# Full Three-Dimensional Inverse Design Method for S-Ducts Using a New Dimensionless Flow Parameter
## 4 Design Example
For each cross-section normal to the centerline, the modified dimensionless pressure of the upper and lower points (Phi = 0, 180) was obtained from Figure 19.3 The modified dimensionless pressures on the other 2-D planner ducts were obtained according to Equation (12). Figure 30 compares the dimensionless wall pressure distributions of the initial and secondary ducts with the modified dimensionless wall pressure distributions on sections A, B, C and D as examples. The areas below all the dimensionless pressure distributions are zero, according to Equation (12). Figure 30 also shows that although the maximum pressure loadings for the modified distributions have decreased compared to those of the upper points in the secondary duct, the pressure loadings of the other points adjacent to the maximum values have increased to compensate the decrease of pressure loading on the symmetry plane. According to Figure 30, sections \"A\" and \"B\" are located at the first bend of the S-shaped duct. In these two sections, the undesirable pressure gradient on the lower wall (\(\varphi=180\)) causes secondary flow. Therefore, the pressure on the bottom wall decreases, and the pressure loading is transferred to the lateral points to decrease the pressure difference of the top and bottom points on the sections \"A\" and \"B\". In sections \"C\" and \"D\", which are located in the second bend of the duct, the situation is opposite. In other words, there is an undesirable pressure gradient in the upper wall of the duct (\(\varphi=0\)) in the second bend. By reducing the pressure in the upper wall, the secondary flow is controlled. The modified dimensionless pressure distributions were applied to the proposed 3-D inverse design method to correct the secondary duct. The design process converged to the favorable geometry only after 50 geometry corrections using the three-dimensionally modified dimensionless pressure contours (Figure 31). Figure 32 compares the pressure contours on the symmetry plane of the initial, secondary, and modified geometries. It is clear that the difference between the pressure along the upper and lower symmetry lines has decreased through the bends of the modified geometry. Figure 33 illustrates the 2-D streamlines inside the symmetry plane and the 3-D streamlines inside the entire duct, which were computed by the RANS flow solver. It is clear that the separation and secondary flow have been eliminated from the three-dimensionally-modified duct. Figure 34 compares the Mach number contours at the outlet of the secondary and three-dimensionally-modified S-ducts with the same aspect ratio of 0.41. The uniformity of the Mach number at the outlet an S-duct can strongly affect the performance of downstream components such as an engine. Figure 34 clearly shows that the outlet Mach number of the three-dimensionally-modified S-duct is more uniform than that of the secondary duct,  
Figure 19.4 Dimensionless pressure distribution of initial geometry in the cross-section normal to the flow. which is the result of modifying the dimensionless pressure contours in high-pressure regions and eliminating the separation and secondary flow inside the duct. Figure 19.5 Modification of the dimensionless pressure distribution along four cross-sections. The uniformity of the total pressure distribution at the outlet of an S-duct is important for the performance of the downstream component. An expression for a distortion coefficient to quantify the flow irregularities or total pressure distortion at the duct outlet is DC (90), which is defined as follows:  
Figure 19.6  
where \(P_{02,ave}\) is the average total pressure at the duct outlet and \(P_{02,ave,90}\) is the average total pressure over a 90 degree sector of the engine face. For the three-dimensionally-modified S-duct, DC (90) is equal to 0.085. In Figure 35, the total pressure contours on different cross-sections are compared for the secondary and three-dimensionally-modified S-ducts.