Background

1.2.1 Grid System in DSMC

The DSMC method has been first developed to compute the hypersonic flow by Bird [1976] in late 1950s. Since then it has become the de-facto computational technique to deal with rarefied gas dynamics, including gas flows around spacecraft [Boyd and Stark, 1990; Boyd et al., 1992; Boyd et al., 1994], vacuum technology [Lee and Lee, 1996a; Lee and Lee, 1996b] and, recently, the micro-scale gas flows [Piekos and Breuer, 1996; Nance et al., 1998; Wu et al., 1999], etc. This method requires the introduction of computational cells (meshes) similar to those in Computational Fluid Dynamics (CFD), however, the cells are used for selecting collision partners, sampling and averaging the macroscopic flow properties. Many physical problems involve very complicated geometry and, hence, the generation of an appropriate mesh becomes a very demanding and time consuming task. Generally, the mesh used for final computation is obtained through trial and errors. In addition,

the sizes of cell used in the DSMC method have to vary according to the density and gradients of flow properties or have to be refined near the body surface to obtain accurate prediction of pressure and heat transfer, however, these are not known as a priori in general.

Most applications of DSMC applied structured meshes [Bird, 1994], in the physical space. For problems with complicated geometry, multi-block meshing techniques were developed first by Bird, which involved two steps: dividing the flow field into several blocks followed by discretizing each block into quadrilateral (2-D) or cubic (3-D) meshes. Subsequent research has been directed to develop alternative meshing techniques such as the coordinate transformation method by Merkle [1958], the body-fitted coordinate system by Shimada and Abe [1989] and the transfinite interpolation method by Olynick et al. All of them still used structured grids. It is much easier to program the code using structured grids; however, it requires tremendous problem specific modification. To alleviate such restriction and the easiness of applying adaptive mesh, unstructured mesh system is one of the best choices, although might be computationally more expensive.

And, unstructured mesh has two major advantages over structured mesh for solving computational problems. First, unstructured mesh able efficient mesh generation around highly complex geometries. Second, appropriate unstructured-grid

data structures facilitate the rapid insertion and deletion of points to allow the mesh to locally adapt to the solution. Boyd and his coworkers [1990; 1992; 1994] has applied such technique to compute the thruster plume produced by spacecraft and found that the results are very satisfactorily. In addition, several studies [Piekos and Breuer, 1996; Wu et al., 1999] have used such technique to compute micro-scale flows such as micro-channel, micro-nozzle and micro-maniflod.

1.2.2 Mesh Refinement in DSMC

For the past decade, the development of CFD using adaptive unstructured meshes has greatly extended the capability of predicting complex flow fields. Several adaptive mesh techniques have been developed to increase the resolution of

“important” region and decrease the resolution of “unimportant” region within flow field [Powell and Roe, 1992; Kallinderis and Vijayan, 1993]. However, the corresponding development in the particle method, such as the DSMC method, is not as popular as expected.

However, it is very often that property variation in rarefied gas dynamics is large, for example, in hypersonic flows or vacuum pumping flows, which requires non-uniform mesh for resolving the flow field. Or equivalently, the cell size in DSMC has to be much smaller than the local mean free path for justifying the decoupling of particle movement and collision inherited in the DSMC method.

Adaptive mesh technique in the DSMC method not only improves the flow field resolution without increasing the computational cost, but also equalizes the statistical uncertainties in the averaging process to obtain macroscopic quantities in each cell.

Among the very few studies on this subject, Wong and Harvey [1994] has first applied solution-based, re-meshing adaptive grid technique in unstructured meshes to study the hypersonic flow field with highly non-uniform density involving shock.

Cybyb et al. [1995] have developed technique using the Monotonic Lagrangian Grid (MLG, hereafter) in the DSMC method, which provides a time-varying grid system that automatically adapts to local number densities within the flow field. However, the application of this MLG technique to external gas flows is not easy due to the problem of particle sorting in the designed scheme. Additionally, this technique highly restricts the size of time step as compared with the traditional DSMC method, which makes the steady-state solution prohibitively high. In addition to the applications in hypersonic flows, several gas flows, for example, in diffusion pumps and micro-nozzle, etc., all involve highly varying density in the flow field.

In our laboratory, C.-H. Kuo [2000] had developed a 2-D and unstructured adaptive mesh successfully, and used it in DSMC code to test a high-speed driven cavity flow and a hypersonic flow over a block. He had proven the 2-D unstuctured adaptive mesh generator could increase the solution accuracy and efficiency. Then,

Fu-Yuan Wu [Wu, 2002] had developed the 3-D unstructured mesh adaptation method (h-refinement) successfully and used in DSMC code to test on several benchmark problems. He also had proof the 3-D unstructured adaptive mesh generator would be efficient and robust.