2 results
Turbulent flow in the bulk of Rayleigh–Bénard convection: aspect-ratio dependence of the small-scale properties
- Matthias Kaczorowski, Kai-Leong Chong, Ke-Qing Xia
-
- Journal:
- Journal of Fluid Mechanics / Volume 747 / 25 May 2014
- Published online by Cambridge University Press:
- 10 April 2014, pp. 73-102
-
- Article
- Export citation
-
Geometrical confinement of turbulent Rayleigh–Bénard convection (RBC) in Cartesian geometries is found to reduce the local Bolgiano length scale in the centre of the cell $L_{B,centre}$ and can therefore be used to study cascade processes in the bulk of RBC. The dependence of $L_{B,centre}$ versus $\varGamma $ suggests a cut-off to the local $L_B$, which depends on the Prandtl number $Pr$ and is of the order of the cell’s smallest dimension. It is also observed that geometrical confinement changes the topology of the flow, causing the turbulent kinetic energy dissipation rate and the temperature variance dissipation rate (averaged over the centre of the cell and normalized by their respective global averages) to exhibit a maximum at a certain $\varGamma $, which roughly coincides with the aspect ratio at which the viscous and thermal boundary layers of the two opposite lateral walls merge. As a result the mean heat flux through the core region also exhibits a maximum. Unlike in the cubic case, we find that geometrical confinement of the flow results in a local balance of the heat flux and the turbulent kinetic energy dissipation rate for $Pr= 4.38$ for all values of the Rayleigh number $Ra$ (up to $10^{10}$), while no balance is observed for $Pr= 0.7$. The need for very high bulk resolution to accurately resolve the gradients of the flow field at high $Ra$ is shown by analysing the second-order structure functions of the vertical velocity and temperature in the bulk of RBC. Under-resolution of the temperature field yields a large error in the dissipative range scaling, which is believed to be an effect of intermittently penetrating thermal plumes. The resolution contrast resulting from the requirement to resolve the thermal plumes and the homogeneous and isotropic background turbulence scales as $\delta _T / \langle \eta _k \rangle _{centre} \sim Ra^{0.1}$ and should therefore be taken into account when tackling very high $Ra$. In the case studied here, under-resolution can have a significant effect on the local heat flux through the centre of the cell.
Turbulent flow in the bulk of Rayleigh–Bénard convection: small-scale properties in a cubic cell
- Matthias Kaczorowski, Ke-Qing Xia
-
- Journal:
- Journal of Fluid Mechanics / Volume 722 / 10 May 2013
- Published online by Cambridge University Press:
- 28 March 2013, pp. 596-617
-
- Article
- Export citation
-
The Rayleigh number ($\mathit{Ra}$) scaling of the global Bolgiano length scale ${L}_{B, global} $ and the local Bolgiano length scale ${L}_{B, centre} $ in the centre region of turbulent Rayleigh–Bénard convection are investigated for Prandtl numbers $\mathit{Pr}= 0. 7$ and $4. 38$ and $3\times 1{0}^{5} \leq \mathit{Ra}\leq 3\times 1{0}^{9} $. It is found that ${L}_{B, centre} $ does not necessarily exhibit the same scaling as ${L}_{B, global} $. While ${L}_{B, global} $ is monotonically deceasing as ${L}_{B, global} \sim {\mathit{Ra}}^{- 0. 10} $ for both $\mathit{Pr}$, ${L}_{B, centre} $ shows a steep increase beyond a certain $\mathit{Ra}$ value. The complex scaling of the local Bolgiano length scale in the centre is a result of the different behaviour of the temperature-variance dissipation rate, ${\epsilon }_{T} $, and the turbulent-kinetic-energy dissipation rate, ${\epsilon }_{u} $. This shows that for sufficiently high $\mathit{Ra}$ the flow is well-mixed and hence temperature is passively advected. It is also observed that the $\mathit{Ra}$-range in which ${L}_{B, centre} $ exhibits the same scaling as the global Bolgiano length scale is increasing with increasing $\mathit{Pr}$. It is further observed that for $\mathit{Pr}= 4. 38$ and $\mathit{Ra}\leq 3\times 1{0}^{7} $ the local vertical heat flux in the centre region is balanced by the turbulent-kinetic-energy dissipation rate. For higher $\mathit{Ra}$ we find that the local heat flux is decreasing. At $\mathit{Pr}= 0. 7$ we do not observe such a balance, as the measured heat flux is between the heat fluxes estimated through the turbulent-kinetic-energy dissipation rate and the temperature-variance dissipation rate. We therefore suggest that the balance of the local heat flux might be Prandtl-number dependent. The conditional average of the local vertical heat flux $\mathop{\langle \mathit{Nu}\vert {\epsilon }_{u} , {\epsilon }_{T} \rangle }\nolimits_{\mathit{centre}} $ in the core region of the flow reveals that the highest vertical heat flux occurs for rare events with very high dissipation rates, while the joint most probable dissipation rates are associated with very low values of vertical heat flux. It is also observed that high values of ${\epsilon }_{u} $ and ${\epsilon }_{T} $ tend to occur together. It is further observed that the longitudinal velocity structure functions approach Kolmogorov K41 scaling. The temperature structure functions appear to approach Bolgiano–Obukhov BO59 scaling for $r\gt {L}_{B, centre} $, while a scaling exponent smaller than the BO59 scaling is observed for separations $r\lt {L}_{B, centre} $. The mixed velocity and temperature structure function for $\mathit{Ra}= 1\times 1{0}^{9} $ and $\mathit{Pr}= 4. 38$ shows a short $4/ 5$-scaling for $r\gt {L}_{B, centre} $. Our results suggest that BO59 scaling might be more clearly observable at higher Prandtl and moderate Rayleigh numbers.