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An Alternative Lattice Boltzmann Model for Incompressible Flows and its Stabilization

Published online by Cambridge University Press:  07 February 2017

Liangqi Zhang,
Zhong Zeng,
Haiqiong Xie,
Zhouhua Qiu,
Liping Yao,
Yongxiang Zhang and
Yiyu Lu
Liangqi Zhang*
Affiliation:
Department of Engineering Mechanics, College of Aerospace Engineering, Chongqing University, Chongqing 400044, P.R. ChinaState Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing University, Chongqing 400044, P.R. China
Zhong Zeng*
Affiliation:
Department of Engineering Mechanics, College of Aerospace Engineering, Chongqing University, Chongqing 400044, P.R. ChinaState Key laboratory of Crystal Material, Shandong University, Jinan 250100, P.R. China
Haiqiong Xie*
Affiliation:
Department of Engineering Mechanics, College of Aerospace Engineering, Chongqing University, Chongqing 400044, P.R. China
Zhouhua Qiu*
Affiliation:
Department of Engineering Mechanics, College of Aerospace Engineering, Chongqing University, Chongqing 400044, P.R. China
Liping Yao*
Affiliation:
College of Engineering and Technology, Southwest University, Chongqing 400716, P.R. China
Yongxiang Zhang*
Affiliation:
Department of Engineering Mechanics, College of Aerospace Engineering, Chongqing University, Chongqing 400044, P.R. China
Yiyu Lu*
Affiliation:
State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing University, Chongqing 400044, P.R. China
*
*Corresponding author.Email addresses:liangqizhang@cqu.edu.cn, liangqizhang@ntu.edu.sg (L. Zhang), zzeng@cqu.edu.cn (Z. Zeng), xiehaiqiong@cqu.edu.cn (H. Xie), qiuzhouhua@cqu.edu.cn (Z. Qiu), lpyao@swu.edu.cn (L. Yao), zhangyx@cqu.edu.cn (Y. Zhang), luyiyu@cqu.edu.cn (Y. Lu)
*Corresponding author.Email addresses:liangqizhang@cqu.edu.cn, liangqizhang@ntu.edu.sg (L. Zhang), zzeng@cqu.edu.cn (Z. Zeng), xiehaiqiong@cqu.edu.cn (H. Xie), qiuzhouhua@cqu.edu.cn (Z. Qiu), lpyao@swu.edu.cn (L. Yao), zhangyx@cqu.edu.cn (Y. Zhang), luyiyu@cqu.edu.cn (Y. Lu)
*Corresponding author.Email addresses:liangqizhang@cqu.edu.cn, liangqizhang@ntu.edu.sg (L. Zhang), zzeng@cqu.edu.cn (Z. Zeng), xiehaiqiong@cqu.edu.cn (H. Xie), qiuzhouhua@cqu.edu.cn (Z. Qiu), lpyao@swu.edu.cn (L. Yao), zhangyx@cqu.edu.cn (Y. Zhang), luyiyu@cqu.edu.cn (Y. Lu)
*Corresponding author.Email addresses:liangqizhang@cqu.edu.cn, liangqizhang@ntu.edu.sg (L. Zhang), zzeng@cqu.edu.cn (Z. Zeng), xiehaiqiong@cqu.edu.cn (H. Xie), qiuzhouhua@cqu.edu.cn (Z. Qiu), lpyao@swu.edu.cn (L. Yao), zhangyx@cqu.edu.cn (Y. Zhang), luyiyu@cqu.edu.cn (Y. Lu)
*Corresponding author.Email addresses:liangqizhang@cqu.edu.cn, liangqizhang@ntu.edu.sg (L. Zhang), zzeng@cqu.edu.cn (Z. Zeng), xiehaiqiong@cqu.edu.cn (H. Xie), qiuzhouhua@cqu.edu.cn (Z. Qiu), lpyao@swu.edu.cn (L. Yao), zhangyx@cqu.edu.cn (Y. Zhang), luyiyu@cqu.edu.cn (Y. Lu)
*Corresponding author.Email addresses:liangqizhang@cqu.edu.cn, liangqizhang@ntu.edu.sg (L. Zhang), zzeng@cqu.edu.cn (Z. Zeng), xiehaiqiong@cqu.edu.cn (H. Xie), qiuzhouhua@cqu.edu.cn (Z. Qiu), lpyao@swu.edu.cn (L. Yao), zhangyx@cqu.edu.cn (Y. Zhang), luyiyu@cqu.edu.cn (Y. Lu)
*Corresponding author.Email addresses:liangqizhang@cqu.edu.cn, liangqizhang@ntu.edu.sg (L. Zhang), zzeng@cqu.edu.cn (Z. Zeng), xiehaiqiong@cqu.edu.cn (H. Xie), qiuzhouhua@cqu.edu.cn (Z. Qiu), lpyao@swu.edu.cn (L. Yao), zhangyx@cqu.edu.cn (Y. Zhang), luyiyu@cqu.edu.cn (Y. Lu)

Abstract

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In this paper, an alternative lattice Boltzmann (LB)model for incompressible flows is proposed. By modifying directly the moments of the equilibrium distribution function (EDF), the continuous expression of the EDF in tensor Hermite polynomials is derived using the moment expansion and then discretizedwith the discrete velocity vectors of the D2Q9 lattice. The present model as well as its counterpart, the incompressible LB model proposed by Guo, reproduces the incompressible Navier-Stokes (N-S) equations for both steady and unsteady flows. Besides, an alternative pressure formula, which represents the pressure as the diagonal part of the stress tensor, is adopted in the present model. Furthermore, in order to enhance the stability of the present LB model, an additional relaxation time pertaining to the non-hydrodynamic mode is added to the BGK collision operator. The present LB model is validated by two benchmark tests: the cavity flow with different Reynolds number (Re) and the flow past an impulsively started cylinder at Re=40 and 550.

Keywords

Lattice Boltzmann methodincompressible flowscavity flowdeviatoric stresscompressibility effect

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 2 , February 2017, pp. 443 - 465
Copyright
Copyright © Global-Science Press 2017 

References

[1] Yong, W.A., Luo, L.-S., Accuracy of the viscous stress in the lattice Boltzmann equation with simple boundary conditions, Phys. Rev. E 86 (2012) 065701(R).Google Scholar
[2] Ladd, A.J.C., Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1 Theoretical foundation, J. Fluid Mech. 271 (1994) 285–309.Google Scholar
[3] Lallemand, P., Luo, L.-S., Lattice Boltzmann method for moving boundaries, J. Comput. Phys. 184 (2003) 406–421.CrossRefGoogle Scholar
[4] Shan, X.W., Chen, H.D., Lattice Boltzmann model for simulating flows with multiple phases and components, Phys. Rev. E 47 (1993) 1815–1819.CrossRefGoogle ScholarPubMed
[5] Shan, X.W., Chen, H.D., Simulation of nonideal gases and liquid-gas phase transitions by the Lattice Boltzmann equation, Phys. Rev. E 49 (1994) 2941–2948.Google Scholar
[6] Lee, T., Liu, L., Lattice Boltzmann simulations of micron-scale drop impact on dry surfaces, J. Comput. Phys. 229 (2010) 8045–8062.Google Scholar
[7] Junk, M., A finite difference interpretation of the lattice Boltzmann method, Numer. Meth. Part D E 17 (2001) 383–402.CrossRefGoogle Scholar
[8] Junk, M., Klar, A., Discretizations for the incompressible Navier-Stokes equations based on the lattice Boltzmann method, A SIAM J. Sci. Comput. 22 (2000) 1–19.Google Scholar
[9] Dellar, P.J., Incompressible limits of lattice Boltzmann equations using multiple relaxation times, J. Comput. Phys. 190 (2003) 351–370.CrossRefGoogle Scholar
[10] Lallemand, P., Luo, L.S., Theory of lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance, and stability, Phys. Rev. E 61 (2000) 6546–6562.CrossRefGoogle ScholarPubMed
[11] Chapman, S., Cowling, T.G., The mathematical theory of non-uniform gases, 3rd edition, Cambridge: Cambridge University Press; 1970.Google Scholar
[12] Dellar, P.J., Bulk and shear viscosities in lattice Boltzmann equations, Phys. Rev. E 64 (2001) 031203.CrossRefGoogle ScholarPubMed
[13] Dellar, P.J., Nonhydrodynamic modes and a priori construction of shallow water lattice Boltzmann equations, Phys. Rev. E 65 (2002) 036309.Google Scholar
[14] He, X.Y., Luo, L.S., Lattice Boltzmann model for the incompressible Navier-Stokes equation, J. Stat. Phys. 88 (1997) 927–944.CrossRefGoogle Scholar
[15] Luo, L.S., Liao, W., Chen, X.W., Peng, Y., Zhang, W., Numerics of the lattice Boltzmann method: effects of collision models on the lattice Boltzmann simulations, Phys. Rev. E 83 (2011) 056710.CrossRefGoogle ScholarPubMed
[16] Guo, Z.L., Shi, B.C., Wang, N.C., Lattice BGK model for the incompressible Navier-Stokes equation, J. Comput. Phys. 165 (2000) 288–306.Google Scholar
[17] Grad, H., On the Kinetic theory of rarefied gases, Commun. Pure Appl. Math. 2 (1949) 331–407.Google Scholar
[18] Shan, X.W., He, X.Y., Discretization of the velocity space in the solution of the Boltzmann equation, Phys. Rev. Lett. 80 (1998) 65–68.Google Scholar
[19] Zhang, L.Q., Zeng, Z., Xie, H.Q., Zhang, Y.X., Lu, Y.Y., Yoshikawa, A., Mizuseki, H., Kawazoe, Y., A comparative study of lattice Boltzmann models for incompressible flow, Computers and Mathematics with Applications 68 (2014) 1446–1466.CrossRefGoogle Scholar
[20] Dellar, P.J., Non-hydrodynamic modes and general equations of state in lattice Boltzmann equations, Phys. A 362 (2006) 132–138.Google Scholar
[21] Benzi, R., Succi, S., Vergassola, M., Turbulence modeling by Nonhydrodynamic variables, Europhys. Lett. 13 (1990) 727–732.Google Scholar
[22] Benzi, R., Succi, S., Vergassola, M., The lattice Boltzmann equation: theory and applications, Phys. Rep. 222 (1992) 145–197.CrossRefGoogle Scholar
[23] d’Humières, D., Generalized lattice-Boltzmann equations. In: Shizgal, BD, Weave, DP, editors. Rarefied Gas Dynamics: Theory and Simulations. Vol. 159 of Prog. Aatronaut. Aeronaut., 1992. p. 450–458.Google Scholar
[24] Succi, S., The Lattice Boltzmann Equation: For Fluid Dynamics and Beyond, Oxford University Press, Oxford, 2001.CrossRefGoogle Scholar
[25] Higuera, F. J., Succi, S., Benzi, R., Lattice gas dynamics with enhanced collisions, Europhys. Lett. 9 (1989) 345–349.CrossRefGoogle Scholar
[26] Montessori, A., La Rocca, M., Falcucci, G., Succi, S., Regularized lattice BGK versus highly accurate spectral methods for cavity flow simulations, Int. J. Mod. Phys. C 25 (2014) 1441003.Google Scholar
[27] Montessori, A., Falcucci, G., Prestininzi, P., La Rocca, M., Succi, S., Regularized lattice Bhatnagar-Gross-Krook model for two- and three-dimensional cavity flow simulations, Phys. Rev. E 89 (2014) 053317.CrossRefGoogle ScholarPubMed
[28] Latt, J., Chopard, B., Lattice Boltzmann method with regularized pre-collision distribution functions, Math. Comp. Simul. 72 (2006) 165–168.Google Scholar
[29] Bhatnagar, P.L., Gross, E.P., Krook, M., Amodel for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev. 94 (1954) 511–525.Google Scholar
[30] Qian, Y.H., d’Humières, D., Lallemand, P., Lattice BGK models for Navier-Stokes equation, Europhys. Lett. 17 (1992) 479–484.CrossRefGoogle Scholar
[31] He, X.Y., Luo, L.S., Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation, Phys. Rev. E 56 (1997) 6811–6817.Google Scholar
[32] Ghia, U., Ghia, K.N., Shin, C.T., High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J. Comput. Phys. 48 (1982) 387–411.CrossRefGoogle Scholar
[33] Hou, S.L., Zou, Q.S., Simulation of cavity flow by the lattice Boltzmann method. J Comput Phys 118 (1995) 329–347.Google Scholar
[34] Guo, Z.L., Zheng, C.G., Shi, B.C., Non-equilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method, Chin. Phys. Soc. 11 (2002) 366–374.Google Scholar
[35] Albensoeder, S., Kuhlmann, H.C., Rath, H.J., Three-dimensional centrifugal-flow instabilities in the lid-driven-cavity problem, Phys. Fluids 13 (2001) 121–135.CrossRefGoogle Scholar
[36] Theofilis, V., Duck, P.W., Owen, J., Viscous linear stability analysis of rectangular duct and cavity flows, J. Fluid Mech. 505 (2004) 249–286.Google Scholar
[37] Chicheportiche, J., Merle, X., Gloerfelt, X., Robinet, J.C., Direct numerical simulation and global stability analysis of three-dimensional instabilities in a lid-driven cavity, C R Mec. 336 (2008) 586–591.Google Scholar
[38] Non, E., Pierre, R., Gervais, J.J., Linear stability of the three-dimensional lid-driven cavity, Phys. Fluids 18 (2006) 084103.CrossRefGoogle Scholar
[39] Bruneau, C.H., Saad, M., The 2D lid-driven cavity problem revisited, Comput. Fluids 35 (2006) 326–348.Google Scholar
[40] Fortin, A., Jardak, M., Gervais, J.J., Pierre, R., Localization of Hopf bifurcations in fluid flow problems, Int. J. Numer. Methods Fluids 24 (1997) 1185–1210.Google Scholar
[41] Sahin, M., Owens, R.G., A novel fully-implicit finite volume method applied to the lid-driven cavity problem, Parts I and II, Int. J. Numer. Methods Fluids 42 (2003) 1.Google Scholar
[42] Auteri, F., Parolini, N., Quartapelle, L., Numerical investigation on the stability of singular driven cavity flow, J. Comput. Phys. 183 (2002) 1.Google Scholar
[43] Pan, T.W., Glowinski, R., A projection/wave-like equation method for the numerical simulation of incompressible viscous fluid flow modeled by the Navier-Stokes equations, Comp. Fluid Dyn. 9 (2000) 2.Google Scholar
[44] Schreiber, R., Keller, H.B., Driven cavity flows by efficient numerical techniques, J. Comput. Phys. 49 (1983) 310–333.Google Scholar
[45] Boppana, V.B.L., Gajjar, J.S.B., Global flow instability in a lid-driven cavity, Int. J. Numer. Methods Fluids 62 (2010) 827–853.Google Scholar
[46] Li, Y.B., Shock, R., Zhang, R.Y., Chen, H.D., Numerical study of flow past an impulsively started cylinder by the lattice-Boltzmann method, J. Fluid Mech. 519 (2004) 273–300.CrossRefGoogle Scholar
[47] Aderson, C., Reider, M., A high order explicit method for the computations of flow about a circular cylinder, J. Comput. Phys. 125 (1996) 207–224.Google Scholar
[48] Collins, W., Dennis, S., Flow past an impulsively started circular cylinder, J. Fluid Mech. 60 (1973) 105–127.Google Scholar
[49] Koumoutsakos, P., Leonard, A., High-resolution simulations of the flow around an impulsively started cylinder using vortex methods, J. Fluid Mech. 296 (1995) 1–38.CrossRefGoogle Scholar
[50] Dupuis, A., Chatelain, P., Koumoutsakos, P., An immersed boundary-lattice-Boltzmann method for simulation of flow past an impulsively started cylinder, J. Comput. Phys. 227 (2008) 4486–4498.Google Scholar
[51] Verschaeve, J.C.G., Müller, B., A curved no-slip boundary condition for the lattice Boltzmann method, J. Comput. Phys. 229 (2010) 6781–6803.Google Scholar
[52] Chopard, B., Droz, M., Cellular automata modeling of physical systems, Cambridge University Press, 1998.Google Scholar
[53] Yu, D.Z., Mei, R.W., Shyy, W., A unified boundary treatment in lattice Boltzmann method, 41st Aerospace Science Meeting and Exhibit, 6-9 January 2003.Google Scholar
[54] Ladd, A.J.C., Numerical simulation of particular suspensions via a discretized Boltzmann equation, Part 2, Numerical results, J. Fluid Mech. 271 (1994), 311–339.Google Scholar
[55] He, X.Y., Doolen, G., Lattice Boltzmann method on curvilinear coordinates system: Flow around a circular cylinder, J. Comput. Phys. 134 (1997) 306.CrossRefGoogle Scholar
[56] Delouei, A.A., Nazari, M., Kayhani, M.H., Succi, S., Non-Newtonian unconned ow and heat transfer over a heated cylinder using the direct-forcing immersed boundarythermal lattice Boltzmann method, Phys. Rev. E 89 (2014) 053312.Google Scholar