Frontiers of Physics

, 14:43602 | Cite as

Nonequilibrium and morphological characterizations of Kelvin–Helmholtz instability in compressible flows

  • Yan-Biao Gan
  • Ai-Guo XuEmail author
  • Guang-Cai Zhang
  • Chuan-Dong Lin
  • Hui-Lin Lai
  • Zhi-Peng Liu
Research Article


We investigate the effects of viscosity and heat conduction on the onset and growth of Kelvin–Helmholtz instability (KHI) via an efficient discrete Boltzmann model. Technically, two effective approaches are presented to quantitatively analyze and understand the configurations and kinetic processes. One is to determine the thickness of mixing layers through tracking the distributions and evolutions of the thermodynamic nonequilibrium (TNE) measures; the other is to evaluate the growth rate of KHI from the slopes of morphological functionals. Physically, it is found that the time histories of width of mixing layer, TNE intensity, and boundary length show high correlation and attain their maxima simultaneously. The viscosity effects are twofold, stabilize the KHI, and enhance both the local and global TNE intensities. Contrary to the monotonically inhibiting effects of viscosity, the heat conduction effects firstly refrain then enhance the evolution afterwards. The physical reasons are analyzed and presented.


Kelvin–Helmholtz instability discrete Boltzmann method thermodynamic nonequilibrium effect morphological characterization 



Y. G., C. L., H. L. and Z. L. acknowledge the support from the National Natural Science Foundation of China (Grant Nos. 11875001, 51806116, and 11602162), Natural Science Foundation of Hebei Province (Grants Nos. A2017409014 and 2018J01654), Natural Science Foundations of Hebei Educational Commission (Grant No. ZD2017001). A. X. and G. Z. acknowledge the support from the National Natural Science Foundation of China (Grant No. 11772064), CAEP Foundation (Grant No. CX2019033), Science Challenge Project (Grant No. JCKY2016212A501), the opening project of State Key Laboratory of Explosion Science and Technology (Beijing Institute of Technology, Grant No. KFJJ19- 01M).


  1. 1.
    S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Oxford University Press, London, 1961zbMATHGoogle Scholar
  2. 2.
    W. D. Smyth and J. N. Moum, Anisotropy of turbulence in stably stratified mixing layers, Phys. Fluids 12, 1327 (2000)ADSzbMATHCrossRefGoogle Scholar
  3. 3.
    Y. Matsumoto and M. Hoshino, Onset of turbulence induced by a Kelvin–Helmholtz vortex, Geophys. Res. Lett. 31, L02807 (2004)ADSCrossRefGoogle Scholar
  4. 4.
    O. Berné and Y. Matsumoto, The Kelvin–Helmholtz instability in orion: A source of turbulence and chemical mixing, Astrophys. J. Lett. 761, L4 (2012)ADSCrossRefGoogle Scholar
  5. 5.
    Z. Xia, Y. Shi, and Y. Zhao, Assessment of the shear-improved Smagorinsky model in laminar-turbulent transitional channel flow, J. Turbul. 16, 925 (2015)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Z. Xia, Y. Shi, and S. Chen, Direct numerical simulation of turbulent channel flow with spanwise rotation, J. Fluid Mech. 788, 42 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Z. Xia, Y. Shi, Q. Cai, M. Wan, and S. Chen, Multiple states in turbulent plane Couette flow with spanwise rotation, J. Fluid Mech. 837, 477 (2018)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    R. P. Drake, High-Energy-Density Physics: Fundamentals, Inertial Fusion and Experimental Astrophysics, Springer, New York, 2006CrossRefGoogle Scholar
  9. 9.
    M. T. Montgomery, V. A. Vladimirov, and P. V. Denissenko, An experimental study on hurricane mesovortices, J. Fluid Mech. 471, 1 (2002)ADSzbMATHCrossRefGoogle Scholar
  10. 10.
    K. Wada and J. Koda, Instabilities of spiral shocks (I): Onset of wiggle instability and its mechanism, Mon. Not. R. Astron. Soc. 349, 270 (2004)ADSCrossRefGoogle Scholar
  11. 11.
    S. N. Borovikov and N. V. Pogorelov, Voyager 1 near the heliopause, Astrophys. J. Lett. 783, L16 (2014)ADSCrossRefGoogle Scholar
  12. 12.
    K. Avinash, G. P. Zank, B. Dasgupta, and S. Bhadoria, Instability of the heliopause driven by charge exchange interactions, Astrophys. J. Lett. 791, 102 (2014)ADSCrossRefGoogle Scholar
  13. 13.
    H. Hasegawa, M. Fujimoto, T. D. Phan, H. Rème, A. Balogh, M. W. Dunlop, C. Hashimoto, and R. Tan- Dokoro, Transport of solar wind into Earth’s magnetosphere through rolled-up Kelvin–Helmholtz vortices, Nature 430, 755 (2004)ADSCrossRefGoogle Scholar
  14. 14.
    C. Foullon, E. Verwichte, V. M. Nakariakov, K. Nykyri, and C. J. Farrugia, Magnetic Kelvin–Helmholtz instability at the Sun, Astrophys. J. Lett. 729, L8 (2011)ADSCrossRefGoogle Scholar
  15. 15.
    X. T. He and W. Y. Zhang, Inertial fusion research in China, Eur. Phys. J. D 44, 227 (2007)ADSCrossRefGoogle Scholar
  16. 16.
    L. Wang, W. Ye, X. He, J. Wu, Z. Fan, C. Xue, H. Guo, W. Miao, Y. Yuan, J. Dong, G. Jia, J. Zhang, Y. Li, J. Liu, M. Wang, Y. Ding, and W. Zhang, Theoretical and simulation research of hydrodynamic instabilities in inertial-confinement fusion implosions, Sci. China-Phys. Mech. Astron. 60, 055201 (2017)ADSCrossRefGoogle Scholar
  17. 17.
    M. Vandenboomgaerde, M. Bonnefille, and P. Gauthier, The Kelvin–Helmholtz instability in National Ignition Facility hohlraums as a source of gold-gas mixing, Phys. Plasmas 23, 052704 (2016)ADSCrossRefGoogle Scholar
  18. 18.
    M. Hishida, T. Fujiwara, and P. Wolanski, Fundamentals of rotating detonations, Shock Waves 19, 1 (2009)ADSzbMATHCrossRefGoogle Scholar
  19. 19.
    V. Bychkov, D. Valiev, V. Akkerman, and C. K. Law, Gas compression moderates flame acceleration in deflagrationto-detonation transition, Combust. Sci. Technol. 184, 1066 (2012)CrossRefGoogle Scholar
  20. 20.
    A. Petrarolo, M. Kobald, and S. Schlechtriem, Understanding Kelvin–Helmholtz instability in paraffin-based hybrid rocket fuels, Exp. Fluids 59, 62 (2018)CrossRefGoogle Scholar
  21. 21.
    H. Takeuchi, N. Suzuki, K. Kasamatsu, H. Saito, and M. Tsubota, Quantum Kelvin–Helmholtz instability in phase-separated two-component Bose–Einstein condensates, Phys. Rev. B 81, 094517 (2010)ADSCrossRefGoogle Scholar
  22. 22.
    D. Kobyakov, A. Bezett, E. Lundh, M. Marklund, and V. Bychkov, Turbulence in binary Bose-Einstein condensates generated by highly nonlinear Rayleigh–Taylor and Kelvin–Helmholtz instabilities, Phys. Rev. A 89, 013631 (2014)ADSCrossRefGoogle Scholar
  23. 23.
    R. V. Coelho, M. Mendoza, M. M. Doria, and H. J. Herrmann, Kelvin–Helmholtz instability of the Dirac fluid of charge carriers on graphene, Phys. Rev. B 96, 184307 (2017)ADSCrossRefGoogle Scholar
  24. 24.
    M. Livio, Astrophysical jets: A phenomenological examination of acceleration and collimation, Phys. Rep. 311, 225 (1999)ADSCrossRefGoogle Scholar
  25. 25.
    L. F. Wang, W. H. Ye, and Y. J. Li, Combined effect of the density and velocity gradients in the combination of Kelvin–Helmholtz and Rayleigh–Taylor instabilities, Phys. Plasmas 17, 042103 (2010)ADSCrossRefGoogle Scholar
  26. 26.
    W. H. Ye, L. F. Wang, C. Xue, Z. F. Fan, and X. T. He, Competitions between Rayleigh–Taylor instability and Kelvin–Helmholtz instability with continuous density and velocity profiles, Phys. Plasmas 18, 022704 (2011)ADSCrossRefGoogle Scholar
  27. 27.
    A. P. Lobanov and J. A. Zensus, A cosmic double helix in the archetypical quasar 3C273, Science 294, 128 (2001)ADSCrossRefGoogle Scholar
  28. 28.
    B. A. Remington, R. P. Drake, and D. D. Ryutov, Experimental astrophysics with high power lasers and Z pinches, Rev. Mod. Phys. 78, 775 (2006)ADSCrossRefGoogle Scholar
  29. 29.
    X. Luo, F. Zhang, J. Ding, T. Si, J. Yang, Z. Zhai, and C. Wen, Long-term effect of Rayleigh–Taylor stabilization on converging Richtmyer–Meshkov instability, J. Fluid Mech. 849, 231 (2018)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    J. J. Tao, X. T. He, and W. H. Ye, and F. H. Busse, Nonlinear Rayleigh–Taylor instability of rotating inviscid fluids, Phys. Rev. E 87, 013001 (2013)ADSCrossRefGoogle Scholar
  31. 31.
    C. Y. Xie, J. J. Tao, and Z. L. Sun, and J. Li, Retarding viscous Rayleigh–Taylor mixing by an optimized additional mode, Phys. Rev. E 95, 023109 (2017)ADSCrossRefGoogle Scholar
  32. 32.
    W. Liu, C. Yu, H. Jiang, and X. Li, Bell-Plessett effect on harmonic evolution of spherical Rayleigh–Taylor instability in weakly nonlinear scheme for arbitrary Atwood numbers, Phys. Plasmas 24, 022102 (2017)ADSCrossRefGoogle Scholar
  33. 33.
    Y. Zhou, Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing (I), Phys. Rep. 720–722, 1 (2017)MathSciNetzbMATHADSGoogle Scholar
  34. 34.
    Y. Zhou, Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing (II), Phys. Rep. 723–725, 1 (2017)MathSciNetzbMATHADSGoogle Scholar
  35. 35.
    L. F. Wang, W. H. Ye, Z. F. Fan, Y. J. Li, X. T. He and M. Y. Yu, Weakly nonlinear analysis on the Kelvin–Helmholtz instability, EPL 86, 15002 (2009)ADSCrossRefGoogle Scholar
  36. 36.
    U. V. Amerstorfer, N. V. Erkaev, U. Taubenschuss, and H. K. Biernat, Influence of a density increase on the evolution of the Kelvin–Helmholtz instability and vortices, Phys. Plasmas 17, 072901 (2010)ADSCrossRefGoogle Scholar
  37. 37.
    M. Zellinger, U. V. Möstl, N. V. Erkaev, and H. K. Biernat, 2.5D magnetohydrodynamic simulation of the Kelvin–Helmholtz instability around Venus-Comparison of the influence of gravity and density increase, Phys. Plasmas 19, 022104 (2012)ADSCrossRefGoogle Scholar
  38. 38.
    H. G. Lee and J. Kim, Two-dimensional Kelvin–Helmholtz instabilities of multi-component fluids, Eur. J. Mech. B. Fluids 49, 77 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    A. Fakhari and T. Lee, Multiple-relaxation-time lattice Boltzmann method for immiscible fluids at high Reynolds numbers, Phys. Rev. E 87, 023304 (2013)ADSCrossRefGoogle Scholar
  40. 40.
    T. A. Howson, I. De Moortel, and P. Antolin, The effects of resistivity and viscosity on the Kelvin–Helmholtz instability in oscillating coronal loops, Astron. Astrophys. 602, A74 (2017)ADSCrossRefGoogle Scholar
  41. 41.
    K. S. Kim and M. Kim, Simulation of the Kelvin–Helmholtz instability using a multi-liquid moving particle semi-implicit method, Ocean Eng. 130, 531 (2017)CrossRefGoogle Scholar
  42. 42.
    R. Zhang, X. He, G. Doolen, and S. Chen, Surface tension effects on two-dimensional two-phase Kelvin–Helmholtz instabilities, Adv. Water Res. 24, 461 (2001)CrossRefGoogle Scholar
  43. 43.
    N. D. Hamlin and W. I. Newman, Role of the Kelvin–Helmholtz instability in the evolution of magnetized relativistic sheared plasma flows, Phys. Rev. E 87, 043101 (2013)ADSCrossRefGoogle Scholar
  44. 44.
    Y. Liu, Z. H. Chen, H. H. Zhang, and Z. Y. Lin, Physical effects of magnetic fields on the Kelvin–Helmholtz instability in a free shear layer, Phys. Fluids 30, 044102 (2018)ADSCrossRefGoogle Scholar
  45. 45.
    W. C. Wan, G. Malamud, A. Shimony, C. A. Di Stefano, M. R. Trantham, S. R. Klein, D. Shvarts, C. C. Kuranz, and R. P. Drake, Observation of single-mode, Kelvin–Helmholtz instability in a supersonic flow, Phys. Rev. Lett. 115, 145001 (2015)CrossRefGoogle Scholar
  46. 46.
    M. Karimi and S. S. Girimaji, Suppression mechanism of Kelvin–Helmholtz instability in compressible fluid flows, Phys. Rev. E 93, 041102(R) (2016)ADSCrossRefGoogle Scholar
  47. 47.
    Y. Gan, A. Xu, G. Zhang, and Y. Li, Lattice Boltzmann study on Kelvin–Helmholtz instability: Roles of velocity and density gradients, Phys. Rev. E 83, 056704 (2011)ADSCrossRefGoogle Scholar
  48. 48.
    L. F. Wang, C. Xue, W. H. Ye, and Y. J. Li, Destabilizing effect of density gradient on the Kelvin–Helmholtz instability, Phys. Plasmas 16, 112104 (2009)ADSCrossRefGoogle Scholar
  49. 49.
    L. F. Wang, W. H. Ye, and Y. J. Li, Numerical investigation on the ablative Kelvin–Helmholtz instability, EPL 87, 54005 (2009)ADSCrossRefGoogle Scholar
  50. 50.
    L. F. Wang, W. H. Ye, W. Don, Z. M. Sheng, Y. J. Li, and X. T. He, Formation of large-scale structures in ablative Kelvin–Helmholtz instability, Phys. Plasmas 17, 122308 (2010)ADSCrossRefGoogle Scholar
  51. 51.
    R. Asthana and G. S. Agrawal, Viscous potential flow analysis of electrohydrodynamic Kelvin–Helmholtz instability with heat and mass transfer, Int. J. Eng. Sci. 48, 1925 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    M. K. Awasthi, R. Asthana, and G. S. Agrawal, Viscous corrections for the viscous potential flow analysis of magnetohydrodynamic Kelvin–Helmholtz instability with heat and mass transfer, Eur. Phys. J. A 48, 174 (2012)ADSCrossRefGoogle Scholar
  53. 53.
    M. K. Awasthi, R. Asthana, and G. S. Agrawal, Viscous correction for the viscous potential flow analysis of Kelvin–Helmholtz instability of cylindrical flow with heat and mass transfe, Int. J. Heat Mass Transfer 78, 251 (2014)CrossRefGoogle Scholar
  54. 54.
    G. Liu, Y. Wang, G. Zang, and H. Zhao, Viscous Kelvin–Helmholtz instability analysis of liquid-vapor two-phase stratified flow for condensation in horizontal tubes, Int. J. Heat Mass Transfer 84, 592 (2015)CrossRefGoogle Scholar
  55. 55.
    Y. Gan, A. Xu, G. Zhang, and S. Succi, Discrete Boltzmann modeling of multiphase flows: Hydrodynamic and thermodynamic non-equilibrium effects, Soft Matter 11, 5336 (2015)ADSCrossRefGoogle Scholar
  56. 56.
    Y. Gan, A. Xu, G. Zhang, Y. Zhang, and S. Succi, Discrete Boltzmann trans-scale modeling of high-speed compressible flows, Phys. Rev. E 97, 053312 (2018)ADSMathSciNetCrossRefGoogle Scholar
  57. 57.
    S. Li and Q. Li, Thermal non-equilibrium effect of smallscale structures in compressible turbulence, Mod. Phys. Lett. B 32, 1840013 (2018)ADSCrossRefGoogle Scholar
  58. 58.
    A. Xu, G. Zhang, Y. Gan, F. Chen and X. Yu, Lattice Boltzmann modeling and simulation of compressible flows, Front. Phys. 7, 582 (2012)CrossRefGoogle Scholar
  59. 59.
    A. Xu, G. Zhang, Y. Ying, and C. Wang, Complex fields in heterogeneous materials under shock: Modeling, simulation and analysis, Sci. China-Phys. Mech. Astron. 59, 650501 (2016)CrossRefGoogle Scholar
  60. 60.
    Y. Gan, A. Xu, G. Zhang, and Y. Yang, Lattice BGK kinetic model for high-speed compressible flows: Hydrodynamic and nonequilibrium behaviors, EPL 103, 24003 (2013)ADSCrossRefGoogle Scholar
  61. 61.
    B. Yan, A. Xu, G. Zhang, Y. Ying, and H. Li, Lattice Boltzmann model for combustion and detonation, Front. Phys. 8, 94 (2013)CrossRefGoogle Scholar
  62. 62.
    C. Lin, A. Xu, G. Zhang, Y. Li, and S. Succi, Polarcoordinate lattice Boltzmann modeling of compressible flows, Phys. Rev. E 89, 013307 (2014)ADSCrossRefGoogle Scholar
  63. 63.
    A. Xu, C. Lin, G. Zhang, and Y. Li, Multiple-relaxationtime lattice Boltzmann kinetic model for combustion, Phys. Rev. E 91, 043306 (2015)ADSCrossRefGoogle Scholar
  64. 64.
    F. Chen, A. Xu, and G. Zhang, Viscosity, heat conductivity, and Prandtl number effects in the Rayleigh–Taylor instability, Front. Phys. 11, 114703 (2016)CrossRefGoogle Scholar
  65. 65.
    H. Lai, A. Xu, G. Zhang, Y. Gan, Y. Ying, and S. Succi, Nonequilibrium thermohydrodynamic effects on the Rayleigh–Taylor instability in compressible flows, Phys. Rev. E 94, 023106 (2016)ADSCrossRefGoogle Scholar
  66. 66.
    C. Lin, A. Xu, G. Zhang, and Y. Li, Double-distributionfunction discrete Boltzmann model for combustion, Combust. Flame 164, 137 (2016)CrossRefGoogle Scholar
  67. 67.
    Y. Zhang, A. Xu, G. Zhang, C. Zhu, and C. Lin, Kinetic modeling of detonation and effects of negative temperature coefficient, Combust. Flame 173, 483 (2016)CrossRefGoogle Scholar
  68. 68.
    C. Lin, A. Xu, G. Zhang, K. H. Luo, and Y. Li, Discrete Boltzmann modeling of Rayleigh–Taylor instability in two-component compressible flows, Phys. Rev. E 96, 053305 (2017)ADSCrossRefGoogle Scholar
  69. 69.
    C. Lin, K. H. Luo, L. Fei, and S. Succi, A multicomponent discrete Boltzmann model for nonequilibrium reactive flows, Sci. Rep. 7, 14580 (2017)ADSCrossRefGoogle Scholar
  70. 70.
    C. Lin and K. H. Luo, MRT discrete Boltzmann method for compressible exothermic reactive flows, Comput. Fluids 166, 176 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    Y. Gan, A. Xu, G. Zhang, and H. Lai, Three-dimensional discrete Boltzmann models for compressible flows in and out of equilibrium, Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci. 232, 477 (2018)CrossRefGoogle Scholar
  72. 72.
    Y. Zhang, A. Xu, G. Zhang, Z. Chen, and P. Wang, Discrete ellipsoidal statistical BGK model and Burnett equations, Front. Phys. 13, 135101 (2018)CrossRefGoogle Scholar
  73. 73.
    A. Xu, G. Zhang, Y. Zhang, P. Wang, and Y. Ying, Discrete Boltzmann model for implosion- and explosion-related compressible flow with spherical symmetry, Front. Phys. 13, 135102 (2018)CrossRefGoogle Scholar
  74. 74.
    C. Lin and K. H. Luo, Mesoscopic simulation of nonequilibrium detonation with discrete Boltzmann method, Combust. Flame 198, 356 (2018)CrossRefGoogle Scholar
  75. 75.
    F. Chen, A. Xu and G. Zhang, Collaboration and competition between Richtmyer–Meshkov instability and Rayleigh–Taylor instability, Phys. Fluids 30, 102105 (2018)ADSCrossRefGoogle Scholar
  76. 76.
    P. Henri, S. S. Cerri, F. Califano, F. Pegoraro, C. Rossi, M. Faganello, O. Šebek, P. M. Tráavnícek, P. Hellinger, J. T. Frederiksen, A. Nordlund, S. Markidis, R. Keppens, and G. Lapenta, Nonlinear evolution of the magnetized Kelvin–Helmholtz instability: From fluid to kinetic modeling, Phys. Plasmas 20, 102118 (2013)ADSCrossRefGoogle Scholar
  77. 77.
    T. Umeda, N. Yamauchi, Y. Wada, and S. Ueno, Evaluating gyro-viscosity in the Kelvin–Helmholtz instability by kinetic simulations, Phys. Plasmas 23, 054506 (2016)ADSCrossRefGoogle Scholar
  78. 78.
    A. Rosenfeld and A. C. Kak, Digital Picture Processing, Academic Press, New York, 1976zbMATHGoogle Scholar
  79. 79.
    V. Sofonea and K. R. Mecke, Morphological characterization of spinodal decomposition kinetics, Eur. Phys. J. B 8, 99 (1999)ADSCrossRefGoogle Scholar
  80. 80.
    Y. Gan, A. Xu, G. Zhang, Y. Li, and H. Li, Phase separation in thermal systems: A lattice Boltzmann study and morphological characterization, Phys. Rev. E 84, 046715 (2011)ADSCrossRefGoogle Scholar
  81. 81.
    Y. Gan, A. Xu, G. Zhang, P. Zhang and Y. Li, Lattice Boltzmann study of thermal phase separation: Effects of heat conduction, viscosity and Prandtl number, EPL 97, 44002 (2012)Google Scholar
  82. 82.
    A. Xu, G. Zhang, X. Pan, P. Zhang and J. Zhu, Morphological characterization of shocked porous material, J. Phys. D 42, 075409 (2009)ADSCrossRefGoogle Scholar
  83. 83.
    R. Machado, On the generalized Hermite-based lattice Boltzmann construction, lattice sets, weights, moments, distribution functions and high-order models, Front. Phys. 9, 490 (2014)CrossRefGoogle Scholar
  84. 84.
    T. Kataoka and M. Tsutahara, Lattice Boltzmann model for the compressible Navier–Stokes equations with flexible specific-heat ratio, Phys. Rev. E 69, 035701(R) (2004)ADSCrossRefGoogle Scholar
  85. 85.
    Y. Zhang, R. Qin, and D. R. Emerson, Lattice Boltzmann simulation of rarefied gas flows in microchannels, Phys. Rev. E 71, 047702 (2005)ADSCrossRefGoogle Scholar
  86. 86.
    Y. H. Zhang, R. S. Qin, Y. H. Sun, R. W. Barber, and D. R. Emerson, Gas flow in microchannels — A lattice Boltzmann method approach, J. Stat. Phys. 121, 257 (2005)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  87. 87.
    B. I. Green and P. Vedula, A lattice based solution of the collisional Boltzmann equation with applications to microchannel flows, J. Stat. Mech: Theory Exp. P07016 (2013)Google Scholar
  88. 88.
    L. H. Holway, New statistical models for kinetic theory: Methods of construction, Phys. Fluids 9, 1658 (1966)ADSCrossRefGoogle Scholar
  89. 89.
    E. M. Shakhov, Generalization of the Krook kinetic relaxation equation, Fluid Dyn. 3, 95 (1968)ADSCrossRefGoogle Scholar
  90. 90.
    G. Liu, A method for constructing a model form for the Boltzmann equation, Phys. Fluids A 2, 277 (1990)ADSzbMATHCrossRefGoogle Scholar
  91. 91.
    X. Shan, Simulation of Rayleigh–Bénard convection using a lattice Boltzmann method, Phys. Rev. E 55, 2770 (1997)ADSCrossRefGoogle Scholar
  92. 92.
    F. Chen, A. Xu, G. Zhang, Y. Li, and S. Succi, Multiplerelaxation-time lattice Boltzmann approach to compressible flows with flexible specific-heat ratio and Prandtl number, EPL 90, 54003 (2010)ADSCrossRefGoogle Scholar
  93. 93.
    F. Chen, A. Xu, G. Zhang, and Y. Wang, Twodimensional MRT LB model for compressible and incompressible flows, Front. Phys. 9, 246 (2014)CrossRefGoogle Scholar
  94. 94.
    R. Machado, On the moment system and a flexible Prandtl number, Mod. Phys. Lett. B 28, 1450048 (2014)ADSMathSciNetCrossRefGoogle Scholar
  95. 95.
    F. M. White, Viscous Fluid Flow, McGraw-Hill, New York, 1974zbMATHGoogle Scholar
  96. 96.
    G. A. Sod, A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys. 27, 1 (1978)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  97. 97.
    P. Woodward and P. Colella, The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys. 54, 115 (1984)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  98. 98.
    G. S. Jiang and C. W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126, 202 (1996)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  99. 99.
    H. X. Zhang, Non-oscillatory and non-free-parameter dissipation difference scheme, Acta Aerodyna. Sinica 6, 143 (1988)Google Scholar
  100. 100.
    U. M. Ascher, S. J. Ruuth, and R. J. Spiteri, Implicitexplicit Runge–Kutta methods for time-dependent partial differential equations, Appl. Numer. Math. 25, 151 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  101. 101.
    Q. Li, Y. L. He, Y. Wang, and W. Q. Tao, Coupled double-distribution-function lattice Boltzmann method for the compressible Navie–Stokes equations, Phys. Rev. E 76, 056705 (2007)ADSMathSciNetCrossRefGoogle Scholar
  102. 102.
    L. M. Yang, C. Shu, and Y. Wang, Development of a discrete gas-kinetic scheme for simulation of two-dimensional viscous incompressible and compressible flows, Phys. Rev. E 93, 033311 (2016)ADSMathSciNetCrossRefGoogle Scholar
  103. 103.
    F. Gao, Y. Zhang, Z. He, and B. Tian, Formula for growth rate of mixing width applied to Richtmyer–Meshkov instability, Phys. Fluids 28, 114101 (2016)ADSCrossRefGoogle Scholar
  104. 104.
    Y. Zhang, Z. He, F. Gao, X. Li, and B. Tian, Evolution of mixing width induced by general Rayleigh–Taylor instability, Phys. Rev. E 93, 063102 (2016)ADSCrossRefGoogle Scholar
  105. 105.
    F. Gao, Y. Zhang, Z. He, L. Li, and B. Tian, Characteristics of turbulent mixing at late stage of the Richtmyer–Meshkov instability, AIP Adv. 7, 075020 (2017)ADSCrossRefGoogle Scholar
  106. 106.
    F. Lei, J. Ding, T. Si, Z. Zhai and X. Luo, Experimental study on a sinusoidal air/SF6 interface accelerated by a cylindrically converging shock, J. Fluid Mech. 826, 819 (2017)ADSCrossRefGoogle Scholar
  107. 107.
    J. Ding, T. Si, J. Yang, X. Lu, Z. Zhai, and X. Luo, Measurement of a Richtmyer–Meshkov instability at an air-SF6 interface in a semiannular shock tube, Phys. Rev. Lett. 119, 014501 (2017)ADSCrossRefGoogle Scholar
  108. 108.
    B. Guan, Z. Zhai, T. Si, X. Lu, and X. Luo, Manipulation of three-dimensional Richtmyer–Meshkov instability by initial interfacial principal curvatures, Phys. Fluids 29, 032106 (2017)ADSCrossRefGoogle Scholar
  109. 109.
    S. Huang, W. Wang, and X. Luo, Molecular-dynamics simulation of Richtmyer–Meshkov instability on a Li-H2 interface at extreme compressing conditions, Phys. Plasmas 25, 062705 (2018)ADSCrossRefGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Yan-Biao Gan
    • 1
    • 2
  • Ai-Guo Xu
    • 3
    • 4
    • 5
    Email author
  • Guang-Cai Zhang
    • 3
  • Chuan-Dong Lin
    • 6
  • Hui-Lin Lai
    • 2
  • Zhi-Peng Liu
    • 7
  1. 1.North China Institute of Aerospace EngineeringLangfangChina
  2. 2.College of Mathematics and Informatics & FJKLMAAFujian Normal UniversityFuzhouChina
  3. 3.Laboratory of Computational PhysicsInstitute of Applied Physics and Computational MathematicsBeijingChina
  4. 4.State Key Laboratory of Explosion Science and TechnologyBeijing Institute of TechnologyBeijingChina
  5. 5.Center for Applied Physics and Technology, MOE Key Center for High Energy Density Physics Simulations, College of EngineeringPeking UniversityBeijingChina
  6. 6.Center for Combustion Energy, Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Energy and Power EngineeringTsinghua UniversityBeijingChina
  7. 7.Department of Physics, School of ScienceTianjin Chengjian UniversityTianjinChina

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