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

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 10¹¹ ≲ Ra ≲ 2 × 10¹⁴ 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 < Ra₁^∗ ≃ 2 × 10¹³ both Re_U and Re_V could be described by a power-law dependence on Ra with an exponent ζ close to 0.44. This exponent is consistent with several other measurements for the classical RBC state at smaller Ra and larger Pr and with the Grossmann-Lohse (GL) prediction for Re_U (Grossmann and Lohse 2000 J. Fluid. Mech. 407 27; Grossmann and Lohse 2001 86 3316; Grossmann and Lohse 2002 66 016305) but disagrees with the prediction ζ ≃ 0.33 by GL (Grossmann and Lohse 2004 Phys. Fluids 16 4462) for Re_V. At Ra = Ra₂^∗ ≃ 7 × 10¹³ the dependence of Re_V on Ra changed, and for larger Ra Re_V ∼ Ra_0.50±0.02, consistent with the prediction for Re_U (Grossmann and Lohse 2000 J. Fluid. Mech. 407 27; Grossmann and Lohse Phys. Rev. Lett. 2001 86 3316; Grossmann and Lohse Phys. Rev. E 2002 66 016305; Grossmann and Lohse 2012 Phys. Fluids 24 125103) in the ultimate state of RBC.