Reynolds numbers and the elliptic approximation near the ultimate state of turbulent Rayleigh-Benard 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-Benard convection (RBC) over the Rayleigh-number range 10(11) less than or similar to Ra less than or similar to 2 x 10(14) and for a Prandtl number Pr similar or equal to 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-1 similar or equal to 2 x 10(13) both Re-U and Re-V could be described by a power-law dependence on Ra with an exponent zeta 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 zeta similar or equal to 0.33 by GL (Grossmann and Lohse 2004 Phys. Fluids 16 4462) for Re-V. At Ra = Ra-2 similar or equal to 7 x 10(13) the dependence of Re-V on Ra changed, and for larger Ra Re-V similar to 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.