We derive scaling relations for the thermal dissipation rate in the bulk and in the boundary layers for moderate and large Prandtl number (Pr) convection. Using direct numerical simulations of Rayleigh-Bénard convection, we show that the thermal dissipation in the bulk is suppressed compared to passive scalar dissipation. The suppression is stronger for large Pr. We further show that the dissipation in the boundary layers dominates that in the bulk for both moderate and large Pr. The probability distribution functions of thermal dissipation rate, both in the bulk and in the boundary layers, are stretched exponential, similar to passive scalar dissipation.
REFERENCES
1.
A. M.
Obukhov
, “Structure of the temperature field in a turbulent flow
,” Isv. Geogr. Geophys. Ser.
13
, 58
–69
(1949
).2.
S.
Corrsin
, “On the spectrum of isotropic temperature fluctuations in an isotropic turbulence
,” J. Appl. Phys.
22
, 469
–473
(1951
).3.
M.
Lesieur
, Turbulence in Fluids
(Springer-Verlag
, Dordrecht
, 2008
).4.
M. K.
Verma
, Physics of Buoyant Flows
(World Scientific
, Singapore
, 2018
).5.
G.
Ahlers
, S.
Grossmann
, and D.
Lohse
, “Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection
,” Rev. Mod. Phys.
81
, 503
–537
(2009
).6.
D.
Lohse
and K.-Q.
Xia
, “Small-scale properties of turbulent Rayleigh–Bénard convection
,” Annu. Rev. Fluid Mech.
42
, 335
–364
(2010
).7.
M. K.
Verma
, A.
Kumar
, and A.
Pandey
, “Phenomenology of buoyancy-driven turbulence: Recent results
,” New J. Phys.
19
, 025012
(2017
).8.
B. I.
Shraiman
and E. D.
Siggia
, “Heat transport in high-Rayleigh-number convection
,” Phys. Rev. A
42
, 3650
–3653
(1990
).9.
R. H.
Kraichnan
, “Turbulent thermal convection at arbitrary Prandtl number
,” Phys. Fluids
5
, 1374
–1389
(1962
).10.
M. K.
Verma
, P. K.
Mishra
, A.
Pandey
, and S.
Paul
, “Scalings of field correlations and heat transport in turbulent convection
,” Phys. Rev. E
85
, 016310
(2012
).11.
S.
Grossmann
and D.
Lohse
, “Scaling in thermal convection: A unifying theory
,” J. Fluid Mech.
407
, 27
–56
(2000
).12.
S.
Grossmann
and D.
Lohse
, “Thermal convection for large Prandtl numbers
,” Phys. Rev. Lett.
86
, 3316
–3319
(2001
).13.
W. V. R.
Malkus
, “The heat transport and spectrum of thermal turbulence
,” Proc. R. Soc. London, Ser. A
225
, 196
–212
(1954
).14.
B.
Castaing
, G.
Gunaratne
, L. P.
Kadanoff
, A.
Libchaber
, and F.
Heslot
, “Scaling of hard thermal turbulence in Rayleigh-Bénard convection
,” J. Fluid Mech.
204
, 1
–30
(1989
).15.
S.
Grossmann
and D.
Lohse
, “Prandtl and Rayleigh number dependence of the Reynolds number in turbulent thermal convection
,” Phys. Rev. E
66
, 016305
(2002
).16.
X.-L.
Qiu
and P.
Tong
, “Temperature oscillations in turbulent Rayleigh-Bénard convection
,” Phys. Rev. E
66
, 026308
(2002
).17.
X.-L.
Qiu
, X.-D.
Shang
, P.
Tong
, and K.-Q.
Xia
, “Velocity oscillations in turbulent Rayleigh-Bénard convection
,” Phys. Fluids
16
, 412
–423
(2004
).18.
E.
Brown
, D.
Funfschilling
, and G.
Ahlers
, “Anomalous Reynolds-number scaling in turbulent Rayleigh–Bénard convection
,” J. Stat. Mech.: Theory Exp.
2007
, P10005
.19.
D.
Funfschilling
, E.
Brown
, A.
Nikolaenko
, and G.
Ahlers
, “Heat transport by turbulent Rayleigh-Bénard convection in cylindrical samples with aspect ratio one and larger
,” J. Fluid Mech.
536
, 145
–154
(2005
).20.
A.
Nikolaenko
, E.
Brown
, D.
Funfschilling
, and G.
Ahlers
, “Heat transport by turbulent Rayleigh-Bénard convection in cylindrical cells with aspect ratio one and less
,” J. Fluid Mech.
523
, 251
–260
(2005
).21.
X.
He
, D.
Funfschilling
, E.
Bodenschadtz
, and G.
Ahlers
, “Heat transport by turbulent Rayleigh-Bénard convection for Pr ≃ 0.8 and 4 × 1011 ≲ Ra ≲ 2 × 1014: Ultimate-state transition for aspect ratio Γ = 1.00
,” New J. Phys.
14
, 063030
(2012
).22.
G.
Ahlers
, X.
He
, D.
Funfschilling
, and E.
Bodenschadtz
, “Heat transport by turbulent Rayleigh-Bénard convection for Pr ≃ 0.8 and 3 × 1012 ≲ Ra ≲ 1015: Aspect ratio Γ = 0.50
,” New J. Phys.
14
, 103012
(2012
).23.
M.
Vial
and R. H.
Hernándes
, “Feedback control and heat transfer measurements in a Rayleigh-Bénard convection cell
,” Phys. Fluids
29
, 074103
(2017
).24.
R.
Verzicco
and R.
Camussi
, “Numerical experiments on strongly turbulent thermal convection in a slender cylindrical cell
,” J. Fluid Mech.
477
, 19
–49
(2003
).25.
J. D.
Scheel
, E.
Kim
, and K. R.
White
, “Thermal and viscous boundary layers in turbulent Rayleigh–Bénard convection
,” J. Fluid Mech.
711
, 281
–305
(2012
).26.
J. D.
Scheel
and J.
Schumacher
, “Local boundary layer scales in turbulent Rayleigh–Bénard convection
,” J. Fluid Mech.
758
, 344
–373
(2014
).27.
F.
Waleffe
, A.
Boonkasame
, and L. M.
Smith
, “Heat transport by coherent Rayleigh-Bénard convection
,” Phys. Fluids
27
, 051702
(2015
).28.
M. K.
Verma
, S. C.
Ambhire
, and A.
Pandey
, “Flow reversals in turbulent convection with free-slip walls
,” Phys. Fluids
27
, 047102
(2015
).29.
W.-F.
Zhou
and J.
Chen
, “Letter: Similarity model for corner roll in turbulent Rayleigh-Bénard convection
,” Phys. Fluids
30
, 111705
(2018
).30.
A.
Pandey
and M. K.
Verma
, “Scaling of large-scale quantities in Rayleigh-Bénard convection
,” Phys. Fluids
28
, 095105
(2016
).31.
A.
Pandey
, A.
Kumar
, A. G.
Chatterjee
, and M. K.
Verma
, “Dynamics of large-scale quantities in Rayleigh-Bénard convection
,” Phys. Rev. E
94
, 053106
(2016
).32.
B. A.
Puthenveettil
and J. H.
Arakeri
, “Plume structure in high-Rayleigh-number convection
,” J. Fluid Mech.
542
, 217
–249
(2005
).33.
B. A.
Puthenveettil
, G.
Ananthakrishna
, and J. H.
Arakeri
, “The multifractal nature of plume structure in high-Rayleigh-number convection
,” J. Fluid Mech.
526
, 245
–256
(2005
).34.
S.
Bhattacharya
, A.
Pandey
, A.
Kumar
, and M. K.
Verma
, “Complexity of viscous dissipation in turbulent thermal convection
,” Phys. Fluids
30
, 031702
(2018
).35.
Y.
Zhang
, Q.
Zhou
, and C.
Sun
, “Statistics of kinetic and thermal energy dissipation rates in two-dimensional turbulent Rayleigh–Bénard convection
,” J. Fluid Mech.
814
, 165
–184
(2017
).36.
S.
Chandrasekhar
, Hydrodynamic and Hydromagnetic Stability
(Oxford University Press
, Oxford
, 2013
).37.
A.
Kumar
and M. K.
Verma
, “Applicability of Taylor’s hypothesis in thermally driven turbulence
,” R. Soc. Open Sci.
5
, 172152
(2018
).38.
H.
Jasak
, A.
Jemcov
, Z.
Tukovic
et al, “OpenFOAM: A C++ library for complex physics simulations
,” in International Workshop on Coupled Methods in Numerical Dynamics
(IUC Dubrovnik
, Croatia
, 2007
), Vol. 1000, pp. 1
–20
.39.
G.
Grötzbach
, “Spatial resolution requirements for direct numerical simulation of the Rayleigh-Bénard convection
,” J. Comput. Phys.
49
, 241
–264
(1983
).40.
N.
Shi
, M. S.
Emran
, and J.
Schumacher
, “Boundary layer structure in turbulent Rayleigh–Bénard convection
,” J. Fluid Mech.
706
, 5
–33
(2012
).41.
G. K.
Batchelor
, “Small-scale variation of convected quantities like temperature in turbulent fluid Part 1. General discussion and the case of small conductivity
,” J. Fluid Mech.
5
, 113
–133
(1959
).42.
S.
Wagner
and O.
Shishkina
, “Aspect-ratio dependency of Rayleigh-Benard convection in box-shaped containers
,” Phys. Fluids
25
, 085110
(2013
).43.
A.
Pandey
, M. K.
Verma
, and P. K.
Mishra
, “Scaling of heat flux and energy spectrum for very large Prandtl number convection
,” Phys. Rev. E
89
, 023006
(2014
).44.
M. S.
Emran
and J.
Schumacher
, “Fine-scale statistics of temperature and its derivatives in convective turbulence
,” J. Fluid Mech.
611
, 13
–34
(2008
).45.
O.
Shishkina
, M. S.
Emran
, S.
Grossmann
, and D.
Lohse
, “Scaling relations in large-Prandtl-number natural thermal convection
,” Phys. Rev. Fluids
2
, 103502
(2017
).46.
M.
Chertkov
, G.
Falkovich
, and I.
Kolokolov
, “Intermittent dissipation of a passive scalar in turbulence
,” Phys. Rev. Lett.
80
, 2121
–2124
(1998
).47.
X.
He
and P.
Tong
, “Measurements of the thermal dissipation field in turbulent Rayleigh-Bénard convection
,” Phys. Rev. E
79
, 026306
(2009
).© 2019 Author(s).
2019
Author(s)
You do not currently have access to this content.