Describing diatomic and polyatomic gases at high temperatures requires a deep understanding of the excitation of molecules to a higher vibrational level. We developed new second-order constitutive models for diatomic and polyatomic gases with vibrational degrees of freedom, starting from the modified Boltzmann–Curtiss kinetic equation. The closing-last balanced closure and cumulant expansion of the calortropy production associated with the Boltzmann collision term are key to the derivation of the second-order models, compatible with the second law of thermodynamics. The topology of the constitutive models showed the presence of highly nonlinear and coupled protruding or sunken regions in the compression branch. It was also shown that the vibrational mode reduces the level of nonlinearity in the topology. In addition, analysis of a strong shock structure highlighted the interplay between the second-order effects in the constitutive relations and the vibrational–translational relaxation. Finally, the analysis showed that the results of the second-order models were in better agreement with the direct simulation Monte Carlo data, when compared with the results of the first-order models, especially in the profiles and slopes of density, velocity, and vibrational temperatures.

1.
C.
Park
,
Nonequilibrium Hypersonic Aerothermodynamics
(
Wiley
,
New York
,
1990
).
2.
C. H.
Kruger
and
W.
Vincenti
,
Introduction to Physical Gas Dynamics
(
John Wiley & Sons
,
New York
,
1965
).
3.
J. D.
Anderson
, Jr.
,
Hypersonic and High Temperature Gas Dynamics
(
McGraw-Hill
,
New York
,
2006
).
4.
G.
Herzberg
,
Molecular Spectra and Molecular Structure
(
Read Books Ltd.
,
Redditch
,
2013
).
5.
J. G.
Kim
and
G.
Park
, “
Thermochemical nonequilibrium parameter modification of oxygen for a two-temperature model
,”
Phys. Fluids
30
,
016101
(
2018
).
6.
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
).
7.
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
).
8.
G. V.
Candler
and
I.
Nompelis
, “
Computational fluid dynamics for atmospheric entry
,” in
Non-Equilibrium Gas Dynamics from Physical Models to Hypersonic Flights
, RTO AVT/VKI Lecture Series (
The von Karman Institute for Fluid Dynamics
,
Belgium
,
2009
).
9.
G. V.
Candler
,
M. J.
Wright
, and
J. D.
McDonald
, “
Data-parallel lower-upper relaxation method for reacting flows
,”
AIAA J.
32
,
2380
(
1994
).
10.
V.
Casseau
,
R.
Palharini
,
T.
Scanlon
, and
R.
Brown
, “
A two-temperature open-source CFD model for hypersonic reacting flows, part one: Zero-dimensional analysis
,”
Aerospace
3
,
34
(
2016
).
11.
G. A.
Bird
,
Molecular Gas Dynamics and the Direct Simulation of Gas Flows
(
Clarendon Press
,
Oxford
,
1994
).
12.
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
).
13.
S. K.
Stefanov
, “
On the basic concepts of the direct simulation Monte Carlo method
,”
Phys. Fluids
31
,
067104
(
2019
).
14.
A.
Alamatsaz
and
A.
Venkattraman
, “
Characterizing deviation from equilibrium in direct simulation Monte Carlo simulations
,”
Phys. Fluids
31
,
042005
(
2019
).
15.
G. A.
Bird
,
The DSMC Method
(
CreateSpace Independent Publishing Platform
,
2013
).
16.
C.
Borgnakke
and
P. S.
Larsen
, “
Statistical collision model for Monte Carlo simulation of polyatomic gas mixture
,”
J. Comput. Phys.
18
,
405
(
1975
).
17.
I. D.
Boyd
, “
Analysis of vibrational-translational energy transfer using the direct simulation Monte Carlo method
,”
Phys. Fluids A
3
,
1785
(
1991
).
18.
G. A.
Bird
, “
The Q-K model for gas-phase chemical reaction rates
,”
Phys. Fluids
23
,
106101
(
2011
).
19.
S. F.
Gimelshein
and
I. J.
Wysong
, “
Bird’s total collision energy model: 4 decades and going strong
,”
Phys. Fluids
31
,
076101
(
2019
).
20.
B. C.
Eu
,
Kinetic Theory and Irreversible Thermodynamics
(
Wiley
,
New York
,
1992
).
21.
B. C.
Eu
, “
A modified moment method and irreversible thermodynamics
,”
J. Chem. Phys.
73
,
2958
(
1980
).
22.
R. S.
Myong
, “
Thermodynamically consistent hydrodynamic computational models for high-Knudsen-number gas flows
,”
Phys. Fluids
11
,
2788
(
1999
).
23.
R. S.
Myong
, “
A computational method for Eu’s generalized hydrodynamic equations of rarefied and microscale gasdynamics
,”
J. Comput. Phys.
168
,
47
(
2001
).
24.
R. S.
Myong
, “
On the high Mach number shock structure singularity caused by overreach of Maxwellian molecules
,”
Phys. Fluids
26
,
056102
(
2014
).
25.
J. W.
Ahn
and
C.
Kim
, “
An axisymmetric computational model of generalized hydrodynamic theory for rarefied multi-species gas flows
,”
J. Comput. Phys.
228
(
11
),
4088
(
2009
).
26.
H.
Xiao
and
Q. J.
He
, “
Aero-heating in hypersonic continuum and rarefied gas flows
,”
Aerosp. Sci. Technol.
82-83
,
566
(
2018
).
27.
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
(
12
),
5252
(
2019
).
28.
Z.
Jiang
,
W.
Zhao
,
W.
Chen
, and
R. K.
Agarwal
, “
Computation of shock wave structure using a simpler set of generalized hydrodynamic equations based on nonlinear coupled constitutive relations
,”
Shock Waves
29
,
1227
(
2019
).
29.
Z.
Jiang
,
W.
Zhao
,
W.
Chen
, and
R. K.
Agarwal
, “
An undecomposed hybrid algorithm for nonlinear coupled constitutive relations of rarefied gas dynamics
,”
Commun. Comput. Phys.
26
(
3
),
880
(
2019
).
30.
S.
Chapman
and
T. G.
Cowling
,
The Mathematical Theory of Non-Uniform Gases
(
Cambridge University Press
,
1990
).
31.
M.
Al-Ghoul
and
B. C.
Eu
, “
Generalized hydrodynamics and shock waves
,”
Phys. Rev. E
56
,
2981
(
1997
).
32.
C. F.
Curtiss
, “
The classical Boltzmann equation of a gas of diatomic molecules
,”
J. Chem. Phys.
75
,
376
(
1981
).
33.
M.
Al-Ghoul
and
B. C.
Eu
, “
Generalized hydrodynamic theory of shock waves in rigid diatomic gases
,”
Phys. Rev. E
64
,
046303
(
2001
).
34.
R. S.
Myong
, “
A generalized hydrodynamic computational model for rarefied and microscale diatomic gas flows
,”
J. Comput. Phys.
195
,
655
(
2004
).
35.
R. S.
Myong
, “
Coupled nonlinear constitutive models for rarefied and microscale gas flows: Subtle interplay of kinematics and dissipation effects
,”
Continuum Mech. Thermodyn.
21
,
389
(
2009
).
36.
H.
Grad
, “
On the kinetic theory of rarefied gases
,”
Commun. Pure Appl. Math.
2
,
331
(
1949
).
37.
H.
Grad
, “
The profile of a steady plane shock wave
,”
Commun. Pure Appl. Math.
5
,
257
(
1952
).
38.
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
(
2014
).
39.
L. P.
Raj
,
S.
Singh
,
A.
Karchani
, and
R. S.
Myong
, “
A super-parallel mixed explicit discontinuous Galerkin method for the second-order Boltzmann-based constitutive models of rarefied and microscale gases
,”
Comput. Fluids
157
,
146
(
2017
).
40.
R. S.
Myong
, “
Gaseous slip models based on the Langmuir adsorption isotherm
,”
Phys. Fluids
16
,
104
(
2004
).
41.
R. S.
Myong
,
J. M.
Reese
,
R. W.
Barber
, and
D. R.
Emerson
, “
Velocity slip in microscale cylindrical Couette flow: The Langmuir model
,”
Phys. Fluids
17
,
087105
(
2005
).
42.
R. S.
Myong
, “
A full analytical solution for the force-driven compressible Poiseuille gas flow based on a nonlinear coupled constitutive relation
,”
Phys. Fluids
23
,
012002
(
2011
).
43.
O.
Ejtehadi
,
A.
Rahimi
,
A.
Karchani
, and
R. S.
Myong
, “
Complex wave patterns in dilute gas–particle flows based on a novel discontinuous Galerkin scheme
,”
Int. J. Multiphase Flow
104
,
125
(
2018
).
44.
O.
Ejtehadi
and
R. S.
Myong
, “
A modal discontinuous Galerkin method for simulating dusty and granular gas flows in thermal non-equilibrium in the Eulerian framework
,”
J. Comput. Phys.
411
,
109410
(
2020
).
45.
S.
Bhola
and
T. K.
Sengupta
, “
Roles of bulk viscosity on transonic shock-wave/boundary layer interaction
,”
Phys. Fluids
31
,
096101
(
2019
).
46.
Y.
Zhu
,
C.
Zhang
,
X.
Chen
,
H.
Yuan
,
J.
Wu
,
S.
Chen
,
C.
Lee
, and
M.
Gad-el-Hak
, “
Transition in hypersonic boundary layers: Role of dilatational waves
,”
AIAA J.
54
,
3039
(
2016
).
47.
S.
Chen
,
X.
Wang
,
J.
Wang
,
M.
Wan
,
H.
Li
, and
S.
Chen
, “
Effects of bulk viscosity on compressible homogeneous turbulence
,”
Phys. Fluids
31
,
085115
(
2019
).
48.
C.
Park
, “
The limits of two-temperature model
,” AIAA Paper 2010-911,
2010
.
49.
I. D.
Boyd
and
E.
Josyula
, “
State resolved vibrational relaxation modeling for strongly nonequilibrium flows
,”
Phys. Fluids
23
,
057101
(
2011
).
50.
J.
Olejniczak
and
G. V.
Candler
, “
Vibrational energy conservation with vibration–dissociation coupling: General theory and numerical studies
,”
Phys. Fluids
7
,
1764
(
1995
).
51.
L.
Landau
, “
Theory of sound dispersion
,”
Phys. Z. Sowjetunion
10
,
34
(
1936
).
52.
R. C.
Millikan
and
D. R.
White
, “
Systematics of vibrational relaxation
,”
J. Chem. Phys.
39
,
3209
(
1963
).
53.
S.
Taniguchi
,
T.
Arima
,
T.
Ruggeri
, and
M.
Sugiyama
, “
Thermodynamic theory of the shock wave structure in a rarefied polyatomic gas: Beyond the Bethe-Teller theory
,”
Phys. Rev. E
89
(
1
),
013025
(
2014
).
54.
T.
Ruggeri
and
M.
Sugiyama
,
Rational Extended Thermodynamics Beyond the Monatomic Gas
(
Springer
,
Cham
,
2015
).
55.
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
(
2016
).
56.
T.
Arima
,
T.
Ruggeri
, and
M.
Sugiyama
, “
Rational extended thermo-dynamics of a rarefied polyatomic gas with molecular relaxation processes
,”
Phys. Rev. E
96
(
4
),
042143
(
2017
).
57.
M.
Bisi
,
T.
Ruggeri
, and
G.
Spiga
, “
Dynamical pressure in a poly-atomic gas: Interplay between kinetic theory and extended thermodynamics
,”
Kinet. Relat. Models
11
(
1
),
71
(
2018
).
58.
A.
Mentrelli
and
T.
Ruggeri
, “
Shock structure in extended thermodynamics with second-order maximum entropy principle closure
,”
Continuum Mech. Thermodyn.
(published online).
59.
M.
Pavic-Colic
,
D.
Madjarevic
, and
S.
Simic
, “
Polyatomic gases with dynamic pressure: Kinetic non-linear closure and the shock structure
,”
Int. J. Non-Linear Mech.
92
,
160
(
2017
).
60.
B.
Rahimi
and
H.
Struchtrup
, “
Capturing non-equilibrium phenomena in rarefied polyatomic gases: A high-order macroscopic model
,”
Phys. Fluids
26
,
052001
(
2014
).
61.
B.
Rahimi
and
H.
Struchtrup
, “
Macroscopic and kinetic modelling of rarefied polyatomic gases
,”
J. Fluid Mech.
806
,
437
(
2016
).
62.
A. S.
Rana
,
V. K.
Gupta
, and
H.
Struchtrup
, “
Coupled constitutive relations: A second law based higher-order closure for hydrodynamics
,”
Proc. R. Soc., A
474
,
20180323
(
2018
).
63.
S.
Kosuge
and
K.
Aoki
, “
Shock-wave structure for a polyatomic gas with large bulk viscosity
,”
Phys. Rev. Fluids
3
(
2
),
023401
(
2018
).
64.
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
).
65.
E.
Kustova
,
M.
Mekhonoshina
, and
A.
Kosareva
, “
Relaxation processes in carbon dioxide
,”
Phys. Fluids
31
,
046104
(
2019
).
66.
T. K.
Mankodi
,
U. V.
Bhandarkar
, and
B. P.
Puranik
, “
Dissociation cross sections for N2 + N → 3N and O2 + O → 3O using the QCT method
,”
J. Chem. Phys.
146
,
204307
(
2017
).
67.
B. C.
Eu
and
Y. G.
Ohr
, “
Generalized hydrodynamics, bulk viscosity, and sound wave absorption and dispersion in dilute rigid molecular gases
,”
Phys. Fluids
13
,
744
(
2001
).
68.
H. K.
Moffatt
,
G.
Zaslavsky
,
P.
Comte
, and
M.
Tabor
,
Topological Aspects of the Dynamics of Fluids and Plasmas
(
Springer Science & Business Media
,
2013
).
69.
L.
Carrión
,
M. A.
Herrada
, and
V. N.
Shtern
, “
Topology changes in a water-oil swirling flow
,”
Phys. Fluids
29
,
032109
(
2017
).
70.
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
).
71.
C.
Truesdell
, “
The present status of the controversy regarding the bulk viscosity of liquids
,”
Proc. R. Soc. London, Ser. A
226
,
59
(
1954
).
72.
W. E.
Meador
,
G. A.
Miner
, and
L. W.
Townsend
, “
Bulk viscosity as a relaxation parameter: Fact or fiction?
,”
Phys. Fluids
8
,
258
(
1996
).
73.
Y.
Wang
,
W.
Ubachs
, and
W.
van de Water
, “
Bulk viscosity of CO2 from Rayleigh-Brillouin light scattering spectroscopy at 532 nm
,”
J. Chem. Phys.
150
,
154502
(
2019
).
74.
H.-S.
Tsien
, “
One-dimensional flows of a gas characterized by van der Waal’s equation of state
,”
J. Math. Phys.
25
,
301
(
1946
).
75.
A. J.
Eggers
, Jr.
, “
One-dimensional flows of an imperfect diatomic gas
,” NACA Report 959,
1949
.
76.
S.
Liu
,
Y.
Yang
, and
C.
Zhong
, “
An extended gas-kinetic scheme for shock structure calculations
,”
J. Comput. Phys.
390
,
1
(
2019
).
77.
M.
Torrilhon
, “
Modeling nonequilibrium gas flow based on moment equations
,”
Annu. Rev. Fluid Mech.
48
,
429
(
2016
).
78.
B.
van Leer
, “
Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method
,”
J. Comput. Phys.
32
,
101
(
1979
).
79.
E. F.
Toro
,
M.
Spruce
, and
W.
Speares
, “
Restoration of the contact surface in the HLL-Riemann solver
,”
Shock Waves
4
,
25
(
1994
).
80.
R. S.
Myong
, “
Analytical solutions of shock structure thickness and asymmetry in Navier-Stokes/Fourier framework
,”
AIAA J.
52
(
5
),
1075
(
2014
).
81.
G. V.
Shoev
,
M. Y.
Timokhin
, and
Y. A.
Bondar
, “
On the total enthalpy behavior inside a shock wave
,”
Phys. Fluids
32
,
041703
(
2020
).
82.
F.
Fei
,
H.
Liu
,
Z.
Liu
, and
J.
Zhang
, “
A benchmark study of kinetic models for shock waves
,”
AIAA J.
58
,
2596
(
2020
).
83.
H.
Alsmeyer
, “
Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam
,”
J. Fluid Mech.
74
,
497
(
1976
).
84.
I.
Wysong
,
S.
Gimelshein
,
Y.
Bondar
, and
M.
Ivanov
, “
Comparison of direct simulation Monte Carlo chemistry and vibrational models applied to oxygen shock measurements
,”
Phys. Fluids
26
,
043101
(
2014
).
85.
L. B.
Ibraguimova
,
A. L.
Sergievskaya
,
V. Y.
Levashov
,
O. P.
Shatalov
,
Y. V.
Tunik
, and
I. E.
Zabelinskii
, “
Investigation of oxygen dissociation and vibrational relaxation at temperatures 4000-10 800 K
,”
J. Chem. Phys.
139
,
034317
(
2013
).
86.
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
(
1998
).
87.
T.
Tokumasu
and
Y.
Matsumoto
, “
Dynamic molecular collision (DMC) model for rarefied gas flow simulations by the DSMC method
,”
Phys. Fluids
11
,
1907
(
1999
).
88.
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
).
89.
B. C.
Eu
,
Nonequilibrium Statistical Mechanics: Ensemble Method
(
Kluwer Academic Publishers
,
1998
).
90.
D.
Jou
,
J.
Casas-Vázquez
, and
G.
Lebon
,
Extended Irreversible Thermodynamics
(
Springer
,
2010
).
You do not currently have access to this content.