Viscosity, heat conductivity, and Prandtl number effects in the Rayleigh–Taylor Instability

Abstract

The two-dimensional Rayleigh–Taylor instability problem is simulated with a multiple-relaxation-time discrete Boltzmann model with a gravity term. Viscosity, heat conductivity, and Prandtl number effects are probed from macroscopic and nonequilibrium viewpoints. In the macro sense, both viscosity and heat conduction show a significant inhibitory effect in the reacceleration stage, which is mainly achieved by inhibiting the development of the Kelvin–Helmholtz instability. Before this, the Prandtl number effect is not sensitive. Viscosity, heat conductivity, and Prandtl number effects on nonequilibrium manifestations and the degree of correlation between the nonuniformity and the nonequilibrium strength in the complex flow are systematically investigated.

Keywords

discrete Boltzmann model/method / multiple-relaxation-time / Rayleigh–Taylor instability / nonequilibrium

References

[1] L. Rayleigh, Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density, Proc. London Math. Soc. s1–14(1), 170 (1882)

[2] G. Taylor, The instability of liquid surfaces when accelerated in a direction perpendicular to their planes (I), P. Roy. Soc. A 201(1065), 192 (1950)

[3] W. H. Ye, W. Y. Zhang, G. N. Chen, C. Q. Jin, and J. Zhang, Numerical simulations of the FCT method on Rayleigh–Taylor and Richtmyer–Meshkov instabilities, Chin. J. Comput. Phys. 15(3), 277 (1998)

[4] X. L. Li, B. X. Jin, and J. Glimm, Numerical study for the three dimensional Rayleigh–Taylor instability through the TVD/AC scheme and parallel computation, J. Comput. Phys. 126(2), 343 (1996)

[5] G. Tryggvason, B. Bunner, A. Esmaeeli, D. Juric, N. Al-Rawahi, W. Tauber, J. Han, S. Nas, and Y. J. Jan, A front-tracking method for the computations of multiphase flow, J. Comput. Phys. 169(2), 708 (2001)

[6] Y. K. Li and A. Umemura, Mechanism of the large surface deformation caused by Rayleigh–Taylor instability at large Atwood number, J. Appl. Math. Phys. 2(10), 971 (2014)

[7] M. S. Shadloo, A. Zainali, and M. Yildiz, Simulation of single mode Rayleigh–Taylor instability by SPH method, Comput. Mech. 51(5), 699 (2013)

[8] L. Duchemin, C. Josserand, and P. Clavin, Asymptotic behavior of the Rayleigh–Taylor instability, Phys. Rev. Lett. 94(22), 224501 (2005)

[9] A. W. Cook and P. E. Dimotakis, Transition stages of Rayleigh–Taylor instability between miscible fluids, J. Fluid Mech. 443, 69 (2001)

[10] A. Celani, A. Mazzino, and L. Vozella, Rayleigh–Taylor turbulence in two dimensions, Phys. Rev. Lett. 96(13), 134504 (2006)

[11] W. Cabot, Comparison of two- and three-dimensional simulations of miscible Rayleigh–Taylor instability, Phys. Fluids 18(4), 045101 (2006)

[12] A. Celani, A. Mazzino, P. Muratore-Ginanneschi, and L. Vozella, Phase-field model for the Rayleigh–Taylor instability of immiscible fluids, J. Fluid Mech. 622, 115 (2009)

[13] R. Betti and J. Sanz, Bubble acceleration in the ablative Rayleigh–Taylor instability, Phys. Rev. Lett. 97(20), 205002 (2006)

[14] M. R. Gupta, L. Mandal, S. Roy, and M. Khan, Effect of magnetic field on temporal development of Rayleigh– Taylor instability induced interfacial nonlinear structure, Phys. Plasmas 17(1), 012306 (2010)

[15] P. K. Sharma, R. P. Prajapati, and R. K. Chhajlani, Effect of surface tension and rotation on Rayleigh–Taylor instability of two superposed fluids with suspended particles, Acta Phys. Pol. A 118(4), 576 (2010)

[16] R. Banerjee, L. K. Mandal, S. Roy, M. Khan, and M. R. Gupta, Combined effect of viscosity and vorticity on single mode Rayleigh–Taylor instability bubble growth, Phys. Plasmas 18(2), 022109 (2011)

[17] 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)

[18] S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics and Beyond, Oxford: Oxford University Press, 2001

[19] R. Benzi, S. Succi, and M. Vergassola, The lattice Boltzmann equation: Theory and applications, Phys. Rep. 222(3), 145 (1992)

[20] A. Xu, G. Gonnella, and A. Lamura, Phase-separating binary fluids under oscillatory shear, Phys. Rev. E 67(5), 056105 (2003)

[21] A. G. Xu, G. Gonnella, and A. Lamura, Morphologies and flow patterns in quenching of lamellar systems with shear, Phys. Rev. E 74(1), 011505 (2006)

[22] A. G. Xu, G. Gonnella, and A. Lamura, Simulations of complex fluids by mixed lattice Boltzmann-finite difference methods, Physica A 362(1), 42 (2006)

[23] X. Shan and H. Chen, Lattice Boltzmann model for simulating flows with multiple phases and components, Phys. Rev. E 47(3), 1815 (1993)

[24] X. Shan and H. Chen, Simulation of nonideal gases and liquid-gas phase transitions by the lattice Boltzmann equation, Phys. Rev. E 49(4), 2941 (1994)

[25] G. Gonnella, E. Orlandini, and J. M. Yeomans, Spinodal decomposition to a lamellar phase: Effects of hydrodynamic flow, Phys. Rev. Lett. 78(9), 1695 (1997)

[26] H. Fang, Z. Wang, Z. Lin, and M. Liu, Lattice Boltzmann method for simulating the viscous flow in large distensible blood vessels, Phys. Rev. E 65(5), 051925 (2002)

[27] Z. Guo and C. Shu, Lattice Boltzmann Method and Its Applications in Engineering (advances in computational fluid dynamics), World Scientific Publishing Company, 2013

[28] A. Xu, G. Zhang, Y. Li, and H. Li, Modeling and simulation of nonequilibrium and multiphase complex systemslattice Boltzmann kinetic theory and application, Prog. Phys. 34(3), 136 (2014)

[29] R. Zhang, Y. Xu, B. Wen, N. Sheng, and H. Fang, Enhanced permeation of a hydrophobic fluid through particles with hydrophobic and hydrophilic patterned surfaces, Sci. Rep. 4, 5738 (2014)

[30] X. B. Nie, Y. H. Qian, G. D. Doolen, and S. Y. Chen, Lattice Boltzmann simulation of the two-dimensional Rayleigh–Taylor instability, Phys. Rev. E 58(5), 6861 (1998)

[31] X. Y. He, S. Y. Chen, and R. Y. Zhang, A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh–Taylor instability, J. Comput. Phys. 152(2), 642 (1999)

[32] X. Y. He, R. Y. Zhang, S. Y. Chen, and G. D. Doolen, On the three-dimensional Rayleigh–Taylor instability, Phys. Fluids 11(5), 1143 (1999)

[33] R. Y. Zhang, X. Y. He, and S. Y. Chen, Interface and surface tension in incompressible lattice Boltzmann multiphase model, Comput. Phys. Commun. 129(1-3), 121 (2000)

[34] Q. Li, K. H. Luo, Y. J. Gao, and Y. L. He, Additional interfacial force in lattice Boltzmann models for incompressible multiphase flows, Phys. Rev. E 85(2), 026704 (2012)

[35] G. J. Liu and Z. L. Guo, Effects of Prandtl number on mixing process in miscible Rayleigh–Taylor instability: A lattice Boltzmann study, Int. J. Numer. Method. H. 23(1), 176 (2013)

[36] H. Liang, B. C. Shi, Z. L. Guo, and Z. H. Chai, Phasefield- based multiple-relaxation-time lattice Boltzmann model for incompressible multiphase flows, Phys. Rev. E 89(5), 053320 (2014)

[37] M. Sbragaglia, R. Benzi, L. Biferale, H. Chen, X. Shan, and S. Succi, Lattice Boltzmann method with self-consistent thermo-hydrodynamic equilibria, J. Fluid Mech. 628, 299 (2009)

[38] A. Scagliarini, L. Biferale, M. Sbragaglia, K. Sugiyama, and F. Toschi, Lattice Boltzmann methods for thermal flows: Continuum limit and applications to compressible Rayleigh–Taylor systems, Phys. Fluids 22(5), 055101 (2010)

[39] L. Biferale, F. Mantovani, M. Sbragaglia, A. Scagliarini, F. Toschi, and R. Tripiccione, Reactive Rayleigh–Taylor systems: Front propagation and non-stationarity, Europhys. Lett. 94(5), 54004 (2011)

[40] A. Xu, G. Zhang, Y. Gan, F. Chen, and X. Yu, Lattice Boltzmann modeling and simulation of compressible flows, Front. Phys. 7(5), 582 (2012)

[41] B. Yan, A. Xu, G. Zhang, Y. Ying, and H. Li, Lattice Boltzmann model for combustion and detonation, Front. Phys. 8(1), 94 (2013)

[42] C. Lin, A. Xu, G. Zhang, and Y. Li, Polar coordinate lattice Boltzmann kinetic modeling of detonation phenomena, Commum. Theor. Phys. 62(5), 737 (2014)

[43] A. Xu, C. Lin, G. Zhang, and Y. Li, Multiple-relaxationtime lattice Boltzmann kinetic model for combustion, Phys. Rev. E 91(4), 043306 (2015)

[44] A. Xu, G. Zhang, and Y. Ying, Progess of discrete Boltzmann modeling and simulation of combustion system, Acta Phys. Sin. 64(18), 184701 (2015)

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

[46] Y. Zhang, A. Xu, G. Zhang, C. Zhu, and C. Lin, Kinetic modeling of detonation and effects of negative temperature coefficient, Combust. Flame (2016) (in press)

[47] Y. Gan, A. Xu, G. Zhang, and S. Succi, Discrete Boltzmann modeling of multiphase flows: Hydrodynamic and thermodynamic non-equilibrium effects, Soft Matter 11 11(26), 5336 (2015)

[48] 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)

[49] F. Chen, A. Xu, G. Zhang, Y. Wang, Two-dimensional MRT LB model for compressible and incompressible flows, Front. Phys. 9(2), 246 (2014)

[50] H. Lai, A. Xu, G. Zhang, Y. Gan, Y. Ying, and S. Succi, Thermo-hydrodynamic non-equilibrium effects on compressible Rayleigh–Taylor instability, arXiv: 1507.01107

[51] D. Layzer, On the instability of superposed fluids in a gravitational field, Astrophys. J. 122, 1 (1955)

[52] S. F. Li, W. H. Ye, Y. Zhang, S. Shu, and A. G. Xiao, High order FD-WENO schemes for Rayleigh–Taylor instability problems, Chin. J. Comput. Phys. 25(4), 379 (2008)

[53] D. Youngs, Numerical simulation of turbulent mixing by Rayleigh–Taylor instability, Physica D 12(1–3), 32 (1984)

[54] Y. D. Zhang, Modeling and research of detonation based on discrete Boltzmann method, A Dissertation Submitted for the Degree of Master, Beihang University, 2015

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