2.1. The convection cell

A sketch of the experimental apparatus is reported in figure 1. The convection cell consists of a Plexiglas cylinder filled with water. The internal diameter of the cylinder is $D=74$ mm, the aspect ratio is $\varGamma =0.5$ (height equal to twice the internal diameter) and the sidewall thickness is $3$ mm. The cylinder is immersed in an octagonal tank, also filled with water.

Figure 1.

Schematic of the experimental apparatus.

The water inside the cylinder is heated from below and cooled from above by thermal sources kept at constant temperature throughout the duration of the experiment via a high-precision thermoelectric controller (TEC). The bottom heating system consists of an electrolytic copper slab connected to a mica-insulated flat heater, which can provide heating power up to $100$ W. The copper slab is made of two parts, glued to each other via a highly conductive epoxy, in-between which a PT100 1/10-DIN RTD sensor is placed. The distance of the sensor from the slab surface in contact with the fluid is $12$ mm; due to the large thermal conductivity of the copper, the temperature drop across this layer is negligible with respect to the sensor accuracy. The copper slab is housed in an appropriate site on the bottom of the tank, and the cylinder is fastened to the copper insert with an interference fit. The cylinder sidewall is in contact with the copper insert for a depth of approximately $20$ mm.

The top cooling system consists of a water heat exchanger obtained by assembling a set of Plexiglas layers. These layers comprise two recirculation chambers with nearly opposite tangential inlet and outlet, delimited by a thin slab with a small central hole. The lower recirculation chamber is separated by the convection cell via an acrylic foil of 0.25 mm thickness, which offers a small thermal resistance. The water is refrigerated by a Peltier thermoelectric cooler and pumped into the lower recirculation chamber. The cooler consists of a copper waterblock coupled with a Peltier element, connected to a heat sink with fan, with a maximum heating/cooling power of $118$ W and a maximum allowable temperature difference between its sides of $\pm 75\,^{\circ }$C. The tubes of the water circuit are coated with neoprene rubber so as to limit heat losses due to heat transfer with the surrounding ambient. The water temperature is measured at the heat exchanger inlet by means of an immersion ultraprecise PT100 1/10-DIN RTD sensor.

Based on the continual measurements from the two resistance temperature detection (RTD) sensors, the thermoelectric controller performs a proportional–integral–derivative control by adjusting the current inputs to the flat heater and the Peltier element and ensures a temperature stability of $0.01\,^{\circ }$C. Since the temperature difference between the top and the bottom plates is $5\,^{\circ }$C in the present experiments, this corresponds to a percentage stability of the imposed unstable temperature gradient of $0.2\,\%$. The laboratory room with the tank is air conditioned, and maintained at constant temperature; the water temperature is measured by means of an immersion PT100 1/10-DIN RTD probe to check that it remains constant and near to the ambient temperature. The latter is equal to $17.5\,^{\circ }$C; being the nominal temperature difference equal to $5\,^{\circ }$C, the nominal temperature of the bottom is $20\,^{\circ }$C, while the nominal temperature of the top is $15\,^{\circ }$C. The tank temperature during the present experiment is $17.35\,^{\circ }$C.

2.2. Imaging system and camera calibration

The imaging system consists of a dual pulse Nd:YAG laser with maximum pulse energy of $200$ mJ and four sCMOS cameras (Andor Zyla) with a resolution of $2560\times 2160$ pixels. The laser light is shaped into a cylindrical beam that is passed through the transparent heat exchanger on the top of the cell and illuminates the entire convection domain. The cylindrical shape is obtained by using an appropriate system of mirrors and lenses, as shown in figure 1.

The four cameras are arranged in a planar configuration with an angular spacing of approximately $40^{\circ }$. In order to focus the whole cylinder interior, each camera is equipped with $28$ mm focal length objectives set at an f-number equal to $22$. The resulting digital resolution is $14$ pixel mm$^{-1}$.

The seeding particles are orange fluorescent polyethylene microspheres (Cospheric UVPMS-BO-1.00); the average particle diameter is $58\ \mathrm {\mu }$m, while the particle density is $1.00$ g cm$^{-3}$, resulting in a relaxation time lower than $1$ ms (which is significantly below the turbulent dissipative time scales of the thermal convection at the currently investigated conditions). The working fluid is seeded before placing the cylinder in situ and it is not possible to add further seeding particles after the beginning of the experiment. Fluorescence of the particles is exploited to reduce the green reflections from the copper base and increase the particle scattering contrast in the recorded images. For this purpose, the camera lenses are equipped with HOYA YA3 orange filters.

Optical calibration of the camera system is a critical point of the present experimental set-up, because of inaccessibility of the cylinder interior and optical distortions caused by the curvature of the sidewall. As is well known, such distortions depend on the ratio of the refractive indexes of the sidewall material (Plexiglas) and the surrounding fluid (water) and vary locally with the viewing direction. Specifically, in the present arrangement, the regions affected by the largest optical distortions are those adjacent to the cylinder sidewall, where classical camera models, such as the pinhole-camera model (Tsai Reference Tsai1987; Heikkila & Silven Reference Heikkila and Silven1997; Zhang Reference Zhang2000), do not provide a satisfactory quality of the tomographic reconstruction of the particle distribution. This issue is obviated by using a perspective camera model with a refraction correction for cylindrical distortion based on Snell's law of refraction. All the relevant details about this method are given in Paolillo & Astarita (Reference Paolillo and Astarita2020). The main strengths of the refractive camera model are the limited number of parameters involved (which all have a clear physical or geometrical meaning and thus are easily checkable) and the possibility of calibrating such parameters without placing a calibration target inside the cylindrical cell (for this purpose, it is sufficient to record images of the target with the cylinder interposed between the cameras and the target itself). The refractive camera model is initially calibrated following the procedure explained in Paolillo & Astarita (Reference Paolillo and Astarita2020); subsequently, both the pinhole camera and the cylinder parameters are refined and optimized via the volume self-calibration technique (Wieneke Reference Wieneke2008; Discetti & Astarita Reference Discetti and Astarita2014).

2.3. Data acquisition and processing

Once the temperature difference between the top and the bottom is set, the system is allowed to settle for approximately two hours before starting the experiment. Then, measurements are carried out over approximately four hours with a sampling frequency of $7.5$ Hz. The image analysis consists essentially of three stages: image preprocessing; time-resolved motion analysis; velocity data post-processing.

Time-resolved motion analysis is based on a combination of the most recent algorithms for particle motion tracking and consists essentially of two steps. Initially, a first set of snapshots (typically 5–10) is processed with the sequential-motion-tracking enhancement (SMTE) algorithm (Lynch & Scarano Reference Lynch and Scarano2015); at this stage, multiple iterations of the SMART (Atkinson & Soria Reference Atkinson and Soria2009) and CSMART (Ceglia et al. Reference Ceglia, Discetti, Ianiro, Michaelis, Astarita and Cardone2014; Castrillo et al. Reference Castrillo, Cafiero, Discetti and Astarita2016) algorithms are carried out with a multiresolution approach (Discetti & Astarita Reference Discetti and Astarita2012a); multipass volumetric cross-correlations are performed via an efficient algorithm using sparse matrices (Discetti & Astarita Reference Discetti and Astarita2012b). In the second phase of the process, the STB method (Schanz, Gesemann & Schröder Reference Schanz, Gesemann and Schröder2016) is used for particle tracking. Particle triangulation is performed by the iterative particle identification method (Wieneke Reference Wieneke2012), while the forward-time projection of particles is based on both the extrapolation of known trajectories and cross-correlation techniques.

The output of the above process consists of the particle tracks. The velocity data processing step is aimed at estimating the instantaneous velocity fields from the particle trajectories. This implies the estimation of the particle velocities and the interpolation of the latter onto a structured grid. In the present experiments, a fourth-order polynomial fitting based on a kernel of seven time positions of the particles was used to calculate the particle velocities. The employed kernel, with the current sampling frequency, corresponds to a time interval of $0.66$ s, which is smaller than the Kolmogorov time scale, estimated to be approximately $1.8$ s for the current experimental conditions. The latter estimate is based on the computation of the time- and volume-averaged turbulent kinetic energy dissipation rate from the theoretical relation $\varepsilon _u=\nu ^{3}H^{-4}(Nu -1)RaPr^{-2}$ (Ahlers et al. Reference Ahlers, Grossmann and Lohse2009), which indeed is exact for cylindrical cells with adiabatic sidewalls. The Nusselt number $Nu$ in the previous formula has been estimated from separate numerical simulations of thermal convection accounting for the presence of the cylinder sidewall and performed with the same code used in Stevens et al. (Reference Stevens, Lohse and Verzicco2014); for all the relevant details about the numerical procedure the reader is also referred to the works of Verzicco & Orlandi (Reference Verzicco and Orlandi1996) and of Verzicco & Camussi (Reference Verzicco and Camussi1999, Reference Verzicco and Camussi2003). As concerns the interpolation of the particle velocities onto a structured grid (i.e. transformation from a Lagrangian reference frame to an Eulerian one), it is based on least-squares polynomial fitting of local data. More specifically, a structured (Cartesian or cylindrical) grid is chosen within the measurement volume; for each point of the grid, particles falling within a fixed search radius are identified and the velocities of such particles are used to determine a local polynomial fitting function which is then evaluated at the location of the grid point. Moreover, to reduce the effects of ghost particles on the determination of the velocity field, only particles with trajectories longer than a fixed number of time instants are used in the above procedure. For the results reported in the following, the employed grid is cylindrical and consists of 71, 38 and 98 nodes in the axial, radial and azimuthal directions, respectively. Each velocity vector in this grid is computed with a second-order polynomial fit on the particle velocities within a search radius of $30$ voxels (approximately $2$ mm) and relying only on the particles with trajectories longer than seven sampling periods. The uncertainty of the present velocity measurements is estimated to be approximately $2\,\%$ of the root mean square of the vertical velocity over the time and the cell volume (which is approximately $0.04w_0$).

2.4. POD

The POD, also known as the Karhunen–Loève method, is a statistical method aimed at finding a basis for modal decomposition from an ensemble of signals (Berkooz, Holmes & Lumley Reference Berkooz, Holmes and Lumley1993). The POD provides a linear decomposition that is optimal from an energetic viewpoint; this means that there is no further linear decomposition that approximates – in a least squares sense – the signal better than the POD when truncated at a specified order (lower than the number of the signal degrees-of-freedom). In the investigation of complex turbulent flows, the POD has been applied for the identification of the most energetic coherent structures since the seminal paper of Lumley (Reference Lumley1967). In 3-D unsteady flows, the POD modes of the velocity field $\underline {u}(\underline {x},t)$ are sought as the functions $\underline {\phi }(\underline {x})$ that maximize their projection on the velocity field itself. This leads to the following eigenvalue problem:

(2.1)\begin{equation} \int_V \overline{\underline{u}(\underline{x},t) \underline{u}(\underline{x}',t)} \underline{\phi}(\underline{x'})\, {\textrm{d}} \underline{x'}=\lambda \underline{\phi}(\underline{x}),\end{equation}

where $V$ is the cell volume, the symbol $\overline {(\cdot )}$ indicates the statistical average, $\lambda$ is the eigenvalue corresponding to the eigenfunction $\underline {\phi }(\underline {x})$ and $\overline { \underline {u}(\underline {x},t) \underline {u}(\underline {x}',t) } \underline {\phi }(\underline {x'})=\sum _{j=1}^{3} \overline { \underline {u}(\underline {x},t)u_j(\underline {x}',t) } \phi _j(\underline {x'})$. When working with discrete measurements, the integral equation (2.1) turns into the following linear eigenvalue problem:

(2.2)\begin{equation} \boldsymbol{C}_{\boldsymbol{U}}\boldsymbol{\phi}= \lambda\boldsymbol{\phi},\end{equation}

where $\boldsymbol {C}_{\boldsymbol {U}}=1/(N_s-1)\boldsymbol {U}\boldsymbol {U}^{\mathrm {T}}$ is the velocity covariance matrix with $\boldsymbol {U}$ being the observation matrix $\boldsymbol {U}=[\boldsymbol {u}_1,\boldsymbol {u}_2,\dots ,\boldsymbol {u}_{N_s}]$. Each observation $\boldsymbol {u}_i$ includes the set of the measurements of the three velocity components for the $i$th snapshot; thus, the size of $\boldsymbol {U}$ is $3 N_p \times N_s$, where $N_p$ is the number of measurement points and $N_s$ the number of snapshots employed for the computation of the POD modes.

Since the number of grid points is significantly larger than the number of snapshots used for the computation of the POD modes (in the present case $N_p=264,404$ and $N_s=2,350$), the eigenvalue problem (2.2) is solved via the so-called method of snapshots, first introduced by Sirovich (Reference Sirovich1987). Such a method allows a reduction of the problem size from $3 N_p\times 3 N_p$ to $N_s \times N_s$ and thus it is computationally advantageous. It is also worth remarking that in the discrete form (2.2), the POD reduces to the singular value decomposition (SVD); in the present case, singular value decomposition algorithms are used to apply the method of snapshots.

The POD has been performed by using a subset of all the available snapshots, which covers the two central hours of the experiment. Moreover, only one in every 25 consecutive snapshots is considered for a total of $2350$ samples. Based on the value of the integral time scales computed from the autocorrelation functions of the three velocity components, the number of statistically independent samples is estimated to be approximately a quarter of the number of employed samples. The uncertainty associated with the POD results reported in the following has been estimated by repeating the same analyses with a different number of samples used for the computation of the POD modes. This resulted in a relative uncertainty up to $7\,\%$.

5.1. Statistical behaviour of the LSC strength

Figure 13 reports the probability density functions (p.d.f.s) and the cumulative distribution functions (c.d.f.s) of the amplitudes $A_j$ for the top, middle and the bottom. With the chosen scaling ($A_j$ normalized by its time-averaged value $\overline {A_j}$), the three p.d.f.s essentially collapse on each other, as already observed in the study of Weiss & Ahlers (Reference Weiss and Ahlers2011b) for $Ra=9\times 10^{10}$, $Pr=4.38$ and $\varGamma =0.5$. However, differently from the latter investigation, in the present case the maximum of the p.d.f. is reached at a value lower than unity for the middle and the top, whereas it is found at $A_j=\overline {A_j}$ for the bottom. In figure 13(b) the dashed line represents the threshold indicated by Weiss & Ahlers (Reference Weiss and Ahlers2011b) to identify an event. The probability that $A_j<0.15\overline {A_j}$ is $3.15\,\%$, $1.73\,\%$ and $2.98\,\%$ for the bottom, middle and top, respectively. These values correspond to frequencies of such events that are, respectively, equal to $112\tau _\nu ^{-1}$, $53\tau _\nu ^{-1}$ and $106\tau _\nu ^{-1}$, with $\tau _\nu =H^{2}/\nu$ being the viscous time. Such values are in agreement with the observations of Xi & Xia (Reference Xi and Xia2007) and Weiss & Ahlers (Reference Weiss and Ahlers2011b) for higher Rayleigh numbers ($>10^{9}$) and lower Prandtl number ($\approx 4.3$). However, a remarkable difference of the present experiment from the latter studies is that the occurrence frequency of the events at midheight is found to be significantly lower than (almost one half) those at the bottom and the top. Moreover, no cessations (i.e. events when all the three amplitudes are below their thresholds) are observed in the present investigation. This is consistent with the experimental observations of Weiss & Ahlers (Reference Weiss and Ahlers2011b), who found a frequency of occurrence of cessations equal to $1.1\tau _\nu ^{-1}$, which turns into a value of approximately 0.2 cessations per hour in the present case, although in disagreement with the study of Xi & Xia (Reference Xi and Xia2007), which reported a considerably higher frequency of cessations in analogous operating conditions (i.e. $Ra$ in the range $10^{10}$–$10^{11}$ and $Pr$ between $4.38$ and $5$).

Figure 13.

Probability distribution of the vertical velocity amplitudes of the LSC at the bottom ($z/H=0.25$), middle ($z/H=0.5$) and top ($z/H=0.75$): (a) p.d.f.; (b) c.d.f.. Each amplitude is normalized by the corresponding time-averaged value.

The statistical distributions of the LSC strengths $S_j$ are reported in figure 14. The behaviour of $S_m$ (black dots) is similar to that reported in the numerical study of Stevens et al. (Reference Stevens, Clercx and Lohse2011) for analogous operating conditions (but with a perfect adiabatic wall condition). Here $S_m$ can be considered as representative of the strength of the LSC in the SRS, while $S_t$ (blue triangles) and $S_b$ (red triangles) are related to both the SRS and the DRS. Indeed, the higher values of the $S_m$ p.d.f. compared with the p.d.f.s of $S_b$ and $S_t$ over the range $0< S_j<0.2$ are essentially related to the occurrence of the DRS which causes $S_m$ to have values statistically smaller than those of $S_t$ and $S_b$. It is also noted that $S_b$ shows values of the p.d.f. greater than those related to $S_t$ in the range $0< S_j<0.05$. This indicates that in the SRS the strength of the LSC in the top half of the cell is statistically greater than that in the bottom half. Such an observation is corroborated by figure 14(b), where it is apparent that the c.d.f. of $S_t$ has lower values than those of $S_m$ and $S_b$. The reason for this behaviour can be found in the influence of the temperature condition at the external side of the sidewall: as aforementioned, hot plumes rising from the bottom are subjected to a stronger heat transfer with the sidewall than the cold plumes and this results in a weakening of the recirculation in the lower half of the cell.

Figure 14.

Probability distribution of the LSC strength at the bottom ($z/H=0.25$), middle ($z/H=0.5$) and top ($z/H=0.75$) of the cell: (a) p.d.f.; (b) c.d.f..

5.2. Statistical occurrence and properties of the SRS and DRS

As commented above, the statistical distribution of the absolute differences $|\delta \varphi _{ij}|$ provides insight in the occurrence of the SRS and the DRS. Figure 15 reports the p.d.f.s of the absolute differences $|\delta \varphi _{tm}|$ (blue triangles), $|\delta \varphi _{bm}|$ (red triangles) and $|\delta \varphi _{tb}|$ (black squares). Such a diagram is obtained using only the samples that satisfy the criterion for the LSC amplitude ($A_j>0.15\overline {A_j}$). The distributions of $|\delta \varphi _{tm}|$ and $|\delta \varphi _{bm}|$ are similar and they exhibit a bell-like shape. This is in very good agreement with the experimental results of Weiss & Ahlers (Reference Weiss and Ahlers2011b) and the numerical results of Stevens et al. (Reference Stevens, Clercx and Lohse2011). On the other side, $|\delta \varphi _{tb}|$ shows higher values of the p.d.f. between $0^{\circ }$ and $60^{\circ }$ and lower values between $120^{\circ }$ and $180^{\circ }$. This suggests that the probability of occurrence of the DRS is slightly lower than that of the SRS. Indeed, the computed values of such probabilities are $27\,\%$ and $29\,\%$, respectively, for the DRS and the SRS; on the other hand, the TSs have a probability of occurrence of $44\,\%$ among states that are not classified as events. Such results are in agreement with the much longer experiments of Weiss & Ahlers (Reference Weiss and Ahlers2011b), who observed that, for $Ra=2\times 10^{8}$ and $Pr=4.38$, the flow state is undefined for approximately $50\,\%$ of the time (in this regard, it is remarked that the events cover only $6\,\%$ of the total time in the present experiments, thus the TSs constitute $41\,\%$ of the total duration).

Figure 15.

Probability distributions for the absolute differences between the orientation of the LSC at the bottom ($z/H=0.25$), middle ($z/H=0.5$) and top ($z/H=0.75$). Only the samples satisfying the criterion for the amplitudes have been considered (see the text for explanation). When exceeding $180^{\circ }$, the difference $|\delta \varphi _{ij}|$ was replaced by $360^{\circ }-|\delta \varphi _{ij}|$. The dashed lines indicate the thresholds for the identification of the SRS ($|\delta \varphi _{ij}|<60^{\circ }$ for any $i\neq j$) and the DRS ($|\delta \varphi _{tb}|>120^{\circ }$) according to Xi & Xia (Reference Xi and Xia2008) and Weiss & Ahlers (Reference Weiss and Ahlers2011b).

Figure 16 shows the p.d.f.s of the LSC orientations at the top, middle and bottom for both the SRS and the DRS. The p.d.f.s for the SRS (figure 16a) are essentially the same at all the three levels and they exhibit a pronounced peak around $\varphi =0^{\circ }$, while the p.d.f.s go to zero towards $\pm 180^{\circ }$. The existence of a preferential orientation of the LSC has already been noted in previous works. Brown & Ahlers (Reference Brown and Ahlers2006a) demonstrated that symmetry-breaking inhomogeneities can be generated by several factors, among which are the Earth's Coriolis force and imperfections of the experimental apparatus such as a slight tilt of the sample, small horizontal thermal gradients in the top and bottom plates and eccentricity of the circular cross-section. However, the studies of Xi & Xia (Reference Xi and Xia2008) and Weiss & Ahlers (Reference Weiss and Ahlers2011b) have shown that the effect of the Earth's Coriolis force seems to be smaller in convection cells with $\varGamma =0.5$ than those with $\varGamma =1$. Interestingly, the p.d.f.s of the LSC orientations for the DRS (figure 16b) are fairly uniform for the top and the bottom, whereas lower values around $\pm 180^{\circ }$ are found for the middle. In conclusion, it is possible to assert that a preferential orientation does not exist for the DRS.

Figure 16.

Probability distributions for the LSC orientation at the bottom ($z/H=0.25$), middle ($z/H=0.5$) and top ($z/H=0.75$) for (a) the SRS and (b) the DRS.

Figure 17 reports the p.d.f. of the LSC angular velocity estimated as the time derivative of the LSC orientation at midheight $\varphi _m$. The p.d.f. computed using all the available samples (blue dots) has a Gaussian shape in the central part with fairly heavy tails, in agreement with the results from the previous investigations of Xi & Xia (Reference Xi and Xia2008) and Weiss & Ahlers (Reference Weiss and Ahlers2011b). In the same diagram, the p.d.f. for the samples satisfying the criterion of Stevens et al. for identification of a coherent LSC based on circulation strength, i.e. $S_m>0.5$, (black squares) and the p.d.f. for the samples with $A_m>\overline {A_m}$ (red diamonds) are also reported. Such curves exhibit an almost identical narrower Gaussian shape, which indicates that the high values of the angular velocity of the LSC are indeed related to states of the turbulent convection with a not-well defined LSC, in which the determination of the LSC orientation is affected by greater uncertainty.

Figure 17.

Probability distribution for the angular velocity of the LSC. The light-blue solid line is the Gaussian fit to the data related to all samples in the range between $\pm 0.05$ rad s$^{-1}$.

5.3. Relationship of the SRS and the DRS with low-order POD modes

In this section, the relationship of the SRS and the DRS with the first two pairs of POD modes is investigated by focusing on the statistical behaviour of the corresponding POD coefficients, which are the projections of the snapshots onto the POD modes. Since the first POD modes appear to be paired as shown above, it is convenient to define an energy contribution related to a single pair of the POD modes and study the statistical distribution of this quantity. In the following, the energy contribution due to the pair of the $i$th and the $j$th POD mode is denoted with the symbol $E^{(i-j)}$.

Figure 18 reports the c.d.f.s of the energy contributions $E^{(1-2)}$ and $E^{(3-4)}$ distinguishing between the samples in the SRS, in the DRS and in a TS. It is worth remarking that a high value of the c.d.f. corresponding to a certain energy level $E^{*}$ means that the probability that $E^{(i-j)}< E^{*}$ is high. Consequently, figure 18(a) shows that the first pair of POD modes statistically contributes more to the SRS than to the DRS. On the other side, figure 18(b) shows that the second pair of POD modes statistically contributes more to the DRS than to the SRS. It is also noted that the energetic contributions of the POD modes’ pairs to the TSs are on an intermediate level between those related to the SRS and the DRS.

Figure 18.

Cumulative probability distributions of the energy contribution due to the (a) the first pair and (b) the second pair of POD modes for the different states of the LSC. Energies are scaled by the maximum value detected over the entire measurement duration.

Another useful perspective on the relationship between the states of the LSC and the POD modes is obtained by representing the c.d.f.s of the energies $E^{(i-j)}$ of the different POD mode pairs for all the samples corresponding to a single state of the LSC. Such a representation is given in figure 19 for the SRS and the DRS; here, energy values are scaled by the mean energy $\overline {E}^{(1-2)}$ contained in the first pair of POD mode in the corresponding state of the LSC. Figure 19(a) shows that the contribution of the first pair of POD modes to the SRS is statistically preponderant with respect to the contributions of the remaining modes. This could seem an obvious consideration since the POD modes retain most of the time-averaged fluctuating kinetic energy. However, it should be remarked that the c.d.f.s in figure 19(a) were constructed by using only the samples in the SRS and not the whole ensemble. When a subset of samples is considered, it is not true in general that the first two POD modes contain the greatest amount of kinetic energy. Indeed, in figure 19(b) it is possible to note that the largest energy contribution to the snapshots in the DRS is provided by the third and the fourth mode.

Figure 19.

Cumulative probability distributions of the mode pair energies for (a) the SRS and (b) the DRS. Energies are scaled by the average energy in the first pair of POD modes related to each state.

The above analysis suggests that the occurrence of the SRS is related to the dominance of the energy contribution from the first two POD modes over the remaining modes, as well as the appearance of the DRS in the unsteady evolution is associated with the energetic prevalence of the third and the fourth POD modes. In order to better show this property and also to discuss the limits of the criteria for the identification of the LSC states used up to this point, in the following focus is given to the unsteady flow evolution over specific time intervals of the experiment.

Figure 20 reports a time sequence related to a transition from the DRS to the SRS. In figure 20(a) the time behaviour of the energy contributions from the POD modes to the total kinetic energy of the fluctuating velocity field is represented, whereas figure 20(b–g) show the 3-D structure of the first-order azimuthal Fourier modes of the instantaneous vertical velocity field corresponding to the time instants marked with yellow lines in figure 20(a). It is worth remarking that the first-order azimuthal Fourier mode is indeed the harmonic regression (with zero mean) of the velocity signal along the azimuthal direction. Therefore, such a mode reflects the current state of the LSC, provided that its energy is larger enough than the higher-order azimuthal modes. Based on the approach of Stevens et al. (Reference Stevens, Clercx and Lohse2011), the relative strength of the first-order azimuthal Fourier mode is evaluated using the following definition:

(5.3)\begin{equation} S(t) = \left.\left( \frac{ \tilde{E}^{(1)}(t) }{\displaystyle \sum_{i=1}^{N} \tilde{E}^{(i)}(t) } -\frac{1}{N} \right) \right/ \left( 1 - \frac{1}{N} \right).\end{equation}

Equation (5.3) is analogous to (5.2), except that here $\tilde {E}^{(1)}(t)$ and $\sum _{i=1}^{N} \tilde {E}^{(i)}(t)$, respectively, define the energy in the first-order Fourier mode and the sum of the energies in all Fourier modes of the 3-D velocity field (not only of the vertical component). In figure 20(a) the behaviour of $S(t)$ is also reported (pink curve). It is worth remarking that, according to Stevens et al. (Reference Stevens, Clercx and Lohse2011), $S(t)>0.5$ indicates a secondary effect of the plumes and/or corner flows on the coherence of the LSC (the first Fourier modes contains at least $50\,\%$ of the energy of the velocity field). With regard to the time sequence of figure 20, $S(t)$ is always above this threshold, thus the flow is organized in a well-defined LSC.

Figure 20.

A time sequence illustrating the transition from the DRS to the SRS. (a) Evolution of the contributions from the POD modes to the total kinetic energy of the flow and of the strength $S$ of the first-order azimuthal Fourier mode of the 3-D velocity field. (b–g) First-order azimuthal Fourier mode of the vertical velocity field at the time instants marked by the yellow vertical lines in panel (a): isosurfaces of the vertical velocity corresponding to the values $\pm 0.01w_0$ (red corresponds to the positive value).

In figure 20(a) the time is normalized by the free-fall time $\tau _0=H/w_0$ (which is approximately $3.87$ s in the present case). A different time scale is also reported based on the convective time $\tau _c=D/u_c$ with $u_c$ being the root mean square of the horizontal velocity over the time and the cell volume. In the literature, the large-scale eddy turnover time is usually estimated from the vertical velocity (e.g. see Bailon-Cuba et al. Reference Bailon-Cuba, Emran and Schumacher2010). The present choice of using the horizontal velocity is motivated by the observation that the dynamics of some LSC oscillatory modes, such as the torsional and the sloshing mode, is characterized by times faster than the LSC turnover time $\tau _{{LSC}}$, and such times depend on the horizontal motions in the cell. An estimate of $\tau _{{LSC}}$ is given by $\tau _{{LSC}}=H/w_c$ with $w_c$ being the root mean square of the vertical velocity over the time and the cell volume. In the present experiments $\tau _{{LSC}}\approx 3\tau _c\approx 52\tau _0$.

Focusing on the behaviour of the energy in the POD modes, from figure 20(a) it is apparent that the energy in the POD modes 3–4, after an initial increase, monotonically decreases across the DRS phase of the flow evolution and reaches a minimum in the TS phase. Conversely, the energy contribution from the POD modes 1–2, despite some fluctuations, undergoes a continual increase up to a maximum at the beginning of the SRS phase. The energy in the POD modes 5–6 reaches relevant values at the turn of the DRS phase and vanishes in the TS one. At the beginning of the sequence (time instant (b)) the prominent value of the energy in the POD modes 3–4 results in a well-defined planar double-roll structure of the flow (the two counter-rotating rolls lie in the same plane), as visible in figure 20(b). The weakening of the contribution from the POD modes 3–4 and the gradual strengthening of that from the POD modes 1–2 during the DRS phase is accompanied by a rotation of the roll in the lower half of the cell. A significant misalignment of the two rolls is, however, observed only after the energy in the POD modes 1–2 has reached a value equal to half of the energy in the POD modes 3–4 (time instant (d), see figure 20c,d). In the TS phase, the flow is much more organized in a twisted domain-filling circulation, as can be inferred from 20(f). It is interesting to note that at the time instant (f), despite the energies in the POD mode pairs 3–4 and 5–6 are in fact insignificant compared with the energy in the POD modes 1–2, the LSC does not have a planar structure. This can be associated with the non-negligible value of the energy in the higher-order modes. Finally, a planar SR structure is attained in the SRS phase when the energy in the POD modes 1–2 is dominant. In conclusion, the above example, on one side, confirms that the occurrence of the SRS and of the DRS is related to the energetic prevalence of the associated POD mode pairs, on the other side, it suggests that a possible mechanism of the transition between the DRS to the SRS is the torsional mode of the LSC.

A distinct time sequence is reported in figure 21. In this case, the flow evolves through an alternation of DRSs and TSs, identified via the criteria introduced above and based on the comparative analysis of the azimuthal velocity profiles at three different heights. The coherence of the LSC, indicated by the value of $S$, is lower than the previous case; however, for most of the time, $S(t)$ is around $0.5$, thus the structure of the first-order azimuthal Fourier mode (figure 21b–g) still provides a reliable description of the flow morphology. Interestingly, the diagram in figure 21 shows that, except in the beginning and in the final part of the selected time interval, the energy in the POD modes 3–4 is considerably smaller than the energies in the POD modes 1–2 and 5–6. Indeed, a well-defined DRS is only detectable in these two phases of the evolution (see figures 21b and 21g). At the other stages (figure 21c–f) the flow tends to organize itself in an SRS, although the LSC does not exhibit an exactly planar structure at any time; indeed, it has a twisted shape (this is associated with the contribution from the POD modes 5–6, which is comparable to that of the POD modes 1–2). It is, therefore, clear that the adopted criteria for the detection of the LSC state are imprecise in the present case. In particular, focusing on the time instant (d), the flow state is classified as DRS based on such criteria, however, an LSC in the SRS is detected both in the first-order azimuthal Fourier mode in figure 21(d) and in the full instantaneous velocity field, which is represented in figure 22. Here, both the isosurfaces (figure 22a) and the $yz$-slice (figure 22b) of the vertical velocity field show the presence of an LSC in an SRS. Stevens et al. (Reference Stevens, Clercx and Lohse2011) noticed that the misinterpretation of the state of the large-scale flow assessed using a cosine fit procedure on the temperature or the velocity profiles can be associated with passing plumes and/or the corner flows: the lower the Rayleigh number and the cell aspect ratio, the more relevant such an issue. Figure 22(c) reports the azimuthal profiles of the vertical velocity at the heights $z/H=0.25,0.5,0.75$ and radial position $r/r_i=0.9$, which have been used in the present case for the determination of the LSC orientation. The diagram clearly shows the low correlation of the cosine fits for the considered profiles; indeed, the corresponding strengths are $S_b=0.11$, $S_m=0.01$ and $S_t=0.11$. Therefore, following the approach of Stevens et al. (Reference Stevens, Clercx and Lohse2011), one might conclude that the state of the LSC is not defined at the selected time instant. Nevertheless, such a misinterpretation is purely related to the choice of the velocity probes employed in the present case. To better show this, in figure 22(d) the azimuthal vertical velocity profiles for the radial location $r/r_i=0.4$ at the bottom, the middle and the top are reported. In this case, the cosine fits show a high goodness; the corresponding strengths are $S_b=0.50$, $S_m=0.81$ and $S_t=0.81$.

Figure 21.

A time sequence illustrating an alternation between the DRS and the TS. (a) Evolution of the contributions from the POD modes to the total kinetic energy of the flow and of the strength $S$ of the first-order azimuthal Fourier mode of the 3-D velocity field. (b–g) First-order azimuthal Fourier mode of the vertical velocity field at the time instants marked by the yellow vertical lines in panel (a): isosurfaces of the vertical velocity corresponding to the values $\pm 0.01w_0$ (red corresponds to the positive value).

Figure 22.

Snapshot of the velocity at the time instant (d) of figure 21: (a) isosurfaces of vertical velocity corresponding to the values $\pm 0.02 w_0$ (red and blue surfaces for the positive and the negative value, respectively); (b) velocity vector map in the $yz$-plane, the velocity magnitude is coded in the vector length, while the colour indicates the value of the vertical velocity (scaled by the free-fall velocity $w_0$); (c,d) azimuthal profiles of the vertical velocity at the heights $z/H=0.25$ (red), $z/H=0.5$ (green) and $z/H=0.75$ (blue) for radial positions (c) $r/r_i=0.9$ and (d) $r/r_i=0.4$. The data points represent the experimental measurements, whereas the solid lines are the cosine fits based on this data.

The above analysis reveals the limitations of the criteria for the identification of the LSC state classically used in the literature. On the other side, the POD analysis appears as a powerful tool for the characterization of the flow state and more reliable criteria can be formulated on such a basis. It is evident that a well-defined LSC (in either the SRS or the DRS) results from the energetic prevalence of the first two pairs of POD modes. Thus, after denoting the energy contribution from these modes as $E^{(1-4)}(t)$, a first requirement for the occurrence of a definite LSC is that $E^{(1-4)}(t)/E_{{tot}}(t)>\varepsilon _{{LSC}}$, where $E_{{tot}}(t)=\sum _i E^{(i)}(t)$ is the total energy in the POD modes (equal to the energy of the instantaneous fluctuating velocity field) and $\varepsilon _{{LSC}}$ is a fixed threshold. On the other side, the occurrence of the SRS or the DRS is related to the relative importance of the energy in the POD mode pairs 1–2 and 3–4, i.e. on the ratio $E^{(3-4)}(t)/E^{(1-2)}(t)$. It is then convenient to define two further thresholds: $\varepsilon _{{SRS}}$ such that the SRS occurs when $E^{(3-4)}(t)/E^{(1-2)}(t)<\varepsilon _{{SRS}}$ and $\varepsilon _{{DRS}}$ such that the DRS occurs when $E^{(3-4)}(t)/E^{(1-2)}(t)>\varepsilon _{{DRS}}$. The TSs are classified as those realizations at any time $t$ such that $\varepsilon _{{SRS}}< E^{(3-4)}(t)/E^{(1-2)}(t)<\varepsilon _{{DRS}}$. Obviously, the above thresholds should be defined in such a way that $0<\varepsilon _{{LSC}}<1$, $0<\varepsilon _{{SRS}}<1$ and $\varepsilon _{{DRS}}>1$.

Figure 23 reports the p.d.f.s of both the quantities $E^{(1-4)}/E_{{tot}}$ and $E^{(3-4)}/E^{(1-2)}$. The p.d.f. of $E^{(1-4)}/E_{{tot}}$ (figure 23a) has a quasi-Gaussian shape with a peak around $0.19$ which is indeed approximately equal to the sum of the percentage energy levels of the first four POD modes ($\approx 21\,\%$), already reported in § 4. In figure 23(a) a threshold value $\varepsilon _{{LSC}}$ is also reported. This has been assumed equal to the sum of the percentage energy levels of the POD modes 3–4, which is $\approx 8.4\,\%$. Although arbitrary, such a choice is based on the idea that, when the energy contribution from the POD modes 1–2 is negligible, the occurrence of the DRS requires the instantaneous contribution from the POD modes 3–4 to be greater than its averaged value. With this threshold, the probability of occurrence of a well-defined LSC is estimated to be $89\,\%$, thus the probability of an event is $11\,\%$. In figure 23(b) the p.d.f. of $E^{(3-4)}/E^{(1-2)}$ is built by considering only the samples where a well-defined LSC is detected (i.e. such that $E^{(1-4)}/E_{{tot}}>\varepsilon _{{LSC}}=8.4\,\%$). It is possible to see that this p.d.f. is monotonically decreasing and the probability that $E^{(3-4)}\gg E^{(1-2)}$ is small. The thresholds values $\varepsilon _{{SRS}}$ and $\varepsilon _{{DRS}}$ are also denoted in the same figure with dashed lines. In particular, $\varepsilon _{{SRS}}$ has been chosen equal to the statistically average value of the quantity $E^{(3-4)}/E^{(1-2)}$, whereas $\varepsilon _{{DRS}}=1+(1-\varepsilon _{{SRS}})$; the resulting values are $\varepsilon _{{SRS}}=0.67$ and $\varepsilon _{{DRS}}=1.33$. With these thresholds, the probability of occurrence for the SRS, the TS and the DRS are, respectively, $49\,\%$, $18\,\%$ and $33\,\%$, among states that are not classified as events. Therefore, compared with the values determined with the criteria classically used in the literature ($29\,\%$, $44\,\%$ and $27\,\%$, respectively, for the SRS, the TS and the DRS), a greater frequency of occurrence of the SRS and lower frequency of occurrence of the TS are found, whereas the frequency of occurrence of the DRS is substantially unchanged. This suggests that the criteria adopted in the literature lead essentially to a misinterpretation of the flow state in the SRS, which is often classified erroneously as a TS.

Figure 23.

Criteria for the identification of the flow state based on POD analysis of the velocity field. (a) P.d.f. of the ratio of the energy in the first four POD modes to the total energy of the fluctuating velocity field. The dashed line indicates the threshold for the identification of a well-defined LSC. (b) P.d.f. of the ratio of the energy in the POD modes 3–4 to the that in the POD modes 1–2. The dashed lines indicate the threshold for the identification of the SRS and the DRS.

The evidence that the first two POD modes contribute more to the SRS allows us to conceive a different way of detecting the LSC orientation, based on the correlation of the snapshots with the first two POD modes. Such a technique and its relationship with the method adopted above is discussed in the next section.

5.4. Identification of LSC orientation based on POD

In this section, an alternative method of detecting the LSC orientation is introduced. Such a method relies on an LOR of snapshots based on only the first two POD modes, which, as aforementioned, contribute more to the SRS. In principle, the following steps have to be performed: reconstructing each snapshot by combining the mean velocity field and the contributions from the first two POD modes; extracting the azimuthal profile of the vertical velocity at midheight; estimating the parameters of the cosine fit model and thus the LSC orientation $\varphi _m$. The main advantage of this method is that the contributions of secondary modes, such as torsion and sloshing, are filtered out by the LOR process, since they are related to the higher-order POD modes. To ease computation, in the second step, it is possible to extract the azimuthal profiles of the vertical velocity at midheight for the first and the second mode and combining them with the corresponding POD mode temporal coefficients to obtain the velocity profile for a selected snapshot. A more robust procedure consists in combining the profiles reported in figure 7, i.e. the azimuthal profiles of the axially and radially averaged velocity for the first and the second mode. This is more accurate since, as shown in § 4, the SR in the first two POD modes has a twisted structure.

Figure 24(a) presents the p.d.f. of the LSC orientation based on the method described above (denoted by the symbol $\varPhi$). Differently from the p.d.f. of $\varphi _m$ (figure 16a), the p.d.f. of $\varPhi$ does not feature a well-defined Gaussian distribution. However, a preferred orientation is still detected, approximately at $\varPhi \approx -12^{\circ }$. This indeed corresponds to the phase angle of the blue curve in figure 7. Such a result is expected since the first POD mode should in principle capture a preferential orientation of the LSC, if any. The fact that the method based on the cosine fit of the azimuthal velocity profile at midheight does not present a prominent peak at the LSC orientation of the first POD mode could indicate a bias in the estimation of the LSC orientation. Indeed, when the p.d.f. of the difference between $\varPhi$ and $\varphi _m$ is plotted (blue dots in figure 24b), a peak at the above-mentioned angle is not observed. However, when restricting to only the samples related to the SRS, a peak appears in the p.d.f. The origin of such a bias is not clear and further investigations are needed on this point.

Figure 24.

Statistical behaviour of the LSC orientation identified by means of the first two POD modes: (a) p.d.f.; (b) comparison with the LSC orientation identified by the cosine fit of the azimuthal velocity profile at $z/H=0.5$. The SRS is identified with criteria based on the analysis of the vertical velocity azimuthal profile.