# Roll Control of Morphing Aircraft with Synthetic Jet Actuators at a High Angle of Attack
## 2 Dynamic Modeling for Morphing Aircraft with SJA
### Modeling for Roll Moment for SJA  
To perform the control effect of the SJA, the NACA 0015 airfoil at an 18\({}^{\circ}\) AoA shown in Figure 2a was chosen, where \(U_{\text{jet}}\) was used to model the blowing/suction type velocity boundary condition at the bottom of the cavity, namely,  
\[U_{\text{jet}}=\overline{U}_{\text{jet}}\sin 2\pi ft, \tag{1}\]  
where \(\overline{U}_{\text{jet}}\) is the blowing magnitude and f is the forcing frequency. A synthetic jet is created by driving the piezoelectric diaphragm at the bottom of the cavity in a periodic manner, as presented in Figure 2b. During oscillation of the membrane, the jet is synthesized and a shear layer is formed by the interaction of pairs of opposite vortex rings at the orifice and the surrounding fluid. The synthetic jet with zero net mass flux, which only requires electrical power, can generate a nonzero momentum [25; 26]. Therefore, it can be used for active flow control at a high AoA [27]. Here, an optimized cavity shape with a better performance proposed in our previous work is employed, as shown in Figure 2b [28]. The unsteady aerodynamic force for SJA is calculated by the commercially available code CFD++, which employs the Reynolds-Averaged Navier-Stokes (RANS) equation and Shear Stress Transport (SST) model. The C-grid about the airfoil and the structured mesh of the cavity are generated as shown in Figure 3. To weaken the influence of numerical reflection from the far-field boundary and get a better solution, the computational domain is set as 40 \(c\)\(\times\) 40 \(c\). Furthermore, the meshes near the wall, trailing edge, and cavity are clustered. The widths of the orifice and bottom are 0.52% \(c\) and 1.56% \(c\), respectively, and the depth of cavity is 1% \(c\). Let the free stream velocity \(U_{\infty}\) be 34 m/s and the Reynolds number Re based on airfoil be 1.9 \(\times\) 10\({}^{6}\). The total number of the grid is about 23,000, the height of the first layer of the grid is 10\({}^{-5}\)\(c\), and the value of \(y+\) is 0.8 in this study. Figure 3: The mesh around NACA 0015 airfoil. (**a**) C-grid detail and (**b**) zoomed-in view near the cavity. Figure 2: Side view of the morphing wing. (**a**) Synthetic jet-based airfoil and (**b**) synthetic jet actuator. In general, the aerodynamic response of synthetic jet-based control is very sensitive to the forcing frequency \(f\), so the aerodynamic lift can be improved by setting the forcing frequency so that it is equal to the vortex-shedding frequency of the airfoil \(f_{\text{s}}\), i.e., \(f=f_{\text{s}}=24.44\) Hz in what follows [28]. The effectiveness of synthetic jet-based control is presented in Figure 4, as one can see that the lift coefficient can be increased greatly due to the delay of the separated flow. Here, the free stream velocity and blowing magnitude of the synthetic jet are 34 m/s. As active flow control, the blowing magnitude \(\overline{U}_{\text{jet}}\) in Equation (1) can be treated as an input for controlling the aerodynamic force. The increment of the lift coefficient for different blowing magnitudes is shown in Figure 5. It indicates that a high blowing magnitude can result in a better aerodynamic performance in the situation of a high AoA. Due to the effect of roll damping, the AoA of the left (right) wing will decrease (increase) as the aircraft rolls along the positive direction of the fuselage-axis, which will generate a change of roll moment, as follows:  
\[