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Atwood ratio dependence of Richtmyer–Meshkov flows under reshock conditions using large-eddy simulations

Published online by Cambridge University Press:  01 February 2011

M. LOMBARDINI ,
D. J. HILL ,
D. I. PULLIN  and
D. I. MEIRON
M. LOMBARDINI*
Affiliation:
Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, USA
D. J. HILL
Affiliation:
Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, USA
D. I. PULLIN
Affiliation:
Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, USA
D. I. MEIRON
Affiliation:
Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, USA
*
†Email address for correspondence: manuel@caltech.edu
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Abstract

We study the shock-driven turbulent mixing that occurs when a perturbed planar density interface is impacted by a planar shock wave of moderate strength and subsequently reshocked. The present work is a systematic study of the influence of the relative molecular weights of the gases in the form of the initial Atwood ratio A. We investigate the cases A = ± 0.21, ±0.67 and ±0.87 that correspond to the realistic gas combinations air–CO2, air–SF6 and H2–air. A canonical, three-dimensional numerical experiment, using the large-eddy simulation technique with an explicit subgrid model, reproduces the interaction within a shock tube with an endwall where the incident shock Mach number is ~1.5 and the initial interface perturbation has a fixed dominant wavelength and a fixed amplitude-to-wavelength ratio ~0.1. For positive Atwood configurations, the reshock is followed by secondary waves in the form of alternate expansion and compression waves travelling between the endwall and the mixing zone. These reverberations are shown to intensify turbulent kinetic energy and dissipation across the mixing zone. In contrast, negative Atwood number configurations produce multiple secondary reshocks following the primary reshock, and their effect on the mixing region is less pronounced. As the magnitude of A is increased, the mixing zone tends to evolve less symmetrically. The mixing zone growth rate following the primary reshock approaches a linear evolution prior to the secondary wave interactions. When considering the full range of examined Atwood numbers, measurements of this growth rate do not agree well with predictions of existing analytic reshock models such as the model by Mikaelian (Physica D, vol. 36, 1989, p. 343). Accordingly, we propose an empirical formula and also a semi-analytical, impulsive model based on a diffuse-interface approach to describe the A-dependence of the post-reshock growth rate.

JFM classification

Instability: Instability Compressible Flows: Shock waves Mixing: Turbulent mixing
Type
Papers
Information
Journal of Fluid Mechanics , Volume 670 , 10 March 2011 , pp. 439 - 480
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Alon, U., Hecht, J., Ofer, D. & Shvarts, D. 1995 Power laws and similarity of Rayleigh–Taylor and Richtmyer–Meshkov mixing fronts at all density ratios. Phys. Rev. Lett. 74 (4), 534–537.CrossRefGoogle ScholarPubMed
Andronov, V. A., Bakhrakh, S. M., Meshkov, E. E., Mokhov, V. N., Nikiforov, V. V., Pevnitskii, A. V. & Tolshmyakov, A. I. 1976 Turbulent mixing at contact surface accelerated by shock waves. Sov. Phys. JETP 44 (2), 424–427.Google Scholar
Arnett, D., Bahcall, J. N., Kirshner, R. P. & Woosler, S. E. 1989 Supernova 1987A. Annu. Rev. Astron. Astrophys. 27, 629–700.CrossRefGoogle Scholar
Berger, M. J. & Colella, P. 1989 Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys. 82 (1), 64–84.CrossRefGoogle Scholar
Berger, M. J. & Oliger, J. 1984 Adaptive mesh refinement for hyperbolic partial-differential equations. J. Comput. Phys. 53 (3), 484–512.CrossRefGoogle Scholar
Blaisdell, G. A. 1991 Numerical simulation of compressible homogeneous turbulence. PhD thesis, Stanford University.Google Scholar
Bond, C., Hill, D. J., Meiron, D. I. & Dimotakis, P. E. 2009 Shock structure in a planar convergent geometry with experiment and simulation. J. Fluid Mech. 641, 297–333.CrossRefGoogle Scholar
Brouillette, M. & Sturtevant, B. 1989 Growth induced by multiple shock waves normally incident on plane gaseous interfaces. Physica D 37, 248–263.CrossRefGoogle Scholar
Brouillette, M. & Sturtevant, B. 1993 Experiments on the Richtmyer–Meshkov instability: small-scale perturbations on a plane interface. Phys. Fluids A 5 (4), 916–930.CrossRefGoogle Scholar
Brouillette, M. & Sturtevant, B. 1994 Experiments on the Richtmyer–Meshkov instability: single-scale perturbations on a continuous interface. J. Fluid Mech. 263, 271–292.CrossRefGoogle Scholar
Burrows, A., Hayes, J. & Fryxell, B. A. 1995 On the nature of core-collapse supernova explosions. Astrophys. J. 450, 830–850.CrossRefGoogle Scholar
Canuto, V. M. & Goldman, I. 1985 Analytical model for large-scale turbulence. Phys. Rev. Lett. 54 (5), 430–433.CrossRefGoogle ScholarPubMed
Charakhchyan, A. A. 2001 Reshocking at the non-linear stage of Richtmyer–Meshkov instability. Plasma Phys. Control. Fusion 43, 1169–1179.CrossRefGoogle Scholar
Collins, B. D. & Jacobs, J. W. 2002 PLIF flow visualization and measurements of the Richtmyer–Meshkov instability of an air/SF6 interface. J. Fluid Mech. 464, 113–136.CrossRefGoogle Scholar
Deiterding, R. 2005 Construction and application of an AMR algorithm for distributed memory computers. In Adaptive Mesh Refinement: Theory and Applications (ed. Plewa, T., Linde, T. & Weirs, V. G.), pp. 361–372. Lecture Notes in Computational Science and Engineering, vol. 41. Springer.CrossRefGoogle Scholar
Deiterding, R., Radovitzky, R., Mauch, S. P., Noels, L., Cummings, J. C. & Meiron, D. I. 2006 A virtual test facility for the efficient simulation of solid material response under strong shock and detonation wave loading. Engng Comput. 22 (3–4), 325–347.CrossRefGoogle Scholar
Dimonte, G. 2004 Dependence of turbulent Rayleigh-Taylor instability on initial perturbations. Phys. Rev. E 69, 056305 (1–14).CrossRefGoogle ScholarPubMed
Dimonte, G., Ramaprabhu, P. & Andrews, M. 2007 Rayleigh–Taylor instability with complex acceleration history. Phys. Rev. E 76, 046313 (1–6).CrossRefGoogle ScholarPubMed
Dimonte, G. & Schneider, M. 2000 Density ratio dependence of Rayleigh–Taylor mixing for sustained and impulsive acceleration histories. Phys. Fluids 12 (2), 304–321.CrossRefGoogle Scholar
Dimotakis, P. E. 2000 The mixing transition in turbulent flows. J. Fluid Mech. 409, 69–97.CrossRefGoogle Scholar
Duff, R. E., Harlow, F. H. & Hirt, C. W. 1962 Effects of diffusion on interface instability between gases. Phys. Fluids 5, 417–425.CrossRefGoogle Scholar
Erez, L., Sadot, O., Oron, D., Erez, G., Levin, L. A., Shvarts, D. & Ben-Dor, G. 2000 Study of the membrane effect on turbulent mixing measurements in shock tubes. Shock Waves 10 (4), 241–251.CrossRefGoogle Scholar
Ghosal, S. 1996 An analysis of numerical errors in large-eddy simulations of turbulence. J. Comput. Phys. 125 (1), 187–206.CrossRefGoogle Scholar
Ghosal, S. 1999 Mathematical and physical constraints on large-eddy simulation of turbulence. AIAA J. 37 (4), 425–433.CrossRefGoogle Scholar
Giordano, J. & Burtschell, Y. 2006 Richtmyer–Meshkov instability induced by shock–bubble interaction: numerical and analytical studies with experimental validation. Phys. Fluids 18 (3), 036102.CrossRefGoogle Scholar
Gottlieb, S., Shu, C.-W. & Tadmor, E. 2001 Strong stability-preserving high-order time discretization methods. SIAM Rev. 43 (1), 89–112.CrossRefGoogle Scholar
Greenough, J. A. & Burke, E. 2004 The effect of initial conditions on late time asymptotics and mixing for multimode Richtmyer–Meshov instability. In Ninth International Workshop on the Physics of Compressible Turbulent Mixing, 19–23 July, Cambridge, UK.Google Scholar
Gupta, M. R., Roy, S., Sarkar, S., Khan, M., Pant, H. C. & Srivastava, M. K. 2007 Effect on Richtmyer–Meshkov instability of deviation from sinusoidality of the corrugated interface between two fluids. Laser Part. Beams 25, 503–510.CrossRefGoogle Scholar
Gutmark, E. & Wygnanski, I. 1976 The planar turbulent jet. J. Fluid Mech. 73 (3), 465–495.CrossRefGoogle Scholar
Hill, D. J., Pantano, C. & Pullin, D. I. 2006 Large-eddy simulation and multi-scale modeling of Richtmyer–Meshkov instability with reshock. J. Fluid Mech. 557, 29–61.CrossRefGoogle Scholar
Hill, D. J. & Pullin, D. I. 2004 Hybrid tuned center-difference-WENO method for large eddy simulations in the presence of strong shocks. J. Comput. Phys. 194 (2), 435–450.CrossRefGoogle Scholar
Holmes, R. L., Dimonte, G., Fryxell, B., Gittings, M. L., Grove, J. W., Schneider, M., Sharp, D. H., Velikovich, A. L., Weaver, R. P. & Zhang, Q. 1999 Richtmyer–Meshkov instability growth: experiment, simulation and theory. J. Fluid Mech. 389, 55–79.CrossRefGoogle Scholar
Honein, A. E. & Moin, P. 2004 Higher entropy conservation and numerical stability of compressible turbulence simulations. J. Comput. Phys. 201 (2), 531–545.CrossRefGoogle Scholar
Houas, L. & Chemouni, I. 1996 Experimental investigation of Richtmyer–Meshkov instability in shock tube. Phys. Fluids 8 (2), 614–627.CrossRefGoogle Scholar
Kosovic, B., Pullin, D. I. & Samtaney, R. 2002 Subgrid-scale modeling for large-eddy simulations of compressible turbulence. Phys. Fluids 14 (4), 1511–1522.CrossRefGoogle Scholar
Latini, M., Schilling, O. & Don, W. S. 2007 Effects of WENO flux reconstruction order and spatial resolution on reshocked two-dimensional Richtmyer–Meshkov instability. J. Comput. Phys. 221 (2), 805–836.CrossRefGoogle Scholar
Layes, G., Jourdan, G. & Houas, L. 2009 Experimental study on a plane shock wave accelerating a gas bubble. Phys. Fluids 21 (7), 074102.CrossRefGoogle Scholar
Leinov, E., Malamud, G., Elbaz, Y., Levin, L. A., Ben-Dor, G., Shvarts, D. & Sadot, O. 2009 Experimental and numerical investigation of the Richtmyer–Meshkov instability under re-shock conditions. J. Fluid Mech. 626, 449–475.CrossRefGoogle Scholar
Lindl, J. D., McCrory, R. L. & Campbell, E. M. 1992 Progress towards ignition and burn propagation in inertial confinement fusion. Phys. Today 45, 32–40.CrossRefGoogle Scholar
Lombardini, M. 2008 Richtmyer–Meshkov instability in converging geometries. PhD thesis, Caltech.Google Scholar
Lombardini, M., Deiterding, R. & Pullin, D. I. 2008 Large-eddy simulation of the Richtmyer–Meshkov instability in a converging geometry. In Quality and Reliability of Large-Eddy Simulations, Proc. of QLES 2007 Intl Symposium (ed. Meyers, J., Geurts, B. J. & Sagaut, P.), ERCOFTAC Series, vol. 12, pp. 23–51. Springer.Google Scholar
Lombardini, M. & Pullin, D. I. 2009 Startup process in the Richtmyer–Meshkov instability. Phys. Fluids 21 (4), 044104.CrossRefGoogle Scholar
Lundgren, T. S. 1982 Strained spiral vortex model for turbulence fine structure. Phys. Fluids 25 (12), 2193–2203.CrossRefGoogle Scholar
Meshkov, E. E. 1969 Instability of the interface of two gases accelerated by a shock wave. Sov. Fluid Dyn. 4 (5), 101–108.CrossRefGoogle Scholar
Mikaelian, K. O. 1989 Turbulent mixing generated by Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Physica D 36, 343–347.CrossRefGoogle Scholar
Mikaelian, K. O. 1991 Density gradient stabilization of the Richtmyer–Meshkov instability. Phys. Fluids A 3 (11), 2638–2643.CrossRefGoogle Scholar
Mikaelian, K. O. 2009 Reshocks, rarefactions, and the generalized Layzer model for hydrodynamic instabilities. Phys. Fluids 21 (2), 024103.CrossRefGoogle Scholar
Misra, A. & Pullin, D. I. 1997 A vortex-based model for large-eddy simulation. Phys. Fluids 9 (8), 2443–2454.CrossRefGoogle Scholar
Moeleker, P. & Leonard, A. 2001 Lagrangian methods for the tensor-diffusivity subgrid model. J. Comput. Phys. 167 (1), 1–21.CrossRefGoogle Scholar
Motl, B., Oakley, J., Ranjan, D., Weber, C., Anderson, M. & Bonazza, R. 2009 Experimental validation of a Richtmyer–Meshkov scaling law over large density ratio and shock strength ranges. Phys. Fluids 21 (12), 126102.CrossRefGoogle Scholar
Niederhaus, J. H. J., Greenough, J. A., Oakley, J. G., Ranjan, D., Anderson, M. H. & Bonazza, R. 2008 A computational parameter study for the three-dimensional shock–bubble interaction. J. Fluid Mech. 594, 85–124.CrossRefGoogle Scholar
Orlicz, G. C., Balakumar, B. J., Tomkins, C. D. & Prestridge, K. P. 2009 A Mach number study of the Richtmyer–Meshkov instability in a varicose, heavy-gas curtain. Phys. Fluids 21 (6), 064102.CrossRefGoogle Scholar
Oron, D., Arazi, L., Kartoon, D., Rikanati, A., Alon, U. & Shvarts, D. 2001 Dimensionality dependence of the Rayleigh–Taylor and Richtmyer–Meshkov instability late-time scaling laws. Phys. Plasmas 8 (6), 2883–2889.CrossRefGoogle Scholar
Pantano, C., Deiterding, R., Hill, D. J. & Pullin, D. I. 2007 A low numerical dissipation patch-based adaptive mesh refinement method for large-eddy simulation of compressible flows. J. Comput. Phys. 221 (1), 63–87.CrossRefGoogle Scholar
Pullin, D. I. 2000 A vortex-based model for the subgrid flux of a passive scalar. Phys. Fluids 12 (9), 2311–2319.CrossRefGoogle Scholar
Read, K. I. 1984 Experimental investigation of turbulent mixing by Rayleigh–Taylor instability. Physica D 12, 45–58.CrossRefGoogle Scholar
Reid, R. C., Prausnitz, J. M. & Polling, B. E. 1987 The Properties of Gases and Liquids. McGraw-Hill.Google Scholar
Richtmyer, R. D. 1960 Taylor instability in shock acceleration of compressible fluids. Commun. Pure Appl. Maths. 13, 297–319.CrossRefGoogle Scholar
Saffman, P. G. & Meiron, D. I. 1989 Kinetic energy generated by the incompressible Richtmyer–Meshkov instability in a continuously stratified fluid. Phys. Fluids A 1 (11), 1767–1771.CrossRefGoogle Scholar
Samtaney, R. & Zabusky, N. J. 1993 Circulation deposition on shock-accelerated planar and curved density-stratified interfaces: models and scaling laws. J. Fluid Mech. 269, 45–78.CrossRefGoogle Scholar
Schilling, O., Latini, M. & Don, W. S. 2007 Physics of reshock and mixing in single-mode Richtmyer–Meshkov instability. Phys. Rev. E 76, 026319.CrossRefGoogle ScholarPubMed
Taccetti, J. M., Batha, S. H., Fincke, J. R., Delamater, N. D., Lanier, N. E., Magelssen, G. R., Hueckstaedt, R. M., Rothman, S. D., Horsfield, C. J. & Parker, K. W. 2005 Richtmyer–Meshkov instability reshock experiments using laser-driven double-cylinder implosions. Astrophys. Space Sci. 298, 327–331.CrossRefGoogle Scholar
Tomkins, C., Kumar, S., Orlicz, G. & Prestridge, K. 2008 An experimental investigation of mixing mechanisms in shock-accelerated flow. J. Fluid Mech. 611, 131–150.CrossRefGoogle Scholar
Vetter, M. & Sturtevant, B. 1995 Experiments on the Richtmyer–Meshkov instability of an air/SF6 interface. Shock Waves 4, 247–252.CrossRefGoogle Scholar
Voelkl, T. & Pullin, D. I. 2000 A physical-space version of the stretched-vortex subgrid-stress model for large-eddy simulations. Phys. Fluids 12 (7), 1810–1825.CrossRefGoogle Scholar
Youngs, D. L. 1984 Numerical simulation of turbulent mixing by Rayleigh–Taylor instability. Physica D 12, 32–44.CrossRefGoogle Scholar
Zabusky, N. J. 1999 Vortex paradigm for accelerated inhomogeneous flows: visiometrics for the Rayleigh–Taylor and Richtmyer–Meshkov environments. Annu. Rev. Fluid Mech. 31, 495–536.CrossRefGoogle Scholar
Zaitsev, S. V., Lazareva, E. V., Chernukha, V. V. & Belyaev, V. M. 1985 An experimental investigation and numerical modeling of turbulent mixing in one-dimensional flows. Sov. Phys. Dokl. 30, 579.Google Scholar
Zhang, T. A., Dahlburg, R. B. & Dahlburg, J. P. 1992 Direct and large-eddy simulations of three-dimensional compressible Navier–Stokes turbulence. Phys. Fluids A 4, 127–140.CrossRefGoogle Scholar