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

We present observations of oscillations in the shape of the temperature profile of the large-scale circulation (LSC) of turbulent Rayleigh-Bénard convection. Temperature measurements are broken down into Fourier moments as a function of $\theta -{\theta }_0$, where $\theta$ is the azimuthal angle in a horizontal plane at midheight, and ${\theta }_0$ is the LSC orientation. The oscillation structure is dominated by a third-order sine moment and third-order cosine moment in a cubic cell. In contrast, these moments are not found to oscillate in a cylindrical cell. This geometry-dependent behavior can be explained by a minimal model that assumes that the heat transported by the LSC is conducted from the thermal boundary layers, and is proportional to the pathlength of the LSC along boundary layers at the top and bottom plates. In a noncircular cross-section cell, oscillations of the LSC orientation ${\theta }_0$ result in an oscillation in the container shape in the reference frame of the LSC, resulting in an oscillation in the pathlength of the LSC at a given $\theta -{\theta }_0$. In a square-cross-section cell, this model predicts the dominant third-order sine moment and third-order cosine moment with magnitudes within 50% of measured values, when using the amplitude of the oscillation of ${\theta }_0$ as input. A cylindrical cell is special in that the pathlength is independent of ${\theta }_0$, and so these oscillating moments are not induced. In a cylindrical cell, the model reproduces the sinusoidal mean temperature profile with a sloshing oscillation dominated by the second-order sine moment, consistent with previous observations in that geometry.

Figure 1

A schematic of the experimental setup. (a) Top view: The thick line represents the central circulation plane of the LSC, with ${\theta }_0$ defined as the orientation relative to a diagonal. Thermistor locations on the sidewall are indicated by solid circles. (b) Side view.

Figure 2

Power spectrum of the Fourier moments $A_n$, as indicated in the legend. Thin lines show the linear background $P_{bg}$ subtracted off when integrating the peak power. Each moment oscillates, except for $A_2$.

Figure 3

Power spectrum of the Fourier moments $B_n$ and other terms of the temperature profile, as indicated in the legend. Each signal oscillates except for $\delta -\langle \delta \rangle$.

Figure 4

Power spectrum of ${\theta }_0-\langle {\theta }_0\rangle$. The oscillations of the Fourier moments have the same frequency as the oscillation of ${\theta }_0$.

Figure 5

Solid line: Illustration of a temperature profile with an $A_3$ moment with amplitude of $0.15\delta$. Dotted line: $cos(\theta -{\theta }_0)$.

Figure 6

Power spectra of moments of the temperature profile as indicated in the legend, for a cylindrical container [24]. $A_2$ is the dominant oscillating mode, corresponding to a sloshing back and forth of the LSC. This lacks the strong $A_3$ and $B_3$ moments found for a cubic container.

Figure 7

Pathlength of the LSC along the top and bottom plates as a function of the orientation ${\theta }_f$ of the flow and coordinate $\theta$ for (a) a circular cross-section cell, and (b) a square cross-section cell. The thick solid lines with arrows indicate the central path of the LSC. The parallel lines indicate pathlengths at different $\theta$.

Figure 8

Thick solid black line: Example of a predicted temperature profile based on the pathlength $\lambda \left(\theta \right)/H$ along the top and bottom plates of a cubic cell for ${\theta }_f=0.3$ rad. Dashed blue line: Cosine fit [Eq. (6)] as a first-order approximation of the temperature profile. Dotted red line: Difference between the predicted temperature profile and first-order cosine term. Thin solid red line: Fit of third-order sine moment ($A_3$) to the difference. The difference is dominated by the $A_3$ moment.

Figure 9

Time series of Fourier moments of the temperature profile predicted from the assumed pathlength in Eqs. (7), (8), and (9). The $A_3$ moment is dominant, followed by $B_3$. The other moments shown in the legend are not excited.

Figure 10

Thick solid black line: Example of the predicted temperature profile at an intermediate phase of the oscillation where ${\theta }_f=0$ rad. Dashed blue line: Cosine fit [Eq. (6)] used for first-order approximation of the temperature profile. Dotted red line: Difference between the predicted temperature profile and first-order cosine term. Thin solid red line: Fit of moment $B_3$ to the difference. The difference has the symmetry of odd-order $B_n$ moments.

Figure 11

Predicted oscillation amplitudes of Fourier moments as a function of the amplitude of the driving oscillation amplitude ${\theta }_{0,p}$.

Figure 12

Predicted oscillation amplitudes of Fourier moments as a function of the offset of the preferred orientation ${\theta }_p$ from the corner for ${\theta }_{0,p}=0.037$ rad. The offset of ${\theta }_p$ leads to an increase in $B_{3,p}$.

Figure 13

Thick solid black line: Example of predicted temperature profile due to the $n=2$ oscillation in a cylinder for ${\theta }_f=0.3$ rad. Thick dashed blue line: Cosine fit [Eq. (6)] as a first-order approximation of the temperature profile. Dashed-dotted red line: Difference between the predicted temperature profile and first-order cosine term. Short-dashed red line: Fit of $A_2$ to the difference. Long-dashed red line: Fit of $A_4$ to the difference. Dotted orange line: Difference between the predicted temperature profile and first-order cosine term smoothed over a range $\pm \pi /4$. Thin solid orange line: Fit of the $A_2$ moment to the smoothed difference. In either case, the difference is dominated by the $A_2$ moment, as observed in experiments. The $A_4$ moment is highly dependent on smoothing.