Application of lattice Boltzmann method to solution of viscous incompressible fluid dynamics problems

The possibilities of simulation of viscous incompressible fluid flows with lattice Boltzmann method are considered. Unlike the traditional discretization approach based on the use of Navier–Stokes equations, the lattice Boltzmann method uses a mesoscopic model to simulate incompressible fluid flows. Macroscopic parameters of a fluid, such as density and velocity, are expressed through the moments of the discrete probability distribution function. Discretization of the lattice Boltzmann equation is carried out using schemes D2Q9 (two-dimensional case) and D3Q19 (three-dimensional case). To simulate collisions between pseudo-particles, the Bhatnaga r–Gross–Crooke approximation with one relaxation time is used. The specification of initial and boundary conditions (no penetration and no-slip conditions, outflow conditions, periodic conditions) is discussed. The patterns of formation and development of vortical flows in a square cavity and cubic cavities are computed. The results of calculations of flow characteristics in a square and cubic cavity at various Reynolds numbers are compared with data available in the literature and obtained based on the finite difference method and the finite volume method. The dependence of the numerical solution and location of critical points on faces of cubic cavity on the lattice size is studied. Computational time is compared with performance of fine difference and finite volume methods. The developed implementation of the lattice Boltzmann method is of interest for the transition to further modeling non-isothermal and high-speed compressible flows.

1. Kruger T., Kusumaatmaja H., Kuzmin A., Shardt O., Silva G., Viggen E.M. The lattice Boltzmann method: Principles and Practice. Springer, 2017, 694 p.

2. Rivet J.P., Boon J.P. Lattice Gas Hydrodynamics. Cambridge, Cambridge University Press, 2001, 289 p.

3. Succi S. The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Oxford, Clarendon Press, 2001, 288 p.

4. Chashchin G.S. Lattice Boltzmann method: simulation of isothermal low-speed flows. Keldysh Institute Preprints, 2021, vol. 99, 31 p. (in Russian). https://doi.org/10.20948/prepr-2021-99

5. Zakharov A.M., Senin D.S., Grachev E.A. Flow simulation by the lattice boltzmann method with multiple-relaxation times. Numerical Methods and Programming, 2014, vol. 15, no. 1, pp. 644–657. (in Russian)

6. Rahmati A.R., Ashrafizaadeh M. A generalized Lattice Boltzmann Method for three-dimensional incompressible fluid flow simulation. Journal of Applied Fluid Mechanics, 2009, vol. 2, no. 1, pp. 71–96. https://doi.org/10.36884/jafm.2.01.11858

7. Anupindi K., Lai W., Frankel S. Characterization of oscillatory instability in lid driven cavity flows using lattice Boltzmann method. Computers and Fluids, 2014, vol. 20, no. 92, pp. 7–21. https://doi.org/10.1016/j.compfluid.2013.12.015

8. Aslan E., Taymaz I., Benim A.C. Investigation of the Lattice Boltzmann SRT and MRT stability for lid driven cavity flow. International. Journal of Materials, Mechanics and Manufacturing, 2014, vol. 2, pp. 317–324. https://doi.org/10.7763/ijmmm.2014. v2.149

9. Yuana K.A., Budiana E.P., Deendarlianto, Indarto. Modeling and simulation of top and bottom lid driven cavity using Lattice Boltzmann Method. IOP Conference Series: Materials Science and Engineering, 2019, vol. 546, no. 5, pp. 052088. https://doi.org/10.1088/1757-899x/546/5/052088

10. Huang T., Lim H.-C. Simulation of lid-driven cavity flow with internal circular obstacles. Applied Sciences, 2020, vol. 10, no. 13, pp. 4583. https://doi.org/10.3390/app10134583

11. Yang J.-Y., Hung L.-H. A three-dimensional semiclassical Lattice Boltzmann Method for lid-driven cubic cavity flows. Journal of Fluid Science and Technology, 2011, vol. 6, no. 5, pp. 780–791. https://doi.org/10.1299/jfst.6.780

12. Belardinelli D., Sbragaglia M., Biferale L., Gross M., Varnik F. Fluctuating multicomponent lattice Boltzmann model. Physical Review E, 2015, vol. 91, no. 2, pp. 023313. https://doi.org/10.1103/ physreve.91.023313

13. Nourgaliev R.R., Dinh T.N., Theofanous T.G., Joseph D. The lattice Boltzmann equation method: theoretical interpretation, numerics and implications. International Journal of Multiphase Flow, 2003, vol. 29, no. 1, pp. 117–169. https://doi.org/10.1016/s0301-9322(02)00108-8

14. Asinari P., Ohwada T. Connection between kinetic methods for fluiddynamic equations and macroscopic finite-difference schemes. Computers and Mathematics with Applications, 2009, vol. 58, no. 5, pp. 841–861. https://doi.org/10.1016/j.camwa.2009.02.009

15. Mattila K., Hyväluoma J., Timonen J., Rossi T. Comparison of implementations of the lattice-Boltzmann method. Computers and Mathematics with Applications, 2008, vol. 55, no. 7, pp. 1514–1524. https://doi.org/10.1016/j.camwa.2007.08.001

16. Szalmás L. Multiple-relaxation time lattice Boltzmann method for the finite Knudsen number region. Physica A: Statistical Mechanics and its Applications, 2007, vol. 379, no. 2, pp. 401–408. https://doi.org/10.1016/j.physa.2007.01.013

17. Tsutahara M., Kataoka T., Shikata K., Takada N. New model and scheme for compressible fluids of the finite difference lattice Boltzmann method and direct simulations of aerodynamic sound. Computers and Fluids, 2008, vol. 37, no. 1, pp. 79–89. https://doi.org/10.1016/j.compfluid.2005.12.002

18. Yu H., Zhao K. Lattice Boltzmann method for compressible flows with high Mach numbers. Physics Review E, 2000, vol. 61, no. 4, pp. 3867–3870. https://doi.org/10.1103/physreve.61.3867

19. Kataoka T., Tsutahara M. Lattice Boltzmann model for the compressible Navier–Stokes equations with flexible specific-heat ratio. Physics Review E, 2004, vol. 69, no. 3, part. 2, pp. 035701. https://doi.org/10.1103/physreve.69.035701

20. Sun C., Hsu A. Multi-level lattice Boltzmann model on square lattice for compressible flows. Computers and Fluids, 2004, vol. 33, no. 10, pp. 1363–1385. https://doi.org/10.1016/j.compfluid.2003.12.001

21. He B., Chen Y., Feng W., Li Q., Song A., Wang Y., Zhang M., Zhang W. Compressible lattice Boltzmann method and applications. International Journal of Numerical Analysis and Modeling, 2012, vol. 9, no. 2, pp. 410–418.

22. Li K., Zhong C. A lattice Boltzmann model for simulation of compressible flows. International Journal for Numerical Methods in Fluids, 2015, vol. 77, no. 6, pp. 334–357. https://doi.org/10.1002/fld.3984

23. Chikatamarla S., Karlin I. Lattices for the lattice Boltzmann method. Physics Review E, 2009, vol. 79, no. 4, pp. 046701. https://doi. org/10.1103/physreve.79.046701

24. Ilyin, O.V. A Method for Simulating the dynamics of rarefied gas based on lattice Boltzmann equations and the BGK equation. Computational Mathematics and Mathematical Physics, 2018, vol. 58, no. 11, pp. 1817–1827. https://doi.org/10.1134/S0965542518110052

25. Ataei M., Shaayegan V., Costa F., Han S., Park C.B., Bussmann M. LBfoam: an open-source software package for the simulation of foaming using the lattice Boltzmann method. Computer Physics Communications, 2021, vol. 259, pp. 107698. https://doi.org/10.1016/j.cpc.2020.107698

26. Latt J., Malaspinas O., Kontaxakis D., Parmigiani A., Lagrava D., Brogi F., Belgacem M.B., Thorimbert Y., Leclaire S., Li S., Marson F., Lemus J., Kotsalos C., Conradin R., Coreixas C., Petkantchin R., Raynaud F., Beny J., Chopard B. Palabos: parallel lattice Boltzmann solver. Computers and Mathematics with Applications, 2021, vol. 81, pp. 334–350. ttps://doi.org/10.1016/j.camwa.2020.03.022

27. Control of Flow Around Bodies with Vortex Cells as Applied to Aircraft of an Integral Configuration (Numerical and Physical Modeling). Ed. by A.V. Ermishin, S.A. Isaev. Moscow, MSU, 2001, 360 p. (in Russian)

28. Volkov K.N., Emelianov V.N. Computational Technologies in Problems of Fluid and Gas Mechanics. Moscow, Fizmatlit, 2012, 468 p. (in Russian)

29. Ghia U., Ghia K.N., Shin C.T. High-Re solutions for incompressible flow using the Navier–Stokes equations and a multigrid method. Journal of Computational Physics, 1982, vol. 48, no. 3, pp. 387–411. https://doi.org/10.1016/0021-9991(82)90058-4

30. Schreiber R., Keller H.B. Driven cavity flows by efficient numerical techniques. Journal of Computational Physics, 1983, vol. 49, no. 2, pp. 310–333. https://doi.org/10.1016/0021-9991(83)90129-8

31. Vanka S.P. Block-implicit multigrid solution of Navier–Stokes equations in primitive variables. Journal of Computational Physics, 1986, vol. 65, no. 1, pp. 138–158. https://doi.org/10.1016/0021-9991(86)90008-2

32. Hou S., Zou Q., Chen S., Doolen G., Cogley A. Simulation of cavity flow by the lattice Boltzmann method. Journal of Computational Physics, 1995, vol. 118, no. 2, pp. 329–347. https://doi.org/10.1006/jcph.1995.1103

33. Gupta M.M., Kalita J.C. A new paradigm for solving Navier–Stokes equations: streamfunction–velocity formulation. Journal of Computational Physics, 2005, vol. 207, no. 1, pp. 52–68. https://doi.org/10.1016/j.jcp.2005.01.002

34. Pandit S.K. On the use of compact streamfunction-velocity formulation of steady Navier–Stokes equations on geometries beyond rectangular. Journal of Science and Computers, 2008, vol. 36, no. 2, pp. 219–242. https://doi.org/10.1007/s10915-008-9186-8

35. Lin L.S., Chen Y.C., Lin C.A. Multi relaxation time lattice Boltzmann simulations of deep lid driven cavity flows at different aspect ratios. Computers and Fluids, 2011, vol. 45, no. 1, pp. 233–240. https://doi.org/10.1016/j.compfluid.2010.12.012

36. Mishra N., Sanyasiraju Y.V.S.S. Exponential compact higher order scheme for steady incompressible Navier–Stokes equations. Engineering Applications of Computational Fluid Mechanics, 2012, vol. 6, no. 4, pp. 541–555. https://doi.org/10.1080/19942060.2012.11015441

37. Bruneau C.-H., Jouron C. An efficient scheme for solving steady incompressible Navier–Stokes equations. Journal of Computational Physics, 1990, vol. 89, no. 2, pp. 389–413. https://doi.org/10.1016/0021-9991(90)90149-u

38. Botella O., Peyret R. Benchmark spectral results on the lid-driven cavity flow. Computers and Fluids, 1998, vol. 27, no. 4, pp. 421–433. https://doi.org/10.1016/s0045-7930(98)00002-4

39. Babu V., Korpela S.A. Numerical solution of the incompressible three-dimensional Navier–Stokes equations. Computers and Fluids, 1994, vol. 23, no. 5, pp. 675–691. https://doi.org/10.1016/0045-7930(94)90009-4

40. Sheu T.W.H., Tsai S.F. Flow topology in a steady three-dimensional lid-driven cavity. Computers and Fluids, 2002, vol. 31, no. 8, pp. 911–934. https://doi.org/10.1016/s0045-7930(01)00083-4

41. Volkov K.N. Topology of an incompressible viscous-fluid flow in a cubic cavity with a moving cover. Journal of Engineering Physics and Thermophysics, 2006, vol. 79, no. 2, pp. 295–300. https://doi.org/10.1007/s10891-006-0100-7