Interdisciplinary Applied Mathematics

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The key theoretical issue here in developing a hybrid description from the atomistic to macroscopic scale is the identification and theoretical justification of a proper set of interface conditions. In past work associated with high-altitude rarefied flows, three different procedures have been proposed and implemented with various degrees of success:


   the Marshak condition,


extrapolating the fluxes, and


   extrapolating the properties.


The Marshak condition is an extension of a concept in radiative heat transfer, and it was first proposed by (Golse, 1989) for higher-atmosphere applications. It employs the half-fluxes at the interface, i.e., the flux of particles with velocity in the half normal velocity range. The total flux is then set to the sum of the half-flux based on the DSMC side and the half-flux based on the Navier-Stokes side. Matching the fluxes by extrapolation leads to a conservative global scheme, but the main difficulty comes from the large scatter of fluxes (i.e., momentum and heat flux) on the DSMC side that cannot match the smooth macroscopic fluxes on the Navier-Stokes side. In low Reynolds number cases, the scatter effect is more pronounced than in the high-altitude rarefied flows, and is particularly large for high-order moments, e.g., the fluxes. Finally, the extrapolation of macroscopic properties at the interface (density, velocity, temperature) does not guarantee monotonicity and is critically dependent on both smoothing and accuracy of extrapolation of these quantities. In addition, a large number of samples is required for averaging in order to obtain the macroscopic properties (an input to the Navier-Stokes solver), which renders this approach inefficient.

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