A numerical method for coupling the BGK model and Euler equations through the linearized Knudsen layer

Abstract

The Bhatnagar-Gross-Krook (BGK) model, a simplification of the Boltzmann equation, in the absence of boundary effect, converges to the Euler equations when the Knudsen number is small. In practice, however, Knudsen layers emerge at the physical boundary, or at the interfaces between the two regimes. We model the Knudsen layer using a half-space kinetic equation, and apply a half-space numerical solver [19, 20] to quantify the transition between the kinetic to the fluid regime. A full domain numerical solver is developed with a domain-decomposition approach, where we apply the Euler solver and kinetic solver on the appropriate subdomains and connect them via the half-space solver. In the nonlinear case, linearization is performed upon local Maxwellian. Despite the lack of analytical support, the numerical evidence nevertheless demonstrate that the linearization approach is promising.

Linear Moment Models to Approximate Knudsen Layers

“…The gas exhibits non-Newtonian behavior in the Knudsen layer, and there is a finite velocity or temperature gap at the surface, known as the velocity slip or temperature jump [21,37]. A better understanding of the Knudsen layer may help design numerical methods for coupling the Boltzmann and Euler equations [13,39], which avoid solving the complex multidimensional Boltzmann equation in the full space.…”