Reynolds numbers and the elliptic approximation near the ultimate state of turbulent Rayleigh–Bénard convection

Abstract

We report results of Reynolds-number measurements, based on multi-point temperature measurements and the elliptic approximation (EA) of He and Zhang (2006 Phys. Rev. E 73 055303), Zhao and He (2009 Phys. Rev. E 79 046316) for turbulent Rayleigh-Bénard convection (RBC) over the Rayleigh-number range Ra 10 2 1 0 11 14≲ ≲ × and for a Prandtl number Pr ≃ 0.8. The sample was a right-circular cylinder with the diameter D and the height L both equal to 112 cm. The Reynolds numbers Re U and Re V were obtained from the mean-flow velocity U and the root-mean-square fluctuation velocity V, respectively. Both were measured approximately at the mid-height of the sample and near (but not too near) the side wall close to a maximum of Re U . A detailed examination, based on several experimental tests, of the applicability of the EA to turbulent RBC in our parameter range is provided. The main contribution to Re U came from a large-scale circulation in the form of a single convection roll with the preferred azimuthal orientation of its down flow nearly coinciding with the location of the measurement probes. First we measured time sequences of Re U (t) and Re V (t) from short (10 s) segments which moved along much longer sequences of many hours. The corresponding probability distributions of Re U (t) and Re V (t) had single peaks and thus did not reveal significant flow reversals. The two averaged Reynolds numbers determined from the entire data sequences were of comparable size. For Ra Ra 2 10 V 0.50 0.02 ∼ ± , consistent with the prediction for = 〈 − 〉 . This was accomplished by using the elliptic approximation (EA) of He and Zhang [1, 2] which permits the determination of the velocities from velocity space-time correlation functions. The EA is based on a second-order Taylor-series expansion of correlation functions which is valid near the origin of the space-time plane. However, He and Zhang postulated that the validity of the EA extends throughout the inertial range of length and time because of flow self-similarity. Analogous derivations and self-similarity assumptions can be applied to a passive scalar, for instance to the temperature in the bulk of turbulent RBC (see [35] and references therein), and thus we were able to make velocity determinations from measurements of the temperature space-time correlation functions. This procedure was used before for smaller Ra and various Pr on several occasions [29,[36][37][38][39].In the present paper, after a description of the apparatus and some measurement procedures in section 2, we first discuss in section 3 measurements of the LSC circulation-plane orientation because this orientation influences U. Then, in section 4, we discuss relevant consequences of the EA. While many of these results have appeared already in the literature, they are scattered among several papers and not always easy to find. Our discussion also includes a derivation of the equivalence between the space and the time domain, which remains valid in the presence of fluctuations and replaces the Taylor frozen-fl...

Azimuthal diffusion of the large-scale-circulation plane, and absence of significant non-Boussinesq effects, in turbulent convection near the ultimate-state transition

“…We also present new measurements of N u which are for the same sample as that used for the study of the azimuthal diffusion. They, as well as Re V measurements reported before (He et al 2015) for a different sample with Γ = 1.00, show the ultimate-state transition and all three physical properties yield consistent results for Ra * 1 and Re * 2 as shown below in Fig. 4.…”

“…The results of Chavanne et al (1997), Chavanne et al (2001), and Roche et al (2010) (the "Grenoble" data, obtained with fluid helium at a temperature of about 5 Kelvin and a pressure of about 2 bars) found transitions in the Ra range from 10 11 to 10 12 , which in our view (but not that of the authors) is too low to correspond to a BL shear instability although the data clearly show a sharp and continuous transition which, we believe, is of unknown origin (see Ahlers et al (2012b)). Measurements of N u and of the Reynolds number Re V = V L/ν (V is the root-mean-square fluctuation velocity) made with compressed sulfur hexafluoride (SF 6 ) at ambient temperatures and pressures up to 19 bars also found a transition ; Ahlers et al (2012b); He et al (2012aHe et al ( , 2015, the "Göttingen" data], but at Ra * ≃ 10 14 in agreement with the theoretical estimate of Grossmann & Lohse (2002).…”

Space-Time Correlations and Dynamic Coupling in Turbulent Flows

“…At the Max Planck Institute for Dynamics and Self-Organization and the University of California, Santa Barbara, He et al (2012bHe et al ( , 2014He et al ( , 2015 extensively tested the validity of the EA model in turbulent RBC, where Taylor's frozen-flow hypothesis is not valid because of large fluctuations (Lohse & Xia 2010). Their experimental results verify that the temperature correlation contours present similar elliptic shapes and the EA model leads to the collapse of all space-time correlations on the normalized space and time separations.…”

On the nature of fluctuations in turbulent Rayleigh–Bénard convection at large Prandtl numbers

“…We wish to scale the spectra so that they become equal to the analogously scaled spatial spectra within the accuracy of the elliptic approximation of He & Zhang (2006). This is achieved by scaling time by a characteristic time scale τ 0 (see § 5.6 below), which is defined by (5.2) and related to the curvature of the autocorrelation function (3.11) at τ = 0 (He et al 2014(He et al , 2015. This time scale plays a role in the time domain that is analogous to that of the Taylor microscale λ 0 in the spatial domain.…”

“…The characteristic time τ 0 plays a role in the time domain that is analogous to that of the Taylor microscale (see e.g. Pope 2000) in the spatial domain (He et al 2014(He et al , 2015. On the basis of the elliptic approximation (EA) of He & Zhang (2006), it can be shown that there is a space-time equivalence when time is scaled with τ 0 and space is scaled with λ 0 , where λ 0 is obtained from a Taylor series expansion, analogous to (5.2), of the spatial autocorrelation function C(z) (z is the spatial displacement; see e.g.…”