Extension of a Multiscale Particle Scheme to Near-Equilibrium Viscous Flows

Abstract

A recently proposed low diffusion (LD) equilibrium particle method based on the direct simulation Monte Carlo (DSMC) method is modified for use with near-equilibrium viscous flows. A finite volume discretization of the viscous terms in the compressible Navier-Stokes equations is used to incorporate diffusive transport effects into the LD particle method, and a velocity and temperature slip wall boundary condition is employed for improved accuracy in the slip flow Knudsen number regime. The modified method is compared with both DSMC and theory for a series of unsteady boundary layer problems, and excellent agreement is observed. The computational cost of the modified LD particle method is shown for a representative near-equilibrium case to be roughly four orders of magnitude lower than that of DSMC, due to reduced scatter as well as less stringent cell size and time step requirements in the LD method. Simulation procedures are outlined for a strongly coupled hybrid algorithm, where the LD particle method is used in continuum flowfield regions and DSMC is employed in nonequilibrium regions. The hybrid scheme is evaluated through a comparison with numerical and experimental data for a flow of N2 through a small convergent-divergent nozzle into a near vacuum, and hybrid simulation results are generally found to agree very well with other available data. I. Introduction AS flows involving a wide range of characteristic length scales appear in a number of different engineering applications, including those related to atmospheric flow around reentry or hypersonic vehicles, high-altitude rocket plumes, flows within and around micro-electro-mechanical systems, and any supersonic flow where internal shock structures are of interest. In these types of flows, a near-equilibrium gas velocity distribution may exist through much of the flowfield, as the equilibrating effect of intermolecular collisions dominates over other processes (such as inhomogeneous diffusive transport or gas-surface interaction) which tend to pull the velocity distribution away from equilibrium. However, some flowfield regions may have characteristic length scales comparable to or smaller than the local mean free path, so that the influence of collisions does not dominate and the velocity distribution diverges considerably from the equilibrium limit. While simulation of near-equilibrium flowfield regions may be efficiently performed using computational fluid dynamics (CFD) techniques based on the Navier-Stokes equations, nonequilibrium regions must be simulated using more expensive techniques based on the Boltzmann equation. The Boltzmann equation is the governing equation for dilute gas flows at arbitrary Knudsen numbers, and its derivation follows from assumptions of molecular chaos and binary intermolecular collisions with no approximations regarding the shape of the velocity distribution. The most mature and commonly used simulation method for the Boltzmann equation is the direct simulation Monte Carlo (DSMC) method, first introduced by Bird. 1 In a DSMC simulation, a large number of representative particles are tracked through a computational grid, and move and collide in a manner consistent with physical arguments underlying the Boltzmann equation. While this method may be applied to both nonequilibrium and continuum flowfield regions, cell size and time step limitations often make it prohibitively expensive for simulating continuum flows where global characteristic length scales are far larger than the mean free path. In practice, such multiscale flows are usually simulated using both DSMC and CFD methods, by applying CFD techniques in near-equilibrium regions where the Navier-Stokes equations are valid, and using DSMC elsewhere in the flow. In the simplest type of hybrid CFD-DSMC approach, a CFD simulation is performed on a domain which 1