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Abstract:

In recent years, computational fluid dynamic (CFD) methods have become essential for the aeronautics industry. These methods are used in conjunction with experimental approaches. They allow, for example, to have a precise understanding of a flow in order to help the design of aircraft components. Thus, these methods are constantly being improved to accurately reproduce the physics of simulated flows. Thanks to its reduced computational cost, its ease of implementation and use, the lattice Boltzmann method (LBM) has gradually emerged as an alternative approach to traditional numerical methods. In addition to its efficiency, this method has the advantages of being inherently unsteady and is perfectly adapted to complex geometries. Unfortunately, in its standard form, it remains limited to the simulation of isothermal and weakly compressible flows, which excludes many aeronautical applications. Nevertheless, compressible versions of this method do exist. Among the most promising ones, one can find the Hybrid method (HLBM) which uses the LBM to compute the mass and momentum equations and a finite difference scheme to computed the energy equation. This thesis focuses on the development and study of hybrid methods for the simulation of compressible flows. First, the standard LBM and its limitations are presented as well as an exhaustive bibliographical study of its compressible versions. Then, the components and general functioning of the hybrid method are shown. A simple model is developed under the Boussinesq approximation and successfully tested. However, its naive extension to the compressible equations is found to be numerically unstable due to: (1) the perfect gas coupling, and (2) the unstable nature of the LBM at high Mach numbers. These two problems are then solved by adopting a more robust collision operator for the LBM part and an entropy equation for the energy part. A stable compressible hybrid model, based on a standard D2Q9 discrete velocity lattice, is thus obtained and validated on subsonic and supersonic test cases including shock waves. Nevertheless, for the simulation of strong shock waves, the model shows conservativity problems. These are mainly attributed to the use of the entropy equation which is non-conservative. Solutions are then proposed. Then, the stability and spectral properties of HLBM schemes are studied using the von Neumann approach. This study highlights the effect of numerous parameters on numerical stability such as the collision model, the choice of the energy equation or the influence of numerical parameters like the CFL number. Instability mechanisms and stabilization methods are shown, confirming at the same time the gain in terms of stability of models based on an entropy equation. Moreover, thanks to a more in-depth analysis, this study allows to explain the observed non-physical behavior of certain models and attributes them to mode transfers.   Finally, the hybrid method is implemented in the ProLB industrial LBM solver. It is then tested on subsonic and supersonic three-dimensional compressible test cases containing boundary conditions and mesh refinements.

Keywords: LBM, compressible, hybrid, von Neumann analysis