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Many-dimensional periodicity and turbulence of Rayleigh–Taylor and Richtmyer–Meshkov instabilities

Published online by Cambridge University Press:  09 March 2009

N.A. Inogamov
N.A. Inogamov
Affiliation:
Landau Institute for Theoretical Physics, Russian Academy of Sciences GSP-1, 117940, Moscow, Kosygin Street 2, Russian Federation
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Abstract

The work begins with a presentation of results concerning periodic Rayleigh-Taylor (RT)and Richtmyer–Meshkov (RM) dynamics. The periodic RT flows are considered in 2D and 3D cases. The periodic RM flows are considered in a 2D case. Nonstationary and steadystate statements of RT and RM problems are given. Nonstationary statements of RT and RM problems are given in the2D case. Steady-state statements of RT and RM problems are considered in the 2D and 3D cases (RT) and in the 2D case (RM). Questions about turbulence are given in the second part of the work. In the nonstationary statement, the beginning of a transition from an equilibrium close to hydrostatics and behavior of a dynamic system when it is close to the steady-state solution are considered. To study the smooth evolution of the dynamic system in the RT case (at last during an initial of order of unity t ≃ 1 stage) we have to cut of f an inverse turbulent cascade incoming from the small scales (inverse means that the typical k decreases in time during the cascade). The cascade starts from some small scale k ≫ 1 (here g = 1) and appears in the scales k ≃ 1 after time t ≃ 1 after start. We have to consider exponentially smoothed initial data to cut it off. This means that amplitudes of initial spectrum of perturbations have to be exponentially decreased in the direction of high k. The steadystates have been studied in two ways. In both of them the spectral decomposition of a velocity field and an elevation of the contact surface has been used. In one of them, corresponding to the nonstationary situation, the ordinary differential system for time dependent spectral components has been integrated. The other solutions of the algebraic equations for constant components have been studied. These results have been compared with a computer simulation that has been done by large particles and artificial compressibility methods. High-order nonlinear terms corresponding to the interaction of a wide spectrum of harmonics are considered in both ways. The spatial structure of the many dimensional periodic solutions is studied. It is known that there are two important features in the flows under consideration: the bubble envelope and the jet “phase.” Spatial structures of the 2D and 3D flows are compared in the report. It is shown that the apices of the bubbles are isolated points and the bubble envelopes in both flows are qualitatively similar; the difference has a quantitative character. On the contrary, the jet “phases” differ qualitatively in the 2D and 3D cases. In the 2D case in the plane picture, the jets are separated from each other. In this picture they are isolated fingers. In the 3D case, near the bubble envelope after some time of development following after cos kx cos qy type initial perturbation, they are not the isolated fingers. They form a structure of the intersecting walls. We see, thus, that the 2D and 3D cases are topologically different. Most powerful ejections of dense fluid are in the points where these wall jets intersect each other.

Type
Research Article
Information
Laser and Particle Beams , Volume 15 , Issue 1 , March 1997 , pp. 53 - 64
Copyright
Copyright © Cambridge University Press 1997

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References

REFERENCES

Abarzhi, S.I. & Inogamov, N.A. 1995 JETP 80, 132.Google Scholar
Anisimov, S.I. et al. 1983 In Shock Waves, Explosions and Detonation, Bowen, J.R., Leyer, J.-C., and Soloukhin, R.I., eds. (AIAA, Washington, DC), p. 218.Google Scholar
Anisimov, S.I. et al. 1993 Russ. J. Comp. Mech. 1, 5.Google Scholar
Anisimov, S.I. & Inogamov, N.A. 1993 Preprint of Landau Inst. for Theor. Phys. (ITF), (Russ. Acad.Sci., Chernogolovka).Google Scholar
Anuchina, N.N. et al. 1988 Voprosi Atomnoi Nauki i Tekhniki 1, 7.Google Scholar
Anuchina, N.N. et al. 1990 Matem. Modelir. 2, 3.Google Scholar
Arnett, D. et al. 1989 ApJ. (Letters) 341, L63.CrossRefGoogle Scholar
Arons, J. & Lea, S.M. 1976 ApJ. 207, 914.CrossRefGoogle Scholar
Atzeni, S. 1990 Las. Part. Beams 8, 227.Google Scholar
Benjamin, R.F. et al. 1992 In Proc. 18th ISSWST, Takayama, K., ed. (Springer Verlag, New York).Google Scholar
Besnard, D. et al. 1989 Phys. D 37D, 229.Google Scholar
Birkhoff, G. & Carter, D. 1957 J. Math. Mech. 6, 769.Google Scholar
Davidov, YU.M. & Panteleev, M.S. 1981 Zhurnal PrikladnoiMekhaniki i Tekhnicheskoi Fiziki 1,117.Google Scholar
Davies, R.M. & Taylor, G.I. 1950 Proc. Roy. Soc. A 200A, 375.Google Scholar
Demchenko, V.V. & Oparin, A.M. 1993 In Proc. 4th Intern. Workshop Phys. Compr. Turb. Mixing, Linden, P.F., Youngs, D.L., and Dalziel, S.B., eds. (Cambridge University Press, Cambridge, UK), p. 207.Google Scholar
Den, M. et al. 1990 KEK Progress Reports 89, 173.Google Scholar
Dibai, E. 1958 Astronomicheskii Zhurnal 35, 469.Google Scholar
Dumitrescu, D.T. 1943 Z. Angew. Math. Mech. 23, 139.CrossRefGoogle Scholar
Ebisuzaki, T. et al. 1989 ApJ. (Letters) 344, L65.CrossRefGoogle Scholar
Elsner, R.F. 1976 Ph.D. thesis. Univ. of Illinois at Urbana-Champaign.Google Scholar
Garabedian, P. 1957 Proc. Roy. Soc. London 241A, 423.Google Scholar
Grebenev, S.A. & Sunyaev, R.A. 1987 Sov. Astron. Lett. 13, 397. (Pis'ma Astronomich. Zhurnal 13, 945).Google Scholar
Gull, S.F. 1975 Monthly Not. Roy. Astron. Soc. 171, 263.CrossRefGoogle Scholar
Haan, S.W. 1991 Phys. Fluids B 3, 2349.CrossRefGoogle Scholar
Hachisu, I. et al. 1990 KEK Progress Reports 89, 185.Google Scholar
Harlow, F.H. & Welch, J.E. 1966 Phys. Fluids 9, 842.CrossRefGoogle Scholar
Hecht, J. et al. 1994 Phys. Fluids 6, 4019.CrossRefGoogle Scholar
Inogamov, N.A. 1978 Sov. Tech. Phys. Lett. 4, 299.Google Scholar
Inogamov, N.A. 1990 In Problems of Dyn. and Stab, of Plasma Lokshin, G.R. et al. eds. (Mosc. Inst. of Phys. Technol. (MFTI), Moscow), p. 115.Google Scholar
Inogamov, N.A. 1992 JETP Lett. 55, 521.Google Scholar
Inogamov, N.A. 1994 Astronomy Lett. 20, 651.Google Scholar
Inogamov, N.A. 1995a JETP Lett. 61, 27.Google Scholar
Inogamov, N.A. 1995b JETP 80, 890.Google Scholar
Inogamov, N.A. & Abarzhi, S.I. 1995 Physica D 87, 339.CrossRefGoogle Scholar
Inogamov, N.A. & Chekhlov, A.V. 1992 Prepr. Landau Inst. for Theor. Phys. (ITF) (Russ. Acad. Sci., Chernogolovka).Google Scholar
Inogamov, N.A. & Chekhlov, A.V. 1993a Dokladi Akademii Nauk 328, 311Google Scholar
Inogamov, N.A. & Chekhlov, A.V. 1993b In Proc. 4th Intern. Workshop Phys. Compr. Turb. Mixing, Linden, P.F., Youngs, D.L., and Dalziel, S.B., eds. (Cambridge University Press, Cambridge, UK), p. 50.Google Scholar
Inogamov, N.A. et al. 1991 In Proc. 3rd Intern. Workshop of Phys. Compr. Turb. Mixing Dautray, R. ed. (Royaumont, France), p. 409 and p. 423.Google Scholar
Kahn, F.D. 1982 In IAU Symp. No. 103 (D. Reidel Publ. Comp.), p. 305.Google Scholar
Klein, R.I. et al. 1993 In Proc. 4th Intern. Workshop Phys. Compr. Turb. Mixing, Linden, P.F., Youngs, D.L., and Dalziel, S.B., eds. (Cambridge University Press, Cambridge, UK), p. 554.Google Scholar
Kucherenko, Yu.A. et al. 1988 Teoreticheskaya i Prikladnaya Fizika 1, 13.Google Scholar
Kull, H.J. 1983 Phys. Rev. Lett. 51, 1434.Google Scholar
Kull, H.J. 1986 Phys. Rev. A 33, 1957.CrossRefGoogle Scholar
Kull, H.J. 1991 Phys. Rep. 206, 197.CrossRefGoogle Scholar
Layzer, D. 1955 ApJ. 122, 1.Google Scholar
Li, X.L. et al. 1993 In Proc. 4th Intern. Workshop Phys. Compr. Turb. Mixing, Linden, P.F., Youngs, D.L., and Dalziel, S.B., eds. (Cambridge University Press, Cambridge, UK), p. 94.Google Scholar
Linden, P. 1979 Geophys. Astroph. Fluid Dyn. 3, 13.Google Scholar
McKee, C.F. 1974 ApJ. 188, 335.CrossRefGoogle Scholar
Meshkov, E.G. 1969 Mekhanika Zhidkosti Gaza 5, 151.Google Scholar
Mikaelian, K.O. 1993 In Proc. 4th Intern. Workshop Phys. Compr. Turb. Mixing, Linden, P.F., Youngs, D.L., and Dalziel, S.B., eds. (Cambridge University Press, Cambridge, UK), p. 240.Google Scholar
Muller, E. et al. 1989 Astron. Astrophys. 220, 167.Google Scholar
Neuvazhayev, V.E. et al. 1991 In Proc. 3rd Intern. Workshop Phys. Compr. Turb. Mixing Dautray, R. ed. (Royaumont, France), p. 477.Google Scholar
Neuvazhayev, V.E. 1994 Phys. Rev. E 50, 1394.CrossRefGoogle Scholar
Ofer, D. et al. 1993 In Proc. 4th Intern. Workshop Phys. Compr. Turb. Mixing, Linden, P.F., Youngs, D.L., and Dalziel, S.B., eds. (Cambridge University Press, Cambridge, UK), p. 119.Google Scholar
Onufriev, A.T. 1967 Zhurnal Prikladnoi Mekhaniki Tekhnicheskoi Fiziki 2, 3.Google Scholar
Polionov, A.V. 1991 In Proc. 3rd Intern. Workshop Phys. Compr. Turb. Mixing Dautray, R. ed. (Royaumont, France), p. 495.Google Scholar
Redondo, J. 1987 J. Hazardous Materials 16, 381.CrossRefGoogle Scholar
Remington, B.A. et al. 1991 Phys. Rev. Lett. 67, 3259.CrossRefGoogle Scholar
Richtmyer, R.D. 1960 Commun. Pure Appl. Math. 13, 297.CrossRefGoogle Scholar
Rodriguez, G. et al. 1993 In Proc. 4th Intern. Workshop Phys. Compr. Turb. Mixing, Linden, P.F., Youngs, D.L., and Dalziel, S.B., eds. (Cambridge University Press, Cambridge, UK), p. 260.Google Scholar
Rosanov, V.B. et al. 1990 Prepr. FIAN, No. 56, Moscow.Google Scholar
Sakagami, H. & Nishihara, K. 1990 Phys. Rev. Lett. 65, 432.CrossRefGoogle Scholar
Schwartz, L.W. & Fenton, J.D. 1982 Ann. Rev. Fluid Mech. 14, 39.Google Scholar
Sharp, D.H. 1984 Phys. D 12D, 3.Google Scholar
Takabe, H. et al. 1985 Phys. Fluids 28, 3676.Google Scholar
Town, R.P.J. & Bell, A.R. 1991 Phys. Rev. Lett. 67, 1863.CrossRefGoogle Scholar
Vanden-Broeck, J.M. 1984 Phys. Fluids 27, 1090.CrossRefGoogle Scholar
Voropayev, S.I. et al. 1993 Phys. Fluids A 5, 2461.CrossRefGoogle Scholar
Yabe, T. et al. 1991 Phys. Rev. A 44 (4).CrossRefGoogle Scholar
Youngs, D. 1993 In Proc. 4th Intern. Workshop Phys. Compr. Turb. Mixing, Linden, P.F., Youngs, D.L., and Dalziel, S.B., eds. (Cambridge University Press, Cambridge, UK), p. 167.Google Scholar
Zaytsev, S. et al. 1991 In Proc. 3rd Intern. Workshop Phys. Compr. Turb. Mixing Dautray, R. ed. (Royaumont, France), p. 57.Google Scholar