Prandtl-number dependence of heat transport in the ultimate regime · revision history

Revision 2

Accepted · core change · record

Changes against revision 1:

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 · changed · core
Revision 1: 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).
Revision 2: 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).
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

Decisions