2.1. Convection cell

The experimental set-up used here is quasi-identical to that used in the earlier work (Schindler et al. Reference Schindler, Eckert, Zürner, Schumacher and Vogt2022). It differs only in that additional excitation coils and magnetic field sensors for performing the CIFT measurements are now installed around the convection cell (see § 2.3). The sidewall of the cylindrical convection cell is made of a fibre glass compound (Keluglas^TM) and has an inner diameter of 320 mm and a height of 640 mm. At the bottom and lid, the cell is bounded by two solid copper plates, which are crossed by spiral cooling channels through which temperature-controlled water flows to achieve constant temperature boundary conditions. The effective thickness of the plates, $d_{P_{Cu}}$, corresponds to the distance between cooling channels and the interface to the fluid. To protect the copper plates from intermetallic reactions and to electrically insulate the liquid metal from the walls, the copper plates were coated with a thin layer of lacquer (thickness $d_{P_l}$). The cylinder is filled with the ternary alloy GaInSn: a liquid metal with an eutectic temperature of 10.5$\,^{\circ}$C. The Prandtl number is $Pr = 0.025$ at room temperature and increases to $Pr = 0.031$ in the measurements with the highest Rayleigh number of ${Ra} = 6.02 \times 10^8$ in which the mean temperature of the fluid heats up to 29$\,^{\circ }$C.

In any experimental investigation of liquid metal convection, the problem of the finite thermal conductivity of the top and bottom plates has to be taken into account. While the thermal conductivity of copper differs from that of water or compressed gases by three or four orders of magnitude, in our case the ratio of the corresponding values for the wall material, $\lambda _P$, and the fluid, $\lambda$ is only around 10. This obviously affects the Biot number $Bi = Nu ({d_{P_{Cu}}}/{\lambda _{P_{Cu}}}+{d_{P_l}}/{\lambda _{P_l}})({\lambda }/{H})$, whose value should be as small as possible to avoid temperature gradients along the plate surface influencing the plume dynamics and the heat transfer (Brown et al. Reference Brown, Nikolaenko, Funfschilling and Ahlers2005b; Ahlers et al. Reference Ahlers, Grossmann and Lohse2009b). In our experiments the values for $Bi$ span a range from 0.19 ($Ra = 9.33 \times 10^6$) to 0.52 ($Ra = 6.02 \times 10^8$). This is clearly less than one but still large enough that a potential influence of non-ideal thermal boundary conditions cannot be completely ruled out. It is difficult to specify the real boundary conditions in an exact way, all the more as the thickness of the lacquer layer might not be perfectly homogeneous. At any rate, one is most likely dealing with a mixture of constant temperature and constant heat flux boundary conditions (Foroozani, Krasnov & Schumacher Reference Foroozani, Krasnov and Schumacher2021). This represents an intrinsic difference compared with experiments in high $Pr$ fluids, where, due to the significantly lower thermal conductivity, it is much easier to achieve small values for $Bi$, and thus to get closer to the requirement for constant temperature boundary conditions on the copper plates.

2.2. Temperature measurements

The convection cell is equipped with 68 temperature measuring points in total. We use type K (NiCr-Ni) thermocouples that are calibrated to a relative accuracy of better than 0.06 K. Twenty thermocouples are installed in the copper plates to determine the plate temperatures, $T_{hot}$ and $T_{cold}$. Eight sensors in each plate are located in the copper in a distance of 4.5 mm to the liquid metal. Additionally, four sensors being in direct contact to the liquid metal are glued in the top plate to measure the temperature gradient in the copper. Following the concept of the multi-thermal-probe approach to identify the LSC structure in the convection cell (Funfschilling & Ahlers Reference Funfschilling and Ahlers2004; Brown, Nikolaenko & Ahlers Reference Brown, Nikolaenko and Ahlers2005a; Brown & Ahlers Reference Brown and Ahlers2006; Xi & Xia Reference Xi and Xia2008; Weiss & Ahlers Reference Weiss and Ahlers2011), 16 thermocouples are arranged in form of three rings at the inner surface of the cylinder sidewall at each of three different heights ($H/4$, $H/2$ and $3H/4$). A sketch of the sensor positions in the convection cell is shown in figure 1(a). Forty-eight holes were drilled into the sidewall at the azimuthal positions $\phi _i = 2{\rm \pi} i/16$ and the thermocouples were passed through so that their tips were exactly flush with the inner wall of the cylinder. This guarantees a perfect thermal contact with the liquid metal. The holes are sealed with epoxy to prevent the liquid metal from leaking out of the cell. All thermocouples are connected to thermal reference junctions which are kept stable at a constant temperature of 0 $^{\circ }$C. The data of all temperature measuring points are read in series with a total sampling interval of 3 s. In some measurements individual thermocouples temporarily failed. In this case, the missing value was determined by linear interpolation between the neighbouring sensors during post-processing.

Figure 1.

Experimental set-up: (a) sketch of the cylindrical convection cell with copper plates at the lid and the bottom and the positions of the thermocouples at the sidewall. (b) Photograph of the convection cell equipped with instrumentation: thermocouples, CIFT excitation coils and the magnetic field sensors (Fluxgate probes).

Following Funfschilling & Ahlers (Reference Funfschilling and Ahlers2004), Brown et al. (Reference Brown, Nikolaenko and Ahlers2005a), Brown & Ahlers (Reference Brown and Ahlers2006) and Xi & Xia (Reference Xi and Xia2008), orientation and strength of the LSC have been derived from fitting the function

(2.1)\begin{equation} T_k(\theta) = T_{m,k} + \delta_k \cos(\theta - \theta_{k}) \end{equation}

to the temperature data recorded by each of the 16 thermocouples in the three sensor rings at the heights $z = H/4$ (bottom), $H/2$ (middle) and $3H/4$ (top). Accordingly, the index $k = bot, mid, top$ distinguishes between the three vertical ring positions. The mean temperature at the respective height is denoted by $T_{m,k}$, while $\delta _k$ and $\theta _k$ stand for the temperature amplitude and the azimuthal orientation, respectively.

2.3. Contactless inductive flow tomography

Contactless inductive flow tomography is a technique for determining global flow structures in electrically conductive fluids based on the principles of magnetohydrodynamics. It has been developed at the Helmholtz-Zentrum Dresden-Rossendorf over the last 20 years and has now reached a level that allows robust measurements under challenging conditions (Stefani et al. Reference Stefani, Gundrum and Gerbeth2004; Wondrak et al. Reference Wondrak, Galindo, Gerbeth, Gundrum, Stefani and Timmel2010; Ratajczak, Wondrak & Stefani Reference Ratajczak, Wondrak and Stefani2016; Wondrak et al. Reference Wondrak, Pal, Stefani, Galindo and Eckert2018; Glavinić et al. Reference Glavinić2022b). Some explanations regarding the mathematical basis for the implementation of CIFT are given in Appendix A.

Figure 2 shows the actual set-up of the CIFT apparatus at the Rayleigh–Bénard experiment. As shown in figure 2(a) a stack of four water-cooled copper coils is constructed, which is integrated into the experimental set-up to surround the convection cell and to generate a nearly homogeneous magnetic field in vertical direction for the actual measurement. All coils are water cooled by a LAUDA thermostat that limits the change in temperature to a value of less than 0.01 K, avoiding any thermal expansion of the coil system during the measurement. The stack of four coils is connected to a Kepco current amplifier (BOP 20-50GL 1 kW-GL) which generates a constant current of 10 A producing a vertical magnetic field of approximately 1 mT. Such a weak magnetic field does not disturb the flow field and is in the measurement range of the used magnetic field sensors. The decisive indicator is the ratio between the electromagnetic and the inertial forces reflected by magnetic interaction parameter $N = \sigma L B_0^2 / (\rho U)$ ($\sigma$ is the electrical conductivity of the fluid, $B_0$ the amplitude of the excitation magnetic field and $\rho$ the density of the fluid). In our experiments presented here, approximate values of $N$ are significantly smaller than 1, in particular $N \thickapprox 0.014\ ({Ra} = 9.33 \times 10^6)$ and $N \thickapprox 0.003\ ({Ra} = 6.02 \times 10^8)$. The volume-averaged velocities measured by CIFT were used for this estimation (see § 6.1). Although the interaction between the applied magnetic field and the convection flow generates electric eddy currents in the fluid and causes measurable changes in the applied magnetic field $\boldsymbol {B_0}$, the resulting electromagnetic forces on the flow are negligible.

Figure 2.

Arrangement of the CIFT measuring system: (a) excitation coil system, (b) sketch of the sensor configuration (6 line arrays of 7 sensors each).

Measurements of the radial component of the flow-induced magnetic field are carried out by 42 Fluxgate sensors distributed evenly over the height (7 levels/100 mm spacing) and along the azimuth (6 angles/60$^\circ$ spacing) with a radial position of 220 mm, see also figure 2(b). The sensors were directly mounted on aluminium bars, in order to avoid positional changes of the magnetic field sensors in respect to the coil system as a consequence of the thermal expansion. In addition, the Earth's magnetic field was also measured in a distance of the experiment in order to compensate for its influence on the measurement result in the case of eventual fluctuations. The Fluxgate probes saturate above 1.2 mT and have a dynamic range of six orders of magnitude, which is indeed required for the reliable measurement of the flow-induced magnetic field. For example, the highest flow-induced magnetic field for the presented measurements are of the order of 100 nT for an applied magnetic field of 1 mT (Wondrak et al. Reference Wondrak, Galindo, Stefani, Schindler, Vogt and Eckert2020; Mitra et al. Reference Mitra, Sieger, Galindo, Stefani, Eckert and Wondrak2022). In order to resolve the interesting dynamics of the flow, a resolution finer than 10 nT is required. In comparison, the Earth's magnetic field is approximately 50 000 nT strong. It is obvious that the smallest of magnetic disturbances, which are caused e.g. by switching on electrical devices, the movement of ferromagnetic parts in the vicinity of the experiment or solar activity, may interfere with the measurement signal and must be filtered out. For this reason, all measurements were performed during the night to avoid disturbances due to other activities in the laboratory. In a series of long-term stability measurements, the shape of the applied magnetic field, the effects of the thermal drift as well as the compensation of fluctuations of the Earth's magnetic field were investigated (Sieger et al. Reference Sieger, Mitra, Schindler, Vogt, Stefani, Eckert and Wondrak2022). It was shown that reliable magnetic field measurements longer than 12 hours with a noise level of $\pm 5$ nT are feasible. For the flow reconstruction, the cylindrical vessel is discretized using 5760 linear hexahedral cells with a mean edge length of 25 mm in vertical and 20 mm in horizontal direction. Therefore, viscous boundary layers are not reconstructed.

6.1. Reynolds number

The Reynolds number $Re ({Ra}, Pr)$ quantifies the system response with respect to the turbulent momentum transport. Therefore, the primary interest of many investigations so far has been the quantification of the corresponding scaling laws $Re \thicksim {Ra}^{\gamma }$. Data for the turbulent liquid-metal convection in cylinders with $\varGamma = 0.5$ and 1 can be found in Schindler et al. (Reference Schindler, Eckert, Zürner, Schumacher and Vogt2022) and Zürner et al. (Reference Zürner, Schindler, Vogt, Eckert and Schumacher2019), respectively. In the following considerations, we focus on the temporal variations of $Re$ during the experiments and investigate whether there is a relationship between the instantaneous values of $Re$ and the flow structures occurring simultaneously.

The governing flow structure in turbulent convection is a coherent large-scale flow (LSC), which is apparently driven by the detaching plumes. It contains the major part of the kinetic energy of the flow and stands out from the background of turbulent motion. Since a highly turbulent and more complex flow structure develops in a convection cell with aspect ratio 0.5 at high $Ra$, the question arises whether the flow dynamics can still be quantitatively described by a uniquely defined Reynolds number. In addition, the experimental determination of a characteristic Reynolds number for the entire fluid volume, $Re_{vol}$ involves specific difficulties (especially in liquid metals), since by means of the measuring methods applied so far the flow velocity is determined only indirectly or the measuring domain is usually limited to single positions or specific subregions of the convection cell. Different characteristic velocities can be used for the calculation of $Re$. A first reasonable approach is to base the definition of $Re$ on quantities that can be directly derived from the LSC, which is assumed to be a large convection roll characterized by a typical turnover time $t_{to}$. Using the path length of the circulation as relevant length scale $L$, a characteristic Reynolds number for the LSC can be defined:

(6.1)\begin{equation} Re_{LSC} = \frac{L^2}{\nu t_{to}}. \end{equation}

Earlier studies (see for example, Heslot, Castaign & Libchaber Reference Heslot, Castaign and Libchaber1987; Takeshita et al. Reference Takeshita, Segawa, Glazier and Sano1996; Cioni et al. Reference Cioni, Ciliberto and Sommeria1997) estimated the mean velocity of the global circulation, $U_{LSC} = L / t_{to}$, from the dominating frequency in the temperature signals under the assumption that the corresponding oscillations arise from perturbations transported by the LSC. Later investigations could assign azimuthal torsional and sloshing modes to this frequency being characteristic for turbulent convection in cylindrical geometries (Funfschilling & Ahlers Reference Funfschilling and Ahlers2004; Funfschilling et al. Reference Funfschilling, Brown and Ahlers2008; Brown & Ahlers Reference Brown and Ahlers2009). Unfortunately, this method is not applicable for our experimental data because in all measurements no dominant oscillation frequencies could be found in either the temperature or velocity signals. This might be a first hint that we are not dealing here with an LSC that has an intact and coherent structure across the entire cell height.

Another approach to determine a characteristic LSC velocity is based on the correlation of temperature signals recorded at various measurement positions (Castaing et al. Reference Castaing, Gunaratne, Heslot, Kadanoff, Libchaber, Thomae, Wu, Zaleski and Zanetti1989; Sano et al. Reference Sano, Wu and Libchaber1989; Takeshita et al. Reference Takeshita, Segawa, Glazier and Sano1996; Chavanne et al. Reference Chavanne, Chilla, Castaing, Hebral, Chabaud and Chaussy1997; Niemela et al. Reference Niemela, Skrbek, Sreenivasan and Donnely2001). Temporary deviations from the temperature occurring successively at two measurement locations are caused by plumes moving across the measurement points. For example, Ahlers, Brown & Nikolaenko (Reference Ahlers, Brown and Nikolaenko2006) extracted the half-turnover time (and finally $Re_{LSC}$) from the peaks of the cross-correlation functions between temperature signals recorded at opposite azimuthal locations. However, such an approach requires a coherent LSC with predictable topology and streamlines. The recent results by Schindler et al. (Reference Schindler, Eckert, Zürner, Schumacher and Vogt2022) already gave a clear indication that such flow conditions are unlikely to exist in the parameter range studied here. In our convection cell, 16 thermocouples are installed on each of three levels on the sidewall. We have randomly calculated the correlation functions of thermocouples located one above the other in the three planes and of azimuthally opposite thermocouples in the centre plane. It turned out that the correlations between the lower and upper levels as well as the azimuthally opposite positions usually do not result in easily evaluable Gaussian peaks (not shown here). Such Gaussian peaks, from which velocities can be derived, appear in the correlation functions between the centre plane and the upper or lower plane. However, these velocity values differ to a considerable extent. This apparent failure of the method in the case of turbulent liquid-metal convection at aspect ratio $\varGamma = 0.5$ can perhaps be attributed to two causes: (i) the structure of the LSC is disturbed in a way that stable correlation relationships are not established over long distances, (ii) the large thermal diffusivity of the liquid metal leads to a rapid decay of temperature gradients. The results of the flow analysis presented below, will demonstrate that the LSC is subject to extreme variations within a few free-fall times. In addition, the structures are highly twisted, with angle and direction also changing over time. Against this background, it is quite understandable that no stable correlations can be obtained from the experimental temperature data. These difficulties emphasize the importance of direct velocity measurements in liquid metals, which is in the focus of this study.

Zürner et al. (Reference Zürner, Schindler, Vogt, Eckert and Schumacher2019) employed an arrangement of multiple ultrasonic velocity probes and compared three different $Re$ definitions based on the measurements of the typical horizontal velocity near the plates, the vertical velocity along the sidewall and the turbulent velocity fluctuations in the centre of the convection cell. Their results demonstrate that, as expected, the different $Re$ vary with respect to their absolute values, but the scaling exponent of $Re({Ra})$ is relatively insensitive to the different characteristic velocities. Moreover, the scaling exponent matches very well with DNS results presented by Scheel & Schumacher (Reference Scheel and Schumacher2017).

The CIFT technique provides access to the entire velocity field throughout the convection cell. On this basis we calculate the Reynolds number for the entire volume of the convection cell, $Re_{vol}$, which takes into account all three velocity components, and compare it with local values of $Re_k$ obtained from the horizontal velocities in three horizontal planes ($k = bot$, mid, top) at the mid-height of the convection cell or near the two plates. Corresponding results are shown in the figures 25 and 26 in Appendix B.

Figure 11 shows time series of $Re_{vol}$ recorded for three data sets of various Rayleigh numbers. The data in figure 11 cover a period of 30 000 seconds each, with the full temperature gradient applied approximately one hour after the start of the recordings (3600 s) and switched off 7 hours later (25 200 s). Approximately one hour after applying the temperature gradient, the experiment reaches a quasi-steady state. This concerns mainly the thermal conditions, while $Re_{vol}$ shows persistent strong and irregular fluctuations within the whole measurement period, which seem to increase with increasing $Ra$. Further, in figure 12, time sections from the same data sets are presented, with instantaneous values of $Re_{vol}$ normalized to the respective time-averaged mean values and the time axis normalized to the free-fall time $t_{f\!f}$ and turnover time $t_{to}$, respectively. The definition of the free-fall time is $t_{f\!f} = H^2 / \sqrt {\nu \kappa {Ra}}$. For the calculation of the turnover time, a single-roll LSC with elliptical streamlines was assumed to fill the entire cylinder. As a realistic measure of the mean velocity of the LSC, we chose the time-averaged horizontal component of the velocity field near the plates and extracted it accordingly from the CIFT data. Figure 12 demonstrates that significant changes of the flow state obviously occur occasionally within a few free-fall time units or less than one turnover of a single-roll LSC. In light of the volatility of the flow in the $\varGamma = 0.5$ cell, it is questionable to what extent the choice of turnover time as characteristic time scale is reasonable here. From the analysis of the energy contributions of the main POD modes in the previous section, the existence of a single-roll LSC can be deduced for all three $Ra$ with a tendency of the SRS to strengthen with increasing $Ra$. However, the use of $t_{to}$ only makes sense if the stability of the SRS is guaranteed at least during this time scale, about which the rapid changes of $Re$ in figure 12(b) raise some doubts. The reconstructions of the 3-D flow structures that we present in § 6.4 will confirm the rapid changes in the flow structure and also show that the shape and size of the SRS is also subject to significant changes even during its lifetime. The dramatic changes in LSC velocity and path length, $L$, imply a high degree of uncertainty in the determination of $t_{to}$. Therefore, in the further course of the paper the free-fall time $t_{f\!f}$ is used as the characteristic time scale.

Figure 11.

Time series of $Re$ recorded for three different $Ra$.

Figure 12.

Time series of $Re_{vol}$ normalized with the time average $\langle Re_{vol} \rangle$, where the time axis is normalized with (a) the free-fall time scale, and (b) the turnover time scale.

Two striking observations from figure 12 are to be noted for all three $Ra$: (i) with respect to the time average, the temporal fluctuations are of the same order of magnitude regardless of the Rayleigh number, reaching quite impressive values of $\pm 50\,\%$; (ii) these fluctuations often occur on very short time scales (less than 10 free-fall times or one turnover time).

Thus, our measurements at small $Pr$ confirm previous observations that the change from a $\varGamma = 1$ cell to smaller aspect ratios destabilizes the SRS–LSC, deteriorates the coherence of the large-scale structures and leads to a much more dynamic behaviour of the flow that apparently results from frequent transitions between different roll states (Xi & Xia Reference Xi and Xia2007, Reference Xi and Xia2008; Weiss & Ahlers Reference Weiss and Ahlers2011; Xi et al. Reference Xi, Zhang, Hao and Xia2016; Zwirner et al. Reference Zwirner, Tilgner and Shishkina2020; Schindler et al. Reference Schindler, Eckert, Zürner, Schumacher and Vogt2022). Numerical simulations by Zwirner et al. (Reference Zwirner, Tilgner and Shishkina2020) in a $\varGamma = 0.2$ cylinder for ${Ra} = 5 \times 10^6$ and $Pr = 0.1$ show significant fluctuations of ${Nu}$ and $Re$ occurring within a few free-fall times. While the changes in the Nusselt number reach values of approximately 30 %, the corresponding $Re$ varies less. We assume that the stronger fluctuations of $Re$ by more than 50 % of the mean that we observe are due to the larger $Ra$ and the smaller $Pr$ in our experiment.

6.2. Characterization of the LSC structure from the temperature measurements at the sidewall

A proved method for an assessment of the LSC structure and its dynamics is the identification of the positions of the warm upflows and the cold downflows along the sidewall on the basis of temperature measurements in the sidewall (Xi & Xia Reference Xi and Xia2008; Brown & Ahlers Reference Brown and Ahlers2009; Weiss & Ahlers Reference Weiss and Ahlers2011). If one can assume that the volume of the convection cell is pervaded by large-scale coherent structures moving primarily along the sidewall, conclusions can be drawn with respect to the underlying flow structure. We also followed this approach and performed temperature measurements within three ring-shaped arrays of 16 thermocouples each on the sidewall. For further considerations, including the results and corresponding analysis to be presented in the following sections, we arbitrarily select two time intervals of 200 free-fall times each for the Rayleigh numbers ${Ra} = 5.31\times 10^7$ and ${Ra} = 6.02\times 10^8$ corresponding to physical time periods of approximately 5800 s and 1750 s, respectively. Figure 13 displays the corresponding space–time plots of the azimuthal temperature distributions at the three rings at $z = H/4$, $H/2$ and $3H/4$ representing the time intervals chosen for ${Ra} = 5.31 \times 10^7$ and ${Ra} = 6.02 \times 10^8$, respectively. These space–time structures result from an interpolation using the measured data of the 16 sensors at each of the three different heights. The red areas indicate an elevation from the mean temperature at that height, which can be associated with plumes rising from the heated bottom, while the blue areas are related to regions of cold fluid sinking downward. As suggested in previous studies, the azimuthal temperature profiles are fitted to a cosine function (see § 2.2) and the line connecting the minimum and maximum temperature values over time is interpreted as the orientation of the LSC plane at the given height. It is noticeable that, near the lid of the cell ($z = 3H/4$), the blue areas look more sharply structured than the red areas (see figure 13a,b). Above the bottom plate ($z = H/4$), the situation is the opposite. Here, the blue areas appear more blurred (see figure 13e, f). This is caused by a widening of the plumes on their way between the plates, where the rising plumes suffer a loss of heat due to thermal diffusion. As a result, both the buoyancy and velocity decrease. This also applies, with the opposite sign, to the descending plumes. As we will see in the next section, this effect, which is particularly pronounced in liquid metals due to their small $Pr$, can also be clearly verified in the velocity reconstructions.

Figure 13.

Time series of the azimuthal temperature distributions measured along the sidewall at ${Ra} = 5.31 \times 10^7$ (a,c,e) and ${Ra} = 6.02 \times 10^8$ (b,d, f) and different heights at $z = 3/4 H$ (a,b), $z = H/2$ (c,d) and $z = 1/4 H$ (e, f).

Looking first at the situation for ${Ra} = 5.31 \times 10^7$ (figure 13a,c,e), we note for the period of approximately the first 80 free-fall times that the ascending plumes are encountered predominantly in an azimuthal range between 100$^{\circ }$ and 300$^{\circ }$. In line with this, values around 200$^{\circ }$ are obtained for the LSC orientation in figure 17(a). According to a first rough estimation, the zones of up-flowing and down-flowing fluid appear clearly separated from each other and similar values for alignment of the LSC plane appear for all three heights, which justifies the assumption that, in a first approximation, we might be dealing with a SRS. However, a closer look reveals that, even under this flow condition, there are always smaller time windows in which the azimuthal temperature distribution has multiple extreme values representing higher Fourier modes. An example becomes manifest in figures 13(a,c) and 14(a) for the planes at $z = 3/4H$ and $H/2$ at $t/t_{f\!f} \approx 436$. Moreover, it is evident that the orientation of the LSC is subject to significant temporal fluctuations. The differences between minimum and maximum values are in the range of approximately 120$^{\circ }$ up to 180$^{\circ }$, i.e. the variations in orientation almost reach the magnitude of a reversal of flow direction. In addition, strong differences in the orientation at the different heights (up to 160$^{\circ }$) are observed at the same time. The temporal rates of change in both directions can be estimated from the temperature curves. Notable values up to $40\cdots 50^{\circ } / t_{f\!f}$ are observed which corresponds to approximately $1.5^{\circ }$/s. These estimates are confirmed by looking at the time derivatives of the orientation of the LSC plane in figure 17 (not shown here). In addition, there are significant discrepancies in the orientation in the upper and lower part of the convection cell and in the dynamics of the re-orientation of the LSC plane at the three different heights. At the time $t/t_{f\!f} \approx 476$, the situation changes dramatically. The entire flow structure starts to rotate in the same direction, but after a few free-fall times the direction of motion is opposite in the upper and lower parts of the cylinder. The LSC plane in the upper half completes approximately one full clockwise rotation, while a counterclockwise rotation can be observed in the lower half. In contrast, the temperature signals on the centre plane do not show a clear rotation, but higher Fourier modes are observed at that time (see figure 14b). While the flow structure in the lower part of the cell carries on to rotate almost continuously until the end of the observation window, the initial positions of the up- and downstreaming plumes seem to be almost re-established on the upper and middle measurement planes in time in the range $510 \lesssim t/t_{f\!f} \lesssim 540$, before there are another abrupt changes in the LSC orientation. Further details will be discussed later in connection with figure 17.

Figure 14.

Azimuthal temperature distributions measured at ${Ra} = 5.31 \times 10^7$ at three heights of $z = 1/4 H$ (bot), $z = H/2$ (mid) and $z = 3/4 H$ (top) for (a) $t/t_{f\!f} = 436$, and (b) $t/t_{f\!f} = 476$.

Let us now consider the temperature field within the time interval at ${Ra} = 6.02 \times 10^8$ (see figure 13b,d, f). As in the case of the smaller $Ra$, we observe similarly strong fluctuations on short time scales of a few free-fall times. These fluctuations are superimposed here by a slow but continuous rotation, with the LSC plane performing a complete 360$^{\circ }$ rotation over a period of approximately 110 free-fall times (equivalent to approximately 960 s) corresponding to a mean rotation speed of approximately 0.37$^{\circ }$/s. As in the case with ${Ra} = 5.31\times 10^7$, the upper and lower parts of the flow structure do not always move in the same direction. This is clearly visible in figure 13(b,d, f) for the period $1710 \lesssim t/t_{f\!f} \lesssim 1745$. While the lower part of the fluid rotates almost uniformly clockwise, the upper part rotates one revolution in the opposite direction in this time domain. In a subsequent longer time interval $1745 \lesssim t/t_{f\!f} \lesssim 1815$, the entire structure rotates clockwise. After that ($1820 \lesssim t/t_{f\!f} \lesssim 1860$), there occur obviously some perturbations of the flow structure in the lower and middle parts of the convection cell. The LSC orientation reveals some jumps before it then seems to continue the rotation in clockwise direction. At approximately the same time, however, the upper part of the flow structure starts moving in the opposite direction. Although the temperature measurements show strong fluctuations, the temperature plots may indicate the existence of a SRS on average within this time period, as clearly demarcated areas for upflows and downflows can be distinguished. However, the assumption, that we are still dealing predominantly with a single-roll LSC here, would indicate an extreme twisting of the LSC plane. We see strong dynamics illustrated by deviations in LSC orientation at the same times at the different heights reaching maximum values of up to 120$^{\circ }$ (see also next section for further discussion on the LSC orientation). Moreover, these fluctuations appear rather erratic and barely correlated with each other. In addition, we observe the appearance of higher Fourier modes in the azimuthal temperature profiles at many points in time. Thus, it can be hardly excluded that the SRS is undergoing disruptive changes, transitions between different roll states or more complex restructurings. In light of the high dynamics of the structural changes, it is not easy to distinguish between a distinct roll state (SRS or DRS) and less well-defined transition states.

Figure 15 contains examples for azimuthal temperature profiles at the three different heights along the inner sidewall. Here, we have chosen a comparison of two moments ($t/t_{f\!f} = 1716$ and 1724) in which the global Reynolds number $Re_{vol}$ passes through a minimum and a maximum, respectively. In figure 16 we contrast these temperature data with corresponding reconstructions of the flow field in the $y$–$z$ plane made by means of CIFT. At the Reynolds number minimum (figure 16a), a flow structure of four rolls with very low flow intensities is revealed. In the lower part, two rolls are stacked one on top of the other, taking up approximately two thirds of the volume of the convection cell. In the upper third part we recognize two rolls that are slightly offset next to each other. At the moment of the $Re$ maximum (figure 16b), a SRS with smaller corner vortices is observed at the bottom left and at the top right. However, this is not reflected in the same way in the azimuthal temperature profiles in figure 15. Although the flow intensities differ greatly from each other, the differences between the temperature minimum and maximum, $\delta _k$ are of the same order of magnitude in all cases. Surprisingly, the temperature distributions in figure 15(a) could still be used to roughly infer the existence of a SRS, although the flow reconstruction delivers a completely different result. Of course, this comparison is not conclusive, since we only consider the velocity distribution in the $y$–$z$ plane in figure 16. However, it should be noted that, even if we do not look specifically at the particular flow structure but at the integral property of the flow intensity in the convection cell, there are some verifiable discrepancies between temperature measurements and the CIFT measurements (see figure 27 in Appendix B). This is a clear indication of the need to analyse the full 3-D reconstructions of the flow field in more detail, which will be done in the following section.

Figure 15.

Azimuthal temperature distributions measured at ${Ra} = 6.02 \times 10^8$ at three heights of $z = 1/4 H$ (bot), $z = H/2$ (mid) and $z = 3/4 H$ (top) for (a) $t/t_{f\!f} = 1716$, and (b) $t/t_{f\!f} = 1724$.

Figure 16.

Flow pattern in the $y$–$z$ plane at ${Ra} = 6.02 \times 10^8$: (a) $t/t_{f\!f} = 1716$, and (b) $t/t_{f\!f} = 1724$ (corresponding azimuthal temperature profiles at the sidewall are shown in figure 15).

6.3. Torsional and sloshing mode

The occurrence of torsional oscillations, in which the orientation of the flow at the plates oscillates almost periodically in opposite phase, is an essential feature of a single-roll LSC in a cylinder with aspect ratio 1. Moreover, the flow in aspect ratios $\varGamma \ge 1$ is characterized by the occurrence of coherent oscillations, whereby this phenomenon is attributed to the torsional and sloshing mode at $\varGamma = 1$ and is associated with the JRV at $\varGamma > 1$. The oscillation periods usually correspond to the turnover of the LSC (Cioni et al. Reference Cioni, Ciliberto and Sommeria1997; Akashi et al. Reference Akashi, Yanagisawa, Tasaka, Vogt, Murai and Eckert2019; Zürner et al. Reference Zürner, Schindler, Vogt, Eckert and Schumacher2019). Thus, it was shown that changes in the orientation of the LSC plane, such as the azimuthal displacement of the horizontal flow direction in the torsional and sloshing modes, move through the convection cell at the characteristic velocity of the LSC (Zürner et al. Reference Zürner, Schindler, Vogt, Eckert and Schumacher2019). The existence of a dominant frequency is then the indicator of a coherent flow structure powered by hot and cold plumes rising and falling in a synchronized manner. So far, we can state, that the large-scale flow in the $\varGamma = 0.5$ cylinder is significantly more volatile compared with $\varGamma = 1$ and that the LSC exists not only as a single-roll pattern, but also occurs as a double- or triple-roll structure. Frequent switching between these roll structures makes the existence of a consistent turnover time questionable. This consideration is apparently supported by our measurements, since a frequency analysis of both the temperature and velocity time series using fast Fourier transformation does not reveal a dominant frequency in the fluctuating signals. Against this background, the question arises whether the torsion mode still exists under these conditions and whether it can be detected in the measured data.

Figure 17 displays the time series of the orientation $\theta _k$ derived from the temperature measurements within the time windows already applied in the previous sections. As already noticed in § 6.2, there are, on the one hand, time intervals characterized by a relative stability in the sense that the orientation changes moderately, with a similar tendency on all three sensor rings on average. Examples of such time periods, in which a certain integrity of the overall structure is reflected, are the intervals $380 \lesssim t/t_{f\!f} \lesssim 460$ in figure 17(a) and $1750 \lesssim t/t_{f\!f} \lesssim 1810$ in figure 17(b). On the other hand, there are periods in which the orientation changes very rapidly and by a whole turn or more. These rotations often do not affect the entire LSC structure, but may be limited to the upper or lower half of the convection cell. Thus, rapid counter-rotations of upper and lower flow structures manifest themselves e.g. in the intervals $480 \lesssim t/t_{f\!f} \lesssim 490 ({Ra} = 5.31 \times 10^7)$ or $1720 \lesssim t/t_{f\!f} \lesssim 1740 ({Ra} = 6.02 \times 10^8)$. Moreover, two rapid turns in the lower part of the convection cell can be observed for $t/t_{f\!f} \approx 1870$ and 1890 in a short time sequence, without the fluid in the upper part following the same movement. However, even during the moderate periods without striking events there are quite strong fluctuations, which we look at in more detail below.

Figure 17.

Time series of the LSC orientation obtained from measurements of the sidewall temperature: (a) ${Ra} = 5.31 \times 10^7$, (b) ${Ra} = 6.02 \times 10^8$.

Figure 18 shows the LSC orientation for the same time windows as in figure 17 with the values calculated using the CIFT data. The horizontal velocity field was obtained from the 3-D reconstruction by spatial averaging in two horizontal planes with a distance of 27 mm to the respective copper plates and the main flow direction was determined in the centre of the plane. For both experiments a very good agreement with respect to the main dynamics of the orientation angle can be observed between the multi-thermal-probe method and the CIFT approach. However, some specific details, such as the above-mentioned two complete revolutions of the fluid in the lower part of the convection cells, are not confirmed here (compare figures 17b and 18c). The following analysis is based on the CIFT results. In order to obtain a more reliable statement regarding the possible occurrence of the torsion and sloshing modes, we adopted a similar approach as suggested by Weiss & Ahlers (Reference Weiss and Ahlers2011). To distinguish between high-frequency and low-frequency changes in orientation, a matching low-frequency filter was applied to the time series in figures 18(a) and 18(c). The resulting curve containing only the low-frequency fluctuations was subtracted from the original signal. The remaining fluctuations of higher frequencies are presented in figures 18(b) and 18(d). The values of $\theta _k$ fluctuate around a mean value with amplitudes up to 80$^{\circ }$, although these fluctuations seem to be somewhat larger for ${Ra} = 5.31 \times 10^7$.

Figure 18.

Time series of the LSC orientation obtained from CIFT measurements at a distance of 27 mm from both plates obtained at ${Ra} = 5.31 \times 10^7$ (a,b) and ${Ra} = 6.02 \times 10^8$ (c,d): (a,c) azimuthal positions of $\theta _{bot}$ and $\theta _{top}$ as a function of time, (b,d) same data as in (a,c), but low-frequency changes (shown as thin red lines) in orientation subtracted.

From figures 18(b) and 18(d), it is difficult to estimate directly whether torsion and sloshing modes are present. Different periods can be found, in which $\theta _{bot}$ and $\theta _{top}$ seem to oscillate in phase or in antiphase. Therefore, in a next step, we calculate the correlation functions. For these calculations, the result of which can be seen in figure 19, the complete data set covering the entire measuring time was used. In particular, in this point we deviated from the approach of Weiss & Ahlers (Reference Weiss and Ahlers2011), who only considered the time intervals in which the flow is supposed to be dominated by the SRS regime. In the next section, however, we will see that this assignment of well-defined flow states in specified time periods is not so straightforward, at least in the parameter ranges studied here. Nevertheless, the result from figure 19 seems clear, as the cross-correlation functions for both $Ra$ exhibit a negative peak at zero time delay, which, according to Weiss & Ahlers (Reference Weiss and Ahlers2011), can be interpreted as an indication for the existence of a remnant of the torsional and sloshing mode.

Figure 19.

Auto-correlation and cross-correlation functions vs delay time $\tau$.

6.4. Identification of flow modes by means of POD

The POD analysis in § 5 demonstrates that diverse multiple roll structures can occur, with the SRS being the major contributor to the kinetic energy of the large-scale flow. In this section, we will now look at what times and how often which POD modes dominate the convection and how long these flow states typically last. Xi & Xia (Reference Xi and Xia2008) and Weiss & Ahlers (Reference Weiss and Ahlers2011) determined the flow regime using the multi-thermal-probe method. For the existence of SRS or DRS it was claimed that the amplitudes of the cosine fit of the azimuthal temperature profiles, $\delta _k$, must be above a certain threshold value. Other flow regimes that do not meet this requirement were classified as transitional states (TS). The relationship between the orientations of the LSC plane in the upper and lower parts of the convection cell, $\Delta \theta = \theta _{top}-\theta _{bot}$, decides the assignment as SRS and DRS, assuming that a single roll is associated with largely the same orientation ($\Delta \theta < 60^{\circ }$), while flows with prevalent opposite orientation ($\Delta \theta > 120^{\circ }$) imply a DRS. We have demonstrated that we are able to resolve single- and diverse multi-roll structures by means of our CIFT measurements. Thus, we follow the approach of Paolillo et al. (Reference Paolillo, Greco, Astarita and Cardone2021) of identifying the flow regimes by the energy contributions of the POD modes.

The time series of the energy contributions of the first eight POD modes are shown in figure 20 for the selected time windows at ${Ra} = 5.31 \times 10^7$ and ${Ra} = 6.02 \times 10^8$ together with the volume-averaged Reynolds number $Re_{vol}$. Animations of the 3-D vector field of velocity in the considered time intervals are provided in supplementary movie 1 available at https://doi.org/10.1017/jfm.2023.794 (${Ra} = 5.31\times 10^7$) and supplementary movie 2 (${Ra} = 6.02\times 10^8$). Individual modes are grouped together according to their basic structures. The same assignments are made here as previously done for figure 10 in § 5. In the case of ${Ra} = 5.31 \times 10^7$, mode 1 represents the SRS, the modes 2, 3, 4 and 7 are related to the DRS, while modes 5, 6 and 8 are considered as contributions to the TRS. For ${Ra} = 6.02 \times 10 ^8$, the modes can be assigned to the main structures in a pairwise manner, where modes 1 and 2 correspond to a SRS, modes 3 and 4 to a DRS and modes 5 and 6 to a TRS. The classification of modes 7 and 8 is not so straightforward. We have summarized these modes in the same way, although they are different toroidal structures of two to four rolls. We will see, however, that the associated energy contribution is very small compared with the other roll structures for the most part.

Figure 20.

Time series of the energy contributions of the first eight POD modes to the total kinetic energy of the flow: (a) ${Ra} = 5.31 \times 10^7$, (b) ${Ra} = 6.02 \times 10^8$.

In both diagrams it is noticeable that the energy inputs of the individual modes fluctuate very strongly on relatively short time scales of a few free-fall times. This corresponds to the behaviour of $Re_{vol}$. Moreover, it is evident that the positions of the respective maximum values of $Re_{vol}$ coincide quite well with the peaks of the energy contributions of SRS and DRS in particular. Such a high volatility of the flow with frequent changes of the flow modes and accompanied by corresponding fluctuations of ${Nu}$ and $Re$ was also found by Zwirner et al. (Reference Zwirner, Tilgner and Shishkina2020) (see especially figure 1 therein). Their numerical simulations for turbulent convection in a $\varGamma = 0.2$ cylinder at $Pr = 0.1$ reveal nearly synchronous fluctuations of both ${Nu}$ and $Re$, with high values reached together with a SRS or a DRS, and multi-roll structures appearing at times of minimum flow intensity. This tendency is confirmed by our measurements. Within the time intervals of high flow velocities, the first modes, especially the SRS and DRS, dominate. It is interesting to observe that, in the case of ${Ra} = 5.31 \times 10^7$ (figure 20a), the maxima are determined in roughly equal parts by an SRS and DRS, respectively, while at the higher Rayleigh number the SRS appears to be numerically clearly superior (figure 20b). This is also to be expected from the increase of the integral energy portion of the SRS (see figure 9 in § 5). The appearance of a TRS with high energy input occurs extremely rarely, in the data shown here exactly once, in figure 20(b) at $t/t_{f\!f} \thickapprox 1817$.

At this point we make another attempt to estimate the statistical frequency of occurrence and duration of certain flow states from the measured data. For this purpose, the respective data sets are divided into time intervals in which the respective fundamental modes provide the highest energy contribution. Figure 21 presents the probability distributions and the cumulative probability distributions of the lifetime of the main roll regimes in free-fall times units. When the energy fraction of each of the three roll modes was found to be less than 5 % of the time-averaged total kinetic energy, the flow was considered undefined and assigned to a TS. Table 1 contains the time fractions of the modes studied as well as the mean lifetime of the individual states for both $Ra$. The comparison of both measurements again reveals a strengthening of the significance of the SRS with increasing $Ra$. According to the analysis performed here, the large-scale structures are found to form more frequently a DRS at ${Ra} = 5.31 \times 10^7$ (figure 21a), while at ${Ra} = 6.02 \times 10^8$ the SRS regime reveals a significantly higher occurrence probability (figure 21b). In the case of the smaller Rayleigh number, the SRS is found for approximately one third of the duration of the experiment, while the DRS dominates for more than half of the measurement time. For the higher $Ra$, this is almost reversed. The following is worth noting: as $Ra$ increases, not only does the percentage for SRS increase, but the time shares of TRS and TS also grow at the expense of the DRS. In addition, the average lifetime in free-fall time units increases for all modes. In the case of the smaller Rayleigh number, the lifetimes of SRS and DRS are approximately the same, but when $Ra$ becomes larger, the lifetime of the SRS in units of free-fall time grows more significantly than for the other modes.

Figure 21.

Probability distributions (a,b) and c.d.f.s (c,d) of the lifetime of the main roll states at ${Ra} = 5.31 \times 10^7$ (a,c) and ${Ra} = 6.02 \times 10^8$ (b,d).

Table 1.

Time shares and average lifetimes of the main flow modes.

6.5. Reconstruction of instantaneous flow structures

In this section, we will illustrate, by means of some examples, which instantaneous 3-D structures actually exist in the convection cell during the respective time intervals. To this end, we have chosen another time window for ${Ra} = 6.02 \times 10^8$ in which we find singular peaks of all the major roll regimes SRS, DRS and TRS, as well as an example in which SRS and DRS reach a local maximum simultaneously. The animation of the 3-D vector field of velocity is provided in supplementary movie 3. Figure 22(a) shows the respective time series of the energy contributions of the POD modes and $Re_{vol}$. The mentioned four selected moments are marked at which the snapshots of the 3-D flow structure are visualized.

Figure 22.

Snapshots of the 3-D flow pattern at ${Ra} = 6.02 \times 10^8$. (a) Time series of the energy contributions of the main POD modes and the volume-averaged Reynolds number, (b) streamlines – point A ($t /t_{f\!f} = 1195$), (c) streamlines – point B ($t /t_{f\!f} = 1209$), (d) streamlines – point C ($t /t_{f\!f} = 1233$), (e) streamlines – point D ($t /t_{f\!f} = 1242$), ( f) vector field - point A, (g) vector field – point B, (h) vector field – point C, (i) vector field – point D. The angular orientation was chosen here so that the structures are best visible. Therefore, deviations from the supplementary movies may occur.

Point A represents one of the short time intervals in which modes 5 and 6 provide the highest energy contribution. The reconstruction of the flow pattern in figure 22(b, f) reveals a TRS accordingly. In this context, it should be noted that flow patterns of three or more rolls exist in all measurements, especially when the Reynolds numbers feature minimum values, while the high-$Re$ periods are dominated by SRS or DRS. A distinct maximum in flow energy is reached in point B of figure 22(a), to which modes 1 and 2 deliver the greatest energy contribution. This flow state follows a DRS of lower intensity at $t / t_{f\!f} = 1201$. With the decay of modes 3 and 4, the energy contribution of modes 1 and 2 starts to grow. The contribution of the DRS regime drops almost to zero initially, but immediately rises again rapidly and reaches a new peak almost simultaneously with the SRS. Transitions from a DRS to an SRS or vice versa have also been examined in previous experiments (Xi & Xia Reference Xi and Xia2008; Weiss & Ahlers Reference Weiss and Ahlers2011; Paolillo et al. Reference Paolillo, Greco, Astarita and Cardone2021). On an appropriate example, that spanned a period of approximately $35 t_{f\!f}$, Paolillo et al. (Reference Paolillo, Greco, Astarita and Cardone2021) likewise reconstructed the behaviour of the POD modes. In their measurement, the initial state is clearly determined by the energy contribution of the DRS. This is followed for approximately $20 t_{f\!f}$ by a transition phase with appreciable contributions of several modes, before the energy contribution of the SRS increases significantly, while all other modes become almost negligible. We find numerous DRS–SRS transitions in our measurements, but they are all often noticeably different in terms of the temporal evolution of the respective energy fractions as well as their 3-D flow patterns. The simultaneous occurrence of the SRS and DRS peaks at point B is reflected in the flow structure in such a way that one recognizes a rather two-roll structure, but in which a significant part of the streamlines are connected over the entire cell height (see figure 22c,g). In the same time window in figure 22(a), however, we also find both an SRS and a DRS peak, in which almost exclusively only modes 1 and 2 (point C) or 3 and 4 (point D) dominate. The corresponding flow structures in the figures 22(d,h) and 22(e,i) show a single-roll and a double-roll LSC. But here too, both structures are not ideally formed. The large circulation of the SRS does not extend over the entire height of the cell and is supplemented by a smaller roll at the bottom, while another corner vortex is also visible at the bottom of the double roll.

Another specific example of the development of a SRS and its transition to a DRS at ${Ra} = 6.02 \times 10^8$ is shown in figure 23. This observation concerns a smaller section of the time window already presented in figure 20(b). This time we also depict the energy contributions of the POD modes together with the difference of the LSC orientation at the lower and upper rings of the temperature sensors, $\Delta \theta = \theta _{top} - \theta _{bot}$. This quantity is suggested as an indicator to distinguish between SRS and DRS assuming a SRS for $\Delta \theta < 60^{\circ }$ and a DRS for $\Delta \theta > 120^{\circ }$ (Xi & Xia Reference Xi and Xia2008; Weiss & Ahlers Reference Weiss and Ahlers2011). At point E, we start from a state of very low kinetic energy and small flow velocities (see figure 23b). At time F, the flow at both plates has initially intensified perceptibly (figure 23c), and at time G, one can observe the emergence of stronger plumes that noticeably drive the flow in the volume (figure 23d). The energy of the SRS and flow velocities reach a distinct maximum at time H as shown in figure 23(e). On the left side two strong rising and falling plumes can be seen, which, however, converge at an almost identical azimuthal position. As a result, the descending flow is obviously deflected from the sidewall and evades into the centre of the convection cell. The ascending flow appears to continue up the sidewall, but changes its azimuthal position counterclockwise with increasing height. This rearrangement of the flow structure is apparently associated with higher dissipation. As a result, the velocities decrease significantly, particularly in the centre of the cell as can be seen at time I in figure 23( f). However, the flows in both the upper and lower regions of the convection cell reduce only to a certain extent and form a new DRS at time J in figure 23(g).

Figure 23.

Snapshots of the entire 3-D vector field and the distribution of the vertical velocity across the horizontal planes at $z/H = 0.25$, 0.5 and 0.75 for ${Ra} = 6.02 \times 10^8$: (a) time series of the energy contributions of the main POD modes and the difference of the orientation of the horizontal flow near the plates ($z/H = 0.1$ and 0.9, respectively), (b) point E, (c) point F, (d) point G, (e) point H, ( f) point I, (g) point J, the colours represent the vertical velocity. The angular orientation was chosen here so that the structures are best visible. Therefore, deviations from the supplementary movies may occur.

Figure 23(a) provides the at first glance astonishing finding that the transition from SRS to DRS seems to have hardly any effect on $\Delta \theta$ in the case at hand, and it also contradicts the impression that can be gained from panels (b–f). At point H, which coincides with the SRS maximum, $\Delta \theta$ shows a value of 47.4$^{\circ }$, while $\Delta \theta = -58.6^{\circ }$ in the DRS maximum at time J. The corresponding azimuthal temperature profiles recorded for both moments in time at $z = 1/4 H$ and $z = 3/4 H$, respectively, are presented in figure 24. In agreement with the energy contributions of the POD modes in figure 23(a), which reveal a more intense flow for the case of the SRS maximum in point H, figure 24(a) shows correspondingly larger amplitudes of the cosine fit of the temperature distribution compared with the DRS maximum in figure 24(b). In the latter case, however, the positions of minimum and maximum temperatures are no longer so straightforward to determine, and the cosine fit method is obviously also more error prone here. A closer look at the flow structures reveals the following issues:

1. (i) The regions of ascending or descending flow on the sidewall vary greatly in terms of their location and extent. For example, crescent-shaped velocity distributions are often observed, with the regions of ascending (descending) flow near the bottom (of the lid) leaning against the sidewall over a wide area, while the oppositely directed flow spreads more toward the centre.

2. (ii) Since the pattern of upflows and downflows is not symmetrical within the horizontal cross-section, the positions of the temperature minimum and maximum are also usually not exactly opposite to each other. This leads to inaccuracies in the application of the cosine function when fitting the azimuthal profiles.

3. (iii) The structures of both the single-roll and the double-roll circulations are often seriously twisted. Due to the complexity of the flow, an exact discrimination between a seriously twisted single roll and a double-roll state is difficult to obtain. A more detailed analysis is foreseen for future work, when the measurement system will be extended by an additional excitation magnetic field in horizontal direction.

Figure 24.

Azimuthal temperature distributions measured at ${Ra} = 6.02 \times 10^8$ at $z = 1/4 H$ (bot) and $z = 3/4 H$ (top) for (a) $t/t_{f\!f} = 1788$ (point H in figure 23a) and (b) $t/t_{f\!f} = 1792$ (point J in figure 23a).

It can also be seen in the vertical velocity distributions in figure 23 that the cross-sectional area of the plume increases significantly with distance from the plate where it has detached. Moreover, the velocity in the plumes successively decreases with increasing distance from their detachment locations. This is obviously due to the high thermal conductivity of the fluid, which causes the superheat of the plumes to spread rather rapidly to the adjacent zones. This observation is in agreement with the qualitative finding of the broadening of the temperature distributions in figure 13. The thermal dissipation of the plumes is supposed to have an influence on the stability and coherence of the LSC (Stringano & Verzicco Reference Stringano and Verzicco2006). The decrease in the temperature difference between the plumes and the surrounding fluid results in a loss of buoyancy of the plumes relative to the surrounding fluid and a corresponding reduction in velocity. A further interesting aspect is that the detaching plumes usually extend over narrow edge regions along the sidewall resulting in crescent-shaped patterns and at certain times tend to fill the central region of the cross-sectional area when arriving at the opposite plate.