Influence of container shape on scaling of turbulent fluctuations in convection

We perform large-eddy simulations of turbulent convection in a cubic cell for Rayleigh numbers, Ra, between ${10}^6$ and ${10}^{10}$ and the molecular Prandtl number, $\text{Pr}=0.7$. The simulations were carried out using a second-order-accurate finite-difference method in which subgrid-scale fluxes of momentum and heat were both parametrized using a Lagrangian and dynamic Smagorinsky model. The scaling of the root-mean-square fluctuations of density (temperature) and velocity measured in the cell center are in excellent agreement with the scaling measured in the laboratory experiments of Daya and Ecke [Phys. Rev. Lett. 87, 184501 (2001)] and differ substantially from that observed in cylindrical cells. We also observe the time-averaged spatial distributions of the local heat flux and density fluctuations, and find that they are strongly inhomogeneous in the horizontal midplane, with the largest density gradients occurring at the corners at the midheight, where hot and cold plumes mix in the form of strong counter-rotating eddies.

Figure 1

Streamlines within the full convection cell, time averaged for more than $5296t_f$. Gray scale (color online) denotes the intensity of the vertical mean velocity, normalized by the free-fall velocity scale $U_f$. Figure (a) shows the diagonal containing the mean wind and (b) shows the opposite diagonal, containing the strong, counter-rotating cells. $\text{Ra}={10}^6;\text{Pr}=0.7;\Gamma =1$.

Figure 2

Contour plots of local heat flux ${\text{Nu}}_w$ at the top and bottom plates, time averaged over $5296t_f$. Gray scale (color online) gives the magnitude of the local heat flux in units of the global Nusselt number. Velocity vectors show the diagonal direction of the LSC for clarity. $\text{Ra}={10}^6;\text{Pr}=0.7;\Gamma =1$.

Figure 3

Normalized fluctuations in (a) density, given by ${\sigma }_{\rho }/\Delta \rho$ and (b) vertical velocity, given by ${\sigma }_{v}H/\nu$, measured at the cell center. The solid lines are the power-law fits to the present data: ${\sigma }_{\rho }/\Delta \rho =49.03{\text{Ra}}^{-0.46}$ (top) and ${\sigma }_{v}H/\nu =0.31{\text{Ra}}^{0.39}$ (bottom). Dashed and dash-dotted lines are power-law fits, respectively, for the near-cubic cell and cylindrical cell of DE01 for comparison.

Figure 4

Time-averaged values of the rms density fluctuations ${\sigma }_{\rho }$. Gray scale (color online) denotes the magnitude of the density fluctuations in units of $\Delta \rho$. Averaging has been performed over $5296t_f$. Figure (a) shows the vertical diagonal plane containing the large-scale circulation and (b) the orthogonal plane to it containing the counter-rotating cells. We have included the velocity vectors for clarity. $\text{Ra}={10}^6$; $\text{Pr}=0.7$; $\Gamma =1$.