Lattice Boltzmann solution of advection- dominated mass transport problem: A comparison

Document Type : Article


1 Department of Civil and Environmental Engineering, Shiraz University of Technology, Shiraz, P.O. Box 71555-313, Iran

2 - Department of Civil and Environmental Engineering, Shiraz University, Shiraz, P.O. Box 7134851156, Iran - Environmental Research and Sustainable Development Center, Shiraz University, Shiraz, Iran

3 Water, Energy, and Environmental Engineering, Research Unit, University of Oulu, Finland, P.O. Box 4300, FI-90014


This article addresses the abilities and limitations of the Lattice Boltzmann (LB) method in solving advection-dominated mass transport problems. Several schemes of the LB method, including D2Q4, D2Q5, and D2Q9, were assessed in the simulation of two-dimensional advection-dispersion equations. The concept of Single Relaxation Time (SRT) and Multiple Relaxation Time (MRT) in addition to linear and quadratic Equilibrium Distribution Functions (EDF) were taken into account. The results of LB models were compared to the well-known Finite Difference (FD) solutions, including Explicit Finite Difference (EFD) and Crank-Nicolson (CN) methods. All LB models are more accurate than the aforementioned FD schemes. The results also indicate the high potency of D2Q5 SRT and D2Q9 SRT in describing advection-controlled mass transfer problems. The numerical artificial oscillations are observed when the Grid Peclet Number (GPN) is greater than 10, 25, 20, 25, and 10 regarding D2Q4 SRT, D2Q5 SRT, D2Q5 MRT, D2Q9 SRT and D2Q9 MRT, respectively, while the corresponding GPN values obtained for the EFD and CN methods were 2 and 5, respectively. Finally, a coupled system of groundwater and solute transport equations were solved satisfactorily using several LB models. Considering computational time, all LB models are much faster than CN method.


Main Subjects

1. Amiri, S., Talebbeydokhti, N., and Baghlani, A. "A two-dimensional well-balanced numerical model for shallow water equations", Sci. Iran., 20(1), pp. 97-107 (2013).
2. Mahdavi, A. and Talebbeydokhti, N. "A hybrid solid boundary treatment algorithm for smoothed particle hydrodynamics", Sci. Iran., 22(4), pp. 1457-1469 (2015).
3. Asadollahfardi, G., Rezaee, M., and Mehrjardi, G.T. "Simulation of unenhanced electrokinetic process for lead removal from kaolinite clay", Int. J. Civ. Eng., 14(4), pp. 263-270 (2016).
4. Alemi, M. and Maia, R. "Numerical simulation of the flow and local scour process around single and complex bridge piers", Int. J. Civ. Eng., 16(5), pp. 475-487 (2018).
5. Hekmatzadeh, A.A., Papari, S., and Amiri, S.M. "Investigation of energy dissipation on various configurations of stepped spillways considering several RANS turbulence models", Ijst-T. Civ. Eng., 42(2), pp. 97- 109 (2018).
6. Mohamad, A.A. and Kuzmin, A. "A critical evaluation of force term in lattice Boltzmann method, natural convection problem", Int. J. Heat Mass Transfer, 53(5), pp. 990-996 (2010).
7. Perumal, D.A. and Dass, A.K. "A review on the development of lattice Boltzmann computation of macro fluid flows and heat transfer", Alexandria Eng. J., 54(4), pp. 955-971 (2015).
8. Yan, Z., Yang, X., Li, S., and Hilpert M. "Tworelaxation- time lattice Boltzmann method and its application to advective-diffusive-reactive transport", Adv. Water Resour., 109, pp. 333-342 (2017).
9. Guo, Z. and Shu, C., Lattice Boltzmann Method and Its Applications in Engineering, World Scientific, Singapore (2013).
10. Gao, J., Xing, H., Tian, Z., Pearce, J.K., Sedek, M., and Golding, S.D. "Reactive transport in porous media for CO2 sequestration: Pore scale modeling using the lattice Boltzmann method", Comput. Geosci., 98, pp. 9-20 (2017).
11. Yang, Y. and Wang, M. "Pore-scale modeling of chloride ion diffusion in cement microstructures", Cement Concrete Comp., 85, pp. 92-104 (2018).
12. Shi, B. and Guo, Z. "Lattice Boltzmann simulation of some nonlinear convection-diffusion equations", Comput. Math. Appl., 61(12), pp. 3443-3452 (2011).
13. Sharma, K.V., Straka, R., and Tavares, F.W. "New cascaded thermal lattice Boltzmann method for simulations of advection-diffusion and convective heat transfer", Int. J. Therm. Sci., 118, pp. 259-277 (2017).
14. Wang, H., Cater, J., Liu, H., Ding, X., and Huang, W. "A lattice Boltzmann model for solute transport in open channel  flow", J. HYDROL., 556, pp. 419-426 (2018).
15. Mohamad, A.A., Lattice Boltzmann Method: Fundamentals and Engineering Applications with Computer Codes, Springer Science & Business Media, Springer (2011).
16. Ibrahem, A.M., El-Amin, M.F., and Mohammadein, A.A. "Lattice Boltzmann technique for heat transport phenomena coupled with melting process", Heat Mass Transfer, 53(1), pp. 213-221 (2017).
17. Xia, Y., Wu, J., and Zhang, Y. "Lattice-Boltzmann simulation of two-dimensional super-diffusion", Eng. Appl. Comp. Fluid., 6(4), pp. 581-594 (2012).
18. Zhou, J.G. "A lattice Boltzmann method for solute transport", Int. J. Numer. Methods Fluids, 61(8), pp. 848-863 (2009).
19. Yoshida, H. and Nagaoka, M. "Lattice Boltzmann method for the convection-diffusion equation in curvilinear coordinate systems", J. Comput. Phys., 257, pp. 884-900 (2014).
20. Perko, J. and Patel, R.A. "Single-relaxation-time lattice Boltzmann scheme for advection-diffusion problems with large diffusion-coefficient heterogeneities and high-advection transport", Phys. Rev. E, 89(5), p. 053309 (2014).
21. Hosseini, R., Rashidi, S., and Esfahani, J.A. "A lattice Boltzmann method to simulate combined radiationforce convection heat transfer mode", J. Braz. Soc. Mech. Sci. Eng., 2017, pp. 1-12 (2017).
22. Zheng, Y., Li, G., Guo, W., and Dong, C. "Lattice Boltzmann simulation to laminar pulsating  flow past a circular cylinder with constant temperature", Heat Mass Transfer, 2017, pp. 1-12 (2017).
23. Bin, D., Bao-Chang, S., and Guang-Chao, W. "A new lattice Bhatnagar-Gross-Krook model for the convection-diffusion equation with a source term", Chin. Phys. Lett., 22(2), pp. 267-270 (2005).
24. Batot, G., Talon, L., Peysson, Y., Fleury, M., and Bauer, D. "Analytical and numerical investigation of the advective and dispersive transport in Herschel- Bulkley  fluids by means of a lattice-Boltzmann tworelaxation- time scheme", Chem. Eng. Sci., 141, pp. 271-281 (2016).
25. Huang, R. and Wu, H. "A modified multiplerelaxation- time lattice Boltzmann model for convection-diffusion equation", J. Comput. Phys., 274, pp. 50-63 (2014).
26. Li, L., Mei, R., and Klausner, J.F. "Lattice Boltzmann models for the convection-diffusion equation: D2Q5 vs D2Q9", Int. J. Heat Mass Transfer, 108, pp. 41-62 (2017).
27. Chai, Z. and Zhao, T. "Nonequilibrium scheme for computing the  flux of the convection-diffusion equation in the framework of the lattice Boltzmann method", Phys. Rev. E, 90(1), p. 013305 (2014).
28. Liu, Q., He, Y.L., Li, Q., and Tao, W.Q. "A multiplerelaxation- time lattice Boltzmann model for convection heat transfer in porous media", Int. J. Heat Mass Transfer, 73, pp. 761-775 (2014).
29. Chopard, B., Falcone, J., and Latt, J. "The lattice Boltzmann advection-diffusion model revisited", Eur. Phys. J. Special Topics, 171(1), pp. 245-249 (2009).
30. Li, Q., Chai, Z., and Shi, B. "Lattice Boltzmann model for a class of convection-diffusion equations with variable coefficients", Comput. Math. Appl., 70(4), pp. 548-561 (2015).
31. Hekmatzadeh, A.A., Karimi-Jashani, A., Talebbeydokhti, N., and Klve, B. "Modeling of nitrate removal for ion exchange resin in batch and fixed bed experiments", Desalination, 284, pp. 22-31 (2012).
32. Zheng, C. and Bennett, G.D., Applied Contaminant Transport Modeling, Wiley-Interscience, New York (2002).
33. Krivovichev, G.V. "Numerical stability analysis of lattice Boltzmann equations for linear diffusion", Appl. Math. Inf. Sci., 9(4), pp. 1687-1692 (2014).
34. Niu, X., Shu, C., Chew, Y.T., and Wang, T.G. "Investigation of stability and hydrodynamics of different lattice Boltzmann models", J. Stat. Phys., 117(3), pp. 665-680 (2004).
35. Servan-Camas, B. and Tsai, F.T.C. "Non-negativity and stability analyses of lattice Boltzmann method for advection-diffusion equation", J. Comput. Phys., 228(1), pp. 236-256 (2009).
36. Sterling, H.D. and Chen, S. "Stability analysis of lattice Boltzmann methods", J. Comput. Phys., 123(1), pp. 196-206 (1996).
37. Suga, S. "Numerical schemes obtained from lattice Boltzmann equations for advection diffusion equations", Int. J. Mod. Phys. C, 17(11), pp. 1563-1577 (2006).
38. Rao, P.R. and Schaefer, L.A. "Numerical stability of explicit off-lattice Boltzmann schemes: A comparative study", J. Comput. Phys., 285, pp. 251-264 (2015).
39. Huang, H., Lu, X., and Sukop, M. "Numerical study of lattice Boltzmann methods for a convection-diffusion equation coupled with Navier-Stokes equations", J. Phys. A: Math. Theor., 44(5), p. 055001 (2011).
40. Li, L., Mei, R., and Klausner, J.F. "Multiplerelaxation- time lattice Boltzmann model for the axisymmetric convection diffusion equation", Int. J. Heat Mass Transfer, 67, pp. 338-351 (2013).
41. Liu, H., Zhou, J.G., Li, M., and Zhao, Y. "Multi-block lattice Boltzmann simulations of solute transport in shallow water flows", Adv. Water Resour., 58, pp. 24- 40 (2013).
42. Seta, T., Takegoshi, E., and Okui, E. "Lattice Boltzmann simulation of natural convection in porous media", Math. Comput. Simul., 72(2), pp. 195-200 (2006).
43. Karamouz, M., Ahmadi, A., and Akhbari, M., Groundwater Hydrology: Engineering, Planning, and Management, CRC Press, Boca Raton (2011).
44. Wang, H. and Anderson, M., Introduction to Groundwater Modeling: Finite Differences and Finite Element Methods, Freeman, San Francisco (1982).
45. Guo, Z. and Shu, C., Lattice Boltzmann Method and Its Applications in Engineering, World Scientific, Singapore (2013).
46. Saadat, S., Hekmatzadeh, A.A., and Karimi-Jashni, A. "Mathematical modeling of the Ni (II) removal from aqueous solutions onto pre-treated rice husk in fixedbed columns: a comparison", Desalin. Water Treat., 57(36), pp. 16907-16918 (2016).
47. Reilly, T.E., Franke, L., and Bennett, G.D. "The principle of superposition and its application in groundwater hydraulics", Techniques of Water-Resources Investigations of the United States Geological Survey (1984).
48. Al-Turbak, A.S. and Al-Muttair, F.F. "Evaluation of dams as a recharge method", Int. J. Water Resour. Dev., 5(2), pp. 119-124 (1989).
49. Abdalla, O.A. and Al-Rawahi, A.S. "Groundwater recharge dams  in arid areas as tools for aquifer replenishment and mitigating seawater intrusion: example of AlKhod, Oman", Environ. Earth. Sci., 69(6), pp. 1951-1962 (2013).