μ-DSMC: a general viscosity method for rarefied flow

A modified DSMC method for rarefied flows is described, by which any viscosity law μ=μ(T) may be simulated. The method is simple to implement. The collision cross-section of a simple collision model, such as the hard sphere or variable hard sphere (VHS) is made to vary from cell to cell, based on the time-averaged cell temperature and the required viscosity at that temperature. The method is here demonstrated for two viscosity laws which fit experimental data better than does the hard sphere or variable hard sphere viscosity laws, but in principle the method can use the experimental data directly. The new method is tested in two different flows: high speed Couette flow and a plane 1D shock. For Couette flow, the shear stress and heat transfer, calculated from the velocity distribution, agree with the theoretical values calculated from the flow gradients and the theoretical transport coefficients. For the plane 1D shock, the new method is compared with the generalized hard sphere (GHS) model. The new method produces profiles of density and temperature within the shock which are generally indistinguishable from the GHS results except for a deviation in the Tx temperature component in a small region ahead of the shock. This deviation depends on the shock Mach number; for the worst case it is 4.6%. The deviation can be reduced by basing the imposed viscosity on the maximum component of kinetic temperature (in this case Tx) rather than the mean kinetic temperature. The new method is shown to be insensitive to the number of simulator particles used in each cell. Three translational degrees of freedom are considered here. However, because μ-DSMC is based on a hard sphere or VHS cross-section, it is compatible with the most commonly used Borgnakke–Larsen energy exchange model for translational–rotational energy exchange.