*4.1. Optimal θ and C<sup>µ</sup> Configuration*

Figure 9 introduces the different values of the temporal averaged *C<sup>l</sup>* , *C<sup>d</sup>* and aerodynamic efficiency *E* = *Cl*/*C<sup>d</sup>* as a function of the different jet injection angles *θ* and momentum coefficients *C<sup>µ</sup>* studied. Observing Figure 9a, it can be stated that regardless of the injection angle studied, high *C<sup>l</sup>* are obtained for momentum coefficients ranging between 0.0005 ≤ *C<sup>µ</sup>* ≤ 0.05. In general the optimum jet inclination angle is *θ* = 40º, although two maximum lift coefficients are obtained at *θ* = 40º *C<sup>µ</sup>* = 0.0005 and *θ* = 20º *C<sup>µ</sup>* = 0.001, respectively. Regarding the results at *θ* = 30º, the graphic presents a much flatter shape than for *θ* = 20º, with an efficiency maximum at about *C<sup>µ</sup>* = 0.005. In fact, regardless of the injection angle studied, the maximum lift coefficients are obtained for momentum coefficients in the range 0.0005 ≤ *C<sup>µ</sup>* ≤ 0.001.

For *C<sup>µ</sup>* values larger than 0.015, regardless of the injection angle, the lift coefficient values significantly decrease. Finally, the *C<sup>l</sup>* curve obtained for *θ* = 40º is the most flat one, indicating the lift obtained at this inclination angle is good and pretty stable for a large range of momentum coefficients. Note that as *θ* increases, *C<sup>l</sup>* tends to be more stable in the range of *C<sup>µ</sup>* between 0.0005 and 0.015, indicating that the effectiveness of *C<sup>µ</sup>* is higher for growing *θ* values. When considering the drag coefficient in Figure 9b, minimum values are obtained for *θ* = 40º, although for *θ* = 30º relatively low drag coefficients are observed. The minimum *C<sup>d</sup>* is obtained at *θ* = 40º and *C<sup>µ</sup>* = 0.003. Comparing Figure 9a with Figure 9b it can be concluded that to maximize lift and minimize drag the optimum jet injection angle is *θ* = 40º, yet, the momentum coefficient to maximize lift is *C<sup>µ</sup>* = 0.0005 while the one minimizing drag needs to be *C<sup>µ</sup>* = 0.003. In order to solve this dilemma the airfoil efficiency must be considered.

**Figure 9.** Aerodynamic coefficients comparison for the different *C<sup>µ</sup>* and *θ* studied. (**a**) Lift coefficient comparison for the different *C<sup>µ</sup>* and *θ* studied. (**b**) Drag coefficient comparison for the different *C<sup>µ</sup>* and *θ* studied. (**c**) Aerodynamic efficiency comparison for the different *C<sup>µ</sup>* and *θ* studied.

Airfoil efficiency *E* is presented in Figure 9c. Maximum aerodynamic efficiencies are obtained for *θ* = 40º followed by *θ* = 30º. The maximum aerodynamic efficiency has been obtained for *θ* = 40º and *C<sup>µ</sup>* = 0.003, with *E* = 12.3291. Comparing this case with the baseline case, it supposes a ∆*C<sup>l</sup>* = 0.51154, ∆*C<sup>d</sup>* = −0.098, and ∆*E* = 7.2043. Consequently, these results have been considered satisfactory in order to proceed with the frequency study.

For completeness, Table A1 is presented in Appendix A, where the results obtained from the different simulations previously introduced in Figure 9 are presented. Note that all values presented in this table correspond to the time averaged values obtained during the last 10 seconds of each simulation.

### *4.2. Forcing Frequency Analysis*

As it has been previously mentioned, maintaining constant the groove position and width as well as the AFC parameters optimized in Section 4.1 (*θ* = 40º and *C<sup>µ</sup>* = 0.003), a set of different *F* + <sup>∗</sup> values (0.5 ≤ *<sup>F</sup>* + <sup>∗</sup> ≤ 7) have been applied in the present subsection. After simulating each *F* + ∗ case for a computational time of 30 s, the resulting time averaged *C<sup>l</sup>* and *C<sup>d</sup>* values, as well as the airfoil efficiency are presented in Table 3. The optimum case corresponds to *F* + <sup>∗</sup> = 4, with a resulting *<sup>C</sup><sup>l</sup>* = 1.8147, *<sup>C</sup><sup>d</sup>* = 0.0942 and *<sup>E</sup>* = 19.2649. It can be concluded that the modification of the pulsating flow frequency has brought an efficiency improvement of around 56% versus the one obtained with the optimum *C<sup>µ</sup>* and *θ*. The increase in lift and decrease in drag corresponds to ∆*C<sup>l</sup>* = 0.0726 and ∆*C<sup>d</sup>* = −0.0471, respectively. When comparing the final AFC optimum properties with the baseline case ones, it is observed an airfoil efficiency increase of 275.8%, being the lift increase and drag decrease respectively of ∆*C<sup>l</sup>* = 0.588 and ∆*C<sup>d</sup>* = −0.1451. From Table 3 there is another condition which is worth to report, this is the maximum lift condition, which happens for *F* + <sup>∗</sup> = 3. It is interesting to note that the AFC parameters to obtain maximum efficiency and maximum lift are almost the same, just the pulsating frequency is slightly different, being four times and three times the natural vortex shedding frequency, respectively.


**Table 3.** Time averaged aerodynamic coefficients and efficiency obtained for the set of *F* + ∗ studied.

The drastic increase in airfoil efficiency is clearly understood when observing the evolution of the pressure coefficient over the airfoil when the optimized AFC parameters are considered. In Figure 10, the pressure coefficient distribution along the chord is presented for the baseline, maximum efficiency and maximum lift cases. For both maximum conditions, a substantial increase of *C<sup>P</sup>* is observed along the airfoil. From *x*/*C* = 0 up to *x*/*C* = 0.6, a huge decrease of the pressure coefficient is seen on the airfoil upper surface, while on the last 40% of the chord the pressure is a bit higher than the one obtained in the baseline case.

To further understand the flow structure around the airfoil, streamlines of the averaged flowfield for the maximum efficiency and maximum lift cases are presented in Figure 11a,b, respectively. A drastic reduction of the vorticular structure generated over the airfoil upper surface with respect to the baseline case solution (see Figure 8) can be observed for both configurations presented. Clearly, for the maximum efficiency condition vortical structures have nearly disappeared over the airfoil. Just a small laminar bubble which appears at about *x*/*C* = 0.7 and disappears before *x*/*C* = 0.8 can be observed, indicating a corresponding boundary layer separation and reattachment at these points. The separation of the boundary layer is delayed to about *x*/*C* = 0.7 for the case of maximum lift, reattaching close to the trailing edge. An elongated vortical structure is therefore generated around the airfoil trailing edge.

**Figure 10.** Pressure coefficient comparison for the baseline case and maximum efficiency and maximum lift configurations.

**Figure 11.** Streamlines of the averaged flowfield for the optimal and maximum lift configurations. (**a**) Streamlines of the averaged flowfield for the optimal configuration. (**b**) Streamlines of the averaged flowfield for the maximum lift configuration.

A good method to understand the flow evolution over the airfoil is via plotting the boundary layer thickness along the airfoil chord. For the baseline, maximum efficiency and maximum lift cases, this is presented in non-dimensional form and every 10% of the chord in Figure 12. The set of profiles for both the maximum efficiency and maximum lift cases show only positive velocity values up to around *x*/*C* = 0.7, entitling no boundary layer separation appears until this streamwise position. For the maximum efficiency case, at around *x*/*C* = 0.7, negative averaged velocities start appearing close to the wall, returning to a completely positive profile at *x*/*C* u 0.8, which matches perfectly well with the small laminar bubble observed in Figure 11a. A slightly different behaviour is observed for the maximum lift case, where Figure 12 shows how the averaged velocities close to the wall become negative for the streamwise positions ranging from *x*/*C* = 0.7 and *x*/*C* = 1. A direct connection between the boundary layer thickness just presented and what it is observed in Figure 11b can be made, the elongated and downstream growing vortex is clearly observed in both figures. When observing the negative velocity distributions associated to the maximum lift case from Figure 12b, and when comparing them with the velocity distributions corresponding to the baseline case, it can be concluded that the intensity associated to the maximum lift case vortex is much smaller than the one corresponding to the baseline case, therefore the maximum lift vortex must rotate with a low angular velocity. The vortex generated for the maximum efficiency configuration at around *x*/*C* = 0.7 has a larger vorticity magnitude than the maximum lift one, but still lower than the vorticity associated to the baseline case vortex. For the baseline case, negative averaged velocity values can be observed from the streamwise position *x*/*C* = 0.1 up to *x*/*C* = 1, indicating a rapid separation of the boundary layer that extends for the whole airfoil. In reality, the negative averaged velocities cannot be clearly seen at streamwise position *x*/*C* = 0.1 in Figure 12a, but the separation point can be easily localized around *x*/*C* = 0.1 in Figure 12a. At *x*/*C* = 1, slightly positive averaged velocities can be observed close to the wall due to the appearance of a counter rotating vortex generated at the trailing edge, as observed in Figure 8.

**Figure 12.** Mean velocity profiles for the baseline, maximum E and maximum L cases, from *x*/*C* = 0.1 to *x*/*C* = 1. (**a**) Velocity profiles of the mean velocity from *x*/*C* = 0.1 to *x*/*C* = 0.5. (**b**) Velocity profiles of the mean velocity from *x*/*C* = 0.6 to *x*/*C* = 1.

### *4.3. Energy Assessment*

In order to find out how effective is the AFC approach employed for the maximum efficiency and maximum lift cases, the power per unit length required by the SJA (*W<sup>j</sup>* ) as well as the power saved after the actuation (*WG*) have to be calculated. The power needed to drive the synthetic jet is given as:

$$\mathcal{W}\_{\dot{\jmath}} = \frac{1}{2} \rho\_{\dot{\jmath}} \mathcal{S}\_{\dot{\jmath}} \sin(\theta) \overline{u\_{\dot{\jmath}}^3} \tag{24}$$

where *S<sup>j</sup>* = *h* ∗ *l* defines the groove cross-sectional area, due to the fact that the airfoil length is equal to unity *l* = 1, the groove and the the jet width *h* are equivalent. The parameter *θ* stands for the jet inclination angle measured versus the wing profile surface.

The definition of the synthetic jet actuator time dependent velocity profile to the power three, *u* 3 *j* , was taken from [6,39].

$$\overline{\mu\_j^3} = \frac{1}{T/2} \int\_0^{T/2} U\_{\text{max}} ^3 \sin^3(2\pi ft) dt = \frac{4}{3\pi} \mathcal{U}\_{\text{max}} ^3 \tag{25}$$

where *Umax* characterizes the jet maximum velocity. The equation representing the power saved when AFC is applied and due to the drag force reduction, takes the form:

$$\mathcal{W}\_{\rm G} = \mathcal{U}\_{\rm co} (D\_{\rm baseline} - D\_{\rm actual}) = \frac{\rho \mathcal{U}\_{\rm os}^{-3} \mathcal{C}}{2} (\mathcal{C}\_{d\_{\rm baseline}} - \mathcal{C}\_{d\_{\rm actual}}) \tag{26}$$

where the drag force and the drag coefficient are respectively given as *D* and *C<sup>d</sup>* . When AFC is applied, the parameter defining the power ratio *P<sup>R</sup>* is represented as:

$$P\_{\rm R} = \frac{W\_{\rm G}}{W\_{\rm j}} \tag{27}$$

Energy saving exist for power ratio values higher than one.

The set up parameters and resulting power ratios are summarised in Table 4. Both maximum efficiency and maximum lift configurations present energy savings, with a power gain two orders of magnitude higher than the power required by the SJA.

This result confirms that the introduction of AFC in an airfoil with separated flow is capable of reducing the drag coefficient and increase the lift coefficient significantly, by keeping the boundary layer attached for a larger portion of the airfoil. Therefore, it can be sated that the implementation of the SJA is energetically efficient for the separated flow case presented.

**Table 4.** Power ratio values characterizing the maximum efficiency and maximum lift configurations. **Cases** *α* ◦ *Umax* **[m/s]** *S<sup>j</sup>* **[m<sup>2</sup> ]** *θ* ◦ *W<sup>j</sup>* **[W]** *W<sup>G</sup>* **[W]** *WG***/***W<sup>j</sup>*


### **5. Conclusions**

This paper presents a parametric analysis on a NACA-8412 airfoil with the aim to optimize three AFC parameters, momentum coefficient, jet inclination angle and pulsating frequency, associated to a SJA. The procedure followed to perform a parametric optimization of any airfoil is established and presented. A maximum airfoil efficiency increase, measured respect to the baseline case, of 276% is obtained for *C<sup>µ</sup>* = 0.003; *F* + <sup>∗</sup> = 4 and *θ* = 40◦ , the groove was located from *x*/*C* = 0.08 to *x*/*C* = 0.09 being its width of 0.01*C*. The maximum airfoil lift was obtained for the same AFC parameters except the pulsating frequency which was of *F* + <sup>∗</sup> = 3. The efficiency increase with respect to the baseline case

was of around 218%. From the comparison of the AFC parameters for maximum lift and maximum efficiency cases, it is proved that the pulsating frequency is capable of highly improving the airfoil efficiency.

**Author Contributions:** Conceptualization, J.M.B. and N.C.; methodology, N.C. and J.M.B.; software, N.C.; validation, N.C.; formal analysis, N.C. and J.M.B.; investigation, N.C. and J.M.B.; data curation, N.C.; writing—original draft preparation, N.C. and J.M.B.; writing—review and editing, J.M.B.; visualization, N.C. and J.M.B.; supervision, J.M.B.; project administration, J.M.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** No funding was associated to this project.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
