Discrete Velocity Boltzmann Schemes with Efficient Multidimensional Models

Abstract

Traditional CFD algorithms have achieved a high degree of sophistication for modelling fluid flows in the past five decades. This sophistication can be seen clearly in one dimensional modelling, with a wide variety of successful algorithms tuned for capturing the discontinuities in the flows, with the focus on accuracy and robustness. The modelling of multidimensional flows, however, has not been as sophisticated. Most of the industrial flow solvers are based on the 1-D models extended to multi-dimensions in a rather simple way, as in the popular cell-centred finite volume methods, with the inherent limitation of locally 1-D modelling. As a result, these algorithms are grid-dependent, with the discontinuities aligned with the grid lines being captured crisply while the discontinuities oblique to the grid lines are diffused. A few good multi-dimensional models introduced by the researchers have not met with the success of industrial applications, as compared to the earlier 1-D physics based models. Thus, there is a need for designing better multi-dimensional models. In this work, a new multi-dimensional kinetic theory based algorithm is introduced for simulating compressible flows. Modelling multidimensional compressible flows at the macroscopic level is beset with the mathematical difficulties of dealing with the non-commuting flux Jacobian matrices, together with the algorithms based strongly on eigen-structure. Alternative modelling based on kinetic theory is thus simpler. The main advantage comes from the linearity of the convection terms in Boltzmann equation (together with the nonlinear collision term), the moments of which lead to the nonlinear hyperbolic conservation equations of the macroscopic model. Thus, developing truly multidimensional algorithms is also expected to be simpler in this elegant framework. A multidimensional kinetic theory based algorithm is proposed in this thesis, with a neat separation of different physical aspects of multidimensional flows and their appropriate numerical treatment.

The new algorithm begins with the separation of fluid and peculiar velocities in the convec tion terms of the Boltzmann equation, which naturally leads to macroscopic convection-pressure splitting. With the identification of the unidirectional information propagation for the fluid velocity part, an appropriate streamline upwinding method is proposed. Based on the multi directional information propagation for the peculiar velocity part (correspondingly the pressure part at the macroscopic level), a fractional update based kinetic flux difference splitting method, which generates an algorithm at the macroscopic level, is introduced. Higher order accuracy is achieved using k-exact reconstruction, which suits the multidimensional features of the scheme well. The new multidimensional kinetic scheme is tested on several typical benchmark test cases for Euler equations and is shown to yield superior results when compared to the corresponding grid-aligned finite volume based kinetic scheme.