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Evaluating the stretching/compression effect of Richtmyer–Meshkov instability in convergent geometries

Published online by Cambridge University Press:  03 August 2022

Jin Ge Open the ORCID record for Jin Ge [Opens in a new window] ,
Haifeng Li Open the ORCID record for Haifeng Li [Opens in a new window] ,
Xinting Zhang  and
Baolin Tian
Jin Ge
Affiliation:
Sino-French Engineer School, Beihang University, Beijing 100191, PR China
Haifeng Li*
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing 100094, PR China
Xinting Zhang
Affiliation:
Sino-French Engineer School, Beihang University, Beijing 100191, PR China
Baolin Tian*
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing 100094, PR China HEDPS, Center for Applied Physics and Technology, and College of Engineering, Peking University, Beijing 100871, PR China
*
†Email addresses for correspondence: tylihaifeng@163.com, tian_baolin@iapcm.ac.cn
†Email addresses for correspondence: tylihaifeng@163.com, tian_baolin@iapcm.ac.cn
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Abstract

Richtmyer–Meshkov (RM) instability in convergent geometries (such as cylinders and spheres) plays a fundamental role in natural phenomena and engineering applications, e.g. supernova explosion and inertial confinement fusion. Convergent geometry refers to a system in which the interface converges and the fluids are compressed correspondingly. By applying a decomposition formula, the stretching or compression (S(C)) effect is separated from the perturbation growth as one of the main contributions, which is defined as the averaged velocity difference between two ends of the mixing zone. Starting from linear theories, the S(C) effect in planar, cylindrical and spherical geometries is derived as a function of geometrical convergence ratio, compression ratio and mixing width. Specifically, geometrical convergence stretches the mixing zone, while fluid compression compresses the mixing zone. Moreover, the contribution of geometrical convergence in the spherical geometry is more important than that in the cylindrical geometry. A series of cylindrical cases with high convergence ratio is simulated, and the growth of perturbations is compared with that of the corresponding planar cases. As a result, the theoretical results of the S(C) effect agree well with the numerical results. Furthermore, results show that the S(C) effect is a significant feature in convergent geometries. Therefore, the S(C) effect is an important part of the Bell–Plesset effect. The present work on the S(C) effect is important for further modelling of the mixing width of convergent RM instabilities.

JFM classification

Instability: Nonlinear instability Convection: Buoyancy-driven instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 946 , 10 September 2022 , A18
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

Barnes, C.W., et al. 2002 Observation of mix in a compressible plasma in a convergent cylindrical geometry. Phys. Plasma 9 (11), 4431–4434.CrossRefGoogle Scholar
Beck, J.B. 1996 The effects of convergent geometry on the ablative Rayleigh–Taylor instability in cylindrical implosions. PhD thesis, Purdue University.Google Scholar
Bell, G.I. 1951 Taylor instability on cylinders and spheres in the small amplitude approximation. Tech. Rep. LA-1321. Los Alamos National Laboratory.Google Scholar
Betti, R. & Hurricane, O.A. 2016 Inertial-confinement fusion with lasers. Nat. Phys. 12 (5), 435–448.CrossRefGoogle Scholar
Brouillette, M. 2002 The Richtmyer–Meshkov instability. Annu. Rev. Fluid Mech. 34 (1), 445–468.CrossRefGoogle Scholar
Chisnell, R.F. 1998 An analytic description of converging shock waves. J. Fluid Mech. 354, 357–375.CrossRefGoogle Scholar
Dimonte, G. 2021 A modal wave-packet model for the multi-mode Richtmyer–Meshkov instability. Phys. Fluids 33 (1), 014108.CrossRefGoogle Scholar
Dimonte, G., et al. 2004 A comparative study of the turbulent Rayleigh–Taylor instability using high-resolution three-dimensional numerical simulations: the alpha-group collaboration. Phys. Fluids 16 (5), 1668–1693.CrossRefGoogle Scholar
Ding, J., Si, T., Yang, J., Lu, X., Zhai, Z. & Luo, X. 2017 Measurement of a Richtmyer–Meshkov instability at an air–${\rm SF}_6$ interface in a semiannular shock tube. Phys. Rev. Lett. 119 (1), 014501.CrossRefGoogle Scholar
El Rafei, M. & Thornber, B. 2020 Numerical study and buoyancy–drag modeling of bubble and spike distances in three-dimensional spherical implosions. Phys. Fluids 32 (12), 124107.CrossRefGoogle Scholar
El Rafei, M., Flaig, M., Youngs, D.L. & Thornber, B. 2019 Three-dimensional simulations of turbulent mixing in spherical implosions. Phys. Fluids 31 (11), 114101.CrossRefGoogle Scholar
Epstein, R. 2004 On the Bell–Plesset effects: the effects of uniform compression and geometrical convergence on the classical Rayleigh–Taylor instability. Phys. Plasma 11 (11), 5114–5124.CrossRefGoogle Scholar
Flaig, M., Clark, D., Weber, C., Youngs, D.L. & Thornber, B. 2018 Single-mode perturbation growth in an idealized spherical implosion. J. Comput. Phys. 371, 801–819.CrossRefGoogle Scholar
Ge, J., Zhang, X., Li, H. & Tian, B. 2020 Late-time turbulent mixing induced by multimode Richtmyer–Meshkov instability in cylindrical geometry. Phys. Fluids 32 (12), 124116.CrossRefGoogle Scholar
Goncharov, V.N. & Li, D. 2005 Effects of temporal density variation and convergent geometry on nonlinear bubble evolution in classical Rayleigh–Taylor instability. Phys. Rev. E 71 (4), 046306.CrossRefGoogle ScholarPubMed
Hsing, W.W., Barnes, C.W., Beck, J.B., Hoffman, N.M., Galmiche, D., Richard, A., Edwards, J., Graham, P., Rothman, S. & Thomas, B. 1997 Rayleigh–Taylor instability evolution in ablatively driven cylindrical implosions. Phys. Plasma 4 (5), 1832–1840.CrossRefGoogle Scholar
Hsing, W.W. & Hoffman, N.M. 1997 Measurement of feedthrough and instability growth in radiation-driven cylindrical implosions. Phys. Rev. Lett. 78 (20), 3876.CrossRefGoogle Scholar
Joggerst, C.C., Nelson, A., Woodward, P., Lovekin, C., Masser, T., Fryer, C.L., Ramaprabhu, P., Francois, M. & Rockefeller, G. 2014 Cross-code comparisons of mixing during the implosion of dense cylindrical and spherical shells. J. Comput. Phys. 275, 154–173.CrossRefGoogle Scholar
Lanier, N.E., Barnes, C.W., Batha, S.H., Day, R.D., Magelssen, G.R., Scott, J.M., Dunne, A.M., Parker, K.W. & Rothman, S.D. 2003 Multimode seeded Richtmyer–Meshkov mixing in a convergent, compressible, miscible plasma system. Phys. Plasma 10 (5), 1816–1821.CrossRefGoogle Scholar
Li, C.K., et al. 2004 Effects of nonuniform illumination on implosion asymmetry in direct-drive inertial confinement fusion. Phys. Rev. Lett. 92 (20), 205001.CrossRefGoogle ScholarPubMed
Li, H., He, Z., Zhang, Y. & Tian, B. 2019 On the role of rarefaction/compression waves in Richtmyer–Meshkov instability with reshock. Phys. Fluids 31 (5), 054102.Google Scholar
Li, H., Tian, B., He, Z. & Zhang, Y. 2021 Growth mechanism of interfacial fluid-mixing width induced by successive nonlinear wave interactions. Phys. Rev. E 103 (5), 053109.CrossRefGoogle ScholarPubMed
Lombardini, M. & Pullin, D.I. 2009 Startup process in the Richtmyer–Meshkov instability. Phys. Fluids 21 (4), 044104.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar
Luo, X., Ding, J., Zhai, Z. & Si, T. 2018 a 16th International Workshop of the Physics of Compressible Turbulent Mixing. Tech. Rep. Advanced Propulsion Laboratory, University of Science and Technology of China.Google Scholar
Luo, X., Li, M., Ding, J., Zhai, Z. & Si, T. 2019 Nonlinear behaviour of convergent Richtmyer–Meshkov instability. J. Fluid Mech. 877, 130–141.CrossRefGoogle Scholar
Luo, X., Zhang, F., Ding, J., Si, T., Yang, J., Zhai, Z. & Wen, C.Y. 2018 b Long-term effect of Rayleigh–Taylor stabilization on converging Richtmyer–Meshkov instability. J. Fluid Mech. 849, 231–244.CrossRefGoogle Scholar
Meshkov, E.E. 1969 Instability of the interface of two gases accelerated by a shock wave. Fluid Dyn. 4 (5), 101–104.CrossRefGoogle Scholar
Mikaelian, K.O. 1990 Stability and mix in spherical geometry. Phys. Rev. Lett. 65 (8), 992.CrossRefGoogle ScholarPubMed
Mikaelian, K.O. 2005 Rayleigh–Taylor and Richtmyer–Meshkov instabilities and mixing in stratified cylindrical shells. Phys. Fluids 17 (9), 094105.CrossRefGoogle Scholar
Plesset, M.S. 1954 On the stability of fluid flows with spherical symmetry. J. Appl. Phys. 25 (1), 96–98.CrossRefGoogle Scholar
Rayleigh, Lord 1882 Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. R. Math. Soc. s1-14 (1), 170–177.CrossRefGoogle Scholar
Reckinger, S.J., Livescu, D. & Vasilyev, O.V. 2016 Comprehensive numerical methodology for direct numerical simulations of compressible Rayleigh–Taylor instability. J. Comput. Phys. 313, 181–208.CrossRefGoogle Scholar
Richtmyer, R.D. 1960 Taylor instability in shock acceleration of compressible fluids. Commun. Pure Appl. Maths 13 (2), 297–319.CrossRefGoogle Scholar
Si, T., Long, T., Zhai, Z. & Luo, X. 2015 Experimental investigation of cylindrical converging shock waves interacting with a polygonal heavy gas cylinder. J. Fluid Mech. 784, 225–251.CrossRefGoogle Scholar
Sutherland, W. 1893 The viscosity of gases and molecular force. Phil. Mag. 36 (223), 507–531.CrossRefGoogle Scholar
Taylor, G.I. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. Proc. R. Soc. Lond. A 201 (1065), 192–196.Google Scholar
Thomas, V.A. & Kares, R.J. 2012 Drive asymmetry and the origin of turbulence in an ICF implosion. Phys. Rev. Lett. 109 (7), 075004.CrossRefGoogle Scholar
Thornber, B., Drikakis, D., Youngs, D.L. & Williams, R.J.R. 2010 The influence of initial conditions on turbulent mixing due to Richtmyer–Meshkov instability. J. Fluid Mech. 654, 99–139.CrossRefGoogle Scholar
Thornber, B., et al. 2017 Late-time growth rate, mixing, and anisotropy in the multimode narrowband Richtmyer–Meshkov instability: the $\theta$-group collaboration. Phys. Fluids 29 (10), 105107.CrossRefGoogle Scholar
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, 429–462.CrossRefGoogle Scholar
Tubbs, D.L., Barnes, C.W., Beck, J.B., Hoffman, N.M., Oertel, J.A., Watt, R.G., Boehly, T., Bradley, D., Jaanimagi, P. & Knauer, J. 1999 Cylindrical implosion experiments using laser direct drive. Phys. Plasma 6 (5), 2095–2104.CrossRefGoogle Scholar
Wang, L.F., Wu, J.F., Guo, H.Y., Ye, W.H., Liu, J., Zhang, W.Y. & He, X.T. 2015 Weakly nonlinear Bell–Plesset effects for a uniformly converging cylinder. Phys. Plasma 22 (8), 082702.Google Scholar
Zhai, Z., Zou, L., Wu, Q. & Luo, X. 2018 Review of experimental Richtmyer–Meshkov instability in shock tube: from simple to complex. Proc. Inst. Mech. Engrs 232 (16), 2830–2849.Google Scholar
Zhang, J., Wang, L.F., Wu, J.F., Ye, W.H., Zou, S.Y., Ding, Y.K., Zhang, W.Y. & He, X.T. 2020 The three-dimensional weakly nonlinear Rayleigh–Taylor instability in spherical geometry. Phys. Plasma 27 (2), 022707.Google Scholar
Zhao, Z., Wang, P., Liu, N. & Lu, X. 2020 Analytical model of nonlinear evolution of single-mode Rayleigh–Taylor instability in cylindrical geometry. J. Fluid Mech. 900, A24.CrossRefGoogle Scholar
Zhou, Y. 2017 a Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. I. Phys. Rep. 720, 1–136.Google Scholar
Zhou, Y. 2017 b Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. II. Phys. Rep. 723, 1–160.Google Scholar
Zhou, Y., et al. 2021 Rayleigh–Taylor and Richtmyer–Meshkov instabilities: a journey through scales. Physica D 423, 132838.CrossRefGoogle Scholar