Thermal convection of liquid sodium in inclined cylinders

The effect of inclination on the low Prandtl number turbulent convection in a cylinder of unit aspect ratio was studied experimentally. The working fluid was sodium (Prandtl number $\mathrm{Pr}=0.0094)$, the measurements were performed for a fixed Rayleigh number $\mathrm{Ra}=(1.47\pm 0.03)\times {10}^7$, and the inclination angle varied from $\beta =0^{\circ }$ (the Rayleigh–Bénard convection, the temperature gradient is vertical) up to $\beta ={90}^{\circ }$ (the applied temperature gradient is horizontal) with a step $\mathrm{\Delta }\beta ={10}^{\circ }$. The effective axial heat flux characterized by the Nusselt number is minimal at $\beta =0^{\circ }$ and demonstrates a smooth growth with the increase of the cylinder inclination, reaching a maximum at angle $\beta \approx {70}^{\circ }$ and decreasing with a further increase of $\beta$. The maximal value of the normalized Nusselt number $\mathrm{Nu}\left(\beta \right)/\mathrm{Nu}\left(0\right)$ was 1.21. In general, the dependence of $\mathrm{Nu}\left(\beta \right)$ in a cylinder with unit aspect ratio is similar to what was observed in sodium convection in inclined long cylinders but is much weaker. The structure of the flow undergoes a significant transformation with inclination. Under moderate inclination $\left(\beta \lesssim {30}^{\circ }\right)$, the fluctuations are strong and are provided by regular oscillations of large-scale circulation (LSC) and by turbulence. Under large inclination $\left(\beta >{60}^{\circ }\right)$, the LSC is regular and the turbulence is weak, while in transient regimes $\left({30}^{\circ }<\beta <{60}^{\circ }\right)$, the LSC fluctuations are weak and the turbulence decreases with inclination. The maximum Nusselt number corresponds to the border of transient and large inclinations. We find the first evidence of strong LSC fluctuations in low Prandtl number convective flow under moderate inclination. The rms azimuthal fluctuations of LSC, about ${27}^{\circ }$ at $\beta =0^{\circ }$, decrease almost linearly up to $\beta ={30}^{\circ }$, where they are about $9^{\circ }$. The angular fluctuations in the vicinity of the end faces are much stronger (about ${37}^{\circ }$ at $\beta =0^{\circ })$ and weakly decrease up to $\beta ={20}^{\circ }$. The strong anticorrelation of the fluctuations in two halves of the cylinder indicates the torsional character of LSC fluctuations. At $\beta ={30}^{\circ }$, the intensity of the oscillations at the periphery of the cylinder falls sharply to the level of oscillations in the central plane and the anticorrelation disappears; the torsional fluctuations vanish.

Figure 1

Experimental setup: 1: convective cell, 2: hot heat exchanger, 3: cold heat exchanger, 4: expansion vessel, 5: inductor.

Figure 2

Convective cell: the bottom view (left); the cross section M-M with five thermocouples for the A to E lines (center); the cross sections N-N with three thermocouples for the B, C, D, and F, G, H lines (right).

Figure 3

Mean vertical velocity, measured along the lines A and E, versus inclination angle. The size of the points corresponds in this plot to error bars.

Figure 4

Standard deviation of temperature versus inclination angle. Large red points show the STD averaged over all thermocouples, small green points show the STD for thermocouple A3, and blue points for thermocouple C2.

Figure 5

Time-averaged azimuthal distribution of axial velocity, cm/s (left column), relative temperature, K (central column), and STD of temperature, K (right column), for all inclination angles examined from $\beta =0^{\circ }$ to $\beta ={90}^{\circ }$ (top-down). The mean axial velocity is estimated by measuring the cross correlations between thermocouples in circles 1 and 3 (red) and between 3 and 5 (blue). The temperature measurements are done in three cylinder cross sections: circle 1: red, circle 3: green, circle 5: blue.

Figure 6

Typical time series of the strength $\delta$ (a) and orientation ${\theta }_0$ (b) of the $m=1$ azimuthal mode, measured in the central cross section (thermocouples A3–H3).

Figure 7

Mean value (a) and STD (b) of the strength $\delta$ versus the inclination angle $\beta$.

Figure 8

STD of the orientation angle ${\theta }_0$ in three cylinder cross sections versus the inclination angle $\beta$.

Figure 9

Power spectral density of the strength $\delta$ (a) and orientation ${\theta }_0$ (b) for some inclination angles $\beta$. Measurements are done in circle 1 (thermocouples A1–H1).

Figure 10

Wavelet spectrograms of the orientation ${\theta }_0$ for inclination angles $\beta =(0^{\circ },{10}^{\circ },{20}^{\circ },{30}^{\circ },{40}^{\circ })$. Measurements are done in circle 1 (thermocouples A1–H1).

Figure 11

Cross-correlation function for orientation angles ${\theta }_0$, measured by thermocouples in circles 1 and 5.

Figure 12

Normalized Nusselt number versus inclination angle.