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

Abstract

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.

Keywords

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

References

[1] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Oxford University Press, London, 1961

[2] W. D. Smyth and J. N. Moum, Anisotropy of turbulence in stably stratified mixing layers, Phys. Fluids 12, 1327 (2000)

[3] Y. Matsumoto and M. Hoshino, Onset of turbulence induced by a Kelvin–Helmholtz vortex, Geophys. Res. Lett. 31, L02807 (2004)

[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)

[5] Z. Xia, Y. Shi, and Y. Zhao, Assessment of the shearimproved Smagorinsky model in laminar-turbulent transitional channel flow, J. Turbul. 16, 925 (2015)

[6] Z. Xia, Y. Shi, and S. Chen, Direct numerical simulation of turbulent channel flow with spanwise rotation, J. Fluid Mech. 788, 42 (2016)

[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)

[8] R. P. Drake, High-Energy-Density Physics: Fundamentals, Inertial Fusion and Experimental Astrophysics, Springer, New York, 2006

[9] M. T. Montgomery, V. A. Vladimirov, and P. V. Denissenko, An experimental study on hurricane mesovortices, J. Fluid Mech. 471, 1 (2002)

[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)

[11] S. N. Borovikov and N. V. Pogorelov, Voyager 1 near the heliopause, Astrophys. J. Lett. 783, L16 (2014)

[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)

[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)

[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)

[15] X. T. He and W. Y. Zhang, Inertial fusion research in China, Eur. Phys. J. D 44, 227 (2007)

[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)

[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)

[18] M. Hishida, T. Fujiwara, and P. Wolanski, Fundamentals of rotating detonations, Shock Waves 19, 1 (2009)

[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)

[20] A. Petrarolo, M. Kobald, and S. Schlechtriem, Understanding Kelvin–Helmholtz instability in paraffin-based hybrid rocket fuels, Exp. Fluids 59, 62 (2018)

[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)

[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)

[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)

[24] M. Livio, Astrophysical jets: A phenomenological examination of acceleration and collimation, Phys. Rep. 311, 225 (1999)

[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)

[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)

[27] A. P. Lobanov and J. A. Zensus, A cosmic double helix in the archetypical quasar 3C273, Science 294, 128 (2001)

[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)

[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)

[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)

[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)

[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)

[33] Y. Zhou, Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing (I), Phys. Rep. 720–722, 1 (2017)

[34] Y. Zhou, Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing (II), Phys. Rep. 723–725, 1 (2017)

[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)

[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)

[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)

[38] H. G. Lee and J. Kim, Two-dimensional Kelvin–Helmholtz instabilities of multi-component fluids, Eur. J. Mech. B. Fluids 49, 77 (2015)

[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)

[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)

[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)

[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)

[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)

[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)

[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)

[46] M. Karimi and S. S. Girimaji, Suppression mechanism of Kelvin–Helmholtz instability in compressible fluid flows, Phys. Rev. E 93, 041102(R) (2016)

[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)

[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)

[49] L. F. Wang, W. H. Ye, and Y. J. Li, Numerical investigation on the ablative Kelvin–Helmholtz instability, EPL 87, 54005 (2009)

[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)

[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)

[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)

[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)

[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)

[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)

[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)

[57] S. Li and Q. Li, Thermal non-equilibrium effect of smallscale structures in compressible turbulence, Mod. Phys. Lett. B 32, 1840013 (2018)

[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)

[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)

[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)

[61] B. Yan, A. Xu, G. Zhang, Y. Ying, and H. Li, Lattice Boltzmann model for combustion and detonation, Front. Phys. 8, 94 (2013)

[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)

[63] A. Xu, C. Lin, G. Zhang, and Y. Li, Multiple-relaxationtime lattice Boltzmann kinetic model for combustion, Phys. Rev. E 91, 043306 (2015)

[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)

[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)

[66] C. Lin, A. Xu, G. Zhang, and Y. Li, Double-distributionfunction discrete Boltzmann model for combustion, Combust. Flame 164, 137 (2016)

[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)

[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)

[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)

[70] C. Lin and K. H. Luo, MRT discrete Boltzmann method for compressible exothermic reactive flows, Comput. Fluids 166, 176 (2018)

[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)

[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)

[73] A. Xu, G. Zhang, Y. Zhang, P. Wang, and Y. Ying, Discrete Boltzmann model for implosion- and explosionrelated compressible flow with spherical symmetry, Front. Phys. 13, 135102 (2018)

[74] C. Lin and K. H. Luo, Mesoscopic simulation of nonequilibrium detonation with discrete Boltzmann method, Combust. Flame 198, 356 (2018)

[75] F. Chen, A. Xu and G. Zhang, Collaboration and competition between Richtmyer–Meshkov instability and Rayleigh–Taylor instability, Phys. Fluids 30, 102105 (2018)

[76] P. Henri, S. S. Cerri, F. Califano, F. Pegoraro, C. Rossi, M. Faganello, O. Šebek, P. M. Tráavníček, 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)

[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)

[78] A. Rosenfeld and A. C. Kak, Digital Picture Processing, Academic Press, New York, 1976

[79] V. Sofonea and K. R. Mecke, Morphological characterization of spinodal decomposition kinetics, Eur. Phys. J. B 8, 99 (1999)

[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)

[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)

[82] A. Xu, G. Zhang, X. Pan, P. Zhang and J. Zhu, Morphological characterization of shocked porous material, J. Phys. D 42, 075409 (2009)

[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)

[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)

[85] Y. Zhang, R. Qin, and D. R. Emerson, Lattice Boltzmann simulation of rarefied gas flows in microchannels, Phys. Rev. E 71, 047702 (2005)

[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)

[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)

[88] L. H. Holway, New statistical models for kinetic theory: Methods of construction, Phys. Fluids 9, 1658 (1966)

[89] E. M. Shakhov, Generalization of the Krook kinetic relaxation equation, Fluid Dyn. 3, 95 (1968)

[90] G. Liu, A method for constructing a model form for the Boltzmann equation, Phys. Fluids A 2, 277 (1990)

[91] X. Shan, Simulation of Rayleigh–Bénard convection using a lattice Boltzmann method, Phys. Rev. E 55, 2770 (1997)

[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)

[93] F. Chen, A. Xu, G. Zhang, and Y. Wang, Twodimensional MRT LB model for compressible and incompressible flows, Front. Phys. 9, 246 (2014)

[94] R. Machado, On the moment system and a flexible Prandtl number, Mod. Phys. Lett. B 28, 1450048 (2014)

[95] F. M. White, Viscous Fluid Flow, McGraw-Hill, New York, 1974

[96] G. A. Sod, A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys. 27, 1 (1978)

[97] P. Woodward and P. Colella, The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys. 54, 115 (1984)

[98] G. S. Jiang and C. W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126, 202 (1996)

[99] H. X. Zhang, Non-oscillatory and non-free-parameter dissipation difference scheme, Acta Aerodyna. Sinica 6, 143 (1988)

[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)

[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)

[102] L. M. Yang, C. Shu, and Y. Wang, Development of a discrete gas-kinetic scheme for simulation of twodimensional viscous incompressible and compressible flows, Phys. Rev. E 93, 033311 (2016)

[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)

[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)

[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)

[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)

[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)

[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)

[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)

RIGHTS & PERMISSIONS

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