BGK Boltzmann models for free-surface water flows and contaminant transport

As an alternative, the mesoscopic method is totally a different approach from conventional methods for modeling free-surface flows and contaminant transport. It is based on the BGK (proposed by Bhatnagar-Gross-Krook in 1954) Boltzmann equation and it has attractive features; especially, the nature of linear advective operator, the simple representative of diffusion or viscous terms by the collision term (which is a simple algebraic difference), the character of its satisfaction of the entropy condition, and the extendable feature to higher dimensional flows.;In this thesis, the second-order numerical models for one- and two-dimensional free-surface water flows and contaminant transport have been established based on the mesoscopic method. In the formulation of these models, the macroscopic variables, such as water depth, momentum and contaminant concentration, are obtained by taking moments (i.e., through the integration) of the BGK Boltzmann difference equation in the particle velocity space. The generic form of the BGK Boltzmann equation and, hence, its difference equation, provides considerable convenience for formulations of different kinds of numerical models.;To demonstrate their performances, the proposed models are applied to solve a range of typical free-surface flow problems and contaminant transport problems, most of which have either analytical solutions or laboratory experimental data. These problems include one- and two-dimensional free-surface flow problems and advection-diffusion transport problems. Numerical results have been compared with analytical solutions, solutions obtained by other numerical schemes, or laboratory experimental data. These comparisons demonstrate the high accuracy, high stability and robustness of the proposed models in simulating free-surface water flows and contaminant transport. It should be emphasized that the proposed models do not need any entropy fixes (or adding artificial damping terms) and all the computed results by the proposed mesoscopic-based models are free of spurious oscillations and non-physical shocks.;Based on the mesoscopic method, in this study, the mass and momentum conservation equations governing free-surface water flows and contaminant transport are obtained as the zeroth and first moments of the BGK Boltzmann equation. This study shows that the conventional shallow water equations and the pure advection transport equation are the special cases of the mesoscopic-based models under the local equilibrium state. The additional terms in the mesoscopic-based models are proved to be viscous terms which ensure that the solutions obtained from these models satisfy the entropy condition.