Propagation and interference of thermal waves in turbulent thermal convection

We report investigation of thermal waves in harmonically modulated turbulent thermal convection in a Rayleigh-Bénard cell. We probe the thermal waves in the working fluid—cryogenic helium gas—by temperature sensors placed in the cell and embedded in the plates. We observe propagation and interference of two coherent thermal waves and describe their behavior in terms of models that explain the weak attenuation and apparent superposition of thermal waves in the bulk, accounting for the temperature profile of convective turbulent flow via a spatially dependent effective thermal diffusivity.

Figure 1

Left: Schematic of the aspect ratio one RBC cell 30 cm in diameter [16]. Both the top and bottom plates are made of 28-mm thick annealed copper of very high thermal conductivity; their temperatures $T_t\left(t\right)$ and $T_b\left(t\right)$ are read by the embedded precisely calibrated Ge sensors [17]. The instantaneous temperature of the working fluid, cryogenic helium gas, is monitored by tiny temperature sensors [18, 19, 20] home-calibrated within 1-mK accuracy against the primary Ge sensors, installed in the cell interior using 0.1-mm constantan wires as shown (distances in millimeters).

Figure 2

Left: Temperature record of 7-Ge sensors in the bulk as indicated, together with their mean (blue) and the pressure trace (red) converted to temperature using the xhepak software [24] under the assumption of constant density ${\rho }_{\mathrm{He}}=3.607$ kg/${\mathrm{m}}^3$. The right axis corresponds to the original pressure trace taken as the difference from the equilibrium value, $p-p_0$, where $p_0=35260$ Pa. Readings of the 7 sensors in the bulk are, within the experimental resolution, identical. Middle and right: Readings of the two-plate thermometers and the (mean) bulk temperature plotted together with results of numerical simulations. The temperature wave in the bulk appears identical, irrespectively, of whether the top (middle panel) or bottom (right panel) plate temperature is modulated. The dotted lines represent calculated responses according to the theoretical Model 1. The offset between calculated and measured bulk values is caused by boundary layer asymmetry as explained in the text. The gray dashed line indicates the mean temperature $T_m\left(t\right)=[T_t\left(t\right)+T_b\left(t\right)]/2$. The inset in the middle panel shows the calculated variations of the thermal wave amplitude and phase with distance from the top plate. The dashed vertical line marks the boundary layer thickness ${\ell }_{BL}$.

Figure 3

Constructive (left panel) and destructive (middle panel) interference of thermal waves propagating coherently from the top and bottom plates obtained by modulating the plate temperatures in phase, or out of phase, respectively. The right panel illustrates interference in a form of beatings of thermal waves propagating from the top plate ($f_{mt}=0.01\mathrm{Hz}$) and bottom plate ($f_{mb}=0.0095$ Hz). Solid lines represent records of thermometers embedded in plates and the mean of sensors placed in the bulk (same as in Fig. 2), whereas dotted lines show values calculated using Model 1. The offset between the measured and calculated data is again due to BLs asymmetry. The insets show the calculated thermal wave amplitudes and phases for constructive and destructive interference.

Figure 4

Illustrative example of the steady temperature profile (red, left axis) and the effective thermal diffusivity (blue, right axis) within the model. The linear background discussed in the text is imperceptible in the temperature profile but imposes an upper limit on the effective diffusivity ${\kappa }_{\mathrm{eff}}\left(z\right)$ in the center of the cell, usually, at least, five orders of magnitude above $\kappa$, approximating the superconducting core suggested by Niemela and Sreenivasan [22].