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 http://dx.doi.org/10.1103/PhysRevE.73.055303 73 http://dx.doi.org/10.1103/PhysRevE.73.055303 ), Zhao and He (2009 Phys. Rev. E http://dx.doi.org/10.1103/PhysRevE.79.046316 79 http://dx.doi.org/10.1103/PhysRevE.79.046316 ) for turbulent Rayleigh–Bénard convection (RBC) over the Rayleigh-number range ${10}^{11}\lesssim {\text{}}{Ra}\lesssim 2\times {10}^{14}$ 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 ${\text{}}{Ra}\lt {\text{}}{{Ra}}_{1}^{*}\simeq 2\times {10}^{13}$ 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. http://dx.doi.org/10.1017/S0022112099007545 407 http://dx.doi.org/10.1017/S0022112099007545 ; Grossmann and Lohse 2001 http://dx.doi.org/10.1103/PhysRevLett.86.3316 86 http://dx.doi.org/10.1103/PhysRevLett.86.3316 ; Grossmann and Lohse 2002 http://dx.doi.org/10.1103/PhysRevE.66.016305 66 http://dx.doi.org/10.1103/PhysRevE.66.016305 ) but disagrees with the prediction $\zeta \simeq 0.33$ by GL (Grossmann and Lohse 2004 Phys. Fluids http://dx.doi.org/10.1063/1.1807751 16 http://dx.doi.org/10.1063/1.1807751 ) for Re _V . At ${\text{}}{Ra}={\text{}}{{Ra}}_{2}^{*}\simeq 7\times {10}^{13}$ the dependence of Re _V on Ra changed, and for larger Ra ${\text{}}{{Re}}_{V}\sim {\text{}}{{Ra}}^{0.50\pm 0.02}$ , consistent with the prediction for Re _U (Grossmann and Lohse 2000 J. Fluid. Mech. http://dx.doi.org/10.1017/S0022112099007545 407 http://dx.doi.org/10.1017/S0022112099007545 ; Grossmann and Lohse Phys. Rev. Lett. 2001 http://dx.doi.org/10.1103/PhysRevLett.86.3316 86 http://dx.doi.org/10.1103/PhysRevLett.86.3316 ; Grossmann and Lohse Phys. Rev. E 2002 http://dx.doi.org/10.1103/PhysRevE.66.016305 66 http://dx.doi.org/10.1103/PhysRevE.66.016305 ; Grossmann and Lohse 2012 Phys. Fluids http://dx.doi.org/10.1063/1.4767540 24 http://dx.doi.org/10.1063/1.4767540 ) in the ultimate state of RBC.

Keywords

• turbulent thermal convection
• Reynolds number
• ultimate regime
• space–time correlation
• elliptic approximation