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? Itis well known that for increasingly rarefied flow fields, the predictions from continuumformulation, such as the Navier-Stokes equations lose accuracy. These inaccuracies areattributed primarily to the linear approximations of the stress and heat flux terms in theNavier-Stokes equations. The inclusion of higher-order terms, such as Burnett or high-order moment equations, could improve the predictive capabilities of such continuumformulations, but there has been limited success in the shock structure calculations, especially for the high Mach number case. Here, after reformulating the viscosity and heatconduction coefficients appropriate for the rarefied flow regime, we will show that theNavier-Stokes-type continuum formulation may still be properly used. The equationswith generalization of the dissipative coefficients based on the closed solution of theBhatnagar-Gross-Krook (BGK) model of the Boltzmann equation, are solved using thegas-kinetic numerical scheme. This paper concentrates on the non-equilibrium shockstructure calculations for both monatomic and diatomic gases. The Landau-Teller-Jeansrelaxation model for the rotational energy is used to evaluate the quantitative differencebetween the translational and rotational temperatures inside the shock layer. Variationsof shear stress, heat flux, temperatures, and densities in the internal structure of theshock 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 theMach number range of 1.2 to 12.9 is found to be satisfactory.
Keywords: Non-equilibrium flow; bulk viscosity; kinetic scheme; shock structure.