Heat-flux scaling in turbulent Rayleigh-Bénard convection with an imposed longitudinal wind

We present a numerical study of Rayleigh-Bénard convection disturbed by a longitudinal wind. Our results show that under the action of the wind, the vertical heat flux through the cell initially decreases, due to the mechanism of plume sweeping, and then increases again when turbulent forced convection dominates over the buoyancy. As a result, the Nusselt number is a nonmonotonic function of the shear Reynolds number. We provide simple models that capture with good accuracy all the dynamical regimes observed. We expect that our findings can lead the way to a more fundamental understanding of the complex interplay between mean wind and plume ejection in the Rayleigh-Bénard phenomenology.

Figure 1

Snapshots of the temperature field for (top panel) the pure RB case (${\text{Re}}_{\tau }=0$) at $\text{Ra}=1.3\times {10}^7$. bottom: mixed convective case $\text{Ra}=1.3\times {10}^7$ and ${\text{Re}}_{\tau }=92$. In the pure-convective case, plumes are detaching from the bottom and top boundary layers and freely travel through the bulk to the opposite walls whereas, in the mixed case, considerable distortion of plumes occur due to the wind.

Figure 2

Nusselt number as a function of ${\text{Re}}_{\tau }$ for $\text{Ra}=0,8.125\times {10}^5,6.5\times {10}^6,1.3\times {10}^7$ (from bottom to top, symbols are, respectively, $▾$, $▴$, $\blacksquare$, $•$). Error bars are estimated from the fluctuations in time of the space average of $\text{Nu}$ (for some cases uncertainties are smaller than the symbol size); typical integration times in the statistically steady state are $\sim 200T_L$ (see Table 1), with $T_L$ being the large eddy turnover time. Notice the nonmonotonic behavior for $\text{Ra}>0$: for low-to-moderate ${\text{Re}}_{\tau }$ a decrease of $\text{Nu}$ is observed due to plume sweeping by the mean wind; at higher ${\text{Re}}_{\tau }$, when buoyancy is basically irrelevant with respect to the channel flow, $\text{Nu}$ increases again, as expected for turbulent forced convection. The solid lines are the predictions from Eq. (16) obtained using $A_1=2.1$ and $A_2=0.045$, values that have been chosen to best fit the case $\text{Ra}=6.5\times {10}^6$ ($\blacksquare$ symbols). The band around solid lines shows the prediction of the model for a variation of the parameters $A_1$ and $A_2$ of $\pm 10\%$. The dashed lines represent the falloff for small ${\text{Re}}_{\tau }$ provided by Eq. (21), with $A_3=1$ for all the three $\text{Ra}$; we also report the $\text{Ri}=1$ curve, giving an estimate of the locus of points [in the ($\text{Ra}$, ${\text{Re}}_{\tau }$) plane] where the crossover between the two regimes occurs. The reader is referred to the text for the details. Inset: Log-log plot of the shear Reynolds number ${\text{Re}}_{\tau }^{*}$ corresponding to the crossover to the $\text{Nu}\sim {\text{Re}}_{\tau }^{-3/2}$ regime [23] as function of Rayleigh number. The dashed line is the ${\text{Re}}_{\tau }^{*}\sim {\text{Ra}}^{1/4}$ power law [Eq. (19)].

Figure 3

Normalized temperature profiles $\Theta \left(z\right)\equiv \frac{\overline{T}\left(z\right)-T_m}{\Delta }$ ($T_m=(T_{\text{hot}}+T_{\text{cold}})/2$ being the mean temperature) for various ${\text{Re}}_{\tau }$ at a fixed $\text{Ra}=6.5\times {10}^6$. Notice the bending of the profile in the bulk in comparison with the usual “thermal shortcut” for ${\text{Re}}_{\tau }=0$ (i.e., no lateral wind). Inset: Kinetic boundary layer thickness ${\lambda }_u$ vs ${\text{Re}}_{\tau }$ (log-log scale) for $\text{Ra}=0$: numerical data (symbols) and ${\text{Re}}_{\tau }^{-1}$ scaling (dashed line).

Figure 4

The anisotropy indicator defined in Eq. (4) as function of the cell height for pure RB (${\text{Re}}_{\tau }=0$) and for ${\text{Re}}_{\tau }=205$ and for two separations in $x$, at $\text{Ra}=6.5\times {10}^6$.