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Title: Prandtl-number dependence of heat transport in the ultimate regime
Plain: 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: 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 $\mathrm{Nu} \sim \mathrm{Pr}^{1/2}\mathrm{Ra}^{1/2}/(\operatorname{ln} \mathrm{Ra})^2$ for $\mathrm{Pr} <\sim 1$ and $\mathrm{Nu} \sim \mathrm{Pr}^{-1/2}\mathrm{Ra}^{1/2}/(\operatorname{ln} \mathrm{Ra})^2$ 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).
Settled by: Heat-transport data at $\mathrm{Ra} > 10^{13}$ for Pr from about 0.02 to 10 compared with the predicted exponents.
Kind: well-posed
Answer type: value
References: (empty)
Literature status: open
Status note: Model proposed in 2024 (Shishkina and Lohse, arXiv:2407.16573); high-Ra data are concentrated near $\mathrm{Pr}\sim 1$.
Posed since: 2024