No-Slip Boundary Conditions for the Lattice Boltzmann Method and High Order Accurate Interface Representation for Volume Tracking

Abstract

The present thesis focuses on two distinct topics of computational fluid dynamics. The first one, treated in part I, focuses on the no-slip boundary conditions for the lattice Boltzmann method, whereas the second one, presented in part II, proposes a high order accurate description of the interface in two-phase flow computations for volume tracking in two dimensions.

Part I of the present thesis presents an important issue in the framework of the lattice Boltzmann method, namely no-slip boundary conditions. Since the central object of the lattice Boltzmann method is the particle distribution function, the implementation of the no-slip boundary condition, although straightforward for continuum Navier Stokes solvers, is more involved. Additional physical arguments for the no-slip boundary condition at straight walls are presented. This leads to an alternative formulation of the no-slip boundary condition for the lattice Boltzmann method. This boundary condition is second order accurate with respect to the grid spacing and conserves mass. The origin of numerical instabilities observed for a variety of other boundary conditions is investigated, and it is shown how these can be removed leading to stable boundary conditions. Some arguments unifying different formulations of the no-slip boundary condition are presented. In addition to straight boundary conditions, the question of curved boundary conditions is treated. These represent an elevated level of complexity, since the lattice Boltzmann method is only defined for equidistant Cartesian grids. The curved boundary condition in the present thesis conserves the second order accuracy of the lattice Boltzmann method.

Due to the complexity of two-phase flow problems, the majority of numerical methods in this field displays a rather low order of accuracy. In part II of the present thesis, a subproblem of the two-phase flow problem, namely the tracking of the interface is treated. Two different interface descriptions allowing a higher order accurate approximation of the interface between two immiscible phases are presented. A central issue has been the conservation of mass. Therefore, the advection of the interface in the present framework is based on the tracking of the volume of each fluid, which makes the method akin to the volume of fluid method. A drawback of the present method compared to traditional volume of fluid methods is that topological changes are not handled automatically. For deforming drops without topological changes, high order of accuracy and mass conservation have been verified for two benchmark test cases.