out of plane properties (Crimaldi 2008). Figure 8a shows the intensity of both Kr and Fl emission signal excited by the laser sheet at a constant temperature at T = 25.4 °C. As can be seen for y = 0 line for both fluorescent dyes the signal distribution is uniform with the overall deviation of 0.2%. For x = 0 line, as the light is absorbed by the fluorescent dye, decay in the emitted light is expected (Karasso and Mungal 1997). However, as shown in Fig. 8b the overall deviation is equal to 1.5% minimizing the laser sheet correction requirement along the light sheet.
Temperature-intensity calibration was performed with the same laser sheet and optical system used to record the Fig. 7 Schematic diagram of the three signals produced to synchronize the two scanning mirrors and the cameras Fig. 8 Variation of the emitted light of Kr (top) and Fl (bottom) along a x∕w = 0 and b y∕h = 0 lines experiment images. This was performed separately for each plane, since the distance from the laser sheet to the collector sensor, in this case a CMOS camera affect the emitted light from the fluorescent dye (Walker 1987). Error bars in Fig. 5b indicate the standard deviation in the intensity emitted from Kr and Fl from each plane for the images of temperature in the range of 5 °C < T < 47 °C. Since a ratiometric approach has been employed in this work, it is necessary to match the field of view of each camera. For this purpose, each camera is set on a microstage having two degree of freedom in moving in x and z-direction. After the setting the field of view of each camera, calibration was performed using a target at 9 different planes by a commercial software (Davis 10.0.5, LaVision GmbH). The result of the camera calibration for each plane showed that the variation in magnification in the depth of the domain was as low as 0.5% at the furthest distance from z∕w = −0.5 to z∕w = 0.5.

Fluid test cell
A slender cell for the RBC with dimensions of 5 × 5 × 50 mm 3 and aspect ratio, Γ = w∕h , equal to 0.1, shown in Fig. 9c was used for this experiment. The test section is made from an acrylic sheet with a thickness of 6.35 mm that has a low thermal conduction coefficient of 0.2 W/mK. The temperature boundary conditions, shown in Fig. 9c, can be described as a constant temperature at the top, T H = 45 °C and bottom, T C = 5 °C and adiabatic at the side walls, T∕ x = T∕ z = 0 . The field-of-view (FOV) of the imaging systems where the temperature investigations were carried out is also highlighted in Fig. 9c. A rendering of the physical set of the slender cell used for this experiment is shown in Fig. 9b. To ensure that the temperatures of the hot and cold heat sources remain constant during the tests, the heat exchangers were connected to thermal baths to control surface temperatures. End heat exchangers feature a copper plate to improve the uniformity of the heat transfer surface. The support bases are made using an SLA 3D printer (Form3, Formlabs Inc.) and printing material, a liquid photopolymer, with a low thermal conduction coefficient of 0.15 W/mK. To ensure that temperature is distributed constantly on the surface of the copper plate, the temperature distribution was visualized by an IR camera (FLIR A35, FLIR Systems Inc.). The temperature distribution on the hot solid boundary is shown in Fig. 9a, showing a smooth constant distribution of the temperature for the whole domain.

3D reconstructed temperature field
The instantaneous temperature field at four different time instances is shown in Fig. 10. To examine and visualize the temperature field variations at the depth of the convection cell, the instantaneous temperature fields are shown in three different depths: at the centre of cell where z∕w = 0 , close to the first vertical boundary, z∕w = −0.3 , and close to the second vertical boundary, z∕w = 0.3 . The instantaneous 3D reconstructed temperature fields are also shown in Fig. 11 using temperature iso-surfaces. From the temperature fields at the first two time instances, t∕t f = 22 and t∕t f = 42 , the number and shape of the hot and cold thermal plumes can be observed. In comparison to the thermal plumes that are formed in a unit aspect ratio enclosures such as those shown in Fig. 1, and in Xi et al. (2004), it can be seen that only one thermal plume from each boundary is formed. It also can be observed that a conical shape is developed instead of the common observed mushroom shape. Although this has not Fig. 9 a Temperature distribution on the copper surface of the hot heat exchanger. b Rendering of the solid model of the slender rectangular RBC cell. c Schematic of the dimensions and boundary conditions of the RBC test cell been reported earlier, it was shown that in Hele-Shaw enclosures, which are severely confined only in one direction, the shape of the thermal plumes is different comparing to the mushroom shape plumes (Letelier et al. 2019;Liu et al. 2020). This implies the effect of the shape of the enclosure on properties of the thermal plumes.
For t∕t f = 57 in Fig. 11, it can be seen that the two plumes do not collide and mix. Instead, they pass each other from a separate side of the enclosure. However, this cannot be inferred from the planar temperature field at z∕w = 0 shown in Fig. 10b. However, by examining the planar temperature fields at the same time instance, t∕t f = 57 , and at z∕w = −0.3 (Fig. 10a) and z∕w = 0.3 (Fig. 10c), it can be seen that at z∕w = −0.3 , the hot region, T * > 0 is dominant and at z∕w = −0.3 the cold region, T * < 0 is dominant. At t∕t f = 97 , from Fig. 11, it can be seen that the cold and hot flow have reached the hot and cold boundary, respectively. At this instant the temperature field is divided to two vertical regions implying the circulation of the flow from the bottom to the top and reverse. This has also been reported recently in Hartmann et al. (2022) and Hartmann et al. (2021) in which the flow properties of RBC in slender cells were investigated by applying DNS.
Similar to t∕t f = 57 , the volumetric temperature field at t∕t f = 97 cannot be found by the temperature field at z∕w = 0 unless the data of the other two planes is known. Although the time-resolved experiment was conducted for Δt∕t f = 284 , to have insight into the heat transfer and temperature field after a long time, data were collected for other instances such as t∕t f = 2571 which is shown in Fig. 11. As can be seen in this figure, the temperature field of the flow has been divided into two zones: a hot zone, T * > 0 at the bottom and a cold zone T * < 0 at the top of the fluid domain. This temperature field is consistent with the one has been reported recently in Hartmann et al. (2022).
To investigate and quantify the temperature evolution, the area of the temperature field, A is divided into two regions of hot, A hot with threshold of T * > 0.025 , and cold, A cold with the threshold of T * < 0.025 where T * = 0 is the initial temperature of the fluid domain. Using the same threshold, the volume of the temperature field, ∀ is divided to a hot region, ∀ hot and a cold region, ∀ cold . The evolution of the hot and cold regions is examined both for 3D temperature field and the planar temperature fields at three planes discussed earlier in Fig. 10. As is shown in Fig. 12, for all the three planes, both hot and cold regions grow linearly and with the same rate after the onset of convection, t∕t f = 0 by rise and fall of the hot and cold thermal plumes. The time in which plumes meet for the very first time is a turning point in these plots. For z∕w = −0.3 , it can be seen that the hot region is significantly dominant during the whole time. For z∕w = 0.3, however, the cold region is mainly dominant. For z∕w = 0 , it can be observed that although hot region is dominant, the cold region is higher than the cold region at z∕w = 0.3 and lower than the cold region at z∕w = −0.3 . Similar moderate behaviour for the hot region at z∕w = 0 can be seen which shows the spatial temperature evolution moving from one side of the vertical boundary to the other, which is consistent with the visualization observed in Fig. 10.
The volumetric evolution of the hot and cold regions is shown in Fig. 13a. As can be seen, the first Δt∕t f = 57 represents the same linear growth of both regions as the planar evolutions at the three different planes. After this time, the evolution of the both hot and cold regions resemble the behaviour of hot and cold regions at z∕w = 0 in Fig. 12b, however, the absolute values are different. From Figs. 12 and 13a, for both planar and volumetric plots, an inverse correlation between the hot and cold variations can be observed. The absolute value of the anti-correlation coefficient, C hc for all the planes has been calculated and plotted in Fig. 13b. As can be seen, the absolute value of the coefficient, C hc for all the planes variates slightly around the mean value of ∼ 51% . This implies a moderate anti-correlation between the evolution of hot and cold region when the temperature The absolute values of the correlation coefficient, C hc of the two hot and cold regions for different planes and the volumetric measurement field is investigated in 2D. For the volumetric measurement, however, as is shown in Fig. 13b, the absolute value of C hc is equal to ∼ 84% which shows a strong anti-correlation specially comparing to the 2D measurements.
The instantaneous temperature fields in Figs. 10 and 11 showed that after the rise and fall of the hot and cold plumes, hot and cold flow develop at each side of the convection cell, z-direction. Since the instances of these two figures does not represent the physics of the whole experiment time interval, it is necessary to examine this temperature pattern for the whole time of Δt∕t f = 284 . Accordingly, temporal evolution of spatial average of the temperature field, ⟨T * ⟩ on each plane ( x − y plane) is plotted in Fig. 14. From this figure, temporal evolution of temperature along with its variation at different depths can be observed. To see more detail, temperature variations for each plane is also plotted in Fig. 14b. As shown in Fig. 14a, b, after development of the thermal plumes, temperature increases at z∕w < 0 and it decreases at z∕w > 0 . Going further in time, at t∕t f ≈ 150 temperature reach to a peak. This increase also affects the other side of the fluid domain though for z∕w > 0.2 temperature is still lower than initial temperature, ⟨T * ⟩ = 0 . After this peak, the whole field temperature decreases leading to a symmetrical temperature distribution in depth of the cell at t∕t f > 250 . It can be inferred that during the whole experiment, after the development of the primary plumes, temperature increases in one side of the fluid flow and it decreases in the other side, however, a certain boundary cannot be define due to the temperature swing.

Nusselt number distribution
The dimensionless heat transfer can be given by the Nusselt number, Nu as: where Q is the heat flux normal to the hot and cold boundaries. Here, using the temperature distribution, Nusselt number is calculated directly based on vertical temperature gradient normal to the hot, y * = 0 , and cold, y * = 1 boundaries as: where T * = (T − T 0 )∕ΔT is the dimensionless temperature based on the initial temperature, T 0 . In this equation, y * is also the vertical dimensionless length normal to the hot or cold boundaries. Using this equation on a structured grid the Nusselt number distribution on both hot and cold boundaries can be obtained as a function of time and space, Nu(x, z, t) (Adrian 2013).
For four different time instances, the instantaneous dimensionless heat transfer, Nu on the hot and cold boundaries are shown in Fig. 15. Reviewing this figure it can be seen that temporal and spatial variations, including in the z-direction, is significant. By moving in time from t∕t f = 22 to t∕t f = 42 , it can be observed that the low heat transfer region grows with the growth of the hot and cold thermal plumes each on a different side of the enclosure (left and right). For t∕t f = 57 , two regions with relatively high and low heat transfer can be seen which can be attributed to the cold and hot thermal plumes approaching to the hot and cold boundaries, respectively. From Fig. 15a at t∕t f = 97 , it can be observed that the high heat transport region in the cold flow area and low heat transport in the hot area is consistent with the previous observations of the 3D temperature field. A similar observation for the cold boundary demonstrates the consistency (6) Nu = Qh∕ ΔT between the heat transport and temperature field. For the first three time instances, it can be observed that the high and low heat transfer regions for hot and cold boundaries are apposite each other. This is due to the direct effect of the hot and cold thermal plumes affecting the cold and hot boundaries, respectively. However, at t∕t f = 97 , this opposite behaviour cannot be observed, which shows the development of the flow with a more complicated flow structure reducing the direct effect of the thermal plumes generated from the boundaries.
The temporal evolution of the spatial averaged Nusselt number, ⟨Nu⟩ x,z for both hot and cold boundaries are plotted in Fig. 16. After the onset of convection for both boundaries, heat transfer decreases and reaches a local minimum. By reaching the cold plume to the hot boundary and the hot plume to the cold boundary the heat transfer increases. Passing through time by reaching to the t∕t f = 225 as was shown in Fig. 14, the temperature increases significantly, which leads to a notable decrease in the heat transfer of the cold boundary and an increase in the heat transfer of the hot boundary.  Fig. 17, the spatial averaged Nusselt number, ⟨Nu⟩ x,z for both hot and cold boundaries is compared to the spatial averaged Nusselt number obtained from the planar temperature field at three location of z∕w = −0.3 , z∕w = 0 , and z∕w = 0 . For the beginning of the convection t∕t f < 75 all the values represent the same behaviour. After that, for both boundaries, the averaged heat transport and its variations at z∕w = 0 follow the same trend as the whole field averaged heat transfer, ⟨Nu⟩ x,z , yet there might be a considerable deviation (maximum ∼ 16% ) between them. This indicates that in case of planar measurements only at the centre of the flow field may provide information about the average heat transfer of the phenomena comparing to the other planes. It also implies that to have a coherent and true understanding of the heat transport evolutions, whole field measurement is necessary.

Conclusion
Application of 4D two-colour laser-induced fluorescence on a laterally confined Rayleigh-Bénard convection with Rayleigh number of Ra = 9.9 × 10 7 and Prandtl number of Pr = 6.1 was investigated employing a scanning laser experimental setup. Two-colour and two dye ratiometric approach was utilized to apply laser-induced fluorescence thermometry. Two temperature sensitive fluorescent dyes, Fluorescein and Kiton red with opposite temperature sensitivity of + 1.3%/°C and − 1.9%/°C were used led to enhance the overall sensitivity to 7.3%/°C. Details of the experimental setup and optical system led to this high temperature sensitivity were also discussed. Using this method the whole field temperature of the Rayleigh-Bénard flow was measured during Δt∕t f = 284 from the onset of convection. From the volumetric temperature field, it was found that both hot and cold thermal plumes, after the onset of convection, develop a conical shape. During the development of these two primary thermal plumes for 0 < t∕t f < 75 it was found that the spatial averaged Nusselt number followed the same trend for both planar and volumetric measurements regardless of the location of the measured 2D plane. However, with further development of the flow, only planar data at the centre of the enclosure follow the behaviour of the spatially averaged Nusselt number obtained from volumetric field. Nevertheless, deviation in the spatial averaged Nusselt number increases to ∼ 16% even for the central plane. This shows that to evaluate the classic and ultimate theories, volumetric measurement is required for a coherent understanding of the physics. From the visualization of the time-resolved 3D temperature field and 2D distribution of the Nusselt number, it was also found that only by volumetric measurement, temporal and spatial variations of the temperature and heat transfer can be evaluated.