Does the ultimate $\mathrm{Nu} \sim \mathrm{Ra}^{1/2}$ regime exist in Rayleigh-Benard convection?
In plain words
When heating from below is strong enough, theory predicts that heat transport should grow as the square root of the heating strength, much faster than at moderate heating. Laboratories using different fluids and cells disagree on whether this 'ultimate' regime has been observed.
Precise statement
System: Oberbeck-Boussinesq Rayleigh-Benard convection between smooth isothermal no-slip plates, Prandtl number $\mathrm{Pr} \sim 1$, aspect ratio $\Gamma \sim 1$. Quantity: Nusselt number $\mathrm{Nu}(\mathrm{Ra})$ as $\mathrm{Ra} \to \infty$. Question: does the effective exponent $\gamma = d \mathrm{ln} \mathrm{Nu} / d \mathrm{ln} \mathrm{Ra}$ cross over from about $1/3$ to the Kraichnan form $\mathrm{Nu} \sim \mathrm{Ra}^{1/2} (\mathrm{ln} \mathrm{Ra})^{-c}$ at finite Ra, or does $\mathrm{Nu} \sim \mathrm{Ra}^{1/3}$ persist? Answer: $\gamma(\mathrm{Ra})$ up to $\mathrm{Ra} \sim 10^{17}$ with non-Boussinesq, sidewall and plate corrections quantified.
What would settle it
Concordant heat-transport measurements in at least two independent cells at $Ra>10^{15}$ with full uncertainty analysis, supported by DNS reaching $Ra\sim 10^{16}$.
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.
Gottingen $\mathrm{SF_6}$ data show a transition between $\mathrm{Ra} \sim 10^{13} \text{ and } 5 \times 10^{14}$ (He et al. 2012) and Grenoble helium data near $10^{11}$ (Chavanne et al. 1997), Oregon and Brno helium data do not; slender-cell DNS give $\gamma \sim 0.33$ up to $\mathrm{Ra} \sim 10^{15}$ (Iyer et al. 2020); a 2026 Brno uncertainty analysis shows data corrections can decide the claim (Urban et al., arXiv:2603.08811).
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