# Full Three-Dimensional Inverse Design Method for S-Ducts Using a New Dimensionless Flow Parameter
## 3 Upgrading and Validating the 3-D BSA for Unique Solution
### Validation of 3D Geometry with Desired Centerline Curvature  
The S-shaped duct with a predefined width and elliptic cross-sectional profiles was chosen as the target geometry for the validation of the upgraded BSA. To do this, a numerical simulation inside the target geometry was accomplished to obtain the walls pressure distributions. This pressure distribution was first nondimensionalized and then applied to the upgraded BSA as the target parameter. Figure 12 shows the initial guess and target geometries with their upper and lower wall pressure contours on their symmetry planes. The initial guess geometry is a straight duct with an arbitrary distribution of cross-sectional area in which only its cross-sectional width is equal to that of the target geometry. The 3-D inverse design of the S-shaped duct is started by replacing the 3-D duct with a large number of 2-D planner ducts, as shown in Figure 3. The centerline curvature is the same for all 2-D planar ducts, and the cross-section profile is modified by the change of the parameter \(a\) for each 2-D planar duct. Because the control points are displaced based on the dimensionless pressure difference, the cross-sections should be considered perpendicular to the duct centerline. Since both parameters \(a\) and \(R\) are unknown for using the dimensionless pressure, the inverse design was performed in two steps, alternately. In this way, if one parameter is assumed to be known, another parameter is modified by the inverse design. First, the height and profile of each cross-section are considered to be constant, and only the duct centerline is deformed according to Equation (10) by applying the difference between the dimensionless pressures on the upper and lower lines of the duct symmetry plane. Each ball is placed on the centerline with a fixed and specified \(x\)-coordinate according to Figure 13. The curvature of the duct is modified for the symmetry plane according to Equation (10), and the cross-sections are placed perpendicular to the centerline along the duct while maintaining the cross-sections' profile and height. Figure 11: Evolution of S-duct geometry with specified curvature centerline (**a**) and their pressure distributions along the upper and lower lines of the symmetry plane (**b**), and dimensionless pressure changes for the cross-sections “C” through the inverse design (**c**). Figure 14 shows how the current wall pressure on the symmetry plane approaches the target wall pressure only after 10 geometry corrections, while its centerline is still a little different compared to the target centerline. This occurs because parameter \(a\) is not the same on the symmetry planes of the initial and target geometry, which means that their parameter \(R\) should not be the same for pressure convergence according to Equation (9). In other words, the ratio \(a/R\), instead of \(R\), changes until the difference of the dimensionless pressure on the upper and lower lines of the symmetry plane in the current geometry equals that in the target geometry. In the second step, the curvature of the centerline is considered to be constant, and only the cross-sections perpendicular to the centerline are modified according to Equation (8). The balls are placed along the upper and lower walls of the 2-D planar ducts. The difference between the current and target dimensionless pressure on each upper and lower line is  
Figure 12: Geometries and wall pressure distributions on the symmetry plane of the initial and target geometry. Figure 13: Evolution of duct centerline curvature in the first step of inverse design. applied to each upper and lower lines of the planar ducts separately to correct the height distribution of the longitudinal sections. Figure 15 shows how the dimensionless pressure and the profiles of the cross-sections change through the second step of the inverse design. Figure 16 compares the geometry after 45 corrections and the target geometry, and their wall pressure on the symmetry plane.