# A Lagrangian fluctuation-dissipation relation for scalar turbulence, III. Turbulent Rayleigh-B\'enard convection

## TL;DR

This paper develops a Lagrangian framework to relate thermal dissipation to mixing times in turbulent Rayleigh-Bénard convection, predicting conditions for the ultimate heat transfer regime and proposing experimental and numerical tests.

## Contribution

It introduces a Lagrangian relation for thermal dissipation in Rayleigh-Bénard convection and proposes a new criterion for the ultimate regime based on near-wall mixing times.

## Key findings

- Ultimate regime occurs if near-wall mixing time is not much longer than free-fall time.
- Scaling laws depend on the behavior of thermal plumes at high Rayleigh numbers.
- Near-wall mixing time can be measured experimentally and numerically.

## Abstract

A Lagrangian fluctuation-dissipation relation has been derived in a previous work to describe the dissipation rate of advected scalars, both passive and active, in wall-bounded flows. We apply this relation here to develop a Lagrangian description of thermal dissipation in turbulent Rayleigh-B\'enard convection in a right-cylindrical cell of arbitrary cross-section, with either imposed temperature difference or imposed heat-flux at the top and bottom walls. We obtain an exact relation between the steady-state thermal dissipation rate and the time for passive tracer particles released at the top or bottom wall to mix to their final uniform value near those walls. We show that an "ultimate regime" with the Nusselt-number scaling predicted by Spiegel (1971) or, with a log-correction, by Kraichnan (1962) will occur at high Rayleigh numbers, unless this near-wall mixing time is asymptotically much longer than the free-fall time, or almost the large-scale circulation time. We suggest a new criterion for an ultimate regime in terms of transition to turbulence of a thermal "mixing zone", which is much wider than the standard thermal boundary layer. Kraichnan-Spiegel scaling may, however, not hold if the intensity and volume of thermal plumes decrease sufficiently rapidly with increasing Rayleigh number. To help resolve this issue, we suggest a program to measure the near-wall mixing time, which we argue is accessible both by laboratory experiment and by numerical simulation.