Small-scale motions in turbulent flows play a significant role in various small-scale processes, such as particle relative dispersion and collision, bubble or droplet deformation, and orientation dynamics of non-sphere particles. Recovering the small-scale flows that cannot be resolved in large eddy simulation (LES) is of great importance for such processes sensitive to the small-scale motions in turbulent flows. This study proposes a subgrid-scale model for recovering the small-scale turbulent velocity field based on the artificial neural network (ANN). The governing equations of small-scale turbulent velocity are linearized, and the pressure gradient and the nonlinear convection term are modeled with the aid of the ANN. Direct numerical simulation (DNS) and filtered direct numerical simulation (FDNS) provide the data required for training and validating the ANN. The large-scale velocity and velocity gradient tensor are selected as inputs for the ANN model. The linearized governing equations of small-scale turbulent velocity are numerically solved by coupling the large-scale flow field information. The results indicate that the model established by the ANN can accurately recover the small-scale velocity lost in FDNS due to filtering operation. With the ANN model, the flow fields at different Reynolds numbers agree well with the DNS results regarding velocity field statistics, flow field structures, turbulent energy spectra, and two-point, two-time Lagrangian correlation functions. This study demonstrates that the proposed ANN model can be applied to recovering the small-scale velocity field in the LES of isotropic turbulent flows at different Reynolds numbers.
REFERENCES
1.
M.
Lesieur
and
O.
Metais
, “
New trends in large-eddy simulations of turbulence
,” Annu. Rev. Fluid Mech.
28
, 45
–82
(1996
).2.
C.
Meneveau
and
J.
Katz
, “
Scale-invariance and turbulence models for large-eddy simulation
,” Annu. Rev. Fluid Mech.
32
, 1
–32
(2000
).3.
C. D.
Pierce
and
P.
Moin
, “
Progress-variable approach for large-eddy simulation of non-premixed turbulent combustion
,” J. Fluid Mech.
504
, 73
–97
(2004
).4.
J.
Smagorinsky
, “
General circulation experiments with the primitive equations: I. The basic experiment
,” Mon. Weather Rev.
91
(3
), 99
–164
(1963
).5.
J.
Bardina
,
J.
Ferziger
, and
W.
Reynolds
, “
Improved subgrid-scale models for large-eddy simulation
,” AIAA Paper No. AIAA-80-1357, 1980
.6.
J.
Bardina
, “
Improved turbulence models based on large eddy simulation of homogeneous, incompressible turbulent flows
,” Ph.D. thesis (
Stanford University
,
Stanford
, 1983
).7.
R. A.
Clark
,
J. H.
Ferziger
, and
W. C.
Reynolds
, “
Evaluation of subgrid-scale models using an accurately simulated turbulent flow
,” J. Fluid Mech.
91
, 1
–16
(1979
).8.
Y.
Zang
,
R. L.
Street
, and
J. R.
Koseff
, “
A dynamic mixed subgrid-scale model and its application to turbulent recirculating flows
,” Phys. Fluids
5
(12
), 3186
–3196
(1993
).9.
M.
Germano
,
U.
Piomelli
,
P.
Moin
, and
W. H.
Cabot
, “
A dynamic subgrid-scale eddy viscosity model
,” Phys. Fluids
3
, 1760
–1765
(1991
).10.
M.
Germano
, “
Turbulence: The filtering approach
,” J. Fluid Mech.
238
, 325
–336
(1992
).11.
B.
Vreman
,
B.
Geurts
, and
H.
Kuerten
, “
Large-eddy simulation of the temporal mixing layer using the Clark model
,” Theor. Comput. Fluid Dyn.
8
(4
), 309
–324
(1996
).12.
Y.
Morinishi
and
O. V.
Vasilyev
, “
A recommended modification to the dynamic two-parameter mixed subgrid scale model for large eddy simulation of wall bounded turbulent flow
,” Phys. Fluids
13
(11
), 3400
–3410
(2001
).13.
Z. X.
Yang
,
G. X.
Cui
,
Z. S.
Zhang
, and
C. X.
Xu
, “
A modified nonlinear sub-grid scale model for large eddy simulation with application to rotating turbulent channel flows
,” Phys. Fluids
24
(7
), 075113
(2012
).14.
Z. X.
Yang
,
G. X.
Cui
,
C. X.
Xu
, and
Z. S.
Zhang
, “
Large eddy simulation of rotating turbulent channel flow with a new dynamic global-coefficient nonlinear subgrid stress model
,” J. Turbul.
13
(48
), N48
(2012
).15.
N. S.
Ghaisas
and
S. H.
Frankel
, “
Dynamic gradient models for the sub-grid scale stress tensor and scalar flux vector in large eddy simulation
,” J. Turbul.
17
(1
), 30
–50
(2016
).16.
G. W.
He
,
G. D.
Jin
, and
Y.
Yang
, “
Space-time correlations and dynamic coupling in turbulent flows
,” Annu. Rev. Fluid Mech.
49
, 51
–70
(2017
).17.
G. W.
He
,
R.
Rubinstein
, and
L. P.
Wang
, “
Effects of subgrid-scale modeling on time correlations in large eddy simulation
,” Phys. Fluids
14
(7
), 2186
–2193
(2002
).18.
G. D.
Jin
,
G. W.
He
, and
L. P.
Wang
, “
Large-eddy simulation of turbulent collision of heavy particles in isotropic turbulence
,” Phys. Fluids
22
(5
), 055106
(2010
).19.
P. L.
Johnson
and
C.
Meneveau
, “
Predicting viscous-range velocity gradient dynamics in large-eddy simulations of turbulence
,” J. Fluid Mech.
837
, 80
–114
(2018
).20.
J.
Ravnik
,
C.
Marchioli
, and
A.
Soldati
, “
Application limits of Jeffery's theory for elongated particle torques in turbulence: A DNS assessment
,” Acta Mech.
229
(2
), 827
–839
(2018
).21.
G. A.
Voth
and
A.
Soldati
, “
Anisotropic particles in turbulence
,” Annu. Rev. Fluid Mech.
49
, 249
–276
(2017
).22.
R.
Jayaram
,
Y. C.
Jie
,
L. H.
Zhao
, and
H. I.
Andersson
, “
Preferential orientation of tracer spheroids in evolving Taylor-green vortex flow
,” Phys. Fluids
34
(4
), 043310
(2022
).23.
A. S.
Ghate
and
S. K.
Lele
, “
Gabor mode enrichment in large eddy simulations of turbulent flow
,” J. Fluid Mech.
903
, A13
(2020
).24.
K. R.
Sreenivasan
and
R. A.
Antonia
, “
The phenomenology of small-scale turbulence
,” Annu. Rev. Fluid Mech.
29
, 435
–472
(1997
).25.
M.
Hausmann
,
F.
Evrard
, and
B.
van Wachem
, “
An efficient model for subgrid-scale velocity enrichment for large-eddy simulations of turbulent flows
,” Phys. Fluids
34
(11
), 0127231
(2022
).26.
S.
Ghosal
,
T. S.
Lund
,
P.
Moin
, and
K.
Akselvoll
, “
A dynamic localization model for large-eddy simulation of turbulent flows
,” J. Fluid Mech.
286
, 229
–255
(1995
).27.
Z. D.
Zhou
,
S. Z.
Wang
, and
G. D.
Jin
, “
A structural subgrid-scale model for relative dispersion in large-eddy simulation of isotropic turbulent flows by coupling kinematic simulation with approximate deconvolution method
,” Phys. Fluids
30
(10
), 105110
(2018
).28.
G. D.
Jin
and
G. W.
He
, “
A nonlinear model for the subgrid timescale experienced by heavy particles in large eddy simulation of isotropic turbulence with a stochastic differential equation
,” New J. Phys.
15
, 035011
(2013
).29.
W. S.
Mcculloch
and
W.
Pitts
, “
A logical calculus of the ideas immanent in nervous activity
,” Bull. Math. Biophys.
5
, 115
–133
(1943
).30.
S.
Athmaja
,
M.
Hanumanthappa
, and
V.
Kavitha
, “A survey of machine learning algorithms for big data analytics,” in 2017 International Conference on Innovations in Information, Embedded and Communication Systems (ICIIECS)
(IEEE, 2017).31.
K.
Hornik
,
M.
Stinchcombe
, and
H.
White
, “
Multilayer feedforward networks are universal approximators
,” Neural Networks
2
(5
), 359
–366
(1989
).32.
H.
Kim
,
J.
Kim
,
S.
Won
, and
C.
Lee
, “
Unsupervised deep learning for super-resolution reconstruction of turbulence
,” J. Fluid Mech.
910
, A29
(2021
).33.
R.
Maulik
,
O.
San
,
A.
Rasheed
, and
P.
Vedula
, “
Subgrid modelling for two-dimensional turbulence using neural networks
,” J. Fluid Mech.
858
, 122
–144
(2019
).34.
X. L. D.
Huang
,
X. I. A.
Yang
, and
R. F.
Kunz
, “
Wall-modeled large-eddy simulations of spanwise rotating turbulent channels-comparing a physics-based approach and a data-based approach
,” Phys. Fluids
31
(12
), 125105
(2019
).35.
J.
Ling
and
J.
Templeton
, “
Evaluation of machine learning algorithms for prediction of regions of high Reynolds averaged Navier–Stokes uncertainty
,” Phys. Fluids
27
(8
), 085103
(2015
).36.
K.
Duraisamy
, “
Perspectives on machine learning-augmented Reynolds-averaged and large eddy simulation models of turbulence
,” Phys. Rev. Fluids
6
(5
), 050504
(2021
).37.
F.
Sarghini
,
G.
de Felice
, and
S.
Santini
, “
Neural networks based subgrid scale modeling in large eddy simulations
,” Comput. Fluids
32
(1
), 97
–108
(2003
).38.
S. L.
Brunton
,
B. R.
Noack
, and
P.
Koumoutsakos
, “
Machine learning for fluid mechanics
,” Annu. Rev. Fluid Mech.
52
, 477
–508
(2020
).39.
A.
Beck
and
M.
Kurz
, “
A perspective on machine learning methods in turbulence modeling
,” GAMM-Mitteilungen
44
(1
), e202100002
(2021
).40.
S. S.
Girimaji
, “
Turbulence closure modeling with machine learning approaches: A perspective
,” arXiv:2312.14902 (2023
).41.
R.
Das
and
S. S.
Girimaji
, “
Data-driven model for lagrangian evolution of velocity gradients in incompressible turbulent flows
,” J. Fluid Mech.
984
, A39
(2024
).42.
S.
Yarlanki
,
B.
Rajendran
, and
H.
Hamann
, “
Estimation of turbulence closure coefficients for data centers using machine learning algorithms
,” in 2012 13th IEEE Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems (ITHERM), San Diego, CA
(IEEE, 2012
), pp. 38
–42
.43.
X. I. A.
Yang
,
S.
Zafar
,
J. X.
Wang
, and
H.
Xiao
, “
Predictive large-eddy-simulation wall modeling via physics-informed neural networks
,” Phys. Rev. Fluids
4
(3
), 034602
(2019
).44.
K.
Fukami
,
K.
Fukagata
, and
K.
Taira
, “
Super-resolution reconstruction of turbulent flows with machine learning
,” J. Fluid Mech.
870
, 106
–120
(2019
).45.
K. E.
Otmani
,
G.
Ntoukas
,
O. A.
Marino
, and
E.
Ferrer
, “
Toward a robust detection of viscous and turbulent flow regions using unsupervised machine learning
,” Phys. Fluids
35
(2
), 027112
(2023
).46.
Y.
Xuan
,
H. Q.
Lyu
,
W.
An
,
J. H.
Liu
, and
X. J.
Liu
, “
A data-driven deep learning approach for predicting separation-induced transition of submarines
,” Phys. Fluids
34
(2
), 024101
(2022
).47.
J.
Steiner
,
R. P.
Dwight
, and
A.
Vire
, “
Data-driven RANS closures for wind turbine wakes under neutral conditions
,” Comput. Fluids
233
, 105213
(2022
).48.
H.
Xiao
,
J. L.
Wu
,
S.
Laizet
, and
L.
Duan
, “
Flows over periodic hills of parameterized geometries: A dataset for data-driven turbulence modeling from direct simulations
,” Comput. Fluids
200
, 104431
(2020
).49.
B.
Zhang
, “
Airfoil-based convolutional autoencoder and long short-term memory neural network for predicting coherent structures evolution around an airfoil
,” Comput. Fluids
258
, 105883
(2023
).50.
Z.
Zhang
,
A. B.
Limaye
, and
A.
Khosronejad
, “
Three-dimensional realizations of flood flow in large-scale rivers using the neural fuzzy-based machine-learning algorithms
,” Comput. Fluids
246
, 105611
(2022
).51.
P.
Veras
,
G.
Balarac
,
O.
Metais
,
D.
Georges
,
A.
Bombenger
, and
C.
Segoufin
, “
Reconstruction of numerical inlet boundary conditions using machine learning: Application to the swirling flow inside a conical diffuser
,” Phys. Fluids
33
(8
), 085132
(2021
).52.
Q.
Wu
,
Y. M.
Zhao
,
Y. P.
Shi
, and
S. Y.
Chen
, “
Large-eddy simulation of particle-laden isotropic turbulence using machine-learned subgrid-scale model
,” Phys. Fluids
34
(6
), 065129
(2022
).53.
C.
Drygala
,
B.
Winhart
,
F.
di Mare
, and
H.
Gottschalk
, “
Generative modeling of turbulence
,” Phys. Fluids
34
(3
), 035114
(2022
).54.
H.
Ghraieb
,
J.
Viquerat
,
A.
Larcher
,
P.
Meliga
, and
E.
Hachem
, “
Single-step deep reinforcement learning for open-loop control of laminar and turbulent flows
,” Phys. Rev. Fluids
6
(5
), 053902
(2021
).55.
P.
Veras
,
O.
Metais
,
G.
Balarac
,
D.
Georges
,
A.
Bombenger
, and
C.
Segoufin
, “
Reconstruction of proper numerical inlet boundary conditions for draft tube flow simulations using machine learning
,” Comput. Fluids
254
, 105792
(2023
).56.
P. M.
Milani
,
J.
Ling
, and
J. K.
Eaton
, “
Turbulent scalar flux in inclined jets in crossflow: Counter gradient transport and deep learning modelling
,” J. Fluid Mech.
906
, A27
(2021
).57.
D.
Xu
,
J.
Wang
,
C. P.
Yu
, and
S. Y.
Chen
, “
Artificial-neural-network-based nonlinear algebraic models for large-eddy simulation of compressible wall-bounded turbulence
,” J. Fluid Mech.
960
, A4
(2023
).58.
A.
Eidi
,
N.
Zehtabiyan-Rezaie
,
R.
Ghiassi
,
X.
Yang
, and
M.
Abkar
, “
Data-driven quantification of model-form uncertainty in Reynolds-averaged simulations of wind farms
,” Phys. Fluids
34
(8
), 085135
(2022
).59.
J. P.
Li
,
D. G.
Tang
,
C.
Yi
, and
C.
Yan
, “
Data-augmented turbulence modeling by reconstructing Reynolds stress discrepancies for adverse-pressure-gradient flows
,” Phys. Fluids
34
(4
), 045110
(2022
).60.
R.
Zangeneh
, “
Data-driven model for improving wall-modeled large-eddy simulation of supersonic turbulent flows with separation
,” Phys. Fluids
33
(12
), 126103
(2021
).61.
Z.
Zhang
,
Y. B.
Mao
,
X. R.
Su
, and
X.
Yuan
, “
Inversion learning of turbulent thermal diffusion for film cooling
,” Phys. Fluids
34
(3
), 035118
(2022
).62.
G.
Narasimhan
,
C.
Meneveau
, and
T. A.
Zaki
, “
Large eddy simulation of transitional channel flow using a machine learning classifier to distinguish laminar and turbulent regions
,” Phys. Rev. Fluids
6
(7
), 074608
(2021
).63.
P. S.
Volpiani
,
M.
Meyer
,
L.
Franceschini
,
J.
Dandois
,
F.
Renac
,
E.
Martin
,
O.
Marquet
, and
D.
Sipp
, “
Machine learning-augmented turbulence modeling for RANS simulations of massively separated flows
,” Phys. Rev. Fluids
6
(6
), 064607
(2021
).64.
A.
Lozano-Duran
and
H. J.
Bae
, “
Machine learning building-block-flow wall model for large-eddy simulation
,” J. Fluid Mech.
963
, A35
(2023
).65.
A.
Beck
,
D.
Flad
, and
C. D.
Munz
, “
Deep neural networks for data-driven LES closure models
,” J. Comput. Phys.
398
, 108910
(2019
).66.
M.
Milano
and
P.
Koumoutsakos
, “
Neural net modeling for near wall turbulent flow
,” J. Comput. Phys.
182
(1
), 1
–26
(2002
).67.
Y. H.
Yin
,
H. R.
Li
,
Y. F.
Zhang
, and
H. X.
Chen
, “
Application of machine learning assisted turbulence modeling in flow separation prediction
,” Acta Aerodyn. Sin.
39
(2
), 23
–32
(2021
).68.
R.
Maulik
and
O.
San
, “
A neural network approach for the blind deconvolution of turbulent flows
,” J. Fluid Mech.
831
, 151
–181
(2017
).69.
R.
Maulik
,
O.
San
,
J. D.
Jacob
, and
C.
Crick
, “
Sub-grid scale model classification and blending through deep learning
,” J. Fluid Mech.
870
, 784
–812
(2019
).70.
A.
Abekawa
,
Y.
Minamoto
,
K.
Osawa
,
H.
Shimamoto
, and
M.
Tanahashi
, “
Exploration of robust machine learning strategy for subgrid scale stress modeling
,” Phys. Fluids
35
(1
), 015162
(2023
).71.
J.
Kim
,
H.
Kim
,
J.
Kim
, and
C.
Lee
, “
Deep reinforcement learning for large-eddy simulation modeling in wall-bounded turbulence
,” Phys. Fluids
34
(10
), 105132
(2022
).72.
A.
Subel
,
A.
Chattopadhyay
,
Y. F.
Guan
, and
P.
Hassanzadeh
, “
Data-driven subgrid-scale modeling of forced Burgers turbulence using deep learning with generalization to higher Reynolds numbers via transfer learning
,” Phys. Fluids
33
(3
), 031702
(2021
).73.
Y. P.
Wang
,
Z. L.
Yuan
,
X. N.
Wang
, and
J. C.
Wang
, “
Constant-coefficient spatial gradient models for the sub-grid scale closure in large-eddy simulation of turbulence
,” Phys. Fluids
34
(9
), 095108
(2022
).74.
Y. P.
Wang
,
Z. L.
Yuan
,
C. Y.
Xie
, and
J. C.
Wang
, “
A dynamic spatial gradient model for the subgrid closure in large-eddy simulation of turbulence
,” Phys. Fluids
33
(7
), 075119
(2021
).75.
Z.
Wang
,
K.
Luo
,
D.
Li
,
J. H.
Tan
, and
J. R.
Fan
, “
Investigations of data-driven closure for subgrid-scale stress in large-eddy simulation
,” Phys. Fluids
30
(12
), 125101
(2018
).76.
P. A.
Srinivasan
,
L.
Guastoni
,
H.
Azizpour
,
P.
Schlatter
, and
R.
Vinuesa
, “
Predictions of turbulent shear flows using deep neural networks
,” Phys. Rev. Fluids
4
(5
), 054603
(2019
).77.
M.
Gamahara
and
Y.
Hattori
, “
Searching for turbulence models by artificial neural network
,” Phys. Rev. Fluids
2
, 054604
(2017
).78.
M.
Kim
,
J.
Park
, and
H.
Choi
, “
Large eddy simulation of flow over a circular cylinder with a neural-network-based subgrid-scale model
,” J. Fluid Mech.
984
, A6
(2024
).79.
J.
Park
and
H.
Choi
, “
Toward neural-network-based large eddy simulation: Application to turbulent channel flow
,” J. Fluid Mech.
914
, A16
(2021
).80.
C. Y.
Xie
,
J. C.
Wang
,
H.
Li
,
M. P.
Wan
, and
S. Y.
Chen
, “
Spatial artificial neural network model for subgrid-scale stress and heat flux of compressible turbulence
,” Theor. Appl. Mech. Lett.
10
(1
), 27
–32
(2020
).81.
Z. D.
Zhou
,
G. W.
He
,
S. Z.
Wang
, and
G. D.
Jin
, “
Subgrid-scale model for large-eddy simulation of isotropic turbulent flows using an artificial neural network
,” Comput. Fluids
195
, 104319
(2019
).82.
C. Y.
Xie
,
J. C.
Yuan
,
Z. L.
Wang
,
H.
Li
,
M. P.
Wan
, and
S. Y.
Chen
, “
Artificial neural network-based subgrid-scale models for large-eddy simulation of turbulence
,” Chin. J. Theor. Appl. Mech.
53
(1
), 1
–16
(2021
).83.
M.
Kang
,
Y.
Jeon
, and
D.
You
, “
Neural-network-based mixed subgrid-scale model for turbulent flow
,” J. Fluid Mech.
962
, A38
(2023
).84.
Z. S.
Zhang
,
G. X.
Cui
, and
C. X.
Xu
, Theory and Modeling of Turbulence
, 2nd ed. (
Tsinghua University Press
,
Beijing
, 2017
).85.
V.
Eswaran
and
S. B.
Pope
, “
An examination of forcing in direct numerical simulations of turbulence
,” Comput. Fluids
16
(3
), 257
–278
(1988
).86.
L. P.
Wang
and
M. R.
Maxey
, “
Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence
,” J. Fluid Mech.
256
, 27
–68
(1993
).87.
Y.
Yang
,
G. W.
He
, and
L. P.
Wang
, “
Effects of subgrid-scale modeling on lagrangian statistics in large-eddy simulation
,” J. Turbul.
9
(8
), N8
(2008
).88.
Y.
Xiong
,
J.
Li
,
Z. h.
Liu
, and
C. G.
Zheng
, “
The influence of sub-grid scale motions on particle collision in homogeneous isotropic turbulence
,” Acta Mech. Sin.
34
(1
), 22
–36
(2018
).89.
P.
Fede
and
O.
Simonin
, “
Numerical study of the subgrid fluid turbulence effects on the statistics of heavy colliding particles
,” Phys. Fluids
18
(4
), 045103
(2006
).90.
B.
Liu
,
J.
Tang
,
H.
Huang
, and
X. Y.
Lu
, “
Deep learning methods for super-resolution reconstruction of turbulent flows
,” Phys. Fluids
32
(2
), 025105
(2020
).91.
J.
Jeong
and
F.
Hussain
, “
On the identification of a vortex
,” J. Fluid Mech.
285
, 69
–94
(1995
).© 2024 Author(s). Published under an exclusive license by AIP Publishing.
2024
Author(s)
You do not currently have access to this content.
