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Lattice Boltzmann approach to simulating a wetting–drying front in shallow flows

Published online by Cambridge University Press:  03 March 2014

H. Liu  and
J. G. Zhou
H. Liu*
Affiliation:
Key Laboratory of Water and Sediment Sciences of Ministry of Education, School of Environment, Beijing Normal University, Beijing 100875, PR China
J. G. Zhou
Affiliation:
School of Engineering, University of Liverpool, Brownlow Hill, Liverpool L69 3GQ, UK
*
†Email address for correspondence: [email protected]
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Abstract

The paper reports a new lattice Boltzmann approach to simulating wetting–drying processes in shallow-water flows. The scheme is developed based on the Chapman–Enskog analysis and the Taylor expansion, which is consistent with the theory of the lattice Boltzmann method. All the forces, such as bed slope and bed friction, are taken into account naturally in determining the wet–dry interface, without the use of either the spurious assumption of a thin water film on a dry bed or the non-physical extrapolation of certain variables such as water depth or velocity. This offers a simple and general model for simulating wetting–drying processes in complex flows involving external forces. Its verification is carried out by modelling several one-dimensional (1D) and two-dimensional (2D) flows: (i) 1D sloshing over a parabolic container; (ii) a 1D tidal wave over three adverse bed slopes; (iii) a 1D solitary wave run up on a plane sloping beach; (iv) a tsunami run up on a plane beach; (v) a 2D stationary case with wet–dry boundaries; (vi) a 2D long-wave resonance over a parabolic basin; and (vii) a 2D solitary wave run up on a conical island. The numerical results agree well with analytical solutions, other numerical results and experimental data, demonstrating the effectiveness and accuracy of the new approach.

JFM classification

Mathematical Foundations: Computational methods Waves/Free-surface Flows: Hydraulics Waves/Free-surface Flows: Solitary waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 743 , 25 March 2014 , pp. 32 - 59
Copyright
© 2014 Cambridge University Press 

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References

Bhatnagar, P., Gross, E. P. & Krook, M. K. 1954 A model for collision processes in gases: I. Small amplitude processes in charged and neutral one-component system. Phys. Rev. A 94, 511–525.Google Scholar
Bradford, S. F. & Sanders, B. F. 2002 Finite-volume model for shallow-water flooding of arbitrary topography. J. Hydraul. Eng. ASCE 128, 289–298.Google Scholar
Briggs, M. J., Synolakis, C. E., Harkins, G. S. & Green, D. R. 1995 Laboratory experiments of tsunami runup on a circular island. Pure Appl. Geophys. 144, 569–593.CrossRefGoogle Scholar
Brufau, P., Vázquez-Cendón, M. E. & García-Navarro, P. 2002 A numerical model for the flooding and drying of irregular domains. Int. J. Numer. Methods Fluids 39, 247–275.Google Scholar
Carrier, G. F., Wu, T. T. & Yeh, H. 2003 Tsunami run-up and draw-down on a plane beach. J. Fluid Mech. 475, 79–99.CrossRefGoogle Scholar
Chen, H., Goldhirsch, I. & Orszag, S. 2008 Discrete rotational symmetry, moment isotropy, and higher order lattice Boltzmann models. J. Sci. Comput. 34, 87–112.CrossRefGoogle Scholar
Delis, A. I., Kazolea, M. & Kampanis, N. A. 2008 A robust high-resolution finite volume scheme for the simulation of long waves over complex domains. Int. J. Numer. Methods Fluids 56, 419–452.Google Scholar
Dellar, P. 2002 Non-hydrodynamic modes and a priori construction of shallow water lattice Boltzmann equations. Phys. Rev. E 65, 036309.CrossRefGoogle Scholar
Frandsen, J. B. 2008 A simple LBE wave runup model. Prog. Comput. Fluid Dyn. 8, 222–232.Google Scholar
Gallardo, J. M., Parés, C. & Castro, M. 2007 On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas. J. Comput. Phys. 227, 574–601.Google Scholar
George, D. L. 2008 Augmented Riemann solvers for the shallow water equations over variable topography with steady states and inundation. J. Comput. Phys. 227, 3089–3113.Google Scholar
Heniche, M., Secretan, Y., Boudreau, P. & Leclerc, M. 2000 A two-dimensional finite element drying–wetting shallow water model for rivers and estuaries. Adv. Water Resources 23, 359–372.Google Scholar
Hibberd, S. & Peregrine, D. H. 1979 Surf and run-up on a beach: a uniform bore. J. Fluid Mech. 95, 323–345.Google Scholar
Hubbard, M. E. & Dodd, N. 2002 A 2d numerical model of wave run-up and overtopping. Coast. Eng. 47, 1–26.Google Scholar
Junk, M., Klar, A. & Luo, L.-S. 2005 Asymptotic analysis of the lattice Boltzmann equation. J. Comput. Phys. 210, 676–704.Google Scholar
Kânoğlu, U. & Synolakis, C. E. 1998 Long wave runup on piecewise linear topographies. J. Fluid Mech. 374, 1–28.CrossRefGoogle Scholar
Kennedy, A. B., Chen, Q., Kirby, J. T. & Dalrymple, R. A. 2000 Boussinesq modeling of wave transformation, breaking and runup. J. Waterway Port Coast. Ocean Eng. 126, 39–47.Google Scholar
Leclerc, M., Bellemare, J.-F., Dumas, G. & Dhatt, G. 1990 A finite element model of Estuarian and river flows with moving boundaries. Adv. Water Resources 13, 158–168.Google Scholar
LeVeque, R. J. & George, D. L. 2008 High-resolution finite volume methods for the shallow water equations with bathymetry and dry states. In Advanced Numerical Models for Simulating Tsunami Waves and Runup (ed. Liu, P., Yeh, H. & Synolakis, C.), Advances in Coastal and Ocean Engineering, vol. 10, pp. 43–73. World Scientific.Google Scholar
Li, Y. & Raichlen, F. 2002 Non-breaking and breaking solitary wave run-up. J. Fluid Mech. 456, 295–318.Google Scholar
Liu, P. L.-F., Cho, Y.-S., Briggs, M.J., Kânoğlu, U. & Synolakis, C. E. 1995 Runup of solitary waves on a circular island. J. Fluid Mech. 302, 259–285.Google Scholar
Liu, H., Zhou, J. G. & Burrows, R. 2010 Lattice Boltzmann simulations of the transient shallow water flows. Adv. Water Resources 33, 387–396.Google Scholar
Liu, H., Zhou, J. G. & Burrows, R. 2012 Inlet and outlet boundary conditions for the lattice-Boltzmann modelling of shallow water flows. Prog. Comput. Fluid Dyn. 12, 11–18.Google Scholar
Lynett, P. J., Wu, T.-R. & Liu, P. L.-F. 2002 Modeling wave runup with depth-integrated equations. Coast. Eng. 46, 89–107.Google Scholar
Madsen, P. A., Sørensen, O. & Schäffer, H. A. 1997 Surf zone dynamics simulated by a Boussinesq type model. Part I. Model description and cross-shore motion of regular waves. Coast. Eng. 32, 255–287.Google Scholar
Mahdavi, A. & Talebbeydokhti, N. 2009 Modeling of non-breaking and breaking solitary wave run-up using FORCE-MUSCL scheme. J. Hydraul. Res. 47, 476–485.Google Scholar
Marche, F., Bonneton, P., Fabrie, P. & Seguin, N. 2007 Evaluation of well-balanced bore-capturing schemes for 2D wetting and drying processes. Int. J. Numer. Methods Fluids 53, 867–894.CrossRefGoogle Scholar
Rogers, B., Fujihara, M. & Borthwick, A. G. L. 2001 Adaptive $Q$ -tree Godunov-type scheme for shallow water equations. Int. J. Numer. Methods Fluids 35, 247–280.3.0.CO;2-E>CrossRefGoogle Scholar
Salmon, R. 1999 The lattice Boltzmann method as a basis for ocean circulation modeling. J. Mar. Res. 57, 503–535.Google Scholar
Sampson, J., Easton, A. & Singh, M. 2006 Moving boundary shallow water flow above parabolic bottom topography. ANZIAM J. 47, C373–C387.Google Scholar
Shafiai, S. H.2011 Lattice Boltzmann method for simulating shallow free surface flows involving wetting and drying. PhD Thesis, University of Liverpool.Google Scholar
Synolakis, C. E.1986 The runup of long waves. PhD Thesis, California Institute of Technology, Pasadena.Google Scholar
Synolakis, C. E. 1987 The runup of solitary waves. J. Fluid Mech. 185, 523–545.Google Scholar
Thacker, W. C. 1981 Some exact solutions to the nonlinear shallow-water wave equations. J. Fluid Mech. 107, 499–508.Google Scholar
Titov, V. V. & Synolakis, C. E. 1995 Modeling of breaking and nonbreaking long-wave evolution and runup using VTCS-2. J. Waterway Port Coast. Ocean Eng. 121, 308–316.Google Scholar
Zelt, J. A. 1991 The run-up of nonbreaking and breaking solitary waves. Coast. Eng. 15, 205–246.Google Scholar
Zhou, J. G. 2002 A lattice Boltzmann model for the shallow water equations with turbulence modeling. Int. J. Mod. Phys. C 13, 1135–1150.CrossRefGoogle Scholar
Zhou, J. G. 2004 Lattice Boltzmann Methods for Shallow Water Flows. Springer.CrossRefGoogle Scholar
Zhou, J. G. 2011 Enhancement of the LABSWE for shallow water flows. J. Comput. Phys. 230, 394–401.CrossRefGoogle Scholar