Development of boundary conditions for Lattice Boltzmann Method applied to non-lattice conforming geometries

This master's thesis focuses on the Lattice Boltzmann Method, an alternative to conventional Computational Fluid Dynamics, which provides significant computational cost savings by employing a regular lattice independent of the geometry considered instead of a structured or unstructured mesh which itself may prove to be a labor and CPU-intensive complex task, especially for complex geometries. The computational efficiency of the LBM, the simplicity of the algorithm, and the ease with which this method allows complex geometries to be dealt with promise numerous advantages in complex industrial flow applications even in the supersonic and hypersonic ranges. This work is a follow-up of the LBMHYPE project, an ESA TRP research project involving a partnership between the von Karman Institute for Fluid Dynamics, École Polytechnique, Université Paris Sud Saclay and CENAERO. The aim of the LBMHYPE project is to extend LBM to supersonic and hypersonic regimes since, at the state of the art, LBM has been limited to the incompressible and weakly compressible range. The major innovation proposed in the LBMHYPE project to simulate compressible flows is the use of innovative vectorial scheme methods, in which each conservation equation is solved by a dedicated Lattice Boltzmann scheme. A practical challenge encountered in LBM which seriously degraded the solution was the application of slip-wall boundary conditions to geometries that do not conform with the Cartesian lattice, such as an arbitrarily inclined wedge surface of a flow over a wedge. The contribution of this work involves implementing a new type of LBM boundary condition, called Bouzidi Bounce Back with Normal, within the Python library "pylbm" which was developed in the course of the LBMHYPE project. This type of boundary condition is an extension of the frequently applied Bouzidi-type LBM boundary condition, which is, in turn, an extension of the better-known LBM boundary condition, namely, the Bounce-Back boundary condition by adding a corrective term, which is a numerical flux, to Bouzidi's relationships. This treatment takes into account the normal and tangential directions of the wall surface and accurately resolves the geometrical orientation of the slip wall and its normal. The Bouzidi Bounce Back with Normal boundary conditions successfully solved the problems faced in the LBMHYPE project regarding the imposition of the slip condition to non-Cartesian coordinate conforming geometries. The results compare more favorably with the analytical and US3D code results than the results obtained in the LBMHYPE project. Furthermore, these conditions have demonstrated good performance in non-rectilinear geometries. In fact, in the last part of this thesis, BCs were tested in the classic problem of a compressible flow around a 2D cylinder. Possible future developments of the present project might involve addressing the numerical diffusion and extension of the LBM vectorial scheme to the Navier-Stokes equations. The LBM method development should also target where it is particularly powerful, in cases involving complex geometries and multiple physics such as multi-phase flows.