# Overset Grid Assembler and Flow Solver with Adaptive Spatial Load Balancing
## 7 CAD Models and Meshes  
The assembler and solver are tested on the generic helicopter configuration [12] consisting of a fuselage, pylon, and four NACA0012-profiled rotor blades. In [12], several collective pitch angles were tested, whereas in this simulation only \(4^{\circ}\) of collective pitch is considered. Together with \(8^{\circ}\) of linear twist, the angle of attack ranges from \(8^{\circ}\) at the tip of a blade to \(12^{\circ}\) at the root cut. A sketch of a blade is shown in Figure 5.  
The shapes of the fuselage and pylon are defined with super-ellipse equations:  
\[ \begin{bmatrix}H(x/l)\\ W(x/l)\\ Z_{0}(x/l)\\ N(x/l)\end{bmatrix} =C_{6}+C_{7}\Bigg{(}C_{1}+C_{2}\Big{|}\frac{x/l+C_{3}}{C_{4 }}\Big{|}^{C_{5}}\Bigg{)}^{1/C_{6}}. \tag{8}\]  
\(x/l\) is the non-dimensional longitudinal coordinate with limits [0, 1.997] for fuselage and [0.400, 1.018] for pylon and \(l\) is the desired length of fuselage which is 78.57 inches. Dimensions given in imperial units in [12] were not converted to metric units to avoid ambiguity. Coefficients (\(C_{1}\) to \(C_{8}\)) are shown in Table 1 in appendices. Cross-sectional coordinates \(y/l\) and \(z/l\) are obtained with  
\[y/l =r\sin\phi\] \[z/l =r\cos\phi+Z_{0}\]  
where \(\phi\) ranges from 0 to \(2\pi\) and \(r\) is defined as  
\[r=\Bigg{[}\frac{\left(\frac{HW}{4}\right)^{N}}{\left|\frac{H}{2}\sin\phi \right|^{N}+\left|\frac{W}{2}\cos\phi\right|^{N}}\Bigg{]}^{1/N}. \tag{9}\]  
Equation (8) and the coefficients provided in [12] are modified since their original counterparts caused problems such as division by zero. Equations (8) and (9) are different  
Figure 5: Sketch of a rotor blade in generic helicopter configuration. aoa is abbreviation for wind angle of attack. from the counterparts by modulus operator around \(\frac{x/l+C_{1}}{C_{4}}\), and \(\frac{H}{2}\sin\phi\) and \(\frac{W}{2}\cos\phi\). The modified coefficients are encircled in Tables 1 and 2. Figure 6 shows CAD models of fuselage, pylon, four rotor blades, and hub modeled with FreeCAD [19]. Leading and trailing faces of fuselage are blunted slightly in order to have satisfactory mesh quality. The hub is modeled as a cylinder with diameter 0.1 m and length 0.05 m, and extensions which connect the hub to the blades have the same profile and angle of attack with the roots of respective blades. The extensions span from the hub such that they overlap 5% of respective blades. CAD models are transferred to the mesh generator, GMSH [20], to generate tetrahedral mesh for each component. Rotor blades have cylindrical meshes with diameter 10 \(\times\) chord and length 20 \(\times\) chord. Fuselage has a spherical mesh with diameter of 20 inches enclosing all blade and hub meshes. The hub also has a spherical mesh with diameter scaling with root cutout length such that \(1.5\times 0.24R\). Figure 7 shows cross-sections of the fuselage, one of the blades, and the hub. Element size \(s\) on fuselage, blade, and hub surfaces are 3, 1, and 2 mm. Volumetric element size is calculated with \(s+d/5\) for all components where \(d\) is the shortest distance from respective wall surface. Table 1 shows the number of cells in each mesh. Figure 8 shows a slice of field cells after donor search. Each color corresponds to a different mesh. In the figure, there are cells that are undesirably larger than the surrounding cells such the one enclosed in a yellow circle. These cells were receptors having candidate donors which themselves were mandatory receptors, or by definition, they were orphans. Since they have to be the donors of mandatory receptors, they are converted to field cells. Figure 9 shows receptors of a slice of fuselage mesh. It is clear that the receptors which only neighbor other receptors do not contribute to flow solution but increase simulation run time by interpolating flow variables across donors. Figure 9 shows the same mesh after exclusion of redundant receptors.