Motivation and Background

1.1.1 Importance of Driven Cavity Flows

The driven cavity flow is one of the fundamental fluid flow problems that was often used as the benchmark test problem in computational fluid dynamics. The rationale behind this should be attributed to its simple geometry but having singular points at the corners, which may cause difficulties in numerical simulations. Although they have been thoroughly studied in the literature, most of them were focused on incompressible or continuum compressible regime [Karniadakis, 2001]. Very few researches have been done in the rarefied and near continuum regimes, where the understanding may become important in micro- and nano-scale gas flows that are often encountered in MEMS and NEMS related devices. Further, it may serve as the benchmarking problem for extending a numerical scheme into flow in these regimes, where standard Navier-Stokes equation fails to describe the flow accurately. Thus, an accurate numerical solution of a driven cavity flow in the rarefied and near-continuum regime is strongly required.

1.1.2 Classification of Rarefaction

Knudsen number (Kn=λ/L) is a standard parameter that is usually used to indicate the degree of rarefaction. Note that the mean free path λ is the average distance traveled by

molecules before collision and L is the flow characteristic length. In general, flows are divided into four regimes as follows traditionally: Kn <0.01 (continuum), 0.01<Kn<0.1 (slip flow), 0.1<Kn<3 (Transitional flow) and Kn>3 (Free molecular flow). Fig. 1.1 (KC Tseng’s thesis) is a sketch adopted from Bird [Bird’s book, 1994] illustrating the various flow regions

and their corresponding solution methods in a dilute gas. In this figure, the local Kn number is defined with L as the scale length of the macroscopic gradient; that is,

dx L d

ρ

= ρ . Firstly,

the lower bar indicates the continuum formulation. When the local Kn number approaches zero, the flow reaches inviscid limit and can be solved by Euler equation. As local Kn increases, molecular nature of the gas becomes dominated. Hence, the Navier-Stoke equation based computational fluid dynamics (CFD) techniques are often used until the Kn approach 0.01. When the Kn larger than 0.01, continuum assumption begins to break down and the particle-based method is necessary. Secondly, the top bar in this figure indicates the validity of the molecular modeling. It is well known that Boltzmann equation is more appropriate for all flow regimes; however it is rarely used to numerically solve the practical problems because of two major difficulties. They include higher dimensionality (up to seven) of the Boltzmann equation and the difficulties of correctly modeling the integral collision term. As the distance between the molecules increases, collisions between the molecules become less

and thus neglected, and the flow can be solved by neglecting the collision term of the collisionless Boltzmann equation.

1.1.3 Direct Simulation Monto Carlo Method

An alternative method, known as Direct Simulation Monte Carlo (DSMC), was proposed by Bird to solve the Boltzmann equation using direct simulation of particle collision kinetics, and the associated monograph was published in 1994 [Bird’s book]. Later on, both Nanbu [1986] and Wagner [1992] were able to demonstrate mathematically that the DSMC method is equivalent to solving the Boltzmann equation as the simulated number of particles becomes large. The DSMC method is a particle method for the simulation of gas flows. The gas is modeled at the microscopic level using simulated particles, which each represents a large number of physical molecules or atoms. The physics of the gas are modeled through the motion of particles and collisions between them. Mass, momentum and energy transports between particles are considered at the particle level. The method is statistical in nature and depends heavily upon pseudo-random number sequences for simulation. Physical events such as collisions are handled probabilistically using largely phenomenological models, which are designed to reproduce real fluid behavior when examined at the macroscopic level. This method has become a widely used computational tool for the simulation of gas flows in the Low Kn flow regime, in which molecular effects become important.

The DSMC method becomes very time-consuming as the flow approaches continuum regime since the sampling cell size has to be much smaller than the local mean free path for the solution to be accurate. Several remedies in speeding up the DSMC computation include:

1) parallel computing [Robinson, 1996-1998]; 2) variable time-step scheme for steady flows [Kannenberg, 2000], and 3) sub-cells within each sampling cell [Bird’s book, 1994]. Details of the “parallel computing” and “variable time-step scheme” can be found in those references cited in the above and are not described here for brevity. Only “sub-cells” concept is described here since it was rarely discussed in detail in the literature. In Bird’s original implementation [see Bird’s book, 1994], number of sub-cells in each sample cell is pre-determined and related sub-cell data are stored in the memory throughout the simulation, which is very costly. This strategy, which enables nearest-neighbor collisions to be enforced, greatly reduces the computational load by increasing the sampling cell size whilst maintaining the same accuracy as compared to that using smaller sampling cell size. Unfortunately, storage of the sub-cell data is memory intensive and inflexible in adjusting the number of sub-cells in each sampling cell during runtime, which may become important in reducing the merit of collision (ratio of mean collision distance to characteristic cell size). Recently, Bird [DS2V code by Bird] has proposed an idea of “transient sub-cells”, which could overcome the above-mentioned shortcomings. However, this idea has been only implemented on structured grids. No report could be found on unstructured grids.