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Contribution of viscosity to the circulation deposition in the Richtmyer–Meshkov instability

Published online by Cambridge University Press:  15 May 2020

Hao-Chen Liu
Shanghai Jiao Tong University, Shanghai, 200240, China
Bin Yu
Shanghai Jiao Tong University, Shanghai, 200240, China
Hao Chen
Shanghai Jiao Tong University, Shanghai, 200240, China
Bin Zhang*
Shanghai Jiao Tong University, Shanghai, 200240, China
Hui Xu
Shanghai Jiao Tong University, Shanghai, 200240, China
Hong Liu
Shanghai Jiao Tong University, Shanghai, 200240, China
Email address for correspondence:


This study focuses on the process of the circulation deposition in the Richtmyer–Meshkov instability (RMI). The growth rate of circulation and its sources are theoretically and numerically studied to reveal the physical mechanism of the viscosity in the circulation deposition process. We derive a predicting model of the circulation rate for RMI. More importantly, all the contributing sources are separately predicted. Particularly, the viscous source, which previously lacked theoretical or numerical investigations, is efficiently predicted. The RMI problems in a large range of initial conditions are simulated with the direct simulation Monte Carlo (DSMC) method to verify our predicting model and further reveal the circulation deposition mechanism. The DSMC simulations provide reliable quantification of the circulation deposition (especially viscous contribution) for RMI due to its molecular nature. Our model predicts the circulation rate, baroclinic and viscous sources accurately for all the cases in comparison with the simulations. A new physical insight into the mechanism of viscosity in RMI is provided. Unlike the previous understandings that nearly all circulation deposition in RMI comes from the baroclinic source, this study reveals the hidden positive contribution of the viscous source, especially for high Mach number conditions (up to 11 % of total circulation rate). For RMI, the large viscosity gradient inside the shock waves plays a crucial role in the circulation deposition even under high Reynolds number conditions. Our study also provides exciting opportunities to further understand the viscous contribution to the vorticity dynamics in the reshocked RMI and shock wave–turbulence interactions.

JFM Papers
© The Author(s), 2020. Published by Cambridge University Press

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Abarzhi, S. I., Gauthier, S. & Sreenivasan, K. R. 2013 Turbulent mixing and beyond: non-equilibrium processes from atomistic to astrophysical scales II. Phil. Trans. R. Soc. A 371 (2003), 20130268.Google ScholarPubMed
Alexander, F. J., Garcia, A. L. & Alder, B. J. 1998 Cell size dependence of transport coefficients in stochastic particle algorithms. Phys. Fluids 10 (6), 15401542.CrossRefGoogle Scholar
Alsmeyer, H. 1976 Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam. J. Fluid Mech. 74 (3), 497513.CrossRefGoogle Scholar
Andreopoulos, Y., Agui, J. H. & Briassulis, G. 2000 Shock waveturbulence interactions. Annu. Rev. Fluid Mech. 32 (1), 309345.CrossRefGoogle Scholar
Balakumar, B. J., Orlicz, G. C., Ristorcelli, J. R., Balasubramanian, S., Prestridge, K. P. & Tomkins, C. D. 2012 Turbulent mixing in a Richtmyer–Meshkov fluid layer after reshock: velocity and density statistics. J. Fluid Mech. 696, 6793.CrossRefGoogle Scholar
Balakumar, B. J., Orlicz, G. C., Tomkins, C. D. & Prestridge, K. P. 2008 Simultaneous particle-image velocimetry–planar laser-induced fluorescence measurements of Richtmyer–Meshkov instability growth in a gas curtain with and without reshock. Phys. Fluids 20 (12), 124103.CrossRefGoogle Scholar
Barber, J. L., Kadau, K., Germann, T. C. & Alder, B. J. 2008 Initial growth of the Rayleigh–Taylor instability via molecular dynamics. Eur. Phys. J. B 64 (2), 271276.CrossRefGoogle Scholar
Barber, J. L., Kadau, K., Germann, T. C. & Alder, B. J. 2007 Simulation of fluid instabilities using atomistic methods. In AIP Conf. Proc., vol. 955, pp. 301304. AIP.Google Scholar
Bird, G. A. 1970 Aspects of the structure of strong shock waves. Phys. Fluids 13 (5), 11721177.CrossRefGoogle Scholar
Bird, G. A. 1994 Molecular Gas Dynamics and the Direct Simulation of Gas Flows, 1st edn, Oxford Engineering Science Series, Appendix A. Oxford University Press; Clarendon.Google Scholar
Boyd, I. D. 2003 Predicting breakdown of the continuum equations under rarefied flow conditions. AIP Conf. Proc. 663, 899906.Google Scholar
Brenner, H. 2005a Kinematics of volume transport. Physica A 349 (1–2), 1159.CrossRefGoogle Scholar
Brenner, H. 2005b Navier–Stokes revisited. Physica A 349 (1–2), 60132.CrossRefGoogle Scholar
Brenner, H. 2009 Bi-velocity hydrodynamics. Physica A 388 (17), 33913398.CrossRefGoogle Scholar
Brouillette, M. 2002 The Richtmyer–Meshkov instability. Annu. Rev. Fluid Mech. 34 (1), 445468.CrossRefGoogle Scholar
Carles, P. & Popinet, S. 2001 Viscous nonlinear theory of Richtmyer–Meshkov instability. Phys. Fluids 13 (7), 18331836.CrossRefGoogle Scholar
Chen, H., Zhang, B. & Liu, H. 2016 Non-Rankine–Hugoniot shock zone of Mach reflection in hypersonic rarefied flows. J. Spacecr. Rockets 53 (4), 619628.CrossRefGoogle Scholar
Chen, H., Zhang, B. & Liu, H. 2019 On the particle discretization in hypersonic nonequilibrium flows with the direct simulation Monte Carlo method. Phys. Fluids 31 (7), 076102.Google Scholar
Fraley, G. 1986 Rayleigh–Taylor stability for a normal shock wave–density discontinuity interaction. Phys. Fluids 29 (2), 376386.CrossRefGoogle Scholar
Gallis, M. A., Koehler, T. P., Torczynski, J. R. & Plimpton, S. J. 2016 Direct simulation Monte Carlo investigation of the Rayleigh–Taylor instability. Phys. Rev. Fluids 1 (4), 043403.CrossRefGoogle Scholar
Gallis, M. A., Koehler, T. P., Torczynski, J. R. & Plimpton, S. J. 2015 Direct simulation Monte Carlo investigation of the Richtmyer–Meshkov instability. Phys. Fluids 27 (8), 084105.CrossRefGoogle Scholar
Garcia, A. L. & Wagner, W. 2000 Time step truncation error in direct simulation Monte Carlo. Phys. Fluids 12 (10), 26212633.CrossRefGoogle Scholar
Gilbarg, D. & Paolucci, D. 1953 The structure of shock waves in the continuum theory of fluids. J. Ration. Mech. Anal. 2, 617642.Google Scholar
Govindarajan, R. & Sahu, K. C. 2014 Instabilities in viscosity-stratified flow. Annu. Rev. Fluid Mech. 46, 331353.CrossRefGoogle Scholar
Greenshields, C. J. & Reese, J. M. 2007 The structure of shock waves as a test of Brenner’s modifications to the Navier–Stokes equations. J. Fluid Mech. 580, 407429.CrossRefGoogle Scholar
Hadjiconstantinou, N. G., Garcia, A. L., Bazant, M. Z. & He, G. 2003 Statistical error in particle simulations of hydrodynamic phenomena. J. Comput. Phys. 187 (1), 274297.CrossRefGoogle Scholar
Hawley, J. F. & Zabusky, N. J. 1989 Vortex paradigm for shock-accelerated density-stratified interfaces. Phys. Rev. Lett. 63 (12), 1241.CrossRefGoogle ScholarPubMed
Hejazialhosseini, B., Rossinelli, D. & Koumoutsakos, P. 2013 Vortex dynamics in 3D shock–bubble interaction. Phys. Fluids 25 (11), 110816.Google Scholar
Henderson, L. F. 1964 On the confluence of three shock waves in a perfect gas. Aeronaut. Q. 15 (2), 181197.CrossRefGoogle Scholar
Henderson, L. F. 1966 The refraction of a plane shock wave at a gas interface. J. Fluid Mech. 26 (3), 607637.CrossRefGoogle Scholar
Hewett, J. S. & Madnia, C. K. 1998 Flame–vortex interaction in a reacting vortex ring. Phys. Fluids 10 (1), 189205.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, 5579.CrossRefGoogle Scholar
Huang, S., Wang, W. & Luo, X. 2018 Molecular-dynamics simulation of Richtmyer–Meshkov instability on a Li-H2 interface at extreme compressing conditions. Phys. Plasmas 25 (6), 062705.CrossRefGoogle Scholar
Kadau, K., Barber, J. L., Germann, T. C. & Alder, B. J. 2008 Scaling of atomistic fluid dynamics simulations. Phys. Rev. E 78 (4), 045301.Google ScholarPubMed
Kadau, K., Barber, J. L., Germann, T. C., Holian, B. L. & Alder, B. J. 2010 Atomistic methods in fluid simulation. Phil. Trans. R. Soc. A 368 (1916), 15471560.CrossRefGoogle ScholarPubMed
Kadau, K., Germann, T. C., Hadjiconstantinou, N. G., Lomdahl, P. S., Dimonte, G., Holian, B. L. & Alder, B. J. 2004 Nanohydrodynamics simulations: an atomistic view of the Rayleigh–Taylor instability. Proc. Natl Acad. Sci. USA 101 (16), 58515855.CrossRefGoogle ScholarPubMed
Kadau, K., Rosenblatt, C., Barber, J. L., Germann, T. C., Huang, Z., Carlès, P. & Alder, B. J. 2007 The importance of fluctuations in fluid mixing. Proc. Natl Acad. Sci. USA 104 (19), 77417745.CrossRefGoogle ScholarPubMed
Kevlahan, N. K.-R. 1997 The vorticity jump across a shock in a non-uniform flow. J. Fluid Mech. 341, 371384.CrossRefGoogle Scholar
Kotelnikov, A. D., Ray, J. & Zabusky, N. J. 2000 Vortex morphologies on reaccelerated interfaces: visualization, quantification and modeling of one-and two-mode compressible and incompressible environments. Phys. Fluids 12 (12), 32453264.CrossRefGoogle Scholar
Lee, D.-K., Peng, G. & Zabusky, N. J. 2006 Circulation rate of change: a vortex approach for understanding accelerated inhomogeneous flows through intermediate times. Phys. Fluids 18 (9), 097102.CrossRefGoogle Scholar
Li, Y., Wang, Z., Yu, B., Zhang, B. & Liu, H. 2019 Gaussian models for late-time evolution of two-dimensional shock–light cylindrical bubble interaction. Shock Waves 30, 169184.CrossRefGoogle Scholar
Liepmann, H. W., Narasimha, R. & Chahine, M. T. 1962 Structure of a plane shock layer. Phys. Fluids 5 (11), 13131324.CrossRefGoogle Scholar
Liu, H., Chen, H., Yu, B., Zhang, B. & Liu, H. 2019 On the shock/step-interface interaction in microscale conditions. In AIP Conf. Proc., vol. 2132, p. 070027. AIP Publishing.Google Scholar
Liu, H., Chen, H., Zhang, B. & Liu, H. 2018 Effects of Mach number on non-Rankine–Hugoniot shock zone of Mach reflection. J. Spacecr. Rockets 56 (3), 761770.CrossRefGoogle Scholar
Livescu, D. 2013 Numerical simulations of two-fluid turbulent mixing at large density ratios and applications to the Rayleigh–Taylor instability. Phil. Trans. R. Soc. Lond. A 371 (2003), 20120185.CrossRefGoogle ScholarPubMed
Lumpkin, F. E. III & Chapman, D. R. 1992 Accuracy of the Burnett equations for hypersonic real gas flows. J. Thermophys. Heat Transfer 6 (3), 419425.CrossRefGoogle Scholar
McFarland, J., Reilly, D., Creel, S., McDonald, C., Finn, T. & Ranjan, D. 2014 Experimental investigation of the inclined interface Richtmyer–Meshkov instability before and after reshock. Exp. Fluids 55 (1), 1640.CrossRefGoogle Scholar
Meshkov, E. E. 1969 Instability of the interface of two gases accelerated by a shock wave. Fluid Dyn. 4 (5), 101104.Google Scholar
Mikaelian, K. O. 1993 Effect of viscosity on Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Phys. Rev. E 47 (1), 375.Google ScholarPubMed
Morgan, R. V., Aure, R., Stockero, J. D., Greenough, J. A., Cabot, W., Likhachev, O. A. & Jacobs, J. W. 2012 On the late-time growth of the two-dimensional Richtmyer–Meshkov instability in shock tube experiments. J. Fluid Mech. 712, 354383.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, 85124.CrossRefGoogle Scholar
Paolucci, S. & Paolucci, C. 2018 A second-order continuum theory of fluids. J. Fluid Mech. 846, 686710.CrossRefGoogle Scholar
Peng, G., Zabusky, N. J. & Zhang, S. 2003 Vortex-accelerated secondary baroclinic vorticity deposition and late-intermediate time dynamics of a two-dimensional Richtmyer–Meshkov interface. Phys. Fluids 15 (12), 37303744.CrossRefGoogle Scholar
Picone, J. M. & Boris, J. P. 1988 Vorticity generation by shock propagation through bubbles in a gas. J. Fluid Mech. 189, 2351.CrossRefGoogle Scholar
Picone, J. M., Oran, E. S., Boris, J. P. & Young, T. R. Jr 1984 Theory of vorticity generation by shock wave and flame interactions. Tech. Rep. Naval Research Lab, Washington DC.Google Scholar
Polyanin, A. D. & Zaitsev, V. F. 2017 Handbook of Ordinary Differential Qquations: Exact Solutions, Methods, and Problems. Chapman and Hall/CRC.CrossRefGoogle Scholar
Quirk, J. J. & Karni, S. 1996 On the dynamics of a shock–bubble interaction. J. Fluid Mech. 318, 129163.CrossRefGoogle Scholar
Ranjan, D., Oakley, J. & Bonazza, R. 2011 Shock–bubble interactions. Annu. Rev. Fluid Mech. 43, 117140.CrossRefGoogle Scholar
Reinaud, J., Joly, L. & Chassaing, P. 2000 The baroclinic secondary instability of the two-dimensional shear layer. Phys. Fluids 12 (10), 24892505.CrossRefGoogle Scholar
Ren, W., Liu, H. & Jin, S. 2014 An asymptotic-preserving Monte Carlo method for the Boltzmann equation. J. Comput. Phys. 276, 380404.CrossRefGoogle Scholar
Renard, P.-H., Thevenin, D., Rolon, J.-C. & Candel, S. 2000 Dynamics of flame/vortex interactions. Prog. Energy Combust. Sci. 26 (3), 225282.CrossRefGoogle Scholar
Richtmyer, R. D. 1960 Taylor instability in shock acceleration of compressible fluids. Commun. Pure Appl. Maths 13 (2), 297319.CrossRefGoogle Scholar
Rudinger, G. & Somers, L. M. 1960 Behaviour of small regions of different gases carried in accelerated gas flows. J. Fluid Mech. 7 (2), 161176.CrossRefGoogle Scholar
Samtaney, R. & Meiron, D. I. 1997 Hypervelocity Richtmyer–Meshkov instability. Phys. Fluids 9 (6), 17831803.CrossRefGoogle Scholar
Samtaney, R., Ray, J. & Zabusky, N. J. 1998 Baroclinic circulation generation on shock accelerated slow/fast gas interfaces. Phys. Fluids 10 (5), 12171230.CrossRefGoogle Scholar
Samtaney, R. & Zabusky, N. J. 1993 On shock polar analysis and analytical expressions for vorticity deposition in shock-accelerated density-stratified interfaces. Phys. Fluids A 5 (6), 12851287.CrossRefGoogle Scholar
Samtaney, R. & Zabusky, N. J. 1994 Circulation deposition on shock-accelerated planar and curved density-stratified interfaces: models and scaling laws. J. Fluid Mech. 269, 4578.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 (2), 026319.Google ScholarPubMed
Sohn, S.-I. 2009 Effects of surface tension and viscosity on the growth rates of Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Phys. Rev. E 80 (5), 055302.Google ScholarPubMed
Uribe, F. J. & Velasco, R. M. 2018 Shock-wave structure based on the Navier–Stokes-Fourier equations. Phys. Rev. E 97 (4), 043117.Google ScholarPubMed
Walchli, B. & Thornber, B. 2017 Reynolds number effects on the single-mode Richtmyer–Meshkov instability. Phys. Rev. E 95 (1), 013104.Google ScholarPubMed
Wang, W. L. & Boyd, I. D. 2003 Predicting continuum breakdown in hypersonic viscous flows. Phys. Fluids 15 (1), 91100.CrossRefGoogle Scholar
Wang, Z., Yu, B., Chen, H., Zhang, B. & Liu, H. 2018 Scaling vortex breakdown mechanism based on viscous effect in shock cylindrical bubble interaction. Phys. Fluids 30 (12), 126103.CrossRefGoogle Scholar
Weber, C. R., Clark, D. S., Cook, A. W., Busby, L. E. & Robey, H. F. 2014 Inhibition of turbulence in inertial-confinement-fusion hot spots by viscous dissipation. Phys. Rev. E 89 (5), 053106.Google ScholarPubMed
Wu, J., Ma, H. & Zhou, M. 2007 Vorticity and Vortex Dynamics, 1st edn, chap. 2. Springer.Google Scholar
Wu, Z., Huang, S., Ding, J., Wang, W. & Luo, X. 2018 Molecular dynamics simulation of cylindrical Richtmyer–Meshkov instability. Sci. China Phys., Mech. Astron. 61 (11), 114712.CrossRefGoogle Scholar
Yang, J., Kubota, T. & Zukoski, E. E. 1991 An analytical and computational investigation of shock-induced vortical flows. In 30th Aerospace Sciences Meeting and Exhibit, p. 316. American Institute of Aeronautics and Astronautics.Google Scholar
Yang, J., Kubota, T. & Zukoski, E. E. 1994 A model for characterization of a vortex pair formed by shock passage over a light-gas inhomogeneity. J. Fluid Mech. 258, 217244.CrossRefGoogle Scholar
Yang, X., Chern, I.-L., Zabusky, N. J., Samtaney, R. & Hawley, J. F. 1992 Vorticity generation and evolution in shock-accelerated density-stratified interfaces. Phys. Fluids A 4 (7), 15311540.CrossRefGoogle Scholar
Zabusky, N. J. & Zeng, S. M. 1998 Shock cavity implosion morphologies and vortical projectile generation in axisymmetric shock–spherical fast/slow bubble interactions. J. Fluid Mech. 362, 327346.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 (1), 495536.CrossRefGoogle Scholar
Zhakhovskii, V. V., Zybin, S. V., Abarzhi, S. I. & Nishihara, K. 2006 Atomistic dynamics of the Richtmyer–Meshkov instability in cylindrical and planar geometries. In AIP Conf. Proc., vol. 845, pp. 433436. AIP.Google Scholar
Zhang, B., Chen, H., Yu, B., He, M. & Liu, H. 2019 Molecular simulation on viscous effects for microscale combustion in reactive shock–bubble interaction. Combust. Flame 208, 351363.CrossRefGoogle Scholar
Zhang, B., Liu, H. & Jin, S. 2016 An asymptotic preserving Monte Carlo method for the multispecies Boltzmann equation. J. Comput. Phys. 305, 575588.CrossRefGoogle Scholar
Zhang, S., Peng, G. & Zabusky, N. J. 2005 Vortex dynamics and baroclinically forced inhomogeneous turbulence for shock-planar heavy curtain interactions. J. Turbul. 6 (6), N3.Google Scholar
Zhou, Y. 2017a Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. I. Phys. Rep. 720–722, 1136.Google Scholar
Zhou, Y. 2017b Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. II. Phys. Rep. 723, 1160.Google Scholar