A Simplified Curved Boundary Condition in Stationary/Moving Boundaries for the Lattice Boltzmann Method
Authors
Abstract
Lattice Boltzmann method is one of computational fluid dynamic subdivisions. Despite complicated mathematics involved in its background, end simple relations dominate on it; so in comparison to the conventional computational fluid dynamic methods, simpler computer programs are needed. Due to its characteristics for parallel programming, this method is considered efficient for the simulation of complex geometry flows, in which a large amount of computational memories is needed. Because of the curved boundaries in the complex geometries, detecting the proper curved boundary condition is unavoidable for the lattice Boltzmann method. For this purpose, more works have been done, and different curved boundary conditions have been proposed. At the present work, first, some curved boundary conditions have been reviewed; then a simplified curved boundary condition is proposed. A computer program based on the lattice Boltzmann method, in FORTRAN language, has been prepared; in this program, the boundary condition along with some others applied on it is proposed. To verify the accuracy and correctness of the proposed boundary condition, 2D cavity flow has been simulated and compared to the available numerical results. Adaptation of the achieved results with those of previous researchers verifies the prepared program correctness. Also, two fluid flows have been simulated, a flow around a stationary cylinder in a 2D channel and one between two stationary and moving cylinders. The results of simulations with the proposed boundary condition, along with the previous boundary conditions, have been compared to the available results. Comparisons demonstrate that solutions with proper accuracy could be obtained by the proposed boundary condition.
Keywords
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2. Naghavi, S. M., "Stirred Tank Fluid Flow Simulation with Two Lattice Boltzmann Methods", Journal of Simulation & Analysis of Novel Technologies in Mechanical Engineering, Vol. 10, pp. 21-33, 2017.
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10. Yu, D., Mei, R., and Shyy, W., "A Unified Boundary Treatment in Lattice Boltzmann Method", in 41st Aerospace Sciences Meeting and Exhibit, p. 953, 2003.
11. Verschaeve, J. C. G. and Müller, B., "A Curved No-Slip Boundary Condition for the Lattice Boltzmann Method", Journal of Computational Physics, Vol. 229, pp. 6781-6803, 2010.
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14. Wolf-Gladrow, D. A., Lattice-gas Cellular Automata and Lattice Boltzmann Models: An Introduction: Springer, 2004.
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17. He, X. and Luo, L. S., "Theory of the Lattice Boltzmann Method: From the Boltzmann Equation to the Lattice Boltzmann Equation", Physical Review E, Vol. 56, pp. 6811-6817, 1997.
18. He, X. and Luo, L. S., "A Priori Derivation of the Lattice Boltzmann Equation", Physical Review E, Vol. 55, pp. 6333-6336, 1997.
19. Guo, Z., Shi, B., and Wang, N., "Lattice BGK Model for Incompressible Navier-Stokes Equation", Journal of Computational Physics, Vol. 165, pp. 288-306, 2000.
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25. Chen, D., Lin, K., and Lin, C., "Immersed Boundary Method Based Lattice Boltzmann Method to Simulate 2D and 3D Complex Geometry Flows", International Journal of Modern Physics C, Vol. 18, pp. 585-594, 2007.
26. Derksen, J. and Van Den Akker, H. E. A., "Large Eddy Simulations on The Flow Driven by a Rushton Turbine", AICHE Journal, Vol. 45, pp. 209-221, 1999.
27. Naghavi, S. M., and Ashrafizaadeh, M., "A Comparison of Two Boundary Conditions for the Fluid Flow Simulation in a Stirred Tank", JCME, Vol. 33, pp. 15-30, 2014 (in persian).
28. Peng, Y. and Luo, L. S., "A Comparative Study of Immersed-Boundary and Interpolated Bounce-Back Methods in LBE", Progress in Computational Fluid Dynamics, an International Journal, Vol. 8, pp. 156-167, 2008.
29. Filippova, O. and Hanel, D., "Grid Refinement for Lattice-BGK Models", Journal of Computational Physics, Vol. 147, pp. 219-228, 1998.
30. Mei, R., Luo, L. S., and Shyy, W., "An Accurate Curved Boundary Treatment in the Lattice Boltzmann Method", Journal of Computational Physics, Vol. 155, pp. 307-330, 1999.
31. Mei, R., Shyy, W., Yu, D., and Luo, L. S., "Lattice Boltzmann Method for 3-D Flows with Curved Boundary", Journal of Computational Physics, Vol. 161, pp. 680-699, 2000.
32. Mei, R., Yu, D., Shyy, W., and Luo, L. S., "Force Evaluation in the Lattice Boltzmann Method Involving Curved Geometry", Physical Review E, Vol. 65, pp. 041203, 2002.
33. Bouzidi, M., Firdaouss, M., and Lallemand, P., "Momentum Transfer of a Boltzmann-Lattice Fluid with Boundaries", Physics of Fluids, Vol. 13, pp. 3452-3459, 2001.
34. 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, Vol. 48, pp. 387-411, 1982.
35. Schafer, M., Turek, S., Durst, F., Krause, E., and Rannacher, R., "Benchmark Computations of Laminar Flow Around a Cylinder", Notes on Numerical Fluid Mechanics, Vol. 52, pp. 547-566, 1996.
2. Naghavi, S. M., "Stirred Tank Fluid Flow Simulation with Two Lattice Boltzmann Methods", Journal of Simulation & Analysis of Novel Technologies in Mechanical Engineering, Vol. 10, pp. 21-33, 2017.
3. Yu, D., Mei, R., Luo, L. S., and Shyy, W., "Viscous Flow Computations with the Method of Lattice Boltzmann Equation", Progress in Aerospace Sciences, Vol. 39, pp. 329-367, 2003.
4. Latt, J., Chopard, B., Malaspinas, O., Deville, M., and Michler, A., "Straight Velocity Boundaries in the Lattice Boltzmann Method", Physical Review E, Vol. 77, pp. 056703, 2008.
5. Chang, C., Liu, C.-H., and Lin, C.-A., "Boundary Conditions for Lattice Boltzmann Simulations with Complex Geometry Flows", Computers & Mathematics with Applications, Vol. 58, pp. 940-949, 2009.
6. Hu, K., Meng, J., Zhang, H., Gu, X.-J., Emerson, D. R., and Zhang, Y., "A Comparative Study of Boundary Conditions for Lattice Boltzmann Simulations of High Reynolds Number Flows", Computers & Fluids, Vol. 156, pp. 1-8, 2017.
7. Zou, Q. and He, X., "On Pressure and Velocity Boundary Conditions for the Lattice Boltzmann BGK Model", Physics of Fluids, Vol. 9, pp 1591-1598, 1997.
8. Lee, H. C., Bawazeer, S., and Mohamad, A. A., "Boundary Conditions for Lattice Boltzmann Method with Multispeed Lattices", Computers & Fluids, Vol. 162, pp. 152-159, 2018.
9. Sanjeevi, S. K. P., Zarghami, A., and Padding, J. T., "Choice of No-Slip Curved Boundary Condition for Lattice Boltzmann Simulations of High-Reynolds-Number Flows", Physical Review E, Vol. 97, pp. 043305, 2018.
10. Yu, D., Mei, R., and Shyy, W., "A Unified Boundary Treatment in Lattice Boltzmann Method", in 41st Aerospace Sciences Meeting and Exhibit, p. 953, 2003.
11. Verschaeve, J. C. G. and Müller, B., "A Curved No-Slip Boundary Condition for the Lattice Boltzmann Method", Journal of Computational Physics, Vol. 229, pp. 6781-6803, 2010.
12. Rohde, M., Kandhai, D., Derksen, J. J., and Van Den Akker, H. E. A., "Improved Bounce-Back Methods for No-Slip Walls in Lattice-Boltzmann Schemes: Theory and Simulations", Physical Review E, Vol. 67, pp. 66703, 2003.
13. Oulaid, O. and Zhang, J., "On the Origin of Numerical Errors in the Bounce-Back Boundary Treatment of the Lattice Boltzmann Method: A Remedy for Artificial Boundary Slip and Mass Leakage", European Journal of Mechanics-B/Fluids, Vol. 53, pp. 11-23, 2015.
14. Wolf-Gladrow, D. A., Lattice-gas Cellular Automata and Lattice Boltzmann Models: An Introduction: Springer, 2004.
15. Succi, S., The lattice Boltzmann Equation: for Fluid Dynamics and Beyond: Oxford University Press, 2001.
16. Mohamad, A. A., Lattice Boltzmann Method: Fundamentals and Engineering Applications with Computer Codes: Springer Science & Business Media, 2011.
17. He, X. and Luo, L. S., "Theory of the Lattice Boltzmann Method: From the Boltzmann Equation to the Lattice Boltzmann Equation", Physical Review E, Vol. 56, pp. 6811-6817, 1997.
18. He, X. and Luo, L. S., "A Priori Derivation of the Lattice Boltzmann Equation", Physical Review E, Vol. 55, pp. 6333-6336, 1997.
19. Guo, Z., Shi, B., and Wang, N., "Lattice BGK Model for Incompressible Navier-Stokes Equation", Journal of Computational Physics, Vol. 165, pp. 288-306, 2000.
20. He, X. and Luo, L. S., "Lattice Boltzmann Model for the Incompressible Navier Stokes Equation", Journal of Statistical Physics, Vol. 88, pp. 927-944, 1997.
21. Skordos, P. A., "Initial and Boundary Conditions for the Lattice Boltzmann Method", Physical Review E, Vol. 48, pp. 4823-4842, 1993.
22. Dellar, P. J., "Incompressible Limits of Lattice Boltzmann Equations Using Multiple Relaxation Times", Journal of Computational Physics, Vol. 190, pp. 351-370, 2003.
23. Guo, Z.-L., Zheng, C.-G., and Shi, B.-C., "Non-Equilibrium Extrapolation Method for Velocity and Pressure Boundary Conditions in the Lattice Boltzmann Method", Chinese Physics, Vol. 11, pp. 366, 2002.
24. Guo, Z.-L., Zheng, C.-G., and Shi, B.-C., "An Extrapolation Method for Boundary Conditions in Lattice Boltzmann Method", Physics of Fluids (1994-present), Vol. 14, pp. 2007-2010, 2002.
25. Chen, D., Lin, K., and Lin, C., "Immersed Boundary Method Based Lattice Boltzmann Method to Simulate 2D and 3D Complex Geometry Flows", International Journal of Modern Physics C, Vol. 18, pp. 585-594, 2007.
26. Derksen, J. and Van Den Akker, H. E. A., "Large Eddy Simulations on The Flow Driven by a Rushton Turbine", AICHE Journal, Vol. 45, pp. 209-221, 1999.
27. Naghavi, S. M., and Ashrafizaadeh, M., "A Comparison of Two Boundary Conditions for the Fluid Flow Simulation in a Stirred Tank", JCME, Vol. 33, pp. 15-30, 2014 (in persian).
28. Peng, Y. and Luo, L. S., "A Comparative Study of Immersed-Boundary and Interpolated Bounce-Back Methods in LBE", Progress in Computational Fluid Dynamics, an International Journal, Vol. 8, pp. 156-167, 2008.
29. Filippova, O. and Hanel, D., "Grid Refinement for Lattice-BGK Models", Journal of Computational Physics, Vol. 147, pp. 219-228, 1998.
30. Mei, R., Luo, L. S., and Shyy, W., "An Accurate Curved Boundary Treatment in the Lattice Boltzmann Method", Journal of Computational Physics, Vol. 155, pp. 307-330, 1999.
31. Mei, R., Shyy, W., Yu, D., and Luo, L. S., "Lattice Boltzmann Method for 3-D Flows with Curved Boundary", Journal of Computational Physics, Vol. 161, pp. 680-699, 2000.
32. Mei, R., Yu, D., Shyy, W., and Luo, L. S., "Force Evaluation in the Lattice Boltzmann Method Involving Curved Geometry", Physical Review E, Vol. 65, pp. 041203, 2002.
33. Bouzidi, M., Firdaouss, M., and Lallemand, P., "Momentum Transfer of a Boltzmann-Lattice Fluid with Boundaries", Physics of Fluids, Vol. 13, pp. 3452-3459, 2001.
34. 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, Vol. 48, pp. 387-411, 1982.
35. Schafer, M., Turek, S., Durst, F., Krause, E., and Rannacher, R., "Benchmark Computations of Laminar Flow Around a Cylinder", Notes on Numerical Fluid Mechanics, Vol. 52, pp. 547-566, 1996.
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