Lattice Boltzmann Simulation for Shallow Water Flow Applications

Abstract

Under the influence of gravity, many free-surface flows can be modelled by the well-known shallowwater equations under the assumption that the vertical scale is much smaller than any typical horizontal scale and the pressure is hydrostatic. These equations can be derived from the depth-averaged incompressible Navier-Stokes equations and usually include continuity and momentum equations. Hence, the applications of depth-averaged models include a wide spectrum of phenomena in hydraulic flows such as ocean circulation modelling Salmon (1999a) and wind-driven ocean circulation Zhong et al. (2005) to name but a few. Simulation of such real-world flow problems is not trivial since the geometry can be complex and the topography irregular. Numerical methods based on the finite difference, the finite volume or the finite element methods have been applied to simulate the shallow water equations, refer to Bermudez & Vazquez (1994); Kurganov & Levy (2002); LeVeque (1998); Stansby & Zhou (1998); Toro (1992); Vazquez-Cendon (1999); Vukovic & Sopta (2002); Xing & Shu (2006); Zhou (1995) among others. For most of these approaches, the treatment of bed slopes and friction forces often cause numerical difficulties in obtaining accurate solutions, see, for example, Bermudez & Vazquez (1994); LeVeque (1998); Vazquez-Cendon (1999). In addition the extension of these schemes to complex geometries is not trivial, refer to Benkhaldoun et al. (2007), for example. Some of these approaches are very expensive if one considers real flows Vukovic & Sopta (2002). Since the problems are posed at a large scale it has been the aim of practitioners to develop a simple and accurate representation of the source terms in order to simulate practical shallow water flows without relying on upwind discretization or Riemann problem solvers, refer to Benkhaldoun et al. (2009; 2010); Benkhaldoun & Seaid (2010) and references therein for these alternatives. The idea of this chapter will be to give the reader a self-contained introduction of the developments and the implementation of the shallow water lattice Boltzmann approach. In this chapter the lattice Boltzmann method will be applied to the simulation of depth-averaged models in flow hydraulics and dispersion Banda et al. (2009); Klar et al. (2008); Seaid & Thommes (2009); Thommes et al. (2007). It can be pointed out that the shallow water equations referred to in this discussion are viscous and also account for the effects of bed slope, bed friction, Coriolis forces and wind stresses in two-dimensional simulations Dellar (2002); Salmon (1999a); Zhou (2002). The practical aspects 11