Logarithmic Spatial Variations and Universal $f^{-1}$ Power Spectra of Temperature Fluctuations in Turbulent Rayleigh-B\'enard Convection

TL;DR

This study reveals universal logarithmic temperature variance profiles and a consistent $f^{-1}$ power spectrum in turbulent Rayleigh-Bénard convection, indicating similarities with shear flow velocity fluctuations.

Contribution

It demonstrates universal temperature fluctuation spectra and profiles in turbulent RBC at high Rayleigh numbers, suggesting an analogy with shear flow turbulence.

Findings

Logarithmic dependence of temperature variance near the sample bottom.

Universal $f^{-1}$ power spectrum for temperature fluctuations.

Spectral exponent varies with height, from 1 to 1.67.

Abstract

We report measurements of the temperature variance $\sigma^2(z,r)$ and frequency power spectrum $P(f,z,r)$ ($z$ is the distance from the sample bottom and $r$ the radial coordinate) in turbulent Rayleigh-B\'enard convection (RBC) for Rayleigh numbers $\textrm{Ra} = 1.6\times10^{13}$ and $1.1\times10^{15}$ and for a Prandtl number $\textrm{Pr} \simeq 0.8$ for a sample with a height $L = 224$ cm and aspect ratio $D/L = 0.50$ ($D$ is the diameter). For $z/L$ less than or similar to $0.1$ $\sigma^2(z,r)$ was consistent with a logarithmic dependence on $z$, and there was a universal (independent of $\textrm{Ra}$, $r$, and $z$) normalized spectrum which, for $0.02$ less than or similar to $f\tau_0$ less than or similar to $0.2$, had the form $P(f\tau_0) = P_0 (f\tau_0)^{-1}$ with $P_0 =0.208 \pm 0.008$ a universal constant. Here $\tau_0 = \sqrt{2R}$ where $R$ is the radius of curvature of the temperature autocorrelation function $C(\tau)$ at $\tau = 0$. For $z/L \simeq 0.5$ the measurements yielded $P(f\tau_0) \sim (f\tau_0)^{-\alpha}$ with $\alpha$ in the range from 3/2 to 5/3. All the results are similar to those for velocity fluctuations in shear flows at sufficiently large Reynolds numbers, suggesting the possibility of an analogy between the flows that is yet to be determined in detail.