Very different types of scaling of the Nusselt number Nu with the Rayleigh number Ra have experimentally been found in the very large Ra regime beyond 1011. We understand and interpret these results by extending the unifying theory of thermal convection [Grossmann and Lohse, Phys. Rev. Lett.86, 3316 (2001)] to the very large Ra regime where the kinetic boundary-layer is turbulent. The central idea is that the spatial extension of this turbulent boundary-layer with a logarithmic velocity profile is comparable to the size of the cell. Depending on whether the thermal transport is plume dominated, dominated by the background thermal fluctuations, or whether also the thermal boundary-layer is fully turbulent (leading to a logarithmic temperature profile), we obtain effective scaling laws of about NuRa0.14, NuRa0.22, and NuRa0.38, respectively. Depending on the initial conditions or random fluctuations, one or the other of these states may be realized. Since the theory is for both the heat flux Nu and the velocity amplitude Re, we can also give the scaling of the latter, namely, ReRa0.42, ReRa0.45, and ReRa0.50 in the respective ranges.

1.
G.
Ahlers
,
S.
Grossmann
, and
D.
Lohse
, “
Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection
,”
Rev. Mod. Phys.
81
,
503
(
2009
).
2.
D.
Lohse
and
K. -Q.
Xia
, “
Small-scale properties of turbulent Rayleigh–Bénard convection
,”
Annu. Rev. Fluid Mech.
42
,
335
(
2010
).
3.
J.
Niemela
and
K. R.
Sreenivasan
, “
The use of cryogenic helium for classical turbulence: Promises and hurdles
,”
J. Low Temp. Phys.
143
,
163
(
2006
).
4.
See the Focus issue on “
New perspectives in high-Rayleigh-number turbulent convection
,”
New J. Phys.
(
2010
)
5.
S.
Grossmann
and
D.
Lohse
, “
Scaling in thermal convection: A unifying view
,”
J. Fluid Mech.
407
,
27
(
2000
).
6.
S.
Grossmann
and
D.
Lohse
, “
Thermal convection for large Prandtl number
,”
Phys. Rev. Lett.
86
,
3316
(
2001
).
7.
S.
Grossmann
and
D.
Lohse
, “
Prandtl and Rayleigh number dependence of the Reynolds number in turbulent thermal convection
,”
Phys. Rev. E
66
,
016305
(
2002
).
8.
S.
Grossmann
and
D.
Lohse
, “
Fluctuations in turbulent Rayleigh-Bénard convection: The role of plumes
,”
Phys. Fluids
16
,
4462
(
2004
).
9.
J.
Niemela
,
L.
Skrbek
,
K. R.
Sreenivasan
, and
R.
Donnelly
, “
Turbulent convection at very high Rayleigh numbers
,”
Nature (London)
404
,
837
(
2000
).
10.
J.
Niemela
and
K. R.
Sreenivasan
, “
Confined turbulent convection
,”
J. Fluid Mech.
481
,
355
(
2003
).
11.
A.
Nikolaenko
and
G.
Ahlers
, “
Nusselt number measurements for turbulent Rayleigh–Bénard convection
,”
Phys. Rev. Lett.
91
,
084501
(
2003
).
12.
D.
Funfschilling
,
E.
Brown
,
A.
Nikolaenko
, and
G.
Ahlers
, “
Heat transport by turbulent Rayleigh–Bénard convection in cylindrical cells with aspect ratio one and larger
,”
J. Fluid Mech.
536
,
145
(
2005
).
13.
J.
Niemela
and
K. R.
Sreenivasan
, “
Turbulent convection at high Rayleigh numbers and aspect ratio 4
,”
J. Fluid Mech.
557
,
411
(
2006
).
14.
G.
Ahlers
,
D.
Funfschilling
, and
E.
Bodenschatz
, “
Transitions in heat transport by turbulent convection at Rayleigh numbers up to 1015
,”
New J. Phys.
11
,
123001
(
2009
).
15.
X.
Chavanne
,
F.
Chilla
,
B.
Castaing
,
B.
Hebral
,
B.
Chabaud
, and
J.
Chaussy
, “
Observation of the ultimate regime in Rayleigh–Bénard convection
,”
Phys. Rev. Lett.
79
,
3648
(
1997
).
16.
X.
Chavanne
,
F.
Chilla
,
B.
Chabaud
,
B.
Castaing
, and
B.
Hebral
, “
Turbulent Rayleigh–Bénard convection in gaseous and liquid He
,”
Phys. Fluids
13
,
1300
(
2001
).
17.
P.
Roche
,
F.
Gauthier
,
B.
Chabaud
, and
B.
Hébral
, “
Ultimate regime of convection: Robustness to poor thermal reservoirs
,”
Phys. Fluids
17
,
115107
(
2005
).
18.
P. E.
Roche
,
G.
Gauthier
,
R.
Kaiser
, and
J.
Salort
, “
On the triggering of the ultimate regime of convection
,”
New J. Phys.
12
,
085014
(
2010
).
19.
R. H.
Kraichnan
, “
Turbulent thermal convection at arbitrary Prandtl number
,”
Phys. Fluids
5
,
1374
(
1962
).
20.
R. J. A. M.
Stevens
,
R.
Verzicco
, and
D.
Lohse
, “
Radial boundary-layer structure and Nusselt number in Rayleigh–Bénard convection
,”
J. Fluid Mech.
643
,
495
(
2010
).
21.
O.
Shishkina
,
R. J. A. M.
Stevens
,
S.
Grossmann
, and
D.
Lohse
, “
Boundary-layer structure in turbulent thermal convection and its consequences for the required numerical resolution
,”
New J. Phys.
12
,
075022
(
2010
).
22.
D.
Funfschilling
,
E.
Bodenschatz
, and
G.
Ahlers
, “
Search for the ultimate state in turbulent Rayleigh–Bénard convection
,”
Phys. Rev. Lett.
103
,
014503
(
2009
).
23.
G.
Ahlers
,
Lecture at the Euromech Colloquium in Les Houches
,
2010
, see www.hirac4.cnrs.fr/HIRAC4_-_Talks_files/Ahlers.pdf;
G.
Ahlers
,
D.
Funfschilling
, and
E.
Bodenschatz
, “
Addendum to Transitions in heat transport by turbulent convection at Rayleigh numbers up to 1015
,”
New J. Phys.
13
,
049401
(
2011
).
24.
L. D.
Landau
and
E. M.
Lifshitz
,
Fluid Mechanics
(
Pergamon
,
Oxford
,
1987
).
25.
C.
Sun
,
Y. H.
Cheung
, and
K. -Q.
Xia
, “
Experimental studies of the viscous boundary-layer properties in turbulent Rayleigh–Bénard convection
,”
J. Fluid Mech.
605
,
79
(
2008
).
26.
Q.
Zhou
and
K. -Q.
Xia
, “
Measured instantaneous viscous boundary-layer in turbulent Rayleigh–Bénard convection
,”
Phys. Rev. Lett.
104
,
104301
(
2010
).
27.
Q.
Zhou
,
R. J. A. M.
Stevens
,
K.
Sugiyama
,
S.
Grossmann
,
D.
Lohse
, and
K. -Q.
Xia
, “
Prandtl–Blasius temperature and velocity boundary-layer profiles in turbulent Rayleigh–Bénard convection
,”
J. Fluid Mech.
664
,
297
(
2010
).
28.
H.
Schlichting
and
K.
Gersten
,
Boundary Layer Theory
, 8th ed. (
Springer
,
Berlin
,
2000
).
29.
D.
Lohse
and
F.
Toschi
, “
The ultimate state of thermal convection
,”
Phys. Rev. Lett.
90
,
034502
(
2003
).
30.
E.
Calzavarini
,
D.
Lohse
,
F.
Toschi
, and
R.
Tripiccione
, “
Rayleigh and Prandtl number scaling in the bulk of Rayleigh–Bénard turbulence
,”
Phys. Fluids
17
,
055107
(
2005
).
31.
B.
Dubrulle
, “
Momentum transport and torque scaling in Taylor–Couette flow from an analogy with turbulent convection
,”
Eur. Phys. J. B
21
,
295
(
2001
).
32.
B. I.
Shraiman
and
E. D.
Siggia
, “
Heat transport in high-Rayleigh number convection
,”
Phys. Rev. A
42
,
3650
(
1990
).
33.
S.
Grossmann
and
D.
Lohse
, in
Proceedings of the 7th International Summer School and Conference “Let’s Face Chaos through Nonlinear Dynamics”
, CAMTP - Center for Applied Mathematics and Theoretical Physics, University of Maribor, Slovenia, 29 June–13 July
2008
, edited by
M.
Robnik
and
V. G.
Romanovski
(
AIP
,
New York
,
2008
), pp.
68
75
.
34.
U.
Hansen
,
D. A.
Yuen
, and
S. E.
Kroening
, “
Transition to hard turbulence in thermal convection at infinite Prandtl number
,”
Phys. Fluids A
2
,
2157
(
1990
).
35.
U.
Hansen
,
D. A.
Yuen
, and
S. E.
Kroening
, “
Mass and heat-transfer in strongly time-dependent thermal convention at infinite Prandtl number
,”
Geophys. Astrophys. Fluid Dyn.
63
,
67
(
1992
).
36.
B.
Eckhardt
,
S.
Grossmann
, and
D.
Lohse
, “
Torque scaling in turbulent Taylor–Couette flow between independently rotating cylinders
,”
J. Fluid Mech.
581
,
221
(
2007
).
37.
D.
van Gils
,
S. G.
Huisman
,
G. W.
Bruggert
,
C.
Sun
, and
D.
Lohse
, “
Torque scaling in turbulent Taylor–Couette flow with co- and counter-rotating cylinders
,”
Phys. Rev. Lett.
106
,
024502
(
2011
).
38.
P.
Cortet
,
A.
Chiffaudel
,
F.
Daviaud
, and
B.
Dubrulle
, “
Experimental evidence of a phase transition in a closed turbulent flow
,”
Phys. Rev. Lett.
105
,
214501
(
2010
).
39.
S.
Weiss
,
R.
Stevens
,
J. -Q.
Zhong
,
H.
Clercx
,
D.
Lohse
, and
G.
Ahlers
, “
Finite-size effects lead to supercritical bifurcations in turbulent rotating Rayleigh–Bénard convection
,”
Phys. Rev. Lett.
105
,
224501
(
2010
);
[PubMed]
R. J. A. M.
Stevens
,
J. -Q.
Zhong
,
H. J. H.
Clercx
,
G.
Ahlers
, and
D.
Lohse
, “
Transitions between turbulent states in rotating Rayleigh–Bénard convection
,”
Phys. Rev. Lett.
103
,
024503
(
2009
).
[PubMed]
40.
F.
Ravelet
,
L.
Marié
,
A.
Chiffaudel
, and
F.
Daviaud
, “
Multistability and memory effect in a highly turbulent flow: Experimental evidence for a global bifurcation
,”
Phys. Rev. Lett.
93
,
164501
(
2004
).
41.
R.
Monchaux
,
M.
Berhanu
,
M.
Bourgoin
,
M.
Moulin
,
P.
Odier
,
J. F.
Pinton
,
R.
Volk
,
S.
Fauve
,
N.
Mordant
,
F.
Petrelis
,
A.
Chiffaudel
,
F.
Daviaud
,
B.
Dubrulle
,
C.
Gasquet
,
L.
Marie
, and
F.
Ravelet
, “
Generation of a magnetic field by dynamo action in a turbulent flow of liquid sodium
,”
Phys. Rev. Lett.
98
,
044502
(
2007
).
42.
F.
Ravelet
,
M.
Berhanu
,
R.
Monchaux
,
S.
Aumaitre
,
A.
Chiffaudel
,
F.
Daviaud
,
B.
Dubrulle
,
M.
Bourgoin
,
P.
Odier
,
N.
Plihon
,
J. F.
Pinton
,
R.
Volk
,
S.
Fauve
,
N.
Mordant
, and
F.
Petrelis
, “
Chaotic dynamos generated by a turbulent flow of liquid sodium
,”
Phys. Rev. Lett.
101
,
074502
(
2008
).
43.
R. D.
Simitev
and
F. H.
Busse
, “
Bistability and hysteresis of dipolar dynamos generated by turbulent convection in rotating spherical shells
,”
Europhys. Lett.
85
,
19001
(
2009
).
44.
This values is taken from Ref. 24—it may also be less, say, 320, as argued in Ref. 10. This uncertainty in Res is still an unresolved issue in the field, as the onset of turbulence in shear flow is not via a linear instability, but of nonlinear-non-normal type and depends on the flow details, cf., e.g., Refs. 45 and 46.
45.
S.
Grossmann
, “
The onset of shear flow turbulence
,”
Rev. Mod. Phys.
72
,
603
(
2000
).
46.
B.
Eckhardt
,
T.
Schneider
,
B.
Hof
, and
J.
Westerweel
, “
Turbulence transition in pipe flow
,”
Annu. Rev. Fluid Mech.
39
,
447
(
2007
).
You do not currently have access to this content.