We adopt the stretched spiral vortex sub-grid model for large-eddy simulation (LES) of turbulent convection at extreme Rayleigh numbers. We simulate Rayleigh–Bénard convection (RBC) for Rayleigh numbers ranging from 106 to 1015 and for Prandtl numbers 0.768 and 1. We choose a box of dimensions 1:1:10 to reduce computational cost. Our LES yields Nusselt and Reynolds numbers that are in good agreement with the direct-numerical simulation (DNS) results of Iyer et al. [“Classical 1/3 scaling of convection holds up to Ra=1015,” Proc. Natl. Acad. Sci. U. S. A. 117, 7594–7598 (2020)] albeit with a smaller grid size and at significantly reduced computational expense. For example, in our simulations at Ra=1013, we use grids that are 1/120 times the grid resolution as that of the DNS [Iyer et al., “Classical 1/3 scaling of convection holds up to Ra=1015,” Proc. Natl. Acad. Sci. U. S. A. 117, 7594–7598 (2020)]. The Reynolds numbers in our simulations span 3 orders of magnitude from 1000 to 1 700 000. Consistent with the literature, we obtain scaling relations for Nusselt and Reynolds numbers as NuRa0.321 and ReRa0.495. We also perform LES of RBC with periodic side walls, for which we obtain the corresponding scaling exponents as 0.343 and 0.477, respectively. Our LES is a promising tool to push simulations of thermal convection to extreme Rayleigh numbers and, hence, enable us to test the transition to the ultimate convection regime.

1.
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
7598
(
2020
).
2.
A. V.
Getling
,
Rayleigh-Bénard Convection: Structures and Dynamics
(
World Scientific
,
Singapore
,
1998
).
3.
M. K.
Verma
,
Physics of Buoyant Flows: From Instabilities to Turbulence
(
World Scientific
,
Singapore
,
2018
).
4.
B.
Castaing
,
G.
Gunaratne
,
F.
Heslot
,
L. P.
Kadanoff
, and
A.
Libchaber
, et al.,
Scaling of hard thermal turbulence in Rayleigh-Bénard convection
,”
J. Fluid Mech.
204
,
1
30
(
1989
).
5.
E. D.
Siggia
, “
High Rayleigh number convection
,”
Annu. Rev. Fluid Mech.
26
,
137
168
(
1994
).
6.
L. P.
Kadanoff
, “
Turbulent heat flow: Structures and scaling
,”
Phys. Today
54
(
8
),
34
39
(
2001
).
7.
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
).
8.
S.
Bhattacharya
,
M. K.
Verma
, and
R.
Samtaney
, “
Revisiting Reynolds and Nusselt numbers in turbulent thermal convection
,”
Phys. Fluids
33
,
015113
(
2021
).
9.
S.
Bhattacharya
,
M. K.
Verma
, and
A.
Bhattacharya
, “
Predictions of Reynolds and Nusselt numbers in turbulent convection using machine-learning models
,”
Phys. Fluids
34
,
025102
(
2022
).
10.
S.
Pandey
,
P.
Teutsch
,
P.
Mäder
, and
J.
Schumacher
, “
Direct data-driven forecast of local turbulent heat flux in Rayleigh–Bénard convection
,”
Phys. Fluids
34
,
045106
(
2022
).
11.
S.
Grossmann
and
D.
Lohse
, “
Scaling in thermal convection: A unifying theory
,”
J. Fluid Mech.
407
,
27
56
(
2000
).
12.
R. H.
Kraichnan
, “
Turbulent thermal convection at arbitrary Prandtl number
,”
Phys. Fluids
5
,
1374
1389
(
1962
).
13.
R.
Verzicco
and
R.
Camussi
, “
Numerical experiments on strongly turbulent thermal convection in a slender cylindrical cell
,”
J. Fluid Mech.
477
,
19
49
(
2003
).
14.
J.
Bailon-Cuba
,
M. S.
Emran
, and
J.
Schumacher
, “
Aspect ratio dependence of heat transfer and large-scale flow in turbulent convection
,”
J. Fluid Mech.
655
,
152
173
(
2010
).
15.
N.
Foroozani
,
D.
Krasnov
, and
J.
Schumacher
, “
Turbulent convection for different thermal boundary conditions at the plates
,”
J. Fluid Mech.
907
,
A27
(
2021
).
16.
O.
Shishkina
and
A.
Thess
, “
Mean temperature profiles in turbulent Rayleigh–Bénard convection of water
,”
J. Fluid Mech.
633
,
449
460
(
2009
).
17.
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
).
18.
G. L.
Kooij
,
M. A.
Botchev
,
E. M.
Frederix
,
B. J.
Geurts
,
S.
Horn
,
D.
Lohse
,
E. P.
van der Poel
,
O.
Shishkina
,
R. J.
Stevens
, and
R.
Verzicco
, “
Comparison of computational codes for direct numerical simulations of turbulent Rayleigh–Bénard convection
,”
Comput. Fluids
166
,
1
8
(
2018
).
19.
J. J.
Niemela
and
K. R.
Sreenivasan
, “
Confined turbulent convection
,”
J. Fluid Mech.
481
,
355
384
(
2003
).
20.
P.
Urban
,
V.
Musilová
, and
L.
Skrbek
, “
Efficiency of heat transfer in turbulent Rayleigh-Bénard convection
,”
Phys. Rev. Lett.
107
,
014302
(
2011
).
21.
A.
Kumar
,
A. G.
Chatterjee
, and
M. K.
Verma
, “
Energy spectrum of buoyancy-driven turbulence
,”
Phys. Rev. E
90
,
023016
(
2014
).
22.
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
).
23.
S.
Wagner
and
O.
Shishkina
, “
Aspect-ratio dependency of Rayleigh-Bénard convection in box-shaped containers
,”
Phys. Fluids
25
,
085110
(
2013
).
24.
K. L.
Chong
,
S.
Wagner
,
M.
Kaczorowski
,
O.
Shishkina
, and
K.-Q.
Xia
, “
Effect of Prandtl number on heat transport enhancement in Rayleigh-Bénard convection under geometrical confinement
,”
Phys. Rev. Fluids
3
,
013501
(
2018
).
25.
M. K.
Verma
,
A.
Kumar
, and
A.
Pandey
, “
Phenomenology of buoyancy-driven turbulence: Recent results
,”
New J. Phys.
19
,
025012
(
2017
).
26.
O.
Shishkina
, “
Rayleigh-Bénard convection: The container shape matters
,”
Phys. Rev. Fluids
6
,
090502
(
2021
).
27.
T. M.
Eidson
, “
Numerical simulation of the turbulent Rayleigh–Bénard problem using subgrid modelling
,”
J. Fluid Mech.
158
,
245
268
(
1985
).
28.
S. J.
Kimmel
and
J. A.
Domaradzki
, “
Large eddy simulations of Rayleigh–Bénard convection using subgrid scale estimation model
,”
Phys. Fluids
12
,
169
184
(
2000
).
29.
N.
Foroozani
,
J. J.
Niemela
,
V.
Armenio
, and
K. R.
Sreenivasan
, “
Reorientations of the large-scale flow in turbulent convection in a cube
,”
Phys. Rev. E
95
,
033107
(
2017
).
30.
D.
Sondak
,
T. M.
Smith
,
R. P.
Pawlowski
,
S.
Conde
, and
J. N.
Shadid
, “
High Rayleigh number variational multiscale large eddy simulations of Rayleigh-Bénard convection
,”
Mech. Res. Commun.
112
,
103614
(
2021
).
31.
M.
Nguyen
,
J. F.
Boussuge
,
P.
Sagaut
, and
J. C.
Larroya-Huguet
, “
Large eddy simulation of a thermal impinging jet using the lattice Boltzmann method
,”
Phys. Fluids
34
,
055115
(
2022
).
32.
S. R. G.
Polasanapalli
and
K.
Anupindi
, “
Large-eddy simulation of turbulent natural convection in a cylindrical cavity using an off-lattice Boltzmann method
,”
Phys. Fluids
34
,
035125
(
2022
).
33.
S.
Vashishtha
,
M. K.
Verma
, and
R.
Samuel
, “
Large-eddy simulations of turbulent thermal convection using renormalized viscosity and thermal diffusivity
,”
Phys. Rev. E
98
,
043109
(
2018
).
34.
W. D.
McComb
,
Homogeneous, Isotropic Turbulence: Phenomenology, Renormalization and Statistical Closures
(
Oxford University Press
,
2014
).
35.
M. K.
Verma
, “
Statistical theory of magnetohydrodynamic turbulence: Recent results
,”
Phys. Rep.
401
,
229
380
(
2004
).
36.
M. K.
Verma
and
S.
Kumar
, “
Large-eddy simulations of fluid and magnetohydrodynamic turbulence using renormalized parameters
,”
Pramana-J. Phys.
63
,
553
561
(
2004
).
37.
S.
Vashishtha
,
R.
Samuel
,
A. G.
Chatterjee
,
R.
Samtaney
, and
M. K.
Verma
, “
Large eddy simulation of hydrodynamic turbulence using renormalized viscosity
,”
Phys. Fluids
31
,
065102
(
2019
).
38.
M. K.
Verma
,
A. G.
Chatterjee
,
R. K.
Yadav
,
S.
Paul
,
M.
Chandra
, and
R.
Samtaney
, “
Benchmarking and scaling studies of pseudospectral code Tarang for turbulence simulations
,”
Pramana-J. Phys.
81
,
617
629
(
2013
).
39.
A.
Misra
and
D. I.
Pullin
, “
A vortex-based subgrid stress model for large-eddy simulation
,”
Phys. Fluids
9
,
2443
2454
(
1997
).
40.
T. S.
Lundgren
, “
Strained spiral vortex model for turbulent fine structure
,”
Phys. Fluids
25
,
2193
2203
(
1982
).
41.
B.
Kosović
,
D. I.
Pullin
, and
R.
Samtaney
, “
Subgrid-scale modeling for large-eddy simulations of compressible turbulence
,”
Phys. Fluids
14
,
1511
1522
(
2002
).
42.
T.
Voelkl
,
D. I.
Pullin
, and
D. C.
Chan
, “
A physical-space version of the stretched-vortex subgrid-stress model for large-eddy simulation
,”
Phys. Fluids
12
,
1810
1825
(
2000
).
43.
D.
Chung
and
D. I.
Pullin
, “
Large-eddy simulation and wall modelling of turbulent channel flow
,”
J. Fluid Mech.
631
,
281
309
(
2009
).
44.
M.
Inoue
and
D. I.
Pullin
, “
Large-eddy simulation of the zero-pressure-gradient turbulent boundary layer up to
Reθ=O(1012),”
J. Fluid Mech.
686
,
507
533
(
2011
).
45.
W.
Cheng
and
R.
Samtaney
, “
A high-resolution code for large eddy simulation of incompressible turbulent boundary layer flows
,”
Comput. Fluids
92
,
82
92
(
2014
).
46.
W.
Cheng
,
D. I.
Pullin
, and
R.
Samtaney
, “
Large-eddy simulation of separation and reattachment of a flat plate turbulent boundary layer
,”
J. Fluid Mech.
785
,
78
108
(
2015
).
47.
W.
Gao
,
W.
Zhang
,
W.
Cheng
, and
R.
Samtaney
, “
Wall-modelled large-eddy simulation of turbulent flow past airfoils
,”
J. Fluid Mech.
873
,
174
210
(
2019
).
48.
W.
Cheng
,
D. I.
Pullin
, and
R.
Samtaney
, “
Large-eddy simulation and modelling of Taylor–Couette flow
,”
J. Fluid Mech.
890
,
A17
(
2020
).
49.
W.
Gao
,
W.
Cheng
, and
R.
Samtaney
, “
Large-eddy simulations of turbulent flow in a channel with streamwise periodic constrictions
,”
J. Fluid Mech.
900
,
A43
(
2020
).
50.
W.
Cheng
,
D.
Pullin
, and
R.
Samtaney
, “
Wall-resolved and wall-modelled large-eddy simulation of plane Couette flow
,”
J. Fluid Mech.
934
,
A19
(
2022
).
51.
D. I.
Pullin
, “
A vortex-based model for the subgrid flux of a passive scalar
,”
Phys. Fluids
12
,
2311
2319
(
2000
).
52.
D.
Chung
and
D. I.
Pullin
, “
Direct numerical simulation and large-eddy simulation of stationary buoyancy-driven turbulence
,”
J. Fluid Mech.
643
,
279
308
(
2010
).
53.
D.
Chung
and
G.
Matheou
, “
Large-eddy simulation of stratified turbulence. Part I: A vortex-based subgrid-scale model
,”
J. Atmos. Sci.
71
,
1863
1879
(
2014
).
54.
D. K.
Lilly
, “
Subgrid closures in large eddy simulation: Part I
,” in
Lecture Notes on Turbulence: Lecture Notes from GTP Summer School, June 1987
, edited by
J. R.
Herring
and
J. C.
McWilliams
(
World Scientific
,
Singapore
,
1989
), pp.
171
218
.
55.
R.
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
).
56.
M. K.
Verma
,
R.
Samuel
,
S.
Chatterjee
,
S.
Bhattacharya
, and
A.
Asad
, “
Challenges in fluid flow simulations using exascale computing
,”
SN Comput. Sci.
1
,
178
(
2020
).
57.
S.
Grossmann
and
D.
Lohse
, “
Thermal convection for large Prandtl numbers
,”
Phys. Rev. Lett.
86
,
3316
3319
(
2001
).
58.
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
).
59.
P. R.
Spalart
,
R. D.
Moser
, and
M. M.
Rogers
, “
Spectral methods for the Navier-Stokes equations with one infinite and two periodic directions
,”
J. Comput. Phys.
96
,
297
324
(
1991
).
60.
Y.
Morinishi
,
T.
Lund
,
O.
Vasilyev
, and
P.
Moin
, “
Fully conservative higher order finite difference schemes for incompressible flow
,”
J. Comput. Phys.
143
,
90
124
(
1998
).
61.
M.
Chandra
and
M. K.
Verma
, “
Dynamics and symmetries of flow reversals in turbulent convection
,”
Phys. Rev. E
83
,
067303
(
2011
).
62.
M.
Chandra
and
M. K.
Verma
, “
Flow reversals in turbulent convection via vortex reconnections
,”
Phys. Rev. Lett.
110
,
114503
(
2013
).
63.
K.
Petschel
,
M.
Wilczek
,
M.
Breuer
,
R.
Friedrich
, and
U.
Hansen
, “
Statistical analysis of global wind dynamics in vigorous Rayleigh-Bénard convection
,”
Phys. Rev. E
84
,
026309
(
2011
).
64.
P. K.
Mishra
,
A. K.
De
,
M. K.
Verma
, and
V.
Eswaran
, “
Dynamics of reorientations and reversals of large-scale flow in Rayleigh–Bénard convection
,”
J. Fluid Mech.
668
,
480
499
(
2011
).
65.
A.
Xu
,
X.
Chen
,
F.
Wang
, and
H.-D.
Xi
, “
Correlation of internal flow structure with heat transfer efficiency in turbulent Rayleigh-Bénard convection
,”
Phys. Fluids
32
,
105112
(
2020
).
66.
K.-Q.
Xia
,
C.
Sun
, and
S.-Q.
Zhou
, “
Particle image velocimetry measurement of the velocity field in turbulent thermal convection
,”
Phys. Rev. E
68
,
066303
(
2003
).
67.
E. P.
van der Poel
,
R. J. A. M.
Stevens
, and
D.
Lohse
, “
Connecting flow structures and heat flux in turbulent Rayleigh-Bénard convection
,”
Phys. Rev. E
84
,
045303
(
2011
).
68.
E. P.
van der Poel
,
R.
Ostilla-Mónico
,
R.
Verzicco
, and
D.
Lohse
, “
Effect of velocity boundary conditions on the heat transfer and flow topology in two-dimensional Rayleigh-Bénard convection
,”
Phys. Rev. E
90
,
013017
(
2014
).
69.
B. A.
Puthenveettil
and
J. H.
Arakeri
, “
Plume structure in high-Rayleigh-number convection
,”
J. Fluid Mech.
542
,
217
249
(
2005
).
70.
D.
Funfschilling
and
G.
Ahlers
, “
Plume motion and large-scale circulation in a cylindrical Rayleigh–Bénard cell
,”
Phys. Rev. Lett.
92
,
194502
(
2004
).
71.
O.
Shishkina
and
C.
Wagner
, “
Analysis of sheet-like thermal plumes in turbulent Rayleigh-Bénard convection
,”
J. Fluid Mech.
599
,
383
404
(
2008
).
72.
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
).
73.
A.
Pandey
and
M. K.
Verma
, “
Scaling of large-scale quantities in Rayleigh-Bénard convection
,”
Phys. Fluids
28
,
095105
(
2016
).
74.
W. V. R.
Malkus
, “
The heat transport and spectrum of thermal turbulence
,”
Proc. R. Soc. London, Ser. A
225
,
196
212
(
1954
).
75.
E. A.
Spiegel
, “
Convection in stars: I. Basic Boussinesq convection
,”
Annu. Rev. Astron. Astrophys.
9
,
323
352
(
1971
).
76.
P.
Orlandi
,
Fluid Flow Phenomena: A Numerical Toolkit
(
Springer Science & Business Media
,
2000
), Vol.
55
.
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