For understanding many real-world problems involving rarefied hypersonic, micro-, and nanoscale gas flows, the primary method may be the direct simulation Monte Carlo (DSMC). However, its computational cost is prohibitive in comparison with the Navier–Stokes–Fourier (NSF) solvers, eclipsing the advantages it provides, especially for situations where flow is in the near continuum regime or three-dimensional applications. This study presents an alternate computational method that bypasses this issue by taking advantage of data-driven modeling and nonlinear coupled constitutive relations. Instead of using numerical solutions of higher-order constitutive relations in conventional partial differential equation-based methods, we build compact constitutive relations in advance by applying deep neural network algorithms to available DSMC solution data and later combine them with the conventional finite volume method for the physical laws of conservation. The computational accuracy and cost of the methodology thus developed were tested on the shock wave inner structure problem, where high thermal non-equilibrium occurs due to rapid compression, for a range of Mach numbers from 2 to 10. The simulation results obtained with the computing time comparable to that of the NSF solver showed almost perfect agreement between the neural network-based combined finite volume method and DSMC and original DSMC solutions. We also present a topology of DSMC constitutive relations that allows us to study how the DSMC topology deviates from the NSF topology. Finally, several challenging issues that must be overcome to become a robust method for solving practical problems were discussed.
REFERENCES
1.
R. S.
Myong
, “
Thermodynamically consistent hydrodynamic computational models for high-Knudsen-number gas flows
,” Phys. Fluids
11
, 2788
–2802
(1999
).2.
R. S.
Myong
, “
On the high Mach number shock structure singularity caused by overreach of Maxwellian molecules
,” Phys. Fluids
26
, 056102
(2014
).3.
N. T. P.
Le
,
H.
Xiao
, and
R. S.
Myong
, “
A triangular discontinuous Galerkin method for non-Newtonian implicit constitutive models of rarefied and microscale gases
,” J. Comput. Phys.
273
, 160
–184
(2014
).4.
A.
Rana
,
R.
Ravichandran
,
J. H.
Park
, and
R. S.
Myong
, “
Microscopic molecular dynamics characterization of the second-order non-Navier–Fourier constitutive laws in the Poiseuille gas flow
,” Phys. Fluids
28
, 082003
(2016
).5.
S.
Singh
,
A.
Karchani
,
T.
Chourushi
, and
R. S.
Myong
, “
A three-dimensional modal discontinuous Galerkin method for the second-order Boltzmann-Curtiss-based constitutive model of rarefied and microscale gas flows
,” J. Comput. Phys.
457
, 111052
(2022
).6.
O.
Ejtehadi
,
R. S.
Myong
,
I.
Sohn
, and
B. J.
Kim
, “
Full continuum approach for simulating plume-surface interaction in planetary landings
,” Phys. Fluids
35
, 043331
(2023
).7.
T. K.
Mankodi
,
O.
Ejtehadi
,
T.
Chourushi
,
A.
Rahimi
, and
R. S.
Myong
, “
nccrFOAM suite: Nonlinear coupled constitutive relation solver in the OpenFOAM framework for rarefied and microscale gas flows with vibrational non-equilibrium
,” Comput. Phys. Commun.
296
, 109024
(2024
).8.
Z.
Jiang
,
W.
Zhao
,
Z.
Yuan
,
W.
Chen
, and
R. S.
Myong
, “
Computation of hypersonic flows over flying configurations using a nonlinear constitutive model
,” AIAA J.
57
, 5252
–5268
(2019
).9.
T. K.
Mankodi
and
R. S.
Myong
, “
Quasi-classical trajectory-based non-equilibrium chemical reaction models for hypersonic air flows
,” Phys. Fluids
31
, 106102
(2019
).10.
T. K.
Mankodi
and
R. S.
Myong
, “
Erratum: ‘Quasi-classical trajectory-based non-equilibrium chemical reaction models for hypersonic air flows’ [Phys. Fluids 31, 106102 (2019)]
,” Phys. Fluids
32
, 019901
(2020
).11.
L. V.
Ballestra
and
R.
Sacco
, “
Numerical problems in semiconductor simulation using the hydrodynamic model: A second-order finite difference scheme
,” J. Comput. Phys.
195
, 320
–340
(2004
).12.
S. K.
Blau
, “
Conduction electrons flow like honey
,” Phys. Today
70
(11
), 22
–22
(2017
).13.
N.
Cagney
and
S.
Balabani
, “
Taylor-Couette flow of shear-thinning fluids
,” Phys. Fluids
31
, 053102
(2019
).14.
R.
Evans
,
D.
Frenkel
, and
M.
Dijkstra
, “
From simple liquids to colloids and soft matter
,” Phys. Today
72
(2
), 38
–39
(2019
).15.
G. A.
Bird
, “
Approach to translational equilibrium in a rigid sphere gas
,” Phys. Fluids
6
, 1518
–1519
(1963
).16.
G. A.
Bird
, Molecular Gas Dynamics and the Direct Simulation of Gas Flows
(
Clarendon Press
,
Oxford
, 1994
).17.
G. A.
Bird
,
M. A.
Gallis
,
J. R.
Torczynski
, and
D. J.
Rader
, “
Accuracy and efficiency of the sophisticated direct simulation Monte Carlo algorithm for simulating noncontinuum gas flows
,” Phys. Fluids
21
, 017103
(2009
).18.
R. S.
Myong
,
A.
Karchani
, and
O.
Ejtehadi
, “
A review and perspective on a convergence analysis of the direct simulation Monte Carlo and solution verification
,” Phys. Fluids
31
, 066101
(2019
).19.
A.
Karchani
and
R. S.
Myong
, “
Convergence analysis of the direct simulation Monte Carlo based on the physical laws of conservation
,” Comput. Fluids
115
, 98
–114
(2015
).20.
I. D.
Boyd
,
G.
Chen
, and
G. V.
Candler
, “
Predicting failure of the continuum fluid equations in transitional hypersonic flows
,” Phys. Fluids
7
, 210
–219
(1995
).21.
T. E.
Schwartzentruber
and
I. D.
Boyd
, “
Progress and future prospects for particle-based simulation of hypersonic flow
,” Prog. Aerosp. Sci.
72
, 66
–79
(2015
).22.
E.
Jun
and
I. D.
Boyd
, “
Assessment of the LD-DSMC hybrid method for hypersonic rarefied flow
,” Comput. Fluids
166
, 123
–138
(2018
).23.
S.
Mallikarjun
,
V.
Casseau
,
W. G.
Habashi
,
S.
Gao
, and
A.
Karchani
, “
Hybrid Navier–Stokes–direct simulation Monte Carlo automatic mesh optimization for hypersonics
,” J. Thermophys. Heat Transfer
37
, 779
–806
(2023
).24.
P. L.
Bhatnagar
,
E. P.
Gross
, and
M.
Krook
, “
A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems
,” Phys. Rev.
94
, 511
–525
(1954
).25.
E. M.
Shakhov
, “
Generalization of the Krook kinetic relaxation equation
,” Fluid Dyn.
3
(5
), 95
–96
(1972
).26.
L. H.
Holway
, Jr., “
New statistical models for kinetic theory: Methods of construction
,” Phys. Fluids
9
, 1658
–1673
(1966
).27.
D.
Burnett
, “
The distribution of molecular velocities and the mean motion in a non-uniform gas
,” Proc. London Math. Soc.
s2-40
, 382
–435
(1936
).28.
C. J.
Greenshields
and
J. M.
Reese
, “
The structure of shock waves as a test of Brenner's modifications to the Navier–Stokes equations
,” J. Fluid Mech.
580
, 407
–429
(2007
).29.
A.
Agrawal
,
A.
Gavasane
, and
R. S.
Jadhav
, “
Improved theory for shock waves using the OBurnett equations
,” J. Fluid Mech.
929
, A37
(2021
).30.
H.
Grad
, “
On the kinetic theory of rarefied gases
,” Commun. Pure Appl. Math.
2
, 331
–407
(1949
).31.
H.
Struchtrup
and
M.
Torrilhon
, “
Regularization of Grad's 13 moment equations: Derivation and linear analysis
,” Phys. Fluids
15
, 2668
–2680
(2003
).32.
D. R.
Emerson
and
X.-J.
Gu
, “
A high-order moment approach for capturing non-equilibrium phenomena in the transition regime
,” J. Fluid Mech.
636
, 177
–216
(2009
).33.
34.
R. S.
Myong
, “
A computational method for Eu's generalized hydrodynamic equations of rarefied and microscale gasdynamics
,” J. Comput. Phys.
168
, 47
–72
(2001
).35.
R. S.
Myong
, “
A generalized hydrodynamic computational model for rarefied and microscale diatomic gas flows
,” J. Comput. Phys.
195
, 655
–676
(2004
).36.
J.
Sirignano
and
J. F.
MacArt
, “
Deep learning closure models for large-eddy simulation of flows around bluff bodies
,” J. Fluid Mech.
966
, A26
(2023
).37.
K.
Duraisamy
,
G.
Iaccarino
, and
H.
Xiao
, “
Turbulence modeling in the age of data
,” Annu. Rev. Fluid Mech.
51
, 357
–377
(2019
).38.
H.
Xiao
,
J. L.
Wu
,
J. X.
Wang
,
R.
Sun
, and
C. J.
Roy
, “
Quantifying and reducing model-form uncertainties in Reynolds-averaged Navier–Stokes simulations: A data-driven, physics-informed Bayesian approach
,” J. Comput. Phys.
324
, 115
–136
(2016
).39.
A.
Kurzawski
,
J.
Ling
, and
J.
Templeton
, “
Reynolds averaged turbulence modelling using deep neural networks with embedded invariance
,” J. Fluid Mech.
807
, 155
–166
(2016
).40.
J.
Zhang
and
W.
Ma
, “
Data-driven discovery of governing equations for fluid dynamics based on molecular simulation
,” J. Fluid Mech.
892
, A5
(2020
).41.
S. H.
Rudy
,
S. L.
Brunton
,
J. L.
Proctor
, and
J. N.
Kutz
, “
Data-driven discovery of partial differential equations
,” Sci. Adv.
3
, e1602614
(2017
).42.
H.
Xing
,
J.
Zhang
,
W.
Ma
, and
D.
Wen
, “
Using gene expression programming to discover macroscopic governing equations hidden in the data of molecular simulations
,” Phys. Fluids
34
, 057109
(2022
).43.
S.
Yao
,
W.
Zhao
,
C.
Wu
, and
W.
Chen
, “
Nonlinear constitutive calculation method of rarefied flow based on deep convolution neural networks
,” Phys. Fluids
35
, 096103
(2023
).44.
W.
Zhao
,
L.
Jiang
,
S.
Yao
, and
W.
Chen
, “
Data-driven nonlinear constitutive relations for rarefied flow computations
,” Adv. Aerodyn.
3
(1
), 19
(2021
).45.
A. S.
Nair
,
J.
Sirignano
,
M.
Panesi
, and
J. F.
MacArt
, “
Deep learning closure of the Navier–Stokes equations for transition-continuum flows
,” AIAA J.
61
, 5484
–5497
(2023
).46.
J. D.
Anderson
, Modern Compressible Flow with Historical Perspective
(
McGraw-Hill
,
New York
, 1982
).47.
M.
Morduchow
and
P. A.
Libby
, “
On a complete solution of the one-dimensional flow equations of a viscous, heat-conducting, compressible gas
,” J. Aeronaut. Sci.
16
, 674
–684
(1949
).48.
H.
Grad
, “
The profile of a steady plane shock wave
,” Commun. Pure Appl. Math.
5
, 257
–300
(1952
).49.
H. M.
Mott-Smith
, “
The solution of the Boltzmann equation for a shock wave
,” Phys. Rev.
82
, 885
–892
(1951
).50.
M.
Al-Ghoul
and
B. C.
Eu
, “
Generalized hydrodynamics and shock waves
,” Phys. Rev. E
56
, 2981
–2992
(1997
).51.
R. S.
Myong
, “
Analytical solutions of shock structure thickness and asymmetry in Navier–Stokes/Fourier framework
,” AIAA J.
52
, 1075
–1081
(2014
).52.
B.
van Leer
, “
Towards the ultimate conservative difference scheme
,” J. Comput. Phys.
135
, 229
–248
(1997
).53.
E. F.
Toro
,
M.
Spruce
, and
W.
Speares
, “
Restoration of the contact surface in the HLL-Riemann solver
,” Shock Waves
4
, 25
–34
(1994
).54.
H. K.
Moffatt
,
G. M.
Zaslavsky
,
P.
Comte
, and
M.
Tabor
, Topological Aspects of the Dynamics of Fluids and Plasmas
(
Springer Netherlands
,
Dordrecht
, 2013
).55.
S.
Singh
,
A.
Karchani
,
K.
Sharma
, and
R. S.
Myong
, “
Topology of the second-order constitutive model based on the Boltzmann–Curtiss kinetic equation for diatomic and polyatomic gases
,” Phys. Fluids
32
, 026104
(2020
).56.
T. K.
Mankodi
and
R. S.
Myong
, “
Boltzmann-based second-order constitutive models of diatomic and polyatomic gases including the vibrational mode
,” Phys. Fluids
32
, 126109
(2020
).© 2024 Author(s). Published under an exclusive license by AIP Publishing.
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