# Full Three-Dimensional Inverse Design Method for S-Ducts Using a New Dimensionless Flow Parameter
###### Abstract
They considered the duct wall as a flexible string that is deformed under the difference between the target and computed pressure distribution. The FSA was developed for inviscid compressible [24] and viscous incompressible internal flow regimes [25]. [PERSON] et al. [26; 27] developed the ball-spine algorithm (BSA) for the quasi-3D design of a centrifugal compressor's impeller. The wall is replaced by virtual balls that freely move along the specified directions called spines. The difference between the target and current pressure distribution at each modification step is applied to each ball as a force that frequently deforms the wall. The BSA was developed for the inverse design of the 90-degree bent between the radial and axial diffuser of a centrifugal compressor [28]. The BSA was also developed for the inverse design of airfoils in subsonic and transonic external flow regimes [29]. [PERSON] et al. [30] incorporated the BSA into the FLUENT software using User Defined Function (UDF) for the inverse design of an axisymmetric duct. [PERSON] et al. [31] applied the BSA for the aerodynamic design of the nozzle of a wind tunnel to achieve uniform airflow with minimum turbulence intensity and flow angle. [PERSON] et al. [32] developed the BSA for swirling viscous flow regime to improve the performance of an axisymmetric 90-degree bend duct between the radial and axial diffuser of a centrifugal compressor. [PERSON] et al. [33] presented a novel iterative physical-based method for solving inverse heat conduction problems. The method extended the BSA concept to inverse heat conduction problems by employing a suitable physical-sense rule. [PERSON] et al. [34] implemented the BSA for the quasi-three-dimensional design of an S-shaped diffuser. They developed this method for the design of two-dimensional blades with blunt [35] and sharp [36; 37] leading edges, and for the quasi-three-dimensional design of rotor and stator blades in axial compressors [38]. They designed a rotor blade with a higher loading coefficient, which increased the outlet pressure of the third stage by 10%. [PERSON] et al. [39] presented a novel shape design method for the numerical solution of inverse heat convection problems (IHCPs) of nanofluids. [PERSON] et al. [40] presented another physical-based algorithm for external flows called Elastic Surface Algorithm (ESA). In their method, the airfoil surface was considered as an elastic beam that was deformed due to the stress provided by the difference between the current and target pressure distributions. The internal stresses in the beam are then set to zero in each modified shape and the process is repeated until the calculated shape satisfies the target pressure distribution along the airfoil surface. [PERSON] et al. [41] upgraded the ESA for the sharp-edged blades in the cascade of axial compressors in a non-viscous subsonic and transonic flow regime, which is capable of reaching sever pressure gradients in the sharp edges. The proposed method was the extension of the BSA inverse method for full 3-D curved ducts. In the present work, the BSA was three-dimensionally developed for the design of 3-D curved ducts with different cross-sectional shapes. For the flow inside a three-dimensional duct, the wall pressure contour is affected by the cross-sectionalarea in addition to the curvature. The original BSA inverse design method diverges if the wall nodes are corrected based on the difference of the target and current pressures. Therefore, the current work defines a new dimensionless pressure distribution using the flow curvature equation to eliminate the area effect on the wall pressure for the convergence of the inverse design. By eliminating the area effect in the new dimensionless pressure, the inverse design of a 3-D curved duct was accomplished by the correction of many 2-D longitudinal planar ducts. To show the capability of the proposed method, a high-deflected 3-D S-shaped diffuser with separation and secondary flow was considered as an initial guess for 3-D inverse design.