Thermal Waves and Heat Transfer Efficiency Enhancement in Harmonically Modulated Turbulent Thermal Convection

We study turbulent Rayleigh-Bénard convection over four decades of Rayleigh numbers $4\times {10}^8<\mathrm{Ra}<2\times {10}^{12}$, while harmonically modulating the temperatures of the plates of our cylindrical cell. We probe the flow by temperature sensors placed in the cell interior and embedded in the highly conducting copper plates and detect thermal waves propagating at modulation frequency in the bulk of the convective flow. We confirm the recent numerical prediction [Yang et al., Phys. Rev. Lett. 125, 154502 (2020)] of the significant enhancement of the Nusselt number and report its dependence on the frequency and amplitude of the temperature modulation of plates.

Figure 1

Left: the $L=30\mathrm{cm}$ tall aspect ratio $G=D/L=1$ RBC cell with 28 mm thick top and bottom plates $D=30\mathrm{cm}$ in diameter made of thermally annealed copper of thermal conductivity ${\lambda }_p=2210\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$ and thermal capacity $c_p=0.144\mathrm{J}{\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}$ at $T_{\mathrm{He}}=(T_T+T_B)/2\approx 5\mathrm{K}$, where $T_T$ and $T_B$ are typical temperatures of the top and bottom plates. From the top plate, most of the heat is removed via the heat exchange chamber to the liquid He vessel above it. The top plate temperature $T_T(t)$ is roughly set by pressure in the heat exchange chamber and more precisely tuned and modulated by the uniformly distributed heater glued in the spiral grove on the upper side of the top plate. A similar heater delivers either steady or harmonically modulated heat to the bottom plate. The temperature of the convective flow at locations as shown (distances in millimeters) is probed by small Ge sensors (numbered 1–12) and that of the plates by the finely calibrated Ge sensors ${\mathrm{T}}_{\mathrm{t}1}$, ${\mathrm{T}}_{\mathrm{t}2}$, ${\mathrm{T}}_{\mathrm{b}1}$, and ${\mathrm{T}}_{\mathrm{b}2}$ embedded in them; see the photograph in the top right, showing their positions and the spiral heater grove.

Figure 2

Top: the records of the imposed harmonic modulation of the top plate $T_T(t)$ and constant $T_B(t)$ plotted together with the temperature records from sensor numbers 2, 6, and 8: their readings are identical within experimental resolution. Middle: examples of phase shifts of the thermal wave plotted versus the distance from the top plate for $f_S<\approx 1$ (left) and $f_S>1$ (right) by the sensors in the bulk as indicated. The solid (dashed) lines are plots of phase shifts in Eq. (1) with ${\kappa }_{\mathrm{eff}}=\text{Nu}\kappa ({\kappa }_{\mathrm{eff}}=\kappa )$. Bottom: the relative temperature amplitude $A_S/A_T$ (left) and the phase shift (right) measured by sensors as indicated at various $f_m$, $A_T$, and Ra as shown in Fig. 3 collapse when plotted versus $f_S$, defined by Eq. (2). The data points measured in Trieste [3, 4] are included for comparison. The lines (blue, red, and green correspond to sensor numbers 8, 2, and 6, respectively) are plots of Eq. (1) with ${\kappa }_{\mathrm{eff}}=\mathrm{Nu}\kappa$. The phase shift saturates at $\pi /2$ as indicated by the arrow.

Figure 3

Enhancement of ${\mathrm{Nu}}_f/{\mathrm{Nu}}_0$ plotted versus non-dimensional amplitude of temperature modulation $A_T$ applied at frequencies $f_m=0.01\mathrm{Hz}$ (left) and 0.003 Hz (right) to the top plate, measured at Ra as indicated. The dashed lines represent the best power law fits.

Figure 4

Top: the relative enhancement ${\mathrm{Nu}}_f/{\mathrm{Nu}}_0$ versus $f_s$, measured for various relative temperature amplitudes $A_T$ applied to the top plate at Ra as indicated. The simulation data of Yang et al. [6] are plotted for comparison. Bottom: phase-averaged bulk dimensionless temperature ${\theta }_c$ measured by sensor numbers 2, 6, and 8 and calculated at $z=L/2$ on the basis of Eq. (1) plotted as black lines. The red (blue) symbols represent the bulk temperature averaged over the half period of modulation when $T_T(t)$ is maximal (minimal). Frequencies $f_S$ used in the top panels of Fig. 2 are indicated by arrows.

Figure 5

Relative enhancement of heat transfer efficiency ${\mathrm{Nu}}_f/{\mathrm{Ra}}^{1/3}$ in harmonically modulated turbulent convection. The data points for modulation amplitudes $A_T=0$, 0.25, 0.5, 0.75, 1 and frequencies $f_m$ as indicated are evaluated based on power law fits of the measured data as shown in Fig. 3. The corresponding steady-state RBC data for constant heat flux to the bottom plate and constant $T_T$ (red circles) [15] and 3D simulation data [6] are shown for comparison.