Shock-accelerated bubbles have long been an intriguing topic for understanding the fundamental physics of turbulence generation and mixing caused by the Richtmyer–Meshkov instability. In this study, the impact of bulk viscosity on the flow morphology of a shock-accelerated cylindrical light bubble in diatomic and polyatomic gases is investigated numerically. An explicit mixed-type modal discontinuous Galerkin scheme with uniform meshes is employed to solve a two-dimensional system of unsteady physical conservation laws derived rigorously from the Boltzmann–Curtiss kinetic equations. We also derive a new complete viscous compressible vorticity transport equation including the bulk viscosity. The numerical results show that, during the interaction between a planar shock wave and a cylindrical light bubble, the bulk viscosity associated with the viscous excess normal stress in diatomic and polyatomic gases plays an important role. The diatomic and polyatomic gases cause significant changes in flow morphology, resulting in complex wave patterns, vorticity generation, vortex formation, and bubble deformation. In contrast to monatomic gases, diatomic and polyatomic gases produce larger rolled-up vortex chains, various inward jet formations, and large mixing zones with strong, large-scale expansion. The effects of diatomic and polyatomic gases are explored in detail through phenomena such as the vorticity generation, degree of nonequilibrium, enstrophy, and dissipation rate. Furthermore, the evolution of the shock trajectories and interface features is investigated. Finally, the effects of bulk viscosity on the flow physics of shock-accelerated cylindrical light bubble are comprehensively analyzed.

1.
R. D.
Richtmyer
, “
Taylor instability in shock acceleration of compressible fluids
,”
Commun. Pure Appl. Math.
13
,
297
319
(
1960
).
2.
E.
Meshkov
, “
Instability of the interface of two gases accelerated by a shock wave
,”
Fluid Dyn.
4
,
101
104
(
1969
).
3.
J. D.
Lindl
,
R. L.
McCrory
, and
E. M.
Campbell
, “
Progress toward ignition and burn propagation in inertial confinement fusion
,”
Phys. Today
45
(
9
),
32
40
(
1992
).
4.
W. D.
Arnett
,
J. N.
Bahcall
,
R. P.
Kirshner
 et al., “
Supernova 1987A
,”
Annu. Rev. Astron. Astrophys.
27
,
629
700
(
1989
).
5.
A. R.
Jamaluddin
,
G. J.
Ball
, and
T. G.
Leighton
, “
Shock/bubble interaction near a rigid boundary in shock wave lithotripsy
,” in
24th International Symposium on Shock Waves
(Springer,
2005
), pp.
1211
1216
.
6.
F. E.
Marble
,
G. J.
Hendricks
, and
E. E.
Zukoski
, “
Progress toward shock enhancement of supersonic combustion processes
,” in
Turbulent Reactive Flows
(
Springer
,
1989
).
7.
Y.
Yang
,
T.
Kubota
, and
E. E.
Zukoski
, “
Applications of shock-induced mixing to supersonic combustion
,”
AIAA J.
31
,
854
862
(
1993
).
8.
D.
Arnett
, “
The role of mixing in astrophysics
,”
Astrophys. J., Suppl. Ser.
127
,
213
(
2000
).
9.
A. M.
Goodbody
,
Cartesian Tensors: With Applications to Mechanics, Fluid Mechanics and Elasticity
(
Ellis Horwood Ltd
.,
Chichester, England
,
1982
).
10.
S. A.
Kinnas
, “
VIScous vorticity equation (VISVE) for turbulent 2-D flows with variable density and viscosity
,”
J. Mar. Sci. Eng.
8
(
3
),
191
(
2020
).
11.
G. H.
Markstein
, “
A shock-tube study of flame front-pressure wave interaction
,”
Symp. (Int.) Combust.
6
,
387
398
(
1957
).
12.
G.
Rudinger
and
L. M.
Somers
, “
Behavior of small regions of different gases carried in accelerated gas flows
,”
J. Fluid Mech.
7
,
161
176
(
1960
).
13.
J. F.
Haas
and
B.
Sturtevant
, “
Interaction of weak shock waves with cylindrical and spherical gas inhomogeneities
,”
J. Fluid Mech.
181
,
41
76
(
1987
).
14.
J. W.
Jacobs
, “
Shock-induced mixing of a light-gas cylinder
,”
J. Fluid Mech.
234
,
629
649
(
1992
).
15.
J. W.
Jacobs
, “
The dynamics of shock accelerated light and heavy gas cylinders
,”
Phys. Fluids
5
,
2239
2247
(
1993
).
16.
G.
Layes
,
G.
Jourdan
, and
L.
Houas
, “
Experimental investigation of the shock wave interaction with a spherical gas inhomogeneity
,”
Phys. Fluids
17
,
028103
(
2005
).
17.
G.
Layes
,
G.
Jourdan
, and
L.
Houas
, “
Experimental study on a plane shock wave accelerating a gas bubble
,”
Phys. Fluids
21
,
074102
(
2009
).
18.
D.
Ranjan
,
J. H. J.
Niederhaus
,
B.
Motl
,
M. H.
Anderson
,
J.
Oakley
, and
R.
Bonazza
, “
Experimental investigation of primary and secondary features in high-Mach-number shock-bubble interaction
,”
Phys. Rev. Lett.
98
,
024502
(
2007
).
19.
D.
Ranjan
,
J. H. J.
Niederhaus
,
J. G.
Oakley
,
M. H.
Anderson
,
R.
Bonazza
, and
J. A.
Greenough
, “
Shock-bubble interactions: Features of divergent shock refraction geometry observed in experiments and simulations
,”
Phys. Fluids
20
,
036101
(
2008
).
20.
J. J.
Quirk
and
S.
Karni
, “
On the dynamics of a shock–bubble interaction
,”
J. Fluid Mech.
318
,
129
163
(
1996
).
21.
N. J.
Zabusky
and
S. M.
Zeng
, “
Shock cavity implosion morphologies and vortical projectile generation in axisymmetric shock-spherical fast/slow bubble interactions
,”
J. Fluid Mech.
362
,
327
346
(
1998
).
22.
A.
Bagabir
and
D.
Drikakis
, “
Mach number effects on shock-bubble interaction
,”
Shock Waves
11
,
209
218
(
2001
).
23.
J. J.
Niederhaus
,
J. A.
Greenough
,
J. G.
Oakley
,
D.
Ranjan
,
M. H.
Anderson
, and
R.
Bonazza
, “
A computational parameter study for the three-dimensional shock-bubble interaction
,”
J. Fluid Mech.
594
,
85
(
2008
).
24.
Y. J.
Zhu
,
G.
Dong
,
B. C.
Fan
, and
Y. X.
Liu
, “
Formation and evolution of vortex rings induced by interactions between shock waves and a low-density bubble
,”
Shock Waves
22
,
495
(
2012
).
25.
Y. J.
Zhu
,
L.
Yu
,
J. F.
Pan
,
Z. H.
Pan
, and
P. G.
Zhang
, “
Jet formation of SF6 bubble induced by incident and reflected shock waves
,”
Phys. Fluids
29
,
126105
(
2017
).
26.
Y. J.
Zhu
,
Z. W.
Yang
,
K. H.
Luo
,
J. F.
Pan
, and
Z. H.
Pan
, “
Numerical investigation of planar shock wave impinging on spherical gas bubble with different densities
,”
Phys. Fluids
31
,
056101
(
2019
).
27.
B.
Rybakin
and
V.
Goryachev
, “
The supersonic shock wave interaction with low-density gas bubble
,”
Acta Astronaut.
94
(
2
),
749
753
(
2014
).
28.
B. P.
Rybakin
,
V. B.
Betelin
,
V. R.
Dushin
,
E. V.
Mikhalchenko
,
S. G.
Moiseenko
,
L. I.
Stamov
, and
V. V.
Tyurenkova
, “
Model of turbulent destruction of molecular clouds
,”
Acta Astronaut.
119
,
131
136
(
2016
).
29.
B. P.
Rybakin
,
V. B.
Betelin
,
N. N.
Smirnov
,
S. G.
Moiseenko
, and
L. I.
Stamov
, “
3D numerical simulation of molecular clouds collision process
,”
J. Phys.: Conf. Ser.
1103
,
012007
(
2018
).
30.
B. P.
Rybakin
,
V. D.
Goryachev
,
L. I.
Stamov
,
E. V.
Mikhalchenko
,
V. V.
Tyurenkova
,
M. N.
Smirnova
,
A. A.
Shamina
,
E. I.
Kolenkina
, and
D. A.
Pestov
, “
Modeling the formation of dense clumps during molecular clouds collision
,”
Acta Astronaut.
170
,
586
591
(
2020
).
31.
Z.
Wang
,
B.
Yu
,
H.
Chen
,
B.
Zhang
, and
H.
Liu
, “
Scaling vortex breakdown mechanism based on viscous effect in shock cylindrical bubble interaction
,”
Phys. Fluids
30
,
126103
(
2018
).
32.
S.
Singh
, “
Role of Atwood number on flow morphology of a planar shock-accelerated square bubble: A numerical study
,”
Phys. Fluids
32
,
126112
(
2020
).
33.
G. G.
Stokes
, “
On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic fluids
,”
Trans. Cambridge Philos. Soc.
8
,
287
305
(
1845
).
34.
G.
Emanuel
, “
Bulk viscosity of a dilute polyatomic gas
,”
Phys. Fluids A
2
,
2252
2254
(
1990
).
35.
G.
Emanuel
, “
Bulk viscosity in the Navier–Stokes equations
,”
Int. J. Eng. Sci.
36
,
1313
1323
(
1998
).
36.
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
).
37.
N.
Carlevaro
and
G.
Montani
, “
Bulk viscosity effects on the early universe stability
,”
Mod. Phys. Lett. A
20
,
1729
1739
(
2005
).
38.
G.
Billet
,
V.
Giovangigli
, and
G. D.
Gassowski
, “
Impact of volume viscosity on a shock–hydrogen-bubble interaction
,”
Combust. Theory Modell.
12
,
221
248
(
2008
).
39.
M. S.
Cramer
, “
Numerical estimates for the bulk viscosity of ideal gases
,”
Phys. Fluids
24
,
066102
(
2012
).
40.
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
3049
(
2016
).
41.
F.
Bahmani
and
M. S.
Cramer
, “
Suppression of shock-induced separation in fluids having large bulk viscosities
,”
J. Fluid Mech.
756
,
1
10
(
2014
).
42.
S.
Pan
and
E.
Johnsen
, “
The role of bulk viscosity on the decay of compressible, homogeneous, isotropic turbulence
,”
J. Fluid Mech.
833
,
717
744
(
2017
).
43.
R. S.
Myong
, “
A generalized hydrodynamic computational model for rarefied and microscale diatomic gas flows
,”
J. Comput. Phys.
195
,
655
(
2004
).
44.
S.
Singh
and
R. S.
Myong
, “
A computational study of bulk viscosity effects on shock-vortex interaction using discontinuous Galerkin method
,”
J. Comput. Fluids Eng.
22
,
86
95
(
2017
).
45.
S.
Singh
,
A.
Karchani
, and
R. S.
Myong
, “
Non-equilibrium effects of diatomic and polyatomic gases on the shock-vortex interaction based on the second-order constitutive model of the Boltzmann–Curtiss equation
,”
Phys. Fluids
30
,
016109
(
2018
).
46.
S.
Singh
, “
Development of a 3D discontinuous Galerkin method for the second-order Boltzmann–Curtiss based hydrodynamic models of diatomic and polyatomic gases
,” Ph.D. thesis (
Gyeongsang National University
, South Korea,
2018
).
47.
B.
Sharma
and
R.
Kumar
, “
Estimation of bulk viscosity of dilute gases using a nonequilibrium molecular dynamics approach
,”
Phys. Rev. E
100
,
013309
(
2019
).
48.
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
).
49.
S.
Singh
and
M.
Battiato
, “
Behavior of a shock accelerated heavy cylindrical bubble under nonequilibrium conditions of diatomic and polyatomic gases
,”
Phys. Rev. Fluids
6
,
044001
(
2021
).
50.
R. S.
Myong
, “
Thermodynamically consistent hydrodynamic computational models for high-Knudsen-number gas flows
,”
Phys. Fluids
11
,
2788
(
1999
).
51.
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
).
52.
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
).
53.
J.
Giordano
and
Y.
Burtschell
, “
Richtmyer–Meshkov instability induced by shock-bubble interaction: Numerical and analytical studies with experimental validation
,”
Phys. Fluids
18
,
028603
(
2006
).
54.
S. K.
Shankar
,
S.
Kawai
, and
S. K.
Lele
, “
Two-dimensional viscous flow simulation of a shock accelerated heavy gas cylinder
,”
Phys. Fluids
23
,
024102
(
2011
).
55.
J. M.
Picone
and
J. P.
Boris
, “
Vorticity generation by shock propagation through bubbles in a gas
,”
J. Fluid Mech.
189
,
23
51
(
1988
).
56.
R.
Samtaney
and
N. J.
Zabusky
, “
Circulation deposition on shock-accelerated planar and curved density-stratified interfaces: Models and scaling laws
,”
J. Fluid Mech.
269
,
45
78
(
1994
).
57.
M.
Latini
and
O.
Schilling
, “
A comparison of two-and three-dimensional single-mode reshocked Richtmyer–Meshkov instability growth
,”
Phys. D
401
,
132201
(
2020
).
58.
C. F.
Curtiss
, “
The classical Boltzmann equation of a gas of diatomic molecules
,”
J. Chem. Phys.
75
,
376
378
(
1981
).
59.
S.
Chapman
and
T. G.
Cowling
,
The Mathematical Theory of Non-Uniform Gases
(
Cambridge University Press
,
Cambridge
,
1970
).
60.
C. H.
Kruger
and
W.
Vincenti
,
Introduction to Physical Gas Dynamics
(
John Wiley and Sons
,
1965
).
61.
J. G.
Parker
, “
Rotational and vibrational relaxation in diatomic gases
,”
Phys. Fluids
2
,
449
(
1959
).
62.
X. D.
Li
,
Z. M.
Hu
, and
Z. L.
Jiang
, “
Continuum perspective of bulk viscosity in compressible fluids
,”
J. Fluid Mech.
812
,
966
990
(
2017
).
63.
M. A.
Saad
,
Compressible Fluid Flow
(
Prentice-Hall
,
1985
).
64.
H. J.
Hanley
and
J. F.
Ely
, “
The viscosity and thermal conductivity coefficients of dilute nitrogen and oxygen
,”
J. Phys. Chem.
2
,
735
756
(
1973
).
65.
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
).
66.
E.
Vogel
, “
Präazisionsmessungen des viskositäatskoeffizienten von stickstoff und den edelgasen zwischen raumtemperatur und 650 K
,”
Ber. Bunsengesellsch. Phys. Chem.
88
,
997
1002
(
1984
).
67.
P.
Kistemaker
,
A.
Tom
, and
A. D.
Vries
, “
Rotational relaxation numbers for the isotopic molecules of N2 and CO
,”
Physica
48
,
414
(
1970
).
68.
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
(
1967
).
69.
Z.
Gu
and
W.
Ubachs
, “
Temperature-dependent bulk viscosity of nitrogen gas determined from spontaneous Rayleigh–Brillouin scattering
,”
Opt. Lett.
38
,
1110
(
2013
).
70.
A.
Meijer
,
A.
de Wijn
,
M.
Peters
,
N.
Dam
, and
W.
van de Water
, “
Coherent Rayleigh–Brillouin scattering measurements of bulk viscosity of polar and nonpolar gases
,”
J. Chem. Phys.
133
,
164315
(
2010
).
71.
G. J.
Prangsma
,
A. H.
Alberga
, and
J.
Beenakker
, “
Ultrasonic determination of the volume viscosity of N2, CO, CH4 and CD4 between 77 and 300 K
,”
Physica
64
,
278
(
1973
).
72.
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. Phys.
157
,
146
163
(
2017
).
73.
S.
Singh
and
M.
Battiato
, “
Strongly out-of-equilibrium simulations for electron Boltzmann transport equation using explicit modal discontinuous Galerkin method
,”
Int. J. Appl. Comput. Math.
6
,
133
(
2020
).
74.
S.
Singh
and
M.
Battiato
, “
Effect of strong electric fields on material responses: The Bloch oscillation resonance in high field conductivities
,”
Materials
13
,
1070
(
2020
).
75.
T.
Chourushi
,
A.
Rahimi
,
S.
Singh
, and
R. S.
Myong
, “
Computational simulations of near-continuum gas flow using Navier–Stokes-Fourier equations with slip and jump conditions based on the modal discontinuous Galerkin method
,”
Adv. Aerodyn.
2
,
8
(
2020
).
76.
S.
Singh
and
M.
Battiato
, “
An explicit modal discontinuous Galerkin method for Boltzmann transport equation under electronic nonequilibrium conditions
,”
Comput. Fluids
224
,
104972
(
2021
).
77.
S.
Singh
,
A.
Karchani
,
T.
Chourushi
, and
R. S.
Myong
, “
A three-dimensional modal discontinuous Galerkin method for second-order Boltzmann-Curtiss constitutive models of rarefied and microscale gas flows
,”
J. Comput. Phys.
(unpublished) (
2021
).
78.
B.
Cockburn
and
C. W.
Shu
, “
The local discontinuous Galerkin method for time dependent convection-diffusion systems
,”
SIAM J. Numer. Anal.
35
,
2440
2463
(
1998
).
79.
B.
Cockburn
and
C. W.
Shu
, “
The Runge–Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems
,”
J. Comput. Phys.
141
,
199
(
1998
).
80.
P. L.
Roe
, “
Approximate Riemann solvers, parameter vectors and difference schemes
,”
J. Comput. Phys.
43
,
357
372
(
1981
).
81.
A.
Marquina
and
P.
Mulet
, “
A flux-split algorithm applied to conservative models for multicomponent compressible flows
,”
J. Comput. Phys.
185
,
120
138
(
2003
).
82.
N. N.
Smirnov
,
V. B.
Betelin
,
R. M.
Shagaliev
,
V. F.
Nikitin
,
I. M.
Belyakov
,
Y. N.
Deryuguin
,
S. V.
Aksenov
, and
D. A.
Korchazhkin
, “
Hydrogen fuel rocket engines simulation using LOGOS code
,”
Int. J. Hydrogen Energy
39
,
10748
10756
(
2014
).
83.
N. N.
Smirnov
,
V. B.
Betelin
,
V. F.
Nikitin
,
L. I.
Stamov
, and
D. I.
Altoukhov
, “
Accumulation of errors in numerical simulations of chemically reacting gas dynamics
,”
Acta Astronaut.
117
,
338
355
(
2015
).
84.
J.
Ding
,
T.
Si
,
M.
Chen
,
Z.
Zhai
,
X.
Lu
, and
X.
Luo
, “
On the interaction of a planar shock with a three-dimensional light gas cylinder
,”
J. Fluid Mech.
828
,
289
(
2017
).
85.
L.
Onsager
, “
Reciprocal relations in irreversible processes
,”
Phys. Rev.
37
,
405
(
1931
).
You do not currently have access to this content.