Chapter 2 Numerical Methods
2.2 General Description of the standard DSMC
In order to the expected rarefaction caused by the rarefied gas flows, the direct simulation Monte Carlo (DSMC) method which is a particle-based method developed by Bird during the 1960s and it is widely used an efficient technique to simulate rarefied gas regime [2, 5]. In the DSMC method, a large number of particles are generated in the flow field to represent real physical molecules rather than a mathematical foundation and it has been proved that the DSMC method is equivalent to solving the Boltzmann equation [14, 19]. The assumptions of molecular chaos and a dilute gas are required by both the Boltzmann formulation and the DSMC method [2, 5]. An important feature of DSMC is that the molecular motion and the intermolecular collisions are uncoupled over the time intervals that are much smaller than the mean collision time. Both the collision between molecules and the interaction between molecules and solid boundaries are computed on a probabilistic basis and,
hence, this method makes extensive random numbers. In most practical applications, the number of simulated molecules is extremely small compared with the number of real molecules. The general procedures of the DSMC method are described in the next section, and the consequences of the computational approximations can be found in Bird [2, 5].
In DSMC, there are three molecular collision models for real physical behavior and imitate the real particle collision, which are the Hard Sphere (HS), Variable Hard Sphere (VHS) and Variable Soft Sphere (VSS) molecular models, in the standard DSMC method [5].
The collision pairs are chosen by the acceptance-rejection method. The no time counter (NTC) method is an efficient method for molecular collision. This method yield the exact collision rate in both simple gases and gas mixtures, and under either equilibrium or non-equilibrium conditions.
Fig. 2.2 is a general flowchart of the standard DSMC method. Important steps of the DSMC method include setting up the initial conditions, moving all the simulated particles, indexing all the particles, colliding between particles and sampling the molecules within cells to obtain the macroscopic quantities. The details of each step will be described in the following:
z Initialization
The first step to use the DSMC method in simulating flows is to set up the geometry and flow conditions. A physical space is discredited into a network of cells and the domain boundaries have to be assigned according to the flow conditions. An important feature has to be noted is the size of the computational cell should be smaller than the mean free path, and the distance of the molecular movement per time step should be smaller than the cell dimension. After the data of geometry and flow conditions have been read in the code, the numbers of each cell is calculated according to the free-stream number density and the current cell volume. The initial particle velocities are assigned to each particle based on the
Maxwell-Boltzmann distribution according to the free-stream velocities and temperature, and the positions of each particle are randomly allocated within the cells.
z Particle Movement
After initialization process, the molecules begin move one by one, and the molecules move in a straight line over the time step if it did not collide with solid surface. For the standard DSMC code by Bird [2, 5], the particles are moved in a structured mesh. There are two possible conditions of the particle movement. First is the particle movement without interacting with solid wall. The particle location can be easy located according to the velocity and initial locations of the particle. Second is the case that the particle collides with solid boundary. The velocity of the particle is determined by the boundary type. Then, the particle continues its journey from the intersection point on the cell surface with its new absolute velocity until it stops. Although it is easier to implement by using structured mesh, it is difficult for those flows with complex geometry.
z Indexing
The location of the particle after movement with respect to the cell is important information for particle collisions. The relations between particles and cells are reordered according to the order of the number of particles and cells. Before the collision process, the collision partner will be chosen by a random method in the current cell. And the number of the collision partner can be easy determined according to this numbering system.
z Gas-Phase Collisions
The other most important phase of the DSMC method is gas phase collision. The current DSMC method uses the no time counter (NTC) method to determine the correct collision rate in the collision cells. The number of collision pairs within volume (area) of the cell V over a time interval C Δt is calculated by the following equation;
1 ( )max /
2N NFN σTcr Δt Vc (2.2)
Where N and N are fluctuating and average number of simulated particles, respectively.
F is the particle weight, which is the number of real particles that a simulated particle N
represents. σT and c are the cross section and the relative speed, respectively. The r collision for each pair is computed with probability
(σTcr)/(σTcr)max (2.3) The collision is accepted if the above value for the pair is greater than a random fraction.
Each cell is treated independently and the collision partners for interactions are chosen at random, regardless of their positions within the cells. The collision process is described sequentially as follows:
1. The number of collision pairs is calculated according to the NTC method, Eq. (2.2), for each cell.
2. The first particle is chosen randomly from the list of particles within a collision cell.
3. The other collision partner is also chosen at random within the same cell.
4. The collision is accepted if the computed probability, Eq. (2.3), is greater than a random number.
5. If the collision pair is accepted then the post-collision velocities are calculated using the mechanics of elastic collision. If the collision pair is not to collide, continue choosing the next collision pair.
6. If the collision pair is polyatomic gas, the translational and internal energy can be redistributed by the Larsen and Borgnakke model [6], which assumes in equilibrium.
The collision process will be finished until all the collision pairs are handled for all
z Sampling
After the particle movement and collision process finish, the particle has updated positions and velocities. The macroscopic flow properties in each cell are assumed to be constant over the cell volume and are sampled from the microscopic properties of each particle within the cell. The macroscopic properties, including density, velocities and temperatures, are calculated in the following equations [2, 5];
=nm Where n, m are the number density and molecule mass, receptively. c, co, and c’ are the total velocity, mean velocity, and random velocity, respectively. In addition, Ttr, Trot, Tv and Ttot are translational, rotational, vibration and total temperature, respectively. εrot andεvare the rotational and vibration energy, respectively. ζrot and ζv are the number of degree of freedom of rotation and vibration, respectively.
If the simulated particle is monatomic gas, the translational temperature is regarded simply as total temperature. Vibration effect can be neglect if the temperature of the flow is low enough.
The flow will be monitored if steady state is reached. If the flow is under unsteady situation, the sampling of the properties should be reset until the flow reaches steady state.
As a rule of thumb, the sampling of particles starts when the number of molecules in the
calculation domain becomes approximately constant.