## 4 Optimization Process
### 2D Optimization Stage  
In the present paper, a multidimensional optimization strategy is proposed in order to reduce the computational time needed. From the initial geometry in Figure 2, it was observed that there was a subdivision in the geometry itself. A first part presented an annular distribution where the primary and the secondary flows were mixed, whereas there was a second transitional tubular section where the flow abandoned its annular character. The results obtained at the inlet of the mixer were used as inputs in the isolated mixer to be optimized. Figure 7: _Cont._  
Figure 7: Evaluation of the different tested methods in terms of area, pressure and velocity. From previous existing simulations of an isolated version of the compressor, the circumferential and radial distributions of pressure, temperature and velocity components were obtained. For both core flow and bypass channel flow, the velocity averaging was performed as:  
\[v_{x}(r_{i})=\frac{1}{2\pi}\int_{-\pi}^{\pi}|\overrightarrow{V_{x}(r_{i},\theta )}|d\theta \tag{4}\]  
\[v_{r}(r_{i})=\frac{1}{2\pi}\int_{-\pi}^{\pi}\overline{(V_{y}(r_{i},\theta),V_{ z}(r_{i},\theta))}\cdot\overline{((cos(\theta),sin(\theta))}d\theta \tag{5}\]  
\[v_{\theta}(r_{i})=\frac{1}{2\pi}\int_{-\pi}^{\pi}\overline{(V_{y}(r_{i},\theta ),V_{z}(r_{i},\theta))}\cdot\overline{((-sin(\theta),cos(\theta))}d\theta \tag{6}\]  
Moreover, the pressure and temperature averaging were, respectively, computed as:  
\[p(r_{i})=\frac{1}{2\pi}\int_{-\pi}^{\pi}P(r_{i},\theta)d\theta \tag{7}\]  
\[T(r_{i})=\frac{1}{2\pi}\int_{-\pi}^{\pi}T(r_{i},\theta)d\theta \tag{8}\]  
Once the pressure, temperature and velocity average distributions were obtained, they needed to be transformed as the original compressor simulation was performed on a revision of the geometry without the real component discharge areas. Because of this, the boundary conditions for the mixer design were adapted as follows:  
1. The updated average Mach number in the core and bypass stations was calculated using the conservation of mass equation assuming isentropic flow. 2. Assuming a conservation of the total temperature, the updated static temperature was calculated from the CFD case total temperature and the updated Mach number from the first step. 3. The updated static pressures were calculated using the isentropic relations and the CFD case total pressure. 4. The updated density was computed with the equation of state for an ideal gas. 5. The flow axial velocity was calculated using the continuity equation and the updated density. The boundary conditions for the CFD cases were obtained as follows. Firstly, all of them were conformally mapped onto the correct radii range from the original compressor CFD radii for the bypass and core streams. Then, each of them was accordingly scaled: the axial velocity, scaled to preserve the average obtained axial inlet speed; the radial and tangential velocity, scaled to preserve the angular momentum of the flow streams; and the static pressure and temperature, scaled by the ratio of the updated area and average value from the original CFD case of static pressure and temperature, respectively. The obtained profiles were used in the following stages of optimization and CFD calculations. Every optimization process is divided in several substages. In the present paper, we opted for a conventional first approximation and the initial step consisted in determining which variables to study. The initial 2D proposed geometry was built from quadratic segment splines with 3 points each. In consequence, following the mixer shape, it led to a geometry of 11 points. Each point, as it was part of the spline, presented 4 variables: X and Y position, angle and curvature. This meant, when using a full factorial design of experiments, a control points set of 55 which, by using a high, a medium and a lower level, would imply a completely unaffordable set of cases.