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Nonlinear evolution of the Richtmyer–Meshkov instability

Published online by Cambridge University Press:  10 October 2008

MARCUS HERRMANN ,
PARVIZ MOIN  and
SNEZHANA I. ABARZHI
MARCUS HERRMANN
Affiliation:
Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZ, USA
PARVIZ MOIN
Affiliation:
Center for Turbulence Research, Stanford, CA, USA
SNEZHANA I. ABARZHI*
Affiliation:
The University of Chicago and Illinois Institute of Technology, Chicago, IL, USA
*
†Author to whom correspondence should be addressed: snezha@stanford.edu
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Abstract

We report analytical and numerical results describing the dynamics of the two-dimensional coherent structure of bubbles and spikes in the Richtmyer–Meshkov instability for fluids with a finite density ratio. The theory accounts for the non-local properties of the interface evolution, and the simulations treat the interface as a discontinuity. Good agreement between the analytical and numerical results is achieved. To quantify accurately the interface dynamics in the simulations, new diagnostics and scalings are suggested. The velocity at which the interface would move if it were ideally planar is used to set the flow time scale as well as the reference point for the bubble (spike) position. The data sampling has high temporal resolution and captures the velocity oscillations caused by sound waves. The bubble velocity and curvature are both monitored, and the bubble curvature is shown to be the relevant diagnostic parameter. According to the results obtained, in the nonlinear regime of the Richtmyer–Meshkov instability the bubbles flatten and decelerate, and the flattening of the bubble front indicates the multiscale character of the coherent dynamics.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 612 , 10 October 2008 , pp. 311 - 338
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Abarzhi, S. I. 2001 Nonlinear asymptotic solutions for RT and RM problems for fluids with finite density contrast. In Proc. 8th Intl Workshop on the Physics of Compressible Turbulent Mixing, p. 104. Pasadena, CA, USA.Google Scholar
Abarzhi, S. I. 2002 Nonlinear evolution of unstable fluid interface. Phys. Rev. E 66, 036301.Google ScholarPubMed
Abarzhi, S. I., Nishihara, K. & Glimm, J. 2003 Rayleigh–Taylor and Richtmyer–Meshkov instabilities for fluids with a finite density ratio. Phys. Lett. A 317, 470–476.CrossRefGoogle Scholar
Aleshin, A. N., Ganalii, E. G. & Zaitsev, S. G. 1988 Study of nonlinear and transition stages of the development of the Richtmyer–Meshkov instability. Prisma V. Zh. Tekh. Fiziki 14, 1063–1067.Google Scholar
Aleshin, A. N., Lazareva, E. & Zaitsev, S. G. 1990 A study of linear, nonlinear, and transition stages of Richtmyer–Meshkov instability. Dokl. Akad. Nauk SSSR 310, 1105–1108.Google Scholar
Alon, U., Hecht, J., Offer, D. & Shvarts, D. 1995 Power laws and similarity of Rayleigh–Taylor and Richtmyer–Meshkov mixing fronts at all density ratios. Phys. Rev. Lett. 74, 534–537.CrossRefGoogle ScholarPubMed
Aref, H. & Tryggvason, G. 1989 Model of Rayleigh–Taylor instability. Phys. Rev. Lett. 62, 749–752.CrossRefGoogle ScholarPubMed
Benjamin, R., Besnard, D. & Haas, J. 1993 Shock and reshock of an unstable interface. LANL Rep. LA-UR 92-1185, Los Alamos National Laboratory.Google Scholar
Chebotareva, E. I., Aleshin, A. N., Zaytsev, S. G. & Sergeev, S. V. 1999 Investigation of interaction between reflected shocks and growing perturbation on an interface. Shock Waves 9, 81–86.CrossRefGoogle Scholar
Collins, B. D. & Jacobs, J. W. 2002 PLIF flow visualization and measurements of the Richtmyer–Meshkov instability. J. Fluid Mech. 464, 113–136.CrossRefGoogle Scholar
Dimonte, G. 2000 Spanwise homogeneous buoyancy-drag model for Rayleigh–Taylor mixing and experimental evaluation. Phys. Plasmas 7, 2255–2269.CrossRefGoogle Scholar
Fedkiw, R., Aslam, T., Merriman, B. & Osher, S. 1999 A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys. 152, 457–492.CrossRefGoogle Scholar
Frisch, U. 1995 Turbulence. Cambridge University Press.CrossRefGoogle Scholar
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., Remington, B. A., Robey, H. F., Turano, E. J., Verdon, C. P. & Zhou, Y. 2003 Effect of shock proximity on Richtmyer—Meshkov growth. Phys. Plasmas 10, 1931–1936.CrossRefGoogle Scholar
Gol'din, L. L., Igoshin, F. F., Kozel, S. M., Kolachevskij, N. N., Mazan'ko, I. P., Noginova, L. V., Radkevich, I. A., Rogozinskij, K. A. & Samarskij, Y. A. 1973 Rukovodstvo k Laboratornym Zanyatiyam po Fizike (Laboratories in General Physics), 2nd edn.Nauka.Google Scholar
Goncharov, V. N. 2002 Analytical model of nonlinear, single-mode, classical Rayleigh–Taylor instability at arbitrary Atwood numbers. Phys. Rev. Lett. 88, 134502.CrossRefGoogle ScholarPubMed
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: A unifying theory. J. Fluid Mech. 407, 27–56.CrossRefGoogle Scholar
Hastie, T., Tibshirani, R. & Friedman, J. 2001 The Elements of Statistical Learning. Springer.CrossRefGoogle Scholar
Holmes, R. L., Dimonte, G., Fryxell, B., Gittings, M. L., Grove, J. W., Schneider, M., Sharp, D. H., Velikovitch, A. L., Weaver, R. P. & Zhang, Q. 1999 Richtmyer–Meshkov instability growth: Experiment, simulation and theory. J. Fluid Mech. 389, 55–79.CrossRefGoogle Scholar
Jacobs, J. W. & Krivets, V. V. 2005 Experiments on the late-time development of single-mode Richtmyer—Meshkov instability. Phys. Fluids 17, (034105).CrossRefGoogle Scholar
Jacobs, J. & Sheeley, J. 1996 Experimental study of incompressible Richtmyer–Meshkov instability. Phys. Fluids 8, 405–415.CrossRefGoogle Scholar
Jiang, G.-S. & Peng, D. 2000 Weighted ENO schemes for Hamilton–Jacobi equations. SIAM J. Sci. Comput. 21, 2126–2143.CrossRefGoogle Scholar
Jones, M. A. & Jacobs, J. W. 1997 A membraneless experiment for the study of Richtmyer–Meshkov instability of a shock-accelerated gas interface. Phys. Fluids 9, 3078–3085.CrossRefGoogle Scholar
Korn, G. A. & Korn, T. M. 1968 Mathematical Handbook for Scientists and Engineers: Definitions, Theorems and Formulas for Reference and Review, 2nd edn.McGraw-Hill.Google Scholar
Kull, F. 1991 Theory of Rayleigh–Taylor instability. Phys. Rep. 206, 197.CrossRefGoogle Scholar
LeVeque, R. J. 1990 Numerical Methods for Conservation Laws. Birkhäuser.CrossRefGoogle Scholar
Matsuoka, C., Nishihara, K. & Fukuda, Y. 2003 Nonlinear evolution of an interface in the Richtmyer–Meshkov instability. Phys. Rev. E 67, (036301).Google ScholarPubMed
Meshkov, E. 1969 Instability of the interface of two gases accelerated by a shock wave. Sov. Fluid Dyn. 4, 101–104.CrossRefGoogle Scholar
Meyer, K. A. & Blewett, P. J. 1972 Numerical investigation of the stability of a shock-accelerated interface between two fluids. Phys. Fluids 15, 753–759.CrossRefGoogle Scholar
Miles, A. R., Edwards, M. J., Blue, B., Hansen, J. F., Robey, H. F., Drake, R. P., Kuranz, C. & Leibrandt, D. R. 2004 The effect of a short-wavelength mode on the evolution of a long-wavelength perturbation driven by a strong blast wave. Phys. Plasmas 11, 5507–5519.CrossRefGoogle Scholar
Oparin, A. & Abarzhi, S. 1999 Three-dimensional bubbles in Rayleigh–Taylor instability. Phys. Fluids 11, 3306–3311.CrossRefGoogle Scholar
Oron, D., Alon, U., Offer, D. & Shvarts, D. 2001 Dimensionality dependence of the Rayleigh–Taylor and Richtmyer–Meshkov instability late-time scaling laws. Phys. Plasmas 8, 2883–2889.CrossRefGoogle Scholar
Osher, S. & Sethian, J. A. 1988 Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79, 12–49.CrossRefGoogle Scholar
Peng, D., Merriman, B., Osher, S., Zhao, H. & Kang, M. 1999 A PDE-based fast local level set method. J. Comput. Phys. 155, 410–438.CrossRefGoogle Scholar
Richtmyer, R. 1960 Taylor instability in shock acceleration of compressible fluids. Comm. Pure Appl. Maths. 13, 297.CrossRefGoogle Scholar
Robey, H. F., Zhou, Y., Buckingham, A. C., Keiter, P., Remington, B. A. & Drake, R. P. 2003 The time scale for the transition to turbulence in a high Reynolds number, accelerated flow. Phys. Plasmas 10, 614–622.CrossRefGoogle Scholar
Schmidt, H. & Klein, R. 2003 A generalized level-set/in-cell-reconstruction approach for accelerating turbulent premixed flames. Combust. Theory Modelling 7, 243–267.CrossRefGoogle Scholar
Schneider, M., Dimonte, G. & Remington, B. 1998 Large and small scale structure in Rayleigh–Taylor mixing. Phys. Rev. Lett. 80, 3507–3510.CrossRefGoogle Scholar
Sharp, D. H. 1984 An overview of Rayleigh–Taylor instability Physica D 12, 3–15.Google Scholar
Smiljanovski, V., Moser, V. & Klein, R. 1997 A capturing–tracking hybrid scheme for deflagration discontinuities. Combust. Theory Modelling 1, 183–215.CrossRefGoogle Scholar
Sreenivasan, K. R. 1999 Fluid turbulence. Rev. Mod. Phys. 71, S383–S395.CrossRefGoogle Scholar
Strang, G. 1967 On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5, 506–517.CrossRefGoogle Scholar
Sussman, M., Smereka, P. & Osher, S. 1994 A level set method for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114, 146.CrossRefGoogle Scholar
Vandenboomgaerde, M., Muegler, C. & Gauthier, S. 1998 Impulsive model for the Richtmyer–Meshkov instability. Phys. Rev. E 58, 1874–1882.Google Scholar
Velikovich, A. L. 1996 Analytic theory of Richtmyer—Meshkov instability for the case of reflected rarefaction wave. Phys. Fluids 8, 1666–1679.CrossRefGoogle Scholar
Wouchuk, J. 2001 a Growth rate of the linear Richtmyer–Meshkov instability when a shock is reflected. Phys. Rev. E 63, 56303.Google Scholar
Wouchuk, J. G. 2001 b Growth rate of the Richtmyer—Meshkov instability when a rarefaction is reflected. Phys. Plasmas 8, 2890–2907.CrossRefGoogle Scholar
Youngs, D. L. 1994 Numerical simulation of mixing by Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Laser Part. Beams 12, 725–750.CrossRefGoogle Scholar
Zabusky, N. J., Kotelnikov, A. D., Gulak, Y. & Peng, G. 2003 Amplitude growth rate of a Richtmyer–Meshkov unstable two-dimensional interface to intermediate times. J. Fluid Mech. 475, 147–162.CrossRefGoogle Scholar
Zhang, Q. & Sohn, S.-I. 1997 Nonlinear theory of unstable fluid mixing driven by shock waves. Phys. Fluids 9, 1106–1124.CrossRefGoogle Scholar