For internal energy relaxation in rarefied gas mixtures, exact relationships are derived between the selection probability P employed in direct simulation Monte Carlo (DSMC) methods and the macroscopic relaxation rates dictated by collision number Z in Jeans’ equation. These expressions apply to the Borgnakke–Larsen model for internal energy exchange mechanics and are not limited to the assumption of constant Z. Although Jeans’ equation leads to adiabatic relaxation curves, which coalesce to a single solution when plotted against the cumulative number of collisions, it is shown that the Borgnakke–Larsen selection probabilities depend upon the intermolecular potential, the number of internal degrees of freedom, and the DSMC selection methodology. Furthermore, simulation results show that the common assumption P=1/Z is invalid, in general, and leads to considerably slower relaxation than stipulated by Z in Jeans’ equation. Moreover, inconsistent definitions of collision rates appearing in the literature can lead to considerable errors in DSMC models. Finally, for general gas mixtures, Borgnakke–Larsen DSMC kinetics match Jeans’ behavior exactly only when using a selection methodology, which prohibits multiple relaxation events during a single collision.
The use of the same symbol, Z, representing a ‘‘collision number’’ for thermal relaxation, has lead to confusion regarding its definition in the context of both continuum and particle simulations. Examination of the relaxation mechanics employed in particle simulations demonstrates that these definitions differ by a numerical factor that depends upon the intermolecular potential. Particle and continuum simulations employing appropriate definitions of Z lead to identical results during isothermal and adiabatic stationary relaxation.
Models are developed to permit direct Monte Carlo techniques to simulate coupled vibration–dissociation (CVD) behavior prevalent in high-temperature gases. This transient thermochemical phenomenon leads to dissociation incubation, reduced quasisteady dissociation rates, and non-Boltzmann distributions of vibrational energy during both dissociation and recombination. Essential for simulation of rarefied gas dynamics, Monte Carlo methods employ discrete particles to simulate molecular interactions directly, but have traditionally incorporated simplistic reaction models which failed to capture CVD behavior. To identify thermochemical collisions within the gas, a new dissociation selection probability is developed as a function of the extent by which the collision energy exceeds the gap between the dissociation threshold and the molecular vibrational energy of bounded anharmonic oscillators. A free parameter φ in the probability function controls the extent of vibrational favoring in dissociation selection. The new model is modified for application to the unbounded simple harmonic oscillator. Simulation of dissociation- and recombination-dominated thermochemical relaxation of O2 reservoirs, as well as the dissociation incubation behavior of N2 behind strong shock waves, demonstrates the ability of the new models to capture CVD behavior. Parameter φ is assessed empirically for O2 and N2 dissociation by comparison to experimental data.
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