Conductive heat flux in measurements of the Nusselt number in turbulent Rayleigh-Bénard convection

We propose a recipe to calculate accurately the Nusselt number $\text{Nu}$ in turbulent Rayleigh-Bénard convection, using the measured total heat flux $q$ and known parameters of the fluid and convection cell. More precisely, we present a method to compute the conductive heat flux $\widehat{q}$, which is a normalization of $q$ in the definition of $\text{Nu}$, for conditions where the fluid parameters may vary strongly across the fluid layer. We show that in the Oberbeck-Boussinesq approximation and also when the thermal conductivity depends exclusively on the temperature, the value of $\widehat{q}$ is determined by simple explicit formulas. For a general non-Oberbeck-Boussinesq (NOB) case we propose an iterative procedure to compute $\widehat{q}$. Using our procedure, we critically analyze some already conducted and some hypothetical experiments and show how $\widehat{q}$ is influenced by the NOB effects.

Figure 1

Phase diagram for the fluids (a) ${\mathrm{SF}}_6$ and (b) helium and the corresponding measurements from [12, 14] and [37], respectively, in these fluids. Each horizontal solid line represents the temperature range $T_t\le T\le T_b$ for each particular experiment. The green bullets in (a) and (b) mark the critical points of ${\mathrm{SF}}_6$ ($T_{\text{crit},{\text{SF}}_6}=318.72\mathrm{K}$ and $P_{\text{crit},{\text{SF}}_6}=37.55$ bars) and He ($T_{\text{crit},\text{He}}=5.195$ K and $P_{\text{crit},\text{He}}=2.276$ bars). Horizontal dotted lines correspond to the pressures considered in Fig. 3.

Figure 2

Relative deviation between $\text{Nu}$ [Eq. (1)] and ${\text{Nu}}_m$ [Eq. (12)] in the measurements in ${\mathrm{SF}}_6$ [12, 14] and helium [37].

Figure 3

Two-dimensional plots of $\mathcal{S}\equiv (\text{Nu}-{\text{Nu}}_m)/{\text{Nu}}_m$ that would take place in hypothetical measurements of the Nusselt number, conducted in RBC cells of the height $H=1.12$ m, (a) using ${\mathrm{SF}}_6$ under a pressure of 18.5 bars fixed at half height of the cell and (b) in a cell of the height $H=0.3$ m, using helium under a pressure of 2.29 bars, as functions of the top temperature $T_t$ and the temperature drop $\mathrm{\Delta }$ between the bottom and top plates. Real experiments conducted under these conditions, namely, those in [14, 37], are marked with black circles in (a) and (b). Note that the color scales in (a) and (b) are very different.