# Full Three-Dimensional Inverse Design Method for S-Ducts Using a New Dimensionless Flow Parameter
###### Abstract  
In this study, a new inverse design method is proposed for the full 3-D inverse design of S-ducts using curvature-based dimensionless pressure distribution as a target function. The wall pressure distribution in a 3-D curved duct is a function of the centerline curvature and the cross-sectional profile and area. A dimensionless pressure parameter was obtained as a function of the duct curvature and height of the cross-sections based on the normal pressure gradient equation. The dimensionless pressure parameter was used to eliminate the effect of the cross-sectional area on the wall pressure distribution. Full 3-D inverse design of an S-shaped duct was carried out by substituting the 3-D duct with a large number of 2-D planar ducts. The ball-spine inverse design method with vertical spins was coupled with the dimensionless pressure parameter as a target function for the design of the planar ducts. The inverse design process was performed in two steps. First, the height of each cross-section was considered constant, and only the duct centerline was allowed to be deformed by applying the difference between the dimensionless pressure on the upper and lower lines of symmetry plane. Then, a constant curvature was considered for each centerline in the equation, and the difference between the current and the target dimensionless pressure was applied to each upper and lower line of the planar sections to correct the heights of the 2-D planar sections, separately. The method was validated by choosing a straight duct as an initial guess, which converges to the target S-shaped duct. The results showed that the method is an efficient physical-based residual-correction method with low computational cost and good convergence rate. The 3-D wall pressure distribution of a high-deflected 3-D S-shaped diffuser was modified to eliminate the separation, secondary flow, and outlet distortion. Finally, the geometry corresponding to the modified pressure was obtained by the proposed 3-D inverse design method, which revealed higher pressure recovery, lower total pressure loss, and lower outlet flow distortion and swirl angle. 3-D inverse design method; 3-D S-shaped duct; streamline curvature; dimensionless pressure parameter; distortion; secondary flow +
Footnote †: journal: Applied Sciences  
and decoupled (iterative) approaches. A coupled approach uses an alternative formulation of the problem, in which the surface coordinates appear explicitly or implicitly as a dependent variable. A fully coupled method reformulates the governing equations in a way that the shape, computational grid, and the fluid state are all updated simultaneously. The main problems of the couple methods are the complexity of the governing equations in viscous flows and complex geometries [1,2]. [PERSON] [3] converted the physical space (x, y) into a computational space (u, w) to regenerate an inverse design from of the Laplace equation in two-dimensional potential flows. [PERSON] [4] developed a prominent three-dimensional version of his method for inverse design in potential flow regimes. [PERSON] [5] developed a similar formulation for the Euler equations for the design of two-dimensional axisymmetric channels. [PERSON] et al. [6] employed 2-D boundary layer calculations combined with inviscid through flow methods to predict the flow without swirl in axisymmetric diffusers. [PERSON] et al. [7] formulated a fully coupled three-dimensional shape design problem for potential flow. [PERSON] et al. [8] applied the developed three-dimensional inverse design method for 3-D ducts. [PERSON] et al. [9] proposed a direct shape design method. They showed that a fully coupled formulation of a shape design problem could be solved in the physical domain using a simple extension of commonly used CFD algorithms. [PERSON] et al. [10] used a direct design approach for duct shape design with Euler equations. [PERSON] [11] extended the application of this method for the design of ducts with nonlinear coupled Euler equations. [PERSON] et al.