Title: A Lagrangian fluctuation-dissipation relation for scalar turbulence, III. Turbulent Rayleigh-Bénard convection,
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-Bénard
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.