Collapse of Coherent Large Scale Flow in Strongly Turbulent Liquid Metal Convection

The large-scale flow structure and the turbulent transfer of heat and momentum are directly measured in highly turbulent liquid metal convection experiments for Rayleigh numbers varied between $4\times {10}^5$ and $\le 5\times {10}^9$ and Prandtl numbers of $0.025\le \mathrm{Pr}\le 0.033$. Our measurements are performed in two cylindrical samples of aspect ratios $\mathrm{\Gamma }=\text{diameter}/\text{height}=0.5$ and 1 filled with the eutectic alloy GaInSn. The reconstruction of the three-dimensional flow pattern by 17 ultrasound Doppler velocimetry sensors detecting the velocity profiles along their beam lines in different planes reveals a clear breakdown of coherence of the large-scale circulation for $\mathrm{\Gamma }=0.5$. As a consequence, the scaling laws for heat and momentum transfer inherit a dependence on the aspect ratio. We show that this breakdown of coherence is accompanied with a reduction of the Reynolds number Re. The scaling exponent $\beta$ of the power law $\mathrm{Nu}\propto {\mathrm{Ra}}^{\beta }$ crosses eventually over from $\beta =0.221$ to 0.124 when the liquid metal flow at $\mathrm{\Gamma }=0.5$ reaches $\mathrm{Ra}\gtrsim 2\times {10}^8$ and the coherent large-scale flow is completely collapsed.

Figure 1

Experimental setup. (a) Photograph of the experimental setup without thermal insulation. (b) Instantaneous large-scale flow visualization by means of UDV (gray sensors) in the measurement at $\mathrm{Ra}=5\times {10}^9$. The velocity magnitude $u$ is given in units of the free-fall velocity $u_{ff}=(g\alpha \mathrm{\Delta }TH{)}^{1/2}$.

Figure 2

Large-scale flow analysis. (a) Determination of the LSC orientation using two crossed UDV sensors (see Ref. [18] for more details). (b) Angular difference $\mathrm{\Delta }{\varphi }_{\mathrm{LSC}}\in [0°,360°]$ of the flow direction at the top and bottom plate vs time normalized by the turnover time $t_{\text{to}}$ for two Ra in the $\mathrm{\Gamma }=1$ cell showing oscillations around a stable mean value of about 180° due to torsion and sloshing of a SR-LSC. (c) Respective data for the $\mathrm{\Gamma }=0.5$ cell revealing nonperiodic frequent changes between various flow states.

Figure 3

Histograms of the angular difference $\mathrm{\Delta }{\mathrm{\Phi }}_{\mathrm{LSC}}\in [0°,180°]$ for all measurements carried out at the same Rayleigh numbers that are exemplary shown in Fig. 2. The histograms are normalized by their respective maximal value. The solid lines represent Gaussian fits.

Figure 4

Averaged values of the $\mathrm{\Delta }{\mathrm{\Phi }}_{\mathrm{LSC}}$ histograms vs Ra. Inset: Corresponding standard deviation $\sigma$.

Figure 5

Global transport laws ($\mathrm{Pr}=0.029$). (a) Reynolds number Re versus Rayleigh number Ra. Inset: Compensated plot of $\mathrm{Re}/{\mathrm{Ra}}^{1/2}$ and corresponding $u_{\mathrm{LSC}}/u_{ff}$ versus Ra. (b) Nusselt number Nu vs Ra. Inset: Compensated plot $\mathrm{Nu}/{\mathrm{Ra}}^{0.124}$ versus Ra.