The Application of the BGK Model in Particle Simulations
14 Jun 2000-
TL;DR: A collision model for the Direct Simulation Monte Carlo (DSMC) method is presented in this article, which is based on the BGK1 equation and makes use of the Cercignani2 ellipsoidal distribution to incorporate the effects of heat conductivity.
Abstract: A collision model for the Direct Simulation Monte Carlo (DSMC) method is presented. The collision model is based on the BGK1 equation and makes use of the Cercignani2 ellipsoidal distribution to incorporate the effects of heat conductivity. Results obtained by the DSMC method and its BGK and BGKC modifications for a 10° wedge and a flat plate are presented and discussed. Introduction An area of interest for particle simulation codes is that of dense near-equilibrium flows in Micro Electromechanical Systems (MEMS) in which the scales of the phenomena make the application of continuum techniques questionable. However, the extension of particle simulation methods to highdensity flows is usually associated with very small time steps needed to resolve the large number of collisions that take place each time step. The computational load that high-density flows impose on particle simulations is such that it challenges even the most modern computer platforms. MEMS flows are typically subsonic the mean velocity is significantly smaller than the thermal velocity of the particles. As a result, a particle may suffer several collisions within a single time step. Simulation of all these collisions is very timeconsuming, and it does not add any information to the flow since these flows are already in near-equilibrium or equilibrium. The same issue appears when simulating charged flows, in which a charged particle may suffer many collisions within a time step. The issue of modeling collisional processes extends back nearly three decades to the early days of digital computers. One of the early approaches to reduce the computational load was to replace the collision operator of the Boltz.mann equation with simpler forms that maintain most of the fimdamental properties of the original equation. The idea behind the simplification of the collision term of the Boltzmann equation is that most of the details of the two-body interactions are not important in reproducing some of me experimentally measured quantities. In this paper extensions of some of these methods, that were initially proposed for the analytic solution of the Boltzmann equatio~ are investigated as possible alternatives to reduce the computational load associated with modeling high-density flows. The BGK’ equation (named tier its authors Bhatnagar, Gross and Krook) is used as the starting point, and related approaches are developed by adding more sophisticated distributions and increasing its physical accuracy. The Boltzmann equation The velocity distribution function~in the sixdimensional phase space (V,7 ) provides a complete description of a gas at the molecular level. The relationship between the velocity distribution fimction and the variables it depends on is given by the Boltzmann equation: [1 ~(nf)+7.~(nf)+F.~= ~(nf)
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TL;DR: In this paper, the ellipsoidal-statistical Bhatnagar-Gross-Krook (ES-BGK) model is investigated for steady gas-phase transport of heat, tangential momentum, and mass between parallel walls.
Abstract: The ellipsoidal-statistical Bhatnagar–Gross–Krook (ES-BGK) kinetic model is investigated for steady gas-phase transport of heat, tangential momentum, and mass between parallel walls (i.e., Fourier, Couette, and Fickian flows). This investigation extends the original study of Cercignani and Tironi, who first applied the ES-BGK model to heat transport (i.e., Fourier flow) shortly after this model was proposed by Holway. The ES-BGK model is implemented in a molecular-gas-dynamics code so that results from this model can be compared directly to results from the full Boltzmann collision term, as computed by the same code with the direct simulation Monte Carlo (DSMC) algorithm of Bird. A gas of monatomic molecules is considered. These molecules collide in a pairwise fashion according to either the Maxwell or the hard-sphere interaction and reflect from the walls according to the Cercignani–Lampis–Lord model with unity accommodation coefficients. Simulations are performed at pressures from near-free-molecular to near-continuum. Unlike the BGK model, the ES-BGK model produces heat-flux and shear-stress values that both agree closely with the DSMC values at all pressures. However, for both interactions, the ES-BGK model produces molecular-velocity-distribution functions that are qualitatively similar to those determined for the Maxwell interaction from Chapman–Enskog theory for small wall temperature differences and moment-hierarchy theory for large wall temperature differences. Moreover, the ES-BGK model does not produce accurate values of the mass self-diffusion coefficient for either interaction. Nevertheless, given its reasonable accuracy for heat and tangential-momentum transport, its sound theoretical foundation (it obeys the H-theorem), and its available extension to polyatomic molecules, the ES-BGK model may be a useful method for simulating certain classes of single-species noncontinuum gas flows, as Cercignani suggested.
57 citations
TL;DR: In this paper, a collision-limiter method, designated as equilibrium direct simulation Monte Carlo (eDSMC), is proposed to extend the DSMC technique to high pressure flows, where equilibrium is enforced in the entire flow by providing a sufficient number of collisions, based on pre-simulation testing.
Abstract: A collision-limiter method, designated as equilibrium direct simulation Monte Carlo (eDSMC), is proposed to extend the DSMC technique to high pressure flows. The method is similar to collision-limiter schemes considered in the past with the important distinction that for inviscid flows, equilibrium is enforced in the entire flow by providing a sufficient number of collisions, based on pre-simulation testing. To test the method with standard DSMC and Navier-Stokes (NS) methods, axi-symmetric nozzle and embedded-channel flows are simulated and compared with experimental temperature data and pre-existing calculations, respectively. The method is shown to agree with third-order Eulerian nozzle flows and first-order channel flows. Chapman-Enskog theory is utilized to predict the range of initial conditions where eDSMC is potentially useful for modeling flows that contain viscous boundary layer regions. Comparison with supersonic nozzle data suggests that the eDSMC method is not adequate for capturing the large variation in flow length scales occurring in supersonic expansions into a vacuum. However, when eDSMC is used in combination with the baseline-DSMC method a near-exact solution is obtained with a considerable computational savings compared to the exact DSMC solution. Viscous flow channel calculations are found to agree well with an exact Navier-Stokes (NS) calculation for a small Knudsen number case as predicted by Chapman-Enskog theory.
44 citations
Cites background from "The Application of the BGK Model in..."
...In recent years, a number of authors have investigated the use of BGK (Bhatnagar, Gross, Krook model) schemes (Gallis and Torczynski 2000, Burt and Boyd 2006) to model dense flows using statistical methods and others have considered hybrid schemes whereby local equilibrium conditions are detected…...
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...The first one (Gallis and Torczynski 2000) uses the background distribution function to define the temperature....
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TL;DR: In this paper, a unified gas-kinetic wave-particle (UGKWP) method for multiscale simulation of continuum and rarefied flow is presented, where the integral solution of kinetic model equation is employed in the construction of UGKWP method to model the flow physics in the cell size and time step scales.
Abstract: In this paper, we present a unified gas-kinetic wave-particle (UGKWP) method on unstructured mesh for multiscale simulation of continuum and rarefied flow. Inheriting from the multicale transport in the unified gas-kinetic scheme (UGKS), the integral solution of kinetic model equation is employed in the construction of UGKWP method to model the flow physics in the cell size and time step scales. A novel wave-particle adaptive formulation is introduced in the UGKWP method to describe the flow dynamics in each control volume. The local gas evolution is constructed through the dynamical interaction of the deterministic hydrodynamic wave and the stochastic kinetic particle. Within the resolution of cell size and time step, the decomposition, interaction, and evolution of the hydrodynamic wave and the kinetic particle depend on the ratio of the time step to the local particle collision time. In the rarefied flow regime, the flow physics is mainly recovered by the discrete particles and the UGKWP method performs as a stochastic particle method. In the continuum flow regime, the flow behavior is solely followed by macroscopic variable evolution and the UGKWP method becomes a gas-kinetic hydrodynamic flow solver for the viscous and heat-conducting Navier--Stokes solutions. In different flow regimes, many numerical test cases are computed to validate the UGKWP method on unstructured mesh. The UGKWP method can get the same UGKS solutions in all Knudsen regimes without the requirement of the time step and mesh size being less than than the particle collision time and mean free path. With an automatic wave-particle decomposition, the UGKWP method becomes very efficient. For example, at Mach number 30 and Knudsen number 0.1, in comparison with UGKS several-order-of-magnitude reductions in computational cost and memory requirement have been achieved by UGKWP.
44 citations
09 Jan 2006
TL;DR: In this article, 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.
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
44 citations
12 Jan 2006
TL;DR: In this paper, 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.
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 method 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.
42 citations
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