# «HIGH FIDELITY OPTIMIZATION OF FLAPPING AIRFOILS AND WINGS A DISSERTATION SUBMITTED TO THE DEPARTMENT OF AERONAUTICS & ASTRONAUTICS AND THE COMMITTEE ...»

## CHAPTER 3. PITCHING & PLUNGING AIRFOIL OPTIMIZATIONS 51

** Figure 3.3: CL versus CD polars for optimal 2D cases.**

The vertical axis represent lift and the horizontal axis represents drag. The left plot shows the polar for the plunging case and the right plot shows the polar for the pitching and plunging case.

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Figure 3.4: History of function evaluations conducted by SNOPT during the optimization of the plunging case. The plot on the left is scaled from zero on the vertical axis, while the plot on the right is scaled to better illustrate convergence behavior.** Figure 3.5: History of function evaluations conducted by SNOPT during the optimization of the pitching and plunging case.**

The plot on the left is scaled from zero on the vertical axis, while the plot on the right is scaled to better illustrate convergence behavior.

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In the pitching and plunging case the leading edge vortex is suppressed and does not detach, and the resulting induced circulation is absorbed by the trailing-edge vortex.

The suppression and absorption of the leading edge vortex likely results in less wasted energy being shed down-stream by the airfoil and is likely a contributing factor to the increased e ciency observed in the pitching/plunging case.

** Figure 3.7 highlights the dynamics of the boundary layer on the downstroke of the ﬂapping cycle, and particularly the formation and detachment of the trailing edge vortex.**

The images sequence begins as the airfoil transitions from pitchingdominated motion at the top of the stroke to plunging-dominated motion during the middle of the downstroke. Regions with a negative x-component of velocity are indicated by the colored contour regions. A negative x-component of velocity suggest ﬂow separation. The plots indicate that there is initially a small region of separation extending roughly from the quarter chord to the trailing edge (image one). As the separated region grows, the trailing edge vortex grows in strength and then detaches (images two and three). After detaching, the addition suction and induced velocity from the trailing edge vortex appear to suppress and then eliminate the separated region.

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** Figure 3.6: Vorticity visualization of the ﬂapping cycle for maximum propulsive efciency.**

The top row depicts the plunging case and the bottom row depicts the pitching and plunging case. Note that the vorticity contour colormaps values have been chosen to illustrate ﬂow features are not not necessarily the same between the top and bottom rows.

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Figure 3.7: Isovorticity contours with areas of negative u-velocity demarcated by the red contour line. Negative u-velocity is an indicator of separated ﬂow.## CHAPTER 3. PITCHING & PLUNGING AIRFOIL OPTIMIZATIONS 56

3.2 Thrust-Constrained Power Minimization A vehicle operating in a state-state cruise condition maintains zero net forces such that thrust equals drag and lift equals vehicle weight. It is often desirable to minimize power consumption in this state in order to maximize range and/or endurance. This can be thought of as a constrained optimization problem where the constraints are target values of lift and thrust and the optimization objective is to minimize the total mechanical power.

We consider several sets of cases of constrained power minimization for the 2D pitching and plunging airfoil. In all cases the mechanical power Pin is minimized with a constraint on the average thrust coe cient. The average angle of attack is also prescribed with values of 0, 5 and 10 to investigate symmetric as well as lift-producing cases.

**The following cases of angle of attack ↵0 and thrust coe cient CT are considered:**

↵0 CT 0 0.0, 0.25, 0.5, 0.75, 1.0 5 0.0, 0.5, 1.0 10 0.0, 0.5, 1.0 Note that in the ↵0 = 0 case more CT constants are considered in order to give better resolution to distinguish trends in the resulting analysis. A coarser range of target thrust coe cients is used for the ↵0 = 5 and ↵0 = 10 cases due to the high computational cost of each optimization.

The parameterization of the pitching and plunging motion is identical to that used

**in the maximization of propulsive e ciency:**

The pitching and plunging NACA0012 airfoil at a Reynolds number of 1850 and mach number of 0.2 is again considered. The numerical solutions again use a 1024 ⇥ 128 mesh advanced through ﬁve periods. The full set of optimal parameters and the

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resulting minimal Pin from the optimization process are given below.The values of minimum power are shown plotted in ﬁgure 3.8. The frequency, pitch amplitude, plunge amplitude and phase are shown plotted in ﬁgure 3.9. Trajectory snapshots of the ﬂapping motions are shown as well in ﬁgures 3.10,3.11 and 3.12, again normalized with respect to frequency. Figure 3.8 indicates that the minimum achievable power increases in a roughly linear fashion with target CT, though the exact nature of the trend cannot be determined without more points. The plot also shows that increasing the angle of attack increases the minimum achievable mechanical power. Figure 3.9 indicates that there are essentially no distinguishable trends in the parameter values for the various optimal cases. There appears to be an increasing

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tendency in pitch amplitude, plunge amplitude and phase, but the parameter values show signiﬁcant variation. There are several possible reasons for this. One is that the parameter values truly show no trend or have low sensitivity in the optimization.Another possibility is that the optimization space is multimodal, resulting in many local minimizers for the gradient-based optimization process to ﬁnd. During the optimization process each case was initialized from several di↵erent sets of parameters to determine if the optimization would converge to the same point every time. In several cases the optimization either reached a di↵erent optimal point or failed to converge to a point that satisﬁed the constrains and optimality conditions. This gives further evidence to the possibility that the constrained power minimization space is multimodal. Unfortunately, optimization algorithms that are able to ﬁnd global optimizers in multimodal spaces are at present far to computationally costly to be applied to this problem.

** Figures 3.13, 3.**

14, 3.15 show the CL vs CD polars for each of the minimal power cases. Note that these plots are scaled to ﬁt each individual plot and thus do not share a common axis range. Figure 3.16 shows the polars from each angle of attack overlaid to give a better sense of their relative magnitudes. All cases with ↵0 = 0 have symmetric polars as is expected of cases with no net lift production. The polars for ↵0 = 0 all display a similar shape and aspect ratio, though there are marked di↵erences in the shapes of the polars as well. For example, the CT = 0 and CT = 0 polars are fairly smooth, whereas the CT = 0.25, CT = 0.5 and CT = 0.75 show more complex polar shapes. A potential cause of the di↵erences in polar shapes is interaction of the airfoil with leading and trailing edge separation/vortices.

The polars for the ↵0 = 5 and ↵0 = 10 cases are asymmetric as a result of net lift production resulting from the angle of attack. In general these polars also display a common shape and aspect ratio in a broad sense, with more subtle variations on a case-by-case basis, however the ↵0 = 5, CT = 1.0 case shows a markedly di↵erent polar shape from the rest. This case also has a lower frequency and much higher pitch and plunge magnitudes then the trend from the other ↵0 = 5 would suggest, and the value of Pi n achieved continues the apparent trend from the prior cases. This evidence further suggests that the optimization space is multimodal.

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Figure 3.17 shows vorticity contours at four points in the ﬂapping cycle for each of the ↵0 = 0 cases. The CT = 0.0 shows essentially no shed vortices in the wake, though there are small-scale vortices are produced downstream of the airfoil as a result of shear layer instabilities. This is expected for this case since the commonlyseen wake vortex patterns are a result of thrust or drag production. Wake vortex structures become more prominent as the target thrust is increased and more energy is transferred to the ﬂow. The wake vortex is composed of many small, closelyspaced vortex cores that are shed multiple times during the ﬂapping cycle. This is in contrast to the maximum propulsive e ciency case that show shedding of one or two large vortices per ﬂapping cycle that correspond to leading and trailing edge suction.In other words, the maximum propulsive e ciency cases maintain attached vortices through the middle portion of the up and down stroke that are shed at the top and bottom of the stroke, whereas the constrained minimum power cases continuously shed small vortices in the wake. The thrust producing cases show the formation of a prominent leading edge vortex that appears to stay attached to during the middle portion of the up and down stroke, which is shed at the top and bottom, similar to what is seen in the plunge-only maximum propulsive e ciency case. This indicates that the trailing edge is the source of the continuous shedding of wake vortices. It is also interesting to note that the leading edge vortex is not suppressed in these cases despite having the pitching degree of freedom. As previously mentioned, the formation and detachment of the leading edge vortex may account for the more complex polar shapes for several cases in ﬁgure 3.13

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Figure 3.8: Plots of the input power requirements resulting from thrust-constrained minimization## CHAPTER 3. PITCHING & PLUNGING AIRFOIL OPTIMIZATIONS 61

** Figure 3.9: Plots of the pitching and plunging motion parameters for each of the thrust-constrained optimization cases.**

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Figure 3.10: Comparison of the relative motion of plunging (top) versus pitching and plunging (bottom). The individual frames of the motion are taken at ten equally spaced intervals in a single period of ﬂapping. Note then that the horizontal axes does not represent either the explicit time or space dimensions.** Figure 3.11: Comparison of the relative motion of plunging (top) versus pitching and plunging (bottom).**

The individual frames of the motion are taken at ten equally spaced intervals in a single period of ﬂapping. Note then that the horizontal axes does not represent either the explicit time or space dimensions.

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Figure 3.12: Comparison of the relative motion of plunging (top) versus pitching and plunging (bottom). The individual frames of the motion are taken at ten equally spaced intervals in a single period of ﬂapping. Note then that the horizontal axes does not represent either the explicit time or space dimensions.## CHAPTER 3. PITCHING & PLUNGING AIRFOIL OPTIMIZATIONS 64

** Figure 3.13: CL versus CD polars for each of the ↵0 = 0 cases.**

The vertical axis represent lift and the horizontal axis represents drag. Note that axis limits di↵er for each plot.

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** Figure 3.14: CL versus CD polars for each of the ↵0 = 5 cases.**

The vertical axis represent lift and the horizontal axis represents drag. Note that axis limits di↵er for each plot.

** Figure 3.15: CL versus CD polars for each of the ↵0 = 10 cases.**

The vertical axis represent lift and the horizontal axis represents drag. Note that axis limits di↵er for each plot.

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** Figure 3.16: CL versus CD polars for optimal 2D cases.**

The vertical axis represent lift and the horizontal axis represents drag. The left plot shows the polar for the plunging case and the right plot shows the polar for the pitching and plunging case.

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** Figure 3.17: Vorticity visualization of the ﬂapping cycle for constrained power minimization cases with ↵0 = 0.**