A Conservative Numerical Method for Solving the Generalized Boltzmann Equation for an Inert Mixture of Diatomic Gases

Abstract

This paper describes the computational methodology for computing hypersonic nonequilibrium shock wave (SW) flows in a mixture of non reacting diatomic gases such as Nitrogen and Oxygen using the Generalized Boltzmann Equation (GBE) at Knudsen numbers in transitional and rarefied flow regimes. In the GBE, the internal and translational degrees of freedom are considered in the framework of quantum and classical mechanics respectively. The computational framework available for the standard Boltzmann equation (for a monoatomic gas with translational degrees of freedom) is extended by including both the rotational and vibrational degrees of freedom in the GBE. The solution of GBE requires modeling of transition probabilities, elastic and inelastic cross-sections etc. of a diatomic gas molecule, needed for the solution of the collision integral. The whole problem that includes both the vibrational-translational (VT) and rotational-translational (RT) energy transfers is solved by applying a three-stage splitting procedure to the GBE. The three stages consist of free molecular transport, VT relaxation, and RT relaxation. For computation of shock structure in a mixture of gases, the GBE needs to be formulated in impulse space instead of the standard velocity space. Furthermore, till now, the computations of SW in neutral gas mixtures have been performed only for 1D problem and they assume cylindrical symmetry of the solution in the velocity space. In this paper, we describe the development of a general numerical method for multicomponent neutral mixtures applicable to 2D and 3D flows. A 3D code has been developed and applied to compute the SW in neutral binary mixture.

Kinetic solution of the structure of a shock wave in a nonreactive gas mixture

“…The eight dimensional collision integral is evaluated on a uniform grid following the method of Tcheremissine. 5,6 The discretized collision operator with a symmetry weighting, ␥ ␣ , is…”

A Rigorous Derivation of a Boltzmann System for a Mixture of Hard-Sphere Gases

“…Similar mixtures have also undergone extensive analysis as possible coolants for thermoacoustic refrigerators [13]. Sound propagation in binary mixtures [16], and hypersonic shockwave analysis for aerospace applications [30,2] have also been studied.…”

“…Global well posedness for mild solutions has been proven in the case of inverse power molecular interactions and small data [20], and in the case of hard potentials also with small data [19]. Stability in various formulations has been extensively studied [12,20,19], in addition to numerous numerical schemes [24,30,8]. Furthermore, rigorous connections between (1.1) and the compressible Navier-Stokes equations for mixtures of fluids have been established [12].…”