PR simulation mesh

A 2D axisymmetric mesh (Figure 2.8) reproduces the interior of the PR device at actual scale in the CFD-ACE+ simulation domain. Dimensions are given in [mm] on the z- and r-axes. The solid regions in the simulation domain include the Al structure (S,grey), Macor insulation (I,dark green), Cu powered electrode (E,brown), and Al2O3 discharge chamber wall (CW, yellow). The fluid regions (aqua) are contiguous, consisting of the plenum (P), discharge chamber (C), and downstream (D, or D1 to D6). Rotating the mesh about the horizontal axis of symmetry renders the cylindrical geometry of PR and a hemispherical downstream region representing the vacuum chamber or space environment.

-18 -33 -30 -21 -15 -11.5 -6.5 -3 0 4 20 50 z 0 20 23 50 r 2.1 3.1 8 10 P C D1 D2 D3 D4 D5 D6 CW E I S Inlet Outlet

Figure 2.8: PR simulation mesh (36,288 cells). Fluid regions (aqua): plenum (P), discharge chamber (C), and downstream (D). Solid regions: Al structure (S, grey), Macor insulation (I, dark green), Cu powered electrode (E, brown), and Al₂O₃ discharge chamber wall (CW,

yellow).

Cis the region of primary interest for resolving fluid, electrostatic, and plasma dynamics, and features a uniform orthogonal square grid consisting of 0.1mm ×0.1mm cells in a structured mesh. The same mesh density is used for the neighbouring CWand D1. In P, the cells smoothly increase in size with increasing distance fromC, up to0.5mm×0.5mm at the top left corner. The size of the cells scales by a hyperbolic tangent function with set initial

and final dimensions. By using larger cells in regions of less importance, the total number of cells in the simulation mesh can be reduced, thus reducing computation time. Orthogonality and zero skew are maintained in theP,C, and D1regions for compatibility with the plasma numerical method employed by CFD-ACE+.

ThePR simulation domain has three notable differences from thePRdevice (Figure 2.2). First, as the external dimensions of S are inconsequential to the flow or plasma behaviour inPR, they have been reduced to decrease the number of cells in the simulation mesh. The reduced thermal mass is not a concern since the structure does not heat up during the∼ms

time scale that the CFD-plasma simulations are modelling. Second, the inlet boundary in thePRsimulation domain is a3mm section at the top right edge of P. When the simulation mesh is rotated about the axis of symmetry, the inlet boundary traces out a cylinder instead of a single hole like in thePRdevice. This difference is also inconsequential as the flow inPis mostly stationary. and the inlet boundary is sufficiently far away fromC. As the flow velocity of the propellant entering through the inlet boundary is axisymmetric and orthogonal to the flow direction in C, it does not carry any axial momentum and therefore does not affect the thrust force.

Finally, D has a hemispherical shape instead of reproducing the exact geometry of the glass expansion tube. At high pressures, the flow behaviour is insensitive to the shape of the downstream region, and the CFD simulation can be directly compared with experiment. At low pressures, the outlet boundary of the downstream region has to be placed sufficiently far away from the exit of the discharge chamber region to mitigate unphysical behaviour. A hemispherical outlet boundary is chosen as it is equidistant from the discharge chamber exit and isotropic, thus eliminating any directional bias and circulation effects that arise due to unequal distances from boundaries, as well as computational anomalies caused by corners. Additionally, it is more versatile as the simulation domain can represent PRbeing mounted directly to a vacuum chamber at a given background pressure p0 [46, 47], or immersed in a space environment [47,53].

Dhas a radius of50mm, and is split into six trapezoidal sub-regions in order to maintain a relatively orthogonal mesh and minimise skew close toD1. D2andD4expand from a linear grid pitch of 0.1mm to 0.15mm over a distance of 16mm, up to 0.5mm at the top right corner of D3. In D5 and D6, the cells gradually increase to a maximum size of 1.25mm×

1.25mm at45° of the hemispherical boundary, but are kept narrow at 1.25mm×0.25mm at the bottom right and top corners as the region near the horizontal and vertical axes are important for resolving axial propellant flow and plasma interactions with the surface of the structure. While this method of manually building the downstream region mesh is rather

labour intensive, following this procedure ensures the accuracy of the simulation results. Creating a mesh by rotating the horizontal edge90° to the vertical position is not ideal since it creates not only a singularity at the axis of rotation but also cells with a high amount of skew. Generating an unstructured mesh is also not ideal as the triangular cells are not orthogonal, and there is less control over the mesh density in the middle of each region.

In total, there are 36,288 cells in thePR simulation mesh shown in Figure 2.8. Overall, the mesh density is significantly higher than what is deemed sufficient by previous mesh independence studies [60,66] performed with an outlet pressure ofp0 = 0.75Torr. Having a higher mesh density is desirable for the CFD-plasma simulations presented in this thesis, as the capabilities of the fluid and plasma numerical techniques are pushed to the limits for modellingp₀ = 0.1Torr and evenp₀ = 0Torr vacuum scenarios.