Lattice Boltzmann simulation of depth-averaged models in flow hydraulics

Abstract

We apply a lattice Boltzmann method (LBM) for the simulation of depth-averaged models in flow hydraulics and dispersion of pollutants. The mathematical equations for these models can be obtained from the incompressible Navier–Stokes equations under the assumptions that the vertical scale is much smaller than any typical horizontal scale and the pressure is hydrostatic. The effects of bed slope, bed friction, Coriolis forces and wind stresses are also accounted for in our simulations. Our aim is to develop a simple and accurate representation of the source terms in order to simulate practical shallow water flows without relying on upwind algorithms or Riemann problem solvers. For the transport of pollutants, a depth-averaged convection–diffusion equation is used. We validate the algorithm in problems where analytical solutions are available. Furthermore, we test the algorithm in the case of a practical application by simulating the tidal flow and pollutant transport in the Strait of Gibraltar. The focus is to examine the performance of the LBM for irregular geometry with complex bathymetry. The method demonstrates its capability to capture the main flow features. Obviously, some of the conclusions in the current work are specific to the employed implementation of the LBM. For instance, the LBM in its implementation as described in the current study failed to approximate numerical solutions for hydraulic problems involving a Froude number larger than the unity.

Keywords:

• lattice Boltzmann method
• shallow water equations
• pollutant transport
• Strait of Gibraltar