One-dimensional non-equilibrium shock wave structure analysis by simplified conventional Burnett equations

Abstract

The inner shock wave structure with non-equilibrium effect is difficult to be accurately simulated due to the great gradient of density and temperature. In this paper, simplified conventional Burnett (SCB) equations were formulated for the study of hypersonic shock wave structure in continuum-transition regime. The conventional Burnett equations were derived by using the second-order Chapman–Enskog expansion of the velocity distribution function in Boltzmann equation. By neglecting conventional Burnett terms which are inversely proportional to Mach number, the constitutive relations in SCB equations were simplified specifically for hypersonic flow. The rotational and vibrational energy balance equations were also introduced into the governing equations to study the non-equilibrium relaxation processes inside shock waves. Meanwhile, generalized Rankine–Hugoniot relations were established to obtain the post-shock flow parameters in non-equilibrium flow. The numerical methods included three-order Runge-Kutta time-splitting method and AUSMPW+ flux-difference splitting method with MUSCL scheme. One-dimensional Nitrogen shock wave structure at different Mach numbers was simulated using SCB and NS equations respectively. The results indicate that the SCB equations can capture the shock waves structures more precisely and the flow variables are in better agreement with the DSMC results than NS equations in high Mach number cases.