Neutral Density Distribution Determination
3.2 DSMC Calculation of Neutral Distribution for an NSTAR Two-Hole ApertureTwo-Hole Aperture
3.2.1 Single Aperture in an Infinite Grid
Using the dimensions appropriate for the NSTAR hole pattern (see Figure 3.2), an axisymmetric simulation domain was created to model the effusion of gas through one aperture in an infinite plane, and is shown in Figure 3.3. The positions of the screen and accel grids are indicated by the shading.
Simulating the entire domain at once was found to require too many simulated particles to obtain a sample of the downstream density without significant statistical error. To reduce the computational time required to obtain a statistical sample of the density downstream of the hole, the simulation was split into two phases. In the first phase the region extending from approximately eleven accel grid hole radii (6 mm) upstream from the screen grid to the aperture exit (5a in the figure) was simulated. There were a total of 16,600 cells that constituted the entire upstream domain used in the first phase. The second phase simulated only the downstream portion extending from the aperture exit (5b in the figure) to approximately 24 accel grid hole radii (14 mm) downstream. The downstream region was made up of 85,000 separate cells.
1 2
2
3
3 4
4
4
5a
5b Screen
Accel
Flow
Rotation Axis
Figure 3.3: Simulation domain for a single aperture in an infinite plane. Both the upstream (left of 5a) and downstream (right of 5b) domains started with zero molecules at the beginning of their respective simulation phases. The upstream flow conditions had a density of 1018cm−3, a temper- ature of 500 K, and zero mean velocity (molecules diffused in through sides 2). The positions and velocities of molecules intersecting 5a during the first phase were recorded, then the particle was removed from the domain. The recorded molecules from the first phase were used as a molecular input (at 5b) to the downstream domain in the second phase. Molecules intersecting sides 4 were removed from the simulation.
The boundary conditions for the domains used for both simulation phases follow.
Boundary 1: Axis of rotation Boundary 2: Free stream boundary
Boundary 3: Diffuse reflecting surface at a temperature of 500 K Boundary 4: Vacuum (intersecting molecules removed from domain) Boundary 5a: Vacuum (molecular output)
Boundary 5b: Molecular input
The free stream was specified to be a flow of argon gas with a density of 1018cm−3, a temperature of 500 K, and zero mean velocity. Over time, the upstream domain became populated with molecules due to diffusion of atoms through boundary 2. Argon was chosen to be the gas used in the simulation, since all atomic parameters were pre-specified for this species, and it is a heavy noble gas like xenon.
The simulation was started with no atoms present within the domain. Once steady state was achieved during the first phase, approximately 160,000 simulation particles, each representing approximately
2,400 real atoms, were in the upstream domain at any time. The position and velocity of any particle that intercepted the surface at 5a, after steady state was achieved, were recorded; the particle was then removed from the domain. Simulation continued until the states of approximately 1.2 million atoms were recorded.
During the second phase, the record from the first was used as the input atom source file for the surface at 5b. The simulation of the particles downstream was started with no atoms in the domain. For this phase, the number of atoms each simulated particle represented was reduced to nine to increase the number of simulated atoms within the domain. As the simulation proceeded the program randomly selected atoms from the input file and added them to the flow at 5b such that the total flux from this surface was the same as that through 5a during the first phase. Once a steady state was achieved, approximately 160,000 simulated molecules were in the domain at any time. Any molecule intersecting boundary 4 was removed from the simulation. The total simulation took approximately nine days on a modern desktop computer.
During both simulation phases, after a steady state had been obtained, a sample of the atom density was recorded at periodic intervals. Each sample comprised a record of the density of atoms within each cell. Over one million samples were obtained during the second phase. The average density, over these one million samples, for each domain cell was provided as output at the conclusion of the simulation. Averaging over many samples reduced any statistical fluctuations that might be present at any particular time.
Measuring from the center of the accel grid hole (where 5b intersects the rotation axis in Fig- ure 3.3), the density at points along curves of constant radius were interpolated from the data output from the simulation. The density for various radii, as a function of the angleχmeasured from the rotation axis, is shown in Figure 3.4 in non-dimensional units. The non-dimensional density is the angular distribution function,f(χ), obtained by rearranging Equation 3.6:
f(χ) = 4n0 ρ0
R
2
. (3.10)
0◦
30◦ 30◦
60◦ 60◦
90◦ 90◦
0.2 0.2
0.4 0.4
0.6 0.6
0.8 0.8
1 1
Cosine /R= 2 /R= 5 /R= 10 /R= 20
f(χ) = 4nρ0
0
R
2
Figure 3.4: Non-dimensional angular density distribution of a single aperture in an infinite plane.
The density at various points along curves of constant radius from the accel grid hole center were interpolated from the averaged data output at the conclusion of the simulation. The non-dimensional angular density distribution for each curve was computed from Equation 3.10 and plotted as a function of the angle measured from the rotation axis, χ. For clarity, axisymmetry allows for only half of each distribution to be shown. The cosine distribution is shown for comparison.
The cosine function, for an infinitely thin hole, is also shown for comparison.
The beaming effect, as a result of an aperture of finite depth, is immediately evident. Despite averaging over 106time steps, fluctuations are still within 5-10% of the maximum. The fluctuations are seen to be greatest at small angles. This is not unexpected. In an axisymmetric case, the volume represented by each domain cell element is proportional to d2, whered is the distance to the cell from the axis. At small angles, the cells are close to the rotation axis, which means the volume of each is relatively small. As the volume of a cell decreases we can expect the probability of molecules to be found within the volume to decrease as well.
As anticipated, the angular distribution is found to approach a single far-field function as the distance from the hole increases. The resulting far-field distribution, from which the reduction in flow rate compared to an infinitely thin hole (Clausing factor) will be found, will be described in more detail in Section 3.3.
1 2
3
3
4 4
5a
5b 6 6
Screen
Accel
Flow
Rotation Axis
Figure 3.5: Simulation domain for a pseudo-periodic aperture pattern. Both the upstream (left of 5a) and downstream (right of 5b) domains started with zero molecules at the beginning of their respective simulation phases. The upstream flow conditions had a density of 1018cm−3, a temperature of 500 K, and zero mean velocity (molecules diffused in through sides 2). The positions and velocities of molecules intersecting 5a during the first phase were recorded, then the particle was removed from the domain. The recorded molecules from the first phase were used as a molecular input (at 5b) to the downstream domain in the second phase. Molecules intersecting sides 4 were removed from the simulation. The width of the upstream domain (distance between sides 1 and 6) is equal to one half of the NSTAR pitch,p/2 = 1.11 mm.