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This thesis is concerned with the application of Lattice Boltzmann Method (LBM) to various steady incompressible fluid-flow and heat-transfer problems in the macro- and micro-flow regimes. The Lattice Boltzmann Method is a means of flow computation based on simplified kinetic models using discrete particle velocities, which exploits the fact that the collective behaviour of the molecules does not explicitly depend on their individual dynamics. Being a relatively new and an alternative method of flow computation, it is still being experimented with to realize its full scope of application and to test its ability to capture untested physics. The present thesis can be considered an attempt in that direction. Several computer programs based on the programming language ‘C’ have been developed to carry out computations to solve a host of problems, some of which are hitherto unexamined. To provide a means of comparison of the LBM results for untested flow configurations, a Finite-Difference-Method (FDM) code has been developed and its results are shown to be highly accurate through a careful codevalidation exercise. In the applications, first a number of two-dimensional (2D) singlelid-driven cavity flow problems are computed through the LBM Single-Relaxation-Time (SRT) and Multi-Relaxation-Time (MRT) methods and the LBM-MRT method is shown to overcome some of the problems faced by the LBM-SRT method - especially in resolving the corner singularities at high-Reynolds number situations. Then a new test problem, namely, ‘two-sided lid-driven square cavity flow’ is proposed and many carefully established qualitative and quantitative results are given for algorithm validation by other workers. It is shown that for parallel motion of the walls there is a ‘free-shear’ layer midway between the moving plates and that near-trailing-edge corner vortices form at a much lower Reynolds number compared with the single-lid-driven cavity flow. Computing flows in two-sided non-facing and four-sided lid-driven square cavities and in a two-sided rectangular cavity with parallel wall motion, it is demonstrated that not only continuum-based methods like the Finite Difference Method and Finite Volume Method (FVM) but also LBM has the ability to capture multiple-steady solutions. It may be noted that the traditional mathematical concept of well-posedness does not apply here and for the first time the ability and accuracy of the Lattice Boltzmann Method to obtain solutions to this peculiar class of problems is demonstrated. LBM computations are also carried out for the single- and two-sided lid-driven cubic and prism cavity flows. For the single-lid-driven case, two-dimensional and three-dimensional (3D) velocity profiles in the symmetry plane are compared to study the end-wall effects. For 3D computations the D3Q19 lattice model is shown to be the most convenient to use. Also LBM in conjunction with the IEDDF approach is used to compute thermally-driven flows in the square and cubic cavities. Comparing the 2D and 3D velocity profiles in the symmetry plane, effect of the end walls with increasing Rayleigh numbers is brought out - probably for the first time. LBM is then applied to compute flows in various micro-geometries. The study reveals many interesting features of micro-couette, micro-channel (pressuredriven) and micro-lid-driven cavity flows and demonstrates the ability of LBM to capture those flow features. The main concern of this thesis is the computation of steady flows. In keeping with this theme, highly accurate LBM computations are also carried out for the flow past a circular cylinder at low Reynolds numbers when the flow is steady and symmetric. To demonstrate the ability of the present method to compute time-varying flows, computations are also carried out for a higher Reynolds number at which the symmetry breaks and the flow becomes time-periodic. All the results presented in the thesis are independent of the lattice size and they are substantiated carefully. Though all the computed flows fall in the laminar regime, difficult flow configurations exemplified by large values of the Reynolds and Rayleigh numbers are also computed. The work successfully demonstrates that LBM has come of age and is now an important alternative solution procedure in computational fluid dynamics.
Supervisor: Anup Kr. Dash