Evaluation of a Particle Method for the Ellipsoidal Statistical Bhatnagar-Gross-Krook Equation

Abstract

A particle method is presented for the numerical simulation of rarefied gas flows, based on the ellipsoidal statistical Bhatnagar-Gross-Krook (ES-BGK) model of the Boltzmann equation. The simulation procedure includes consideration of rotational nonequilibrium, and enforces exact momentum and energy conservation for a mixture involving monatomic and diatomic species. This method is applied to the simulation of a nozzle flow of the type associated with cold-gas spacecraft thrusters, and flowfield characteristics are compared with experimental data as well as results from direct simulation Monte Carlo (DSMC) and Navier-Stokes simulations of the same flow. The ES-BGK method is shown to allow for a relatively high degree of accuracy in transitional flow regimes, while avoiding the intermolecular collision calculations which typically make the DSMC simulation of low Knudsen number flows prohibitively expensive. I. Introduction n te im the design and performance analysis of low-thrust spacecraft propulsion systems, various numerical simulation chniques may be employed to determine efficiency, thrust characteristics, or the potential for plume pingement and contamination on spacecraft surfaces. A particular challenge in the simulation of small thrusters involving a chemically inert neutral gas, such as electro-thermal or cold-gas thrusters, is the accurate consideration of a wide range of Knudsen number regimes. In a typical thruster of this type, gas is expelled through a convergentdivergent (Laval) nozzle into a near vacuum, with subsonic near-equilibrium flow in the convergent section of the nozzle. As the gas accelerates through the divergent section beyond the nozzle throat, a subsonic boundary layer grows along the wall, while a supersonic core-flow region exists around the nozzle centerline. The gas density continues to decrease with downstream distance through the nozzle, and rarefaction effects become more prominent within both the supersonic and subsonic regions. Here the gas velocity distribution begins to diverge significantly from the equilibrium limit, and thermal energy is increasingly distributed non-uniformly among the translational and internal degrees of freedom. Beyond the nozzle exit plane, these nonequilibrium effects continue to increase as the gas rapidly expands and thermal energy is converted into energy associated with bulk motion of the exhaust flow. Rotational freezing occurs due to the large gradients and low collision frequency, and the flow approaches the free molecular limit within a short distance of the nozzle exit, particularly at points far from the nozzle centerline. I The simulation of highly rarefied flows, as described above for the divergent nozzle region and plume, is typically performed using the direct simulation Monte Carlo (DSMC) method of Bird. 1 This method approximates a numerical solution to the Boltzmann equation – the governing equation for dilute gas flows based on a statistical representation of molecular velocities – by decoupling in time the collision and advection terms in the equation. A large number of particles, each representing a large number of atoms or molecules, are tracked through a computational grid, and are sorted into cells according to their location. During each time step, some fraction of the particles in a cell collide with each other, and probabilistic techniques are used for calculations of individual collisions. All particles are then moved through the grid according to assigned velocities, and particles are created or removed at inflow and outflow boundaries. Finally, macroscopic quantities are sampled by averaging various particle properties in each cell, and the process is then repeated at the next time step. The DSMC method has been shown to provide accurate solutions for highly rarefied nozzle and plume flows, 2