Implementation of curved wall boundary and absorbing open boundary conditions for the D3Q19 lattice Boltzmann method for simulation of in compressible fluid flows

Document Type : Article


Aerospace Engineering Group, Dept. of New Technologies Engineering, Shahid Beheshti University, Tehran, Iran.


In this work, a three-dimensional lattice Boltzmann method is developed for numerical simulation of the fluid flows around the arbitrary geometries in the wide range of Reynolds numbers. For efficient simulation of high Reynolds number flow structures in the turbulent regime, a large eddy simulation (LES) approach with the Smagorinsky subgrid turbulence model is employed. An absorbing boundary condition based on the concept of sponge layer is improved and implemented to damp the vorticity fluctuations near the open boundaries and regularize the numerical solution by significantly reducing the spurious reflections from the open boundaries. An off-lattice scheme with a polynomial interpolation is used for implementation of curved boundary conditions for the arbitrary geometries. The efficiency and accuracy of the numerical approach presented are examined by computing the low to high Reynolds number flows around the practical geometries, including the flow past a sphere in a range of Reynolds numbers from 102 to 104 and flow around the NACA0012 wing section in two different flow conditions. The present results are in good agreement with the numerical and experimental data reported in the literature. The study demonstrates the present computational technique is robust and efficient for solving flow problems with practical geometries.


Main Subjects

1. Ginzbourg, I. and Adler, P.M. Boundary ow condition analysis for the three-dimensional lattice Boltzmann model", Journal de Physique, 4, pp. 191-214 (1994). 2. Shu, C., Liu, N., and Chew, Y. A novel immersed boundary velocity correction-lattice Boltzmann method and its application to simulate ow past a circular cylinder", Journal of Computational Physics, 226, pp. 1607-1622 (2007). 3. Navidbakhsh, M. and Rezazadeh, M. An immersed boundary-lattice Boltzmann model for simulation of malaria-infected red blood cell in micro-channel", Scientia Iranica B, 19(5), pp. 1329-1336 (2012). 4. Filippova, O. and Hanel, D. Boundary _tting and local grid re_nement for lattice-BGK models", International Journal of Modern Physics C, 9, pp. 1271- 1279 (1998). 5. Mei, R., Luo, L.-S., and Shyy, W. An accurate curved boundary treatment in the lattice Boltzmann method", Journal of Computational Physics, 155, pp. 307-330 (1999). 6. Guo, Z., Zheng, C., and Shi, B. An extrapolation method for boundary conditions in lattice Boltzmann method", Physics of Fluids, 14(6), pp. 2007-2010 (2002). 7. Latt, J., Chopard, B., Malaspinas, O., Deville, M., and Michler, A. Straight velocity boundaries in the lattice Boltzmann method", Physical Review E, 77(5), p. 056703 (2008). 8. Verschaeve, J. and Muller, B. A curved no-slip boundary condition for the lattice Boltzmann method", Journal of Computational Physics, 229, pp. 6781-6803 (2010). 9. Guo, Z. and Zhao, T.S. Explicit _nite-di_erence lattice Boltzmann method for curvilinear coordinates", Physical Review E, 67, p. 066709 (2003). 10. Hejranfar, K. and Ezzatneshan, E. A high-order compact _nite-di_erence lattice Boltzmann method for simulation of steady and unsteady incompressible ows", Int. J. Numer. Meth. Fluids, 75, pp. 713-746 (2014). 11. Patil, V. and Lakshmisha, K.N. Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh", Journal of Computational Physics, 228, pp. 5262-5279 (2009). 12. He, X. and Luo, L.-S. Theory of the lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation", Physical Review E, 53, pp. 6811-6817 (1997). 13. Hejranfar, K. and Ezzatneshan, E. Implementation of a high-order compact _nite-di_erence lattice Boltzmann method in generalized curvilinear coordinates", Journal of Computational Physics, 267, pp. 28-49 (2014). 14. McCracken, M.E. and Abaham, J. Multiplerelaxation- time lattice-Boltzmann model for multiphase ow", Physical Review E, 71, p. 036701 (2005). 15. Du, R., Shi, B., and Chen, X. Multi-relaxationtime lattice Boltzmann model for incompressible ow", Physics Letters A, 359, pp. 564-572 (2006). 16. Jung, N., Seo, H.W., and Yoo, C.S. Two-dimensional characteristic boundary conditions for open boundaries in the lattice Boltzmann methods", Journal of Computational Physics, 302, pp. 191-199 (2015). DOI: 10.1016/ 17. hompson, K.W. Time dependent boundary conditions for hyperbolic systems", Journal of Computational Physics, 68(1), pp. 1-24 (1987). 18. Heubes, D., Bartel, A., and Ehrhardt, M. Characteristic boundary conditions in the lattice Boltzmann method for uid and gas dynamics", Journal of Computational and Applied Mathematics, 262, pp. 51-61 (2014). 19. Hu, F. A stable perfectly matched layer for linearized Euler equations in unsplit physical variables", Journal of Computational Physics, 173, pp. 455-480 (2001). 20. Naja_-Yazdi, A. and Mongeau, L. An absorbing boundary condition for the lattice Boltzmann method based on the perfectly matched layer", Computers & Fluids, 68, pp. 203-218 (2012). 21. Bogey, C. and Bailly, C. Three-dimensional nonreactive boundary conditions for acoustic simulations: far _eld formulation and validation test cases", Acta Acustica United, 88, pp. 463-471 (2002). 22. Vergnault, E., Malaspinas, O., and Sagaut, P. A lattice Boltzmann method for nonlinear disturbances around an arbitrary base ow", Journal of Computational Physics, 231(24), pp. 8070-8082 (2012). 23. Koda, Y. and Lien, F.S. The lattice Boltzmann method implemented on the GPU to simulate the turbulent ow over a square cylinder con_ned in a channel", Flow, Turbulence and Combustion, 94(3), pp. 495-512 (2015). 24. Bhatnagar, P., Gross, E.P., and Krook, M. A model for collision processes in gases i: small amplitude processes in charged and neutral one-component systems", Physical Review E, 94, pp. 511-525 (1954). 25. Teixeira, C.M. Incorporating turbulence models into the lattice-Boltzmann method", Int. J. Modern Physics C, 9(8), pp. 1159-1175 (1998). 26. Krafczyk, M., Tolke, J., and Luo, L.S. Large-eddy simulations with a multi-relaxation-time LBE model", Int. J. Modern Physics B, 17(1), pp. 33-39 (2003). 27. Freitas, R.K., Schroder, W., and Meinke, M. Investigation of lattice boltzmann methods for LES", Progress in Turbulence II, 109, pp. 279-283 (2007). E. Ezzatneshan/Scientia Iranica, Transactions B: Mechanical Engineering 26 (2019) 2329{2341 2341 28. Verschaeve, J.C.G. Analysis of the lattice Boltzmann Bhatnagar-Gross-Krook no-slip boundary condition: ways to improve accuracy and stability", Physical Review E, 80, p. 036703 (2009). 29. Taneda, S. Experimental investigation of the wake behind a sphere at low Reynolds numbers", J. Phys. Soc. Japan, 11, pp. 1104-1108 (1956). 30. Taneda, S. Visual observations of the ow past a sphere at Reynolds numbers between 104 and 106", J. Fluid Mech., 85, pp. 187-192 (1978). 31. Magarvey, R.H. and Bishop, R.L. Transition ranges for three-dimensional wakes", Can. J. Phys., 39, pp. 1418-1422 (1961). 32. Tomboulides, A.G. Direct and large-eddy simulation of wake ows: ow past a sphere", PhD Thesis, Princeton University (1993). 33. Magnaudet, J., Rivero, M., and Fabre, J. Accelerated ows past a rigid sphere or a spherical bubble, Part 1: Steady straining ow", J. Fluid Mech., 284, pp. 97-135 (1961). 34. Johnson, T.A. and Patel, V.C. Flow past a sphere up to a Reynolds number of 300", J. Fluid Mech., 378, pp. 19-70 (1999). 35. Le Clair, B.P., Hamielec, A.E., and Pruppacher, H.R. A numerical study of the drag on a sphere at low and intermediate Reynolds numbers", J. Atmos. Sci., 27, pp. 308-315 (1970). 36. Sakamoto, H. and Haniu, H. A study of vortex shedding from spheres in uniform ow", J. Fluids Eng., 112, pp. 386-393 (1990). 37. Achenbach, E. Experiments on the ow past spheres at very high Reynolds numbers", J. Fluid Mech., 54, pp. 565-575 (1972). 38. Constantinescu, G.S. and Squires, K. LES and DES investigations of turbulent ow over a sphere at Re 10000", Flow Turbulence and Combustion, 70, pp. 267- 298 (2003). 39. Mei, R., Yu, D., Shyy, W., and Luo, L.-S. Force evaluation in the lattice Boltzmann method involving curved geometry", Physical Review E, 65(4), p. 041203 (2002). 40. Roos, F.W. and Willmarth, W.W. Some experimental results on sphere and disk drag", AIAA J., 9, pp. 285- 291 (1971). 41. Clift, R., Grace, J.R., and Weber, M.E., Bubbles, Drops and Particles, Academic Press, New York (1978). 42. Tabata, M. and Itakura, K. A precise computation of drag coe_cients of a sphere", Int. J. Comput. Fluid Dynam., 9, p. 303 (1998). 43. Ploumhans, P., Winckelmans, G.S., Slamon, J.K., Leonard, A., and Warren, M.S. Vortex methods for direct numerical simulation of three-dimensional blu_ body ow: Application to the sphere at Re = 300, 500, and 1000", Journal of Computational Physics, 178, pp. 427-463 (2002). 44. Poon, E.K.W., Iaccarino, G., Ooi, A.S.H., and Giacobello, M. Numerical studies of high Reynolds number ow past a stationary and rotating sphere", 7th International Conference on CFD in the Minerals and Process Industries, CSIRO, Melbourne, Australia (2009). 45. Constantinescu, G.S., Chapelet, M.C., and Squires, K.D. Prediction of turbulent ow over a sphere", AIAA J., 41, pp. 1733-1742 (2003). 46. Kim, S.E. Large eddy simulation using unstructured meshes and dynamic subgrid- scale turbulence models", 34th AIAA Fluid Dynamics Conference and Exhibit, Portland, Oregon, AIAA paper number 2004- 2548 (2004). 47. Meyers, J. and Sagaut, P. On the model coe_- cients for the standard and the variational multi-scale smagorinsky model", J. Fluid Mechanics, 569, pp. 287-319 (2006). 48. Pope, S.B., Turbulent Flows, Cambridge University Press, Cambridge, UK (2000). 49. Hafez, M., Shatalov, A., and Nakajima, M. Improved numerical simulations of incompressible ows based on viscous/inviscid interaction procedures", Computers & Fluids, 36, pp. 1588-1591 (2007). 50. Ho_man, J., Jansson, J., and Jansson, N. Simulation of 3D unsteady incompressible ow past a NACA 0012 wing section", Computational Technology Laboratory, Scienti_c and Technical Information Division, Report KTH-CTL-4023 (2012).