Prandtl-number dependence of heat transport in the ultimate regime
This text mentions work from 2024 on. Those parts were partly written without a live literature search and have not been checked by an editor yet.
In plain words
How strongly heat transport depends on the fluid's ratio of viscosity to heat diffusivity, the Prandtl number, in the ultimate regime is unknown. This matters because liquid metals, stellar plasma and water have very different Prandtl numbers.
Precise statement
Within the ultimate regime of Rayleigh-Benard convection, determine the exponents in Nu ~ Pr^g1 Ra^g2 (up to logarithms) for $\mathrm{Pr} << 1$ and $\mathrm{Pr} >> 1$. Test the 2024 model prediction $\mathrm{Nu} \sim \mathrm{Pr}^{1/2}\mathrm{Ra}^{1/2}/(\ln \mathrm{Ra})^2$ for $\mathrm{Pr} <\sim 1$ and $\mathrm{Nu} \sim \mathrm{Pr}^{-1/2}\mathrm{Ra}^{1/2}/(\ln \mathrm{Ra})^2$ for $\mathrm{Pr} >\sim 1$, and its consistency with the rigorous bound $\mathrm{Nu} <\sim \mathrm{Ra}^{1/3}$ (up to logarithms) for $\mathrm{Pr} >\sim \mathrm{Ra}^{1/3}$ (Choffrut, Nobili, Otto 2016).
What would settle it
Heat-transport data at $\mathrm{Ra} > 10^{13}$ for Pr from about 0.02 to 10 compared with the predicted exponents.
Status in the literature
Unverified note: it names recent work and was partly written without a live literature search; it awaits a check by an editor.
Model proposed in 2024 (Shishkina and Lohse, arXiv:2407.16573); high-Ra data are concentrated near $\mathrm{Pr}\sim 1$.
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