# 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
### Definition of Curvature-Based Dimensionless Pressure Parameter for Urgrading the 3-D BSA  
For an inviscid flow along a curved streamline with the curvature of (1/R),  
\[\frac{\partial p}{\partial n}=\rho\frac{U^{2}}{R} \tag{3}\]  
where \(\partial p/\partial n\) is the pressure gradient perpendicular to the streamline in the \(s\)-\(n\) coordinate system. This is why the cross-sections are considered perpendicular to the diffuser centerline according to Figure 7.  
Figure 6: Results of inverse design with different cross-sectional profiles for the initial guess and target geometry after 200 geometry corrections. Figure 7: (**a**) S-shaped diffuser with longitudinal section and cross-sections, (**b**) longitudinal section ‘A’, (**c**) cross section ‘B’. For each longitudinal section of the S-shaped diffuser, such as section \"A\" Figure 7b, the pressure difference between the upper and lower walls is due to the curvature and mean velocity at each cross-section Figure 7b according to Equation (3). By discretizing Equation (3) along the centerline of each longitudinal section, the following equation is obtained:  
\[\frac{p_{up}-p_{down}}{2a}=\rho\frac{U^{2}}{R} \tag{4}\]  
where \(p_{up}\) and \(p_{down}\) are respectively the upper and lower pressures of the corresponding points, which are obtained by intersection of the longitudinal section \"A\" and cross-section \"B\". In Equation (4), \(a\) is half of the distance between these two corresponding points Figure 7c, and \(R\) is the curvature of the centerline at each longitudinal section. The average pressure for the two corresponding points is defined as:  
\[P_{ave}=\frac{P_{up}+P_{down}}{2} \tag{5}\]  
The dynamic pressure can be written as:  
\[\rho U^{2}=\gamma PM^{2}=\gamma P\frac{2}{\gamma-1}\left[\left(\frac{P_{0}}{P }\right)^{\frac{1}{35}}-1\right] \tag{6}\]  
Substituting Equations (5) and (6) into Equation (4) yields:  
\[\frac{P_{up}-P_{ave}}{\gamma P_{ave}\frac{2}{\gamma-1}\left[\left(\frac{P_{0}} {P_{ave}}\right)^{\frac{1}{35}}-1\right]}=\frac{a}{R} \tag{7}\]  
The left side of Equation (7) is defined as dimensionless pressure, which is a function of the two parameters \(a\) and \(R\). The dimensionless pressure eliminates the effect of the cross-sectional area. Figure 8 shows the target geometry and the two different initial guesses. The centerline curvature and the width of cross-sections are the same for all three geometries. As can be seen, the second and target geometries have the same cross-sectional areas while the first and target geometries have the same symmetry planes. Figure 8: Variations of cross-sectional area along the duct for the target and two initial geometries. Figures 9 and 10, respectively, compare the wall pressure and the dimensionless pressure distributions on the symmetry planes of these three geometries. The wall pressure distributions on the symmetry planes of the first and target geometry are different in Figure 9 due to the difference in their cross-sectional areas. However, their dimensionless pressures are the same in Figure 10 because both the centerline curvature and the parameter \(a\) on the symmetry planes of the first and target geometry are the same. Therefore, if the parameters \(a\) and \(R\) (centerline curvature) are equal for two different ducts, their dimensionless pressure distributions will be the same, as indicated in Equation (7). Figure 10: Dimensionless pressure distribution of the upper and lower lines of the symmetry plane for the target and the two initial geometries. Figure 7.0 Pressure distribution of the upper and lower lines of the symmetry plane for the target and the two initial geometries. Figure 9 also shows that the wall pressure distributions of the second and target geometries on their symmetry plane are almost the same, which is the reason of their equal distributions of cross-sectional area (\(A\)) and centerline curvatures (\(R\)).