Non-uniform grid based LBM for the Saint-Venant equations

A novel form of the lattice Boltzmann method for the Saint-Venant equations (LABSVE) is formulated and implemented. To eliminate equidistant property of the lattice structure being the major drawback of the LBM, and thus establish the LB model as tool for practical application (natural water bodies) fully competitive compared to the presently available computational models, new form of equilibrium distribution function (EDF) based on geometrically transformed Saint-Venant equations (SVE) is derived. Introducing the metric relation function between the physical and computational system, the non-uniform spatial discretization is managed by solving the transformed SVE rather than altering the symmetrical properties of lattice structure. Since the geometrical correlation between the two corresponding systems is introduced into the transformed SVE, the very basic structure of the LBM in terms of simplicity in implementing the boundary conditions and computational efficiency (owing to the parallel structure) is maintained and further improved by removing the limitations imposed by the equidistant lattice condition. Furthermore, the discharge-water level definition of the SVE as more applicable form when natural watercourses are considered, is applied. In this way the bed slope term is completely eliminated from the equations, establishing therefore a model more suitable for natural systems characterized by arbitrary cross sectional geometry. In order to test the proposed non-equidistant LABSVE model, steady flow in prismatic channel, unsteady flow in non-prismatic channel of simple cross section geometry, and unsteady flow along a Danube section is modelled. Additional simulations by using the HEC-RAS modelling tool are conducted in order to produce comparison basis for both unsteady simulations, while in case of the Danube River section actual water level and discharge measurements are additionally used for calibration and verification of the model. Very good agreement between compared results is obtained showing that the proposed non-equidistant form of the LABSVE is capable of solving complex environmental problems, maintaining the order of accuracy, simplicity and efficiency of the basic LBM.