Heat transfer scaling in natural convection with shear due to rotation

Heat transport in natural convection commonly encountered in natural and engineering flows is affected by buoyancy forces and shear. The effect of rotational shear on heat transport is examined here. Direct numerical simulation of Rayleigh-Bénard convection (RBC) with a rotating lid is performed at various rotation Reynolds numbers (Re) and Rayleigh numbers (Ra). The simulations are done for a range $\text{Ra}\in [2\times {10}^3,2\times {10}^7]$ and $\text{Re}\in [300,3000]$. The rotating lid induces a shear into the flow resulting in two prominent flow regimes: a rotation-dominated (RD) regime and a convection-dominated (CD) regime. The RD and CD regimes are identified based on the heat transport scaling exponent of Ra. The flow topology in these two regimes is distinct, with the CD flow showing more RBC-like flow structures while the RD regime shows a central axial vortex flow. The variations of boundary layer thickness with Ra and Re also demonstrate the regimes. A parameter $\chi \left(\text{Ra,Re}\right)$ is developed to demarcate the two regimes.

Figure 1

Instantaneous isosurfaces of $u_z$ and $T$ with the streamlines (black) at the two regimes for $\text{Re}=2200$. At (a) $\text{Ra}=2\times {10}^4$ (RD), streamlines swirl around the axis, a recirculation called a vortex breakdown bubble is shown as isosurface $u_z=0$ (cyan); and (b) $\text{Ra}=2\times {10}^7$ (CD), the streamlines form RBC-like convection rolls. Temperature isosurfaces at $T=0.9$ (red) and 0.05 (blue) indicate hot plumes at the bottom and cold plumes at the top for the two regimes. The direction of the top lid rotation is also indicated.

Figure 2

Nu plotted against nondimensional time $t$ for (a) $\text{Re}=800$ and (b) $\text{Re}=3000$. The corresponding Ra are  $\text{Ra}=0$,  $\text{Ra}=2\times {10}^4$,  $\text{Ra}=2\times {10}^5$,  $\text{Ra}=2\times {10}^6$, and  $\text{Ra}=2\times {10}^7$. The inset in (b) shows the zoomed signal with a window size of 40 time units.

Figure 3

Variation of ${\text{Nu}}_{\text{Ra}=0}$ for various Re. Note that $\text{Ra}=0$ implies that buoyancy force is neglected, and the boundary conditions $T=0$ at the top and $T=1$ at the bottom plates ensure a passive thermal gradient in the domain. The heat transfer is due to swirling motion (Re), and the Nu follows a power law $\sim {\text{Re}}^{1.1}$.

Figure 4

Variation of Nu with Ra for various Re. Transition from rotational- to convection-dominated regimes is evident from the change in slopes for various Re.

Figure 5

${\text{Ra}}_t$ is identified using three different ways. The red circle is the ${\text{Ra}}_t$ obtained from method-1, the square from method-2, and the triangle from method-3. All horizontal lines are joining the Nu in the RD regime for various Re as given Fig. 4. The dashed black line is RBC scaling $\text{Nu}\sim {\text{Ra}}^{0.33}$.

Figure 6

Variation of transitional Rayleigh number ${\text{Ra}}_t$ with Re. ${\text{Ra}}_t$ identified from method-1 (red circles) is used to get the least-squares fit (dashed line) given as ${\text{Ra}}_t=5\times {10}^{-5}{\text{Re}}^{3.3}$. Blue squares and black triangles are ${\text{Ra}}_t$ from method-2 and method-3, respectively, and the deviation from least-squares fit is shown as the error in evaluating the exponent.

Figure 7

Normalized Nusselt number plotted against the normalized Rayleigh number. The circles indicate rotation dominated while the upper triangles indicate convection-dominated regimes, identified from Fig. 4. The red dashed lines indicate upper- and lower-bound values for the RD regime. The black dashed line indicates RBC correlation $\text{Nu}\sim {\text{Ra}}^{0.33}$, which is attained in the CD regime. The insets show plumes identified by the temperature isosurface value of 0.9 for various Ra and Re, indicated by the top and bottom values in the insets, respectively. Note the distinct plume shapes in the RD and CD regimes.

Figure 8

The temperature contours with streamlines for various Ra and Re. The streamline patterns show the transition from RD to CD (from left to right) with the increase in Ra. Large-scale circulation, which is a signature of RBC observed in the convection-dominated regime. Also seen are the axial vortex and vortex breakdown bubbles in the RD regimes [far left, (e) and (f)].

Figure 9

The temperature profiles for varying Ra at $\text{Re}=2200$. The corresponding Ra are  $\text{Ra}=2\times {10}^4$,  $\text{Ra}=2\times {10}^5$,  $\text{Ra}=2\times {10}^6$,  $\text{Ra}=4\times {10}^6$,  $\text{Ra}=6\times {10}^6$, and  $\text{Ra}=2\times {10}^7$. The inset shows a schematic of the thermal boundary-layer thickness. The height at which the linear extrapolation at the bottom plate intersects with $T_c$ is defined as ${\delta }_{{\theta }_B}$.

Figure 10

The thermal boundary layer thickness ${\delta }_{\theta }$ for the cases shown in Fig. 9. The ${\delta }_{{\theta }_T}$ (square) and ${\delta }_{{\theta }_B}$ (circle) at $\text{Re}=2200$ for various Ra.

Figure 11

Variation of the bottom thermal boundary layer thickness, ${\delta }_{{\theta }_B}$ (black open triangles), for varying Ra and Re, compared with the inverse of the Nusselt number (red filled circles). The subplot (a) has all the data points from the CD regime while (b) has all the data points from the RD regime. Open green squares are top thermal boundary layer thickness, and the dashed lines indicate the slopes with power-law exponents as shown.

Figure 12

The contour plot of the nondimensional number, $\chi$, plotted as $ln\left(\chi \right)$. Data from the simulations are plotted over the contour. The circles correspond to RD and triangles correspond to CD. The thick line demarcates the two regimes at $ln\left(\chi \right)=10$. The contour line of $ln\left({\chi }_1\right)=9$ is shown to include the effect of error in evaluating ${\text{Ra}}_t$. For comparison, $\text{Ri}=1$ is plotted to show that the demarcation between the regimes is captured better by $\chi$.

Figure 13

Instantaneous isosurfaces of $u_z$ and $T$ along with streamlines for various Ra and Re. For the RD regime, $\text{ln}\left(\chi \right)\gtrsim 10$, streamlines swirl around the axis and form a recirculation called a vortex breakdown bubble [37], which is shown as isosurface $u_z=0$ (cyan). For the CD regime $\text{ln}\left(\chi \right)\lesssim 10$, the streamlines form RBC-like convection rolls. The temperature isosurface at $T=0.9$ (red) and 0.05 (blue) indicates hot plumes at the bottom and cold plumes at the top for the two regimes. A single axial plume dominates for the RD regime while line plumes dominate for the CD regime.