Computation of Hypersonic Flow of a Diatomic Gas in Rotational Non-Equilibrium past a Blunt Body Using the Generalized Boltzmann Equation

Abstract

The results of 2-D numerical simulations of non-equilibrium hypersonic flow of a diatomic gas, e.g., nitrogen past a 2-D blunt body at low to high Knudsen Numbers are presented. The flow field is computed using the Generalized Boltzmann (or the Wang-Chang Uhlenbeck [1]) Equation (GBE) for Kn varying from 0.1 to 10. In the GBE [2], the internal and translational degrees of freedom are considered in the framework of quantum and classical mechanics respectively. The computational framework available for the classical Boltzmann equation (for a monoatomic gas with translational degrees of freedom) [3] is extended by including the rotational degrees of freedom in the GBE. The general computational methodology for the solution of the GBE for a diatomic gas is similar to that for the classical BE except that the evaluation of the collision integral becomes significantly more complex due to the quantization of rotational energy levels. 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. An efficient computational methodology has been developed for the solution of GBE for computing the flow field in diatomic gases at high Mach numbers. There are two main difficulties encountered in computation of high Mach number flows of diatomic gases with rotational degrees of freedom using the GBE: (1) a large velocity domain is needed for accurate numerical description of molecular velocity distribution function resulting in enormous computational effort in calculation of the collision integral, and (2) about 50 to 70 energy levels are needed for accurate representation of the rotational spectrum of the gas. These two problems result in very large CPU and memory requirements for shock wave computations at high Mach numbers (> 6). Our computational methodology has addressed these problems, and as a result efficiency of calculations has increased by several orders of magnitude. The code has been parallelized on a SGI Origin 2000, 64 R12000 MIPS processor supercomputer.

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

“…Let m = (m 1 , m 2 ) ∈ N 2 + be the number of particles of each type, ǫ = (ǫ 1 , ǫ 2 ) ∈ (0, ∞) 2 be their diameters, and recall from Section 2.1 that we denote the vectors of all positions and velocities by X m = X (1,0) m1 , X (0,1)…”

“…(ǫ1,ǫ2) , f (N1,N2) (0) = f (N1,N2),0 (1.9) where Z * (N1,N2) is the post-collisional configuration related to the pre-collisional configuration Z (N1,N2) by the collisional laws given in (1.3) -(1.8). 1 We note that the above boundary condition is defined for a full surface measure subset of the boundary (see Section 3).…”