(untitled) (Advances in Applied Mathematics and Mechanics, 2025)
The lattice Boltzmann method is a relatively modern method. It has been widely applied to industrial flows (for small scales) governed by the Navier Stokes equations, but little applied to large-scale flows, such as geophysical flows. In this paper, we then attempt to apply the lattice Boltzmann method to coastal flows governed by the Saint-Venant equations (also called Shallow water equations) via an application to the real case of the Nador lagoon (Moroccan Mediterranean coast). The establishment of the equations of the model and their discretization is given here. The introduction of turbulence by the large eddy simulations was also presented. The verification of the developed code concerns two test cases in hydraulic engineering. The first verification test deals with complex flow in a 180^◦ bent channel. For this case, the model has faithfully reproduced all the characteristics of this flow (pressure/depression zone and the recirculation zone after the bend). Similarly, the results of this test case have been successfully compared to measurements available in the literature. The second verification test analyzes the flow around a spur-dike. Finally, the developed code has been adapted for its application to the real case of the Nador lagoon whose flow is generated by the superposition of the tidal waves M_2 and S_2. The model reproduced with good accuracy the dynamics of the flow and more particularly the two vortices observed on both sides of the lagoon inlet. The founding results are successfully compared with the results of previous studies published for the lagoon.