This paper investigates three different particle-based continuum methods, the ellipsoidal statistical Bhatnagar-Gross-Krook (ESBGK) and Fokker-Planck (ESFP) methods and the Low Diffusion (LD) method, for a coupling with the direct simulation Monte Carlo (DSMC) method. After a short description of the methods and their implementation, including the coupling concept for the LD-DSMC, simulation results of a nozzle expansion are compared with available experimental measurements and a DSMC simulation. Excellent agreement between ESBGK, ESFP, and DSMC can be observed in the throat of the nozzle, while the LD method fails to predict the correct velocity, temperature, and density profile. Further downstream, only the DSMC and the coupled ESBGK/ESFP-DSMC simulations are able to reproduce the measured rotational temperature profiles. A performance comparison shows the possible computational savings of a coupled ESBGK/ESFP-DSMC simulation, where a speedup of four orders of magnitude can be observed compared to a regular DSMC simulation.

1.
G. A.
Bird
,
Molecular Gas Dynamics and the Direct Simulation of Gas Flows
, 2nd ed. (
Oxford Engineering Science
,
1994
).
2.
J.
Zhang
,
B.
John
,
M.
Pfeiffer
,
F.
Fei
, and
D.
Wen
, “
Particle-based hybrid and multiscale methods for nonequilibrium gas flows
,”
Adv. Aerodyn.
1
,
12
(
2019
).
3.
M. A.
Gallis
and
J. R.
Torczynski
, “
Investigation of the ellipsoidal-statistical Bhatnagar-Gross-Krook kinetic model applied to gas-phase transport of heat and tangential momentum between parallel walls
,”
Phys. Fluids
23
,
030601
(
2011
).
4.
M.
Pfeiffer
, “
Particle-based fluid dynamics: Comparison of different Bhatnagar-Gross-Krook models and the direct simulation Monte Carlo method for hypersonic flows
,”
Phys. Fluids
30
,
106106
(
2018
).
5.
M. H.
Gorji
and
P.
Jenny
, “
An efficient particle Fokker–Planck algorithm for rarefied gas flows
,”
J. Comput. Phys.
262
,
325
343
(
2014
).
6.
E.
Jun
,
M. H.
Gorji
,
M.
Grabe
, and
K.
Hannemann
, “
Assessment of the cubic Fokker–Planck–DSMC hybrid method for hypersonic rarefied flows past a cylinder
,”
Comput. Fluids
168
,
1
13
(
2018
).
7.
J.
Mathiaud
and
L.
Mieussens
, “
A Fokker–Planck model of the Boltzmann equation with correct Prandtl number
,”
J. Stat. Phys.
162
,
397
414
(
2016
).
8.
E.
Jun
,
M.
Pfeiffer
,
L.
Mieussens
, and
M. H.
Gorji
, “
Comparative study between cubic and ellipsoidal Fokker–Planck kinetic models
,”
AIAA J.
57
2524
(
2019
).
9.
J. M.
Burt
and
I. D.
Boyd
, “
A low diffusion particle method for simulating compressible inviscid flows
,”
J. Comput. Phys.
227
,
4653
4670
(
2008
).
10.
J. M.
Burt
and
I. D.
Boyd
, “
A hybrid particle approach for continuum and rarefied flow simulation
,”
J. Comput. Phys.
228
,
460
475
(
2009
).
11.
A.
Mirza
,
P.
Nizenkov
,
M.
Pfeiffer
, and
S.
Fasoulas
, “
Three-dimensional implementation of the low diffusion method for continuum flow simulations
,”
Comput. Phys. Commun.
220
,
269
278
(
2017
).
12.
D. E.
Rothe
, “
Electron-beam studies of viscous flow in supersonic nozzles
,”
AIAA J.
9
,
804
811
(
1971
).
13.
I. D.
Boyd
,
P. F.
Penko
,
D. L.
Meissner
, and
K. J.
DeWitt
, “
Experimental and numerical investigations of low-density nozzle and plume flows of nitrogen
,”
AIAA J.
30
,
2453
2461
(
1992
).
14.
C.-H.
Chung
,
S. C.
Kim
,
R. M.
Stubbs
, and
K. J.
De Witt
, “
Low-density nozzle flow by the direct simulation Monte Carlo and continuum methods
,”
J. Propul. Power
11
,
64
70
(
1995
).
15.
J.
Burt
and
I.
Boyd
, “
Evaluation of a particle method for the ellipsoidal statistical Bhatnagar-Gross-Krook equation
,” in
44th AIAA Aerospace Sciences Meeting and Exhibit
(
American Institute of Aeronautics and Astronautics
,
Reston, VA
,
2006
), pp.
1
15
.
16.
R.
Kumar
,
E. V.
Titov
,
D. A.
Levin
,
N. E.
Gimelshein
, and
S. F.
Gimelshein
, “
Assessment of Bhatnagar-Gross-Krook approaches for near continuum regime nozzle flows
,”
AIAA J.
48
,
1531
1541
(
2010
).
17.
M.
Pfeiffer
and
P.
Nizenkov
, “
Coupled ellipsoidal statistical BGK-DSMC simulations of a nozzle expansion
,”
AIP Conf. Proc.
(to be published).
18.
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
).
19.
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
).
20.
P.
Andries
and
B.
Perthame
, “
The ES-BGK model equation with correct Prandtl number
,”
AIP Conf. Proc.
585
,
30
36
(
2001
).
21.
M.
Pfeiffer
, “
Extending the particle ellipsoidal statistical Bhatnagar-Gross-Krook method to diatomic molecules including quantized vibrational energies
,”
Phys. Fluids
30
,
116103
(
2018
).
22.
N. E.
Gimelshein
,
S. F.
Gimelshein
, and
D. A.
Levin
, “
Vibrational relaxation rates in the direct simulation Monte Carlo method
,”
Phys. Fluids
14
,
4452
(
2002
).
23.
C.
Zhang
and
T. E.
Schwartzentruber
, “
Inelastic collision selection procedures for direct simulation Monte Carlo calculations of gas mixtures
,”
Phys. Fluids
25
,
106105
(
2013
).
24.
Q.
Sun
and
I. D.
Boyd
, “
Evaluation of macroscopic properties in the direct simulation Monte Carlo method
,”
J. Thermophys. Heat Transfer
19
,
329
335
(
2005
).
25.
B. L.
Haas
,
D. B.
Hash
,
G. A.
Bird
,
F. E. I.
Lumpkin
, and
H. A.
Hassan
, “
Rates of thermal relaxation in direct simulation Monte Carlo methods
,”
Phys. Fluids
6
,
2191
(
1994
).
26.
C.-D.
Munz
,
M.
Auweter-Kurtz
,
S.
Fasoulas
,
A.
Mirza
,
P.
Ortwein
,
M.
Pfeiffer
, and
T. K. M.
Stindl
, “
Coupled particle-in-cell and direct simulation Monte Carlo method for simulating reactive plasma flows
,”
C. R. Mec.
342
,
662
670
(
2014
).
27.
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
).
28.
O.
Tumuklu
,
Z.
Li
, and
D. A.
Levin
, “
Particle ellipsoidal statistical Bhatnagar–Gross–Krook approach for simulation of hypersonic shocks
,”
AIAA J.
54
,
3701
3716
(
2016
).
29.
P.
Jenny
,
M.
Torrilhon
, and
S.
Heinz
, “
A solution algorithm for the fluid dynamic equations based on a stochastic model for molecular motion
,”
J. Comput. Phys.
229
,
1077
1098
(
2010
).
30.
A. L.
Garcia
and
B. J.
Alder
, “
Generation of the Chapman–Enskog distribution
,”
J. Comput. Phys.
140
,
66
70
(
1998
).
31.
E.
Jun
,
J. M.
Burt
, and
I. D.
Boyd
, “
AII-particle multiscale computation of hypersonic rarefied flow
,”
AIP Conf. Proc.
1333
,
557
562
(
2011
).
32.
F.
Fei
,
J.
Zhang
,
J.
Li
, and
Z.
Liu
, “
A unified stochastic particle Bhatnagar-Gross-Krook method for multiscale gas flows
,” e-print arXiv:1808.03801 (
2018
).
33.
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
).
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