Study of Non-Equilibrium Flow Behind Normal Shock

Abstract

Normal shock problems in high enthalpy flows are of special interests to aerodynamicists and fluid dynamicists. When the shock Mach number become hypersonic and increasing further, the gas passing through the shock is compressed resulting in increase in temperature and pressure. As the Mach number increases the internal degrees of freedom of the diatomic molecules are activated to an increasing extent when it crosses the shock resulting dissociation especially for high enthalpy flows. Hence dissociation of diatomic molecules must be taken into account in the determination of some of the aerodynamic parameters. This thermal and chemical process can be divided into three types such as nearly frozen, non-equilibrium and nearly non-equilibrium depending on the rates of reaction and excitation. For typical re-entry conditions of spacecrafts into a planets atmosphere, dissociation reactions of the molecules is dominant in the stagnation flow. Further in the stagnation region of the flow field one of the most important parameter that characterizes the flow field is the shock stand-off distance. This parameter is often employed for validation purposes of numerical methods as well as for non-reactive and reactive gases. For high Mach number flows the shock is very close to the body hence experimental determination of shock stand-off distance is very difficult and there would be relatively large errors. Therefore the theoretical determination of this parameter is of great significance in the discussion of this physical phenomenon. There are some works which presents how the dissociation behind shock affects the shock stand-off distance. Thus the dissociation behind the shock is a very important process which has great impact in aerodynamic flight and design. In this present work we studied how dissociation of diatoms occur behind a normal shock. Treanor and Marrone (1962) proposed CVD(coupled vibration-dissociation) model for diatoms by assuming diatom as a harmonic oscillator with a cut-off level. But actually diatoms are not harmonic oscillator, because spectroscopic data of energy level spacing is not like harmonic oscillator. For this reason, Treanor, Rich, and Rehm(1968) used anharmonic oscillator model for diatoms to study vibrational relaxation. Taking the anharmonicity of diatom, Philip Morse(1929) gave a formula for potential energy levels for diatoms, which is known to express the experimental values quite accurately. Unlike the energy levels of the harmonic oscillator potential, which are evenly spaced , the Morse potential level spacing decreases as the energy approaches the dissociation energy and then it is continuous. So it is quite accurate to take Morse oscillator theory for diatomic dissociation instead of harmonic oscillator with a cut-off level. We have used Morse oscillator theory to derive a dissociation-recombination reaction rate equation for diatom. To derive the rate equation we have used the transition probability between different vibrational energy levels . The rate equation is numerically solved to get the different flow variables behind the shock. The result of the present work has been compared with some of the previous work. Some of the flow variables are well matching with the previous work and some has discrepancy near the shock but well matching after few distance from the shock. We have also studied under what conditions the post shock flow shows self-similar behavior in its scaling relations. It is shown that as far as there is no dissociation, we could expect to obtain self-similar solutions. However, when there is dissociation, the non-equillibrium nature of the phenomenon disrupts the self-similar nature of the flow.