Lattice Boltzmann Model Using Two Relaxation Times for Shallow-Water Equations
Publication: Journal of Hydraulic Engineering
Volume 142, Issue 2
Abstract
A lattice Boltzmann method with two relaxation times for shallow-water equations () without turbulence is proposed. The described model is validated through simulations of three typical cases with laminar flows: 1D steady flow over a bump, 2D unsteady dam-break flow, and flow around circular cylinder. Good agreement between prediction and analytical or experimental solutions are obtained. In addition, the performance of the and the lattice Boltzmann method for shallow-water equations using a single relaxation time (LABSWE) is compared in detail. Studies have shown that the former is more stable than the latter for 2D cases. Different combinations of the two relaxation times are also studied and the optimized one is recommended for good numerical stability. This study demonstrates that the additional relaxation time in the can improve the stability of simulations for laminar shallow flows using a procedure almost as simple as that in the LABSWE.
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Acknowledgments
The first author acknowledges the financial support provided by the National Key Basic Research Program of China (Grant No. 2013CB036401) and National Natural Science Foundation of China (Grant No. 51409183). The second author acknowledges the financial support provided by National Natural Science Foundation of China (Grant No. 51279118). The third author acknowledges the financial support provided by the open funding from the State Key Laboratory of Hydraulics and Mountain River Engineering in Sichuan University, China (Grant No. 1004).
References
Begnudelli, L., and Sanders, B. F. (2007). “Conservative wetting and drying methodology for quadrilateral grid finite-volume models.” J. Hydraul. Eng., 312–322.
Bradford, S. F., and Sanders, B. F. (2002). “Finite-volume model for shallow-water flooding of arbitrary topography.” J. Hydraul. Eng., 289–298.
Chen, H., Chen, S., and Matthaeusm, W. H. (1992). “Recovery of the Navier-Stokes equations using a lattice gas Boltzmann method.” Phys. Rev. A, 45(8), R5339–R5342.
Chen, S. Y., and Doolen, G. D. (1998). “Lattice Boltzmann method for fluid flows.” Annu. Rev. Fluid Mech., 30(1), 329–364.
Dellar, P. J. (2002). “Non-hydrodynamic modes and a priori construction of shallow water lattice Boltzmann equations.” Phys. Rev. E, Stat. Nonlinear Soft Matter Phys., 65(036309), 1–12.
d’Humieres, D., Ginzburg, I., Krafczyk, M., Lallemand, P., and Luo, L.-S. (2002). “Multiple-relaxation-time lattice Boltzmann models in three dimensions.” Phil. Trans. R Soc. Lond. A, 360(1792), 437–451.
d’Humiμeres, D. (1992). “Generalized lattice Boltzmann equations.” Rarefied gas dynamics: Theory and simulations, Progress in astronautics and aeronautics, B. D. Shizgal and D. P. Weaver, eds., Vol. 159, AIAA Press, Washington, DC, 450–458.
Ginzburg, I. (2005). “Equilibrium-type and link-type lattice Boltzmann models for generic advection and anisotropic-dispersion equation.” Adv. Water Resour., 28(11), 1171–1195.
Ginzburg, I., d’Humières, D., and Kuzmin, A. (2010). “Optimal stability of advection-diffusion lattice Boltzmann models with two relaxation times for positive/negative equilibrium.” J. Stat. Phys., 139(6), 1090–1143.
Goutal, N., and Maurel, F., eds. (1997). “Proceedings of the 2nd Workshop on Dam-break Wave Simulation.”, Département Laboratoire National d’Hydraulique, Groupe Hydraulique Fluviale, Electricité de France, France.
Hammou, H., Ginzburg, I., and Boulerhcha, M. (2011). “Two-relaxation-times Lattice Boltzmann schemes for solute transport in unsaturated water flow, with a focus on stability.” Adv. Water Resour., 34(6), 779–793.
Kuzmin, A., Ginzburg, I., and Mohamad, A. A. (2011). “The role of the kinetic parameter in the stability of two-relaxation-time advection-diffusion lattice Boltzmann schemes.” Comput. Math. Appl., 61(12), 3417–3442.
Lai, Y. G. (2010). “Two-dimensional depth-averaged flow modeling with an unstructured hybrid mesh.” J. Hydraul. Eng., 12–23.
Lallemand, P., and Luo, L. S. (2000). “Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance and stability.” Phys. Rev. E, 61(6), 6546–6562.
Liu, H., Zhou, J. G., and Burrows, R. (2009a). “Multi-block lattice Boltzmann simulations of subcritical flow in open channel junctions.” Comput. Fluids, 38(6), 1108–1117.
Liu, H., Zhou, J. G., and Burrows, R. (2009b). “Numerical modeling of turbulent compound channel flow using the lattice Boltzmann method.” Int. J. Numer. Methods Fluids, 59(7), 753–765.
Liu, H., Zhou, J. G., and Burrows, R. (2010). “Lattice Boltzmann simulations of the transient shallow water flows.” Adv. Water Resour., 33(4), 387–396.
Macchione, F., and Morelli, M. A. (2003). “Practical aspects in comparing the shock-capturing schemes for dam break problems.” J. Hydraul. Eng., 187–195.
Peng, Y., Zhou, J. G., and Burrows, R. (2011a). “Modelling solute transport in shallow water with the lattice Boltzmann method.” Comput. Fluids, 50(1), 181–188.
Peng, Y., Zhou, J. G., and Burrows, R. (2011b). “Modelling the free surface flow in rectangular shallow basins by lattice Boltzmann method.” J. Hydraul. Eng., 1680–1685.
Qian, Y., d’Humiμeres, D., and Lallemand, P. (1992). “Lattice BGK models for Navier-Stokes equation.” Europhys. Lett., 17(6), 479–484.
Salmon, R. (1999). “The lattice Boltzmann method as a basis for ocean circulation modeling.” J. Mar. Res., 57(3), 503–535.
Servan-Camas, B., and Tsai Frank, T.-C. (2008). “Lattice Boltzmann method with two relaxation times for advection-diffusion equation: Third order analysis and stability analysis.” Adv. Water Resour., 31(8), 1113–1126.
Servan-Camas, B., and Tsai Frank, T.-C. (2009). “Saltwater intrusion modeling in heterogeneous confined aquifers using two-relaxation-time lattice Boltzmann method.” Adv. Water Resour., 32(4), 620–631.
Stecca, G., Siviglia, A., and Toro, E. F. (2012). “A finite volume upwind-biased centred scheme for hyperbolic systems of conservation laws. Application to shallow water equations.” Commun. Comput. Phys., 12(4), 1183–1214.
Talon, L., Bauer, D., Gland, N., Youssef, S., Auradou, H., and Ginzburg, I. (2012). “Assessment of the two relaxation time lattice-Boltzmann scheme to simulate Stokes flow in porous media.” Water Resour. Res., 48(4), W04526.
Yoon, T. H., and Kang, S. K. (2004). “Finite volume model for two-dimensional shallow water flows on unstructured grids.” J. Hydraul. Eng., 678–688.
Yu, C. S., and Duan, J. (2014). “Two-dimensional hydrodynamic model for surface-flow routing.” J. Hydraul. Eng., 04014045.
Yulistiyanto, B., Zech, Y., and Graf, W. H. (1998). “Flow around a cylinder: Shallow-water modeling with diffusion-dispersion.” J. Hydraul. Eng., 419–429.
Zhou, J. G. (2002). “A lattice Boltzmann model for the shallow water equations.” Comput. Methods Appl. Mech. Eng., 191(32), 3527–3539.
Zhou, J. G. (2004). Lattice Boltzmann methods for shallow water flows, Springer, Berlin.
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Published In
Journal of Hydraulic Engineering
Volume 142 • Issue 2 • February 2016
Copyright
© 2015 American Society of Civil Engineers.
History
Received: May 25, 2014
Accepted: Jun 8, 2015
Published online: Aug 19, 2015
Discussion open until: Jan 19, 2016
Published in print: Feb 1, 2016
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