We theoretically study the effect of pressure fluctuations on the Richtmyer-Meshkov (RM) unstable interface in approximation of ideal incompressible immiscible fluids and two-dimensional flow. Pressure fluctuations are treated as an effective acceleration directed from the heavy to light fluid with inverse square time dependence. The group theory approach is applied to analyze large-scale coherent dynamics, solve the complete set of the governing equations, and find regular asymptotic solutions describing RM bubbles. A strong effect is found, for the first time to our knowledge, of pressure fluctuations on the interface morphology and dynamics. In the linear regime, a nearly flat bubble gets more curved, and its velocity increases for strong pressure fluctuations and decreases otherwise. In the nonlinear regime, solutions form a one-parameter family parameterized by the bubble front curvature. For the fastest stable solution in the family, the RM bubble is curved for strong pressure fluctuations and is flattened otherwise. The flow is characterized by the intense motion of the fluids in the vicinity of the interface, effectively no motion away from the interface, and presence of shear at the interface leading to formation of smaller scale vortical structures. Our theoretical results agree with and explain existing experiments and simulations and identify new qualitative and quantitative characteristics to evaluate the strength of pressure fluctuations in experiments and simulations.
References
1.
Abarzhi
, S. I.
, “The stationary periodic flows in Rayleigh–Taylor instability: Solutions multitude and its dimension
,” Phys. Scr., T
66
, 238
–242
(1996
).2.
Abarzhi
, S. I.
, “Stable steady flows in Rayleigh–Taylor instability
,” Phys. Rev. Lett.
81
, 337
–340
(1998
).3.
Abarzhi
, S. I.
, “Length scale for bubble problem in Rayleigh-Taylor instability
,” Phys. Fluids
11
, 940
(1999
).4.
Abarzhi
, S. I.
, “Regular and singular late-time asymptotes of potential motion of fluid with a free-boundary
,” Phys. Fluids
12
, 3112
–3120
(2000
).5.
Abarzhi
, S. I.
, “Low-symmetric bubbles in Rayleigh-Taylor instability
,” Phys. Fluids
13
, 2181
–2189
(2001
).6.
Abarzhi
, S. I.
, “A new type of the evolution of the bubble front in the Richtmyer–Meshkov instability
,” Phys. Lett. A
294
, 95
–100
(2002
).7.
Abarzhi
, S. I.
, “Review on nonlinear coherent dynamics of unstable fluid interface: Conservation laws and group theory
,” Phys. Scr., T
132
, 297681
(2008a
).8.
Abarzhi
, S. I.
, “Coherent structures and pattern formation in the Rayleigh-Taylor turbulent mixing
,” Phys. Scr.
78
, 015401
(2008b
).9.
Abarzhi
, S. I.
, “Review of theoretical modeling approaches of Rayleigh-Taylor instabilities and turbulent mixing
,” Philos. Trans. R. Soc., A
368
, 1809
(2010
).10.
Abarzhi
, S. I.
, Nishihara
, K.
, and Glimm
, J.
, “Rayleigh–Taylor and Richtmyer–Meshkov instabilities for fluids with a finite density ratio
,” Phys. Lett. A
317
, 470
(2003
).11.
Abarzhi
, S. I.
, Nishihara
, K.
, and Rosner
, R.
, “A multi-scale character of the large-scale coherent dynamics in the Rayleigh–Taylor instability
,” Phys. Rev. E
73
, 036310
(2006
).12.
Alon
, U.
, Hecht
, J.
, Offer
, D.
, and Shvarts
, D.
, “Power-laws and similarity of Rayleigh–Taylor and Richtmyer–Meshkov mixing fronts at all density ratios
,” Phys. Rev. Lett.
74
, 534
(1995
).13.
Anisimov
, S. I.
, Drake
, R. P.
, Gauthier
, S.
, Meshkov
, E. E.
, and Abarzhi
, S. I.
, “What is certain and what is not so certain in our knowledge of Rayleigh-Taylor mixing?
,” Philos. Trans. R. Soc., A
371
, 20130266
(2013
).14.
Arnett
, D.
, Supernovae and Nucleosynthesis: An Investigation of the History of Matter from the Big Bang to the Present
(Princeton University Press
, 1996
), ISBN: 9780691011479.15.
Barenblatt
, G. I.
, Similarity, Self-Similarity and Intermediate Asymptotics
(Consultants Bureau
, New York
, 1979
).16.
Chandrasekhar
, S.
, Hydrodynamic and Hydromagnetic Stability
, 3rd ed. (Dover
, New York
, 1981
), pp. 428
–477
.17.
Davies
, R. M.
and Taylor
, G. I.
, “The mechanics of large bubbles rising through extended liquids and through liquids in tubes
,” Proc. R. Soc. London, Ser. A
200
, 375
(1950
).18.
Dimonte
, G.
, Youngs
, D. L.
, Dimits
, A.
, Weber
, S.
, Marinak
, M.
, Wunsch
, S.
, Garasi
, C.
, Robinson
, A.
, Andrews
, M. J.
, Ramaprabhu
, P.
, Calder
, A. C.
, Fryxell
, B.
, Biello
, J.
, Dursi
, L.
, MacNeice
, P.
, Olson
, K.
, Ricker
, P.
, Rosner
, R.
, Timmes
, F.
, Tufo
, H.
, Young
, Y. N.
, and Zingale
, M.
, “A comparative study of the turbulent Rayleigh-Taylor instability using high-resolution three-dimensional numerical simulations: The Alpha-Group collaboration
,” Phys. Fluids
16
, 1668
–1693
(2004
).19.
Dell
, Z.
, Stellingwerf
, R. F.
, and Abarzhi
, S. I.
, “Effect of initial perturbation amplitude on Richtmyer-Meshkov flows induced by strong shocks
,” Phys. Plasmas
22
, 092711
(2015
).20.
Fermi
, E.
and von Neumann
, J.
, Taylor Instability of an Incompressible Liquid
In Fermi E 1962 Collected papers (Chicago: The University of Chicago Press
), Vol 2, p. 816
.21.
Garabedian
, P. R.
, “On steady-state bubbles generated by Taylor instability
,” Proc. R. Soc. A
241
, 423
(1957
).22.
Glendinning
, S. G.
, Bolstad
, J.
, Braun
, D. G.
, Edwards
, M. J.
, Hsing
, W. W.
, Lasinski
, B. F.
, Louis
, H.
, Miles
, A.
, Moreno
, J.
, Peyser
, T. A.
, et al., “Effect of shock proximity on Richtmyer-Meshkov growth
,” Phys. Plasmas
10
, 1931
(2003
).23.
Goncharov
, V. N.
, “Analytical model of nonlinear, single-mode, classical Rayleigh-Taylor instability
,” Phys. Rev. Lett.
88
, 134502
(2002
).24.
25.
Haan
, S. W.
, Lindl
, J. D.
, Callahan
, D. A.
, Clark
, D. S.
, Salmonson
, J. D.
, Hammel
, B. A.
, Atherton
, L. J.
, Cook
, R. C.
, Edwards
, M. J.
, Glenzer
, S.
, et al., “Point design targets, specifications, and requirements for the 2010 ignition campaign on the National Ignition Facility
,” Phys. Plasmas
18
, 051001
(2011
).26.
Hahn
, T.
, International Tables for Crystallography
, Vol. A: Space Group Symmetry Series, 5th revision 2002, Corr. 2nd printing, 2005 (Springer
, 2002
).27.
Hecht
, J.
, Alon
, U.
, and Shvarts
, D.
, “Potential flow models of Rayleigh-Taylor and Richtmyer-Meshkov bubble fronts
,” Phys. Fluids
6
, 4019
(1994
).28.
Herrmann
, M.
, Moin
, P.
, and Abarzhi
, S. I.
, “Nonlinear evolution of the Richtmyer–Meshkov instability
,” J. Fluid Mech.
612
, 311
–338
(2008
).29.
Holmes
, R. L.
, Dimonte
, G.
, Fryxell
, B.
, Gittings
, M. L.
, Grove
, J. W.
, Schneider
, M.
, Sharp
, D. H.
, Velikovich
, A. L.
, Weaver
, R. P.
, and Zhang
, Q.
, “Richtmyer–Meshkov instability growth: Experiment, simulation and theory
,” J. Fluid Mech.
389
, 55
–79
(1999
).30.
Hurricane
, O. A.
, Callahan
, D. A.
, Casey
, D. T.
, Dewald
, E. L.
, Dittrich
, T. R.
, Döppner
, T.
, Barrios Garcia
, M. A.
, Hinkel
, D. E.
, Berzak Hopkins
, L. F.
, Kervin
, P.
, et al., “The high-foot implosion campaign on the National Ignition Facility
,” Phys. Plasmas
21
, 056314
(2014
).31.
Inogamov
, N. A.
, “Higher order Fourier approximations and exact algebraic solutions in the theory of hydrodynamic Rayleigh–Taylor instability
,” JETP Lett.
55
, 521
–525
(1992
); see http://www.jetpletters.ac.ru/ps/1276/article_19296.shtml.32.
Inogamov
, N. A.
and Abarzhi
, S. I.
, “Dynamics of fluid surface in multi-dimensions
,” Physica D
87
, 339
(1995
).33.
Jacobs
, J. W.
and Krivets
, V. V.
, “Experiments on the late-time development of single-mode Richtmyer–Meshkov instability
,” Phys. Fluids
17
, 034105
(2005
).34.
Johnson
,B. M.
and Schilling
,O.
, “Reynolds-averaged Navier-Stokes model predictions of linear instability. i. Buoyancy- and shear-driven flows
,” J. Turbul.
12
(36
), 1
(2011
);Johnson
, B. M.
and Schilling
, O.
, “Reynolds-averaged Navier-Stokes model predictions of linear instability. ii. Shock-driven flows,” J. Turbul.
12
(37
), 1
(2011
).35.
Kovalev
, O. V.
, Irreducible and Induced Representations and Co-Representations of Fedorov Groups
(Nauka
, Moscow
, 1986
).36.
Landau
,L. D.
and Lifshitz
,E. M.
, Fluid Mechanics
, Course of Theoretical Physics Vol. VI (Pergamon Press
, New York
, 1987
);Landau
,L. D.
andLifshitz
,E. M.
, Statistical Physics
, Course of Theoretical Physics Vol. V (Pergamon Press
, New York
, 1987
);Landau
, L. D.
and Lifshitz
, E. M.
, Mechanics
, Course of Theoretical Physics Vol. I (Pergamon Press
, New York
, 1987
).37.
Layzer
, D.
, “On the instability of superposed fluids in a gravitational field
,” Astrophys. J
122
, 1
(1955
).38.
Lindl
, J. D.
, Amendt
, P. A.
, Berger
, R. L.
, Glendinning
, S. G.
, Glenzer
, S. H.
, Haan
, S. W.
, Kauffman
, R. L.
, Landen
, O. L.
, and Suter
, L. J.
, “The physics basis for ignition using indirect-drive targets on the National Ignition Facility
,” Phys. Plasmas
11
, 339
(2004
).39.
Matsuoka
, C.
and Nishihara
, K.
, “Vortex core dynamics and singularity formations in incompressible Richtmyer-Meshkov instability
,” Phys. Rev. E
73
, 026304
(2006
).40.
Meshkov
, E. E.
, “Instability of the interface of two gases accelerated by a shock wave
,” Fluid Dyn.
4
, 101
(1969
).41.
Meshkov
, E. E.
, Studies of Hydrodynamic Instabilities in Laboratory Experiments
(FGYC-VNIIEF
, Sarov
, 2006
), ISBN: 5-9515-0069-9 (in Russian).42.
Mikaelian
, K. O.
, “Limitations and failures of the Layzer model for hydrodynamic instabilities
,” Phys. Rev. E
78
, 015303
(2008
).43.
Mikaelian
, K. O.
, “Reshock, rarefaction and the generalized Layzer model for hydrodynamic instabilities
,” Phys. Fluids
21
, 024103
(2009
).44.
Motl
, B.
, Oakley
, J.
, Ranjan
, D.
, Weber
, C.
, Anderson
, M.
, and Bonazza
, R.
, “Experimental validation of a Richtmyer–Meshkov scaling law over large density ratio and shock strength ranges
,” Phys. Fluids
21
, 126102
(2009
).45.
Nishihara
, K.
, Wouchuk
, J. G.
, Matsuoka
, C.
, Ishizaki
, R.
, and Zhakhovsky
, V. V.
, “Richtmyer-Mehskov instability: Theory of linear and nonlinear evolution
,” Philos. Trans. R. Soc., A
368
, 1769
(2010
).60.
Rayleigh Lord Strutt
, J. W.
, “Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density
,” Proc. London Math. Society
14
, 170
–177 (1883
).46.
Remington
, B. A.
, Drake
, R. P.
, and Ryutov
, D. D.
, “Experimental astrophysics with high power lasers and Z-pinches
,” Rev. Mod. Phys.
78
, 755
(2006
).47.
Richtmyer
, R. D.
, “Taylor instability in shock acceleration of compressible fluids
,” Pure Appl. Math.
13
, 297
(1960
).48.
Saffman
, P. G.
and Taylor
, G. I.
, “The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid
,” Proc. R. Soc. London, Ser. A
245
(1242
), 312
–329
(1958
).49.
Sedov
,L. I.
, “Propagation of strong shock waves
,” J. Appl. Math. Mech.
10
, 241
(1946
)50.
Sedov
, L. I.
, Similarity and Dimensional Methods in Mechanics
, 10th ed. (CRC Press
, 1993
).51.
Shubnikov
, A. V.
and Koptsik
, V. A.
, Symmetry in Science and Art
(Springer
, 1974
), ISBN: 13 978-1468420692, ISBN: 10 1468420690.52.
Stanyukovich
, K. P.
, Non-Steady Motion of Continuous Media
(Oxford University Press
, Oxford, UK
, 1960
).53.
Stanic
, M.
, Stellingwerf
, R. F.
, Cassibry
, J. T.
, and Abarzhi
, S. I.
, “Scale coupling in Richtmyer-Meshkov flows induced by strong shocks
,” Phys. Plasmas
19
, 082706
(2012
).54.
Stanic
, M.
, Stellingwerf
, R. F.
, Cassibry
, J. T.
, McFarland
, J.
, Ranjan
, D.
, Bonazza
, R.
, Greenough
, J. A.
, and Abarzhi
, S. I.
, “Non-uniform volumetric structures in Richtmyer-Meshkov flows
,” Phys. Fluids
25
, 106107
(2013
).55.
Swisher
, N.
, Kuranz
, C.
, Arnett
, W. D.
, Hurricane
, O.
, Robey
, H.
, Remington
, B. A.
, and Abarzhi
, S. I.
, “Rayleigh-Taylor mixing in supernova experiments
,” Phys. Plasmas
22
, 102707
(2015
).56.
Taylor
, G. I.
, “The formation of a blast wave by a very intense explosion. II. The atomic explosion of 1945
,” Philos. Trans. R. Soc., A
201
(1065
), 175
–186
(1950
).57.
Velikovich
, A. L.
, Herrmann
, M.
, and Abarzhi
, S. I.
, “Perturbation theory and numerical modeling of weakly and moderately nonlinear dynamics of the classical Richtmyer-Meshkov instability
,” J. Fluid Mech.
751
, 432
–479
(2014
).58.
Wouchuk
,J. G.
, “Growth rate of the linear Richtmyer–Meshkov instability when a shock is reflected
,” Phys. Rev. E
63
, 056303
(2001
);Wouchuk
, J. G.
, “Growth rate of the Richtmyer–Meshkov instability when a rarefaction is reflected
,” Phys. Plasmas
8
, 2890
–2907
(2001
).59.
Zeldovich
, Y. B.
and Raizer
, Y. P.
, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena
, 2nd Engl. ed. (Dover
, New York
, 2002
).© 2016 Author(s).
2016
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