Rotating spherical gap convection in the GeoFlow International Space Station (ISS) experiment

Thermal convection in a rotating spherical gap is investigated using numerical simulations and compared with results of the GeoFlow ISS experiment. To induce convection, a radial buoyancy force field is established by using the dielectrophoretic effect from a high-frequency alternating electric field. Two heating sources are implemented. One source is a temperature difference across the gap and the other is the internal dielectric heating of the working fluid. To distinguish both heating sources a heating parameter, $\lambda$, is introduced that is varied together with the electric Rayleigh number, $L$, and Ekman number, Ek $\approx {10}^{-3}$. The governing thermoelectro hydrodynamic equations are analyzed via a linear stability analysis and by three-dimensional numerical simulations. The results are compared with experimental data of the GeoFlow experiment which show that the threshold of convection and the occurrence of global columnar cells agreed with the theoretical predictions. In addition the observed fluid flow showed non-Gaussian characteristics which are described by the quasinormal approximation. The overall flow phenomena are based on polar plumes and equatorial confined columnar cells and in addition are influenced by internal dielectric heating.

Figure 1

Schematic cross section of the GeoFlow experimental setup showing a numerically calculated temperature distribution over the gap.

Figure 2

Analytic solutions for (a) radial temperature distribution, (b) dielectrophoretic acceleration, and (c) Brunt-Väisälä frequency for varying $\lambda$ in conductive and nonrotating case for $\eta =0.5$. Red crosses (“x”) present solutions for a three-dimensional numerical simulation with $\lambda =11$. Values for $\lambda$ are chosen according to the GeoFlow experiment. (d) Sketch of convection in the spherical gap geometry when $N^2<0$ in the complete gap and $\lambda <1$). (e) Sketch of layered convection in the spherical gap geometry where $N^2\ge 0$ at the middle of the gap and $N^2<0$ elsewhere with $\lambda >1$.

Figure 3

(a) experimental interferogram for $\mathrm{Ek}=5.2\times {10}^{-3}$ and $\lambda =1$ for the conductive case. The base flow structures are outlined in yellow, with a sinusoidal distortion around the North Pole labeled with a green dot. The equator is in the lower part of the interferogram. (b) Numerical interferograms with the same parameters evaluated by a three-dimensional simulation. (c) Vertical plane of the temperature distribution through both poles.

Figure 4

Experimental points (EP) of the GeoFlow experiment in the $\lambda -L$ plane for (a) $\text{Ek}=7.6\times {10}^{-3}, \text{Pr}=175$; (b) $\text{Ek}=3.8\times {10}^{-3}, \text{Pr}=175$; (c) $\text{Ek}=5.2\times {10}^{-3}, \text{Pr}=125$; and (d) $\text{Ek}=2.6\times {10}^{-3}, \text{Pr}=125$. Dark gray dots represent the conductive cases, red dots the columnar flows, blue dots the transition and green dots the turbulent cases. Black lines with $\square$ symbols mark the onset of convection, and with $▿$ symbols the transition to the turbulent regime.

Figure 5

Columnar cells visualized by automatic pattern recognition, Zaussinger et al. [13]. The North Pole is located in the center and the equator at the outline. The colors refer to angles of identified fringe lines. (a) Spiraling columnar cells with $m=4$ for $L=8849, \mathrm{Ek}=7.6\times {10}^{-3}$, and $\lambda =0.16$, (b) slightly spiraling columnar cells with $m=5$ for $L=7217, \mathrm{Ek}=5.2\times {10}^{-3}$, and $\lambda =0.33$, (c) almost straight cells with $m=6$ for $L=12243, \mathrm{Ek}=5.2\times {10}^{-3}$, and $\lambda =0.19$. Animations of all three cases are available as Supplemental Material files in Refs. [46, 47, 48]. (d) Sketch of columnar cells aligned with the rotation axis in the spherical shell.

Figure 6

Averaged thermal properties for simulations with $L/L_c=1.15$ (black) and $L/L_c=2.25$ (red), for $\lambda <1$ and $\lambda >1$, respectively. (a) The mean temperature, (b) the temperature variance, (c) the skewness of the temperature, (d) the kurtosis of temperature, (e) the dielectrophoretic acceleration, (f) the Brunt-Väisälä frequency, (g) the a 3D simulation for $L/L_c=2.25$ and $\lambda =2.5$, and (h) a 3D simulation for $L/L_c=1.15$ and $\lambda =0.16$.

Figure 7

Averaged (a) temperature, (b) temperature variance, (c) skewness of temperature, and (d) kurtosis of temperature for $\lambda \le 1$ and $1.4\times {10}^4<L<2.4\times {10}^4$.

Figure 8

Averaged (a) radial component of dielectrophoretic acceleration, (b) meridional component of dielectrophoretic acceleration, (c) azimuthal component of dielectrophoretic acceleration, and (d) Brunt-Väisälä frequency for $\lambda <1$ and $1.4\times {10}^4<L<2.4\times {10}^4$.

Figure 9

Averaged (a) temperature, (b) temperature variance, (c) skewness of temperature, and (d) kurtosis of temperature for $\lambda \ge 1$ and $2.0\times {10}^3<L<3.5\times {10}^4$.

Figure 10

Interferograms of the GeoFlow experiment with the North Pole at the center of the circle: (a) Remnants of columnar cells for $L=2.4\times {10}^4$ and $\lambda =0.7$. (b) Equatorial, pointwise plumes for $L=6000$ and $\lambda =60$ with highlighted structures after postprocessing.

Figure 11

Averaged (a) radial component of dielectrophoretic acceleration and (b) Brunt-Väisälä frequency for $\lambda \ge 1$ and $2.0\times {10}^3<L<3.5\times {10}^4$.