TY - JOUR
T1 - A Lagrangian fluctuation-dissipation relation for scalar turbulence. Part III. Turbulent Rayleigh-Bénard convection
AU - Eyink, Gregory L.
AU - Drivas, Theodore D.
N1 - Funding Information:
We would like to thank C. Doering and D. Goluskin for useful discussions, and the comments of several anonymous reviewers that have helped to improve the paper. We would also like to thank M. Cencini for bringing to our attention the prior work of Celani et al. (2004). Finally, we thank the Institute for Pure and Applied Mathematics (IPAM) at UCLA, where this paper was partially completed during the fall 2014 long program on ‘Mathematics of Turbulence’. G.L.E. is partially supported by a grant from NSF CBET-1507469 and T.D.D. was partially supported by the Duncan Fund and a Fink Award from the Department of Applied Mathematics and Statistics at the Johns Hopkins University.
Publisher Copyright:
© 2017 Cambridge University Press.
PY - 2018/2/10
Y1 - 2018/2/10
N2 - 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 τmix 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 (Annu. Rev. Astron., vol. 9, 1971, p. 323) or, with a log correction, by Kraichnan (Phys. Fluids, vol. 5 (11), 1962, pp. 1374-1389) will occur at high Rayleigh numbers, unless this near-wall mixing time is asymptotically much longer than the free-fall time free. Precisely, we show that τmix=τfree =RaPr1/2/Nu; with Ra the Rayleigh number, Pr the Prandtl number, and Nu the Nusselt number. 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 τmix, which is precisely defined in the paper and which we argue is accessible both by laboratory experiment and by numerical simulation.
AB - 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 τmix 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 (Annu. Rev. Astron., vol. 9, 1971, p. 323) or, with a log correction, by Kraichnan (Phys. Fluids, vol. 5 (11), 1962, pp. 1374-1389) will occur at high Rayleigh numbers, unless this near-wall mixing time is asymptotically much longer than the free-fall time free. Precisely, we show that τmix=τfree =RaPr1/2/Nu; with Ra the Rayleigh number, Pr the Prandtl number, and Nu the Nusselt number. 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 τmix, which is precisely defined in the paper and which we argue is accessible both by laboratory experiment and by numerical simulation.
KW - Bénard convection
KW - convection
KW - turbulent convection
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U2 - 10.1017/jfm.2017.788
DO - 10.1017/jfm.2017.788
M3 - Article
AN - SCOPUS:85039723869
SN - 0022-1120
VL - 836
SP - 560
EP - 598
JO - Journal of Fluid Mechanics
JF - Journal of Fluid Mechanics
ER -