Spatio-temporal dynamics of superstructures and vortices in turbulent Rayleigh–Bénard convection

Understanding turbulent thermal convection is essential for modeling many natural phenomena. This study investigates the spatiotemporal dynamics of the vortical structures in the mid-plane of turbulent Rayleigh – B (cid:1) enard convection in SF 6 via experiments. For this, a Rayleigh – B (cid:1) enard cell of aspect ratio 10 is placed inside a pressure vessel and pressurized up to 1, 1.5, and 2.5bar in order to reach Rayleigh numbers of Ra ¼ 9 : 4 (cid:2) 10 5 ; 2 : 0 (cid:2) 10 6 , and 5 : 5 (cid:2) 10 6 , respectively. For all three cases, the Prandtl number is Pr ¼ 0 : 79 and D T (cid:3) 7 K. Then, stereoscopic particle image velocimetry is conducted to measure the three velocity components in the horizontal-mid-plane for 5 : 78 (cid:2) 10 3 free fall times. For the given aspect ratio, the flow is no longer dominated by the side walls of the cell and turbulent superstructures that show a two-dimensional repetitive organization form. These superstructures show diverse shapes with faster dissipation rates as Ra increases. Out-of-plane vortices are the main feature of the flow. As Ra increases, the number of these vortices also increases, and their size shrinks. However, their total number is almost constant for each Ra through the measurement period. Furthermore, their occurrence is random and does not depend on whether the flow is upward-heated, downward-cooled, or horizontally directed. Vortex tracking was applied to measure lifetime, displacement, and traveled distance of these structures. The relation between lifetime and traveled distance is rather linear. Interestingly, in the vortex centers, the out-of-plane momentum transport is larger in comparison to the bulk flow. Therefore, these vortices will play a major role in the heat transport in such flows.


I. INTRODUCTION
Buoyancy-driven thermal convection plays a crucial role in many natural phenomena and engineering applications, 1,2 including but not limited to Earth's mantle, 3 Sun, 4 oceans, 5 atmosphere, 6 and metalproduction processes. 7We cannot ignore the fact that some of the most crucial environmental issues, such as the movement of the Earth's magnetic poles or solar magnetic storms, are among the environmental concerns for mankind's survival, and thermal convection is at the heart of them all. 8Consequently, thermal convection has been the subject of many recent studies.There is, however, a need for further research in a variety of directions, including experimental investigations of thermal convection in the air, which is the subject of the current study.
A well-known idealized model to study thermal convection is the Rayleigh-B enard convection (RBC), 9,10 where fluid is confined in a box with a certain height h and length l, which is heated from below (with a plate with constant temperature T H ) and cooled from above (with a plate with constant temperature T C ) such that T C < T H , and the side walls are adiabatic.It has been well established by numerous studies that the characteristics of RBC are dependent on Rayleigh and Prandtl numbers (Ra and Pr), and the aspect ratio of the cell C ¼ l=h.The latter plays a critical role in transforming from large-scale convection rolls into so-called superstructures.For aspect ratios around one, the theory on the turbulent flow behavior is attributed to a change from laminar into turbulent boundary layers for high enough Ra numbers. 11However, for large aspect ratio systems, the boundary layers may not play such an important role as no preferred flow direction for longer time spans exists so that boundary layers may evolve.In order to compare the studies to nature, a sufficiently large C is crucial as usually C > 10 for convection in the mantle or the sun and raises to even higher values for convention in the atmosphere or oceans. 12he Rayleigh number (Ra ¼ aDTgh 3 =j) is the main characterizing factor in RBC which relates the buoyancy forces to the friction.Here, a stands for the thermal expansion coefficient, DT ¼ T H À T C , is the kinematic viscosity, and j is the thermal diffusivity of the fluid.The onset of convection starts in Ra !1708, 13 and large-scale circulations further evolve as the Ra increases, finally reaching a fully turbulent flow.5][16][17] As a rule of thumb, superstructures should be visualized with a time averaging of around 100 Â t f .Here, t f ¼ h= ffiffiffiffiffiffiffiffiffiffiffiffiffi aDTgh p is the free fall time, representing the timescale required for fast convective motion.However, the dissipative reorganization timescale for the superstructures is suggested to be in the range of 1500 Â t f .For typical engineering problems, Ra ranges somewhere between 10 6 and 10 8 , while due to large h values, the magnitudes are even higher in nature.With increasing Ra, these superstructures exhibit more complex temporal and spatial dynamics.Thus, the investigation of RBC in high Ra is very much in demand.It is, however, challenging to simulate flow for those ranges of Ra since it takes a long time and is computationally very expensive as the effort increases with Ra.For experimental setups, the most convenient way to increase Ra is an increase in h as it scales with the third power.
However, the cell's aspect ratio (C) plays a crucial role in the characteristics of the superstructures; 18,19 meanwhile, the C values are very large for the thermal convection in nature (e.g., earth atmosphere or mantle convection with l ) h).While in the onset of convection superstructures of unique Prandtl number-independent wavelength of k c % 2 appear in the flow, 13,20 Stevens et al. 15 reported convergence of horizontally averaged statistics in C % 8 for Ra ¼ 10 9 .However, this value should be smaller for lower Rayleigh numbers, as the size of superstructures increases with the increase in the Rayleigh number. 162][23][24][25] Pandey et al. 16 simulated the RBC for 0:005 < Pr < 70 and concluded that the character of convective turbulence drastically changes from highly inertia-dominated Kolmogorov type turbulence to a finely structured convection as Prandtl number increases.This is specifically important when one shifts from the investigation of RBC in water to the air (or other gases) and consequently from Pr ¼ 7 to Pr ¼ 0.7.Therefore, RBC in the air always accompanies more complex superstructures and richer dynamics for the same Ra in general.
Although numerous numerical studies have focused on RBC recently, experiments are necessary to validate the observations in simulations and provide a benchmark for faster and cheaper investigation of the flow.In this regard, Moller et al. 26,27 have recently investigated RBC in water for a cell with C ¼ 25 in the Rayleigh number range of 2 Â 10 5 Ra 7 Â 10 5 .9][30] The results were generally in line with comparable numerical investigations and revealed the existence of fine superstructures in the flow.Weiss et al. 17 experimentally investigated the threedimensional velocity field in the entire volume of the cell for RBC flow in water using the Shake-The-Box Lagrangian particle tracking algorithm.The experiments were conducted in a cell with C ¼ 25 for Ra ¼ 1:1 Â 10 6 .The results show superstructures of a similar size to the numerical studies.Furthermore, proper orthogonal decomposition was suggested as a possible method also to reveal the superstructures in the turbulent flow.
Although experimental investigations in large aspect ratio RBC in water have added promising information to the community, when fluid is changed from water to gas, the flow dynamics drastically changes due to the change in the Prandtl number, and more complex flow structures are faced.Therefore, it is necessary to provide similar experimental measurements for the gases like air or, in the case of this study, SF 6 .2][33] Recently, initial SPIV experiments in the SCALEX over an RBC with C ¼ 10 were conducted by Valori et al. 34 However, due to the experimental challenges, the experiments were limited to 1:7 Â 10 4 Ra 5:1 Â 10 5 , and the PIV measurements only covered 2:9h Â 2:2h of the horizontal-mid-plane of the cell.As the field of view should be larger than the expected superstructures, the motivation of the current study is to provide the necessary experimental data of the RBC flow for low Prandtl number Pr ¼ 0.79 using SF 6 as the working fluid for a large field of view of A ¼ 7h Â 5:8h of the horizontal-mid-plane section of the cell and extend the Ra range about one order of magnitude.

II. EXPERIMENTAL SETUP
Experiments for Rayleigh-B enard Convection (RBC) in high Rayleigh numbers are demanded due to the higher degree of complexity in the turbulent flow and comparability with convection in nature.In experiments, however, it is difficult to reach high Rayleigh numbers (Ra ¼ aDTgh 3 =j).The temperature difference DT is limited to ensure the Boussinesq approximation 35 and adiabatic side walls in the cell.When one aims to have a large aspect ratio C RBC, h will also be limited.Any increase in h should be accompanied by an increase in the cell width L, resulting in spacing issues and minimizing the resolution in the measurements.Fluid properties, thermal expansion coefficient a, kinematic viscosity , and thermal diffusivity j are what remain to be adjusted.However, a does not vary significantly for moderate pressures, while and j scale inversely with the pressure, and, thus, an increase in pressure will result in a higher Rayleigh number.
Therefore, experiments in pressurized fluids are necessary in order to reach high Rayleigh numbers.In this regard, the Scaled Convective Airflow Laboratory Experiment (SCALEX) at Technische Universit€ at Ilmenau was designed to conduct experiments for pressurized gases. 31The pressure vessel is cylindrical and has a diameter of 1100 mm and a length of 2200 mm.In total, 35 windows are distributed over its surface in order to provide optical access for the experiments.A working pressure of up to 10 bar above ambient is allowed in the vessel, and it can use air or SF 6 as the working fluid.Compared to air, SF 6 has the advantage of providing higher Rayleigh numbers (roughly 10 times more) in the same pressure ranges while maintaining the desired low Prandtl numbers the same as the air (0:7 < Pr < 1:3).For a further detailed explanation of the SCALEX facility, please see Ref. 31.
The Rayleigh-B enard Cell (RBCell) for the current experiment has a height of h ¼ 30 mm and a length of l ¼ 300 mm, resulting in an aspect ratio of C ¼ 10.This aspect ratio is large enough to ensure that the size of turbulent superstructures is independent of the RBCell's lateral dimension. 15The nontransparent cooling plate is made from aluminum, and a recirculating cooler (Medingen C20 KB10) moderates its temperature, while the transparent heating plate 31 is made from a glass pane with a thin layer of coated Indium Tin Oxide (ITO) in its outer surface to provide Ohmic heating using an electric current that is applied by copper flat bands pushed on the ITO surface.As discussed earlier, this careful arrangement ensures that the thermal boundary conditions are close to ideal isothermal walls.The side walls are made from 4 mm thick polycarbonate, and considering the temperature difference range of DT % 7 K for the current experiments, they can be assumed adiabatic.Ideal thermal boundary conditions are always challenging to maintain in experiments.However, they can be classified by the Biot number Bi, a measure of the ratio between convective heat transfer at a body's surface and pure heat conduction inside it, where k is the respective thermal conductivity of the fluid and plate and d p and H stand for the plate thickness and RBCell height, respectively.For the current experiments for both heating and cooling plates, Bi < 0.1, and, thus, isothermal boundary conditions can be well approximated.
In order to conduct stereoscopic particle image velocimetry (SPIV), it is necessary to seed the RBCell with particles that are small enough to follow the flow faithfully without short sedimentation time but large enough to provide particle images for processing in SPIV.For the current experiment, DEHS (di(2-ethylhexyl) sebacate) was vaporized via a Laskin nozzle, generating particles of a mean diameter of 1 lm.The Stokes number (St ¼ d p q p u f =18lg) for the particles in the current experiment is calculated to be St < 0.1, and, thus, the particles are assumed to follow the flow faithfully.Then, the generated particles were settled in a settling chamber and pushed to the RBCell via a fan gently from six inlets (three on the left and three on the right of the figures for the current study). 31Due to the momentum addition during the seeding particle injection to the RBCell, all experiments are conducted at least 15 min after the particle injection.
Figure 1 shows the SCALEX facility and a schematic sketch of the SCALEX facility, RBCell, and the related optical systems to do SPIV.A laser light sheet enters the facility from the transparent window on the left and illuminates the mid-horizontal-plane of the RBCell.The light sheet is created by a double pulse laser (Quantel Q-smart Twins 850) with approximately 300 mJ pulse energy, equipped with light sheet optics to provide a 3 mm thick light sheet.Consequently, images of the illuminated particles are captured by two sCMOS cameras (LaVision Imager sCMOS) within the pressure vessel looking at the RBCell through the heating plate via a mirror, as shown in Fig. 1.The cameras have been tested under vacuum (not working) and up to 6 bar pressure (working) with no problems.The cameras have 6.5 Â 6.5 lm 2 pixel size and are equipped with Zeiss Distagon f 50 mm objective lenses.The stero angle was % 40 .The calibration was performed via a 3D calibration plate (200 Â 200 mm 2 , LaVision GmbH) positioned in the center of the RBCell.In order to further reduce the calibration errors, stero self-calibration was conducted as well. 36able I shows the conditions in which current experiments are conducted.SF 6 is used as the working fluid in all experiments to reach higher Rayleigh numbers compared to air.Three experiments were conducted in total.The DT % 7 and T H % 24:6 C in all, while pressure was p ¼ 1; 1:5; and 2.5 bar.This results in three Rayleigh numbers of Ra ¼ 9:4 Â 10 5 ; 2:0 Â 10 6 , and 5:5 Â 10 6 .The Prandtl number was Pr ¼ 0.79 for all three experiments.This is due to the fixed DT  between the plates and independence of the ratio of and j to the pressure changes in the range of the current experiments, which is a very comparable value with the Prandtl number of the air, where Pr ¼ 0.7.The measurements are all conducted at 10 Hz double frame recording for 2000 s resulting in 20 000 snapshots of the velocity field.Typically, in RBC studies, time is reported by the units of free fall time t f ¼ h= ffiffiffiffiffiffiffiffiffiffiffiffiffi aDTgh p in order to be able to compare different studies.For the current experiments, due to the similar DT and T H values, free fall time is t f ¼ 0:35 s for all three Rayleigh numbers.Consequently, all three experiments covered 5:78 Â 10 3 t f temporally.The field of view covered A ¼ 7h Â 5:8h in the center of the RBCell.
The SPIV processing was conducted using an advanced cross correlation evaluation via DaVis 10 software (LaVision GmbH) for an initial rectangular interrogation window size of 96 Â 96 pixels with 50% overlap and a final circular Gaussian window weighting of 48 Â 48 pixels with 75% overlap.The number of passes of the initial and final interrogation windows was two and three, respectively.This results in 174 Â 208 vectors in the flow field, which translates to a spatial resolution of 1 Â 1 mm 2 .The number of outlier vectors for all experiments is less than one percent in a normalized median test. 37For the highest Rayleigh number, the dynamic spatial and velocity ranges are [0-8] pixels and [0-35] mm/s, respectively.Assuming the absolute error to be 0.1 pixels in a well-adjusted experiment, 38 this results in a minimum of 1.2% in the regions with maximum displacements.

III. RESULTS
As already mentioned, SPIV experiments provide the three components of velocity in addition to out-of-plane vorticity in the horizontal-mid-plane. Figure 2 shows the instantaneous horizontal u, vertical v, and out-of-plane w velocity components for the three Rayleigh numbers in the current study.It should be noted that for w, the positive direction is from the heating plate toward the cooling plate, and, thus, the red colors represent the upward hot flow.For all three Ra, all three velocity components have magnitudes in the range of [À0.5u f , 0.5u f ], where free fall velocity u f is equal for all three Ra, as shown in Table I.This is due to the identical thermal boundary conditions for all experiments at different pressures.Compared to similar experimental studies in water in similar Ra ranges, 26 the velocity fields are far more turbulent, and there are no coherent structures visible at the first glance.However, a closer look at Fig. 2 shows that as the Ra increases, the size of the velocity clusters decreases clearly.This will be further discussed in Sec.III B when talking about the out-of-plane vortices.Figure 3 shows the probability density function (PDF) estimates for the results.As already discussed, the range of the velocity values is similar for all The fluctuations can be characterized by their respective total turbulent kinetic energy (TKE) as follows: Higher TKE values mean higher fluctuations around the average values in each grid point in experiments.which, as will be shown later, also have slower movements and stay for longer periods of time.Therefore, although the amplitudes of the velocity fluctuations are the same for all three Rayleigh numbers, the periods in which values stay in those amplitudes are more extended for lower Ra numbers, and, thus, they have higher mean TKE values.

A. Spatio-temporal dynamics
One of the main features of the dynamics of the turbulent RBC flow, in large aspect ratios, is the superstructures.Pandey et al. 16 have extensively characterized the variation in the size of these superstructures with respect to Ra and Pr numbers.Moller et al. 27 compared the superstructure sizes in numerical and experimental studies for water for Ra ¼ 2 À 7 Â 10 5 and concluded that superstructures are larger in experiments.This has to do with the effect of smaller aspect ratios compared to numerics and non-ideal isothermal boundary conditions in the experiments. 39Hence, a similar comparison is intended for the current study as well.First, an estimate of the speed at which the structures in the out-of-plane velocity field w are degrading should be obtained, so that a suitable averaging time interval can be selected.
In this regard, the average correlation decay of the instantaneous w fields for the all three Ra numbers are shown at the left of Fig. 5.This shows the average correlation decay of each time step as the w field evolves and changes its shape.Consequently, faster correlation decay means a faster rate of evolution of the w field and, therefore, of superstructures.It is evident that correlations reach their minimum after roughly 20 s ¼ 57t f , and flat lines continue thereafter.Therefore, it can be argued that after 50t f , the structures present in instantaneous w fields are replaced with new shapes for all three Ra numbers, and therefore, this period is suitable to average the w field to reveal the superstructures underneath.Now, the same average correlation decay values are shown for time averaged w fields in a period of 50t f w 50tf at the right of Fig. 5.As the Ra increases, the minimum correlation values decrease, and the time in which the correlations approach their minimum increases.This is consistent with the larger average values of w for lower Ra numbers in Fig. 24, indicating a weak but present embedded structure beneath the flow that prevents the correlations from decreasing beyond a certain value.For the initial time interval after each snapshot, despite the sudden drop in correlations for both w and w 50tf for all three Ra numbers, when zoomed in, some trends with respect to Ra values can be seen.Figure 6 is plotted to better represent these differences.At the top, correlation values after DT ¼ 0:3s ¼ 0:87t f are shown for 500 s.The correlations are always higher when the Ra is lower, even though the values fluctuate.This difference is not as prominent for the correlation values of w 50tf after DT ¼ 8:7s ¼ 25t f at the bottom of Fig. 6, but still the general trend is the same.Now that DT ¼ 50t f is chosen as a suitable time interval to average the flow to reveal the superstructures, it will be of specific interest to plot the variation of w 50tf fields in time.Figure 7 shows the respective velocity fields for a period of 3000t f in every 500t f .For all three Ra numbers, the magnitudes are limited to [À0.25u f < w 50tf < 0.25], this almost half the magnitude of w fields in Fig. 2. For Ra ¼ 5:5 Â 10 6 , the superstructures are changing the shape significantly in every 500t f , and the shapes are diverse and different.For Ra ¼ 2:0 Â 10 6 , the speed in which the superstructures restructure, and the diversity of their shapes is decreased.Finally, for Ra ¼ 9:4 Â 10 5 , the flow is almost always upward in the center while tilting to the left and right as time passes.This is in line with the average w field in Fig. 24 in the Appendix.
The Fourier spectral analysis (FFT) can be used to determine the characteristic length scale of the superstructures (k). 16,27In this method, first, a FFT is applied to the time-averaged w fields, and then, the azimuthally averaged profiles of the power spectrum are calculated to find the respective k for the data.
Table II shows the characteristic length scale of the superstructures for the current study along with the respective numerical counterparts for Pr ¼ 0.7 and Ra ¼ 10 6 and 10 7 in a study by Pandey et al. 16 For Ra ¼ 9:4 Â 10 5 and 2:0 Â 10 6 ; k=h ¼ 4:9 was calculated, which is larger than the comparable numerical counterpart for Ra ¼ 10 6 , where k=h ¼ 4.Then, as the Ra increases to Ra ¼ 5:5 Â 10 6 , the characteristic length scale reduces to k=h ¼ 3:8.For this Ra number, there are no close numerical results to compare with.However, the values for Ra ¼ 10 6 and 10 7 are both higher in comparison (k=h ¼ 4:9 and 4, respectively).In general, while the experimental wavelength k seems to deviate from numerical counterparts, however, the values are quite close, and the difference can be attributed to the fact that the experiments are conducted in SF 6 while the numerical results for air, thus As the final part of this section, the correlation map of the w 50tf fields for the entire time steps is shown in Fig. 8.In this way, one can determine how often similar structures will occur in the measurement period once a structure is formed in the flow.The high value in diagonal is basically the correlation of each w 50tf field with itself and few time steps before and after.In general, all three Ra numbers have the same thickness for these diagonal high values, but closer inspection reveals that there are some time steps at which the structure of the flow remains unchanged for longer periods.The next interesting point is that, as already shown in Fig. 5, the correlation values are, in general, larger for lower Ra numbers.However, still some zero correlation values are present even in the lowest Ra.For Ra ¼ 9:4 Â 10 5 and 2:0 Â 10 6 , these low correlations form some kind of chess-board like small square structures in the flow.This shows that there is some kind of periodicity in the occurrence of such structures in the flow.Moreover, once they occur, they never repeat during the measurement.However, for Ra ¼ 5:5 Â 10 6 , this time it is the high correlation values that form such chess-board like structures.
A closer look at Ra ¼ 2:0 Â 10 6 in the middle shows that its correlation map is a kind of mixture of its lower and higher Ra counterparts in its left and right.While in the four edges of the map, similar structures to Ra ¼ 9:4 Â 10 5 with high value squares of correlations separated by low value parallel lines are visible, in the middle low value, squares bounded by parallel high values similar to Ra ¼ 5:5 Â 10 6 are visible.This shows that as the Ra number increases, more diverse structures appear in the flow, while for the mediator Ra ¼ 2:0 Â 10 6 , the turbulent flow is in variation between acting like a low Ra where structures are quite similar in shape and high Ra where the structures are very diverse and non-repetitive.

B. Vortices
Vortices are one of the important features of turbulent convection.Out-of-plane vortices, in particular, have been associated with thermal plumes detaching from top or bottom plates, and their presence in the flow is one of the main indicators of transition from so-called soft to hard turbulent convection. 40,41Hard turbulence is characterized by intense fluctuations far away from average in the temporal and spatial domains, and energy cascades into small scales, the latter of which is explicitly connected to the appearance of vortices and their dynamics in the flow. 42Thunderstorms and dust devils are the most iconic examples of such vortices in the atmosphere. 43Many have characterized the appearance of these vortices as one of the primary differences in the transition from convection from small aspect ratio cells to large aspect ratios. 44Thus, although the analysis of the three velocity components of the flow provided valuable insight into the flow structure in the previous sections, additional discussion of the out-of-plane vortices x z is represented to further analyze the dynamics of the turbulent flow in the current study.In this regard, Fig. 9 shows the PDF distributions of x z for the three Rayleigh numbers.In general, the distributions are similar irrespective of the change in the Rayleigh number.However, as the Ra increases, the PDF distributions slightly rise in the tails.Figure 10 shows the respective x z fields for the three Rayleigh numbers.Separating actual vortices from shear has always been a challenge when considering the vortex field.For this purpose, the green values are those that are close to zero in both positive and negative directions.It is evident that as Ra increases, the vortices become smaller, but the number of them increases.That is why PDF distributions remain relatively unchanged as Ra increases.
To further explore the distribution of vortices, Fig. 11 shows the enstrophy fields.Enstrophy is simply the (x 2 z ), and it has been represented as a measure of the energy cascade in turbulent flow. 45Thus, in the enstrophy field, one can see the vortices and their power irrespective of their direction.Once again, it is apparent that the number of these vortices increases as Ra increases, but their size decreases.
This shows that the in-plane velocity components most have smaller clusters for higher Ra numbers, which is in line with what has been shown already in Fig. 2. Additionally, a closer look at the enstrophy fields reveals a non-homogeneous distribution of vortices, with certain areas in the field having a higher number of vortices while others are nearly empty.This suggests that there might be some relationship between the orientation of superstructures in the flow and the possibility of vortex occurrence in different parts of the field.
Figure 12 shows the average sum of the enstrophy for the entire field in each time step for all three Ra numbers.Clearly, average enstrophy values increase with the increase in Rayleigh number.This is consistent with the fact that enstrophy is a representation of energy cascade in turbulence, and higher Rayleigh numbers contain higher energy levels that can be dissipated, but TKE is less, so the energy seems to be more organized instead of randomly distributed on smaller scales.In addition, for all three Ra numbers, the values fluctuate around certain constant values over time.This indicates that the available energy for dissipation in the current experiments remains constant during the measurement time of 2000 s or 5780 free fall times.The mean of the average values is 9.6, 10.9, and 12.6 1=s 2 , whereas the standard deviations are 0.50, 0.53, and 0.54 for Ra ¼ 9:4 Â 10 5 ; 2:0 Â 10 6 , and 5:5 Â 10 6 , respectively.
The number of local maxima in the enstrophy field represents the approximate number of vortices in the field.Therefore, the total numbers of local maxima in the enstrophy field in each time step is shown in Fig. 13.The average number of total maxima rises from almost 600 to around 900 as Ra increases from Ra ¼ 9:4 Â 10 5 to Ra ¼ 5:5 Â 10 6 .The values are quite stable in time with minimal fluctuations; thus, one can argue that they are not random in time but represent in a mean way the turbulent state of the flow.In fact, fluctuations of the number are relatively far less compared to average enstrophy values in Fig. 12.Now that the higher number of vortices in higher Ra numbers is obvious, the period in which vortices stay in the field before dissipating, and the velocity at which they move and the reorganization of the whole field should also be investigated.Thus, in Fig. 14, the correlation decay  of the x z fields is shown.It is clear that for higher Ra numbers, the correlation values are significantly lower.Therefore, vortices are more stationary in the lower Ra numbers.This can be either because of the longer lifetime of vortices in lower Ra numbers or because of their less movement in the flow during their lifetime or due to their appearance with the superstructures that show a similar trend with respect to Ra.However, when compared with the correlation decay of the w fields, one can see that the sensitivity to Ra variation is much higher for the x z field.Thus, it can be concluded that actually, the dynamics of the vortices is even more sensitive to the Rayleigh number.So far, the out-of-plane vortices have been only analyzed statistically.Thus, in order to analyze them dynamically, the vortex centers (VC) are tracked through the measurements using methods of advanced particle tracking algorithms. 46,47In order to distinguish between shear flow and actual vortices, only those local maxima of enstrophy fields are considered as VCs that satisfy the k 2 criterion.Figure 15 shows the remaining number of VCs when the k 2 criterion is applied.It is evident that nearly half of the local maxima satisfy the k 2 criterion when applied.From now on, they going to be referred to simply as VCs.Before analyzing the results of vortex trackings, the distribution of the VCs in the field and their PDFs can add valuable insight into their general behavior.A more detailed discussion in this regard can be found in the Appendix.However, later, we see that they are related to regions of large magnitude of the out-of-plane velocities.
By vortex tracking, the aim is to characterize the lifetime, displacement, and traveled distance (the total distance a VC is traveling even if it ends at the same place) of VCs in general against each other and also with respect to variation of the Rayleigh number.Figure 16 FIG.15.The number of local maxima in the enstrophy (x 2 z ) field, which satisfies Lamdba 2 criterion with respect to time.(Multimedia view) shows an exemplary vortex tracking for Ra ¼ 9:4 Â 10 5 .The circles are centered in the vortex center of the tracked vortex in its final position before disappearance.The black path line represents the vortex's movement from its initial position in the flow to its final position.Figure 17 shows the PDFs of lifetime, displacement, and traveled distance of VCs at left, middle, and right, respectively.It is quite obvious that vortices with a long lifetime are quite rare in flow.However, higher Ra numbers have the same chance of having VCs with a long lifetime.This is quite counterintuitive, as one might expect that vortices might break down faster in higher Ra numbers due to the more chaotic nature of the flow.Also, the turnover time of the superstructures is much faster (see discussion on correlations), but w.r.t to the vortex lifetime, the turn-over time is still larger.Maybe the more persistent nature of these vortices is the main reason behind the greater complexity of the flow in the higher Ra numbers.Next is the PDF of displacements of the VCs from the first snapshot that they appear in the flow till the one that they disappear.Again, the higher Ra has the same displacements as the lower Ra numbers, just like life times.Finally, the PDF of traveled distances of the VCs is shown on the right, with the same similar distributions for all three Ra numbers.In general, it is very significant that a considerable fraction of the vortices can have displacements and traveled distances of up to 1 and 1.2 Â h in the flow.
Figure 18 shows the correlation between the lifetime of vortices and their traveled distances with respect to Ra numbers.The lines in the plots show the linear relation between these two variables.Apparently, for all Ra numbers, lifetime % 4.5 Â traveled distance.However, there are many vortices with long life times and small traveled distances and vice versa, but still, there are limits for them as well.For instance, there are no vortices with 12 t f lifetime and lower than 2 h traveled distances.So it can be concluded that in order to have long lifetime, a vortex has no choice but to travel in the flow, and stationary vortices are basically not possible.
Finally, Fig. 19 shows the lifetime of VCs vs their displacements (top) and the displacements vs traveled distances at the bottom.In both cases, the relationship between the variables is more complex compared to lifetime vs traveled distance.Apparently, there are vortices with almost zero displacement with quite a long lifetime (up to six FIG. 17. Probability density function of lifetime, displacement, and traveled distance of out-of-plane vortices at left, middle, and right, respectively.The statistics are for 100,000 vortices in each Rayleigh number.

FIG. 18.
Scatter plot of lifetime of out-of-plane vortices with respect to their traveled distance for Ra ¼ 9:4 Â 10 5 ; 2:0 Â 10 6 , and 5:5 Â 10 6 at left, middle, and right, respectively.The statistics are for 100 000 vortices in each Rayleigh number.free fall times) in the flow, and this shows that although all vortices are dancing, but some end up in their initial location in which they started to travel in the flow.However, there are also vortices with short life spans that can have large displacements in the flow.Therefore, in this regard, the behaviors of vortices are quite divergent, and this divergence is equally present in all three Ra numbers.However, as the Ra increases, more vortices with extreme lifetime and displacement appear in the flow.This is due to inherently higher number of vortices that are present in larger Ra numbers.In Fig. 19 (bottom), there are vortices with displacements equal to their traveled distances, which are, in other words, direct movement in the flow.Again, there are some vortices that end up in their initial position and, thus, have zero displacement and large traveled distances.
Last but not least, it is worth analyzing the distribution of VCs in the field with respect to their lifetime to see whether there exist specific criteria in the field for hosting such long-lasting vortices.Figure 23 shows the variation in the PDF of w with respect to the lifetime of VCs along with the PDF of w for the entire field.It is clear that parts of the flow with more extreme w values are more probable to have long-lasting VCs.For Ra ¼ 9:4 Â 10 4 , the distributions have a shoulder-like form for moderate values as well.However, for higher Ra values, these shoulders disappear, and the only difference remains in the far tails.For all three Ra values, long-lasting VCs tend to appear in the upward-heated flow slightly more than the downwardcooled flow, and this difference becomes more apparent when vortices with life times longer than 7t f (the red lines) are observed.Figures 20-22 show the tracks of vortices with life times larger than 5 t f for a time interval of 50t f in the flow for Ra ¼ 9:4 Â10 5 ; 2:0 Â 10 6 , and 5:5 Â 10 6 , respectively.In the background, the time-averaged w for the respective 50t f is shown as well.The indicating circles have diameters proportional to their jx z j levels.Each vortex track is represented with a different random color in order to distinguish between vortices.For Ra ¼ 9:4 Â 10 5 , it is clear that these long-lasting vortices have a slightly higher presence in the middle of the flow field where there is an upward-directed superstructure.Despite this fact, there are also vortex trajectories that cross from positive to negative values in the averaged w-field (see, e.g., x=h ¼ À2:8 and y=h ¼ 1:5 in Fig. 20).This finding suggests that the vortices are somewhat in between the time scales of the instantaneous fluctuation with t f and the long time scales of the turbulent superstructure with $ 50t f .However, one should be careful about coming to bold conclusions about that.It should be considered that for the edges of the flow field, it is much less probable to catch such longlasting vortices, as they might escape the field of view.For Ra ¼ 2:0 Â 10 6 , the same observations are true as well.However, for Ra ¼ 5:5 Â 10 6 , one can argue that long-lasting vortices are mean value for these long lasting vortices is about 20% higher than the value for the entire measurement field (% 0:1).This indicates that a higher share of out-of-plane momentum transport (and consequently heat transport) is related to the vortices.

IV. CONCLUSIONS
In this study, experiments were conducted to study the spatiotemporal dynamics of the turbulent flow in Rayleigh-B enard Convection (RBC).For this, a Rayleigh-B enard cell (RBCell) with an  aspect ratio of C ¼ 10 was placed inside a pressure vessel (SCALEX 31 ) to conduct experiments in pressurized SF 6 .Three experiments in three different pressures of 1, 1.5, and 2.5 bar were conducted, resulting in Rayleigh numbers of Ra ¼ 9:4 Â 10 5 ; 2:0 Â 10 6 , and 5:5 Â 10 6 , respectively.Since the Prandtl number in the current study (Pr ¼ 0.79) was close to that of the air (0.7), it can also be regarded as a representation of the RBC in the air as well.Due to the similar DT % 7 K, between the heating and cooling plates, all three experiments had identical measurement duration of 5:78 Â 10 3 t f .The horizontal-mid-plane of the RBCell was illuminated via a laser sheet, and SPIV was conducted in a large field of view of A ¼ 7h Â 5:8h to measure the three velocity components.
The results indicate that higher Ra results in more complexity of the turbulent flow field.As Ra increases, average velocity values for the measurement period decrease, and velocities appear in smaller clusters.Superstructures form more diverse shapes with faster dissipation rates in higher Ra numbers.Thus, lower Ra numbers have slower correlation decay for their out-of-plane velocity fields, and the correlation values approach higher minima.The characteristic length scales of the superstructures in the current study are different in comparison to similar numerical results but still quite close.The PDF distributions of the out-of-plane vortices have slightly higher probabilities for extreme values in higher Ra numbers.
The results indicate that the out-of-plane vortices are one of the main features of the RBC in SF 6 and consequently in the air.The main difference in the vortex fields is the number of vortex cores and their movement.Higher Ra numbers have more vortices with smaller sizes that move faster compared to lower Ra values.The total number of these vortices in the field is constant and stable through measurement for each Ra number.However, the appearance of the vortices in the fields is random, and there is no preference to occur in upward-heated, downward-cooled, or horizontally directed flows.Therefore, the PDF distributions of the velocity components in the vortex centers are similar to the rest of the flow.However, it was shown that long lasting vortices are slightly more likely to occur in upward-heated flow in general.The results of the vortex tracking show that lifetime, displacement, and traveled distance of the vortices have similar probability distributions for all three Ra numbers.The mean lifetimes of the vortices for all three Ra values are around one free fall time, and, thus, they are smaller than the typical time scales of the turbulent superstructures.There is a quite linear relation between the lifetime and the traveled distance of the vortices, and there are no stationary vortices in the flow.The relations between the lifetime vs displacement and displacement vs traveled distance are more complex with very diverse possibilities.For instance, there are vortices that end up in their initial position in the flow after a long travel and then dissipate, or they are vortices with direct path (a traveled distance equal to displacement) in the flow, which is quite odd given the extremely turbulent flow present in the field.Finally, there is a slightly higher probability for long-lasting vortices to appear in parts of the flow with more extreme out-of-plane velocity values.
The results of the current study open many avenues to proceed in the investigation of RBC, particularly in gases.Of course, the most obvious one is to expand the range of Rayleigh and Prandtl numbers.The latter is very important, as it is the reason why vortices appear in the flow when fluid changes from water to SF 6 or air.These out-ofplane vortices need to be further investigated in detail.A threedimensional measurement can be very insightful in this regard. 48As another suggestion, porosity could also be added to the cooling and heating plates so as to see its effect in the presence of out-of-plane vortices; a much denser oil over the surface of the heating plate can have a similar effect and make a more realistic simulation of convection over, for example, oceans where the surface beneath is not solid.The data which is gathered in the current study can be used for analysis of vortex interaction in the flow independent of the topic of convection.Moreover, the data are a very suitable complex example of turbulence, which can be used for machine learning to see how far AI algorithms can proceed in understanding, modeling, and predicting of such complex dynamics.This is actually the intention of the authors in a broader context of their investigation of Machine learning application in turbulence. 49,50G. 26.Probability density function comparison between w and w vc (the out-of-plane velocities only in the vortex centers) for Ra ¼ 9:4 Â 10 5 ; 2:0 Â 10 6 , and 5:5 Â 10 6 at left, middle, and right, respectively.
Figure 24 shows the time-averaged velocity field for the entire measurement time of 20000 snapshots or 5:78 Â 10 3 t f as mentioned in Table I.For Ra ¼ 5:5 Â 10 6 , there are no significant structures in neither of the three velocity components.However, for Ra ¼ 9:4 Â 10 5 and 2:0 Â 10 6 , time averaged w values range between [À0:1u f < w < 0:1u f ].Although the values are much less than the instantaneous fields, however, they are higher compared to the Ra ¼ 5:5 Â 10 6 case.This indicates that for the lower Rayleigh numbers, either the measurement time is not long enough to completely wash away the presence of any preferred direction in the w fields, or the aspect ratio of the current experiments C ¼ 10 results in preferred orientations of the superstructures in the cell, regardless of measurement duration.However, this should not be mistaken with slow dynamics of the turbulent flow for these Rayleigh numbers, as it was shown that the flow dynamics is rich and comparable for all three Ra.

Randomness of appearance of vortices
Figure 25 shows the sum of the VCs that every grid point experiences (RN) in the entire measurements divided by the average N for the entire grid points in each Ra.The RN=Avg varies around 0.7 and 1.3 approximately.However, the distributions are quite random.Figure 26 compares the PDF distributions of the three velocity components of the VCs and the entire flow.Evidently, there is no particular difference between the VCs and the rest of the flow.Thus, there is no preference for VCs to occur in upward-heated, downward-cooled, or horizontally directed flows, which is very important because it shows that VC can occur in any of these situations.This is in line with what was observed in Fig. 25 regarding the randomness of VCs occurrence.

FIG. 1 .
FIG.1.SCALEX facility (left), and a schematic sketch of the SCALEX facility and its convection cell, shown from the front (right).31

FIG. 22 .FIG. 23 .FIG. 24 .FIG. 25 .
FIG.22.Tracks of vortices with at least 5 t f lifetime (black lines) for 50 t f of the measurement along with the respective timeaveraged (50t f ) out-of-plane velocity (w 50tf ) for Ra ¼ 5:5 Â 10 6 .The circles that indicate the VC positions have diameters proportional to their jx z j levels.

TABLE I .
Parameters for the current experiments in the Rayleigh-B enard cell.