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

Document Type : Article

Authors

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 of Technology, Shiraz, P.O. Box 71555-313, Iran.University of Technology, Shiraz, Iran

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

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

Abstract

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.

Keywords

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  ow 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 con_gurations  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  uid ows and heat transfer", Alexandria Eng. J.,  54(4), pp. 955{971 (2015).  A.A. Hekmatzadeh et al./Scientia Iranica, Transactions A: Civil Engineering 27 (2020) 625{638 637  8. Yan, Z., Yang, X., Li, S., and Hilpert M. Tworelaxation-  time lattice Boltzmann method and its  application to advective-di_usive-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 Scienti_c,  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 di_usion in cement microstructures", Cement  Concrete Comp., 85, pp. 92{104 (2018).  12. Shi, B. and Guo, Z. Lattice Boltzmann simulation of  some nonlinear convection-di_usion 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-di_usion 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 ow", 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-di_usion", 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-di_usion 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-di_usion problems  with large di_usion-coe_cient 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 ow 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-di_usion 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 uids by means of a lattice-Boltzmann tworelaxation-  time scheme", Chem. Eng. Sci., 141, pp.  271{281 (2016).  25. Huang, R. and Wu, H. A modi_ed multiplerelaxation-  time lattice Boltzmann model for  convection-di_usion equation", J. Comput. Phys.,  274, pp. 50{63 (2014).  26. Li, L., Mei, R., and Klausner, J.F. Lattice Boltzmann  models for the convection-di_usion 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 ux of the convection-di_usion 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-di_usion 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-di_usion equations with  variable coe_cients", Comput. Math. Appl., 70(4), pp.  548{561 (2015).  31. Hekmatzadeh, A.A., Karimi-Jashani, A., Talebbeydokhti,  N., and Kl_ve, B. Modeling of nitrate removal  for ion exchange resin in batch and _xed 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 di_usion", 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 di_erent  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-di_usion 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).  638 A.A. Hekmatzadeh et al./Scientia Iranica, Transactions A: Civil Engineering 27 (2020) 625{638  37. Suga, S. Numerical schemes obtained from lattice  Boltzmann equations for advection di_usion equations",  Int. J. Mod. Phys. C, 17(11), pp. 1563{1577  (2006).  38. Rao, P.R. and Schaefer, L.A. Numerical stability of  explicit o_-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-di_usion  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 di_usion 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 ows", 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 Di_erences and Finite Element  Methods, Freeman, San Francisco (1982).  45. Guo, Z. and Shu, C., Lattice Boltzmann Method  and Its Applications in Engineering, World Scienti_c,  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 _xedbed  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).