When a planar shock hits a corrugated interface between two fluids, the Richtmyer–Meshkov instability (RMI) occurs. Vortices are generated in bulk behind the transmitted and reflected shocks in RMI. As the shock intensity becomes larger, the stronger bulk vortices are created. The nonlinear evolution of RMI is investigated within the vortex sheet model (VSM), taking the nonlinear interaction between the interface and the vortices into account. The fluid becomes incompressible as the shocks move away from the interface, and VSM can then be applied. The vorticity and position of the bulk vortices obtained from the compressible linear theory [F. Cobos-Campos and J. G. Wouchuk, Phys. Rev. E93, 053111 (2016)] are applied as initial conditions of the bulk point vortices in VSM. The suppression of RMI due to the bulk vortices is observed in the region such that the corrugation amplitude is less than one-tenth of the wavelength, and the reduction of the growth is quantitatively evaluated and compared with the compressible linear theory. In the nonlinear stage, the interaction between the interface and the bulk vortices strongly affects the interfacial shape and the dynamics of bulk vortices, e.g., the creation of a vortex pair is observed. Strong bulk vortices behind the transmitted shock enhance the growth of spike, supplying flow from spike root to its top and mushroom umbrella in the fully nonlinear stage.
References
1.
R. D.
Richtmyer
, “Taylor instability in shock acceleration of compressible fluids
,” Commun. Pure Appl. Math.
13
, 297
–319
(1960
).2.
E. E.
Meshkov
, “Instability of the interface of two gases accelerated by a shock wave
,” Fluid Dyn.
4
, 101
–108
(1972
).3.
K.
Nishihara
, J. G.
Wouchuk
, C.
Matsuoka
, R.
Ishizaki
, and V. V.
Zhakhovskii
, “Richtmyer-Meshkov instability: Theory of linear and nonlinear evolution
,” Philos. Trans. R. Soc. A
368
, 1769
–1807
(2010
).4.
Y.
Zhou
, “Rayleigh-Taylor and Richtmyer-Meshkov instability induced flow, turbulence, and mixing. I
,” Phys. Rep.
720–722
, 1
–136
(2017
).5.
S. I.
Abarzhi
, A. K.
Bhowmick
, A.
Naveh
, A.
Pandian
, N. C.
Swisher
, R. F.
Stellingwerf
, and W. D.
Arnett
, “Supernova, nuclear synthesis, fluid instabilities, and interfacial mixing
,” Proc. Natl. Acad. Sci. U. S. A.
116
, 18184
–18192
(2019
).6.
R. Y. T.
Inoue
and S.
Inutsuka
, “Turbulence and magnetic field amplification in supernova remnants: Interactions between a strong shock wave and multiphase interstellar medium
,” Astrophys J.
695
, 825
(2009
).7.
T.
Sano
, K.
Nishihara
, C.
Matsuoka
, and T.
Inoue
, “Magnetic field amplification associated with the Richtmyer-Meshkov instability
,” Astrophys. J.
758
, 126
(2012
).8.
T.
Sano
, T.
Inoue
, and K.
Nishihara
, “Critical magnetic field strength for suppression of the Richtmyer-Meshkov instability in plasmas
,” Phys. Rev. Lett.
111
, 205001
(2013
).9.
C.
Matsuoka
, K.
Nishihara
, and T.
Sano
, “Nonlinear dynamics of non-uniform current-vortex sheets in magnetohydrodynamic flows
,” J. Nonlinear Sci.
27
, 531
–572
(2017
).10.
G.
Dimonte
, C. E.
Frerking
, M.
Schneider
, and B.
Remington
, “Richtmyer-Meshkov instability with strong radiatively driven shocks
,” Phys. Plasmas
3
, 614
–630
(1996
).11.
J.
Lindl
, Inertia Confinement Fusion: The Quest for Ignition and High Gain Using Indirect Drive
(Springer/AIP
, New York
, 1997
).12.
R. L.
Holmes
, G.
Dimonte
, B.
Fryxell
, M. L.
Gittings
, J. W. W.
Grove
, H.
Schneider
, D.
Sharp
, A. L.
Velikovich
, R.
Weaver
, and Q.
Zang
, “Richtmyer-Meshkov instability growth: Experiment, simulation and theory
,” J. Fluid Mech.
389
, 55
–79
(1999
).13.
A. L.
Velikovich
, J. G.
Wouchuk
, C. H. R.
de Lira
, N.
Melzler
, S.
Zalesak
, and A. J.
Schmitt
, “Shock front distortion and Richtmyer-Meshkov-type growth caused by a small preshock nonuniformity
,” Phys. Plasmas
14
, 072706
(2007
).14.
J. G.
Wouchuk
and K.
Nishihara
, “Asymptotic growth in the linear Richtmyer-Meshkov instability
,” Phys. Plasmas
4
, 1028
–1038
(1997
).15.
C.
Matsuoka
, K.
Nishihara
, and Y.
Fukuda
, “Nonlinear evolution of an interface in the Richtmyer-Meshkov instability
,” Phys. Rev. E
68
, 036301
(2003
).16.
J. G.
Wouchuk
, C. H. R.
de Lira
, and A. L.
Velikovich
, “Analytical linear theory for the interaction of a planar shock wave with an isotropic turbulent vorticity field
,” Phys. Rev. E.
79
, 066315
(2009
).17.
F.
Cobos-Campos
and J. G.
Wouchuk
, “Analytic solution for the zero-order postshock profiles when an incident planar shock hits a planar contact surface
,” Phys. Rev. E
100
, 033107
(2019
).18.
J. G.
Wouchuk
, “Growth rate of the linear Richtmyer-Meshkov instability when a shock is reflected
,” Phys. Rev. E
63
, 056303
(2001
).19.
G.
Fraley
, “Rayleigh-Taylor stability for a normal shock wave-density discontinuity interaction
,” Phys. Fluids
29
, 376
–386
(1986
).20.
M.
Stanic
, R. F.
Stellingwerf
, J. T.
Cassibry
, and S. I.
Abarzhi
, “Scale coupling in Richtmyer-Meshkov flows induced by strong shocks
,” Phys. Plasmas
19
, 082706
(2012
).21.
M.
Stanic
, R. F.
Stellingwerf
, J. T.
Cassibry
, and S. I.
Abarzhi
, “Non-uniform volumetric structures in Richtmyer-Meshkov flows
,” Phys. Fluids
25
, 106107
(2013
).22.
M.
Vandenboomgaerde
, D.
Souffland
, C.
Mariani
, L.
Biamino
, G.
Jourdan
, and L.
Houas
, “An experimental and numerical investigation of the dependency on the initial conditions of the Richtmyer-Meshkov instability
,” Phys. Fluids
26
, 024109
(2014
).23.
Z.
Dell
, R. F.
Stellingwerf
, and S. I.
Abarzhi
, “Effect of initial perturbation amplitude on Richtmyer-Meshkov flows induced by strong shocks
,” Phys. Plasmas
22
, 092711
(2015
).24.
F.
Cobos-Campos
and J. G.
Wouchuk
, “Analytical scalings of the linear Richtmyer-Meshkov instability when a shock is reflected
,” Phys. Rev. E
93
, 053111
(2016
).25.
J. G.
Wouchuk
and K.
Nishihara
, “Linear perturbation growth at a shocked interface
,” Phys. Plasmas
3
, 3761
–3776
(1996
).26.
F.
Cobos-Campos
and J. G.
Wouchuk
, “Analytical scalings of the linear Richtmyer-Meshkov instability when a rarefaction is reflected
,” Phys. Rev. E
96
, 013102
(2017
).27.
28.
C.
Matsuoka
and K.
Nishihara
, “Vortex core dynamics and singularity formations in incompressible Richtmyer-Meshkov instability
,” Phys. Rev. E
73
, 026304
(2006
).29.
C.
Matsuoka
and K.
Nishihara
, “Fully nonlinear evolution of a cylindrical vortex sheet in incompressible Richtmyer-Meshkov instability
,” Phys. Rev. E
73
, 055304(R)
(2006
).30.
C.
Matsuoka
and K.
Nishihara
, “Analytical and numerical study on a vortex sheet in incompressible Richtmyer-Meshkov instability in cylindrical geometry
,” Phys. Rev. E
74
, 066303
(2006
).31.
C.
Matsuoka
and K.
Nishihara
, “Nonlinear interaction between bulk point vortices and an unstable interface with non-uniform velocity shear such as Richtmyer-Meshkov instability
,” Phys. Plasmas
27
, 052305
(2020
).32.
R.
Krasny
, “A study of singularity formation in a vortex sheet by the point vortex approximation
,” J. Fluid Mech.
167
, 65
–93
(1986
).33.
R.
Krasny
, “Computation of vortex sheet roll-up in the Trefftz plane
,” J. Fluid Mech.
184
, 123
–155
(1987
).34.
G.
Baker
, D. I.
Meiron
, and S. A.
Orszag
, “Generalized vortex methods for free surface flow problems
,” J. Fluid Mech.
123
, 477
–501
(1982
).35.
S.-I.
Sohn
, D.
Yoon
, and W.
Hwang
, “Long-time simulations of the Kelvin-Helmholtz instability using an adaptive vortex method
,” Phys. Rev. E
82
, 046711
(2010
).36.
S.-I.
Sohn
, “Late-time vortex dynamics of Rayleigh-Taylor instability
,” J. Phys. Soc. Jpn.
80
, 084401
(2011
).37.
C.
Matsuoka
, “Vortex sheet motion in incompressible Richtmyer-Meshkov and Rayleigh-Taylor instabilities with surface tension
,” Phys. Fluids
21
, 092107
(2009
).38.
C.
Matsuoka
, K.
Nishihara
, and T.
Sano
, “Nonlinear interfacial motion in magnetohydrodynamic flows
,” High Energy Density Phys.
31
, 19
–23
(2019
).39.
G.-H.
Cottet
and P. D.
Koumoutsakos
, Vortex Methods: Theory and Practice
(Cambridge University Press
, Cambridge
, 2000
).40.
G.
Birkhoff
, “Helmholtz and Taylor instability
,” Proc. Symp. Appl. Math. Soc.
13
, 55
–76
(1962
).41.
N.
Rott
, “Diffraction of a weak shock with vortex generation
,” J. Fluid Mech.
1
, 111
–128
(1956
).42.
43.
M. J.
Shelley
, “A study of singularity formation in vortex-sheet motion by a spectrally accurate vortex method
,” J. Fluid Mech.
244
, 493
–526
(1992
).44.
C.
Matsuoka
, “Renormalization group approach to interfacial motion in incompressible Richtmyer-Meshkov instability
,” Phys. Rev. E
82
, 036320
(2010
).45.
D. W.
Moore
, “The spontaneous appearance of a singularity in the shape of an evolving vortex sheet
,” Proc. R. Soc. A
365
, 105
–119
(1979
).46.
A. L.
Velikovich
, “Analytic theory of Richtmyer-Meshkov instability for the case of reflected rarefaction wave
,” Phys. Fluids
8
, 1666
–1679
(1996
).47.
N.
Zabusky
, “Vortex paradigm for accelerated inhomogeneous flows: Visiometrics for the Rayleigh-Taylor and Richtmyer-Meshkov environments
,” Annu. Rev. Fluid Mech.
31
, 495
–536
(1999
).48.
A. L.
Velikovich
, J. P.
Dahlburg
, A. J.
Schmitt
, J. H.
Gardner
, and L.
Phillips
, “Richtmyer-Meshkov-like instabilities and early-time perturbation growth in laser targets and z-pinch loads
,” Phys. Plasmas
7
, 1662
–1671
(2000
).49.
Y.
Yang
, Q.
Zhang
, and D. H.
Sharp
, “Small amplitude theory of Richtmyer-Meshkov instability
,” Phys. Fluids
6
, 1856
–1873
(1994
).50.
51.
D. I.
Pullin
, “Numerical studies of surface-tension effects in nonlinear Kelvin-Helmholtz and Rayleigh-Taylor instability
,” J. Fluid Mech.
119
, 507
–532
(1982
).52.
R. M.
Kerr
, “Simulation of Rayleigh-Taylor flows using vortex blobs
,” J. Comput. Phys.
76
, 48
–84
(1988
).53.
J. G.
Wouchuk
, “Growth rate of the Richtmyer-Meshkov instability when a rarefaction is reflected
,” Phys. Plasmas
8
, 2890
–2907
(2001
).54.
J. G.
Wouchuk
and F.
Cobos-Campos
, “Kinetic energy of the rotational flow behind an isolated rippled shock wave
,” Phys. Scr.
93
, 094003
(2018
).55.
A.
Sidi
and M.
Israeli
, “Quadrature methods for periodic singular and weakly singular Fredholm integral equations
,” J. Sci. Comput.
3
, 201
–231
(1988
).56.
G.
Baker
and A.
Nachbin
, “Stable methods for vortex sheet motion in the presence of surface tension
,” SIAM J. Sci. Comput.
19
, 1737
–1766
(1998
).57.
S. I.
Abarzhi
, K.
Nishihara
, and J.
Glimm
, “Rayleigh-Taylor and Richtmyer-Meshkov instabilities for fluids with a finite density ratio
,” Phys. Lett. A
317
, 470
–476
(2003
).58.
M.
Herrmann
, P.
Moin
, and S.
Abarzhi
, “Nonlinear evolution of the Richtmyer-Meshkov instability
,” J. Fluid Mech.
612
, 311
–338
(2008
).59.
J. W.
Jacobs
and J. M.
Sheeley
, “Experimental study of incompressible Richtmyer-Meshkov instability
,” Phys. Fluids
8
, 405
–415
(1996
).© 2020 Author(s).
2020
Author(s)
You do not currently have access to this content.
