1.1 Motivation and Background
1.1.1 Importance of Mesh Refinement
The mesh refinement scheme is very important problem for numerical method. The Direct Simulation Monte Carlo (DSMC) method is a computational tool for the simulation of flows in which effects at the molecular scale become significant [5]. Cells are used for particle collisions and sampling of macroscopic properties in a DSMC simulation, in which the sizes have to be much smaller than the local mean free path. Unfortunately, it is generally impossible to know the distribution of local mean free path before the simulation. In order to obtain better resolution of space and physics, we develop and verify a new mesh refinement module, based on a new concept, in a parallelized direct simulation Monte Carlo code (PDSC).
In the past, we had develop had developed two- and three-dimensional adaptive mesh refinement modules for triangular- and tetrahedral cells, respectively [24, 25], based on the concept of h-refinement similar to those employed in computational fluid dynamics. However, particle tracing on the refined unstructured mesh becomes inefficient and mesh quality is generally difficult to maintain. In this thesis, we will utilize the concept of transient adaptive sub-cells (TAS) proposed by Tseng et al. [18] and propose a new type of mesh refinement on unstructured grids for DSMC simulation. This method is a two-level virtual mesh refinement, in which the background mesh is refined based on an initial DSMC simulation. Finally, several 2D test cases including triangular, quadrilateral and mixed triangular-quadrilateral mesh will be demonstrated in the thesis to show the robustness of this new mesh-refining algorithm.
1.1.2 Classification of Flow Region
Knudsen number (Kn=λ/L) is usually used to indicate the degree of rarefaction. The mean free path λ is the average distance traveled by molecules before collision and L is the flow characteristic length. In general, flow are divided into four regimes and three solutions.
When the local Knudsen number approaches zero, the flow reaches inviscid limit and can be solved by Euler equation. When the flow is close to the continuum regime (Kn approach 0.01), the well known Navier-Stokes equation may be applied to obtain accurate result for engineering purposes. When Kn is the larger than 0.01, assumption of continuum begins to break down and the particle-based method is necessary and a kinetic approach, based on the Boltzmann equation [7]. It is important to note that the kinetic approach is valid in the whole range of the gas rarefaction. However, it is rarely used to numerically solve the practical problems because of two major difficulties. They are included higher dimensionality (up to seven) of the Boltzmann equation and the difficulties of correctly modeling the integral collision term. The well known direct simulation Monte Carlo (DSMC) method [5] is also a powerful computational scheme.
1.1.3 Direct Simulation Monte Carlo Method
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 [5]. Later on, both Nanbu [14] and Wagner [19] were able to demonstrate mathematically that the DSMC method is equivalent to solving the Boltzmann equation as the simulated number of particles become large. The DSMC method is a particle-based method for the simulation of flow of gas. The gas is modeled at the microscopic level using simulated particles, which each represents a large number of physical molecules or atoms. And gas dynamics are modeled through between the motion of particles
and collisions. The 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 had been widely used computational tool for the simulation of flow of gas in the rarefied regime, in which molecular effects become important.
1.2 Literature Survey
Mesh criterion is an important factor to the DSMC simulation [5] because the domain is discreted into mesh for particle movements and collisions. Several mesh refinement algorithms are proposed to conquer the mesh issue, including h-refinement, re-mesh and moving mesh. In the past, we had developed two- and three-dimensional adaptive mesh refinement modules for triangular- and tetrahedral cells, respectively [24, 25], based on the concept of h-refinement similar to those employed in computational fluid dynamics. Several inherent problems arise, which include: 1) Refined cell becomes skewed due to hanging-node removal, which makes particle tracking more difficult; 2) Particles are tracked on refined unstructured grids, which is slow as compared to structured grids; 3) Hanging-node removal algorithm becomes very complicated, especially, in three-dimensional case [24]; 4) Difficult to parallelize due to complicated data structure [24]; and 5) Increasing memory as compared to the original gird. Thus, an alternative algorithm of mesh refinement on unstructured grids, which is free of the above problems, is critical in applying unstructured grids in the parallel DSMC method [25].
At present, Tseng et al. [18] had proposed a sub-cell module that named transient adaptive sub-cell (TAS) module to ensure to obtain the better collision behavior. A new
module with same idea is proposed to virtually refine cells, which is named two-level virtual mesh refinement (VMR) algorithm. This proposed module uses the virtual cells for particles collision and sampling. It is supposed can obtain accurate results without scarifying the memory cost h-refinement method and difficult to particle tracing.
1.3 Specific Objectives of the Thesis
The current objectives of this thesis are summarized as follows:
1. To develop and verify the virtual mesh refinement module for a Parallel DSCM code on unstructured grids.
2. To simulate 2-D hypersonic flow over a block with different size of cells, including Mach number=12 argon of the upstream speed, temperature T∞ = 4 0K and density
ρ∞ = 8.6043E-5 kg/m . 3
3. To simulate 2-D hypersonic flow over a cylinder with different size of cells, including Mach number=10 argon of the upstream speed, temperature T∞ = 2 0 0 K and density
ρ∞ = 2.8327E-5 kg/m . 3
4. To verify and discuss the effects of virtual mesh refinement module in PDSC.
The organization of the thesis is stated as follows: Chapter 1 describes the Introduction, Chapter 2 describes the Numerical Method, Chapter 3 describes the verification of virtual mesh refinement module, and followed by the Results and Discussion. Finally Chapter 4 describes the Conclusions.