This article continues our cycle devoted to comprehensive investigation of the diatomic molecule collision process. In this paper, we focus particularly on the in-depth study of the rotational–translational (R–T) energy exchange process and Borgnakke–Larsen (BL) energy exchange model used in the direct simulation Monte Carlo method. The present study, which was performed on several levels of description (molecular, microscopic, and macroscopic), is based mainly on the highly detailed dataset (around 1011 configurations) of binary N2–N2 collisions, obtained via the classical trajectory calculation (CTC) method. This dataset, along with the explicit mathematical representation of the Borgnakke–Larsen model derived in the present paper, allowed us to obtain new results regarding the R–T energy exchange process: (1) we present an ab initio method to derive physically accurate expressions for inelastic collision probability pr in the BL model directly from CTC data; (2) we present a new two-parametric model for pr and compared it to the previously known models, including the recent nonequilibrium-direction-dependent model of Zhang et al. [“Nonequilibrium-direction-dependent rotational energy model for use in continuum and stochastic molecular simulation,” AIAA J. 52(3), 604 (2014)]; (3) it showed that apart from the well-known dependence of the rotational relaxation rate on “direction to equilibrium” (ratio between translational and rotational temperatures), on molecular scale, rotationally over-excited molecule pairs demonstrate almost zero energy transfer to the translational energy mode (even in the case of very significant discrepancies between translational and rotational energies); (4) it was also shown that the Borgnakke–Larsen approach itself may require reassessment since it fails to give a proper description of distribution of post-collision energies. Throughout this paper, we also tried to put together and analyze the existing works studying the rotational relaxation process and estimating the rotational collision number Zrot by performing reviews and assessment of (1) numerical approaches to simulate non-equilibrium problems, (2) models for inelastic collision probabilities pr, (3) approaches to estimate Zrot, and (4) intermolecular potentials used for molecular dynamics and CTC simulations. The corresponding conclusions are given in this paper.

1.
V.
Kosyanchuk
and
A.
Yakunchikov
, “
An atomic-level study of the N2-N2 collision process at temperatures up to 2000 K
,”
Phys. Fluids
32
,
056109
(
2020
).
2.
A.
Yakunchikov
,
V.
Kosyanchuk
,
A.
Kroupnov
,
M.
Pogosbekian
,
I.
Bryukhanov
, and
A.
Iuldasheva
, “
Potential energy surface of interaction of two diatomic molecules for air flows simulation at intermediate temperatures
,”
Chem. Phys.
536
,
110850
(
2020
).
3.
G. M.
Kremer
,
O. V.
Kunova
,
E. V.
Kustova
, and
G. P.
Oblapenko
, “
The influence of vibrational state-resolved transport coefficients on the wave propagation in diatomic gases
,”
Physica A
490
,
92
113
(
2018
).
4.
V. A.
Istomin
and
E. V.
Kustova
, “
State-specific transport properties of partially ionized flows of electronically excited atomic gases
,”
Chem. Phys.
485
,
125
139
(
2017
).
5.
V. A.
Istomin
and
E. V.
Kustova
, “
Transport coefficients and heat fluxes in non-equilibrium high-temperature flows with electronic excitation
,”
Phys. Plasmas
24
,
022109
(
2017
).
6.
N. S.
Parsons
,
T.
Zhu
,
D. A.
Levin
, and
A. C.
Van Duin
, “
Development of dsmc chemistry models for nitrogen collisions using accurate theoretical calculations
,” in
52nd Aerospace Sciences Meeting
(
American Institute of Aeronautics and Astronautics
,
2014
), p.
1213
.
7.
N.
Parsons
,
D. A.
Levin
,
A. C. T.
van Duin
, and
T.
Zhu
, “
Modeling of molecular nitrogen collisions and dissociation processes for direct simulation Monte Carlo
,”
J. Chem. Phys.
141
,
234307
(
2014
).
8.
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
).
9.
T. E.
Schwartzentruber
,
M. S.
Grover
, and
P.
Valentini
, “
Direct molecular simulation of nonequilibrium dilute gases
,”
J. Thermophys. Heat Transfer
32
,
892
903
(
2018
).
10.
I.
Borges Sebastião
and
A.
Alexeenko
, “
Consistent post-reaction vibrational energy redistribution in DSMC simulations using TCE model
,”
Phys. Fluids
28
,
107103
(
2016
).
11.
P.
Valentini
,
T. E.
Schwartzentruber
,
J. D.
Bender
, and
G. V.
Candler
, “
Direct molecular simulation of high-temperature nitrogen dissociation due to both N-N2 and N2-N2 collisions
,” in
45th AIAA Thermophysics Conference
(
American Institute of Aeronautics and Astronautics
,
2015
), p.
3254
.
12.
P.
Valentini
,
T. E.
Schwartzentruber
,
J. D.
Bender
,
I.
Nompelis
, and
G. V.
Candler
, “
Direct molecular simulation of nitrogen dissociation based on an ab initio potential energy surface
,”
Phys. Fluids
27
,
086102
(
2015
).
13.
I. V.
Adamovich
, “
Three-dimensional analytic probabilities of coupled vibrational-rotational-translational energy transfer for DSMC modeling of nonequilibrium flows
,”
Phys. Fluids
26
,
046102
(
2014
).
14.
D. A.
Andrienko
and
I. D.
Boyd
, “
High fidelity modeling of thermal relaxation and dissociation of oxygen
,”
Phys. Fluids
27
,
116101
(
2015
).
15.
F.
Esposito
,
E.
Garcia
, and
A.
Laganà
, “
Comparisons and scaling rules between N+N2 and N2+N2 collision induced dissociation cross sections from atomistic studies
,”
Plasma Sources Sci. Technol.
26
,
045005
(
2017
).
16.
J. G.
Kim
and
I. D.
Boyd
, “
Monte Carlo simulation of nitrogen dissociation based on state-resolved cross sections
,”
Phys. Fluids
26
,
012006
(
2014
).
17.
M.
Kulakhmetov
,
M.
Gallis
, and
A.
Alexeenko
, “
Ab initio-informed maximum entropy modeling of rovibrational relaxation and state-specific dissociation with application to the O2 + O system
,”
J. Chem. Phys.
144
,
174302
(
2016
).
18.
Z.
Li
,
N.
Parsons
, and
D. A.
Levin
, “
A study of internal energy relaxation in shocks using molecular dynamics based models
,”
J. Chem. Phys.
143
,
144501
(
2015
).
19.
T. K.
Mankodi
,
U. V.
Bhandarkar
, and
B. P.
Puranik
, “
An ab initio chemical reaction model for the direct simulation Monte Carlo study of non-equilibrium nitrogen flows
,”
J. Chem. Phys.
147
,
084305
(
2017
).
20.
A.
Munafò
,
Y.
Liu
, and
M.
Panesi
, “
Modeling of dissociation and energy transfer in shock-heated nitrogen flows
,”
Phys. Fluids
27
,
127101
(
2015
).
21.
P.
Valentini
,
C.
Zhang
, and
T. E.
Schwartzentruber
, “
Molecular dynamics simulation of rotational relaxation in nitrogen: Implications for rotational collision number models
,”
Phys. Fluids
24
,
106101
(
2012
).
22.
C.
Zhang
,
P.
Valentini
, and
T. E.
Schwartzentruber
, “
Nonequilibrium-direction-dependent rotational energy model for use in continuum and stochastic molecular simulation
,”
AIAA J.
52
,
604
617
(
2014
).
23.
Z.
Chavis
and
R. G.
Wilmoth
, “
Plume modeling and application to Mars 2001 Odyssey aerobraking
,”
J. Spacecr. Rockets
42
,
450
456
(
2005
).
24.
J.
Zhong
,
T.
Ozawa
, and
D. A.
Levin
, “
Modeling of stardust reentry ablation flows in the near-continuum flight regime
,”
AIAA J.
46
,
2568
2581
(
2008
).
25.
I. D.
Boyd
,
K. A.
Trumble
, and
M. J.
Wright
, “
Modeling of stardust entry at high altitude, Part 1: Flowfield analysis
,”
J. Spacecr. Rockets
47
,
708
717
(
2010
).
26.
I. D.
Boyd
and
P.
Jenniskens
, “
Modeling of stardust entry at high altitude, Part 2: Radiation analysis
,”
J. Spacecr. Rockets
47
,
901
909
(
2010
).
27.
P. W.
Erdman
,
E. C.
Zipf
,
P.
Espy
,
C. L.
Howlett
,
D. A.
Levin
,
R. J.
Collins
, and
G. V.
Candler
, “
Measurements of ultraviolet radiation from a 5-km/s bow shock
,”
J. Thermophys. Heat Transfer
8
,
441
446
(
1994
).
28.
A. A.
Shevyrin
,
Y. A.
Bondar
,
S. T.
Kalashnikov
,
V. I.
Khlybov
 et al., “
Direct simulation of rarefied high-enthalpy flow around the RAM C-II capsule
,”
High Temp.
54
,
383
389
(
2016
).
29.
J. N.
Moss
and
G. A.
Bird
, “
Direct simulation Monte Carlo simulations of hypersonic flows with shock interactions
,”
AIAA J.
43
,
2565
2573
(
2005
).
30.
C. H.
Kruger
and
W.
Vincenti
,
Introduction to Physical Gas Dynamics
(
John Wlley & Sons
,
1965
).
31.
R. C.
Millikan
and
D. R.
White
, “
Systematics of vibrational relaxation
,”
J. Chem. Phys.
39
,
3209
3213
(
1963
).
32.
P.
Norman
,
P.
Valentini
, and
T.
Schwartzentruber
, “
GPU-accelerated classical trajectory calculation direct simulation Monte Carlo applied to shock waves
,”
J. Comput. Phys.
247
,
153
167
(
2013
).
33.
M.
Panesi
,
R. L.
Jaffe
,
D. W.
Schwenke
, and
T. E.
Magin
, “
Rovibrational internal energy transfer and dissociation of N2(1Σg+)−N(4Su) system in hypersonic flows
,”
J. Chem. Phys.
138
,
044312
(
2013
).
34.
M.
Panesi
,
A.
Munafò
,
T.
Magin
, and
R.
Jaffe
, “
Nonequilibrium shock-heated nitrogen flows using a rovibrational state-to-state method
,”
Phys. Rev. E
90
,
013009
(
2014
).
35.
P.
Skovorodko
,
A.
Ramos
,
G.
Tejeda
,
J.
Fernández
, and
S.
Montero
, “
Experimental and numerical study of supersonic jets of N2, H2, and N2 + H2 mixtures
,”
AIP Conf. Pro.
1501
,
1228
1235
(
2012
).
36.
J.
Lengrand
,
V.
Prikhodko
,
P.
Skovorodko
,
I.
Yarygin
, and
V.
Yarygin
, “
Outflow of gas from nozzle with screen into vacuum
,”
AIP Conf. Proc.
1084
,
1158
1163
(
2008
).
37.
C.
Xie
, “
Characteristics of micronozzle gas flows
,”
Phys. Fluids
19
,
037102
(
2007
).
38.
M.
Darbandi
and
E.
Roohi
, “
Study of subsonic-supersonic gas flow through micro/nanoscale nozzles using unstructured DSMC solver
,”
Microfluid. Nanofluid.
10
,
321
335
(
2011
).
39.
F.
La Torre
,
S.
Kenjereš
,
J.-L.
Moerel
, and
C. R.
Kleijn
, “
Hybrid simulations of rarefied supersonic gas flows in micro-nozzles
,”
Comput. Fluids
49
,
312
322
(
2011
).
40.
M.
Vargas
,
S.
Naris
,
D.
Valougeorgis
,
S.
Pantazis
, and
K.
Jousten
, “
Time-dependent rarefied gas flow of single gases and binary gas mixtures into vacuum
,”
Vacuum
109
,
385
396
(
2014
).
41.
S.
Montero
, “
Molecular description of steady supersonic free jets
,”
Phys. Fluids
29
,
096101
(
2017
).
42.
M.
Sabouri
and
M.
Darbandi
, “
Numerical study of species separation in rarefied gas mixture flow through micronozzles using DSMC
,”
Phys. Fluids
31
,
042004
(
2019
).
43.
N. Y.
Bykov
and
V. V.
Zakharov
, “
Binary gas mixture outflow into vacuum through an orifice
,”
Phys. Fluids
32
,
067109
(
2020
).
44.
A. A.
Alexeenko
,
D. A.
Fedosov
,
S. F.
Gimelshein
,
D. A.
Levin
, and
R. J.
Collins
, “
Transient heat transfer and gas flow in a MEMS-based thruster
,”
J. Microelectromech. Syst.
15
,
181
194
(
2006
).
45.
H.
Laribou
,
C.
Fressengeas
,
D.
Entemeyer
,
V.
Jeanclaude
,
R.
Pesci
, and
A.
Tazibt
, “
Effects of the impact of a low temperature nitrogen jet on metallic surfaces
,”
Proc. R. Soc. A
468
,
3601
3619
(
2012
).
46.
B. M.
Smirnov
, “
Metal nanostructures: From clusters to nanocatalysis and sensors
,”
Phys.-Usp.
60
,
1236
(
2017
).
47.
Y. K.
Mishra
,
N. A.
Murugan
,
J.
Kotakoski
, and
J.
Adam
, “
Progress in electronics and photonics with nanomaterials
,”
Vacuum
146
,
304
307
(
2017
).
48.
B.
Kim
,
R. N.
Candler
,
M. A.
Hopcroft
,
M.
Agarwal
,
W.-T.
Park
, and
T. W.
Kenny
, “
Frequency stability of wafer-scale film encapsulated silicon based MEMS resonators
,”
Sens. Actuators, A
136
,
125
131
(
2007
).
49.
M. H.
Ghayesh
,
H.
Farokhi
, and
M.
Amabili
, “
Nonlinear behaviour of electrically actuated MEMS resonators
,”
Int. J. Eng. Sci.
71
,
137
155
(
2013
).
50.
H.
Bhugra
and
G.
Piazza
, “Piezoelectric MEMS resonators” (
Springer
,
2017
).
51.
M. M. J.
Opgenoord
and
P. C.
Caplan
, “
Aerodynamic design of the hyperloop concept
,”
AIAA J.
56
,
4261
4270
(
2018
).
52.
J.
Braun
,
J.
Sousa
, and
C.
Pekardan
, “
Aerodynamic design and analysis of the hyperloop
,”
AIAA J.
55
,
4053
4060
(
2017
).
53.
Y.
Ben-Ami
and
A.
Manela
, “
Nonlinear thermal effects in unsteady shear flows of a rarefied gas
,”
Phys. Rev. E
98
,
033121
(
2018
).
54.
J.-S.
Oh
,
T.
Kang
,
S.
Ham
,
K.-S.
Lee
,
Y.-J.
Jang
,
H.-S.
Ryou
, and
J.
Ryu
, “
Numerical analysis of aerodynamic characteristics of hyperloop system
,”
Energies
12
,
518
(
2019
).
55.
P.
Zhou
,
J.
Zhang
,
T.
Li
, and
W.
Zhang
, “
Numerical study on wave phenomena produced by the super high-speed evacuated tube maglev train
,”
J. Wind Eng. Ind. Aerodyn.
190
,
61
70
(
2019
).
56.
R.
Hruschka
and
D.
Klatt
, “
In-pipe aerodynamic characteristics of a projectile in comparison with free flight for transonic Mach numbers
,”
Shock Waves
29
,
297
306
(
2019
).
57.
K.
van Goeverden
,
D.
Milakis
,
M.
Janic
, and
R.
Konings
, “
Analysis and modelling of performances of the HL (hyperloop) transport system
,”
Eur. Transp. Res. Rev.
10
,
41
(
2018
).
58.
Hyperloop webpage, https://hyperloop-one.com/; accessed 25 November 2019.
59.
T. G.
Elizarova
,
A. A.
Khokhlov
, and
S.
Montero
, “
Numerical simulation of shock wave structure in nitrogen
,”
Phys. Fluids
19
,
068102
(
2007
).
60.
A. V.
Chikitkin
,
B. V.
Rogov
,
G. A.
Tirsky
, and
S. V.
Utyuzhnikov
, “
Effect of bulk viscosity in supersonic flow past spacecraft
,”
Appl. Numer. Math.
93
,
47
60
(
2015
).
61.
D.
Bruno
and
V.
Giovangigli
, “
Relaxation of internal temperature and volume viscosity
,”
Phys. Fluids
23
,
093104
(
2011
).
62.
B.
Rahimi
and
H.
Struchtrup
, “
Capturing non-equilibrium phenomena in rarefied polyatomic gases: A high-order macroscopic model
,”
Phys. Fluids
26
,
052001
(
2014
).
63.
B.
Rahimi
and
H.
Struchtrup
, “
Refined Navier-Stokes-Fourier equations for rarefied polyatomic gases
,” in
International Conference on Nanochannels, Microchannels, and Minichannels
(
American Society of Mechanical Engineers (ASME)
,
2014
), Vol. 46278, p.
V001T01A001
.
64.
E. V.
Kustova
and
G. P.
Oblapenko
, “
Reaction and internal energy relaxation rates in viscous thermochemically non-equilibrium gas flows
,”
Phys. Fluids
27
,
016102
(
2015
).
65.
E.
Kustova
,
E.
Nagnibeda
,
G.
Oblapenko
,
A.
Savelev
, and
I.
Sharafutdinov
, “
Advanced models for vibrational-chemical coupling in multi-temperature flows
,”
Chem. Phys.
464
,
1
13
(
2016
).
66.
B.
Rahimi
and
H.
Struchtrup
, “
Macroscopic and kinetic modelling of rarefied polyatomic gases
,”
J. Fluid Mech.
806
,
437
505
(
2016
).
67.
M.
Bisi
and
M. J.
Cáceres
, “
A BGK relaxation model for polyatomic gas mixtures
,”
Commun. Math. Sci.
14
,
297
325
(
2016
).
68.
G.
Cao
,
H.
Liu
, and
K.
Xu
, “
Physical modeling and numerical studies of three-dimensional non-equilibrium multi-temperature flows
,”
Phys. Fluids
30
,
126104
(
2018
).
69.
S.
Singh
,
A.
Karchani
,
K.
Sharma
, and
R.
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
).
70.
Z.
Yuan
,
Z.
Jiang
,
W.
Zhao
, and
W.
Chen
, “
Multiple temperature model of nonlinear coupled constitutive relations for hypersonic diatomic gas flows
,”
AIP Adv.
10
,
055023
(
2020
).
71.
K.
Aoki
,
M.
Bisi
,
M.
Groppi
, and
S.
Kosuge
, “
Two-temperature Navier-Stokes equations for a polyatomic gas derived from kinetic theory
,”
Phys. Rev. E
102
,
023104
(
2020
).
72.
J. O.
Hirschfelder
,
C. F.
Curtiss
,
R. B.
Bird
, and
M. G.
Mayer
,
Molecular Theory of Gases and Liquids
(
Wiley
,
New York
,
1964
), Vol. 165.
73.
S.
Taniguchi
,
T.
Arima
,
T.
Ruggeri
, and
M.
Sugiyama
, “
Overshoot of the non-equilibrium temperature in the shock wave structure of a rarefied polyatomic gas subject to the dynamic pressure
,”
Int. J. Non-Linear Mech.
79
,
66
75
(
2016
).
74.
T.
Arima
,
T.
Ruggeri
, and
M.
Sugiyama
, “
Rational extended thermodynamics of a rarefied polyatomic gas with molecular relaxation processes
,”
Phys. Rev. E
96
,
042143
(
2017
).
75.
T.
Arima
,
T.
Ruggeri
, and
M.
Sugiyama
, “
Extended thermodynamics of rarefied polyatomic gases: 15-field theory incorporating relaxation processes of molecular rotation and vibration
,”
Entropy
20
,
301
(
2018
).
76.
T.
Ruggeri
,
Extended Thermodynamics
, Springer Tracts in Natural Philosophy (
Springer
,
1993
), Vol. 37.
77.
I.
Müller
and
T.
Ruggeri
,
Rational Extended Thermodynamics
(
Springer Science & Business Media
,
2013
), Vol. 37.
78.
T.
Ruggeri
and
M.
Sugiyama
,
Rational Extended Thermodynamics beyond the Monatomic Gas
(
Springer
,
2015
).
79.
G. A.
Bird
and
J.
Brady
,
Molecular Gas Dynamics and the Direct Simulation of Gas Flows
(
Clarendon Press Oxford
,
1994
), Vol. 5.
80.
l. D.
Boyd
, “
Analysis of rotational nonequilibrium in standing shock waves of nitrogen
,”
AIAA J.
28
,
1997
1999
(
1990
).
81.
I. D.
Boyd
, “
Rotational-translational energy transfer in rarefied nonequilibrium flows
,”
Phys. Fluids A
2
,
447
452
(
1990
).
82.
K.
Koura
, “
Statistical inelastic cross‐section model for the Monte Carlo simulation of molecules with discrete internal energy
,”
Phys. Fluids A
4
,
1782
1788
(
1992
).
83.
K.
Koura
, “
Statistical inelastic cross‐section model for the Monte Carlo simulation of molecules with continuous internal energy
,”
Phys. Fluids A
5
,
778
780
(
1993
).
84.
K.
Koura
, “
A set of model cross sections for the Monte Carlo simulation of rarefied real gases: Atom-diatom collisions
,”
Phys. Fluids
6
,
3473
3486
(
1994
).
85.
K.
Koura
, “
A generalization for Parker rotational relaxation model based on variable soft sphere collision model
,”
Phys. Fluids
8
,
1336
1337
(
1996
).
86.
I. J.
Wysong
and
D. C.
Wadsworth
, “
Assessment of direct simulation Monte Carlo phenomenological rotational relaxation models
,”
Phys. Fluids
10
,
2983
2994
(
1998
).
87.
C.
Borgnakke
and
P. S.
Larsen
, “
Statistical collision model for Monte Carlo simulation of polyatomic gas mixture
,”
J. Comput. Phys.
18
,
405
420
(
1975
).
88.
K.
Koura
, “
Monte Carlo direct simulation of rotational relaxation of diatomic molecules using classical trajectory calculations: Nitrogen shock wave
,”
Phys. Fluids
9
,
3543
3549
(
1997
).
89.
K.
Koura
, “
Monte Carlo direct simulation of rotational relaxation of nitrogen through high total temperature shock waves using classical trajectory calculations
,”
Phys. Fluids
10
,
2689
2691
(
1998
).
90.
T.
Tokumasu
and
Y.
Matsumoto
, “
Dynamic molecular collision (DMC) model for rarefied gas flow simulations by the DSMC method
,”
Phys. Fluids
11
,
1907
1920
(
1999
).
91.
B. J.
Alder
and
T. E.
Wainwright
, “
Studies in molecular dynamics. I. General method
,”
J. Chem. Phys.
31
,
459
466
(
1959
).
92.
B. J.
Alder
and
T. E.
Wainwright
, “
Studies in molecular dynamics. II. Behavior of a small number of elastic spheres
,”
J. Chem. Phys.
33
,
1439
1451
(
1960
).
93.
P.
Valentini
and
T. E.
Schwartzentruber
, “
A combined event-driven/time-driven molecular dynamics algorithm for the simulation of shock waves in rarefied gases
,”
J. Comput. Phys.
228
,
8766
8778
(
2009
).
94.
V. R.
Akkaya
and
I.
Kandemir
, “
Event-driven molecular dynamics simulation of hard-sphere gas flows in microchannels
,”
Math. Prob. Eng.
2015
,
1
.
95.
V.
Kovalev
,
A.
Yakunchikov
, and
V.
Kosiantchouk
, “
Study of gas separation by the means of high-frequency membrane oscillations
,”
Acta Astronaut.
116
,
282
285
(
2015
).
96.
V.
Kosyanchuk
,
A.
Yakunchikov
,
I.
Bryukhanov
, and
S.
Konakov
, “
Numerical simulation of novel gas separation effect in microchannel with a series of oscillating barriers
,”
Microfluid. Nanofluid.
21
,
116
(
2017
).
97.
A.
Yakunchikov
and
V.
Kosyanchuk
, “
Application of event-driven molecular dynamics approach to rarefied gas dynamics problems
,”
Comput. Fluids
170
,
121
127
(
2018
).
98.
A.
Yakunchikov
and
V.
Kosyanchuk
, “
Numerical investigation of gas separation in the system of filaments with different temperatures
,”
Int. J. Heat Mass Transfer
138
,
144
151
(
2019
).
99.
A.
Yakunchikov
and
V.
Kosyanchuk
, “
A new principle of separation of gas mixtures in non-stationary transitional flows
,”
Acta Astronaut.
163
,
120
125
(
2019
).
100.
A.
Yakunchikov
,
V.
Kosyanchuk
, and
A.
Iuldasheva
, “
Rotational relaxation model for nitrogen and its application in free jet expansion problem
,”
Phys. Fluids
32
,
102006
(
2020
).
101.
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
(
1954
).
102.
T. F.
Morse
, “
Kinetic model for gases with internal degrees of freedom
,”
Phys. Fluids
7
,
159
169
(
1964
).
103.
K.
Xu
and
L.
Tang
, “
Nonequilibrium Bhatnagar-Gross-Krook model for nitrogen shock structure
,”
Phys. Fluids
16
,
3824
3827
(
2004
).
104.
K.
Xu
,
X.
He
, and
C.
Cai
, “
Multiple temperature kinetic model and gas-kinetic method for hypersonic non-equilibrium flow computations
,”
J. Comput. Phys.
227
,
6779
6794
(
2008
).
105.
F.
Bernard
,
A.
Iollo
, and
G.
Puppo
, “
BGK polyatomic model for rarefied flows
,”
J. Sci. Comput.
78
,
1893
1916
(
2019
).
106.
C.
Baranger
,
Y.
Dauvois
,
G.
Marois
,
J.
Mathé
,
J.
Mathiaud
, and
L.
Mieussens
, “
A BGK model for high temperature rarefied gas flows
,”
Eur. J. Mech.: B/Fluids
80
,
1
12
(
2020
).
107.
J.
Mathiaud
and
L.
Mieussens
, “
BGK and Fokker-Planck models of the Boltzmann equation for gases with discrete levels of vibrational energy
,”
J. Stat. Phys.
178
,
1076
1095
(
2020
).
108.
L. H.
Holway
, Jr.
, “
New statistical models for kinetic theory: Methods of construction
,”
Phys. Fluids
9
,
1658
1673
(
1966
).
109.
P.
Andries
,
P.
Le Tallec
,
J.-P.
Perlat
, and
B.
Perthame
, “
The Gaussian-BGK model of Boltzmann equation with small Prandtl number
,”
Eur. J. Mech.: B/Fluids
19
,
813
830
(
2000
).
110.
S.
Brull
and
J.
Schneider
, “
On the ellipsoidal statistical model for polyatomic gases
,”
Continuum Mech. Thermodyn.
20
,
489
508
(
2009
).
111.
Z.
Cai
and
R.
Li
, “
The NRxx method for polyatomic gases
,”
J. Comput. Phys.
267
,
63
91
(
2014
).
112.
S.
Kosuge
,
K.
Aoki
, and
T.
Goto
, “
Shock wave structure in polyatomic gases: Numerical analysis using a model Boltzmann equation
,”
AIP Conf. Proc.
1786
,
180004
(
2016
).
113.
V. A.
Titarev
and
A. A.
Frolova
, “
Application of model kinetic equations to calculations of super-and hypersonic molecular gas flows
,”
Fluid Dyn.
53
,
536
551
(
2018
).
114.
Y.
Jiang
,
Z.
Gao
, and
C.-H.
Lee
, “
Particle simulation of nonequilibrium gas flows based on ellipsoidal statistical Fokker-Planck model
,”
Comput. Fluids
170
,
106
120
(
2018
).
115.
C.
Klingenberg
,
M.
Pirner
, and
G.
Puppo
, “
A consistent kinetic model for a two-component mixture of polyatomic molecules
,”
Commun. Math. Sci.
17
,
149
173
(
2019
).
116.
S.-B.
Yun
, “
Ellipsoidal BGK model for polyatomic molecules near Maxwellians: A dichotomy in the dissipation estimate
,”
J. Differ. Equations
266
,
5566
5614
(
2019
).
117.
S.
Kosuge
,
H.-W.
Kuo
, and
K.
Aoki
, “
A kinetic model for a polyatomic gas with temperature-dependent specific heats and its application to shock-wave structure
,”
J. Stat. Phys.
177
,
209
251
(
2019
).
118.
V.
Rykov
, “
A model kinetic equation for a gas with rotational degrees of freedom
,”
Fluid Dyn.
10
,
959
966
(
1975
).
119.
V. A.
Rykov
,
V. A.
Titarev
, and
E. M.
Shakhov
, “
Shock wave structure in a diatomic gas based on a kinetic model
,”
Fluid Dyn.
43
,
316
326
(
2008
).
120.
I. N.
Larina
and
V. A.
Rykov
, “
Kinetic model of the Boltzmann equation for a diatomic gas with rotational degrees of freedom
,”
Comput. Math. Math. Phys.
50
,
2118
2130
(
2010
).
121.
S.
Liu
,
P.
Yu
,
K.
Xu
, and
C.
Zhong
, “
Unified gas-kinetic scheme for diatomic molecular simulations in all flow regimes
,”
J. Comput. Phys.
259
,
96
113
(
2014
).
122.
L.
Wu
,
C.
White
,
T. J.
Scanlon
,
J. M.
Reese
, and
Y.
Zhang
, “
A kinetic model of the Boltzmann equation for non-vibrating polyatomic gases
,”
J. Fluid Mech.
763
,
24
50
(
2015
).
123.
Z.
Wang
,
H.
Yan
,
Q.
Li
, and
K.
Xu
, “
Unified gas-kinetic scheme for diatomic molecular flow with translational, rotational, and vibrational modes
,”
J. Comput. Phys.
350
,
237
259
(
2017
).
124.
C. A.
Brau
, “
Kinetic theory of polyatomic gases: Models for the collision processes
,”
Phys. Fluids
10
,
48
55
(
1967
).
125.
M. H.
Gorji
and
P.
Jenny
, “
A Fokker-Planck based kinetic model for diatomic rarefied gas flows
,”
Phys. Fluids
25
,
062002
(
2013
).
126.
Y.
Jiang
and
C.-Y.
Wen
, “
On the conservative property of particle-based Fokker-Planck method for rarefied gas flows
,”
Phys. Fluids
32
,
127108
(
2020
).
127.
J.
Mathiaud
and
L.
Mieussens
, “
A Fokker–Planck model of the Boltzmann equation with correct Prandtl number for polyatomic gases
,”
J. Stat. Phys.
168
,
1031
1055
(
2017
).
128.
F.
Tcheremissine
, “
Direct numerical solution of the Boltzmann equation
,”
AIP Conf. Proc.
762
,
677
685
(
2005
).
129.
F. G.
Tcheremissine
, “
Method for solving the Boltzmann kinetic equation for polyatomic gases
,”
Comput. Math. Math. Phys.
52
,
252
268
(
2012
).
130.
R.
Agarwal
and
F.
Tcherimmissine
, “
Computation of hypersonic shock wave flows of diatomic gases and gas mixtures using the generalized Boltzmann equation
,” in
48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition
(
American Institute of Aeronautics and Astronautics
,
2010
), p.
813
.
131.
Y. A.
Anikin
,
O. I.
Dodulad
,
Y. Y.
Kloss
, and
F. G.
Tcheremissine
, “
Method of calculating the collision integral and solution of the Boltzmann kinetic equation for simple gases, gas mixtures and gases with rotational degrees of freedom
,”
Int. J. Comput. Math.
92
,
1775
1789
(
2015
).
132.
G.
Qian
and
R. K.
Agarwal
, “
Computations of rarefied hypersonic blunt body flow in binary inert gas mixtures using the generalized Boltzmann equation
,”
AIP Conf. Proc.
2132
,
100011
(
2019
).
133.
T.
Schwartzentruber
and
I.
Boyd
,
Detailed analysis of a hybrid CFD-DSMC method for hypersonic non-equilibrium flows
,” in
38th AIAA Thermophysics Conference
(
American Institute of Aeronautics and Astronautics
,
2005
), p.
4829
.
134.
T. R.
Deschenes
,
T. D.
Holman
, and
I. D.
Boyd
, “
Effects of rotational energy relaxation in a modular particle-continuum method
,”
J. Thermophys. Heat Transfer
25
,
218
227
(
2011
).
135.
K. A.
Stephani
,
D. B.
Goldstein
, and
P. L.
Varghese
, “
A non-equilibrium surface reservoir approach for hybrid DSMC/Navier-Stokes particle generation
,”
J. Comput. Phys.
232
,
468
481
(
2013
).
136.
P. G.
Kistemaker
,
A.
Tom
, and
A. E.
De Vries
, “
Rotational relaxation numbers for the isotopic molecules of N2 and CO
,”
Physica
48
,
414
424
(
1970
).
137.
T. G.
Winter
and
G. L.
Hill
, “
High-temperature ultrasonic measurements of rotational relaxation in hydrogen, deuterium, nitrogen, and oxygen
,”
J. Acoust. Soc. Am.
42
,
848
858
(
1967
).
138.
R. N.
Healy
and
T. S.
Storvick
, “
Rotational collision number and eucken factors from thermal transpiration measurements
,”
J. Chem. Phys.
50
,
1419
1427
(
1969
).
139.
G.
Ganzi
and
S. I.
Sandler
, “
Determination of thermal transport properties from thermal transpiration measurements
,”
J. Chem. Phys.
55
,
132
140
(
1971
).
140.
E. H.
Carnevale
,
C.
Carey
, and
G.
Larson
, “
Ultrasonic determination of rotational collision numbers and vibrational relaxation times of polyatomic gases at high temperatures
,”
J. Chem. Phys.
47
,
2829
2835
(
1967
).
141.
A. P.
Malinauskas
, “
Thermal transpiration. Rotational relaxation numbers for nitrogen and carbon dioxide
,”
J. Chem. Phys.
44
,
1196
1202
(
1966
).
142.
J. G.
Parker
, “
Rotational and vibrational relaxation in diatomic gases
,”
Phys. Fluids
2
,
449
462
(
1959
).
143.
J. A.
Lordi
and
R. E.
Mates
, “
Rotational relaxation in nonpolar diatomic gases
,”
Phys. Fluids
13
,
291
308
(
1970
).
144.
C.
Nyeland
and
G. D.
Billing
, “
Transport coefficients of diatomic gases: Internal-state analysis for rotational and vibrational degrees of freedom
,”
J. Phys. Chem.
92
,
1752
1755
(
1988
).
145.
G. D.
Billing
and
L.
Wang
, “
Semiclassical calculations of transport coefficients and rotational relaxation of nitrogen at high temperatures
,”
J. Phys. Chem.
96
,
2572
2575
(
1992
).
146.
C.
Wang-Chang
and
G.
Uhlenbeck
, “
Transport phenomena in polyatomic gases
,”
Research Report No. CM-681
,
University of Michigan Engineering
,
1951
.
147.
F. E.
Lumpkin
 III
,
B. L.
Haas
, and
I. D.
Boyd
, “
Resolution of differences between collision number definitions in particle and continuum simulations
,”
Phys. Fluids A
3
,
2282
2284
(
1991
).
148.
B. L.
Haas
,
D. B.
Hash
,
G. A.
Bird
,
F. E.
Lumpkin
 III
, and
H. A.
Hassan
, “
Rates of thermal relaxation in direct simulation Monte Carlo methods
,”
Phys. Fluids
6
,
2191
2201
(
1994
).
149.
A.
Belikov
,
I. Y.
Solov'ev
,
G.
Sukhinin
, and
R.
Sharafutdinov
, “
Rotational relaxation time of nitrogen
,”
J. Appl. Mech. Tech. Phys.
29
,
630
636
(
1988
).
150.
J. C.
Maxwell
, “
VII. On stresses in rarified gases arising from inequalities of temperature
,”
Philos. Trans. R. Soc. London
170
,
231
256
(
1879
).
151.
D. P.
Weaver
and
B. D.
Shizgal
, “
Direct simulation Monte Carlo method for internal-translational energy exchange in nonequilibrium flow
,”
Rarified Gas Dynamics: Theory and Simulations
(American Institute of Aeronautics and Astronautics, 1994), pp.
103
113
.
152.
M.
Epstein
, “
A model of the wall boundary condition in kinetic theory
,”
AIAA J.
5
,
1797
1800
(
1967
).
153.
A.
Van der Avoird
,
P. E. S.
Wormer
, and
A. P. J.
Jansen
, “
An improved intermolecular potential for nitrogen
,”
J. Chem. Phys.
84
,
1629
1635
(
1986
).
154.
M. S. H.
Ling
and
M.
Rigby
, “
Towards an intermolecular potential for nitrogen
,”
Mol. Phys.
51
,
855
882
(
1984
).
155.
M. H.
Karimi-Jafari
and
M.
Ashouri
, “
Quantifying the anisotropy of intermolecular potential energy surfaces: A critical assessment of available N2-N2 potentials
,”
Phys. Chem. Chem. Phys.
13
,
9887
9894
(
2011
).
156.
J. D.
Bender
,
P.
Valentini
,
I.
Nompelis
,
Y.
Paukku
,
Z.
Varga
,
D. G.
Truhlar
,
T.
Schwartzentruber
, and
G. V.
Candler
, “
An improved potential energy surface and multi-temperature quasiclassical trajectory calculations of N2 + N2 dissociation reactions
,”
J. Chem. Phys.
143
,
054304
(
2015
).
157.
L.
Gomez
,
B.
Bussery-Honvault
,
T.
Cauchy
,
M.
Bartolomei
,
D.
Cappelletti
, and
F.
Pirani
, “
Global fits of new intermolecular ground state potential energy surfaces for N2-H2 and N2-N2 van der Waals dimers
,”
Chem. Phys. Lett.
445
,
99
107
(
2007
).
158.
Y.
Paukku
,
K. R.
Yang
,
Z.
Varga
, and
D. G.
Truhlar
, “
Global ab initio ground-state potential energy surface of N4
,”
J. Chem. Phys.
139
,
044309
(
2013
).
159.
R.
Hellmann
, “
Ab initio potential energy surface for the nitrogen molecule pair and thermophysical properties of nitrogen gas
,”
Mol. Phys.
111
,
387
401
(
2013
).
160.
D.
Cappelletti
,
F.
Vecchiocattivi
,
F.
Pirani
,
E. L.
Heck
, and
A. S.
Dickinson
, “
An intermolecular potential for nitrogen from a multi-property analysis
,”
Mol. Phys.
93
,
485
499
(
1998
).
161.
P.
Strąk
and
S.
Krukowski
, “
Molecular nitrogen-{N2} properties: The intermolecular potential and the equation of state
,”
J. Chem. Phys.
126
,
194501
(
2007
).
162.
V.
Aquilanti
,
M.
Bartolomei
,
D.
Cappelletti
,
E.
Carmona-Novillo
, and
F.
Pirani
, “
The N2-N2 system: An experimental potential energy surface and calculated rotovibrational levels of the molecular nitrogen dimer
,”
J. Chem. Phys.
117
,
615
627
(
2002
).
163.
A.
Yakunchikov
, “
Potential energy surface of interaction of two diatomic molecules for air flows simulation
,” Harvard Dataverse, V3 (Harvard Dataverse,
2019
).
164.
C. A.
Brau
and
R. M.
Jonkman
, “
Classical theory of rotational relaxation in diatomic gases
,”
J. Chem. Phys.
52
,
477
484
(
1970
).
165.
C.
Zhang
and
T. E.
Schwartzentruber
, “
Inelastic collision selection procedures for direct simulation Monte Carlo calculations of gas mixtures
,”
Phys. Fluids
25
,
106105
(
2013
).
166.
N. E.
Gimelshein
,
S. F.
Gimelshein
, and
D. A.
Levin
, “
Vibrational relaxation rates in the direct simulation Monte Carlo method
,”
Phys. Fluids
14
,
4452
4455
(
2002
).
167.
B.-Y.
Cao
,
J.
Sun
,
M.
Chen
, and
Z.-Y.
Guo
, “
Molecular momentum transport at fluid-solid interfaces in MEMS/NEMS: A review
,”
Int. J. Mol. Sci.
10
,
4638
4706
(
2009
).
168.
N.
Yamanishi
,
Y.
Matsumoto
, and
K.
Shobatake
, “
Multistage gas-surface interaction model for the direct simulation Monte Carlo method
,”
Phys. Fluids
11
,
3540
3552
(
1999
).
169.
C.
Cercignani
and
M.
Lampis
, “
Kinetic models for gas-surface interactions
,”
Transp. Theory Stat. Phys.
1
,
101
114
(
1971
).
170.
R. G.
Lord
, “
Some extensions to the Cercignani-Lampis gas-surface scattering kernel
,”
Phys. Fluids A
3
,
706
710
(
1991
).
171.
I.
Choquet
, “
Thermal nonequilibrium modeling using the direct simulation Monte Carlo method: Application to rotational energy
,”
Phys. Fluids
6
,
4042
4053
(
1994
).
172.
R. M.
Berns
and
A.
van der Avoird
, “
N2-N2 interaction potential from ab initio calculations, with application to the structure of (N2)2
,”
J. Chem. Phys.
72
,
6107
6116
(
1980
).
173.
A.
Riganelli
,
F. V.
Prudente
, and
A. J. C.
Varandas
, “
On the rovibrational partition function of molecular hydrogen at high temperatures
,”
J. Phys. Chem. A
105
,
9518
9521
(
2001
).
174.
P.
Valentini
,
P.
Norman
,
C.
Zhang
, and
T. E.
Schwartzentruber
, “
Rovibrational coupling in molecular nitrogen at high temperature: An atomic-level study
,”
Phys. Fluids
26
,
056103
(
2014
).
175.
F.
Robben
and
L.
Talbot
, “
Experimental study of the rotational distribution function of nitrogen in a shock wave
,”
Physi. Fluids
9
,
653
662
(
1966
).
176.
I. F.
Golubev
,
Viscosity of Gases and Gas Mixtures: A Handbook
(
Israel Program for Scientific Translations; US Department
,
1970
).
177.
V. V.
Voevodin
,
A. S.
Antonov
,
D. A.
Nikitenko
,
P. A.
Shvets
,
S. I.
Sobolev
,
I. Y.
Sidorov
,
K. S.
Stefanov
,
V. V.
Voevodin
, and
S. A.
Zhumatiy
, “
Supercomputer Lomonosov-2: Large scale, deep monitoring and fine analytics for the user community
,”
Supercomput. Front. Innovations
6
,
4
11
(
2019
).
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