The task of this diploma thesis is the Lattice Boltzmann method (LBM). LBM is a mesoscopic method describing the particle motion in a fluid by the Boltzmann equation, where the distribution function is involved. The Chapman-Enskog expansion shows the connection with the macroscopic Navier-Stokes equations of conservation laws. In this process the Hermite polynoms are used. The Lattice Boltzmann equation is derived by the discretisation of velocity, space and time which is concluding to the numerical algorithm. This algorithm is applied at two problems of fluid flow: the two-dimensional square cavity and a flow arround obstacles. In both cases were the results of velocities compared to results calculated by finite volume method (FVM). The relative errors are in order of multiple 1 %.

This dissertation includes a theoretical and numerical development to simulate liquid-vapor flows and the applications to microchannels. First, we obtain a consistent non-local pressure equation for simulating liquid-vapor interfacial flows using mean-field free-energy theory. This new pressure equation is shown to be the general form of the classical van der Waals" square-gradient theory. The new equation is implemented in two-dimensional (2D) D2Q7, D2Q9, and three-dimensional (3D) D3Q19 lattice Boltzmann method (LBM). The three LBM models are validated successfully in a number of analytical solutions of liquid-vapor interfacial flows. Second, we have shown that the common bounceback condition in the literature leads to an unphysical velocity at the wall in the presence of surface forces. A few new consistent mass and energy conserving velocity-boundary conditions are developed for D2Q7, D2Q9, and D3Q19 LBM models, respectively. The three LBM models are shown to have the capabilities to successfully simulate different wall wettabilities, the three typical theories or laws for moving contact lines, and liquid-vapor channel flows. Third, proper scaling laws are derived to represent the physical system in the framework of the LBM. For the first time, to the best of the author's knowledge, we obtain a flow regime map for liquid-vapor channel flows with a numerical method. Our flow map is the first flow regime map so far for submicrochannel flows, and also the first iso-thermal flow regime map for CO₂ mini- and micro-channel flows. Our results show that three major flow regimes occur, including dispersed, bubble/plug, and liquid strip flow. The vapor and liquid dispersed flows happen at the two extremities of vapor quality. When vapor quality increases beyond a threshold, bubble/plug patterns appear. The bubble/plug regimes include symmetric and distorted, submerged and non-wetting, single and train bubbles/plugs, and some combination of them. When the Weber number<10, the bubble/plug flow regime turns to a liquid strip pattern at the increased vapor quality of 0.5~0.6. When the Weber number>10, the regime transition occurs around a vapor quality of 0.10~0.20. In fact, when an inertia is large enough to destroy the initial flow pattern, the transition boundary between the bubble and strip regimes depends only on vapor quality and exists between x=0.10 and 0.20. The liquid strip flow regimes include stratified strip, wavy-stratified strip, intermittent strip, liquid lump, and wispy-strip flow. We also find that the liquid-vapor interfaces become distorted at the Weber number of 500~1000, independent of vapor quality. The comparisons of our flow maps with two typical experiments show that the simulations capture the basic and important flow mechanisms for the flow regime transition from the bubble/plug regimes to the strip regimes and from the non-distorted interfaces to the distorted interfaces. Last, our available results show that the flow regimes of both 2D and 3D fall in the same three broad categories with similar subdivisions of the flow regimes, even though the 3D duct produces some specific 3D corner flow patterns. The comparison between 2D and 3D flows shows that the flow map obtained from 2D flows can be generally applied to a 3D situation, with caution, when 3D information is not available. In addition, our 3D study shows that different wettabilities generate different flow regimes. With the complete wetting wall, the flow pattern is the most stable. / Ph. D.