1. National Key Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, P.O. Box 8009-26, Beijing 100088, China 2. Center for Applied Physics and Technology, MOE Key Center for High Energy Density Physics Simulations, College of Engineering, Peking University, Beijing 100871, China 3. Key Laboratory of Transient Physics, Nanjing University of Science and Technology, Nanjing 210094, China
To kinetically model implosion- and explosion-related phenomena, we present a theoretical framework for constructing a discrete Boltzmann model (DBM) with spherical symmetry in spherical coordinates. To achieve this goal, a key technique is to use localCartesian coordinates to describe the particle velocity in the kinetic model. Therefore, geometric effects, such as divergence and convergence, are described as a “force term”. To better access the nonequilibrium behavior, even though the corresponding hydrodynamic model is one-dimensional, the DBM uses a discrete velocity model (DVM) with three dimensions. A new scheme is introduced so that the DBM can use the same DVM regardless of whether or not there are extra degrees of freedom. As an example, a DVM with 26 velocities is formulated to construct the DBM at the Navier–Stokes level. Via the DBM, one can study simultaneously the hydrodynamic and thermodynamic nonequilibrium behaviors in implosion and explosion processes that are not very close to the spherical center. The extension of the current model to a multiple-relaxation-time version is straightforward.
L. F. Wang, W. H. Ye, X. T. He, J. F. Wu, Z. F. Fan, C. Xue, H. Y. Guo, W. Y. Miao, Y. T. Yuan, J. Q. Dong, G. Jia, J. Zhang, Y. J. Li, J. Liu, M. Wang, Y. K. Ding, and W. Y. Zhang, Theoretical and simulation research of hydrodynamic instabilities in inertial-confinement fusion implosions, Sci. China Phys. Mech. Astron. 60(5), 055201 (2017) https://doi.org/10.1007/s11433-017-9016-x
3
A. G. Xu, G. C. Zhang, Y. J. Ying, and C. Wang, Complex fields in heterogeneous materials under shock: modeling, simulation and analysis, Sci. China Phys. Mech. Astron. 59(5), 650501 (2016) https://doi.org/10.1007/s11433-016-5801-0
4
Z. H. Li and H. X. Zhang, Study on gas kinetic unified algorithm for flows from rarefied transition to continuum, J. Comput. Phys. 193(2), 708 (2004) https://doi.org/10.1016/j.jcp.2003.08.022
5
Z. H. Li, A. P. Peng, H. X. Zhang, and J. Y. Yang, Rarefied gas flow simulations using high-order gaskinetic unified algorithms for Boltzmann model equations, Prog. Aerosp. Sci. 74, 81 (2015) https://doi.org/10.1016/j.paerosci.2014.12.002
6
A. P. Peng, Z. H. Li, J. L. Wu, X. Y. Jiang, Validation and analysis of gas-kinetic unified algorithm for solving Boltzmann model equation with vibrational energy excitation, Acta Physica Sinica 66(20), 204703 (2017)
7
Z. H. Li, A. P. Peng, F. Fang, S. X. Li, and S. Y. Zhang, Gas-kinetic unified algorithm for hypersonic aerothermodynamics covering various flow regimes solving Boltzmann model equation, Acta Physica Sinica 64(22), 204703 (2015)
8
Z. Li, X. Jiang, J. Wu, A. Peng, Gas-kinetic unified algorithm for Boltzmann model equation in rotational nonequilibrium and its application to the whole range flow regimes, Chin. J. Theor. Appl. Mech. 46(3), 336 (2014)
9
W. Lei, J. M. Reese, and Y. Zhang, Solving the Boltzmann equation deterministically by the fast spectral method: Application to gas microflows, J. Fluid Mech. 746(746), 53 (2014)
10
L. Wu, J. Zhang, J. M. Reese, and Y. Zhang, A fast spectral method for the Boltzmann equation for monatomic gas mixtures, J. Comput. Phys. 298(C), 602 (2015) https://doi.org/10.1016/j.jcp.2015.06.019
11
J. Li, C. Zhong, Y. Wang, and C. Zhuo, Implementation of dual time-stepping strategy of the gas-kinetic scheme for unsteady flow simulations, Phys. Rev. E 95(5), 053307 (2017) https://doi.org/10.1103/PhysRevE.95.053307
12
Y. Zhu, C. Zhong, and K. Xu, Implicit unified gaskinetic scheme for steady state solutions in all flow regimes, J. Comput. Phys. 315, 16 (2016) https://doi.org/10.1016/j.jcp.2016.03.038
13
G. H. Tang, Y. H. Zhang, X. J. Gu, and D. R. Emerson, Lattice Boltzmann modelling Knudsen layer effect in non-equilibrium flows, EPL 83(4), 40008 (2008) https://doi.org/10.1209/0295-5075/83/40008
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G. H. Tang, Y. H. Zhang, and D. R. Emerson, Lattice Boltzmann models for nonequilibrium gas flows, Phys. Rev. E 77(4), 046701 (2008) https://doi.org/10.1103/PhysRevE.77.046701
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G. H. Tang, X. J. Gu, R. W. Barber, D. R. Emerson, and Y. H. Zhang, Lattice Boltzmann simulation of nonequilibrium effects in oscillatory gas flow, Phys. Rev. E 78(2), 026706 (2008) https://doi.org/10.1103/PhysRevE.78.026706
16
J. Meng, Y. Zhang, Kinetic diffuse boundary condition for high-order lattice Boltzmann model with streamingcollision mechanism, J. Comput. Phys. 258, 601 (2014) https://doi.org/10.1016/j.jcp.2013.10.057
17
J. Meng and Y. Zhang, Gauss-Hermite quadratures and accuracy of lattice Boltzmann models for nonequilibrium gas flows, Phys. Rev. E 83(3 Pt 2), 036704 (2011) https://doi.org/10.1103/PhysRevE.83.036704
18
J. Meng, Y. Zhang, and X. Shan, Multiscale lattice Boltzmann approach to modeling gas flows, Phys. Rev. E 83(4 Pt 2), 046701 (2010)
19
J. P. Meng, N. Dongari, J. M. Reese, and Y. Zhang, A kinetic switching criterion for hybrid modelling of multiscale gas flows, J. Phys. Confer. Ser. 362(1), 12006 (2012) https://doi.org/10.1088/1742-6596/362/1/012006
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J. Meng, Y. Zhang, N. G. Hadjiconstantinou, G. A. Radtke, and X. Shan, Lattice ellipsoidal statistical BGK model for thermal non-equilibrium flows, J. Fluid Mech. 718(3), 347 (2013) https://doi.org/10.1017/jfm.2012.616
21
K. Xu and J. C. Huang, A Unified Gas-Kinetic Scheme for Continuum and Rarefied Flows, Academic Press Professional, Inc., 2010
22
K. Xu and J. C. Huang, An improved unified gas-kinetic scheme and the study of shock structures, IMA J. Appl. Math. 76(5), 698 (2011) https://doi.org/10.1093/imamat/hxr002
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J. C. Huang, K. Xu, and P. Yu, A unified gas-kinetic scheme for continuum and rarefied flows ii: multidimensional cases, Commun. Comput. Phys. 12(03), 662 (2012) https://doi.org/10.4208/cicp.030511.220911a
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A. Xu, G. Zhang, and Y. Zhang, Discrete Boltzmann modeling of compressible flows, Chapter 2, in: Kinetic Theory, edited by G. Z. Kyzas and A. C. Mitropoulos, Croatia: InTech, 2018 https://doi.org/10.5772/intechopen.70748
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A. G. Xu, G. C. Zhang, and Y. J. Ying, Progress of discrete Boltzmann modeling and simulation of combustion system, Acta Physica Sinica 64 64(18), 184701 (2015)
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C. Lin, A. Xu, G. Zhang, Y. Li, and S. Succi, Polarcoordinate lattice Boltzmann modeling of compressible flows, Phys. Rev. E 89(1), 013307 (2014) https://doi.org/10.1103/PhysRevE.89.013307
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A. Xu, C. Lin, G. Zhang, and Y. Li, Multiple-relaxationtime lattice Boltzmann kinetic model for combustion, Phys. Rev. E 91(4), 043306 (2015) https://doi.org/10.1103/PhysRevE.91.043306
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Y. Gan, A. Xu, G. Zhang, and S. Succi, Discrete Boltzmann modeling of multiphase flows: Hydrodynamic and thermodynamic non-equilibrium effects, Soft Matter 11(26), 5336 (2015) https://doi.org/10.1039/C5SM01125F
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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(2), 023106 (2016) https://doi.org/10.1103/PhysRevE.94.023106
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) https://doi.org/10.1016/j.combustflame.2016.04.003
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Y. Zhang, A. Xu, G. Zhang, and Z. Chen, Discrete Boltzmann method with Maxwell-type boundary condition for slip flow, Commum. Theor. Phys. 69(1), 77 (2018) https://doi.org/10.1088/0253-6102/69/1/77
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Y. Gan, A. Xu, G. Zhang, and H. Lai, Threedimensional discrete Boltzmann models for compressible flows in and out of equilibrium, Proc. Inst. Mech. Eng. C: J. Mech. Eng. Sci. 232(3), 477 (2018) https://doi.org/10.1177/0954406217742181
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C. Lin, A. Xu, G. Zhang, K. Luo, and Y. Li, Discrete Boltzmann modeling of Rayleigh-Taylor instability in bi-component compressible flows, Phys. Rev. E 96(5), 053305 (2017) https://doi.org/10.1103/PhysRevE.96.053305
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C. Lin, K. Luo, L. Fei, and S. Succi, A multi-component discrete Boltzmann model for nonequilibrium reactive flows, Sci. Rep. 7(1), 14580 (2017) https://doi.org/10.1038/s41598-017-14824-9
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Y. Zhang, A. Xu, G. Zhang, Z. Chen, and P. Wang, Discrete ellipsoidal statistical BGK model and Burnett equations, Front. Phys. 13(3), 135101 (2018) https://doi.org/10.1007/s11467-018-0749-3
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M. La Rocca, A. Montessori, P. Prestininzi, and S. Succi, A multispeed discrete Boltzmann model for transcritical 2d shallow water flows, J. Comput. Phys. 284(C), 117 (2015) https://doi.org/10.1016/j.jcp.2014.12.029
38
S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics and Beyond, New York: Oxford University Press, 2001
39
S. Toppaladoddi, S. Succi, and J. S. Wettlaufer, Roughness as a route to the ultimate regime of thermal convection, Phys. Rev. Lett. 118(7), 074503 (2017) https://doi.org/10.1103/PhysRevLett.118.074503
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X. Shan, X. Yuan, and H. Chen, Kinetic theory representation of hydrodynamics: A way beyond the Navier- Stokes equation, J. Fluid Mech. 550, 413 (2006) https://doi.org/10.1017/S0022112005008153
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J. Meng, Y. Zhang, N. G. Hadjiconstantinou, G. A. Radtke, and X. Shan, Lattice ellipsoidal statistical BGK model for thermal non-equilibrium flows, J. Fluid Mech. 718(3), 347 (2013) https://doi.org/10.1017/jfm.2012.616
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D. Sun, M. Zhu, S. Pan, and D. Raabe, Lattice Boltzmann modeling of dendritic growth in a forced melt convection, Acta Mater. 57(6), 1755 (2009) https://doi.org/10.1016/j.actamat.2008.12.019
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Y. Wang, C. Shu, H. B. Huang, and C. J. Teo, Multiphase lattice Boltzmann flux solver for incompressible multiphase flows with large density ratio, J. Comput. Phys. 280(C), 404 (2015) https://doi.org/10.1016/j.jcp.2014.09.035
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Z. Chai, C. Huang, B. Shi, and Z. Guo, A comparative study on the lattice Boltzmann models for predicting effective diffusivity of porous media, Int. J. Heat Mass Transfer 98, 687 (2016) https://doi.org/10.1016/j.ijheatmasstransfer.2016.03.065
L. Chen, L. Zhang, Q. Kang, H. S. Viswanathan, J. Yao, and W. Tao, Nanoscale simulation of shale transport properties using the lattice Boltzmann method: Permeability 445 and diffusivity, Sci. Rep. 5(1), 8089 (2015) https://doi.org/10.1038/srep08089
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C. Zhuo, C. Zhong, and J. Cao, Filter-matrix lattice Boltzmann model for incompressible thermal flows, Phys. Rev. E 85(4), 046703 (2012) https://doi.org/10.1103/PhysRevE.85.046703
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J. Meng and Y. Zhang, Diffuse reflection boundary condition for high-order lattice Boltzmann models with streaming-collision mechanism, J. Comput. Phys. 258(C), 601 (2014) https://doi.org/10.1016/j.jcp.2013.10.057
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L. Wang, G. Zhou, X. Wang, Q. Xiong, and W. Ge, Direct numerical simulation of particle-fluid systems by combining time-driven hard-sphere model and lattice Boltzmann method, Particuology 8(4), 379 (2010) https://doi.org/10.1016/j.partic.2010.07.003
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A. Doostmohammadi, T. N. Shendruk, K. Thijssen, and J. M. Yeomans, Onset of meso-scale turbulence in active nematics, Nat. Commun. 8, 15326 (2017) https://doi.org/10.1038/ncomms15326
51
A. G. Xu, G. C. Zhang, Y. B. Gan, F. Chen, and X. J. Yu, Lattice Boltzmann modeling and simulation of compressible flows, Front. Phys. 7(5), 582 (2012) https://doi.org/10.1007/s11467-012-0269-5
52
M. Watari and M. Tsutahara, Possibility of constructing a multispeed Bhatnagar–Gross–Krook thermal model of the lattice Boltzmann method, Phys. Rev. E 70(1), 016703 (2004) https://doi.org/10.1103/PhysRevE.70.016703
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H. Liu, W. Kang, Q. Zhang, Y. Zhang, H. Duan, and X. T. He, Molecular dynamics simulations of microscopic structure of ultra strong shock waves in dense helium, Front. Phys. 11(6), 115206 (2016) https://doi.org/10.1007/s11467-016-0590-5
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H. Liu, Y. Zhang, W. Kang, P. Zhang, H. Duan, and X. T. He, Molecular dynamics simulation of strong shock waves propagating in dense deuterium, taking into consideration effects of excited electrons, Phys. Rev. E 95, 023201 (2017) https://doi.org/10.1103/PhysRevE.95.023201
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H. Liu, W. Kang, H. Duan, P. Zhang, and X. He, Recent progresses on numerical investigations of microscopic structure of strong shock waves in fluid, Scientia Sinica Physica Mechanica and Astronomica 47(7), 070003 (2017) https://doi.org/10.1360/SSPMA2016-00405
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F. Chen, A. G. Xu, G. C. Zhang, and Y. L. Wang, Two-dimensional MRT LB model for compressible and incompressible flows, Front. Phys. 9(2), 246 (2014) https://doi.org/10.1007/s11467-013-0368-y
