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

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          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.

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          Author and article information

          Journal
          02 May 2014
          Article
          10.1103/PhysRevLett.112.174501
          1405.0432
          1f7d7c69-6f32-4417-9690-05765bd1ff06

          http://creativecommons.org/licenses/by-nc-sa/3.0/

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          Custom metadata
          Phys. Rev. Lett. 112, 174501 (2014)
          5 pages, 4 figures, 1 supplementary material
          physics.flu-dyn

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