Lattice kinetic scheme for generalized coordinates and curved spaces

We present a new lattice kinetic method to simulate fluid dynamics in curvilinear geometries and curved spaces. A suitable discrete Boltzmann equation is solved in contravariant coordinates, and the equilibrium distribution function is obtained by a Hermite polynomials expansion of the Maxwell-Boltzmann distribution, expressed in terms of the contravariant coordinates and the metric tensor. To validate the model, we calculate the critical Reynolds number for the onset of the Taylor-Couette instability between two concentric cylinders, obtaining excellent agreement with the theory. In order to extend this study to more general geometries, we also calculate the critical Reynolds number for the case of two concentric spheres, finding good agreement with experimental data, and the case of two concentric tori, where we have found that it is around 10% larger than the respective values for the two concentric cylinders. © 2014 World Scientific Publishing Company.