# Check-Valve Design in Enhancing Aerodynamic Performance of Flapping Wings
## 3 Results
### Identification of Cruising Condition  
The following section describes the procedure used to determine the cruising condition:  
* Select a curve in Figure 8 or Figure 9 with respect to the specific driving voltage and inclined angle;
* Identify the cruising speed by observing the intercepted point with zero net thrust on the net thrust diagram (marked with \(\bigstar\) or \(\bigstar\));
* Find the corresponding cruising lift from the identified condition on the lift diagram (marked with \(\bigstar\) or \(\bigstar\)). Table 3 shows the cruising speed and cruising lift values with respect to different driving voltages, various inclined angles for both with and without check valve cases that  
Figure 7: (**Left**) Lift wave forms without check-valves subject to inclined angle of 30\({}^{\circ}\) and driving voltage of 3.7 V: (**a1**) 1.5 m/s; (**b1**) 2.0 m/s; (**c1**) 3.0 m/s. (**Right**) Lift wave forms with check-valves subject to inclined angle of 30\({}^{\circ}\) and driving voltage of 3.7 V: (**a2**) 1.5 m/s; (**b2**) 2.0 m/s; (**c2**) 3.0 m/s. are for all cruising conditions derived from the aerodynamic data shown in Figures 8 and 9. In the third and fourth column of Table 3, sometimes neither cruising speed nor lift corresponding to the zero net thrust are visible; or, the deduced cruising lift is too small (!10 gf) to support the body weight of the FWMAV (\"cannot fly\".) We prepared a remark column which mentions they cannot fly, all negative net thrust, improper trend, and all positive net thrust conditions. The reasons for failure in cruising condition selection are clarified as follows:  
Figure 8: The lift and net thrust, subject to the wing without check-valves, for various driving voltages: (**a**) 3.0 V; (**b**) 3.4 V; (**c**) 3.7 V.  
* Improper trend: In the cases of 3.4 V-30\({}^{\circ}\) and V-60\({}^{\circ}\), and 3.7 V-50\({}^{\circ}\) for the check-valve, the net thrust increases with the increase in wind speed. However, this trend is irrational because, in general, drag increases with wind speed. Hence, the cruising condition is not found. * All negative (\(-\)) net thrust: In the case of 3.0 V-70\({}^{\circ}\) without a check-valve, the net thrust data are all negative. Therefore, neither the intercept of zero net thrust nor the cruising condition can be found. A similar issue also occurs in the case of 3.4 V-20\({}^{\circ}\) with the check-valve. * All positive (\(+\)) net thrust: In the case of 3.4 V-70\({}^{\circ}\) with a check-valve, the net thrust data are all positive. Therefore, neither the intercept of zero net thrust nor the cruising condition can be found. Figure 9: The lift and net thrust subjected to the wing with check-valves for various driving voltages: (**a**) 3.0 V; (**b**) 3.4 V; (**c**) 3.7 V.  
Finally, we identified that, for the case of (3.0 V-30\({}^{\circ}\)-1.6 m/s), the check-valve case generates 14 gf lift, which is 27% higher than the 11 gf lift of the membrane wing without a check-valve. In addition, for the case of (3.7 V-30\({}^{\circ}\)-1.5-1.75 m/s), the check-valve case generates 17 gf lift, which is 68% higher than the 10.1 gf lift of the membrane wing without a check-valve. The corresponding 2D quasi steady COMSOL flow simulation is shown in Figure 10 with a freestream flow speed of 1.6 m/s, a flapping frequency of 14 Hz and a stroke angle of 90\({}^{\circ}\) (the total incident velocity is 4.68 m/s). The computational fluid dynamics (CFD) are conducted and the velocity vectors over the airfoil for the inclined angle of 30\({}^{\circ}\) (Figure 10a,b) and inclined angle of 90\({}^{\circ}\) during the upstroke (Figure 10c) are depicted.