Description:
<jats:title>Abstract</jats:title><jats:p>The unifying theory of scaling in thermal convection (Grossmann & Lohse, <jats:italic>J. Fluid. Mech.</jats:italic>, vol. 407, 2000, pp. 27–56; henceforth the GL theory) suggests that there are no pure power laws for the Nusselt and Reynolds numbers as function of the Rayleigh and Prandtl numbers in the experimentally accessible parameter regime. In Grossmann & Lohse (<jats:italic>Phys. Rev. Lett.</jats:italic>, vol. 86, 2001, pp. 3316–3319) the dimensionless parameters of the theory were fitted to 155 experimental data points by Ahlers & Xu (<jats:italic>Phys. Rev. Lett.</jats:italic>, vol. 86, 2001, pp. 3320–3323) in the regime <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S002211201300298X_inline1" /><jats:tex-math>$3\times 1{0}^{7} \leq \mathit{Ra}\leq 3\times 1{0}^{9} $</jats:tex-math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S002211201300298X_inline2" /><jats:tex-math>$4\leq \mathit{Pr}\leq 34$</jats:tex-math></jats:alternatives></jats:inline-formula> and Grossmann & Lohse (<jats:italic>Phys. Rev.</jats:italic> E, vol. 66, 2002, p. 016305) used the experimental data point from Qiu & Tong (<jats:italic>Phys. Rev.</jats:italic> E, vol. 64, 2001, p. 036304) and the fact that <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S002211201300298X_inline3" /><jats:tex-math>$\mathit{Nu}(\mathit{Ra}, \mathit{Pr})$</jats:tex-math></jats:alternatives></jats:inline-formula> is independent of the parameter <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S002211201300298X_inline4" /><jats:tex-math>$a$</jats:tex-math></jats:alternatives></jats:inline-formula>, which relates the dimensionless kinetic boundary thickness with the square root of the wind Reynolds number, to fix the Reynolds number dependence. Meanwhile the theory is, on the one hand, well-confirmed through various new experiments and numerical simulations; on the other hand, these new data points provide the basis for an updated fit in a much larger parameter space. Here we pick four well-established (and sufficiently distant) <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S002211201300298X_inline5" /><jats:tex-math>$\mathit{Nu}(\mathit{Ra}, \mathit{Pr})$</jats:tex-math></jats:alternatives></jats:inline-formula> data points and show that the resulting <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S002211201300298X_inline6" /><jats:tex-math>$\mathit{Nu}(\mathit{Ra}, \mathit{Pr})$</jats:tex-math></jats:alternatives></jats:inline-formula> function is in agreement with almost all established experimental and numerical data up to the ultimate regime of thermal convection, whose onset also follows from the theory. One extra <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S002211201300298X_inline7" /><jats:tex-math>$\mathit{Re}(\mathit{Ra}, \mathit{Pr})$</jats:tex-math></jats:alternatives></jats:inline-formula> data point is used to fix <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S002211201300298X_inline8" /><jats:tex-math>$\mathit{Re}(\mathit{Ra}, \mathit{Pr})$</jats:tex-math></jats:alternatives></jats:inline-formula>. As <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S002211201300298X_inline9" /><jats:tex-math>$\mathit{Re}$</jats:tex-math></jats:alternatives></jats:inline-formula> can depend on the definition and the aspect ratio, the transformation properties of the GL equations are discussed in order to show how the GL coefficients can easily be adapted to new Reynolds number data while keeping <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S002211201300298X_inline10" /><jats:tex-math>$\mathit{Nu}(\mathit{Ra}, \mathit{Pr})$</jats:tex-math></jats:alternatives></jats:inline-formula> unchanged.</jats:p>