The Standard DS MC Procedures

Figure 2.1 is a general flowchart of the DSM C method. Important steps of the DSM C method include setting up the initial conditions, moving all the simulated particles, indexing (or sorting) all the particles, colliding between particles and sampling

the molecules within cells to determine the macroscopic quantities. The details of each step will be described in the following subsections.

2.3.1 Initialization

The first step to use the DSM C method in simulating flows is to set up the geometry and flow conditions. A physical space is discretized into a network of cells and the domain boundaries have to be assigned according to the flow conditions. A point has to be noted is the cell dimension 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 M axwell-Boltzmann distribution according to the free-stream velocities and temperature, and the positions of each particle are randomly allocated within the cells.

2.3.2 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 DSM C code by Bird [10, 11], 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.

2.3.3 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 identities of the collision partners can be easy determined according to this numbering system.

2.3.4 Gas-Phase Collisions

The other most important phase of the DSM C method is gas phase collision. The current DSM C method uses the no time counter (NTC) method to determine the correct collision rate in the collision cells. The number of collision pairs within a cell of volume V_c over a time interval ∆ is calculated by the following equation; t

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. σ_T and c are the cross section and the relative speed, respectively. The _r collision for each pair is computed with probability

)max

/(

)

(σ_Tc_r σ_Tc_r (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-Borgnakke model [72], which assumes in equilibrium.

The collision process will be finished when all the collision pairs are handled for all cells and then progress to the next step.

2.3.5 S ampling

After the particle movement and collision process finishes, 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 [10, 11];

=nm

n, m are the number density and molecule mass, receptively. c, c_o, and c’ are the total velocity, mean velocity, and random velocity, respectively. In addition, T_tr, T_rot, T_v and T_tot are translational, rotational, vibrational and total temperature, respectively. ε_rot and εv are the rotational and vibrational energy, respectively. ζ_rot and ζ_v are the number of degrees of freedom of rotation and vibration, respectively. If the simulated particle is monatomic gas, the translational temperature is regarded simply as the total temperature.

Vibrational 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.