## 3 Results  
As shown in Figure 6a,b, an additional plane (denoted as \"Plane+\") was used to capture the velocity, pressure and temperature contours inside the foams. The plane was inclined of 45\({}^{\circ}\) with respect to the horizontal axis in order to comprehensively investigate the hydraulic and thermal performances of the air crossing the periodic aluminum foams. To analyze the effect of the elliptical fiber orientation on the heat transfer and pressure drops with respect to the reference circular strut (subscript \"0\"), the ratios of Colbourn numbers \(\frac{j}{j_{0}}\) and friction factors \(\frac{f}{j_{0}}\) as function of the air mass velocity (\(G\)) were used, respectively. The friction factor \(f\) was calculated by Equation (4), while the Colbourn number \(j\) was obtained by means of the following equation:  
\[j=\frac{HTC}{\rho_{f}c_{p,f}t}\left(\frac{c_{p,f}\mu_{f}}{k_{f}}\right)^{\frac {2}{3}} \tag{6}\]  
Figure 5: Model validation: (**a**) Friction factor and (**b**) Thermal power. Figure 6: Plane+ (**a**) 2D and (**b**) 3D sections. _HTC_ (W m\({}^{-2}\) K\({}^{-1}\)) is the interstitial convective heat transfer coefficient defined as the ratio between the thermal power \(Q\) (see Equation (5)) and the total heat transfer area \(A\) multiplied by the logarithmic mean temperature difference \(\Delta T_{log}\), as shown in Equation (7):  
\[HTC=\frac{Q}{\Delta T_{log}A} \tag{7}\]  
\[\Delta T_{log}=\frac{T_{out}-T_{in}}{\ln\frac{T_{b}-T_{in}}{T_{b}-T_{out}}} \tag{8}\]  
\(G\) (kg m\({}^{-2}\) s\({}^{-1}\)), mass velocity, is defined as the product between the air density and the air velocity:  
\[G=\rho_{f}u \tag{9}\]  
As shown in Figure 7, results for E2-1, E1-2, and E2-2 were reported. To avoid redundancy, graphs for E1-1 foams were not considered due to their strong geometrical similarity and thermal-hydraulic behavior with E2-1 ones (see Table 1). Figure 7: _Cont._  
Focusing on the heat transfer performance, it is interesting to notice that the elliptical E2-1 foams (Figure 7a) outperformed the reference circular ones for all the investigated angles of attack at all the air mass velocities. In fact, the ratio _j/j\({}_{0}\)_ is always greater than 1. However, it must be underlined that while 0-15-30\({}^{\circ}\) show a substantial decrease of the rate of heat transfer enhancement as the velocity increases, the 45\({}^{\circ}\) cases show a growing enhancement rate only up to 3 m s\({}^{-1}\) and then they start decreasing with a slope comparable with other fibers' inclination. Hence, inlet velocity equals to 3 m s\({}^{-1}\) can be considered as the point of maximum. The same trends were obtained for E1-2 (Figure 7c) and E2-2 (Figure 7e). This phenomenon can be explained by investigating the contours of air velocity inside the foams before the 3 m s\({}^{-1}\) (i.e., 2 m s\({}^{-1}\) inlet velocity) (Figure 8a,c,e,g) and this value (i.e., 4 m s\({}^{-1}\) inlet velocity) (Figure 8b,d,f,h). At 2 m s\({}^{-1}\), the highest value of the velocity through the fibers grows as the orientation increases. In fact, it can be noticed that from 0\({}^{\circ}\) (Figure 8a) to 45\({}^{\circ}\) (Figure 8g), the maximum velocity increases from 5.15 m s\({}^{-1}\) to 6.17 m s\({}^{-1}\). Therefore, the increasing tortuosity could justify the enhancement of the heat transfer due to the continuous mixing of the fluid. Nevertheless, considering 4 m s\({}^{-1}\) inlet velocity something different can be inferred. All the foams show longer stagnation zones behind  
Figure 7: Colbourn numbers ratio for (**a**) E2-1, (**c**) E1-2, (**e**) E2-2. Friction factors ratio for (**b**) E2-1, (**d**) E1-2, (**f**) E2-2. the fibers, i.e., dead zone. The stagnation zones can be detrimental for the heat transfer, because extremely low velocities cause the convective heat transfer coefficient to decrease. Moreover, for 4 m s\({}^{-1}\), the local maximum velocity is not completely developed as it was for 2 m s\({}^{-1}\).