Oscillation in the temperature profile of the large-scale circulation of turbulent convection induced by a cubic container

The design of the experimental setup is the same apparatus used in Ref. Bai et al. (2016), where it is described in more detail. We describe it briefly here. The flow chamber is nearly cubic, with height $H=20.32$ cm, and horizontal dimensions of $20.04$ cm and $19.99$ cm, as shown in Fig. 1. The top and bottom surface are aluminum plates, with double-spiral water-cooling channels. To control the plates at a temperature difference of $\Delta T$, with the bottom plate hotter than the top plate, water is circulated through each plate by a temperature-controlled water bath. The sidewalls are made of plexiglas, and three out of four have a thickness of 0.55 cm. The fourth sidewall has thickness $0.90$ cm, and is shared with another identical flow chamber to be used in future experiments, but the two chambers are designed to be thermally insulated from each other for the experiments reported here.

The working fluid is degassed and deionized water with mean temperature of 23.0${}^{\circ}\mathrm{C}$, for a Prandtl number $Pr=\nu/\kappa=6.4$, where $\nu=9.4\times 10^{-7}$ m²/s is the kinematic viscosity and $\kappa$ is the thermal diffusivity. The Rayleigh number is given by $Ra=g\alpha\Delta TH^{3}/\kappa\nu$ where $g$ is the acceleration of gravity and $\alpha$ is the thermal expansion coefficient. We report measurements at $\Delta T=18.35$ ^∘C for $Ra=2.62\times 10^{9}$.

The alignment of the LSC is partially locked along a diagonal to prevent corner-switching with a small tilt of the cell relative to gravity, as is commonly done Krishnamurti and Howard (1981). We report data for a tilt angle of $\beta=(1.0\pm 0.2)^{\circ}$, at an orientation within $0.03$ rad of the diagonal where we define $\theta_{0}=0$.

To measure the LSC, thermistors were mounted in the sidewalls. There are 3 rows thermistors at heights $-H/4$, $0$, and $H/4$ relative to the mid-height of the container, as shown in Fig. 1b. In each row there are 8 thermistors equally spaced in the azimuthal angle $\theta$ measured in a horizontal plane, as shown in Fig. 1a. In this manuscript, we only report measurements from the middle row of thermistors, unless otherwise stated. Temperatures were recorded every 9.7 s at each thermistor.

The LSC orientation was obtained using the same methods as Ref. Brown and Ahlers (2006). As the LSC pulls hot fluid from near the bottom plate up one side and cold fluid from near the top plate down the other side, a temperature difference is detected along a horizontal direction at the mid-height of the container. We fit the 8 thermistor temperatures at the middle row to the function

$$T=T_{0}+\delta cos(\theta-\theta_{0})$$

(1)

to get the orientation $\theta_{0}$ and strength $\delta$ of the LSC at each time step.

We first fit Eq. 1 to $T(\theta)$ to obtain $\theta_{0}$ at each time step. This allows viewing of the temperature profile relative to the LSC orientation as $T(\theta-\theta_{0})$ to separate out meandering and oscillations of the LSC orientation from changes in the temperature profile. We then express perturbations on the sinusoidal temperature profile as the Fourier series

$$T=T_{0}+\delta\cos(\theta-\theta_{0})+\sum_{n=2}^{4}A_{n}\sin(n(\theta-\theta_{0}))\\ +\sum_{n=2}^{4}B_{n}\cos(n(\theta-\theta_{0}))\ ,$$

(2)

where the Fourier moments are calculated as

$$A_{n}=\frac{1}{4}\sum_{i=1}^{8}{[T_{i}-T_{0}-\delta\cos(\theta-\theta_{0})]\sin[n(\theta-\theta_{0})]}$$

(3)

and

$$B_{n}=\frac{1}{4}\sum_{i=1}^{8}{[T_{i}-T_{0}-\delta\cos(\theta-\theta_{0})]\cos[n(\theta-\theta_{0})]}\ .$$

(4)

Here the sum over $i$ corresponds to the sum over different thermistors. We can only calculate the first 4 moments due to the Nyquist limit with 8 measured temperatures. We don’t include $B_{1}$ since that was already defined as $\delta$, and $A_{1}$ is trivially zero since we first fit to Eq. 1.

To check for consistency of data with expected symmetries, we measured phase shifts between different oscillating moments by calculating correlation functions (not shown) as in Ji and Brown (2020). Notably, the oscillation of moment $A_{4}$ is found to be in phase with $\theta_{0}$. An in-phase superposition of these two moments results in an asymmetry between the hot and cold sides of the LSC at the mid-height where the hot side oscillates with a larger amplitude than the cold side. This asymmetry between the hot and cold sides of the flow appears to be a form of non-Boussinesq asymmetry, similar to an observation of a very weak oscillation of $A_{2}$ out-of-phase by $\pi$ rad with $\theta_{0}$ Ji and Brown (2020). To test whether this is a non-Boussinesq effect due to the change in material parameter values with the temperature on opposite sides of the cell, or due to some other asymmetry of the setup, we compare data from a similar dataset where the cell was nearly level but tilted slightly to cancel out asymmetries so that the LSC orientation sampled all 4 corners of the container Bai et al. (2016). When calculating the phase shift between $A_{4}$ and $\theta_{0}$ separately for each potential well, we find them to be in phase in 2 wells and $\pi$ out of phase in the other 2 wells. This difference between potential wells suggests that the existence of $A_{4}$ in the long-term average is likely due to an asymmetry of the experimental setup. An alternative possibility is that it is a long-lived spontaneous-symmetry-breaking mode that did not have time to sample all of the possible phase shifts, although it was observed in a dataset that was 10 days long.

In a cell without circular symmetry, changes in the orientation $\theta_{0}$ lead to a change in the pathlength with both $\theta$ and $\theta_{f}$ as the shape of the container changes in the reference frame of the LSC. The $n=1$ advected oscillation mode was observed in experiments with the same cubic cell that we report data from in this paper Ji and Brown (2020). The preferred flow orientation is aligned along a diagonal, with oscillations in $\theta_{0}$ centered on a diagonal. While $\theta_{0}=0$ at the center of the top and bottom plates in the $n=1$ mode, assuming the LSC runs in a rectangular path along the inner wall of the container, the peak amplitude of the oscillation in $\theta_{0}$ at the edges of the plates is still expected to be $\sin[\pi/2\times\sqrt{2}/(1+\sqrt{2})]=0.79$ times the amplitude at the mid-height. These hot and cold spots advect upward and downward, respectively, so that the thermal profile obtained at the top and bottom plates passes the mid-height at the same time. The hot and cold spots are also predicted to advect azimuthally Brown and Ahlers (2009), and azimuthal flow has been observed Vasiliev et al. (2019). Because the displacements in $\theta_{0}$ are in phase at the top and bottom plates for the $n=1$ mode, the restoring force pushes them in the same direction such that they shift the entire temperature profile so that $\theta_{f}$ is closer to zero at the mid-height. Thus, even with this azimuthal advection, the predicted temperature profile at the mid-height is the same as the pathlength across the plate in terms of $T(\theta-\theta_{0}$).

For a square cross-section with origin $\theta=0$ in a corner as shown in Fig. 7, the pathlength at the top and bottom plates is given by

$$\frac{\lambda(\theta)}{H}=\frac{1}{\cos(\pi/4-|\theta_{f}|)}$$

(7)

for $0<\theta<\pi/2$ and $[1+\tan(\theta-\pi/4)]/2>\tan(\pi/4-|\theta_{f}|)$, which is a constant independent of $\theta$, and

$$\frac{\lambda(\theta)}{H}=\frac{1-\tan(\theta-\pi/4)}{2\sin(\pi/4-|\theta_{f}|)}$$

(8)

for $0<\theta<\pi/2$ and $[1+\tan(\theta-\pi/4)]/2<\tan(\pi/4-|\theta_{f}|)$, and

$$\frac{\lambda(\theta)}{H}=\frac{1+\tan(\theta-\pi/4)}{2\cos(\pi/4-|\theta_{f}|)}\ .$$

(9)

for $-\pi/2<\theta<0$. This profile is repeated periodically every $\pi$ rad. An example plot is shown in Fig. 8.

To determine the proportionality coefficient in the temperature profile of Eq. 6 for a non-circular cross-section, we use the assumption of the Brown-Ahlers model that the flow velocity at the sidewall in the direction of the LSC (and thus $\delta)$ does not vary with small changes in diameter Brown and Ahlers (2008a). This choice had a significant consequence in determining the form of the potential in $\theta_{0}$, for example with potential minima at the corners Brown and Ahlers (2008a). In contrast, allowing $\delta$ to vary with the pathlength along $\theta_{0}$ would have led to a different potential, with a prediction that corners would no longer be the most preferred orientations, which disagrees with observations Zimin and Ketov (1978); Zocchi et al. (1990); Qiu and Xia (1998); Valencia et al. (2007); Liu and Ecke (2009); Bai et al. (2016); Foroozani et al. (2017); Giannakis et al. (2018); Vasiliev et al. (2016, 2019); Ji and Brown (2020). To keep consistent with the previous model assumption Brown and Ahlers (2008a) and experimental observations, the temperature profiles should be normalized by $\delta$ to remove variations in $\delta$ with $\theta_{0}$, so the proportionality coefficient should be $\langle\delta\rangle^{2}/\delta$, to retain the result that the mean value of the coefficient is $\langle\delta\rangle$. However, since $\delta$ is calculated from the temperature profile, this normalization is done after the shape of the temperature profile is calculated and $\delta$ obtained from a fit of the temperature profile. In practice, this leads to a correction of less than 0.3% in the rms amplitudes of Fourier moments in the range of measured parameters shown in the following Figs. 11 and 12, but more importantly it eliminates a large oscillation in $\delta$ that would occur if we instead assumed the proportionality in Eq. 6 were a constant.

In principle, oscillations in the shape of the temperature profile can occur if the pathlength of the LSC oscillates over time. Since we can analyze oscillation modes without adding noise, which would just obfuscate the results, we include the effect of the oscillation in $\theta_{0}$ as a deterministic time series $\theta_{f}(t)=\theta_{f,max}\cos(\omega t)$, where $\theta_{f,max}$ is the amplitude of the oscillation in $\theta_{f}$ at the edge of the top and bottom plates.

We calculate time series for the temperature profile from Eq. 6 using the pathlength of Eqs. 7, 8, and 9 for a square cross-section and oscillatory $\theta_{f}(t)$. The temperature was sampled at 8 equally spaced orientations, mimicking our thermistor measurements. The resulting temperature profile was fit at each time step to obtain the LSC orientation $\theta_{0}$ using Eq. 1. Fourier moments were then calculated at each time step using Eqs. 3 and 4. An example time series of Fourier moments is shown in Fig. 9 for $\langle\delta\rangle=0.431$ K and $\theta_{f,max}=0.3$ rad. The $A_{3}$ moment has a dominant oscillation, followed by the $B_{3}$ moment. The $A_{3}$ moment also has the dominant oscillation in the experimental data, followed by the $B_{3}$ moment (not counting $A_{4}$ which violates symmetry expectations).

The dominance of the $A_{3}$ moment in the temperature profile is illustrated by comparing the predicted temperature profile (Eq. 6) at the phase where $\theta_{f}$ peaks to the 1st-order approximation of the cosine profile (Eq. 1) in Fig. 8. The difference between the predicted temperature profile and 1st-order cosine profile is shown as the dotted line in Fig. 8. A fit of the $A_{3}$ moment to the difference in Fig. 8 shows that it is dominated by that moment. If $\theta_{f}$ oscillates, then Eq. 6 leads to oscillations of $A_{3}$ as is shown in Fig. 9. Thus, the dominant $A_{3}$ moment can be understood as a result of the change in pathlength in the reference frame of the LSC as it oscillates around the corner of a cubic cell.

To visualize the $B_{3}$ moment, the predicted temperature profile for the intermediate phase where $\theta_{f}=0$ is shown in Fig. 10. It has the symmetry of odd-order $B_{n}$ moments (antisymmetric around $\theta=\pm\pi/2$ and symmetric around $\theta=0$). With 8 thermistors, $B_{3}$ is the only odd-order $B_{n}$ moment that we can detect (shown by the fit in Fig. 10) due to the Nyquist limit. The $B_{3}$ moment does not capture the full difference between the predicted temperature profile and the cosine fit shown in Fig. 10, and a $B_{5}$ moment with similar magnitude to the $B_{3}$ moment is apparent in the predicted profile by inspection.

The details of the time series shown in Fig. 9 vary with the number and location of temperature measurements. For example, at the phase shown in Fig. 10, the arrangement of our thermistors yields a simulated measurement of $B_{3}=0$ at that phase even though the complete temperature profile in Fig. 10 leads to a maximum of $B_{3}$ at this phase. However, this moment is still detected in the time series at other intermediate phases, suggesting different thermistor arrangements will still detect the same moments, but with different amplitudes. The model temperature profile actually predicts $B_{3}$ is the strongest moment for small amplitude oscillations of $\theta_{0}$ (not shown), but our thermistors are not arranged to detect this moment optimally. At least 10 evenly-spaced thermistors would be required to detect the predicted $B_{5}$ moment.

A feature of note is that the fit of the temperature profile in Fig. 8 yields the amplitude of the oscillation in $\theta_{0}$ to be $0.18$ rad instead of the input $\theta_{f,max}=0.3$ rad, indicating that $\theta_{0}$ from the cosine fit of the temperature profile underestimates the flow orientation $\theta_{f}$ in cubic cells. $\theta_{0}$ and $\theta_{f}$ still oscillate together, and there is a one-to-one relationship between $\theta_{0}$ and $\theta_{f}$, so $\theta_{0}$ can work as a proxy for the flow direction as long as it is acknowledged that there is a conversion function between them in non-circular cross-section cells.

Phase shifts between different signals can be identified from correlation functions, many of which were already reported for the cubic cell dataset Ji and Brown (2020). The observed in-phase behavior of $A_{3}$ at all 3 rows of thermistors is expected from the model, as it depends on the driving $\theta_{0}$-oscillation which is predicted to be in-phase at all 3 rows of thermistors in the $n=1$ advected mode Ji and Brown (2020) . The observation that $A_{3}$ is $\pi$ rad out of phase with $\theta_{0}$ Ji and Brown (2020) is consistent with the picture that azimuthal advection shifts $\theta_{f}$ in Fig. 8 to 0 at the mid-height of the container so that $\theta_{0}$ is slightly negative while $A_{3}$ is positive. This extra contribution to the measured $\theta_{0}$ superposed on the driving oscillation in $\theta_{0}$ may also explain why $\theta_{0}$ was not measured to be in-phase at all 3 rows of thermistors as predicted for the $n=1$ advected mode Ji and Brown (2020).

To test whether the model quantitatively captures the observed moment amplitudes reported in Table 1, we varied the input oscillation amplitude $\theta_{f,max}$ and compare amplitudes of oscillating signals. Figure 11 shows the root-mean-square (rms) amplitudes of the oscillating signals vs. the rms of the oscillating component of the LSC orientation $\theta_{0,p}$ for $\langle\delta\rangle=0.431$ K (matching the measured value of $\langle\delta\rangle$). Generally, $A_{3,p}$ is dominant, followed by $B_{3,p}$, and both moments increase nonlinearly with $\theta_{0,p}$. We find an input value of $\theta_{f,max}=0.074$ rad matches the observation of $\theta_{0,p}=0.037$ rad from Table 1. For these input parameter values, the model yields the amplitudes $A_{3,p}=17$ mK and $B_{3,p}=1.4$ mK, somewhat smaller than measured root-mean-squared values of $A_{3,p}=28$ mK and $B_{3,p}=7$ mK for the middle row.

We tried smoothing the pathlength function as in Ref. Song et al. (2014), and found that it did not introduce any different oscillating moments, and has less than 1% effect on the magnitudes on moment amplitudes.

The preceding model assumes that the oscillation of $\theta_{0}$ is centered around the corner of the cell. In actual experiments, the oscillation is not centered around the corner, with the preferred orientation at an angle $\theta_{p}=0.05$ rad away from the corner, mainly due to a nonuniform temperature profile in the heating and cooling plates Ji and Brown (2020). This offset can change the induced temperature profile in the coordinate $\theta-\theta_{0}$, leading to different Fourier moment amplitudes than found in Sec. V.4. To determine the effect of this offset on the Fourier moments, we added the preferred orientation to the model for the oscillation of the flow direction: $\theta_{f}(t)=\theta_{f,max}\cos(\omega t)+\theta_{p}$. Figure 12 shows rms amplitudes of Fourier moments as a function of the preferred orientation $\theta_{p}$ for $\theta_{0,p}=0.037$ rad. $B_{3,p}$ increases significantly with $\theta_{p}$, and no new moments are observed. At the measured offset of $\theta_{p}=0.05$ rad, the model yields $A_{3,p}=17$ mK and $B_{3,p}=3.9$ mK. This is within 50% of the observed magnitudes of 28 mK and 7 mK, respectively.

Since the pathlength model can predict the existence of the new $A_{3}$ and $B_{3}$ moments, it is worth checking if it can recover the temperature profiles of other known oscillations in temperature profiles, in particular the sloshing oscillation in a circular cross-section container. The observed sloshing mode has peak amplitude at the mid-height Xi et al. (2009); Zhou et al. (2009); Brown and Ahlers (2009), and an oscillating $A_{2}$ moment Brown and Ahlers (2009). This is part of the $n=2$ advected oscillation mode (with 2 oscillation periods per LSC rotation) which also has a twisting component corresponding to oscillations in $\theta_{0}$ at the top and bottom rows of thermistors out of phase with each other Funfschilling and Ahlers (2004); Brown and Ahlers (2009). The twisting oscillation is $\pi/2$ rad out of phase with the sloshing oscillation.

The pathlength-based model predicts that the cosine-shaped pathlengths on the hot and cold sides of the cell are advected upward and downward, respectively, to meet at the middle row. Both cosine profiles at the hot and cold sides are advected azimuthally by a displacement $\theta_{f,max}$, but for the $n=1$ mode they advect in opposite directions in $\theta$ due to the twist displacements being on opposite sides at the hot and cold plates. This results in a temperature profile where the hot and cold extrema are pushed closer together with the antisymmetry of the sloshing mode.

We show an example of this temperature profile in Fig. 13 for $\theta_{f}=0.3$ rad. At the interfaces where the hot and cold sides are pushed together or pulled apart, we do not know how the temperature profile evolves, so as a simplest model we assume there is no advection past $\theta=\pi/2$ and the temperature around $-\pi/2$ is close to the average $T_{0}$. The non-zero moments of the temperature profile for $n\leq 4$ are $A_{2,p}\approx 0.56\langle\delta\rangle\theta_{f,max}$ and $A_{4,p}\approx 0.21\langle\delta\rangle\theta_{f,max}$. The difference between the predicted temperature profile and its 1st-order cosine fit, as well as the $A_{2}$ and $A_{4}$ moments, are shown in Fig. 13. This produces a dominant $A_{2}$ moment as observed, but also a significant $A_{4}$ moment, which is not observed in experiments Brown and Ahlers (2009).

Since a non-smooth temperature profile is unrealistic, as there will be some turbulent mixing to smooth the temperature profile, we considered the effects of smoothing in $\theta$. Smoothing in general will have a tendency to reduce the contribution of higher order moments in the temperature profile. For example, smoothing the difference in the predicted temperature profile from the 1st-order cosine fit over a range of $\pm\pi/4$ rad is shown in Fig. 13. This smoothing is even more dominated by the $A_{2}$ moment with $A_{2,p}\approx 0.39\langle\delta\rangle\theta_{f,max}$ and $A_{4,p}\approx 0.04\langle\delta\rangle\theta_{f,max}$. Due to the approximate nature of the model, we can only predict orders of magnitude for the effect of thermal mixing. Since vertical temperature gradients are comparable to horizontal temperature gradients in the bulk Brown and Ahlers (2007), the turbulent thermal mixing which is solely responsible for the vertical temperature gradient is comparable in magnitude to effects that generate the azimuthal temperature profile, and thus we cannot give a precise prediction of how much smoothing is expected from mixing. Even without a precise prediction of the effects of thermal mixing, we can still say that the temperature profile is a combination of $A_{2}$ and potentially $A_{4}$ terms. Note that including the effects of thermal mixing more generally in the model is expected to similarly smooth the temperature profile, reducing the magnitudes of higher order moments more severely than lower order moments, but would not fundamentally change which moments can be excited by the geometry, or the order of magnitude estimates.

To compare the magnitudes of the predicted moments with experimental measurements, we measure an amplitude from the peak power of $\theta_{b,p}=0.094$ rad at the bottom row of thermistors to represent the oscillation in $\theta_{0}$ for data from the cylindrical container. The amplitude of the advected oscillation in $\theta_{0}$ at the bottom row of thermistors is not as large as at the edge of the bottom plate. If we assume a rectangular pathlength around the inner edge of the aspect ratio 1 container, then $\theta_{f,max}=\theta_{b,p}\sin(\pi/4)/\sin(\pi/8)=0.173$ rad. For this $\theta_{f,max}$ and the observed $\langle\delta\rangle=0.255$ K, Eq. 6 predicts $A_{2,p}=17-24$ mK and $A_{4}=2-9$ mK, where the range is given from zero smoothing to smoothing by $\pm\pi/2$. This overestimates the measured $A_{2,bot}=11$ mK at the mid-height in the cylindrical container by about a factor of 2, and is 1-4 times the resolution limit for measurements of the $A_{4}$ moment, so is reasonably consistent with the measurement of $A_{2}$ and lack of observed $A_{4}$ for an approximate model.

The situation is expected to be similar for the sloshing aspect of the $n=1$ advected oscillation mode in non-circular cross sections. The $n=1$ mode includes sloshing at the top and bottom rows of thermistors, but not at the mid-height Brown and Ahlers (2009); Ji and Brown (2020). We expect horizontal advection of the temperature profile will move $\theta_{f}$ from the in-phase twist displacement near the top and bottom plates by different amounts due to the different times it takes to reach the top and bottom rows of thermistors from the top and bottom plates, so that the two temperature extrema have moved closer together to produce a slosh displacement. Due to the conceptual similarity with the slosh profile for the cylinder, it is not shown here. Due to the ambiguity of how to define the temperature profile near the border of the hot and cold sides, any breakdown of the slosh profile into $A_{2}$ and $A_{4}$ components would be again highly dependent on assumptions about smoothing, and our measurement of $A_{4}$ is plagued by large asymmetries of the setup, any quantitative results we could show would not be very informative.

The jumprope oscillation is the other mode that is known to cause an oscillation in the shape of the temperature profile, specifically with the symmetries of even order $B_{n}$ moments and $T_{0}$ Vogt et al. (2018). However, since this is known to consist of the core of the LSC oscillating around in the central plane of the LSC, it does not involve changes in pathlengths of parallel paths along the top and bottom plates in the sense discussed here, so that temperature profile oscillation should be described by a different mechanism.