A general, approximate expression is described that can be used to predict the thermophoretic force on a free-molecular, motionless, spherical particle suspended in a quiescent gas with a temperature gradient. The thermophoretic force is equal to the product of an order-unity coefficient, the gas-phase translational heat flux, the particle cross-sectional area, and the inverse of the mean molecular speed. Numerical simulations are used to test the accuracy of this expression for monatomic gases, polyatomic gases, and mixtures thereof. Both continuum and noncontinuum conditions are examined; in particular, the effects of low pressure, wall proximity, and high heat flux are investigated. The direct simulation Monte Carlo (DSMC) method is used to calculate the local molecular velocity distribution, and the force-Green's-function method is used to calculate the thermophoretic force. The approximate expression is found to predict the calculated thermophoretic force to within 10% for all cases examined.
A Micro Electro Mechanical System (MEMS) typically consists of micron-scale parts that move through a gas at atmospheric or reduced pressure. In this situation, the gas-molecule mean free path is comparable to the geometric features of the microsystem, so the gas flow is noncontinuum. When mean-free-path effects cannot be neglected, the Boltzmann equation must be used to describe the gas flow. Solution of the Boltzmann equation is difficult even for the simplest case because of its sevenfold dimensionality (one temporal dimension, three spatial dimensions, and three velocity dimensions) and because of the integral nature of the collision term. The Direct Simulation Monte Carlo (DSMC) method is the method of choice to simulate high-speed noncontinuum flows. However, since DSMC uses computational molecules to represent the gas, the inherent statistical noise must be minimized by sampling large numbers of molecules. Since typical microsystem velocities are low (< 1 m/s) compared to molecular velocities ({approx}400 m/s), the number of molecular samples required to achieve 1% precision can exceed 1010 per cell. The Discrete Velocity Gas (DVG) method, an approach motivated by radiation transport, provides another way to simulate noncontinuum gas flows. Unlike DSMC, the DVG method restricts molecular velocities to have only certain discrete values. The transport of the number density of a velocity state is governed by a discrete Boltzmann equation that has one temporal dimension and three spatial dimensions and a polynomial collision term. Specification and implementation of DVG models are discussed, and DVG models are applied to Couette flow and to Fourier flow. While the DVG results for these benchmark problems are qualitatively correct, the errors in the shear stress and the heat flux can be order-unity even for DVG models with 88 velocity states. It is concluded that the DVG method, as described herein, is not sufficiently accurate to simulate the low-speed gas flows that occur in microsystems.
An improved model for the gas damping of out-of-plane motion of a microbeam is developed based on the Reynolds equation (RE). A boundary condition for the RE is developed that relates the pressure at the beam perimeter to the beam motion. The two coefficients in this boundary condition are determined from Navier-Stokes (NS) simulations with the slip boundary condition for small slip lengths (relative to the gap height) and from Direct Simulation Monte Carlo (DSMC) molecular gas dynamics simulations for larger slip lengths. This boundary condition significantly improves the accuracy of the RE for cases where the beam width is only slightly greater than the gap height.
An approach is presented to compute the force on a spherical particle in a rarefied flow of a monatomic gas. This approach relies on the development of a Green's function that describes the force on a spherical particle in a delta-function molecular velocity distribution function. The gas-surface interaction model in this development allows incomplete accommodation of energy and tangential momentum. The force from an arbitrary molecular velocity distribution is calculated by computing the moment of the force Green's function in the same way that other macroscopic variables are determined. Since the molecular velocity distribution function is directly determined in the DSMC method, the force Green's function approach can be implemented straightforwardly in DSMC codes. A similar approach yields the heat transfer to a spherical particle in a rarefied gas flow. The force Green's function is demonstrated by application to two problems. First, the drag force on a spherical particle at arbitrary temperature and moving at arbitrary velocity through an equilibrium motionless gas is found analytically and numerically. Second, the thermophoretic force on a motionless particle in a motionless gas with a heat flux is found analytically and numerically. Good agreement is observed in both situations.
Direct Simulation Monte Carlo (DSMC) and Navier-Stokes calculations are performed for a Mach 11 25 deg.-55 deg. spherically blunted biconic. The conditions are such that flow is laminar, with separation occurring at the cone-cone juncture. The simulations account for thermochemical nonequilibrium based on standard Arrhenius chemical rates for nitrogen dissociation and Millikan and White vibrational relaxation. The simulation error for the Navier-Stokes (NS) code is estimated to be 2% for the surface pressure and 10% for the surface heat flux. The grid spacing for the DSMC simulations was adjusted to be less than the local mean-freepath (mfp) and the time step less than the cell transient time of a computational particle. There was overall good agreement between the two simulations; however, the recirculation zone was computed to be larger for the NS simulation. A sensitivity study is performed to examine the effects of experimental uncertainty in the freestream properties on the surface pressure and heat flux distributions. The surface quantities are found to be extremely sensitive to the vibrational excitation state of the gas at the test section, with differences of 25% found in the surface pressure and 25%-35% for the surface heat flux. These calculations are part of a blind validation comparison and thus the experimental data has not yet been released.
Icarus is a 2D Direct Simulation Monte Carlo (DSMC) code which has been optimized for the parallel computing environment. The code is based on the DSMC method of Bird[11.1] and models from free-molecular to continuum flowfields in either cartesian (x, y) or axisymmetric (z, r) coordinates. Computational particles, representing a given number of molecules or atoms, are tracked as they have collisions with other particles or surfaces. Multiple species, internal energy modes (rotation and vibration), chemistry, and ion transport are modeled. A new trace species methodology for collisions and chemistry is used to obtain statistics for small species concentrations. Gas phase chemistry is modeled using steric factors derived from Arrhenius reaction rates or in a manner similar to continuum modeling. Surface chemistry is modeled with surface reaction probabilities; an optional site density, energy dependent, coverage model is included. Electrons are modeled by either a local charge neutrality assumption or as discrete simulational particles. Ion chemistry is modeled with electron impact chemistry rates and charge exchange reactions. Coulomb collision cross-sections are used instead of Variable Hard Sphere values for ion-ion interactions. The electro-static fields can either be: externally input, a Langmuir-Tonks model or from a Green's Function (Boundary Element) based Poison Solver. Icarus has been used for subsonic to hypersonic, chemically reacting, and plasma flows. The Icarus software package includes the grid generation, parallel processor decomposition, post-processing, and restart software. The commercial graphics package, Tecplot, is used for graphics display. All of the software packages are written in standard Fortran.
A new scheme to simulate elastic collisions in particle simulation codes is presented. The new scheme aims at simulating the collisions in the highly collisional regime, in which particle simulation techniques typically become computationally expensive. The new scheme is based on the concept of a grid-based collision field. According to this scheme, the particles perform a single collision with the background grid during a time step. The properties of the background field are calculated from the moments of the distribution function accumulated on the grid. The collision operator is based on the Langevin equation. Based on comparisons with other methods, it is found that the Langevin method overestimates the collision frequency for dilute gases.
This paper presents an investigation of a technique for using two-dimensional bodies composed of simple polygons with a body-decoupled uniform Cartesian grid in the Direct Simulation Monte Carlo method (DSMC). The method employs an automated grid preprocessing scheme beginning from a CAD geometry definition file, and is based on polygon triangulation using a trapezoid algorithm. A particle-body intersection time comparison is presented between the Icarus DSMC code using a body-fitted structured grid, and using a structured body-decoupled Cartesian grid with both linear and logarithmic search techniques. A comparison of neutral flow over a cylinder is presented using the structured body fitted grid, and the Cartesian body de-coupled grid.