Robustness of heat transfer in confined inclined convection at high Prandtl number

We investigate the dependency of the magnitude of heat transfer in a convection cell as a function of its inclination by means of experiments and simulations. The study is performed with a working fluid of large Prandtl number, $\mathrm{Pr}\simeq 480$, and at Rayleigh numbers $\mathrm{Ra}\simeq {10}^8$ and $\mathrm{Ra}\simeq 5\times {10}^8$ in a quasi-two-dimensional rectangular cell with unit aspect ratio. By changing the inclination angle $\left(\beta \right)$ of the convection cell, the character of the flow can be changed from moderately turbulent, for $\beta =0^{\circ }$, to laminar and steady at $\beta ={90}^{\circ }$. The global heat transfer is found to be insensitive to the drastic reduction of turbulent intensity, with maximal relative variations of the order of $20\%$ at $\mathrm{Ra}\simeq {10}^8$ and $10\%$ at $\mathrm{Ra}\simeq 5\times {10}^8$, while the Reynolds number, based on the global root-mean-square velocity, is strongly affected with a decay of more than $85\%$ occurring in the laminar regime. We show that the intensity of the heat flux in the turbulent regime can be only weakly enhanced by establishing a large-scale circulation flow by means of small inclinations. However, in the laminar regime the heat is transported solely by a slow large-scale circulation flow which exhibits large correlations between the velocity and temperature fields. For inclination angles close to the transition regime in-between the turbulentlike and laminar state, a quasiperiodic heat-flow bursting phenomenon is observed.

Figure 1

A sketch of the rectangular convection cell and the coordinate frame adopted in this study. The height, length and width of the cell are denoted by $H,L_x$, and $L_z$, respectively. $\beta$ is the inclination angle relative to the horizontal plane. The $x$ and $y$ coordinate axes align with the bottom plate and the sidewall respectively. Circles in the top bottom plates indicate the position of the 12 thermistors used to monitor the global thermal gap $\mathrm{\Delta }T$, while $T_{1,2}$ show the positions of the two thermal probes employed for the measurement of the intensity of the local flow.

Figure 2

Visualizations of the instantaneous flow pattern in the CIC cell at $\mathrm{Ra}=5\times {10}^8$ from numerics and experiments for various inclination angles. Panels of the first raw (a)–(d) report visualizations both of the velocity and the temperature field in false colours (red-hot, green-intermediate, blue-cold) for the angles $\beta =0^{\circ }$ (a), $\beta ={25}^{\circ }$ (b), $\beta ={40}^{\circ }$ (c), $\beta ={75}^{\circ }$ (d). The second raw (e–h) shows the shadowgraph images of a vertical central plane in the CIC system from the experiments at $\beta =0^{\circ }$ (e), $\beta ={25}^{\circ }$ (f), $\beta ={47}^{\circ }$ (g), $\beta ={75}^{\circ }$ (h).

Figure 3

(a) The fluctuation intensity of the turbulent flow $I_u\left(\beta \right)$ in inclined thermal convection for $\mathrm{Ra}={10}^8$ (squares) and $\mathrm{Ra}=5\times {10}^8$ (diamonds). Inset: the phase diagram of the flow pattern in the $\mathrm{Ra}$ and $\beta$ plane, the black diamonds refer to the laminar regime, the red squares refer to the turbulent regime and the pink circles refer to the bursting phenomenon regime. (b) The fluctuation intensity of the heat transfer $I_{Nu}\left(\beta \right)$ in the inclined thermal convection for $\mathrm{Ra}={10}^8$ (squares) and $\mathrm{Ra}=5\times {10}^8$ (diamonds).

Figure 4

(a) Time series of instantaneous (volume-averaged) Nusselt number at $\beta ={42}^{\circ }$ for $\mathrm{Ra}=5\times {10}^8$ in simulation. The averaged period of the bursting phenomenon in simulation is about $5\tau$. (b) Time series of the area fraction of plumes $I_p$ at $\beta ={47}^{\circ }$ for $\mathrm{Ra}\simeq 5\times {10}^8$ in experiment. The averaged period of the bursting phenomenon is close to $6\tau$.

Figure 5

Nusselt number $\mathrm{Nu}\left(\beta \right)$ as a function of the inclination angle $\beta$, normalized by its value for $\beta =0$, from experiments (circles) and numerics (triangles) at (a) $\mathrm{Ra}={10}^8$ and (b) $\mathrm{Ra}=5\times {10}^8$. In the experiments the error bars are estimated from ${\delta }_{\frac{\text{Nu}\left(\beta \right)}{\text{Nu}\left(0\right)}}=\frac{\mathrm{Nu}\left(\beta \right)}{\mathrm{Nu}\left(0\right)}\sqrt{{\left[\frac{\delta \text{Nu}\left(\beta \right)}{\text{Nu}\left(\beta \right)}\right]}^2+{\left[\frac{\delta \text{Nu}\left(0\right)}{\text{Nu}\left(0\right)}\right]}^2}$.

Figure 6

Reynolds numbers as functions of the inclination angle $\beta$: (a) for $\mathrm{Ra}={10}^8$. (b) for $\mathrm{Ra}=5\times {10}^8$. In (b) we report also the local Reynolds number ${\mathrm{Re}}_{\mathrm{l}}$ from experiments is (red diamonds). In the insets: the superposed graphs for $\mathrm{Re}\left(\beta \right)/\mathrm{Re}\left(0\right)$ (squares) and $\mathrm{Nu}\left(\beta \right)/\mathrm{Nu}\left(0\right)$ (triangles) from DNS.

Figure 7

The vertical $U_y\left(x\right)$ and horizontal $U_x\left(y\right)$ velocity profiles associated to the LSC flow structure as derived from the conditionally time averaged velocity fields ${\overline{u}}_x$ and ${\overline{u}}_y$ at different inclination angles for $\mathrm{Ra}=5\times {10}^8$. The abscissa and ordinate scales are the dimensionless width and height. The upper scales show the horizontal velocity profiles nondimensionalized with $\nu /H$, i.e., ${\text{Re}}_x\left(y\right)$; the right scales show the vertical velocity's $U_y\left(x\right)$ profiles, also nondimensionalized by $\nu /H$.

Figure 8

Correlation coefficient of $y$-axis velocity $u_y$ and temperature $T,C_{u_{y}T}$, in inclined convection as a function of inclination angle $\beta$ for $\mathrm{Ra}={10}^8$ (squares) and $\mathrm{Ra}=5\times {10}^8$ (diamonds).

Figure 9

Normalized Nusselt number $\mathrm{Nu}\left(\beta \right)/\mathrm{Nu}\left(0\right)$ (triangles) and fraction of heat transfer, ${\mathrm{Nu}}_{\mathrm{c}}\left(\beta \right)/\mathrm{Nu}\left(0\right)$, carried by the coherent structure (diamonds) for (a) $\mathrm{Ra}={10}^8$ and (b) $\mathrm{Ra}=5\times {10}^8$ as functions of the inclination angle $\beta$.