2.1. Experimental set-up

The set-up employed for the experiments is almost identical to that used in previous studies (Tasaka et al. Reference Tasaka, Igaki, Yanagisawa, Vogt, Zuerner and Eckert2016; Vogt et al. Reference Vogt, Ishimi, Yanagisawa, Tasaka, Sakuraba and Eckert2018b; Akashi et al. Reference Akashi, Yanagisawa, Tasaka, Vogt, Murai and Eckert2019; Tasaka et al. Reference Tasaka, Yanagisawa, Fujita, Miyagoshi and Sakuraba2021; Vogt et al. Reference Vogt, Yang, Schindler and Eckert2021; Yang, Vogt & Eckert Reference Yang, Vogt and Eckert2021) to which the reader is referred for a detailed description. A noteworthy difference from our previous study (Akashi et al. Reference Akashi, Yanagisawa, Tasaka, Vogt, Murai and Eckert2019) is that additional measuring positions for the velocity sensors have been placed at the mid-height of the fluid container in order to better compare the flow structures found here with the results reported by Vogt et al. (Reference Vogt, Horn, Grannan and Aurnou2018a). Figure 1 shows schematics of the container and the arrangement of the ultrasonic measurement lines. The fluid layer has a square horizontal cross-section of $L \times L = 200 \times 200\ \textrm {mm}^{2}$ and a height $H$ of 40 mm, resulting in an aspect ratio of $\varGamma = 5$. In the vertical direction the fluid layer is bordered by two copper plates, whose temperatures, $T_{bot}$ and $T_{top}$, are kept constant by water circulation in branched channels inside the plates. The water temperatures are controlled by external thermostats. The vertical temperature difference $\Delta T=T_{bot}-T_{top}$ is varied between 3.0 and 15.5 K corresponding to the $Ra$ range from $6.7 \times 10^{4}$ to $3.5 \times 10^{5}$. The sidewalls are made of electrically non-conducting polyvinyl chloride. The entire fluid layer is encased in 30 mm closed-cell foam to minimize heat loss. The working fluid is the eutectic alloy $\textrm {Ga}^{67}\textrm {In}^{20.5}\textrm {Sn}^{12.5}$ that is liquid at room temperature. Temperature-dependent data of the thermo-physical properties can be found in Plevachuk et al. (Reference Plevachuk, Sklyarchuk, Eckert, Gerbeth and Novakovic2014). In particular, for a temperature of $25\,^{\circ } \textrm {C}$ the following values are reported: density $\rho = 6360\ \textrm {kg}\ \textrm {m}^{-3}$, thermal expansion coefficient $\alpha = 1.24 \times 10^{-4}\ \textrm {K}^{-1}$, kinematic viscosity $\nu = 3.4 \times 10^{-7}\ \textrm {m}^{2} \ \textrm {s}^{-1}$ and thermal diffusivity $\kappa = 1.03 \times 10^{-5} \textrm {m}^{2}\ \textrm {s}^{-1}$. This gives a Prandtl number of $Pr = 0.033$. The thermal diffusion time for the fluid layer, $t_\kappa = H^{2}/\kappa$, is $155$ s. Table 1 presents the $Ra$ investigated in the experiments together with the corresponding values for the free-fall velocity $u_{ff}= \sqrt {g\alpha \Delta TH}$ and the free-fall time $t_{ff} =\sqrt {H/(g\alpha \Delta T)}$ resulting from the respective temperature difference $\Delta T$.

Figure 1. Experimental set-up and arrangement of measurement lines: (a) top view and (b) side view, where light blue lines indicate the ultrasonic measurement lines.

Table 1. Parameters for the experiments including the Rayleigh number $Ra$, the Prandtl number $Pr$, the free-fall velocity $u_{ff}$ and the free-fall time $t_{ff}$.

Due to the limited possibilities for measuring fluid velocities in liquid metals, the majority of previous experimental studies of liquid metal convection were largely restricted to the description of heat transfer in the system and the measurement of temperatures and their fluctuations (Takeshita et al. Reference Takeshita, Segawa, Glazier and Sano1996; Cioni, Cilberto & Sommeria Reference Cioni, Cilberto and Sommeria1997; Segawa, Naert & Sano Reference Segawa, Naert and Sano1998; Horanyi, Krebs & Müller Reference Horanyi, Krebs and Müller1999; Burr & Müller Reference Burr and Müller2001). The development of the ultrasonic Doppler technique for flow measurements in liquid metals (Takeda Reference Takeda1987; Eckert & Gerbeth Reference Eckert and Gerbeth2002) enables an experimental determination of flow structures in low-$Pr$ liquid metal convection (Cramer, Zhang & Eckert Reference Cramer, Zhang and Eckert2004; Mashiko et al. Reference Mashiko, Tsuji, Mizuno and Sano2004; Yanagisawa et al. Reference Yanagisawa, Yamagishi, Hamano, Tasaka, Yoshida, Yano and Takeda2010,  Reference Yanagisawa, Hamano, Miyagoshi, Yamagishi, Tasaka and Takeda2013; Tasaka et al. Reference Tasaka, Igaki, Yanagisawa, Vogt, Zuerner and Eckert2016; Zürner et al. Reference Zürner, Schindler, Vogt, Eckert and Schumacher2019; Tasaka et al. Reference Tasaka, Yanagisawa, Fujita, Miyagoshi and Sakuraba2021; Vogt et al. Reference Vogt, Yang, Schindler and Eckert2021; Yang et al. Reference Yang, Vogt and Eckert2021). This powerful technique was successfully used in the previous study by Akashi et al. (Reference Akashi, Yanagisawa, Tasaka, Vogt, Murai and Eckert2019) to identify the individual flow regimes and is also employed here. It provides instantaneous profiles of the velocity component projected onto each measurement line, $u_x(x,t)$ and $u_y(y,t)$, respectively. The spatial resolution of the velocity measurements is 1.4 mm in the direction of the ultrasonic beam line and about 5 to 8 mm in the direction perpendicular to it. The velocity resolution is about $0.5\ \textrm {mm} \ \textrm {s}^{-1}$ and a temporal resolution of 0.6 s is achieved. The measurement instrument (DOP 3010, Signal Processing SA) is applied in combination with ultrasonic transducers having a basic frequency of 8 MHz and a piezo-element with an active diameter of 5 mm. These sensors, which are mounted horizontally inside holes drilled into the sidewalls of the container, are in direct contact with the liquid metal. In general, velocity probes can be installed at sidewalls at different heights. In the present study, measurement results are obtained along three measurement lines (shown as light blue lines in figure 1), designated as $Middle \textit {1}$, $Middle \textit {2}$ and $Bottom$. $Middle \textit {1}$ and $Middle \textit {2}$ are at the mid-height of the container ($z = 0.5H$) and $Bottom$ is situated close to the bottom plate ($z = 0.25H$). The velocity profiles along each measurement line are acquired sequentially by multiplexing.

2.2. Numerical scheme

For the numerical simulation, a Boussinesq fluid in a cuboid container with a square horizontal cross-section and an aspect ratio of $\varGamma = 5$ is considered. Here, we used the non-magnetic version of a code originally developed for magnetohydrodynamic simulations in Yanagisawa, Hamano & Sakuraba (Reference Yanagisawa, Hamano and Sakuraba2015). Cartesian coordinates $(x, y, z)$ are used with the $z$ axis in the upward direction and the origin located at one of the bottom corners of the container. We solve the governing equations for the non-dimensional velocity ${\boldsymbol {u}}$, temperature $T$ and pressure $p$ as follows:

(2.1a,b)\begin{gather} \frac{\partial {\boldsymbol{u}}}{\partial t}={-}({\boldsymbol{u}}\boldsymbol{\cdot}\boldsymbol{\nabla}){\boldsymbol{u}}-\boldsymbol{\nabla} p+PrRa ({{T}}-\bar{T}(z)) {\boldsymbol{k}}_z+Pr\nabla^{2}{\boldsymbol{u}} , \quad \bar{T}(z)=1-z , \end{gather}

(2.2)\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot}{\boldsymbol{u}}=0 , \end{gather}

(2.3)\begin{gather}\frac{\partial T}{\partial t}={-}({\boldsymbol{u}}\boldsymbol{\cdot}\boldsymbol{\nabla})T+\nabla^{2} T , \end{gather}

where ${\boldsymbol {k}}_z$ is the unit vector in the $z$ direction. Length and time are respectively non-dimensionalized by the layer thickness $H$ and the thermal diffusion time $t_\kappa =H^{2}/\kappa$. Accordingly, the velocity is normalized by $\kappa /H$. The temperature satisfies the isothermal boundary conditions at the top ($T_{top} = 0$) and bottom ($T_{bot} = 1$), while adiabatic conditions are assumed for the sidewalls. No-slip conditions are applied for the velocity at all boundaries. The second-order-accurate staggered-grid finite-difference method is used with a uniform grid interval in each direction. The time integration was carried out explicitly by means of the third-order Runge–Kutta method. This procedure modifies the equation of continuity to

(2.4)\begin{equation} \varepsilon^{2}{\partial p}/{\partial t}={-}\boldsymbol{\nabla}\boldsymbol{\cdot}{\boldsymbol{u}}, \end{equation}

where $\varepsilon$ is a small parameter from Chorin (Reference Chorin1967). In our numerical simulations a value of $\varepsilon =0.002$ is used (Yanagisawa et al. Reference Yanagisawa, Hamano and Sakuraba2015). The code is parallelized with MPI in combination with OpenMP.

To check the resolution of the simulation, we compared results from calculations made with different grid resolutions: $n_z = 80$, 128, 256 and $512$ (see table 2). For these pre-calculations, the parameters are set as $Pr = 0.025$, which is a representative value for liquid metals, and $Ra = 1.0\times 10^{5}$, $3.0\times 10^{5}$ and $6.0\times 10^{5}$. All simulations performed were able to reconstruct the same qualitative characteristics of the flow field: a cellular pattern with quasi-periodic oscillations. Quantitative comparisons are carried out using the values of the Nusselt number and the Reynolds number which were averaged both temporally and spatially. In our equations $\langle \cdot \rangle _{S, t}$ stands for the time–surface average on the top or bottom boundary and $\langle \cdot \rangle _{V, t}$ stands for the time–volume average. Nusselt numbers are calculated both at the top boundary $Nu_{top}$ and at the bottom boundary $Nu_{bot}$:

(2.5a,b)\begin{equation} Nu_{top}={-}\left \langle \left (\frac{\partial T}{\partial z} \right)_{z=1} \right \rangle_{S, t}, \quad Nu_{bot}={-}\left \langle \left(\frac{\partial T}{\partial z} \right)_{z=0} \right \rangle_{S, t}. \end{equation}

An alternative way to determine the Nusselt number is to integrate the upward heat transport in the entire volume (e.g. Pandey et al. Reference Pandey, Scheel and Schumacher2018). For comparison we also use this definition for the Nusselt number $Nu_{vol}$:

(2.6)\begin{equation} Nu_{vol}=1 +\langle u_z T \rangle_{V, t}. \end{equation}

Table 2. Pre-tests of numerical simulations conducted with different numbers of grid points: the Rayleigh number $Ra$, the Prandtl number $Pr$, the number of grid points $n_x$, $n_y$ and $n_z$, the time-averaged Nusselt number $Nu$, the standard deviation of Nusselt number $Nu_{std}$, the number of grid points in the thermal boundary layer $\lambda _{\varTheta }$ estimated by the relation $Nu = 1/{(2\lambda _{\varTheta })}$, $n_{\lambda _{\varTheta }}$, the time-averaged Reynolds number $Re$, the standard deviation of Reynolds number $Re_{std}$ and the ratio of the Kolmogorov scale $\eta _{K}$ to the spatial resolution $l_z$.

When thermally balanced states are achieved after adequate time integrations, applying (2.5a,b) and (2.6) to our data yields a very good agreement. Therefore, in the further course of the paper only the values of $Nu_{top}$ will be presented as $Nu$.

The Reynolds number is calculated using the root-mean-square (r.m.s.) velocities $U_{rms}$ determined for the entire volume as characteristic velocity scale:

(2.7)\begin{equation} Re = \frac{U_{rms}}{Pr} \quad \mbox{with} \ U_{rms}=\sqrt{\langle {u_x}^{2}+{u_y}^{2}+{u_z}^{2} \rangle_{V, t}}. \end{equation}

The quantitative comparison of the pre-tests is shown in table 2. The agreement of the time-averaged Nusselt numbers $Nu$ is satisfactory within 0.4 %, and the Reynolds numbers $Re$ are almost the same within 0.8 %. As is shown below, quasi-periodicity is a dominant feature of the pattern; hence, fluctuations of the Nusselt number are also important. The fluctuations are evaluated in the form of the standard deviation and are listed in table 2 as $Nu_{std}$.

In addition, we checked the spatial resolution of our simulations by comparing them with the Kolmogorov scale $\eta _{K}$. Using the results of $Nu$ for given values of $Pr$ and $Ra$, $\eta _{K}$ can be estimated as (Scheel, Emran & Schumacher Reference Scheel, Emran and Schumacher2013)

(2.8)\begin{equation} \eta_{K} =\left(\frac{Pr^{2}}{(Nu-1)Ra}\right)^{{1}/{4}}. \end{equation}

Table 2 contains the ratio of $\eta _{K}$ to the spatial resolution $l_z= 1/n_z$. A ratio larger than one indicates a sufficient resolution in terms of direct numerical simulation; hence, these values are used as references. As shown in table 2, simulations with the highest grid resolution satisfy this criterion for the $Ra$ range considered here. Calculations of lower resolution are conducted in the case of long simulation times. Due to the high diffusivity of temperature, we assume that this is nevertheless sufficient to adequately evaluate $Nu$, $Re$, temperature distribution and flow pattern. Our independence grid study shows that the values of the quantities we focus on in this paper do not change significantly on a finer grid (see above), so we can assume their validity. Higher resolutions of the grid are necessary if details within the velocity boundary layers are to be examined, but this is outside the scope of this study.

Based on these comparisons and considering available computational resources, we performed simulations mainly with the resolution $n_z=128$. To check the long-term stability of the quasi-periodic behaviour and to perform long-time averaging over 100 thermal diffusion times (${\sim }300$ oscillations), we utilized the resolution of $n_z=80$. The parameters chosen for the numerical simulations in this study are listed in table 3. The numerical simulations cover the parameter range that is investigated in the experiments.

Table 3. Parameters for the numerical simulations of the experimental configuration including the Rayleigh number $Ra$, the number of grid points $n_x$, $n_y$ and $n_z$, the free-fall velocity $u_{ff}$ and the free-fall time $t_{ff}$.

3.1. Three-dimensional cellular regime

Previous experiments carried out by Akashi et al. (Reference Akashi, Yanagisawa, Tasaka, Vogt, Murai and Eckert2019) using the same set-up show a clear and reproducible dependence on $Ra$ with respect to the occurrence of different flow regimes. While the range of smaller $Ra$ is associated with various regimes of unsteady convection rolls, at $Ra$ above $6.5\times 10^{4}$ a new, stable flow pattern occurs, which is characterized by a high symmetry with respect to the square base area of the fluid container. This newly discovered flow pattern was designated as the cellular flow regime. We are aware that the term of a cellular flow structure in thermal convection is often associated with stationary cell patterns observed, for example, in the slightly supercritical state after the onset of convection. This is explicitly not the case here. Accompanying measurements of the temperature fluctuations indicate that the stage of developed thermal turbulence is reached in the $Ra$ range where the cellular regime occurs (Akashi et al. Reference Akashi, Yanagisawa, Tasaka, Vogt, Murai and Eckert2019). Our point of view is supported by results from Bailon-Cuba et al. (Reference Bailon-Cuba, Emran and Schumacher2010) who found extended convection rolls and pentagon-like cells in their direct numerical simulation data for a turbulent flow in a $\varGamma = 12$ cylinder at $Ra = 10^{7}$ and $Pr = 0.7$. After filtering out the small-scale turbulence, these resulting patterns of turbulent convection resemble the weakly nonlinear regime just above the onset of convection. However, the observation of the cellular regime by Akashi et al. (Reference Akashi, Yanagisawa, Tasaka, Vogt, Murai and Eckert2019) is also surprising because the appearance of such cell-like structures is anticipated to be more prevalent in fluids with larger $Pr \gtrsim 1$, while the convection pattern at low $Pr$ is supposed to be dominated by roll-like structures (Breuer et al. Reference Breuer, Wessling, Schmalzl and Hansen2004; Pandey et al. Reference Pandey, Scheel and Schumacher2018). In order to analyse the cellular flow regime in more detail, the investigations in this study are focused on the $Ra$ range $6.7 \times 10^{4} \leqslant Ra \leqslant 3.5 \times 10^{5}$. The upper limit of $Ra$ is determined by the technical capabilities of the experimental equipment.

Figure 2(a) shows the spatio-temporal map of the velocity component $u_x$ at $Ra=1.2\times 10^{5}$ recorded by the ultrasonic sensor $Bottom$ at a height of $z = 0.25 H$. A negative velocity (blue) indicates a flow toward the transducers, while a positive velocity (red) is associated with the direction away from the sensor. In all results presented in the following, the velocity is given dimensionless with respect to the free-fall velocity $u_{ff}= \sqrt {g\alpha \Delta T H}$. In the convection cell considered here, the free-fall velocity reaches a value of $16.48\ \textrm {mm}\ \textrm {s}^{-1}$ at $Ra=1.2\times 10^{5}$. The dimensionless time is given in units of the free-fall time $t_{ff} =\sqrt { H/(g\alpha \Delta T)} = 2.43$ s.

Figure 2. (a) Spatio-temporal velocity map at $Ra=1.2\times 10^{5}$ recorded for $u_{x}$ along the $x$ direction (sensor $Bottom$ at $z = 0.25H$). (b) A schematic illustration of the cell structure with the sensor positions.

The velocity map obtained by the sensor $Bottom$ (figure 2a) reproduces a flow structure almost identical to those presented in the recent study by Akashi et al. (Reference Akashi, Yanagisawa, Tasaka, Vogt, Murai and Eckert2019). The flow pattern divides the measured profile into two parts with respect to the centre of the container. The strength of the measured velocity is subject to strong quasi-periodic fluctuations, which at some points in time (e.g. $t/t_{ff} = 100, 150, 270$) even lead to short-term reversals of the flow direction. If the average behaviour of the flow is considered, then a fluid motion from the sidewall towards the centre is clearly dominant, which means that an ascending flow in the centre of the cell can be assumed. Based on the evaluation of the measured velocity profiles along six different measurement lines within a horizontal plane above the heated bottom, Akashi et al. (Reference Akashi, Yanagisawa, Tasaka, Vogt, Murai and Eckert2019) obtained a rough reconstruction of the time-averaged velocity patterns in the convection cell. For the case of the cellular regime, they revealed a three-dimensional structure having upwelling flows at the centre and the four corners of the container as shown in figure 2(b) (Akashi et al. Reference Akashi, Yanagisawa, Tasaka, Vogt, Murai and Eckert2019). Once this flow pattern is established, it is stable for long measurement times of the order of a few thousand free-fall times and does not change spontaneously. Repeating the measurements several times, inverse flow patterns with a downward flow in the centre of the fluid container were also observed. These structures, which differ principally in the sign of the velocity, have similar probabilities of occurrence. Which variant is realized in the experiment apparently depends only on random asymmetries in the initial conditions.

Inspired by the work of Vogt et al. (Reference Vogt, Horn, Grannan and Aurnou2018a), here we additionally monitor the velocities in the horizontal centre plane of our container. If the convection pattern corresponds to a single circulation, which fills the space between the copper plates, one would expect that in the measuring plane at the mid-height the vertical component clearly dominates and only marginal parts of a horizontal flow can be found here. As shown in figure 3(b,c), it is surprising to observe pronounced flows also in the centre plane, their intensity only slightly below that detected along the measuring lines at $z = 0.25H$ and $z = 0.75H$. Moreover, these velocity plots are characterized by the same pronounced oscillations. In contrast to the measurement at $z = 0.25H$, however, an inversion of the flow direction occurs in every period where no dominance of one of the two directions can be determined. Figure 3 also contains a corresponding dataset from Vogt et al. (Reference Vogt, Horn, Grannan and Aurnou2018a) (see figure 3d,e). For better comparability, the time axis and the velocity are normalized with the free-fall time and the free-fall velocity, respectively. The comparison of both flow patterns shows striking similarities with respect to the manifestation of the dominant oscillations in terms of both the amplitude of the velocity and the time scale of the oscillations. This raises the question as to whether we are dealing with the same phenomenon in the $\varGamma =5$ box that is referred to as JRV and has already been observed in the $\varGamma =2$ cylinder. To answer this question, we take a closer look at the flow structure and its changes during the oscillations. Since the experiment reaches its limits here due to the small numbers of measuring sensors, we performed numerical simulations in parallel.

Figure 3. Spatio-temporal velocity distributions measured at the mid-height of the fluid container for $Ra = 1.2 \times 10^{5}$. (a) Sketch of the measuring line in the $\varGamma = 5$ box ($Pr = 0.033$). (b) Velocity pattern for the the full-length measuring line. (c) The same dataset as in (b) plotted for only one half of the measuring line. (d) Sketch of the measuring line in the $\varGamma = 2$ cylinder ($Pr = 0.027$). (e) Velocity pattern in the $\varGamma = 2$ cylinder. (Adapted from Vogt et al. Reference Vogt, Horn, Grannan and Aurnou2018a.)

Figure 4(a,b) contains the numerical counterpart of the measurements as presented in figures 2 and 3. The qualitative agreement of the flow patterns proves the reliability of the numerical approach. The numerical velocity distributions are also dominated by strong oscillations, with periodic changes of direction in the centre plane of the container.

Figure 4. Results of the numerical simulations showing a reconstruction of the spatio-temporal velocity maps at $Ra=1.2\times 10^{5}$ in the cuboid container for (a) $u_x$ according to the measuring line of sensor $Bottom$ at $z = 0.25$ and (b) $u_y$ according to the measuring line of sensor $Middle \textit {1}$ at $z = 0.5$. The measurement positions are shown in figure 2(b).

Snapshots of temperature isosurfaces and velocity distributions in figure 5 provide a three-dimensional visualization of the convection pattern. Figures 5(a) and 5(b) display instantaneous isosurfaces of the temperature for the values $T = 0.2$ and $T = 0.8$. Figures 5(c) and 5(d) show the instantaneous distribution of the vertical velocity $u_z$ at different heights of $z = 0.75$ and $z = 0.25$. The colour indicates the direction of flow, meaning that blue (red) signifies a downwelling (upwelling) motion. The temperature surface appears to be smoother than the velocity fields, which is naturally due to the fact that low-$Pr$ fluids feature a high thermal diffusivity. Apart from that, temperature field and vertical velocity show an evident conformity. Previous studies of turbulent superstructures raised the question as to what extent the temperature distribution and the vertical velocity structure represent the same coherent structure (Pandey et al. Reference Pandey, Scheel and Schumacher2018; Stevens et al. Reference Stevens, Blass, Zhu, Verzicco and Lohse2018; Krug et al. Reference Krug, Lohse and Stevens2020). Sakievich et al. (Reference Sakievich, Peet and Adrian2016) showed that the thermal updrafts and downdrafts match the three-dimensional field to a high degree in a $\varGamma =6.3$ cylinder. Further analysis by Sakievich et al. (Reference Sakievich, Peet and Adrian2020) revealed a high correlation in the temporal dynamics of the low-order Fourier modes of temperature and vertical velocity. Based on this clear similarity and the findings of Krug et al. (Reference Krug, Lohse and Stevens2020), Sakievich et al. (Reference Sakievich, Peet and Adrian2020) hypothesize similar principles of spatial organization of the coherent patterns in a $\varGamma =6.3$ cylinder and superstructures occurring in larger aspect ratios.

Figure 5. Results of the numerical simulations performed for $Ra=1.2\times 10^{5}$ representing snapshots of the temperature isosurface for (a) $T=0.2$ and (b) $T=0.8$, and the distribution of the vertical velocity $u_z$ at the height of (c) $z = 0.75$ and (d) $z = 0.25$. The colour scale of $u_z$ is the same as that in figure 4. See also supplementary movies 1–4 available at https://doi.org/10.1017/jfm.2021.996.

The temperature plots in figure 5 show that the cellular structure is characterized by an ascending flow in its centre and at the four corners of the container (see figure 5b), while the shape of the areas where the fluid descends resembles a rhombus (see figure 5a). It can also be seen that the zones of ascending warm fluids in the middle and the four corners are connected by diagonally running ridges (figure 5b). This rough characterization of the flow structure is consistent with the distributions of the vertical velocity component in figures 5(c) and 5(d). The appearance of finer structures in the velocity contours illustrates the character of inertia-dominated fluid turbulence and is in line with previous studies (Pandey et al. Reference Pandey, Scheel and Schumacher2018; Vogt et al. Reference Vogt, Horn, Grannan and Aurnou2018a; Krug et al. Reference Krug, Lohse and Stevens2020; Sakievich et al. Reference Sakievich, Peet and Adrian2020) which demonstrated the occurrence of more small-scale velocity fluctuations compared with the temperature field. Thus, it can be stated that the results of the numerical simulations confirm the mean flow structure of the cellular regime suggested from the experiment based on a limited number of velocity sensors (Akashi et al. Reference Akashi, Yanagisawa, Tasaka, Vogt, Murai and Eckert2019). A striking property of this cellular convection is the occurrence of pronounced periodic oscillations, which consequently must be reflected in some form of periodic changes of the flow topology.

3.2. Dominant oscillation frequencies

The dominant frequency $f_{OS}$ of the periodic oscillations in the spatio-temporal velocity maps is determined by calculating the spatially averaged power spectral densities (PSDs) from velocity time series. The velocity time series are divided into multiple time intervals of 60 free-fall times. The spectra are calculated by fast Fourier transform for each time interval and for each measurement point. These interim results are finally averaged both spatially along the respective measuring line and temporally over the duration of the measurement.

Figures 6(a) and 6(b) present the PSDs obtained from the velocity maps along the measurement lines covered by the sensors $Bottom$ and $Middle \textit {1}$. The frequency is normalized by the thermal diffusion frequency $f_\kappa = \kappa /H^{2}$. The dominant oscillation frequency can be easily identified by the clear peaks in the PSDs. The same frequency is found at both measuring positions indicating a certain coherence of the flow structure. The spectra in both figures 6(a) and 6(b) show similar magnitudes over the whole frequency range and an identical slope in the inertial domain for $f/f_\kappa > 10^{1}$. There is a slight deviation between experiment and numerical simulation concerning the values of the normalized frequency $f/f_\kappa = 3.3$ and $f/f_\kappa = 3.1$, respectively.

Figure 6. Spatially averaged PSDs calculated from the spatio-temporal velocity maps for $Ra = 1.2\times 10^{5}$ (a) measured in the experiments by the sensors $Bottom$ and $Middle \textit {1}$ and (b) obtained from numerical simulations. (c) Oscillation frequency $f_{OS}$ as a function of $Ra$ from experimental and numerical data, where the solid line represents the least-squares fit of the experimental results, $f_{OS}H^{2}/\kappa = 0.016Ra^{0.46}$, and the dotted line shows the fitting of the results of numerical simulations, $f_{OS}H^{2}/\kappa = 0.011Ra^{0.48}$. (d) The period length of the oscillations normalized with the turnover time $\tau _{TO}$ versus $Ra$, where the the dotted line represents a value of one.

The oscillation frequency $f_{OS}$ increases with increasing $Ra$ as shown in figure 6(c). A power-law fit leads to a scaling of $f_{OS}/f_\kappa =0.016Ra^{0.46\pm 0.02}$ for the velocity measurements in the experiments and $f_{OS}/f_\kappa =0.011Ra^{0.48\pm 0.02}$ for the numerical data. The agreement is satisfactory, where the exponents agree within the error bars. The slight discrepancy may result from uncertainties in the material properties, which in turn could affect the exact determination of $Pr$ and $Ra$ in the experiment. In the paper by Plevachuk et al. (Reference Plevachuk, Sklyarchuk, Eckert, Gerbeth and Novakovic2014), from which the numerical values for the thermal diffusivity, viscosity and density of GaInSn were taken, the accuracy of these measured values is specified in the range from $\pm 1.5\,\%$ to $\pm 7\,\%$, which might be sufficient to explain the above mentioned deviations.

Previous results reported by Akashi et al. (Reference Akashi, Yanagisawa, Tasaka, Vogt, Murai and Eckert2019) for experiments in the same container reported a scaling law of $f_{OS}/f_\kappa = 0.031Ra^{0.40\pm 0.02}$. The discrepancy can be explained by the fact that the range in which the power-law fit was calculated in Akashi et al. (Reference Akashi, Yanagisawa, Tasaka, Vogt, Murai and Eckert2019) also includes smaller $Ra$, where unstable rolls occur instead of the cellular structure. This also indicates that the two separate flow patterns cause different exponents in the scaling law. Experimental (Tsuji et al. Reference Tsuji, Mizuno, Mashiko and Sano2005; Vogt et al. Reference Vogt, Horn, Grannan and Aurnou2018a; Zürner et al. Reference Zürner, Schindler, Vogt, Eckert and Schumacher2019) and numerical (Scheel & Schumacher Reference Scheel and Schumacher2017) studies in cylindrical cells with $\varGamma = 1$ and $\varGamma =2$ come to smaller exponents. Zürner et al. (Reference Zürner, Schindler, Vogt, Eckert and Schumacher2019) found $f_{OS}/f_\kappa = (0.010\pm )\times Ra^{0.40\pm 0.02}$ at $Pr = 0.029$. The direct numerical simulation by Schumacher & Scheel (Reference Schumacher and Scheel2016) at $Pr=0.021$ results in a scaling $f_{OS}/f_\kappa = (0.08\pm )\times Ra^{0.42\pm 0.02}$. Another scaling law, $f_{OS}/f_\kappa = (0.027\pm )\times Ra^{0.419\pm 0.006}$, is given by Vogt et al. (Reference Vogt, Horn, Grannan and Aurnou2018a).

Akashi et al. (Reference Akashi, Yanagisawa, Tasaka, Vogt, Murai and Eckert2019) demonstrated that the ratio between the oscillation period $\tau _{OS} = 1/f_{OS}$ and the turnover time $\tau _{TO}$ is close to unity. The turnover time $\tau _{TO}$ describes the time required for a fluid parcel to complete a full circulation within the LSC structure. Except for the movement in the corners, the fluid elements follow in good approximation the path of an ellipse with the semi-axes $L/4$ and $H/2$. Using a simple approximation formula to determine the circumference of an ellipse, the turnover time can be calculated as follows:

(3.1)\begin{equation} \tau_{TO} = \frac{\pi \sqrt{2((L/4)^{2} + (H/2)^{2})}}{u_{rms}}. \end{equation}

The velocity $u_{rms}$ is calculated from the velocity data recorded by the sensor $Bottom$ as

(3.2)\begin{equation} u_{rms}=\overline{\sqrt{\frac{1}{L}\int^{L}_0u_x^{2}(x,t)\,{\textrm{d}}\kern0.06em x}}. \end{equation}

The ratio $\tau _{OS}$ to $\tau _{TO}$ is plotted versus $Ra$ in figure 6(d). All data points are in a close proximity to the value one.

The occurrence of characteristic frequency peaks is observed in many experimental studies (Castaing et al. Reference Castaing, Gunaratne, Heslot, Kadanoff, Libchaber, Thomae, Wu, Zaleski and Zanetti1989; Siggia Reference Siggia1994; Qiu & Tong Reference Qiu and Tong2001) and is apparently a characteristic feature of coherent large-scale structures occurring in thermal convection. Several studies construe these oscillations as an indication of the transport of individual plumes by the LSC (Cioni et al. Reference Cioni, Cilberto and Sommeria1997; Grossmann & Lohse Reference Grossmann and Lohse2000,  Reference Grossmann and Lohse2002; Stevens et al. Reference Stevens, van der Poel, Grossmann and Lohse2013). Another perspective is to associate these oscillations with the advection of deviations from an ideal two-dimensional LSC structure or other substructures by the LSC. This in a natural way explains an inherent relationship between the frequency of the oscillations and the turnover time. Thus, the torsion and sloshing phenomenon could also be understood as a deformation of the two-dimensional single-roll LSC that travels with a recurrence rate $1/\tau _{TO}$ throughout the fluid container (Zürner et al. Reference Zürner, Schindler, Vogt, Eckert and Schumacher2019). A strong correlation between the turnover time and the periodicity of predominant oscillations has been demonstrated in cylinders with $\varGamma = 1$ (Brown & Ahlers Reference Brown and Ahlers2009; Zürner et al. Reference Zürner, Schindler, Vogt, Eckert and Schumacher2019) or $\varGamma = 2$ (Vogt et al. Reference Vogt, Horn, Grannan and Aurnou2018a). In these geometries, where a single-roll LSC exists, the oscillatory behaviour is due to torsion and sloshing at $\varGamma = 1$ or to JRV dynamics at $\varGamma = 2$. In our case of the $\varGamma = 5$ box, a torsional movement of single-roll LSC can be excluded. Nevertheless, it is reasonable to assume that perturbations of the flow pattern or instabilities of the boundary layers are transported at the characteristic velocity of circulation within the coherent structures. This interpretation was also adopted by Cioni et al. (Reference Cioni, Cilberto and Sommeria1997) who assumed that the dominating frequency of the oscillations reflected in their temperature signal is associated with the velocity of the global circulation and path length of the LSC. This assumption leads to the relation ${f_{OS}H^{2}}/{\kappa } \varpropto RePr$ (Cioni et al. Reference Cioni, Cilberto and Sommeria1997). In the next section it becomes evident that this applies also for the flow configuration studied here, since the Reynolds number and the oscillation frequency follow the same power law as a function of $Ra$.

3.3. Transport of momentum and heat

The Reynolds number $Re$ is the characteristic parameter to quantify the turbulent momentum transport in the convection cell. Our definition of $Re$ from the experiment relies on the r.m.s. velocity $u_{rms}$ determined from the linear velocity profile along the measuring line $Bottom$:

(3.3)\begin{equation} Re = \frac{u_{rms}H}{\nu}. \end{equation}

A simultaneous acquisition of all three velocity components in the entire container is difficult to achieve in the experiment. With the limitation to one velocity component on one single measuring line, $Re$ derived from this appears relatively arbitrary and the question arises as to what extent it is representative for the total flow and the momentum transfer. To check this, two $Re$ values are determined by means of the numerical simulations, one corresponding to the experiment with $u_{rms}$ determined on the $Bottom$ measurement line and the other with the r.m.s. velocity $U_{rms}$ averaged over the entire volume in accordance with (2.7). To compensate for the difference that all three components of the velocity were considered in the calculation of $U_{rms}$ instead of only one in $u_{rms}$, the corresponding values of $Re$ have been corrected by a factor of $\sqrt {1/3}$.

The scaling behaviour of the resulting $Re$ is plotted in figure 7. There is no significant difference between the two results of numerical simulation, which suggests that the time-averaged velocities determined on the $Bottom$ measurement line are representative for the global flow. A power-law fit reveals a scaling law of $Re = 3.43Ra^{0.44}$ for velocity measurements and $Re = 2.83Ra^{0.45}$ for the numerical simulations. The direct comparison between experiment and simulation reveals a slight difference in the scaling exponents, but the absolute values agree very well.

Figure 7. Scaling of the Reynolds number $Re$ versus the Rayleigh number $Ra$. Results of numerical simulations are calculated for two characteristic velocities $u_{rms}$ and $U_{rms}$ representing the r.m.s. velocity for the measuring line of sensor $Bottom$ and the entire volume, respectively. Least-squares approximation leads to $Re = 3.43Ra^{0.44}$ (grey solid line) for the experimental data, $Re = 2.83Ra^{0.45}$ for numerical simulations (grey dashed line) and $Re = 2.56Ra^{0.46}$ for the entire volume (grey dotted line).

A comparison with the previous study by Akashi et al. (Reference Akashi, Yanagisawa, Tasaka, Vogt, Murai and Eckert2019) also reveals a slightly different scaling exponent, $u_{rms}(H/\kappa ) = 0.059Ra^{0.50\pm 0.02}$. In Akashi et al. (Reference Akashi, Yanagisawa, Tasaka, Vogt, Murai and Eckert2019) also smaller $Ra$, where unsteady roll arrangements are observed but no cellular flow regime occurs, were included into the power-law fit. Thus, $Re$ in the cellular regime apparently shows a measurably more gradual increase with $Ra$ than in a roll regime. There are no differences between the current measurements and the results of Akashi et al. (Reference Akashi, Yanagisawa, Tasaka, Vogt, Murai and Eckert2019), as long as only the range of the cellular regime is considered there.

Scaling exponents from $0.42$ to $0.53$ were reported by other experimental studies (Takeshita et al. Reference Takeshita, Segawa, Glazier and Sano1996; Cioni et al. Reference Cioni, Cilberto and Sommeria1997; Yanagisawa et al. Reference Yanagisawa, Hamano, Miyagoshi, Yamagishi, Tasaka and Takeda2013,  Reference Yanagisawa, Hamano and Sakuraba2015; Vogt et al. Reference Vogt, Horn, Grannan and Aurnou2018a; Zürner et al. Reference Zürner, Schindler, Vogt, Eckert and Schumacher2019) and numerical simulations (Scheel & Schumacher Reference Scheel and Schumacher2017) for low-$Pr$ thermal turbulence. Yanagisawa et al. (Reference Yanagisawa, Hamano, Miyagoshi, Yamagishi, Tasaka and Takeda2013) found a higher velocity magnitude and increased scaling exponent, $u_{rms}(H/\kappa ) = 0.08Ra^{0.53}$, in a square container with the same geometry filled with pure gallium corresponding to $Pr = 0.025$. Zürner et al. (Reference Zürner, Schindler, Vogt, Eckert and Schumacher2019) defined three different characteristic velocities: (1) the typical horizontal velocity magnitude near the plates $v_{LSC}$, (2) the typical vertical velocity magnitude of the LSC along the sidewall $v_{vert}$ and (3) the turbulent velocity fluctuations in the centre of the cell $v_{centre}$. This is reflected in slightly varying scaling exponents: (1) $v_{LSC} \propto Ra^{0.42\pm 0.03}$, (2) $v_{vert}\propto Ra^{0.42\pm 0.04}$ and (3) $v_{centre}\propto Ra^{0.46\pm 0.04}$ (Zürner et al. Reference Zürner, Schindler, Vogt, Eckert and Schumacher2019). The smaller scaling exponent for the horizontal and vertical LSC is a result of probing different parts of the complex three-dimensional flow structure. It suggests that the $Ra$ dependence of the representative velocities depends on how and where it is measured in the convection cell. It seems plausible that the option of calculating the r.m.s. velocity over the entire cell can be considered as the most reliable method for evaluating the representative convection velocity. For instance, Scheel & Schumacher (Reference Scheel and Schumacher2017) found a scaling $U_{rms}\propto Ra^{0.45\pm 0.01}$ in a cylinder with $\varGamma = 1$ at $Pr = 0.021$, which matches our results almost perfectly.

The heat transport of the three-dimensional cell structure is evaluated in numerical simulations by considering the Nusselt number dependence on the Rayleigh number, $Nu(Ra)$. Respective results are shown in figure 8. We find a scaling of $Nu= 0.14\times Ra^{0.26\pm 0.01}$ which is in agreement with the outcome of measurements and simulations in a cylindrical cell with $\varGamma = 1$ (Scheel & Schumacher Reference Scheel and Schumacher2017; Zürner et al. Reference Zürner, Schindler, Vogt, Eckert and Schumacher2019). The laboratory experiments conducted by Zürner et al. (Reference Zürner, Schindler, Vogt, Eckert and Schumacher2019) for $Pr = 0.029$ provide $Nu\cong (0.12\pm 0.04)\times Ra^{(0.27\pm 0.04)}$ while the simulations by Scheel & Schumacher (Reference Scheel and Schumacher2017) suggest $Nu\cong (0.13\pm 0.04)\times Ra^{(0.27\pm 0.01)}$ for $Pr = 0.021$. Recent experiments conducted in the same container by Vogt et al. (Reference Vogt, Yang, Schindler and Eckert2021) show a scaling of $Nu = 0.166\times Ra^{0.25}$ and thus a satisfactory agreement with the result presented here.

Figure 8. Scaling of the Nusselt number $Nu$ versus $Ra$ obtained by numerical simulations, where the error bars represent the standard deviation of the temporal fluctuations and the grey line shows the fitted curve, $Nu = 0.14\times Ra^{0.26}$ (numerical simulation).

4.1. Procedure of phase averaging

As shown by the flow measurements and the numerical simulations presented in figures 2, 3 and 4, the flow patterns are characterized by a variety of large- and small-scale fluctuations, which indicate the presence of both a periodically oscillating coherent flow filling the entire container and inertia-dominated turbulence on smaller scales. We are interested in the dynamics of the coherent flow structure and want to separate it from the background of the turbulent fluctuations by a suitable averaging method. Investigations of turbulent superstructures in very large aspect ratios are confronted with the same problem. Pandey et al. (Reference Pandey, Scheel and Schumacher2018) propose an averaging procedure where the averaging time to be used is in a range between the free-fall time $t_{ff}$ and the effective dissipation time $t_d = \max (H^{2}/\nu, H^{2}/\kappa )$. In the case of turbulent superstructures in very large aspect ratios, the patterns are slowly changing. The structure under consideration here undergoes significant oscillations with a constant periodicity. For this reason it is useful to apply a phase averaging to our numerical data. Such a procedure has already been suggested by Vogt et al. (Reference Vogt, Horn, Grannan and Aurnou2018a) to reveal the nature of the dynamic flow structures of the periodic oscillations of velocity and temperature in the $\varGamma = 2$ cylinder. The analysis covers one complete oscillation period $\tau _{OS} = 1/f_{OS}$, which in our case also corresponds to the turnover time for a respective fluid parcel (see figure 6). It is based on simulation results with a computation time comprising 16 oscillation periods with a total number of 5536 snapshots. The entire dataset is divided into 16 parts, each corresponding exactly to one oscillation period. This time period of one oscillation is again divided into 16 equidistant intervals, also called phases. Each interval or phase includes 21 snapshots of temperature, velocity and pressure. In the next step, these snapshots are averaged. Finally, for the distributions of temperature, velocity and pressure obtained for all oscillation periods are conditionally averaged with the respective distributions belonging to the same phase.

Vogt et al. (Reference Vogt, Horn, Grannan and Aurnou2018a) also had to adequately consider the orientation of the LSC in a convection cell for each time step, since the LSC symmetry plane in the cylindrical $\varGamma = 2$ cylinder wanders in the azimuthal direction. This did not have to be taken into account in our case, since the flow is approximately fixed in its orientation by the square geometry of the convection cell.

4.2. Variations of temperature and flow field during the oscillation period

The results presented in this section are restricted to $Ra = 1.2\times 10^{5}$ as an example. The flow at other $Ra$ shows a qualitatively similar behaviour. The numerical calculations of temperature and flow fields are performed for a duration of 16 oscillations which corresponds to about 300 free-fall times $t_{ff}$ and 5 thermal diffusion times $t_\kappa = H^{2}/\kappa$ with a grid resolution of $n_x \times n_y \times n_z = 400 \times 400 \times 80$. Figures 9(a) and 9(b) show the phase-averaged temperature distribution for the horizontal plane at the mid-height of $z = 0.5$ at two different phases. The temperature scale refers to the mean temperature $T_m = 0.5$ in the considered plane in such a way that the regions of the warm fluid marked by red coloration have a temperature above $T_m$, while colder fluid appears in blue coloration in the plots. We observe a distribution of temperature that resembles a checkerboard pattern, where the cold fluid occurs in the centre and the four corners of the container, while the warm fluid is located in centrally situated areas at the four sidewalls. This downwelling flow pattern is opposite to the upwelling one shown in the figure 5. The simulations reveal that the boundaries between the areas of warm and cold fluid which are identical to the isolines $T = T_m = 0.5$ are subject to significant deformation during the oscillations (see also supplementary movie 5). To quantify this behaviour in a simple way, we follow the displacements of some specially defined locations in the temperature distribution during one oscillation period. For this purpose we have selected the following positions: the intersection points of the $T_m$ isolines with the central lines of the centre plane along the $x$ axis (white triangles) and the $y$ axis (yellow triangles) as well as at the intersection points of the $T_m$ isolines with the sidewalls. The latter are marked by the grey and white squares in the $x$ direction and by the blue and white circles along the $y$ direction. Each of these marker points on the isolines can be assigned an $x$ coordinate $X_{T_m}$ and a $y$ coordinate $Y_{T_m}$. In the following, the movements of these points along the $x$ and $y$ centre lines and along the sidewalls are recorded. Our analysis takes into account that all marker points have only one degree of freedom and can change their position either exclusively along the $x$ or $y$ direction. The two snapshots in figures 9(a) and 9(b) show in a certain sense two extreme states in which the respective pairs of points have either a minimum or maximum distance to each other. The oscillation of the flow pattern is reflected accordingly in the coordinates of these markers.

Figure 9. Phase-averaged temperature distribution for $Ra = 1.2\times 10^{5}$ in the centre plane at (a) $t = 0$ and (b) $t = 0.5\tau _{OS}$; (c) time variations of the $x$ and $y$ positions of the mean temperature of $T_m = 0.5$ on the centre lines and the sidewalls of the container in the temperature distribution shown in (a,b) (numerical simulation).

This is demonstrated in figure 9(c) which displays the varying locations of the marker points during one oscillation period. A sinusoidal behaviour is detected, where the coordinates of each point accomplish a full oscillation in one cycle. It is noticeable that the selected pairs of markers always move in opposite directions; this concerns the white and yellow triangles on the centre lines as well as the points on the differently aligned side edges. If, for example, one considers the associated deformation of the fluid regions of higher temperature, this means that the approximately rectangular fields alternately retract to the area in front of the sidewall or extend into the central zone. For further consideration we have defined the phase shown in figure 9(a) as the starting point $t = 0$ of the oscillation period. Because of the symmetry of the configuration, both states in figures 9(a) and 9(b) are quasi-identical, since the marker points reach their extreme positions here. The distances between all marker points are nearly equal at intermediate times $t = 0.25\tau _{OS}$ and $t = 0.75\tau _{OS}$.

Figures 10(a)–10(d) show the cross-section of the phase-averaged temperature distribution, streamlines and velocity vectors in two vertical centre planes of the container at $y = 2.5$ and $x = 2.5$. The colour map represents the temperature of the fluid. The length of the velocity vectors indicates the magnitude of the velocity. The streamlines and the velocity vectors visualize two LSCs lined up side by side, where the fluid sinks in the centre of the cell and rises on the sidewalls. Size and location of the two dominating vortices are not constant but subject to permanent changes. The positions of the rotational axes of the vortices are not stationary, but move in the shown sectional planes. Naturally, this has significant consequences for the distribution of cold and warm fluid, which is reflected in the temperature colour plot (see also supplementary movie 6). The changes in the size of the respective zones of cold and warm fluids are clearly visible in the plots and confirm the dynamics already described in the context of figure 9. It should also be pointed out here that the temperature distribution does not only change in the centre of the convection box, but that significant alterations also occur near the bottom and the lid. We focus on this aspect in more detail later in § 4.4.

Figure 10. Phase-averaged temperature distribution with velocity vectors and streamlines for $Ra = 1.2\times 10^{5}$ in the cross-section of the centre lines of $y = 2.5$ (left column) and $x = 2.5$ (right column) at different phases: (a) $t=0$, (b) $t = 0.25\tau _{OS}$, (c) $t = 0.5\tau _{OS}$ and (d) $t = 0.75\tau _{OS}$. (e) Temporal change of representative velocity $u_{rms}$ calculated along the measurement lines of $z = 0.25$ and $z = 0.75$ in the cross-sections of $y = 2.5$ and $x = 2.5$ shown in (a–d) (numerical simulation).

For further evaluation, we consider the temporal behaviour of the r.m.s. velocities $u_{rms}$ recorded along two horizontal lines at $z = 0.25$ and $z = 0.75$ in both cross-sections. Time variations of $u_{rms}$ during one oscillation period are plotted as shown in figure 10(e). The velocities $u_{rms}$ monitored along the various horizontal lines reach their maximum (minimum) values at $t = 0.25\tau _{OS}$ or half a period later at $t= 0.75\tau _{OS}$. As already seen in the evaluation of the temperature distribution in figure 9(c), a sinusoidal behaviour of the velocities is observed. Furthermore, it is immediately noticeable that the velocities at the different heights in one plane and the velocities at the same height in different planes vary in opposite directions. This corresponds in an excellent way with the velocity measurements in the experiment (see Akashi et al. Reference Akashi, Yanagisawa, Tasaka, Vogt, Murai and Eckert2019). The periodic up and down of the velocity magnitude near the lid or bottom is related to an oscillatory shift of the centre of the dominating vortices in both vertical and horizontal directions. In figures 10(a)–10(d) it can be seen that smaller secondary vortices can form temporarily on the sidewalls if the axes of the larger vortices are located near the centre of the convection container. The vertical shift of the vortex positions lets the zone around the horizontal centre line alternately come into the effective range of the horizontal flow pointing towards the centre or towards the sidewalls, respectively. This explains the flow patterns with periodic reversals of flow direction found in the mid-height of the convection cell by both measurements and simulations (see figures 3b, 3c and 4b).

4.3. Delineation of the three-dimensional flow structure

In the previous section, it was shown that the location and properties of dominant vortices filling the entire height of the fluid layer change significantly during a period of oscillation.The position of a vortex centre can be determined by its axis of rotation, which is characterized by minimum pressure. The phase-averaged pressure distribution has been calculated for this purpose. Snapshots of the pressure fields in both vertical sectional planes are displayed in figures 11(a) and 11(b) for $t = 0.25\tau _{OS}$. Here, the positions of the minimum pressure are marked with dark blue coloured circles. The trajectories of these vortex axes during one oscillation period are presented in the figures 11(c) and 11(d). In addition to the information regarding the position of the pressure minimum in the respective sectional plane, the colouring of the circles in this diagram also indicates the time when the vortex centre was at which position. The trajectories of the axes of the LSCs form the central segment of a diagonal line running approximately between the lower corners of the convection cell and the centre of the upper plate, with the forward and return paths not exactly on the same line. It becomes also obvious that the two circulation centres always move in opposite directions in their respective sectional planes with an offset of half a period.

Figure 11. Phase-averaged pressure contour for $Ra = 1.2\times 10^{5}$ in the cross-section of (a) $y = 2.5$ and (b) $x = 2.5$ at the phase of $t = 0.25\tau _{OS}$. The minimum pressure points in the phase-averaged pressure contour during one oscillation period in the cross-sections of (c) $y = 2.5$ and (d) $x = 2.5$ (numerical simulation).

Figure 12 is intended to illustrate the movement of the circulation areas in three dimensions by means of the streamlines in the vicinity of the vortex centres. To visualize the third component in the vertical direction, the streamlines are coloured according to the vertical position of the structure. A first thing to notice is that four recirculating swirls exist in the convection cell. These four connected swirls are arranged as two pairs, with the axes of the respective pair aligned on average mainly along the $x$ and $y$ directions, respectively. Figures 12(a) and 12(b) show two snapshots at $t = 0.25\tau _{OS}$ and $t = 0.75\tau _{OS}$, respectively. These are the moments when the swirls are close to either the bottom or the top of the convection cell (see figures 10 and 11). A corresponding animation of these results (supplementary movies 7–9) during one oscillation period shows, in fact, that we are dealing here with the phenomenon of a three-dimensional flow structure that shows the same characteristics as JRV, which was observed and described first by Vogt et al. (Reference Vogt, Horn, Grannan and Aurnou2018a). The main difference from the geometry of a $\varGamma = 2$ cylinder, in which the dynamics of a JRV extends over almost the entire convection cell, is that four such structures exist in our $\varGamma = 5$ configuration, whose movements are restricted to each partial area of the fluid layer. Supplementary movie 10 shows that the phase-averaged temperature and velocity fields in the centre plane and the vertical cross-sections of the limited area are in very good agreement with the conditionally averaged temperature and velocity fields in the centre plane, the LSC symmetry plane and the perpendicular plane of the LSC symmetry plane in the cylinder of $\varGamma = 2$ (Vogt et al. Reference Vogt, Horn, Grannan and Aurnou2018a). The direction of motion of the vortex centre is clockwise, whereas the direction of flow is counterclockwise, and vice versa. It is interesting to note that the revolution of these swirls in the convection cell occurs in the opposite direction to the main flow of the LSCs. The direction of motion of the vortex centre is clockwise, whereas the direction of flow is counterclockwise, and vice versa. Another peculiarity is that the trajectories of the circulation centres obviously do not follow a circle, as one would normally expect for a typical jump rope, but through a tilted ellipse-like trajectory as visualized in figure 11(c,d). One reason for this might be the fact that the limited height of the fluid layer imposes a geometric constraint on the dynamics of the JRV.

Figure 12. Three-dimensional visualization of phase-averaged streamlines at (a) $t = 0.25\tau _{OS}$ and (b) $t = 0.75\tau _{OS}$ for $Ra = 1.2\times 10^{5}$. The dashed lines mark the two centre planes $x = 2.5$ and $y = 2.5$, which also represent the symmetry axes of the flow pattern. The colour represents the vertical position of the structure in the container (numerical simulation).

4.4. Impact on heat transport

The temperature distributions presented in § 4.2 demonstrate that considerable redistribution of cold and warm fluid occurs in the course of the three-dimensional oscillations. In this section, we consider the extent to which this has measurable effects on the heat transport.

To verify the steadiness of the heat transport, Nusselt numbers are calculated from the phase-averaged temperature distributions. Figure 13 presents the phase-averaged $Nu$ numbers during one oscillation period. The results reveal that $Nu$ is not constant but rather shows an oscillatory behaviour with a verifiable difference between the maximum and minimum value. Moreover, at the first glance it is a very surprising finding that the period of the $Nu$ oscillation is only half as long as that of the fluctuations in the temperature field or the horizontal velocities.

Figure 13. Phase-averaged values of the Nusselt number $Nu$ during one oscillation period for for $Ra = 1.2\times 10^{5}$ (numerical simulation).

Figure 14 shows contour plots of phase-averaged local heat flux. These plots represent local heat flux calculated at the top boundary $F_{z, top}=- ({\partial T}/{\partial z} )_{z=1}$ and the bottom boundary $F_{z, bot}=- ({\partial T}/{\partial z} )_{z=0}$. Here, the average number of local heat flux for an entire plate means the Nusselt number of the convection. The red-coloured zones feature the local heat flux that is above the mean value of the entire area and period ($Nu \approx 3.0$), while local heat flux below the mean value is marked in blue colour. Here, the contrasting behaviour of the local heat flux at the bottom and the top is obvious which is caused by the flow that either impinges or detaches in the corresponding areas of the plates. The local heat flux distributions change during the oscillation period and agree to a certain degree with the temperature distributions presented in figure 9. According to figure 13, $Nu$ reaches a maximum value at time points $t = 0$ and $t = 0.5\tau _{OS}$, while minima occur at $t = 0.25\tau _{OS}$ and $t = 0.75\tau _{OS}$. The corresponding local heat flux distributions on the plates are geometrically similar. This is in agreement with the periodicity of the $Nu$ number oscillations.

Figure 14. Phase-averaged local heat flux at the top boundary $F_{z, top}$ (a–d) and at the bottom boundary $F_{z, bot}$ (e–h) for $Ra = 1.2\times 10^{5}$, at different phases: (a,e) $t=0$, (b,f) $t = 0.25\tau _{OS}$, (c,g) $t = 0.5\tau _{OS}$ and (d,h) $t = 0.75\tau _{OS}$. The contour interval is 0.5 (numerical simulation).

The heat transfer of the system is mainly determined by the conductivity and stability of the thermal boundary layers (Ahlers et al. Reference Ahlers, Grossmann and Lohse2009; Iyer et al. Reference Iyer, Scheel, Schumacher and Sreenivasan2020). Our results presented in figure 10 suggest that the turbulent fluid motion might have a considerable impact on the extent of the thermal boundary layers. Indeed, our data also show that the instantaneous thermal boundary layer thickness at both the bottom and the top plates $\lambda _{\varTheta }$ oscillates with the same frequency as the JRV. Owing to the relation $Nu = 1/(2\lambda _{\varTheta })$, this is reflected accordingly in the changes of $Nu$ illustrated in figure 13.

Figure 15(a) compares time series of $Nu$ with time series of the horizontal velocity $u_{rms}$ obtained along the line $Bottom$ and the volume-averaged vertical velocity for the entire container $U_{z}$ obtained by numerical simulations. All data show quasi-periodic oscillations. The signals of $Nu$ and $U_{z}$ are synchronized with each other, i.e. they oscillate with the same frequency, while $u_{rms}$ exhibits twice the oscillation period. This is confirmed by the corresponding PSDs presented in figure 15(b). Both PSDs of $Nu$ and $U_{z}$ show dominant peaks at the same frequency, which is almost exactly twice the oscillation frequency $f_{OS}$. Here we see an effect that, on closer inspection, is already apparent in figure 10. Figure 10(e) reveals that the approach of the centre of the circulation to one of the horizontal interfaces suppresses the horizontal flow there, while the horizontal flow on the opposite side simultaneously reaches a maximum. The evaluation of the numerical data in the whole convection cell shows that the vertical flow component, on the other hand, becomes greatest when the centre of the circulation moves either from bottom to top or in the opposite direction. This vertical momentum transport thus additionally contributes to the heat transport.Accordingly, figure 15(a) demonstrates a simultaneous occurrence of maximal values for $Nu$ and $U_z$.

Figure 15. (a) Time variations of $Nu$, the volume average of the vertical velocity component $U_z$ and the horizontal velocity $u_{rms}$ obtained by the line $Bottom$ at $z = 0.25$ and (b) the PSDs calculated from the temporal fluctuations of $Nu$, $U_z$ and the PSD calculated from the horizontal velocity $u_{rms}$ obtained by the line $Bottom$ for $Ra = 1.2\times 10^{5}$ (numerical simulation).