Are extensions to continuum formulations for solving fluid dynamic problems in the transition-to-rarefied regimes viable alternatives to particle methods? It
is well known that for increasingly rarefied flow fields, the predictions from continuum
formulation, such as the Navier-Stokes equations lose accuracy. These inaccuracies are
attributed primarily to the linear approximations of the stress and heat flux terms in the
Navier-Stokes equations. The inclusion of higher-order terms, such as Burnett or high-order moment equations, could improve the predictive capabilities of such continuum
formulations, but there has been limited success in the shock structure calculations, especially for the high Mach number case. Here, after reformulating the viscosity and heat
conduction coefficients appropriate for the rarefied flow regime, we will show that the
Navier-Stokes-type continuum formulation may still be properly used. The equations
with generalization of the dissipative coefficients based on the closed solution of the
Bhatnagar-Gross-Krook (BGK) model of the Boltzmann equation, are solved using the
gas-kinetic numerical scheme. This paper concentrates on the non-equilibrium shock
structure calculations for both monatomic and diatomic gases. The Landau-Teller-Jeans
relaxation model for the rotational energy is used to evaluate the quantitative difference
between the translational and rotational temperatures inside the shock layer. Variations
of shear stress, heat flux, temperatures, and densities in the internal structure of the
shock waves are compared with, (a) existing theoretical solutions of the Boltzmann solution, (b) existing numerical predictions of the direct simulation Monte Carlo (DSMC)
method, and (c) available experimental measurements. The present continuum formulation for calculating the shock structures for monatomic and diatomic gases in the
Mach number range of 1.2 to 12.9 is found to be satisfactory.