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Refined modelling of the single-mode cylindrical Richtmyer–Meshkov instability

Published online by Cambridge University Press:  03 December 2020

Jinxin Wu Open the ORCID record for Jinxin Wu [Opens in a new window] ,
Han Liu  and
Zuoli Xiao Open the ORCID record for Zuoli Xiao [Opens in a new window]
Jinxin Wu
Affiliation:
State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing100871, PR China
Han Liu
Affiliation:
State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing100871, PR China
Zuoli Xiao*
Affiliation:
State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing100871, PR China HEDPS and Center for Applied Physics and Technology, College of Engineering, Peking University, Beijing100871, PR China Beijing Innovation Center for Engineering Science and Advanced Technology, Peking University, Beijing100871, PR China
*
†Email address for correspondence: z.xiao@pku.edu.cn
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Abstract

Evolution of the two-dimensional single-mode Richtmyer–Meshkov (RM) instability in a cylindrical geometry is numerically investigated through direct numerical simulation. A proper decomposition of the measured initial perturbation amplitude is found to be crucial for a comparative study between the numerical simulation and benchmark experiment. A refined compressible model is proposed based on the Bell equation by taking the premixed width of the initial interface into consideration. The modified model can accurately reproduce the development history of a single-mode perturbed gaseous interface between the first shock-interface interaction and reshock based on the evolution data of the unperturbed interface under the same premixing condition. The detailed effects of the RM instability, Rayleigh–Taylor stabilization and compressibility coupled with the Bell–Plesset effect are also specified with the aid of this model. It turns out that the refined Bell model can be further applied to the post-reshock stage of the RM instability before the appearance of strong nonlinearity.

JFM classification

Compressible Flows: Shock waves Mixing: Turbulent mixing
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 908 , 10 February 2021 , A9
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Bell, G. I. 1951 Taylor instability on cylinders and spheres in the small amplitude approximation. Report LA-1321. Los Alamos National Laboratory.Google Scholar
Betti, R. & Hurricane, O. A. 2016 Inertial-confinememnt fusion with lasers. Nat. Phys. 12, 435–448.Google Scholar
Biamino, L., Jourdan, G., Mariani, C., Houas, L., Vandenboomgaerde, M. & Souffland, D. 2015 On the possibility of studying the converging Richtmyer–Meshkov instability in a conventional shock tube. Exp. Fluids 56, 26.Google Scholar
Brouillette, M. 2002 The Richtmyer–Meshkov instability. Annu. Rev. Fluid Mech. 34 (34), 445–468.Google Scholar
Dimonte, G., Frerking, C. E. & Schneider, M. 1995 Richtmyer–Meshkov instability in the turbulent regime. Phys. Rev. Lett. 74 (24), 4855–4858.Google ScholarPubMed
Dimotakis, P. E. & Samtaney, R. 2006 Planar shock cylindrical focusing by a perfect-gas lens. Phys. Fluids 18, 031705.Google Scholar
Ding, J., Si, T., Yang, J., Lu, X., Zhai, Z. & Luo, X. 2017 Measurement of a Richtmyer–Meshkov instability at an air-SF$_6$ interface in a semiannular shock tube. Phys. Rev. Lett. 119, 014501.Google Scholar
Dutta, S., Glimma, J., Grove, J. W., Sharp, D. H. & Zhang, Y. 2004 Spherical Richtmyer–Meshkov instability for axisymmetric flow. Maths Comput. Simul. 65, 417–430.Google Scholar
Fincke, J. R., Lanier, N. E., Batha, S. H., Hueckstaedt, R. M., Magelssen, G. R., Rothman, S. D., Parker, K. W. & Horsfield, C. J. 2004 Postponement of saturation of the Richtmyer–Meshkov instability in a convergent geometry. Phys. Rev. Lett. 93 (11), 115003.Google Scholar
Glimm, J., Grove, J., Zhang, Y. & Dutta, S. 2002 Numerical study of axisymmetric Richtmyer–Meshkov instability and azimuthal effect on spherical mixing. J. Stat. Phys. 107, 241–260.Google Scholar
Graves, R. E. & Argrow, B. M. 1999 Bulk viscosity: past to present. J. Thermophys. Heat Transfer 13 (3), 337–342.Google Scholar
Groom, M. & Thornber, B. 2019 Direct numerical simulation of the multimode narrowband Richtmyer–Meshkov instability. Comput. Fluids 194, 1–12.Google Scholar
Guderley, G. 1942 Starke kugelige und zylindrische Verdichtungsstösse in der nähe des kugelmittelpunktes bzw. der Zylinderachse. Luftfahrtforschung 19, 302–312.Google Scholar
Hosseini, S. H. R. & Takayama, K. 2005 Experimental study of Richtmyer–Meshkov instability induced by cylindrical shock waves. Phys. Fluids 17, 084101.Google Scholar
Jacobs, J. W. & Krivets, V. V. 2005 Experiments on the late-time development of single-mode Richtmyer–Meshkov instability. Phys. Fluids 17, 034105.Google 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 (10), 3078–3085.Google Scholar
Jourdan, G. & Houas, L. 2005 High-amplitude single-mode perturbation evolution at the Richtmyer–Meshkov instability. Phys. Rev. Lett. 95, 204502.Google ScholarPubMed
Kane, J., Drake, R. P. & Remington, B. A. 1999 An evaluation of the Richtmyer–Meshkov instability in supernova remnant formation. Astrophys. J. 511, 335–340.Google Scholar
Kawai, S. & Lele, S. K. 2008 Localized artificial diffusivity scheme for discontinuity capturing on curvilinear meshes. J. Comput. Phys. 227 (22), 9498–9526.Google Scholar
Lei, F. 2017 Experimental and theoretical study on converging Richtmyer–Meshkov instability. PhD thesis, University of Science and Technology of China.Google Scholar
Lei, F., Ding, J., Si, T., Zhai, Z. & Luo, X. 2017 Experimental study on a sinusoidal air/SF$_6$ interface accelerated by a cylindrically converging shock. J. Fluid Mech. 826, 819–829.Google Scholar
Liu, H. & Xiao, Z. 2016 Scale-to-scale energy transfer in mixing flow induced by the Richtmyer–Meshkov instability. Phys. Rev. E 93 (5), 053112.Google ScholarPubMed
Lombardini, M. & Pullin, D. I. 2009 Small-amplitude perturbations in the three-dimensional cylindrical Richtmyer–Meshkov instability. Phys. Fluids 21, 114103.Google Scholar
Lombardini, M., Pullin, D. I. & Meiron, D. I. 2014 a Turbulent mixing driven by spherical implosions. Part 1. Flow description and mixing-layer growth. J. Fluid Mech. 748, 85–112.Google Scholar
Lombardini, M., Pullin, D. I. & Meiron, D. I. 2014 b Turbulent mixing driven by spherical implosions. Part 2. Turbulence statistics. J. Fluid Mech. 748, 113–142.Google Scholar
Luo, X., Ding, J., Wang, M., Zhai, Z. & Si, T. 2015 A semi-annular shock tube for studying cylindrically converging Richtmyer–Meshkov instability. Phys. Fluids 27, 091702.Google Scholar
Luo, X., Zhang, F., Ding, J., Si, T., Yang, J., Zhai, Z. & Wen, C.-Y. 2018 Long-term effect of Rayleigh–Taylor stabilization on converging Richtmyer–Meshkov instability. J. Fluid Mech. 849, 231–244.Google Scholar
Mariani, C., Vandenboomgaerde, M., Jourdan, G., Souffland, D. & Houas, L. 2008 Investigation of the Richtmyer–Meshkov instability with stereolithographed interfaces. Phys. Rev. Lett. 100, 254503.Google ScholarPubMed
Meshkov, E. E. 1969 Instability of the interface of two gases accelerated by a shock wave. Fluid Dyn. 4 (5), 101–104.Google Scholar
Plesset, M. S. 1954 On the stability of fluid flows with spherical symmetry. J. Appl. Phys. 25 (1), 96–98.Google Scholar
Ranjan, D., Oakley, J. & Bonazza, R. 2011 Shock-bubble interactions. Annu. Rev. Fluid Mech. 43, 117–140.Google Scholar
Rayleigh, Lord 1883 Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Lond. Math. Soc. 14 (1), 170–177.Google Scholar
Richtmyer, R. D. 1960 Taylor instability in shock acceleration of compressible fluids. Commun. Pure Appl. Maths 13, 297–319.Google Scholar
Rupert, V. 1992 Shock-interface interaction: current research on the Richtmyer–Meshkov problem. In Shock Waves (ed. K. Takayama), pp. 83–94. Springer.CrossRefGoogle Scholar
Sadot, O., Erez, L., Alon, U., Oron, D., Levin, L. A., Erez, G., Ben-Dor, G. & Shvarts, D. 1998 Study of nonlinear evolution of single-mode and two-bubble interaction under Richtmyer–Meshkov instability. Phys. Rev. Lett. 80 (8), 1654–1657.Google Scholar
Schilling, O., Latini, M. & Don, W. S. 2007 Physics of reshock and mixing in single-mode Richtmyer–Meshkov instability. Phys. Rev. Lett. 76, 026319.Google ScholarPubMed
Shankar, S. K., Kawai, S. & Lele, S. K. 2011 Two-dimensional viscous flow simulation of a shock accelerated heavy gas cylinder. Phys. Fluids 23 (2), 024102.Google Scholar
Si, T., Zhai, Z. & Luo, X. 2014 Experimental study of Richtmyer–Meshkov instability in a cylindrical converging shock tube. Laser Part. Beams 32 (3), 343–351.Google Scholar
Taylor, G. I. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. Proc. R. Soc. Lond. 201, 192–196.Google Scholar
Thornber, B. & Zhou, Y. 2012 Energy transfer in the Richtmyer–Meshkov instability. Phys. Rev. E 86 (5), 056302.Google ScholarPubMed
Tritschler, V. K., Olson, B. J., Lele, S. K., Hickel, S., Hu, X. Y. & Adams, N. A. 2014 On the Richtmyer–Meshkov instability evolving from a deterministic multimode planar interface. J. Fluid Mech. 755 (1), 429–462.Google Scholar
Wu, J., Liu, H. & Xiao, Z. 2019 A numerical investigation of Richtmyer–Meshkov instability in spherical geometry. Adv. Appl. Maths Mech. 11 (3), 583–597.Google Scholar
Youngs, D. L. & Williams, R. J. R. 2008 Turbulent mixing in spherical implosions. Intl J. Numer. Meth. Fluids 56, 1597–1603.Google Scholar
Zhai, Z., Zhang, F., Zhang, Z., Ding, J. & Wen, C. 2019 Numerical study on Rayleigh–Taylor effect on cylindrically converging Richtmyer–Meshkov instability. Sci. China-Phys. Mech. Astron. 62, 124712.Google Scholar
Zhang, Q., Deng, S. & Guo, W. 2018 Quantitative theory for the growth rate and amplitude of the compressible Richtmyer–Meshkov instability at all density ratios. Phys. Rev. Lett. 121, 174502.Google ScholarPubMed
Zhang, Q. & Graham, M. J. 1997 Scaling laws for unstable interfaces driven by strong shocks in cylindrical geometry. Phys. Rev. Lett. 79 (14), 2674–2677.Google Scholar
Zhang, Q., Liu, H., Ma, Z. & Xiao, Z. 2016 Preferential concentration of heavy particles in compressible isotropic turbulence. Phys. Fluids 28, 055104.Google Scholar
Zhang, Q. & Xiao, Z. 2018 Single-particle dispersion in compressible turbulence. Phys. Fluids 30, 040904.Google Scholar
Zheng, J. G., Lee, T. S. & Winoto, S. H. 2008 Numerical simulation of Richtmyer–Meshkov instability driven by imploding shocks. Maths Comput. Simul. 79, 749–762.Google Scholar
Zhou, Y. 2017 a Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. I. Phys. Rep. 720–722, 1–136.Google Scholar
Zhou, Y. 2017 b Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. II. Phys. Rep. 723–725, 1–160.Google Scholar