# Full Three-Dimensional Inverse Design Method for S-Ducts Using a New Dimensionless Flow Parameter
## 4 Design Example
curvature of the duct leads to an increase in wall pressure gradients along the upper and lower lines of all 2-D planar ducts, which increases the possibility of flow separation and outlet flow distortion. The increase of aspect ratio also increases the difference between the pressure along the upper and lower lines through the first and second bends, which induces a secondary flow and increases the flow distortion at the duct outlet. Figure 23: Symmetry plane of the S-duct designed by quasi-3-D inverse design. Figure 24: Streamlines through the symmetry plane of the initial 3-D S-duct. Figure 25: Symmetry plane of the initial (AR = 0.34) and secondary (AR = 0.41) S-ducts. Figure 26 shows the pressure distributions along the upper and lower lines of the symmetry plane. The viscous flow solver numerically calculated the flow field inside the secondary 3-D S-duct. Figure 27 shows the 3-D streamlines inside the secondary duct, which indicates that the increase of aspect ratio induces secondary flow and separation inside the duct and increases the flow distortion at the outlet. According to Equation (4), an increase in the duct curvature increases the dimensionless pressure difference between the upper and lower lines of the duct, in addition to the increase of the corresponding pressure difference. To eliminate the separation and reduce the secondary flow in the secondary duct with the aspect ratio of 0.41, the dimensionless pressure distributions on all wall lines in the high-positive-pressure-gradient regions were modified and then applied to the proposed 3-D design method to correct the cross-sectional profiles. Figure 27: 3D streamlines inside the secondary S-duct (AR = 0.41). Figure 19.0 Pressure distribution along the upper and lower lines of the symmetry plane. The width of each cross-section was considered unchanged during the geometry corrections as a design constraint. Equation (8) was used to modify the cross-sections because the curvature of the centerline was known and considered equal to that of the secondary duct. First, the dimensionless pressure distributions on the upper and lower symmetry lines were modified to match the corresponding dimensionless pressure distribution of the initial geometry in the region with a high positive pressure gradient. Figure 28 compares the modified dimensionless pressure distributions on the upper and lower symmetry lines with those of the initial and secondary ducts. The increase of the aspect ratio in the secondary duct increases the dimensionless pressure difference between the upper and lower symmetry lines. Figure 28 also shows that the modified dimensionless pressure distributions on the upper and lower symmetry lines in the high-positive-pressure-gradient regions match with the corresponding dimensionless pressure distributions of the initial geometry that does not have separation. In other words, the pressure loading on the symmetry plane decreases in the regions with a high positive pressure gradient. Because the curvature of the centerline was considered equal to that of the secondary duct, the decrease of the modified pressure loading on the symmetry plane should be compensated by increasing the pressure loading on the other 2-D planar planes in the high-positive-pressure-gradient regions. To do this, a correlation was needed to obtain a constraint for the dimensionless pressure distribution of each cross-section. If the dimensionless pressure is plotted on a plane normal to the flow of an S-shaped duct, according to Equation (12), the sum of the dimensionless pressures is zero. In other words, on the plane normal to the flow, the integral of the dimensionless pressure graph becomes zero. Figure 29 shows the dimensionless pressure of the initial geometry in section \"A\". Figure 19.1  
Figure 19.2 Modified dimensionless pressure distribution along the upper and lower lines of the symmetry plane compared to the initial and secondary geometry.