Structure of shock waves in a partially ionized gas. Ph.D. Thesis

Abstract

A unified picture of the structure of steady one-dimensional shock waves in a partially ionized gas in the absence of externally applied electric or magnetic field is presented. The investigation is based on a two-temperature, three-fluid continuum approach using the Navier–Stokes equations as a model. The self-induced electric field is given by Poisson's equation. All the dissipative and transfer terms are included. Charge neutrality and zero velocity slip are assumed everywhere except across the Atom–Ion Shock. It is also assumed that the rates of atom–atom and electron–atom inelastic collisional ionization processes are governed by the rate of excitation of the atoms from the ground level to the first level. The analysis of the governing equations is based on the Matched Asymptotic Technique which results in the following three layers: (1) A broad Thermal Layer dominated by electron thermal conduction; (2) Atom–Ion (A–I) Shock structured by the heavy particle-atoms and ions-collisional dissipative phenomena; and (3) Ionization Relaxation Layer (IRL) wherein electron–atom inelastic collisions dominate. The small parameter of the asymptotic analysis has been identified to be the ratio of energy fluxes due to heavy particle viscous dissipation and electron thermal conduction and has been designated by ?=?la?/le?\delta = \epsilon l'_a / l'_e?=?la??/le??, where ?\epsilon? is the ratio of electron mass to atom mass and la?l'_ala?? and le?l'_ele?? are the atom–atom and electron–electron collisional mean free paths evaluated at free stream conditions. By expanding the flow variables in terms of this parameter, the governing equations are simplified in each layer and then solved numerically. The analysis has been restricted to the first two orders of ?\delta? and the results obtained have been presented in the form of Tables and Figures for Argon for both complete thermodynamic equilibrium and ionizational nonequilibrium conditions at the upstream infinity. In the Thermal Layer, whose thickness decreases with increasing free stream Mach number (M?M_\inftyM??) at constant atom number density (na?n_a^\inftyna??) or increasing na?n_a^\inftyna?? at constant M?M_\inftyM??, only the electron temperature and electric potential show any marked increases and the changes in other flow variables are very small. In the A–I Shock the changes in all the variables are significant. In contrast to the frozen ionization case, the induced electric field is found to decrease with increasing M?M_\inftyM?? at constant na?n_a^\inftyna?? or decreasing na?n_a^\inftyna?? at constant M?M_\inftyM??. The shock thickness based on the maximum velocity gradient is in general of the order of three atom–atom elastic collisional mean free paths and is found to increase with increasing M?M_\inftyM?? or decreasing na?n_a^\inftyna??, the other remaining constant. Inclusion of electron thermal conduction terms has been observed to introduce numerical instabilities in the analysis of the IRL. A new small parameter ?A=?i?\delta_A = \alpha_i \delta?A?=?i??, ?i\alpha_i?i? being the initial degree of ionization and ?i\Theta_i?i? the nondimensional characteristic temperature for first ionization, has been identified. The numerical troubles are circumvented by adopting a regular perturbation scheme with respect to ?A\delta_A?A? and obtaining solutions for the first two orders of ?A\delta_A?A?. Electron thermal conduction has been found to introduce oscillations in the electron temperature profiles in the beginning of the IRL. Towards the end of the IRL, besides the electron thermal conduction, heavy particle dissipative mechanisms have also been found to bring about marked changes in the flow variables thus emphasising the importance of their inclusion in the governing equations for a better understanding of the shock structure in ionized gases. Such sharp changes in the flow variables induce a strong electric field in the IRL and the consequent rise in the electric potential across the layer is substantial. The characteristic ionization relaxation length and relaxation time based on Petschek and Byron’s definition have been found to decrease with increasing M?M_\inftyM?? or decreasing na?n_a^\inftyna??. Irrespective of the nature of the free stream conditions, be it complete thermodynamic equilibrium or ionizational nonequilibrium, the analysis has been found to be valid for only ??10?2\delta \sim 10^{-2}??10?2.