Solute transport in shallow water flows using the coupled curvilinear Lattice Boltzmann method

A novel form of coupled curvilinear lattice Boltzmann model (CCLBM) for solving the solute transport in shallow water flows, imposed by the complex geometry, is developed and applied. Shallow water equations, transformed into curvilinear coordinate system, are solved using the lattice Boltzmann equation with multiple-relaxation-time (MRT-LBM), while the curvilinear form of the 2D advection-diffusion equation is solved applying the Bhatnagar-Gmss-Krook (BGK-LBM) approach. Corresponding forms of equilibrium distribution function for both the shallow water and the advection-diffusion equation is derived using the D2Q9 lattice. The physical flow domain of arbitrary geometry in the horizontal plane is covered with adequate curvilinear mesh, while the calculation procedure for flow and transport is carried out in the D2Q9 square lattice, applying the basic form of the boundary condition method on water-solid and open boundaries as well. The coupled flow and pollution transport is tested using a straight inclined channel, irrigation channel with a parabolic cross section in a 90 degrees bend, and a segment of the Danube River. In the cases of the straight inclined channels analytic solution is used as a comparative base, while for the bent channel previously obtained velocity measurements are utilized for flow calibration, where a mathematical model based on traditional CFD procedures is used for pollution transport validation. Unsteady flow and salinity transport in the Danube River is calibrated and modelled using velocity and salinity measurements, along with the results obtained by the traditionally based CFD model. The high level of agreement between the results obtained by the proposed model and the corresponding analytical values and measurements indicates that the presented curvilinear form of the LBM is capable of solving very complex environmental problems.