Isogeometric schemes in rarefied gas dynamics context

Abstract

Geometrical complexity is typical of missile and reentry systems, small-scale devices with intrinsic milli-to-micro scale features, or in general topology-rich systems. Additionally, at high altitudes or at microscales, when the flow becomes rarefied, the conventional continuum/macroscopic equations produce unsatisfactory results. A microscopic/kinetic description becomes necessary. In this work, we introduce isogeometric (IG) kinetic schemes (KS) for the full Boltzmann equation and related kinetic models (Bhatnagar–Gross–Krook (BGK), Ellipsoidal statistical BGK, and Shakov). These schemes are specifically aimed for modeling flows in topologically complex “multi-patch” geometries such as those constructed using computer-aided geometric design tools; high order accurate; unconditionally time-stable (as per our tests) following generalized-α method; non-linearly stable following streamlined upwind Petrov–Galerkin method applied at every patch level with patch-to-patch coupling enforced either strongly by constraining the interface using single degree of freedom, or weakly using numerical fluxes as in discontinuous Galerkin (useful because the underlying system is first order in space); amenable to nearly-linear parallel efficiency owing to the six-dimensional nature of Boltzmann equation, wherein the convection and collision can be, respectively, performed simultaneously over velocity and physical spaces. In particular, since the surface representation is exact in IG, when surface properties are to be computed, which is often the case in aerodynamic applications, these schemes perform well. A series of verification tests for 1D/2D/3D-3V initial–boundary value and boundary-value flow problems, involving momentum, energy, and diffusive transport, including problems posed on non-conforming domains/patches, is performed to illustrate the stability and accuracy of the proposed method.