Discrete-Velocity Models and Numerical Schemes for the Boltzmann-BGK Equation in Plane and Axisymmetric Geometries

We present new numerical models for computing transitional or rarefied gas flows as described by the Boltzmann-BGK and BGK-ES equations. We first propose a new discrete-velocity model, based on the entropy minimization principle. This model satisfies the conservation laws and the entropy dissipation. Moreover, the problem of conservation and entropy for axisymmetric flows is investigated. We find algebraic relations that must be satisfied by the discretization of the velocity derivative appearing in the transport operator. Then we propose some models that satisfy these constraints. Owing to these properties, we obtain numerical schemes that are economic, in terms of discretization, and robust. In particular, we develop a linearized implicit scheme for computing stationary solutions of the discrete-velocity BGK and BGK-ES models. This scheme is the basis of a code which can compute high altitude hypersonic flows, in 2D plane and axisymmetric geometries. Our results are analyzed and compared to other methods.