Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T19:17:53.093Z Has data issue: false hasContentIssue false
  • English
  • Français

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:[email protected]
Email:[email protected]
Email:[email protected]
Email:[email protected]
Get access

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

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.)

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