# Full Three-Dimensional Inverse Design Method for S-Ducts Using a New Dimensionless Flow Parameter
## 4 Design Example  
In order to show the capability of the proposed 3-D inverse design method, a high-deflected 3-D S-shaped diffuser was first considered as the initial guess for 3-D inverse design. After modifying the 3-D pressure contour on the wall, the 3-D geometry of the duct was improved by applying the modified pressure contour to the upgraded 3-D inverse design code. To evaluate the general parameters of the flow such as pressure recovery, total pressure loss, and outlet flow distortion for the initial and designed ducts, the Reynolds  
Figure 18: Flowchart of up-grated ball-spine algorithm (BSA). Average Navier Stokes (RANS) equations with the k-\(\omega\) SST turbulence model were solved by Ansys Fluent 2020R2 software to compare their overall aerodynamic performances. The short-offset S-duct inherently have complex three-dimensional flows, which is a challenge for CFD to accurately simulate the secondary flow and flow separation. The accuracy of CFD solutions mainly depend on the turbulence model that is employed. Previous studies show that \(k-\omega\) SST is mostly used for S-duct numerical simulation [17; 18; 45]. [PERSON] et al. [46] used \(k-\omega\) SST turbulence model for S-duct numerical simulation and the results agreed well with experimental data. [PERSON] and [PERSON] [47] compared the simulation results of diffusing S-duct using four different turbulence models. They found that \(k-\omega\) SST turbulence model gave best agreement with the experimental data in predicting the pressure distribution along the duct, the separated flow region, and the secondary flow. The computational mesh that was used for the RANS flow solver was a hybrid mesh. It consisted of the boundary layer mesh near the duct surface and the unstructured mesh elements in the interior of the duct Figure 19. Fine grids were used near the wall (with a first layer thickness of \(5\times 10^{-6}\)\(m\)) so that the value of \(y^{+}\) was less than 2 to accurately capture the boundary layer (Figure 20). Several grid sizes were generated to ensure that the simulation results were sufficiently grid-independent. Figure 21 shows how the pressure recovery and DC (90) changed in terms of the number of mesh elements. It indicates that the pressure recovery and outlet flow distortion became stable and remained nearly unchanged for the values greater than 1.13 million mesh elements. Figure 19: Cross section mesh topology. Figure 20: Value of \(y^{+}\) parameters plotted on S-duct surface. In order to generate the initial geometry, the quasi-3-D design of an S-shaped duct was first accomplished for the symmetry plane of the duct using the reported method [34]. In the quasi-3-D design, all cross-sections were considered elliptical with a constant width, and a suitable pressure distribution was applied on the upper and lower lines of the symmetry plane [34]. Figure 22 shows the suitable pressure distribution along the upper and lower lines of the symmetry plane [34]. Figure 23 shows the symmetry plane of the initial S-duct, which was designed by the quasi-3-D inverse design method. The height-to-axial-length ratio (aspect ratio) of the initially designed S-duct was equal to 0.34. Figure 24 shows the 2-D streamlines on the symmetry plane of the initial 3-D duct, indicating no separation inside the duct. The aspect ratio of the initially designed S-duct was increased by 20% to generate the secondary geometry as a high-deflected 3-D S-shaped diffuser. The secondary geometry was actually obtained by scaling the upper and lower symmetry lines of the initial duct and using elliptical cross-sections with constant width. Figure 25 shows the symmetry plane of the secondary duct, which had an aspect ratio of 0.41. The increase in aspect ratio and  
Figure 21: Pressure recovery (**a**) and distortion (**b**) as a function of number of mesh elements. Figure 22: Target pressure distribution along the upper and lower line of the symmetry plane of S-duct (AR = 0.34).