Lattice Boltzmann method for groundwater flow in non-orthogonal structured lattices. (English) Zbl 1443.76166
Summary: The efficiency of the lattice Boltzmann method (LBM) in modeling isotropic groundwater flow in domains of arbitrary geometry has been investigated. The Poisson equation was transformed in general curvilinear coordinates. The corresponding equilibrium function for the D2Q9 square lattice based on metric function between the physical and the computational domain has been established. The resulting LBM was checked on examples having higher generality; flows in confined and unconfined aquifers, in vertical and horizontal plane have been considered. In addition, the phreatic water table representing upper boundary in the vertical plane was determined by the dynamic \(\sigma\)-stretching approach, not requiring complex concepts for dealing with the free surface (like the volume of fluid method). The accuracy and stability of the model is controlled by the adaptive mesh concept. This allows application of higher density grid in critical areas with high pressure and velocity gradients, and vice versa. The number of computation points is significantly reduced without loosing accuracy. The basic characteristics of the LBM including features like parallelization and simplicity, are maintained. The advantages of the proposed curvilinear LBM in modeling groundwater flow in domains of complex shape over the former published methods is demonstrated by three examples.
MSC:
| 76M28 | Particle methods and lattice-gas methods |
| 65M75 | Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs |
| 76S05 | Flows in porous media; filtration; seepage |
Keywords:
groundwater flow; lattice Boltzmann method; non-orthogonal structured lattices; complex geometry; confined and unconfined aquifer; horizontal and vertical planeReferences:
| [1] | Rivet, J. P.; Boon, J. P., Lattice Gas Hydrodynamics (2001), Cambridge University Press: Cambridge University Press London · Zbl 1034.82056 |
| [2] | Wolf-Gladrow, D. A., Lattice-Gas Cellular Automata and Lattice Boltzmann Models—An Introduction (2005), Springer: Springer Berlin · Zbl 0999.82054 |
| [3] | Gladrow, D. W., A lattice boltzmann equation for diffusion, J. Stat. Phys., 79, 1023-1032 (1994) · Zbl 1106.82363 |
| [4] | Zhou, G. J., A lattice boltzmann model for groundwater flows, Int. J. Mod. Phys. C, 18, 6, 973-991 (2007) · Zbl 1194.76238 |
| [5] | Zhou, G. J., A rectangular lattice boltzmann method for groundwater flows, Modern Phys. Lett. B, 21, 531-542 (2007) · Zbl 1144.82329 |
| [6] | Zhou, G. J., Lattice Boltzmann Method for Shallow Water Flows (2004), Springer-Verlag: Springer-Verlag Berlin · Zbl 1052.76002 |
| [7] | Dellar, P. J., Non-hydrodynamic modes and a priori construction of shallow water lattice Boltzmann equations, Phys. Rev. E, 65, Article 036309 pp. (2002), (12 pages) |
| [8] | Liu, H.; Li, M.; Shu, A., Large eddy simulation of turbulent shallow water flows using multi-relaxation-time lattice Boltzmann model, Internat. J. Numer. Methods Fluids, 70, 1573-1589 (2012) · Zbl 1412.76048 |
| [9] | Thömmes, G.; Seaïd, M.; Banda, M. K., Lattice Boltzmann methods for shallow water flow applications, Internat. J. Numer. Methods Fluids, 55, 673-692 (2007) · Zbl 1127.76052 |
| [10] | Liu, H.; Zhou, G. J.; Burrows, R., Lattice Boltzmann model for shallow water flows in curved and meandering channels, Int. J. Comput. Fluid Dyn., 23, 209-220 (2009) · Zbl 1184.76798 |
| [11] | Linhao, Z.; Shide, F.; Shouting, G., Wind-driven ocean circulation in shallow water lattice boltzmann model, Adv. Atmos. Sci., 22, 349-358 (2005) |
| [12] | Li, Y.; Huang, P., A coupled lattice Boltzmann model for the shallow water-contamination system, Internat. J. Numer. Methods Fluids, 59, 195-213 (2009) · Zbl 1394.76101 |
| [13] | Banda, M. K.; Seaïd, M.; Thömmes, G., Lattice Boltzmann simulation of dispersion in two-dimensional tidal flows, Internat. J. Numer. Methods Engrg., 77, 878-900 (2009) · Zbl 1156.76426 |
| [14] | Peng, Y.; Zhou, J. G.; Burrows, R., Modelling solute transport in shallow water with the lattice Boltzmann method, Comput. & Fluids, 50, 181-188 (2011) · Zbl 1271.76270 |
| [15] | Janßen, C.; Krafczyk, M., Free surface flow simulations on GPGPUs using the LBM, Comput. Math. Appl., 61, 3549-3563 (2011) · Zbl 1225.76230 |
| [16] | Attar, E.; Körner, C., Lattice Boltzmann model for thermal free surface flows with liquid-solid phase transition, Int. J. Heat Fluid Flow, 32, 156-163 (2011) |
| [17] | Janssen, C.; Krafczyk, M., A lattice Boltzmann approach for free-surface-flow simulations on non-uniform block-structured grids, Comput. Math. Appl., 59, 2215-2235 (2010) · Zbl 1193.76088 |
| [18] | Filippova, O.; Hänel, D., Grid refinement for lattice-BGK models, J. Comput. Phys., 147, 219-228 (1998) · Zbl 0917.76061 |
| [19] | He, X.; Luo, L. S.; Dembo, M., Some progress in lattice Boltzmann method: Part I. Nonuniform mesh grids, J. Comput. Phys., 129, 357-363 (1996) · Zbl 0868.76068 |
| [20] | Shu, C.; Chew, Y. T.; Niu, X. D., Least-squares-based lattice Boltzmann method: A meshless approach for simulation of flows with complex geometry, Phys. Rev. E, 64, 1-4 (2001) |
| [21] | Qian, Y. H.; d’Humières, D.; Lallemand, P., Lattice BGK models for Navier-Stokes equation, Europhys. Lett., 17, 6, 479-484 (1992) · Zbl 1116.76419 |
| [22] | Zhou, G. J., Rectangular lattice Boltzmann method, Phys. Rev. E, 81, 1-10 (2010) |
| [23] | Budinski, Lj., Lattice Boltzmann method for 2D flows in curvilinear coordinates, J. Hydroinf., 14, 3, 772-783 (2012) |
| [24] | Ginzburg, I.; Verhaeghe, F.; d’Humiéres, D., Two-relaxation-time lattice Boltzmann scheme: About parametrization, velocity, pressure and mixed boundary conditions, Commun. Comput. Phys., 3, 427-478 (2008) |
| [25] | Ginzburg, I., Equilibrium-type and link-type lattice Boltzmann models for generic advection and anisotropic-dispersion equation, Adv. Water Resour., 28, 1171-1195 (2005) |
| [26] | Lallemand, P.; Luo, L. S., Theory of the lattice Boltzmann method: dispersion,dissipation, isotropy, Galilean invariance, and stability, Phys. Rev. E, 61, 6546-6562 (2000) |
| [27] | Bouzidi, M.; d’Humiéres, D.; Lallemand, P.; Luo, L. S., Lattice Boltzmann equation on a two-dimensional rectangular grid, Comput. Phys., 170, 704-717 (2001) · Zbl 1028.76040 |
| [28] | Budinski, Lj., MRT Lattice Boltzmann method for 2D flows in curvilinear coordinates, Comput. & Fluids, 96, 288-301 (2014) · Zbl 1391.76096 |
| [29] | Janßen, C. F.; Grilli, S. T.; Krafczyk, M., On enhanced non-linear free surface flow simulations with a hybrid LBM-VOF model, Comput. Math. Appl., 65, 211-229 (2013) · Zbl 1268.76046 |
| [30] | Thürey, N.; Rüde, U., Stable free surface flows with the lattice Boltzmann method on adaptively coarsened grids, Comput. Vis. Sci., 12, 5, 247-263 (2009) · Zbl 1426.76055 |
| [31] | Zhao, Z.; Huang, P.; Li, Y.; Li, J., A lattice Boltzmann method for viscous free surface waves in two dimensions, Internat. J. Numer. Methods Fluids, 71, 2, 223-248 (2013) · Zbl 1430.76158 |
| [32] | Bear, J., Hydraulics of Groundwater (1979), McGraw Hill: McGraw Hill New York |
| [33] | Simmonds, J. G., A Brief on Tensor Analysis (1994), Springer: Springer New York · Zbl 0790.53014 |
| [34] | Zhou, G. J., Lattice Boltzmann method for advection and anisotropic dispersion equation, J. Appl. Mech., 78, 1-5 (2011) |
| [35] | Mohamad, A. A., Lattice Boltzmann Method: Fundamentals and Engineering Applications with Computer Codes (2011), Springer-Verlag: Springer-Verlag London · Zbl 1247.80003 |
| [36] | Van Genuchten, M. Th., A closed-formed equation for predicting the hydraulic conductivity of unsaturated soils, Soil Sci. Soc. Am. J., 44, 5, 892-898 (1980) |
| [37] | Thiem, G., Hydrologische Methoden (1906), Gebhardt: Gebhardt Leipzig |
| [38] | Wang, H. F.; Anderson, M. P., Introduction to Groundwater Modeling (1982), Academic Press: Academic Press California |
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