Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-24T00:55:43.767Z Has data issue: false hasContentIssue false
  • English
  • Français

A Fully Implicit Finite Volume Lattice Boltzmann Method for Turbulent Flows

Part of: Basic methods in fluid mechanics Turbulence Incompressible viscous fluids

Published online by Cambridge University Press:  21 June 2017

Fatih Çevik  and
Kahraman Albayrak
Fatih Çevik*
Affiliation:
Middle East Technical University, Mechanical Engineering Department, Çankaya, Ankara, Turkey Aselsan Inc., MGEO Division, Unmanned and Autonomous Systems Design Department, Akyurt Ankara, Turkey
Kahraman Albayrak*
Affiliation:
Middle East Technical University, Mechanical Engineering Department, Çankaya, Ankara, Turkey
*
*Corresponding author. Email addresses:[email protected] (F. Çevik), [email protected] (K. Albayrak)
*Corresponding author. Email addresses:[email protected] (F. Çevik), [email protected] (K. Albayrak)
Get access

Abstract

Almost all schemes existed in the literature to solve the Lattice Boltzmann Equation like stream & collide, finite difference, finite element, finite volume schemes are explicit. However, it is known fact that implicit methods utilizes better stability and faster convergence compared to the explicit methods. In this paper, a method named herein as Implicit Finite Volume Lattice Boltzmann Method (IFVLBM) for incompressible laminar and turbulent flows is proposed and it is applied to some 2D benchmark test cases given in the literature. Alternating Direction Implicit, an approximate factorization method is used to solve the obtained algebraic system. The proposed method presents a very good agreement for all the validation cases with the literature data. The proposed method shows good stability characteristics, the CFL number is eased. IFVLBM has about 2 times faster convergence rate compared with Implicit-Explicit Runge Kutta method even though it possesses a computational burden from the solution of algebraic systems of equations.

Keywords

Implicit finite volumelattice Boltzmann methodturbulent flow

MSC classification

Secondary: 76M12: Finite volume methods 76M28: Particle methods and lattice-gas methods 76D99: None of the above, but in this section 76F99: None of the above, but in this section
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 2 , August 2017 , pp. 393 - 421
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Communicated by Boo-Cheong Khoo

References

[1] 2d zero pressure gradient flat plate verification case. http://turbmodels.larc.nasa.gov/flatplate.html. Accessed: 2015.02.20.Google Scholar
[2] Ascher, Uri M, Ruuth, Steven J, and Spiteri, Raymond J. Implicit-explicit runge-kutta methods for time-dependent partial differential equations. Applied Numerical Mathematics, 25(2-3):151–167, November 1997.CrossRefGoogle Scholar
[3] Bhatnagar, P. L., Gross, E. P., and Krook, M.. A model for collision processes in gases. i. small amplitude processes in charged and neutral one-component systems. Phys. Rev., 94:511–525, May 1954.CrossRefGoogle Scholar
[4] Blazek, J.. Computational Fluid Dynamics: Principles and Applications. Elsevier, Sep 2005.Google Scholar
[5] Rajani, Sekhar Majumdar B.N., Kandasamy, A.. Numerical simulation of laminar flow past a circular cylinder. Applied Mathematical Modelling, 33:1228–1247, February 2008.CrossRefGoogle Scholar
[6] Cao, Nianzheng, Chen, Shiyi, Jin, Shi, and Martinez, Daniel. Physical symmetry and lattice symmetry in the lattice boltzmann method. Phys. Rev. E, 55:R21–R24, Jan 1997.CrossRefGoogle Scholar
[7] Chew, Y. T., Shu, C., and Niu, X. D.. A new differential lattice boltzmann equation and its application to simulate incompressible flows on non-uniform grids. Journal of Statistical Physics, 107(April):329–342, 2002.CrossRefGoogle Scholar
[8] Coutanceau, Madeleine and Bouard, Roger. Experimental determination of themain features of the viscous flow in the wake of a circular cylinder in uniform translation. part 1. steady flow. Journal of Fluid Mechanics, 79:231–256, 2 1977.CrossRefGoogle Scholar
[9] Crouse, Bernd, Rank, Ernst, Krafczyk, Manfred, and Tölke, Jonas. a lb-based approach for adaptive flow simulations. International Journal of Modern Physics B, 17(01n02):109–112, January 2003.CrossRefGoogle Scholar
[10] Ding, H. and Shu, C.. A stencil adaptive algorithm for finite difference solution of incompressible viscous flows. Journal of Computational Physics, 214(1):397–420, 2006.CrossRefGoogle Scholar
[11] Dupuis, Alexandre and Chopard, Bastien. Theory and applications of an alternative lattice boltzmann grid refinement algorithm. Physical review. E, Statistical, nonlinear, and soft matter physics, 67(6 Pt 2):066707, June 2003.CrossRefGoogle Scholar
[12] Filippova, O. and Hänel, D.. Boundary-fitting and local grid refinement for lattice-bgk models. International Journal of Modern Physics C, 09:1271–1279, 1998.CrossRefGoogle Scholar
[13] Frisch, U., Hasslacher, B., and Pomeau, Y.. Lattice-gas automata for the navier-stokes equation. Physical Review Letters, 56:1505–1508, 1986.CrossRefGoogle ScholarPubMed
[14] Ghia, U, Ghia, K.N, and Shin, C.T. High-re solutions for incompressible flow using the navier-stokes equations and a multigrid method. Journal of Computational Physics, 48:387–411, 1982.CrossRefGoogle Scholar
[15] Guo, Zhaoli and Shu, Chang. Lattice Boltzmann Method and Its Applications in Engineering. World Scientific Publishing Company, Mar 2013.CrossRefGoogle Scholar
[16] Guo, Zhaoli and Zhao, T S. Explicit finite-difference lattice boltzmann method for curvilinear coordinates. Physical review. E, Statistical, nonlinear, and soft matter physics, 67(April):066709, 2003.CrossRefGoogle ScholarPubMed
[17] Guo, Zhaoli, Zheng, Chuguang, and Shi, Baochang. An extrapolation method for boundary conditions in lattice boltzmann method. Physics of Fluids, 14(6):2007–2010, 2002.CrossRefGoogle Scholar
[18] Guo, Zhaoli, Zheng, Chuguang, and Shi, Baochang. Domain-decomposition technique in lattice boltzmann method. International Journal of Modern Physics B, 17(01n02):129–133, January 2003.CrossRefGoogle Scholar
[19] Guzel, Goktan and Koc, Ilteris. Simulation of turbulent flows using a finite-volume based lattice boltzmann flow solver. Communications in Computational Physics, 17:213–232, 1 2015.CrossRefGoogle Scholar
[20] He, X and Doolen, G D. Lattice-boltzmann method on curvilinear coordinates system: flow around a circular cylinder. Journal of Computational Physics, 134:306–315, 1997.CrossRefGoogle Scholar
[21] He, Xiaoyi and Luo, Li-Shi. Lattice boltzmann model for the incompressible navier stokes equation. Journal of Statistical Physics, 88:927–944, 1997.CrossRefGoogle Scholar
[22] He, Xiaoyi, Luo, Li-Shi, and Dembo, Micah. Some progress in lattice boltzmann method. part i. nonuniform mesh grids. Journal of Computational Physics, 129:357–363, 1996.CrossRefGoogle Scholar
[23] He, Xiaoyi, Luo, Li-Shi, and Dembo, Micah. Some progress in the lattice boltzmann method: Reynolds number enhancement in simulations. Physica A: Statistical Mechanics and its Applications, 239(13):276–285, 1997.CrossRefGoogle Scholar
[24] Imamura, Taro, Suzuki, Kojiro, Nakamura, Takashi, and Yoshida, Masahiro. Acceleration of steady-state lattice boltzmann simulations on non-uniform mesh using local time step method. Journal of Computational Physics, 202:645–663, 2005.CrossRefGoogle Scholar
[25] Kwak, D., Kiris, C. C. Computation of Viscous Incompressible Flows. Springer, 2011.CrossRefGoogle Scholar
[26] Lee, Taehun and Lin, Ching-Long. A characteristic galerkin method for discrete boltzmann equation. Journal of Computational Physics, 171(1):336–356, 2001.CrossRefGoogle Scholar
[27] Li, Yusong, LeBoeuf, Eugene J., and Basu, P. K.. Least-squares finite-element lattice boltzmann method. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 69(62):1–4, 2004.CrossRefGoogle ScholarPubMed
[28] McNamara, Guy R. and Zanetti, Gianluigi. Use of the boltzmann equation to simulate lattice-gas automata. Physical Review Letters, 61(20):2332–2335, 1988.CrossRefGoogle ScholarPubMed
[29] Mei, Renwei and Shyy, Wei. On the finite difference-based lattice boltzmann method in curvilinear coordinate. Journal of Computational Physics, 143:426–448, 1998.CrossRefGoogle Scholar
[30] Min, Misun and Lee, Taehun. A spectral-element discontinuous galerkin lattice boltzmann method for nearly incompressible flows. Journal of Computational Physics, 230(1):245–259, 2011.CrossRefGoogle Scholar
[31] Mittal, S.. Excitation of shear layer instability in flow past a cylinder at low reynolds number. International Journal for Numerical Methods in Fluids, 49(10):1147–1167, 2005.CrossRefGoogle Scholar
[32] Nannelli, Francesca and Succi, Sauro. The lattice boltzmann equation on irregular lattices. Journal of Statistical Physics, 68(3-4):401–407, 1992.CrossRefGoogle Scholar
[33] Nieuwstadt, F. and Keller, H.B.. Viscous flow past circular cylinders. Computers & Fluids, 1(1):59–71, 1973.CrossRefGoogle Scholar
[34] Niu, X.D., Chew, Y.T., and Shu, C.. Simulation of flows around an impulsively started circular cylinder by taylor series expansion- and least squares-based lattice boltzmann method. Journal of Computational Physics, 188(1):176–193, 2003.CrossRefGoogle Scholar
[35] Patil, D. V. and Lakshmisha, K. N.. Finite volume tvd formulation of lattice boltzmann simulation on unstructured mesh. Journal of Computational Physics, 228:5262–5279, August 2009.CrossRefGoogle Scholar
[36] Peng, Gongwen, Xi, Haowen, Duncan, Comer, and Chou, So-Hsiang. Lattice boltzmann method on irregular meshes. Physical Review E, 58(4):R4124, 1998.CrossRefGoogle Scholar
[37] Peng, Gongwen, Xi, Haowen, Duncan, Comer, and Chou, So-Hsiang. Finite volume scheme for the lattice boltzmann method on unstructured meshes. Physical Review E, 59(4):4675, 1999.CrossRefGoogle Scholar
[38] Piziali, R. A.. 2-d and 3-d oscillating wing aerodynamics for a range of angles of attack including stall. NASA Technical Memorandum 4362, NASA, September 1994.Google Scholar
[39] Posdziech, O. and Grundmann, R.. A systematic approach to the numerical calculation of fundamental quantities of the two-dimensional flow over a circular cylinder. Journal of Fluids and Structures, 23:479–499, April 2007.CrossRefGoogle Scholar
[40] Dhiman Ravi Golani, A. K.. Fluid flow and heat transfer across a circular cylinder in the unsteady flow regime. The International Journal Of Engineering And Science, 3(3):08–19, 2014.Google Scholar
[41] Shu, C., Chew, Y. T., and Niu, X. D.. Least-squares-based lattice boltzmann method: A meshless approach for simulation of flows with complex geometry. Phys. Rev. E, 64:045701, Sep 2001.CrossRefGoogle Scholar
[42] Shu, C., Niu, X. D., and Chew, Y. T.. Taylor-series expansion and least-squares-based lattice boltzmann method: Two-dimensional formulation and its applications. Phys. Rev. E, 65:036708, Mar 2002.CrossRefGoogle ScholarPubMed
[43] Spalart, S. R., Allmaras, P.R., Spalart, P R, and Allmaras, S R. sa.pdf. AIAA Pap. 92–0439, Reno, NV, January 1992.Google Scholar
[44] Stiebler, Maik, Tlke, Jonas, and Krafczyk, Manfred. An upwind discretization scheme for the finite volume lattice boltzmann method. Computers & Fluids, 35(8-9):814–819, 2006. Proceedings of the First International Conference for Mesoscopic Methods in Engineering and Science.CrossRefGoogle Scholar
[45] Tölke, Jonas, Freudiger, Sören, and Krafczyk, Manfred. An adaptive scheme using hierarchical grids for lattice boltzmann multi-phase flow simulations. Computers & Fluids, 35(8-9):820–830, September 2006.CrossRefGoogle Scholar
[46] Ubertini, Stefano and Succi, Sauro. Recent advances of lattice boltzmann techniques on unstructured grids. Progress in Computational Fluid Dynamics, an International Journal, 5(1):85–96, January 2005.CrossRefGoogle Scholar
[47] Wang, Y., He, Y. L., Zhao, T. S., Tang, G. H., and Tao, W. Q.. Implicit-explicit finite-difference lattice boltzmann method for compressible flows. International Journal of Modern Physics C, 18(12):1961–1983, 2007.CrossRefGoogle Scholar
[48] White, Frank. Viscous Fluid Flow (Mcgraw Hill Series in Mechanical Engineering). McGraw-Hill Science/Engineering/Math, 2005.Google Scholar
[49] Wilcox, David C.. Turbulence Modeling for CFD. DCW Industries, Jan 2006.Google Scholar
[50] Wu, J. and Shu, C.. A solution-adaptive lattice boltzmann method for two-dimensional incompressible viscous flows. Journal of Computational Physics, 230(6):2246–2269, 2011.CrossRefGoogle Scholar
[51] Xi, H, Peng, G, and Chou, S H. Finite-volume lattice boltzmann method. Physical Review Ew E, 59(5):6202–6205, 1999.CrossRefGoogle ScholarPubMed
[52] Xi, H, Peng, G, and Chou, S H. Finite-volume lattice boltzmann schemes in two and three dimensions. Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 60(3):3380–3388, 1999.CrossRefGoogle ScholarPubMed
[53] Yu, Dazhi, Yu, Dazhi, Mei, Renwei, Mei, Renwei, Shyy, Wei, and Shyy, Wei. A multi-block lattice boltzmann method for viscous fluid flows. International Journal for Numerical Methods in Fluids, 39(October 2001):99–120, 2002.CrossRefGoogle Scholar
[54] Yu, Zhao and Fan, Liang-Shih. An interaction potential based lattice boltzmann method with adaptive mesh refinement (amr) for two-phase flow simulation. Journal of Computational Physics, 228(17):6456–6478, 2009.CrossRefGoogle Scholar
[55] Zarghami, A., Maghrebi, M. J., Ghasemi, J., and Ubertini, S.. Lattice boltzmann finite volume formulation with improved stability. Communications in Computational Physics, 12(1):42–64, 2012.CrossRefGoogle Scholar