Another comment on claims of a transition to the ultimate regime of heat transfer

TL;DR

This paper critically examines recent claims of observing a transition to the ultimate heat transfer regime in 2D Rayleigh-Bénard convection simulations, questioning their validity and interpretation.

Contribution

It provides a critical analysis challenging the evidence and conclusions of recent studies claiming to observe the ultimate regime in 2D convection.

Findings

Questions the validity of the claimed transition to the ultimate regime

Highlights potential issues in the interpretation of simulation results

Suggests further scrutiny is needed for these claims

Abstract

Claims made by Zhu et al., PRL 120, 144502, (2018), that they had found evidence of a transition to the so called "ultimate regime" in 2D simulations of Rayleigh-B\'enard convection, have recently been repeated by Lohse & Shishkina, Rev. Mod. Phys. 96, 03501 (2024). The author questions the validity of these claims.

Equations

$$ϵ=\nu {\kappa }^{2}Ra(Nu-1)/H^4,$$

$$|\frac{\text{mbox}dE}{\text{mbox}dt}|\ll R{a}^{-1/2}Nu,$$

Full text

Comment on evidence of a transition to the ultimate regime of heat transfer

Erik Lindborg

Department of Engineering Mechanics, KTH, Osquars backe 18, SE-100 44, Stockholm, Sweden

Zhu et al. Zhu carried out DNS of 2D Rayleigh-Bénard convection (RBC) up to Rayleigh number $Ra=10^{14}$ and reported evidence of a transition to the ‘ultimate regime’ of heat transfer predicted by Kraichnan62 for 3D RBC, with Nusselt number dependence $Nu\sim Ra^{\gamma}$, where $\gamma>1/3$ for high $Ra$. Doering et al. Doering analysed the results of Zhu and concluded that they should rather be interpreted as evidence of absence of a transition. Zhu et al. Zhu2 carried out two more simulations at $Ra>10^{14}$ and claimed that they had now collected ‘overwhelming evidence’ of a transition.

The author of this comment would like to point out that none of the simulations at $Ra>10^{10}$ presented in Zhu reached a statistically stationary state. A sensitive indicator of stationarity is the development of the mean kinetic energy, $E$. In requesting information from two of the authors of Zhu (Detlef Lohse and Xiaojue Zhu), the author was informed that $E$ was still growing in all simulations at $Ra>10^{10}$, when they were ended. For $Ra\leq 10^{13}$ the simulations were all ended at $t=1000$, where time is measured in $H/u_{f}$, $H$ being the height of the domain and $u_{f}$ the free fall velocity. Two simulations were carried out at $10^{13}<Ra<10^{14}$, ending at $t=500$, and one simulations at $Ra=10^{14}$, ending at $t=250$. No information was provided in Zhu2 on how long time the two simulations at $Ra>10^{14}$ were run. Lohse & Zhu sent the author a figure depicting the time evolution of the four simulations 7, 8, 9 and 10 listed in the supplementary material of Zhu . The simulations had been continued after publication to check the convergence of $E$. Unfortunately, the figure cannot be shown, because Lohse & Zhu do not grant the author permission to publish it. The figure shows that in the two simulations 7 and 10 ($Ra=10^{10}$ and $Ra=10^{11}$), $E$ reaches approximate stationarity at $t_{s}\approx 1000$ and $t_{s}\approx 3000$, with stationary values $E\approx 0.25$ and $E\approx 0.48\approx 0.5$, in each case respectively. The simulation at $Ra=10^{11}$ was far from stationarity when it was ended at $t=1000$, with $E\approx 0.38$. Assuming that $E$ continues to double and $t_{s}$ continues to triple when $Ra$ is increased by a factor of ten, the simulation at $Ra=10^{14}$ would reach stationary first at $t_{s}\approx 80000$ with $E\approx 4$. Since this simulations was ended at $t=250$ with $E\approx 0.2$, the Nusselt number was evaluated in a state that was, indeed, very far from stationarity.

A cornerstone of scaling theories of RBC, for example the theory of Grossmann & Lohse (2000), is the exact expression for the mean kinetic energy dissipation rate in a statistically stationary state,

[TABLE]

where $\nu$ is the kinematic viscosity and $\kappa$ the diffusivity. For $Pr=\nu/\kappa\sim 1$, a condition for this relation to be satisfied is

[TABLE]

where the time derivative on the left hand side is nondimensionalized by $u_{f}^{3}/H$. The high $Ra$ simulations of Zhu were far from satisfying this condition in the state where the Nusselt number was evaluated. As pointed out by Ahlers et al. (2022): ‘One can only start to collect statistics when the flow is fully developed and has attained a statistically stationary state.’ In conclusion, the issue regarding the scaling of $Nu$ in high $Ra$ 2D RBC is not settled yet.