3.1 Long-term dynamics of the large-scale flow structure

Figure 2.

Calculation of the orientation angle $\unicode[STIX]{x1D703}_{top}$ of the large-scale flow close to the top plate. The UDV sensor $T_{0}$ at $\unicode[STIX]{x1D719}=0^{\circ }$ measures the radial $v_{x}$ profile (left) and $T_{90}$ at $\unicode[STIX]{x1D719}=90^{\circ }$ measures the radial $v_{y}$ profile (right). The mean velocities $\bar{v}_{x}$ and $\bar{v}_{y}$ are determined over the central shaded profile intervals. These values are used as components of the horizontal velocity vector of the LSC which is denoted as $\boldsymbol{v}_{top}$ .

We now discuss the long-time evolution of the LSC for a measurement at our highest Rayleigh number $Ra=6\times 10^{7}$ . Times are given in units of the free-fall time $\unicode[STIX]{x1D70F}_{ff}=\sqrt{H/(g\unicode[STIX]{x1D6FC}\unicode[STIX]{x0394}T)}$ . Variable $g$ is the acceleration due to gravity. For the example data set at $Ra=6\times 10^{7}$ , the free-fall time is $\unicode[STIX]{x1D70F}_{ff}=2.3$  s and the total duration of the time series is $1700\unicode[STIX]{x1D70F}_{ff}$ . Figure 2 illustrates how the orientation angle of the flow in the centre close to the top plate is calculated. The velocity profiles $v_{x}(x,t)|_{y=0}$ and $v_{y}(y,t)|_{x=0}$ are measured by UDV sensors $T_{0}$ and $T_{90}$ , respectively, 10 mm below the top plate. At the central crossing point of the ultrasonic beams the horizontal velocity vector $\boldsymbol{v}_{top}=(\bar{v}_{x},\bar{v}_{y})$ and the resulting orientation angle are given by

(3.1) $$\begin{eqnarray}\bar{v}_{i}(t)=\langle v_{i}(i,t)\rangle _{-D/8\leqslant i\leqslant D/8},\quad \text{and thus}~\unicode[STIX]{x1D703}_{top}=\arctan \left(\frac{\bar{v}_{y}}{\bar{v}_{x}}\right),\end{eqnarray}$$

with $i=x,y$ . The orientation angle at the bottom plate $\unicode[STIX]{x1D703}_{bot}$ is calculated analogously using UDV sensors $B_{0}$ and $B_{90}$ . Figure 3(a) shows a time series of $\unicode[STIX]{x1D703}_{top}+180^{\circ }$ and $\unicode[STIX]{x1D703}_{bot}$ . The two angles match very well which implies that in this measurement they always maintain a mean azimuthal offset of $180^{\circ }$ . This validates the presence of a single coherent LSC roll in the cell. Figure 3(f,h) presents a detailed view of the data in figure 3(a). It can now be seen that the orientation angles oscillate around a common mean at an oscillation time of $\unicode[STIX]{x1D70F}_{osc}\sim 10\unicode[STIX]{x1D70F}_{ff}$ . Furthermore, the angles oscillate in anti-phase, which is a clear indication of the torsion mode (Funfschilling & Ahlers Reference Funfschilling and Ahlers2004; Xie, Wei & Xia Reference Xie, Wei and Xia2013; Khalilov et al. Reference Khalilov, Kolesnichenko, Pavlinov, Mamykin, Shestakov and Frick2018). The LSC flow is thus characterized by a torsion in agreement with DNS at similar $Pr$ by Scheel & Schumacher (Reference Scheel and Schumacher2016, Reference Scheel and Schumacher2017) and experiments in liquid sodium at $Pr=0.0094$ (Khalilov et al. Reference Khalilov, Kolesnichenko, Pavlinov, Mamykin, Shestakov and Frick2018). We come back to this point in § 3.2. Figure 3(a) shows clearly the additional slow drift of the mean orientation angle by more than $180^{\circ }$ over the full measurement time period of $\unicode[STIX]{x1D70F}_{total}\approx 1700\unicode[STIX]{x1D70F}_{ff}$ , a motion that is due to the statistical azimuthal symmetry in the cylindrical set-up. The characteristic time scale  $\unicode[STIX]{x1D70F}_{drift}$ of this motion can be estimated to be of the order of the thermal diffusion time scale  $\unicode[STIX]{x1D70F}_{drift}\sim \unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D705}}=H^{2}/\unicode[STIX]{x1D705}$ . Using $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D705}}=\sqrt{RaPr}\,\unicode[STIX]{x1D70F}_{ff}$ , this can be expressed in terms of the free-fall time to be $\unicode[STIX]{x1D70F}_{drift}\sim 170\,\unicode[STIX]{x1D70F}_{ff}$ to $1300\,\unicode[STIX]{x1D70F}_{ff}$ for our $Ra$ range of $10^{6}<Ra<6\times 10^{7}$ . A quantitative measurement of this time scale would require a Fourier time series analysis of even longer data records than the ones we could obtain.

Figure 3.

Large-scale flow structure for $Ra=6\times 10^{7}$ ( $\unicode[STIX]{x1D70F}_{ff}=2.3$  s) measured by UDV sensors near the plates (velo.) and thermocouples at mid-height (temp.). (a) The LSC orientation angles at the top ( $\unicode[STIX]{x1D703}_{top}$ ) and bottom ( $\unicode[STIX]{x1D703}_{bot}$ ) plates (see figure 2). The black dashed line is the smoothed average of the top and bottom angles. (b) Temperatures near the wall at half the height of the cell over time and azimuthal position  $\unicode[STIX]{x1D719}$ . The dashed black line is identical to the one in (a). (c–e) Detailed views of time series (b) for a time interval of $40\unicode[STIX]{x1D70F}_{ff}$ . (f–h) Detailed views of time series (a) for same time interval.

Most experiments on the large-scale flow structure in turbulent RBC use temperature measurements close to the sidewalls of the cell to infer the azimuthal profile of the vertical flow direction from their temperature imprint: up-welling fluid from the hot bottom plate is detected as a high-temperature signal, while down-welling fluid from the cold top plate gives a low-temperature signal. This principle is employed here as well, however at a very fine resolution. Figure 3(b) shows a colour plot of the temperature time series taken simultaneously by 11 thermocouples which are arranged in a semicircle at the sidewall at half-height (see figure 1 c). This arrangement allows us to present a space–time plot of the temperature with details never obtained before in a liquid metal flow experiment. Temperatures below the average fluid temperature $\bar{T}$ are coloured in blue and temperatures above $\bar{T}$ are shown in orange. The black dashed line re-plots the LSC orientation $\unicode[STIX]{x1D703}_{LSC}=(\unicode[STIX]{x1D703}_{top}+180^{\circ }+\unicode[STIX]{x1D703}_{bot})/2$ from figure 3(a). The profile has been additionally smoothed over time using a moving average filter over five successive measurements. It confirms the coherence of the average LSC: temperature at half-height and velocity dynamics at top/bottom are in perfect synchronization and drift slowly as a common, single LSC roll.

Figures 3(c) and 3(e) show magnified sections of figure 3(b) at finer temporal resolution that correspond to those in figures 3(f) and 3(h), respectively. In both cases, hot rising and cold falling plumes at the sidewall bounce together and move away from each other again. This periodic motion is known as a sloshing mode of the LSC (Brown & Ahlers Reference Brown and Ahlers2009; Zhou et al. Reference Zhou, Xi, Zhou, Sun and Xia2009). We detect $\unicode[STIX]{x1D70F}_{slosh}=\unicode[STIX]{x1D70F}_{osc}$ from these two pairs of panels. Previous experiments in water (Brown & Ahlers Reference Brown and Ahlers2009; Zhou et al. Reference Zhou, Xi, Zhou, Sun and Xia2009) report that the up- and down-welling flows come as close as $45^{\circ }$ . In our measurements this minimal azimuthal distance is much smaller, regularly reaching the order of our azimuthal resolution of $18^{\circ }$ . This extreme sloshing amplitude seems to be a property of low- $Pr$ convection and the high inertia of liquid metals.

Figure 4.

Large-scale flow for (a–d) $Ra=10^{6}$ ( $\unicode[STIX]{x1D70F}_{ff}=17$  s) and (e–h) $Ra=10^{7}$ ( $\unicode[STIX]{x1D70F}_{ff}=5.4$  s). Quantities and the colour scale are the same as in figure 3. Panels (c) and (d) are detailed views of (a) and (b), respectively. Panels (e) and (f) are detailed views of (g) and (h), respectively. The time span of the detailed views is $40\unicode[STIX]{x1D70F}_{ff}$ .

A rare event in the large-scale flow dynamics is shown in figure 3(d,g). Here, the LSC rapidly changes its orientation by about $90^{\circ }$ within less than $10\unicode[STIX]{x1D70F}_{ff}$ ( $940<t/\unicode[STIX]{x1D70F}_{ff}<950$ ). At the same time, the characteristic sloshing pattern in the temperature plot is disrupted and over most of the circumference the mean value of the temperature is detected which is in line with the absence of the coherent pattern up- or down-welling flow. Only after the sudden orientation change does the sloshing pattern reappear, now shifted by $90^{\circ }$ in the azimuthal direction. This observation suggests that a cessation has taken place – an event which has also been observed in experiments in water (Brown & Ahlers Reference Brown and Ahlers2006) and fluorinert FC-77 electronic liquid (Xie et al. Reference Xie, Wei and Xia2013). Cessations consist of a breakdown of the coherent LSC into an incoherent flow state and a subsequent re-establishment of the coherence of the LSC with a different orientation. These events are rare; in water experiments they occurred at rates of the order of days $^{-1}$ (Brown & Ahlers Reference Brown and Ahlers2006; Xie et al. Reference Xie, Wei and Xia2013). Our measurement series did not reach such time durations. Consequently, a statistical analysis of cessations in the present liquid metal convection experiments was not possible.

In figure 4, two additional experiments – one at the lowest Rayleigh number  $Ra=10^{6}$ and one at an intermediate $Ra=10^{7}$ – are presented. The time series cover again more than 1000 free-fall times in both cases. Just as for $Ra=6\times 10^{7}$ in figure 3, the temperature and velocity measurements show the presence of the LSC, as well as the sloshing and torsion modes in the magnifications. Differences can mainly be seen with respect to the temperature magnitude when comparing the data to figure 3. With increasing $Ra$ , the amplitude of the fluid temperature at mid-height of the cell decreases steadily indicating an enhanced mixing of the scalar field due to an increasingly inertial fluid turbulence. Furthermore, the hot and cold patches of the up- and down-wellings are thicker and more washed out for lower Rayleigh numbers. This can be seen best for the lowest $Ra=10^{6}$ in figure 4(d). We have found that the magnitude of the total long-term azimuthal drift does not show a dependence on the Rayleigh number in the accessible range. To summarize this part, our experimental study shows clearly that the LSC dynamics has to be considered as a superposition of multiple processes with different characteristic time scales which can be reconstructed from the time series.

3.2 Rayleigh number dependence of oscillation frequencies

Frequency spectra of $\unicode[STIX]{x1D703}_{top}$ are shown for three Rayleigh numbers $Ra$ in figure 5(a). A clear peak gives the oscillation frequency $f_{osc}$ of the torsional mode. The characteristic frequency value is extracted by fitting the function $A(f)$ to the spectra

(3.2) $$\begin{eqnarray}A(f)=a\exp \left(-\frac{(f-f_{osc})^{2}}{2\unicode[STIX]{x0394}f^{2}}\right)+b\,f^{\unicode[STIX]{x1D6FE}}.\end{eqnarray}$$

This function models the spectra as a Gaussian peak at  $f_{osc}$ with a width of  $\unicode[STIX]{x0394}f$ on top of an algebraic power-law background. The fitting parameters are $a$ , $b$ , $\unicode[STIX]{x1D6FE}$ , $f_{osc}$ and  $\unicode[STIX]{x0394}f$ .

Figure 5.

(a) Frequency spectra of $\unicode[STIX]{x1D703}_{top}$ for different $Ra$ , normalized by the thermal diffusion frequency  $f_{\unicode[STIX]{x1D705}}$ . The spectra at $Ra=10^{6}$ and $Ra=6\times 10^{7}$ are shifted for better visibility by a factor of 0.1 and 20, respectively. The dashed lines are fits of (3.2) to the spectra to determine the characteristic oscillation frequency $f_{osc}$ . (b) Scaling of the frequency $f_{osc}$ with $Ra$ . The error bars are taken from the standard deviation  $\unicode[STIX]{x0394}f$ of the Gaussian fit (3.2). (c) Comparison of $f_{osc}$ with the turnover frequency  $f_{to}$ over  $Ra$ .

Figure 5(b) shows the $Ra$ dependence of the frequency $f_{osc}$ , normalized by the thermal diffusion frequency

(3.3) $$\begin{eqnarray}f_{\unicode[STIX]{x1D705}}=\frac{\unicode[STIX]{x1D705}}{H^{2}}.\end{eqnarray}$$

The error bars correspond to the standard deviation  $\unicode[STIX]{x0394}f$ of the fitted Gaussian peaks. A power law is fitted to the data using orthogonal distance regression (Boggs & Rogers Reference Boggs and Rogers1990) to incorporate the errors of both the abscissa and the ordinate. The error estimation of the fit is outlined in the Appendix. This procedure is used for all following power-law fits in this work if not noted otherwise. The fit results in a scaling of $f_{osc}/f_{\unicode[STIX]{x1D705}}\simeq (0.10\pm 0.04)Ra^{0.40\pm 0.02}$ , indicated as a dashed line in figure 5(b). Dimensional arguments suggest a scaling of the oscillation frequency with the free-fall time  $\unicode[STIX]{x1D70F}_{ff}$ of $f_{osc}\propto Ra^{0.5}$ for constant material properties, since $1/\unicode[STIX]{x1D70F}_{ff}=\sqrt{\unicode[STIX]{x1D708}\unicode[STIX]{x1D705}Ra}/H^{2}$ . The exponent of 0.4 indicates that the underlying time scale is indeed close to the free-fall time, but that the inertial character of the fluid turbulence in the low-Prandtl-number flow affects the turbulent momentum transfer. We will return to this point in § 4.2.

The DNS by Schumacher et al. (Reference Schumacher, Bandaru, Pandey and Scheel2016) at $Pr=0.021$ result in a slightly stronger scaling of $f_{osc}/f_{\unicode[STIX]{x1D705}}\simeq (0.08\pm 0.05)Ra^{0.42\pm 0.02}$ when their data are corrected from radians to units of cycles per diffusive time (open circles in figure 5 b). Previous experiments by Tsuji et al. (Reference Tsuji, Mizuno, Mashiko and Sano2005) in mercury at a Prandtl number of $Pr=0.024$ coincide with our results (crosses in figure 5 b). For liquid gallium ( $Pr=0.027$ ) in a $\unicode[STIX]{x1D6E4}=2$ cell, Vogt et al. (Reference Vogt, Horn, Grannan and Aurnou2018a ) found the same scaling exponent as for the $\unicode[STIX]{x1D6E4}=1$ case, but with a lower magnitude: $f_{osc}/f_{\unicode[STIX]{x1D705}}\simeq 0.027Ra^{0.419\pm 0.006}$ . The scaling exponents and absolute values for measurements in larger-Prandtl-number fluids are generally higher. For example, in water at $Pr\sim 5.4$ scaling laws of $f_{osc}/f_{\unicode[STIX]{x1D705}}\simeq 0.2Ra^{0.46}$ (Qiu & Tong Reference Qiu and Tong2001), $f_{osc}/f_{\unicode[STIX]{x1D705}}\simeq 0.167Ra^{0.47}$ (Qiu et al. Reference Qiu, Shang, Tong and Xia2004) and $f_{osc}/f_{\unicode[STIX]{x1D705}}\simeq 0.12Ra^{0.49}$ (Zhou et al. Reference Zhou, Xi, Zhou, Sun and Xia2009) are found. Experiments in methanol at $Pr=6.0$ (Funfschilling & Ahlers Reference Funfschilling and Ahlers2004) give a similar result of $f_{osc}/f_{\unicode[STIX]{x1D705}}\simeq 0.126Ra^{0.460\pm 0.012}$ . Considering that the thermal diffusivity of water (Çengel Reference Çengel2008) is about 70 times lower than that of GaInSn at $35\,^{\circ }\text{C}$ , the absolute frequencies $f_{osc}$ in water at $Ra=10^{7}$ are about 14 times smaller than our values in GaInSn. This higher intensity of the flow dynamics in liquid metals is to be expected, as low- $Pr$ liquids are known to drive a more turbulent convective flow than high- $Pr$ fluids at the same $Ra$ (Breuer et al. Reference Breuer, Wessling, Schmalzl and Hansen2004; Schumacher et al. Reference Schumacher, Götzfried and Scheel2015).

The averaged velocity components (3.1) are also used to calculate the velocity amplitude of the LSC:

(3.4) $$\begin{eqnarray}v_{LSC}=\left\langle \frac{|\boldsymbol{v}_{top}|+|\boldsymbol{v}_{bot}|}{2}\right\rangle _{t},\end{eqnarray}$$

where $\langle \cdot \rangle _{t}$ denotes a time average. Using this velocity, the turnover time  $\unicode[STIX]{x1D70F}_{to}$ of the LSC can be defined as $\unicode[STIX]{x1D70F}_{to}=\unicode[STIX]{x03C0}H/v_{LSC}$ . Here, a roll shape in the form of a circle of diameter  $H$ has been assumed as the LSC path. The turnover frequency is then

(3.5) $$\begin{eqnarray}f_{to}=\frac{v_{LSC}}{\unicode[STIX]{x03C0}H}.\end{eqnarray}$$

Figure 5(c) shows the ratio $f_{osc}/f_{to}$ , which is close to unity for all Rayleigh numbers. This implies that one period of the torsional mode – and thus also of the sloshing mode – takes one turnover time of the LSC. If, alternatively, the length $2H+2D$ is used for the LSC path instead of the circumference of a circle, $f_{to}$ decreases by a factor of $\unicode[STIX]{x03C0}/4\approx 0.79$ . However, the relation of $f_{osc}\propto f_{to}$ would still be valid.

3.3 Interplay of the torsion and sloshing modes

In this section, we investigate how the sloshing and torsion modes described in § 3.1 coexist and build a single coherent flow structure. The basic connection of the two modes is that the flow directions at the top and bottom plate, $\unicode[STIX]{x1D703}_{top}$ and $\unicode[STIX]{x1D703}_{bot}$ , indicate the azimuthal position where the up- and down-welling flows will appear. This can be monitored by the temperature sensors at mid-height. We have already shown that on average the top and bottom flows are anti-parallel with $\langle \unicode[STIX]{x1D703}_{bot}\rangle _{t}-\langle \unicode[STIX]{x1D703}_{top}\rangle _{t}\sim 180^{\circ }$ (see figure 3 a). If one assumes that these flows will be deflected in different azimuthal directions by an angle $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}$ (which represents the effect of the LSC torsion) then the orientation angles will get closer to one another. With $\unicode[STIX]{x1D703}_{top/bot}=\langle \unicode[STIX]{x1D703}_{top/bot}\rangle _{t}\pm \unicode[STIX]{x0394}\unicode[STIX]{x1D703}$ , the azimuthal distance of the up- and down-welling flows is given by

(3.6) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{bot}-\unicode[STIX]{x1D703}_{top}\sim 180^{\circ }-2\unicode[STIX]{x0394}\unicode[STIX]{x1D703}.\end{eqnarray}$$

With the torsion displacement $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}$ at top and bottom getting closer to $90^{\circ }$ , the hot up-welling and cold down-welling flows will also get closer to each other. This is exactly the behaviour of the sloshing mode detected in figure 3(b).

Figure 6.

Correlation of temperatures at mid-height with flow orientations at the top and bottom plates. (a) Illustration of the possible correlations. (b) Example of correlations which are normalized by their maxima and plotted over the time shift  $\unicode[STIX]{x1D70F}$ in units of the oscillation frequency  $f_{osc}$ . The data record is taken from the measurement in figure 3 at $Ra=6\times 10^{7}$ . (c) Time shift  $\unicode[STIX]{x1D70F}_{max}$ of the maximal correlation over $Ra$ .

However, the flow orientation at the plates and the respective vertical flow at half the height of the cell do not coincide in time. We have established that one oscillation period is the same as one turnover of the LSC. From this point, we would expect that the flow orientation propagates with the same speed, e.g. a given flow direction at the top plate would result in the same azimuthal position of the down-flow a quarter turnover later. To investigate this behaviour more closely, we calculate a temporal correlation of the top and bottom LSC angles with the temperatures measured at mid-height (see figure 6 a). As an example, we correlate $\unicode[STIX]{x1D703}_{top}$ with the cold down-flow at mid-height, which we denote as Top $\rightarrow$ Cold. All positive values of the temperature profile are clipped in figure 3(b) to zero, in order to include the cold signature of the down-flow only. This is equivalent to clipping the temperatures to the mean fluid temperature $\bar{T}$ . Next, we construct a pseudo-temperature profile ${\mathcal{T}}$ from the time series of $\unicode[STIX]{x1D703}_{top}$ which is given by

(3.7) $$\begin{eqnarray}{\mathcal{T}}(\unicode[STIX]{x1D719},t)=\left\{\begin{array}{@{}ll@{}}A\cos ^{2}(\unicode[STIX]{x1D719}-\unicode[STIX]{x1D703}_{top}(t)),\quad & \text{for }\unicode[STIX]{x1D703}_{top}-90^{\circ }<\unicode[STIX]{x1D719}<\unicode[STIX]{x1D703}_{top}+90^{\circ },\\ 0,\quad & \text{else}.\end{array}\right.\end{eqnarray}$$

This profile emulates the temperature at mid-height if the position of the down-flow would follow the flow orientation at the top plate instantaneously. The amplitude is set to $A=-1$ in this particular case since we correlate with the negative values of the measured temperature profile. In the case of a correlation with the up-welling flow, an amplitude $A=1$ is taken. The function ${\mathcal{T}}$ is evaluated at the azimuthal positions of the mid-height thermocouples and correlated with the clipped temperature profile over time. The result is normalized by its maximal value and plotted in figure 6(b) as a solid black line. The correlation time shift $\unicode[STIX]{x1D70F}$ is normalized by the oscillation frequency $f_{osc}$ .

We observe that the maximum correlation is shifted to $\unicode[STIX]{x1D70F}f_{osc}>0$ , i.e. the down-flow at mid-height lags behind the flow orientation at the top. The time lag  $\unicode[STIX]{x1D70F}_{max}$ of the maximum correlation is calculated by fitting a quadratic polynomial around the maximal value. In the case Top $\rightarrow$ Cold the time shift is $\unicode[STIX]{x1D70F}_{max}f_{osc}=0.32$ , which is larger than the expected value of $1/4$ . The correlations of the three other cases Cold $\rightarrow$ Bottom, Bottom $\rightarrow$ Hot and Hot $\rightarrow$ Top are displayed in figure 6(b) as well. It can be seen that the vertical flows at mid-height have a time lag towards the flows at the respective plate, which is larger than a quarter of one oscillation period (Top $\rightarrow$ Cold and Bottom $\rightarrow$ Hot). Furthermore, the horizontal flows at the plates follow the mid-height flows with a lag shorter than a quarter oscillation period. We repeated this analysis for all our experiments. The corresponding time shifts display this behaviour (see figure 6 c) in all data sets. On average the time lags  $\unicode[STIX]{x1D70F}_{max}f_{osc}$ are: $0.33\pm 0.01$ for Top $\rightarrow$ Cold, $0.19\pm 0.01$ for Cold $\rightarrow$ Bottom, $0.30\pm 0.01$ for Bottom $\rightarrow$ Hot and $0.16\pm 0.01$ for Hot $\rightarrow$ Top. The sum of all four time lags gives on average the expected value of one, here $0.98\pm 0.03$ , and thus the propagation of the flow orientation requires the same amount of time as one oscillation period or one turnover. From the present data, the flow can be understood as independent parcels of the fluid circulating in the cell. Each fluid parcel circulates on average in a vertical plane which it does not leave. The observed flow modes are then a result of the collective motion of these fluid parcels, which are phase-shifted in time and the azimuthal orientation of their plane. This interpretation of our data does not predict significant azimuthal velocity components as part of the mode dynamics of the LSC. This cannot be verified with the current set-up, but future experiments could aim to include the measurement of azimuthal velocities.

A similar analysis as in figure 6 is found in Qiu et al. (Reference Qiu, Shang, Tong and Xia2004). Their data reveal a shift close to $\unicode[STIX]{x1D70F}_{osc}/4$ between fluctuations of the velocity at the bottom centre and of the temperature at half-height near the up-welling flow. This would correspond to our Bottom $\rightarrow$ Hot case. An exact shift value is however not discernible from their plots. Other data on the phase shift between torsion and sloshing modes can be grouped into three categories: (i) correlation of the top and bottom flow orientations, which gives the characteristic torsion phase shift of $\unicode[STIX]{x1D70F}_{osc}/2$ ; see experiments by Funfschilling & Ahlers (Reference Funfschilling and Ahlers2004) in methanol at $Pr=6.0$ , by Xie et al. (Reference Xie, Wei and Xia2013) in fluorinert FC-77 electronic liquid at $Pr=19.4$ or by Khalilov et al. (Reference Khalilov, Kolesnichenko, Pavlinov, Mamykin, Shestakov and Frick2018) in liquid sodium at $Pr=0.0094$ ; (ii) correlation of the hot and cold temperature signatures at half-height, resulting in the characteristic sloshing phase shift of $\unicode[STIX]{x1D70F}_{osc}/2$ ; see data by Qiu & Tong (Reference Qiu and Tong2001) and Xi et al. (Reference Xi, Zhou, Zhou, Chan and Xia2009) in water at $Pr\sim 5.3$ ; and (iii) correlation of the mean LSC orientation at half-height and the top or bottom flow orientation yielding a phase shift of $\unicode[STIX]{x1D70F}_{osc}/4$ ; see measurements by Zhou et al. (Reference Zhou, Xi, Zhou, Sun and Xia2009) in water at $Pr=5.3$ . These results are consistent with our data: (i) and (ii) are equivalent to the sum of two successive phase shifts in figure 6 (e.g. Top $\rightarrow$ Cold plus Cold $\rightarrow$ Bottom) and (iii) is equal to an average of two successive phase shifts in figure 6. Each of these analysis steps averages out the deviations of individual time lags from $\unicode[STIX]{x1D70F}_{osc}/4$ and results in phase shifts of $\unicode[STIX]{x1D70F}_{osc}/2$ for (i) and (ii), and $\unicode[STIX]{x1D70F}_{osc}/4$ for (iii). All these data show that the torsion and sloshing modes are present for a wide range of Prandtl numbers. However, whether the specific time lags found in our system are a general feature needs to be clarified in the future.

Figure 7.

Averaged and normalized horizontal UDV velocity profiles along the LSC at (a) the top and (b) the bottom plate for different $Ra$ . The experiments correspond to the ones shown in figure 5(a).

The deviations of the individual lags from a quarter oscillation period indicate that the LSC path is asymmetric. Such distortions might be caused by corner vortices of the LSC which are observed in simulations. Here, we introduce a horizontal axis  $\unicode[STIX]{x1D709}$ that is aligned in the direction of the LSC orientation. To calculate an averaged horizontal velocity profile of the LSC at the top (bottom) plate, we select the velocity profiles of the UDV sensors $T_{0}$ , $T_{45}$ and $T_{90}$ ( $B_{0}$ , $B_{45}$ and $B_{90}$ ) at time instants when the LSC orientation $\unicode[STIX]{x1D703}_{top}$ ( $\unicode[STIX]{x1D703}_{bot}$ ) is aligned within $\pm 5^{\circ }$ with the azimuthal position of a sensor. The averages of those velocity profiles $\bar{v}_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D709})$ are normalized by their maximum magnitude and shown in figure 7 for three different $Ra$ . It should be noted that the noisy or missing profiles for $\unicode[STIX]{x1D709}>80$  mm are due to an inaccessible zone close to the UDV sensors caused by the ringing of the piezo-crystal in the transducer. The velocities at the top plate are predominantly positive (flow to the right) and the bottom velocities are negative (flow to the left). Inverted velocities can only be seen at the positions where the up- and down-welling flows are impinging on the plates (left at the top, right at the bottom). These profiles indicate indeed the presence of recirculation vortices in the cell corners, which seem to become smaller with increasing $Ra$ . This challenges our supposition at the beginning of this paragraph that corner vortices might be responsible for the different time lags in figure 6(c). With increasing Rayleigh number the fluid turbulence becomes more vigorous. This means that the vortical structures are still there, but get averaged out more effectively.

A further explanation for the phase-shift deviations from the expected value of $\unicode[STIX]{x1D70F}_{osc}/4$ could be traced to a varying LSC velocity. The fluid would have to be slower when leaving the plates (Top $\rightarrow$ Cold and Bottom $\rightarrow$ Hot) and accelerate while approaching the opposite plate (Cold $\rightarrow$ Bottom and Hot $\rightarrow$ Top). This point has to be left for future studies on this subject.

4.1 Heat transport

This section discusses the global transport properties in the liquid metal convection flow and compares the results with other experiments and simulations. We first analyse the turbulent heat transport in the experiment. The heat flux through the fluid layer is characterized by the Nusselt number  $Nu$ which is calculated at the cooled plate, $Nu=\dot{Q}/\dot{Q}_{cond}$ . Here, $\dot{Q}_{cond}=\unicode[STIX]{x1D706}\unicode[STIX]{x03C0}R^{2}\unicode[STIX]{x0394}T/H$ is the purely conductive heat flux. The data are plotted over $Ra$ in figure 8 and we find $Nu\simeq (0.12\pm 0.04)Ra^{0.27\pm 0.02}$ . This agrees excellently with measurements by Cioni et al. (Reference Cioni, Ciliberto and Sommeria1997) in mercury ( $Pr=0.025$ ) and matches the numerical results of Scheel & Schumacher (Reference Scheel and Schumacher2017) of $Nu\simeq (0.13\pm 0.04)Ra^{0.27\pm 0.01}$ for $Pr=0.021$ with only a small shift.

Figure 8.

Turbulent heat transfer in liquid metal convection flow. The scaling of $Nu$ versus $Ra$ is displayed. The Nusselt number $Nu$ is determined from the temperature difference of the cooling water for $T_{out}-T_{in}>0.2$  K. Our data are compared with a numerical simulation (circles), four laboratory experiments and the GL theory at $Pr=0.029$ , as indicated in the legend.

The results of Takeshita et al. (Reference Takeshita, Segawa, Glazier and Sano1996) in mercury ( $Pr=0.024$ ) agree with respect to the scaling exponent, but give a somewhat higher Nusselt number magnitude with $Nu\simeq 0.155Ra^{0.27\pm 0.02}$ . King & Aurnou (Reference King and Aurnou2013, Reference King and Aurnou2015) found the same magnitude of $Nu$ in gallium ( $Pr\sim 0.025$ ), but their scaling exponent is smaller, $Nu\simeq (0.19\pm 0.01)Ra^{0.249\pm 0.004}$ . All of the above experiments and simulations were conducted in cylindrical cells with $\unicode[STIX]{x1D6E4}=1$ . Finally, Glazier et al. (Reference Glazier, Segawa, Naert and Sano1999) conducted experiments in mercury ( $Pr\sim 0.025$ ) for multiple aspect ratios. Their data at $\unicode[STIX]{x1D6E4}=1$ are closer to the results of Takeshita et al. (Reference Takeshita, Segawa, Glazier and Sano1996) in magnitude (see figure 8). A least-squares fit gives a slower scaling $Nu\propto Ra^{0.242\pm 0.002}$ in the considered range. Including all their data for $\unicode[STIX]{x1D6E4}=0.5$ and 2, they found a scaling $Nu\propto Ra^{0.29\pm 0.01}$ over a large range of $10^{5}<Ra<10^{11}$ . The deviation of their $\unicode[STIX]{x1D6E4}=1$ data from this exponent is attributed to a strong bulk circulation.

The Grossmann–Lohse (GL) theory prediction (Stevens et al. Reference Stevens, van der Poel, Grossmann and Lohse2013) is in very good agreement with our data as seen by the grey line in figure 8. It can also be seen that the GL theory predicts an increase in the exponent of $Nu(Ra)$ for higher $Ra$ . At smaller Rayleigh numbers ( $Ra\lesssim 5\times 10^{5}$ ), Rossby (Reference Rossby1969) found $Nu\simeq 0.147Ra^{0.257\pm 0.004}$ for $Pr=0.025$ and $\unicode[STIX]{x1D6E4}\sim 22$ and 7.4. Kek & Müller (Reference Kek and Müller1993) reported $Nu\simeq 0.20Ra^{0.20}$ at $Pr=0.006$ and $\unicode[STIX]{x1D6E4}\sim 11$ .

4.2 Momentum transport

The turbulent momentum transport of the convection flow is quantified by the Reynolds number  $Re$ . In contrast to the quantification of the turbulent heat transfer by a Nusselt number, the Reynolds number is not uniquely defined since different characteristic velocities can be employed for its definition. With the multiple probes at hand, we can define three different $Re$ with three different corresponding characteristic velocities: (1) the typical horizontal velocity magnitude near the plates, (2) the typical vertical velocity magnitude of the LSC along the sidewall and (3) the turbulent velocity fluctuations in the centre of the cell,

(4.1a-c ) $$\begin{eqnarray}Re_{LSC}=\frac{v_{LSC}H}{\unicode[STIX]{x1D708}},\quad Re_{vert}=\frac{v_{vert}H}{\unicode[STIX]{x1D708}}\quad \text{and}\quad Re_{centre}=\frac{v_{centre}H}{\unicode[STIX]{x1D708}}.\end{eqnarray}$$

This circumstance gives the opportunity to directly compare the sensitivity of the scaling exponent with respect to these different characteristic velocities.

First, the large-scale flow is characterized by the velocity magnitude $v_{LSC}$ as done in § 3.2. The resulting Reynolds number $Re_{LSC}=v_{LSC}H/\unicode[STIX]{x1D708}$ is plotted in figure 9 (filled circles). A power-law fit reveals a scaling of $Re_{LSC}\simeq (8.0\pm 4.4)Ra^{0.42\pm 0.03}$ . The exponent is close to the scaling of $f_{to}$ and $f_{osc}$ in figure 5(b) since $Re_{LSC}=f_{to}\unicode[STIX]{x03C0}H^{2}/\unicode[STIX]{x1D708}$ .

Second, the vertical velocity of the LSC is measured at radial position $r/R=0.8$ by the UDV sensor $V_{0}$ . Due to the sloshing mode, the up- or down-welling flows move periodically towards and away from the sensor measuring volume (see figure 9 b). Additionally, the LSC is slowly rotating as a whole. Since a pronounced vertical flow of the LSC is of interest only, we estimate the characteristic vertical velocity by calculating the average velocity profile $v_{z}(t)$ over an interval $H/4$ centred around the mid-plane for every time step, similar to (3.1). The corresponding standard deviation ( $\text{std}$ ) is added to the average of the resulting velocity magnitude to accommodate for the fluctuations of this signal. Thus, $v_{vert}=\langle |v_{z}(t)|\rangle _{t}+\text{std}(|v_{z}(t)|)$ . This velocity gives a vertical Reynolds number $Re_{vert}=v_{vert}H/\unicode[STIX]{x1D708}$ . The values are shown in figure 9(a) as triangles and give a scaling of $Re_{vert}\simeq (9.5\pm 9.4)Ra^{0.42\pm 0.04}$ .

Figure 9.

Analysis of the turbulent momentum transfer in liquid metal convection flow based on the present UDV measurements. (a) Scaling of Reynolds number  $Re$ with Rayleigh number  $Ra$ . Uncertainties of the present measurements were removed for better visibility. Their relative values are about 20 % to 30 % due to the turbulent fluctuations. (b) Colour-plot of $v_{z}$ measured by UDV sensor $V_{0}$ for $Ra=6\times 10^{7}$ . The data correspond to the measurement shown in figure 3.

Finally, the turbulent velocity fluctuations are considered in the centre of the cell. Here, the three crossing UDV sensors $M_{0}$ , $M_{90}$ and $V_{c}$ can measure all three components of the velocity vector $\boldsymbol{v}_{centre}$ (see figure 1 b). The components are determined by taking the root-mean-square (r.m.s.) value of, again, the central $H/4$ interval of the velocity profiles. The r.m.s. time average of the velocity magnitude $v_{centre}=\langle (\boldsymbol{v}_{centre}(t))^{2}\rangle _{t}^{1/2}$ is used to calculate $Re_{centre}=v_{centre}H/\unicode[STIX]{x1D708}$ . It is plotted as diamonds in figure 9(a), along with the power-law fit of $Re_{centre}\simeq (3.1\pm 3.2)Ra^{0.46\pm 0.04}$ .

The scaling exponent of $Re_{centre}(Ra)$ agrees very well with the result of DNS by Scheel & Schumacher (Reference Scheel and Schumacher2017) (open circles in figure 9 a). In the numerical simulations, the Reynolds number was calculated from the r.m.s. velocity over the whole cell for $Pr=0.021$ and gave a scaling of $Re\simeq (6.5\pm 0.6)Ra^{0.45\pm 0.01}$ . The absolute values of $Re_{centre}$ are about half as large as the results of the DNS, since the average over the whole cell volume also includes the high-velocity components of the LSC outside the centre region. The absolute values of $Re_{LSC}$ and $Re_{vert}$ match the DNS results more closely, but have a somewhat smaller exponent of 0.42.

How do our results compare to previous laboratory experiments? Measurements of a vertical Reynolds number in mercury by Takeshita et al. (Reference Takeshita, Segawa, Glazier and Sano1996) were also taken at half-height and $r/R=0.8$ . They show a higher velocity magnitude with a scaling of $Re\simeq 6.24Ra^{0.46\pm 0.02}$ . In the $\unicode[STIX]{x1D6E4}=2$ case, Vogt et al. (Reference Vogt, Horn, Grannan and Aurnou2018a ) found an increased scaling exponent for a horizontal $Re$ at the cell centre: $Re\simeq 5.662Ra^{0.483}$ . Comparing all these results underlines the dependence of the momentum transport on the specific velocity that enters the Reynolds number definition. The somewhat smaller scaling exponent for the horizontal and vertical LSC velocities in comparison to previous experiments or the DNS can thus be considered as a result of probing different parts of the complex three-dimensional flow structure that we analysed before, as well as using different measurement techniques and procedures of calculating the characteristic velocities. Interestingly, the turbulent fluctuations seem to be the best indication of the global momentum transport scaling as reported by DNS, albeit being smaller in their absolute magnitude.

We also show the Reynolds number predicted by the GL theory (Stevens et al. Reference Stevens, van der Poel, Grossmann and Lohse2013) in figure 9 a. Compared to $Re_{LSC}$ and $Re_{vert}$ , the GL theory underpredicts the Reynolds number by up to a factor of 2. Stevens et al. (Reference Stevens, van der Poel, Grossmann and Lohse2013) used here a measurement at $Pr=5.4$ to fix the values of $Re$ . A least-squares power-law fit of the GL results in figure 9 a gives an exponent of $Re\propto Ra^{0.44}$ , which lies within the accuracy of our experimental data.