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Lattice Boltzmann Finite Volume Formulation with Improved Stability

Published online by Cambridge University Press:  20 August 2015

A. Zarghami ,
M. J. Maghrebi ,
J. Ghasemi  and
S. Ubertini
A. Zarghami*
Affiliation:
Department of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran
M. J. Maghrebi*
Affiliation:
Department of Mechanical Engineering, Ferdowsi University ofMashhad, Mashhad, Iran
J. Ghasemi*
Affiliation:
Faculty of Engineering, Zanjan University, Zanjan, Iran
S. Ubertini*
Affiliation:
Department of Technologies, University of Naples “Parthenope”, Naples, Italy
*
Corresponding author.Email:ahad.zarghami@gmail.com
Email:mjmaghrebi@um.ac.ir
Email:j.ghasemi@znu.ac.ir
Email:Stefano.ubertini@uniparthenope.it
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Abstract

The most severe limitation of the standard Lattice Boltzmann Method is the use of uniform Cartesian grids especially when there is a need for high resolutions near the body or the walls. Among the recent advances in lattice Boltzmann research to handle complex geometries, a particularly remarkable option is represented by changing the solution procedure from the original “stream and collide” to a finite volume technique. However, most of the presented schemes have stability problems. This paper presents a stable and accurate finite-volume lattice Boltzmann formulation based on a cell-centred scheme. To enhance stability, upwind second order pressure biasing factors are used as flux correctors on a D2Q9 lattice. The resulting model has been tested against a uniform flow past a cylinder and typical free shear flow problems at low and moderate Reynolds numbers: boundary layer, mixing layer and plane jet flows. The numerical results show a very good accuracy and agreement with the exact solution of the Navier-Stokes equation and previous numerical results and/or experimental data. Results in self-similar coordinates are also investigated and show that the time-averaged statistics for velocity and vorticity express self-similarity at low Reynolds numbers. Furthermore, the scheme is applied to simulate the flow around circular cylinder and the Reynolds number range is chosen in such a way that the flow is time dependent. The agreement of the numerical results with previous results is satisfactory.

Keywords

47.20.Ky47.11.DfLattice Boltzmann equationfinite volumestabilitycell-centered schemefree shear flows
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 1 , July 2012 , pp. 42 - 64
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]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 (1954), 511.CrossRefGoogle Scholar
[2]Qian, Y., d’Humie’res, D. and Lallemand, P., Lattice BGK models for Navier-Stokes equation, Europhys. Lett., 17 (1992), 479.Google Scholar
[3]Rivet, J. P. and Boon, J. P., Lattice Gas Hydrodynamics, Cambridge University Press, 2000.Google Scholar
[4]Benzi, R., Succi, S. and Vergassola, M., The lattice Boltzmann equation: theory and applications, Phys. Rep., 222 (1992), 145.Google Scholar
[5]Boghosian, B. M., Alexander, F. J. and Coveney, P., Proceedings of conference on discrete models for fluid mechanics, Int. J. Mod. Phys. C, 8(4) (1997), 637.CrossRefGoogle Scholar
[6]Chen, S. and Doolen, G. D., Lattice Boltzmann method for fluid flows, Ann. Rev. Fluid Mech., 30 (1998), 329.Google Scholar
[7]Falcucci, G., Ubertini, S., Biscarini, C., Francesco, S. Di, Chiappini, D., Palpacelli, S., De Maio, A. and Succi, S., Lattice Boltzmann methods for multiphase flow simulations across scales, Commun. Comput. Phys., 9 (2011), 269–296.Google Scholar
[8]Chiappini, D., Bella, G., Succi, S., Toschi, F. and Ubertini, S., Improved lattice Boltzmann without parasitic currents for Rayleigh-Taylor instability, Commun. Comput. Phys., 7 (2010), 423–444.CrossRefGoogle Scholar
[9]Succi, S., The Lattice Boltzmann Equation for Fluid Dynamics and Beyond, Oxford University Press, 2001.Google Scholar
[10]Wolf-Gladrow, D. A., Lattice-Gas Cellular Automata and Lattice Boltzmann Models: An In-troduction, Springer, 2000.Google Scholar
[11]He, X., Luo, L. S. and Dembo, M., Some progress in lattice Boltzmann method: part I: nonuniform mesh grids, J. Comput. Phys., 129 (1996), 357.CrossRefGoogle Scholar
[12]Nannelli, F. and Succi, S., The lattice Boltzmann equation on irregular lattices, J. Stat. Phys., 68 (1992), 401.Google Scholar
[13]Peng, G., Xi, H., Duncan, C. and Chou, S. H., Finite volume scheme for the lattice Boltzmann method on unstructured meshes, Phys. Rev. E, 59 (1999), 4675.Google Scholar
[14]Xi, H., Peng, G. and Chou, S. H., Finite volume lattice Boltzmann method, Phys. Rev. E, 59 (1999), 6202.Google Scholar
[15]Ubertini, S., Succi, S. and Bella, G., Lattice Boltzmann method on unstructured grids: further developments, Phys. Rev. E, 68 (2003), 016701.CrossRefGoogle ScholarPubMed
[16]Stiebler, M., Tolke, J. and Krafczyk, M., An upwind discretization scheme for the finite volume lattice Boltzmann method, Comput. Fluids, 35 (2006), 814.Google Scholar
[17]Chen, H., Volumetric formulation of the lattice Boltzmann method for fluid dynamics: basic concept, Phys. Rev. E, 58 (1998), 3955.Google Scholar
[18]Ubertini, S. and Succi, S., Recent advances of Lattice Boltzmann techniques on unstructured grids, Prog. Comput. Fluid Dyn., 5 (2005), 85.CrossRefGoogle Scholar
[19]Ubertini, S. and Succi, S., A generalised lattice Boltzmann equation on unstructured grids, Commun. Comput. Phys., 3 (2008), 342.Google Scholar
[20]Hirsch, C., Numerical Computation of Internal and External Flows, Vol. I: Fundamentals of Numerical Discretization, Wiley, 1988.Google Scholar
[21]Ghasemi, J. and Razavi, S. E., On the finite volume lattice boltzmann modeling of thermo-hydrodynamics, Comput. Math. Appl., 60 (2010), 1135.Google Scholar
[22]Tanehill, J. C., Anderson, D. A and Pletcher, R. H., Computational Fluid Mechanics and Heat Transfer, Taylor &Francis, 1997.Google Scholar
[23]Zou, Q. and He, X., On pressure and velocity boundary conditions for the lattice Boltzmann BGK model, Phys. Fluids, 9 (1997), 1591.Google Scholar
[24]Chen, S., Martinez, D. and Mei, R., On boundary conditions in lattice Boltzmann methods, Phys. Fluids, 8 (1996), 2527.Google Scholar
[25]Guo, Z., Zheng, C. and Shi, B., An extrapolation method for boundary conditions in lattice Boltzmann method, Phys. Fluids, 14 (2007), 2002.Google Scholar
[26]Chopard, B. and Dupuis, A., A mass conserving boundary condition for lattice Boltzmann models, Int. J. Mod. Phys. B, 17 (2003), 103.CrossRefGoogle Scholar
[27]Schlichting, H., Boundary Layer Theory, Springer, 2005.Google Scholar
[28]Lowery, P. S. and Reynolds, W., Numerical simulation of spatially developing forced plane mixing layer, NASA Technical Report NCC2-015, 1986.Google Scholar
[29]Townsend, A. A. R., The Structure of Turbulent Shear Flow, Cambridge University Press, 1956.Google Scholar
[30]Oster, D. and Wygnanski, I., The forced mixing layer between paralell streams, J. Fluid Mech., 123 (1982), 91.Google Scholar
[31]Coutanceau, M. and Bouard, R., Experimental determination of the main features of the viscous flow in the wake of a circular cylinder in uniform translation: part 1: steady flow, part 2: unsteady flow, J. Fluid Mech., 79 (1977), 231.Google Scholar
[32]Dennis, S. C. R. and Chang, G. Z., Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100, J. Fluid Mech., 42 (1970), 471.Google Scholar
[33]Nieuwstadt, F. and Keller, H. B., Viscous flow past circular cylinders, Comput. Fluids, 1 (1973), 59.CrossRefGoogle Scholar
[34]He, X. and Doolen, G., Lattice Boltzmann method on curvilinear coordinates system: flow around a circular cylinder, J. Comput. Phys., 134 (1997), 306.Google Scholar
[35]Patil, D. V. and Lakshmisha, K. N., Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh, J. Comput. Phys., 228 (2009), 5262.CrossRefGoogle Scholar
[36]Fornberg, B., A numerical study of steady viscous flow past a ircular cylinder, J. Fluid Mech., 98 (1980), 819.Google Scholar
[37]Calhoun, D., A cartesian grid method for solving the two-dimensional streamfunction-vorticity equations in irregular regions, J. Comput. Phys., 176 (2002), 231.Google Scholar
[38]Ye, T., Mittal, R., Udaykumar, H. and Shyy, W., An accurate cartesian grid method for viscous incompressible flow with complex immersed boundaries, J. Comput.l Phys. 156 (1999), 209.Google Scholar
[39]Ubertini, S., Succi, S. and Bella, G., Lattice Boltzmann Scheme without Coordinates, Phil. Trans. R. Soc. Lond. A, 362 (2004), 1763.Google Scholar
[40]Ding, H., Shu, M. and Cai, Q. D., Application of stencil-adaptive finite difference method to incompressible viscous flows with curved boundary, Comput. Fluids, 36 (2007), 786.Google Scholar
[41]Liu, C., Zheng, X. and Sung, S. H., Preconditioned multi-grid methods for unsteady incompressible flows, J. Comput. Phys., 139 (1998), 35.CrossRefGoogle Scholar
[42]Braza, M., Chassaing, P., and Ha Minh, H., Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder, J. Fluid Mech., 165 (1986), 79.Google Scholar