Lattice Boltzmann shallow water equations for large scale hydraulic analysis

Abstract

This thesis presents a new lattice Boltzmann model for both steady and unsteady two-dimensional shallow water equations. Throughout this work, the usage of different collision operators (CO) for the lattice Boltzmann solution of shallow water equations is proposed and investigated: BGK linear CO based on a single relaxation time (SRT), cascaded and cumulant CO with a multiple relaxation times approach (MRT). The motivation in using a MRT collision operator instead of the standard BGK, was to introduce the maximum number of adjustable parameters, which leads to an improvement of both stability and accuracy. The thesis focuses on the development, validation and applications of the aforementioned CO for shallow water flows. The cascaded LBM is based on the use of central moments as basis; it overcomes the defects in Galilean invariance of the original MRT method and it has been shown to further improve stability. An adaptation of the original formulation proposed for a single-phase fluid is therefore proposed and developed to reproduce shallow free surface flow. Furthermore, an alternative and more concise approach is based on the use of a cumulant collision operator, which relaxes, in the collision step, quantities (i.e. cumulants) that are Galilean invariant by construction. In the first part of the thesis, a convergence study of the different approaches, based on the use of the Taylor Green Vortex as test case, is performed, to compare conventional and innovative solution methods from stability and accuracy point of view. Then, the second part is devoted to analyzing different strategies to introduce, in the innovative models, the treatment of external forces term and various kinds of boundary conditions, that maintain the accuracy characteristics of the model. Special attention is due to the wet-dry front in shallow flows; in fact, a correct simulation of such processes plays a crucial role in practical engineering studies. The proposed methodologies are tested and validated through the use of analytical solutions and experimental solutions, taken as benchmarks throughout the thesis. Finally, the suitability of the proposed mathematical model for hydraulic engineering applications is discussed through the modelling of a real flood event.