Numerical simulations of thermal convection on a hemisphere

In this paper we present numerical simulations of two-dimensional turbulent convection on a hemisphere. Recent experiments on a half soap bubble located on a heated plate have shown that such a configuration is ideal for studying thermal convection on a curved surface. Thermal convection and fluid flows on curved surfaces are relevant to a variety of situations, notably for simulating atmospheric and geophysical flows. As in experiments, our simulations show that the gradient of temperature between the base and the top of the hemisphere generates thermal plumes at the base that move up from near the equator to the pole. The movement of these plumes gives rise to a two-dimensional turbulent thermal convective flow. Our simulations turn out to be in qualitative and quantitative agreement with experiments and show strong similarities with Rayleigh-Bénard convection in classical cells where a fluid is heated from below and cooled from above. To compare to results obtained in classical Rayleigh-Bénard convection in standard three-dimensional cells (rectangular or cylindrical), a Nusselt number adapted to our geometry and a Reynolds number are calculated as a function of the Rayleigh number. We find that the Nusselt and Reynolds numbers verify scaling laws consistent with turbulent Rayleigh-Bénard convection: $\text{Nu}\propto {\text{Ra}}^{0.31}$ and $\text{Re}\propto {\text{Ra}}^{1/2}$. Further, a Bolgiano regime is found with the Bolgiano scale scaling as ${\text{Ra}}^{-1/4}$. All these elements show that despite the significant differences in geometry between our simulations and classical 3D cells, the scaling laws of thermal convection are robust.

Figure 1

Temperature field in the equatorial plane at $\text{Ra}=3\times {10}^9$ and $\text{Pr}=7$. The Cartesian grid is defined on a square larger than the bubble, and the corners are taken into consideration thanks to a penalization method.

Figure 2

A staggered cell. The pressure and the temperature are computed at the center of the cell. The velocity components are computed in the middle of the cell borders.

Figure 3

Evolution of the thermal (left) and kinetic (right) total energies with time for various values of the stabilization parameters $F$ and $S$ at $\text{Ra}=3\times {10}^8$ and $\text{Pr}=7$. Computations with $\delta t=5\times {10}^{-4}$ with outputs every 2000 time steps. Total time steps : $200\times 2000=400000$.

Figure 4

Temperature (left) and vorticity (right) fields in the beginning of the simulation at $\text{Ra}=3\times {10}^8$ and $\text{Pr}=7$. Mushroom shape structures are created at the equator.

Figure 5

Temperature (left) and vorticity (right) fields in the stationary state at $\text{Ra}=3\times {10}^8$ and $\text{Pr}=7$. Plumes move up on the bubble and the structures start to interact one with each other.

Figure 6

Global Nusselt number (left) and Reynolds number (right) versus Rayleigh number at $\text{Pr}=7$ for $F=S=0,F=S=0.03,\text{and}F=S=0.06$.

Figure 7

Global Nusselt number (left) and Reynolds number (right) versus Rayleigh number at $\text{Pr}=7$.

Figure 8

Global Nusselt (left) and Reynolds (right) numbers versus Rayleigh number at $\text{Pr}=7$. Scaling laws can be clearly observed.

Figure 9

Local Nusselt numbers versus latitude (radians) at $\text{Pr}=7$.

Figure 10

Vorticity fields at $\text{Pr}=7$ for $\text{Ra}={10}^6$ (left) and $\text{Ra}={10}^{10}$ (right). Many more structures can be observed in the more convective numerical experiment.

Figure 11

Temperature fields at $\text{Pr}=7$ for $\text{Ra}={10}^6$ (left) and $\text{Ra}={10}^{10}$ (right). Many more structures can be observed in the more convective numerical experiment.

Figure 12

Averaged temperature as a function of the latitude (rad.) for various Rayleigh number simulations at $\text{Pr}=7$. Evolution of the boundary layer length can be clearly observed.

Figure 13

RMS of the temperature fluctuations as a function of the polar angle for various Rayleigh numbers at $\text{Pr}=7$. It is used as a proxy for the thermal boundary layer.

Figure 14

Boundary layer size measured by various methods, as a function of the Rayleigh number at $\text{Pr}=7$. All the methods lead to the same scaling of the boundary layer with respect to the Rayleigh number.

Figure 15

Zoom on RMS of the temperature fluctuations as a function of the polar angle for $\text{Ra}=3\times {10}^9$ and $\text{Ra}={10}^{10}$ at $\text{Pr}=7$.

Figure 16

Energy spectra for various Rayleigh number simulations (from $\text{Ra}={10}^6$ to $\text{Ra}={10}^{10})$ at $\text{Pr}=7$. A Bolgiano regime can be clearly observed for high Rayleigh number values.

Figure 17

Enstrophy spectra for various Rayleigh number simulations (from $\text{Ra}={10}^6$ to $\text{Ra}={10}^{10})$ at $\text{Pr}=7$. A Bolgiano regime can be clearly observed for high Rayleigh number values.

Figure 18

Evolution of the energy spectrum with time from 0.2 to 3.6 (with a 0.2 nondimensional time step) at $\text{Ra}={10}^{10}$ and $\text{Pr}=7$. Red spectrum: stationary state spectrum.

Figure 19

Entropy spectra for various Rayleigh number simulations (from $\text{Ra}={10}^6$ to $\text{Ra}={10}^{10})$ at $\text{Pr}=7$. A dual branch due to the averaged temperature profile can be observed.

Figure 20

Temperature fluctuations spectra for various Rayleigh number simulations (from $\text{Ra}={10}^6$ to $\text{Ra}={10}^{10})$.

Figure 21

Absolute value of the scaling exponent of the temperature fluctuations spectra as a function of the Rayleigh number. Slope of the linear fit $\approx 1/6$.

Figure 22

Temperature fluctuations spectra computed in the pole area from 0 to $\pi /6$ rad at $\text{Pr}=7$. The $k^{-7/5}$ scaling for high Rayleigh numbers confirms the presence of the Bolgiano regime.

Figure 23

Energy fluxes for various Rayleigh number simulations (from $\text{Ra}={10}^6$ to $\text{Ra}={10}^{10})$ at $\text{Pr}=7$. The zero-crossing of the energy flux can be used as an indicator of the behavior of the Bolgiano scale with respect to the Rayleigh number.

Figure 24

Enstrophy fluxes for various Rayleigh number simulations (from $\text{Ra}={10}^6$ to $\text{Ra}={10}^{10})$ at $\text{Pr}=7$. The zero-crossing of the enstrophy flux can be used as an indicator of the behavior of the Bolgiano scale with respect to the Rayleigh number.

Figure 25

Entropy fluxes for various Rayleigh number simulations (from $\text{Ra}={10}^6$ to $\text{Ra}={10}^{10})$ at $\text{Pr}=7$. The zero-crossing of the entropy flux can be used as an indicator of the behavior of the Bolgiano scale with respect to the Rayleigh number.

Figure 26

Scales $S_z$ of the zero crossing points for the energy, enstrophy, and entropy fluxes for various Rayleigh number simulations (from $\text{Ra}={10}^6$ to $\text{Ra}={10}^{10})$ at $\text{Pr}=7$. These parameters can be used as proxies for the Bolgiano scale. A clear scaling in function of the Rayleigh number can be observed.

Figure 27

Stereographic projection: the point P of coordinates $(x_0,y_0,z_0)$ on the sphere is projected on ${\mathrm{P}}^{\prime }$ of coordinates $(x,y)$ on the plane. N and S denote, respectively, the north and south poles.

Figure 28

Entropy spectra for various functions and extensions. At the top the odd modes of a Heaviside function (black stars) are superimposed to the straight line of slope $-2$, the odd modes of the 1-padding spectrum are also superimposed on this line up to $k\simeq 70$ and then decrease (green circles). In the middle, the even modes of the 1-padding spectrum (green circles), of the 0-padding spectrum (red crosses), and of the symmetrical padding spectrum (blue pluses) are superimposed. Below, the odd modes of the 0-padding extension are shown (in red). In the bottom of the figure, the even modes of Heaviside function and the odd modes of the symmetrical padding spectrum are very small and near zero. The spectra are shown for the mean temperature field for $\text{Ra}=3\times {10}^9$ with different padding schemes.