Continuum Formulation for Non-Equilibrium Shock Structure Calculation (Communications in Computational Physics, 2006)
Are extensions to continuum formulations for solving fluid dynamic problems in the transition-to-rarefied regimes viable alternatives to particle methods? It\ris well known that for increasingly rarefied flow fields, the predictions from continuum\rformulation, such as the Navier-Stokes equations lose accuracy. These inaccuracies are\rattributed primarily to the linear approximations of the stress and heat flux terms in the\rNavier-Stokes equations. The inclusion of higher-order terms, such as Burnett or high-order moment equations, could improve the predictive capabilities of such continuum\rformulations, 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\rconduction coefficients appropriate for the rarefied flow regime, we will show that the\rNavier-Stokes-type continuum formulation may still be properly used. The equations\rwith generalization of the dissipative coefficients based on the closed solution of the\rBhatnagar-Gross-Krook (BGK) model of the Boltzmann equation, are solved using the\rgas-kinetic numerical scheme. This paper concentrates on the non-equilibrium shock\rstructure calculations for both monatomic and diatomic gases. The Landau-Teller-Jeans\rrelaxation model for the rotational energy is used to evaluate the quantitative difference\rbetween the translational and rotational temperatures inside the shock layer. Variations\rof shear stress, heat flux, temperatures, and densities in the internal structure of the\rshock waves are compared with, (a) existing theoretical solutions of the Boltzmann solution, (b) existing numerical predictions of the direct simulation Monte Carlo (DSMC)\rmethod, and (c) available experimental measurements. The present continuum formulation for calculating the shock structures for monatomic and diatomic gases in the\rMach number range of 1.2 to 12.9 is found to be satisfactory.