In this paper, we develop a multivariate regression model and a neural network model to predict the Reynolds number (Re) and Nusselt number in turbulent thermal convection. We compare their predictions with those of earlier models of convection: Grossmann–Lohse [Phys. Rev. Lett. 86, 3316 (2001)], revised Grossmann–Lohse [Phys. Fluids 33, 015113 (2021)], and Pandey–Verma [Phys. Rev. E 94, 053106 (2016)] models. We observe that although the predictions of all the models are quite close to each other, the machine-learning models developed in this work provide the best match with the experimental and numerical results.

1.
S.
Chandrasekhar
,
Hydrodynamic and Hydromagnetic Stability
(
Dover Publications
,
Oxford
,
1981
).
2.
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
).
3.
F.
Chillà
and
J.
Schumacher
, “
New perspectives in turbulent Rayleigh-Bénard convection
,”
Eur. Phys. J. E
35
,
58
(
2012
).
4.
E. D.
Siggia
, “
High Rayleigh number convection
,”
Annu. Rev. Fluid Mech.
26
,
137
168
(
1994
).
5.
K.-Q.
Xia
, “
Current trends and future directions in turbulent thermal convection
,”
Theor. Appl. Mech. Lett.
3
,
052001
(
2013
).
6.
M. K.
Verma
,
Physics of Buoyant Flows: From Instabilities to Turbulence
(
World Scientific
,
Singapore
,
2018
).
7.
W. V. R.
Malkus
, “
The heat transport and spectrum of thermal turbulence
,”
Proc. R. Soc. London, Ser. A
225
,
196–212
(
1954
).
8.
B. I.
Shraiman
and
E. D.
Siggia
, “
Heat transport in high-Rayleigh-number convection
,”
Phys. Rev. A
42
,
3650
3653
(
1990
).
9.
S.
Cioni
,
S.
Ciliberto
, and
J.
Sommeria
, “
Strongly turbulent Rayleigh–Bénard convection in mercury: Comparison with results at moderate Prandtl number
,”
J. Fluid Mech.
335
,
111
140
(
1997
).
10.
J. D.
Scheel
and
J.
Schumacher
, “
Predicting transition ranges to fully turbulent viscous boundary layers in low Prandtl number convection flows
,”
Phys. Rev. Fluids
2
,
123501
(
2017
).
11.
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
).
12.
X.
Chavanne
,
F.
Chillà
,
B.
Castaing
,
B.
Hebral
,
B.
Chabaud
, and
J.
Chaussy
, “
Observation of the ultimate regime in Rayleigh-Bénard convection
,”
Phys. Rev. Lett.
79
,
3648
3651
(
1997
).
13.
S.
Horn
,
O.
Shishkina
, and
C.
Wagner
, “
On non-Oberbeck–Boussinesq effects in three-dimensional Rayleigh–Bénard convection in glycerol
,”
J. Fluid Mech.
724
,
175
202
(
2013
).
14.
S.
Wagner
and
O.
Shishkina
, “
Aspect-ratio dependency of Rayleigh-Bénard convection in box-shaped containers
,”
Phys. Fluids
25
,
085110
(
2013
).
15.
M.
Kaczorowski
and
K.-Q.
Xia
, “
Turbulent flow in the bulk of Rayleigh–Bénard convection: Small-scale properties in a cubic cell
,”
J. Fluid Mech.
722
,
596
617
(
2013
).
16.
J. J.
Niemela
and
K. R.
Sreenivasan
, “
Confined turbulent convection
,”
J. Fluid Mech.
481
,
355
384
(
2003
).
17.
D.
Funfschilling
,
E.
Brown
,
A.
Nikolaenko
, and
G.
Ahlers
, “
Heat transport in turbulent Rayleigh-Bénard convection in cylindrical samples with aspect ratio one and larger
,”
J. Fluid Mech.
536
,
145
154
(
2005
).
18.
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
507
(
2010
).
19.
D.-L.
Dong
,
B.-F.
Wang
,
Y.-H.
Dong
,
Y.-X.
Huang
,
N.
Jiang
,
Y.-L.
Liu
,
Z.-M.
Lu
,
X.
Qiu
,
Z.-Q.
Tang
, and
Q.
Zhou
, “
Influence of spatial arrangements of roughness elements on turbulent Rayleigh-Bénard convection
,”
Phys. Fluids
32
,
045114
(
2020
).
20.
U.
Madanan
and
R. J.
Goldstein
, “
High-Rayleigh-number thermal convection of compressed gases in inclined rectangular enclosures
,”
Phys. Fluids
32
,
017103
(
2020
).
21.
M.
Vial
and
R. H.
Hernández
, “
Feedback control and heat transfer measurements in a Rayleigh-Bénard convection cell
,”
Phys. Fluids
29
,
074103
(
2017
).
22.
K.-Q.
Xia
,
S.
Lam
, and
S.-Q.
Zhou
, “
Heat-flux measurement in high-Prandtl-number turbulent Rayleigh-Bénard convection
,”
Phys. Rev. Lett.
88
,
064501
(
2002
).
23.
R.
Verzicco
and
R.
Camussi
, “
Prandtl number effects in convective turbulence
,”
J. Fluid Mech.
383
,
55
73
(
1999
).
24.
J. J.
Niemela
,
L.
Skrbek
,
K. R.
Sreenivasan
, and
R. J.
Donnelly
, “
The wind in confined thermal convection
,”
J. Fluid Mech.
449
,
169
178
(
2001
).
25.
S.
Lam
,
X.-D.
Shang
,
S.-Q.
Zhou
, and
K.-Q.
Xia
, “
Prandtl number dependence of the viscous boundary layer and the Reynolds numbers in Rayleigh-Bénard convection
,”
Phys. Rev. E
65
,
066306
(
2002
).
26.
M. S.
Emran
and
J.
Schumacher
, “
Fine-scale statistics of temperature and its derivatives in convective turbulence
,”
J. Fluid Mech.
611
,
13
34
(
2008
).
27.
G.
Silano
,
K. R.
Sreenivasan
, and
R.
Verzicco
, “
Numerical simulations of Rayleigh–Bénard convection for Prandtl numbers between 10–1 and 104 and Rayleigh numbers between 105 and 109
,”
J. Fluid Mech.
662
,
409
446
(
2010
).
28.
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
).
29.
A.
Pandey
and
M. K.
Verma
, “
Scaling of large-scale quantities in Rayleigh-Bénard convection
,”
Phys. Fluids
28
,
095105
(
2016
).
30.
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
).
31.
E.
Brown
,
D.
Funfschilling
, and
G.
Ahlers
, “
Anomalous Reynolds-number scaling in turbulent Rayleigh–Bénard convection
,”
J. Stat. Mech. Theor. Exp.
2007
,
P10005
.
32.
R. H.
Kraichnan
, “
Turbulent thermal convection at arbitrary Prandtl number
,”
Phys. Fluids
5
,
1374
1389
(
1962
).
33.
D.
Lohse
and
F.
Toschi
, “
Ultimate state of thermal convection
,”
Phys. Rev. Lett.
90
,
034502
(
2003
).
34.
S. S.
Pawar
and
J. H.
Arakeri
, “
Two regimes of flux scaling in axially homogeneous turbulent convection in vertical tube
,”
Phys. Rev. Fluids
1
,
042401(R)
(
2016
).
35.
L. E.
Schmidt
,
E.
Calzavarini
,
D.
Lohse
, and
F.
Toschi
, “
Axially homogeneous Rayleigh-Bénard convection in a cylindrical cell
,”
J. Fluid Mech.
691
,
52
68
(
2012
).
36.
P.-E.
Roche
,
B.
Castaing
,
B.
Chabaud
, and
B.
Hebral
, “
Observation of the 1/2 power law in Rayleigh-Bénard convection
,”
Phys. Rev. E
63
,
045303(R)
(
2001
).
37.
G.
Ahlers
,
X.
He
,
D.
Funfschilling
, and
E.
Bodenschatz
, “
Heat transport by turbulent Rayleigh-Bénard convection for Pr0.8 and 3×1012Ra1015: Aspect ratio T = 0.50
,”
New J. Phys.
12
,
103012
(
2012
).
38.
X.
He
,
D.
Funfschilling
,
H.
Nobach
,
E.
Bodenschatz
, and
G.
Ahlers
, “
Transition to the ultimate state of turbulent Rayleigh-Bénard convection
,”
Phys. Rev. Lett.
108
,
024502
(
2012
).
39.
J. J.
Niemela
,
L.
Skrbek
,
K. R.
Sreenivasan
, and
R. J.
Donnelly
, “
Turbulent convection at very high Rayleigh numbers
,”
Nature
404
,
837
840
(
2000
).
40.
P.
Urban
,
P.
Hanzelka
,
T.
Kralik
,
V.
Musilová
,
A.
Srnka
, and
L.
Skrbek
, “
Effect of boundary layers asymmetry on heat transfer efficiency in turbulent Rayleigh-Bernard convection at very high Rayleigh numbers
,”
Phys. Rev. Lett.
109
,
154301
(
2012
).
41.
S.
Grossmann
and
D.
Lohse
, “
Scaling in thermal convection: A unifying theory
,”
J. Fluid Mech.
407
,
27
56
(
2000
).
42.
S.
Grossmann
and
D.
Lohse
, “
Thermal convection for large Prandtl numbers
,”
Phys. Rev. Lett.
86
,
3316
3319
(
2001
).
43.
S.
Bhattacharya
,
M. K.
Verma
, and
R.
Samtaney
, “
Revisiting Reynolds and Nusselt numbers in turbulent thermal convection
,”
Phys. Fluids
33
,
015113
(
2021
).
44.
S.
Bhattacharya
,
A.
Pandey
,
A.
Kumar
, and
M. K.
Verma
, “
Complexity of viscous dissipation in turbulent thermal convection
,”
Phys. Fluids
30
,
031702
(
2018
).
45.
S.
Bhattacharya
,
R.
Samtaney
, and
M. K.
Verma
, “
Scaling and spatial intermittency of thermal dissipation in turbulent convection
,”
Phys. Fluids
31
,
075104
(
2019
).
46.
E. J.
Parish
and
K.
Duraisamy
, “
A paradigm for data-driven predictive modeling using field inversion and machine learning
,”
J. Comput. Phys.
305
,
758
774
(
2016
).
47.
E.
Fonda
,
A.
Pandey
,
J.
Schumacher
, and
K. R.
Sreenivasan
, “
Deep learning in turbulent convection networks
,”
Proc. Natl. Acad. Sci. U.S.A.
116
,
8667
8672
(
2019
).
48.
S.
Pandey
and
J.
Schumacher
, “
Reservoir computing model of two-dimensional turbulent convection
,”
Phys. Rev. Fluids
5
,
113506
(
2020
).
49.
S.
Pandey
,
J.
Schumacher
, and
K. R.
Sreenivasan
, “
A perspective on machine learning in turbulent flows
,”
J. Turbul.
21
,
567
584
(
2020
).
50.
S. L.
Brunton
,
B. R.
Noack
, and
P.
Koumoutsakos
, “
Machine learning for fluid mechanics
,”
Annu. Rev. Fluid Mech.
52
,
477
508
(
2020
).
51.
I.
Goodfellow
,
Y.
Bengio
, and
A.
Courville
,
Deep Learning
(
The MIT Press
,
Cambridge, MA
,
2016
).
52.
T.
Hastie
,
R.
Tibshirani
, and
J.
Friedman
,
The Elements of Statistical Learning
(
Springer
,
New York
,
2009
).
53.
A.
Burkov
,
The Hundred-Page Machine Learning Book
(
Andriy Burkov
,
Québec
,
2019
).
54.
R. J. A. M.
Stevens
,
E. P.
van der Poel
,
S.
Grossmann
, and
D.
Lohse
, “
The unifying theory of scaling in thermal convection: The updated prefactors
,”
J. Fluid Mech.
730
,
295
308
(
2013
).
55.
M.
Lesieur
,
Turbulence in Fluids
(
Springer-Verlag
,
Dordrecht
,
2008
).
56.
M. K.
Verma
,
Energy Trasnfers in Fluid Flows: Multiscale and Spectral Perspectives
(
Cambridge University Press
,
Cambridge
,
2019
).
57.
S.
Bhattacharya
,
S.
Sadhukhan
,
A.
Guha
, and
M. K.
Verma
, “
Similarities between the structure functions of thermal convection and hydrodynamic turbulence
,”
Phys. Fluids
31
,
115107
(
2019
).
58.
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
).
59.
N.
Shi
,
M. S.
Emran
, and
J.
Schumacher
, “
Boundary layer structure in turbulent Rayleigh–Bénard convection
,”
J. Fluid Mech.
706
,
5
33
(
2012
).
60.
R. J.
Samuel
,
S.
Bhattacharya
,
A.
Asad
,
S.
Chatterjee
,
M. K.
Verma
,
R.
Samtaney
, and
S. F.
Anwer
, “
SARAS: A general-purpose PDE solver for fluid dynamics
,”
J. Open Source Software
6
,
2095
(
2021
).
61.
M. K.
Verma
,
R. J.
Samuel
,
S.
Chatterjee
,
S.
Bhattacharya
, and
A.
Asad
, “
Challenges in fluid flow simulations using exascale computing
,”
SN Comput. Sci.
1
,
178
(
2020
).
62.
S.
Bhattacharya
,
M. K.
Verma
, and
R.
Samtaney
, “
Prandtl number dependence of the small-scale properties in turbulent Rayleigh-Bénard convection
,”
Phys. Rev. Fluids
6
,
063501
(
2021
).
63.
C. F.
Jekel
and
G.
Venter
, see https://github.com/cjekel/piecewise_linear_fit_py for “pwlf: A python library for fitting 1D continuous piecewise linear functions (
2019
)”
64.
E.
Frank
,
M.
Hall
,
G.
Holmes
,
R.
Kirkby
,
B.
Pfahringer
,
I. H.
Witten
, and
L.
Trigg
, “
Weka-A machine learning workbench for data mining
,” in
Data Mining and Knowledge Discovery Handbook
(
Springer
,
2009
), pp.
1269
1277
.
65.
K.
Hornik
,
M.
Stinchcombe
, and
H.
White
, “
Multilayer feedforward networks are universal approximators
,”
Neural Networks
2
,
359
366
(
1989
).
66.
A.
Gulli
and
S.
Pal
,
Deep Learning with Keras
(
Packt Publishing
,
Birmingham
,
2017
).
67.
C. M.
Bishop
,
Pattern Recognition and Machine Learning
(
Springer
,
Singapore
,
2006
).
68.
X.
Zhu
,
V.
Mathai
,
R. J. A. M.
Stevens
,
R.
Verzicco
, and
D.
Lohse
, “
Transition to the ultimate regime in two-dimensional Rayleigh-Bénard convection
,”
Phys. Rev. Lett.
120
,
144502
(
2018
).
69.
K. P.
Iyer
,
J. D.
Scheel
,
J.
Schumacher
, and
K. R.
Sreenivasan
, “
Classical 1/3 scaling of convection holds up to Ra
=1015,”
Proc. Natl. Acad. Sci. U.S.A
117
,
7594
(
2020
).

Supplementary Material

You do not currently have access to this content.