Temperature Fluctuation Profiles in Turbulent Thermal Convection: A Logarithmic Dependence versus a Power-Law Dependence

We report an experimental measurement of the rms temperature (${\sigma }_T$) profiles in two regions inside a large aspect ratio ($\mathrm{\Gamma }=4.2$) rectangular convection cell. It is found that, in the region where the boundary layer is sheared by a large-scale wind, ${\sigma }_T$ has a power-law dependence on the vertical distance ($z$) from the plate, whereas in the region where plumes are abundant, ${\sigma }_T$ has a logarithmic dependence on $z$. The power-law profile may be understood by balancing the inertia force and the viscous force in the equations of motion, and the logarithmic profile may be understood in terms of the balance between the buoyancy and the inertia forces. When normalized by a convective temperature scale, ${\theta }_{*}$, the profiles of ${\sigma }_T$ collapse onto a single curve for different values of the Rayleigh number. This shows that the convective temperature first proposed by Deardorff is the suitable temperature scale outside the thermal boundary layer for both logarithmic and power-law profiles. Our finding suggests a strong connection between plumes and the logarithmic rms temperature profile. The present Letter reveals that multiple force balance mechanisms can coexist in the bulk of highly turbulent flows.

Figure 1

Schematic drawing of experimental apparatus. $A$: shear-dominated region; $B$: plume-abundant region.

Figure 2

(a)–(d) Temperature time series measured at $\mathrm{Ra}\approx 1.1\times 10^8$. Top panel: in the wind-shearing region $A$. Middle panel: in the plume-abundant region $B$. (e) and (f) rms temperature versus Ra; circles: data taken in region $A$; squares: data taken in region $B$.

Figure 3

Probability density function of the normalized temperature fluctuations for all Ra values. Upper panel: in the shear-dominated region $A$. Lower panel: in the plume-abundant region $B$. Left panel: data taken at $z/{\lambda }_{\text{th}}\approx 2$. Right panel: at $z/{\lambda }_{\text{th}}\approx 20$.

Figure 4

Measured rms temperature profiles in the shear-dominated region $A$: (a) Log-log plot of rms temperature ${\sigma }_T$ versus vertical position, $z$. The solid symbols are data points used in the fitting. From top to bottom, the corresponding Ra are as follows: $1.94\times {10}^8$, $1.64\times {10}^8$, $1.29\times {10}^8$, $1.06\times {10}^8$, $8.25\times {10}^7$, $5.08\times {10}^7$, and $3.25\times {10}^7$. (b) Log-log plot of normalized rms temperature ${\sigma }_T/{\theta }_{*}$ versus normalized vertical position, $z/H$. (c) Log-log plot of the normalized rms temperature averaged over Ra, $⟨{\sigma }_T/{\theta }_{*}{⟩}_{\mathrm{Ra}}$, versus $z/H$. (d) The same data as in (c) but in a semilog scale.

Figure 5

Measured rms temperature profiles in plume-dominated region: (a) Semilog plot of rms temperature ${\sigma }_T$, versus vertical position, $z$. The solid symbols are data points used in the fitting. The legends for different Ra are the same as in Fig. 4. (b) Semilog plot of normalized rms temperature ${\sigma }_T/{\theta }_{*}$ versus normalized vertical position, $z/H$. (c) Semilog plot of the normalized rms temperature averaged over Ra, $⟨{\sigma }_T/{\theta }_{*}{⟩}_{\mathrm{Ra}}$, versus $z/H$. (d) The same data as in (c) but in log-log plot.