Kinetic Flux–Vector Splitting for the Navier–Stokes Equations

Before a hybrid scheme can be developed combining the direct simulation Monte Carlo (DSMC) method and a Navier–Stokes (NS) representation, one must have access to compatible kinetic-split fluxes from the NS portion of the hybrid scheme. The kinetic theory basis is given for the development of the required fluxes from the Chapman–Enskog velocity distribution function for a simple gas; and these are then extended to a polyatomic gas by use of the Eucken approximation. The derived fluxes are then used to implement boundary conditions at solid surfaces that are based on concepts associated with kinetic theory and the DSMC method. This approach is shown to lead to temperature slip and velocity slip as a natural outcome of the new formulation, a requirement for use in the near-continuum regime where DSMC and NS must be joined. Several different flows, for which solid boundaries are not present, are computed using the derived fluxes, together with a second-order finite-volume scheme, and the results are shown to agree well with several established numerical schemes for the NS equations.