Extension of the normal shock wave relations for calorically imperfect gases

An extension to the normal shock relations for a thermally perfect, calorically imperfect gas, modelling the vibrational excitation with an anharmonic oscillator model and including the influence of electronic modes, is derived and studied. Such additional considerations constitute an extension to the work achieved in the past, which modelled the caloric imperfections with a harmonic oscillator for vibrational energy and did not consider the effect of electronic energy. Additionally, the newly derived expressions provide physical insights into the limitations of experimentation for replicating flight conditions, which is demonstrated through providing solutions at different upstream temperatures. The results are compared with direct simulation Monte Carlo simulations for nitrogen and air, with the extent of the caloric imperfection of the gas showing excellent agreement. For low upstream temperatures, the extended relations are found to be in good agreement with the original normal shock wave expressions, but the results diverge for higher upstream temperatures that would be more representative of real flows. The results show that the new expressions depart from ideal gas theory for Mach numbers in excess of 4.9 at wind-tunnel conditions and for any Mach number above 3.0 at flight conditions. It is also shown that the traditional harmonic oscillator model and the anharmonic oscillator model begin to diverge at Mach number 3.0 for molecular oxygen gas and at Mach number 5.0 for an air mixture at flight conditions.