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

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The resulting motion is identical to the 1–1–0 case and the sweep amplitude has been driven to 0. This indicates that the addition of sweep produces no improvement in attainable propulsive e ciency. The optimization was initialized from several additional starting points to verify the result and in each case there was no improvement. This result is somewhat unexpected because many birds and insects show some degree of sweep in the flapping motions, leading to a figure-eighttype wing-tip trajectory [29].

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4.3 Wing Thickness and Planform E↵ects This section examines the influence of wing thickness and planform shape on achievable propulsive e ciency. These parameters are essentially static through the flapping cycle in contrast to the kinematic parameters that vary sinusoidally. It is generally thought that thin wings o↵er superior performance over the thicker wings typical of human-scale aircraft, and indeed small birds and insects have thin, membranous wings. This is largely due to the fundamentally increased susceptibility of the laminar boundary layer to stall and separation. Evidence also suggests that planform shape can have significant impact on performance through the mechanism of leading-edge vortex capture. Small birds and insects tend to have elliptic wings with pointed tips, and it is though that these shapes minimize interference from the wing-tip vortex and allow a stable wing-tip vortex to be maintained during the middle portion of the stroke.

In this section the achievable propulsive e ciency of the rectangular wing with a NACA0012 (12% thickness) airfoil used in the previous section is compared to one with a NACA0002 (2% thickness) airfoil these comparisons are made using both the 1–1–0 and 2–1–0 parameterizations. Then, the rectangular NACA0012 section wing is compared with two semi-elliptic planform wings with a tip chords of 50% and 25% of the root chord using the 1–1–0 parameterization. The semi-elliptic wing planform shapes are illustrated in figure 4.9. Note that changing the wing planform shape alters the power requirements independent of the aerodynamic performance by changing the moment of inertia of the wing itself. The planform shapes shown in figure 4.9 will have a lower inertial moment and thus will require less power to flap in application, however these e↵ects are not taken into account here. Only the aerodynamic advantage is considered. Table 4.10 presents the parameters and propulsive e ciency attained from the 2% thickness wing optimizations compared with the 12% thickness wing results.

Table 4.11 presents additional non-dimensional parameters computed from the values in table 4.

10. Figure 4.10 shows the span-wise twist distributions from the 12% and 2% thickness optimizations. Table 4.12 presents the parameters and propulsive e ciency attained from semi-elliptic planforms compared with the rectangular wing

CHAPTER 4. FLAPPING WING OPTIMIZATIONS 88

results. Table 4.13 presents additional non-dimensional parameters computed from the values in table 4.12.

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2 14.0 0.239 28.0 49.5% 2 12.6 0.225 25.2 53.8% Table 4.11: Derived parameters for the 2% thick airfoil cases compared with the 12% thick cases Figure 4.10: Plots of the span-wise twist distribution for the linear and non-linear twist cases. Distributions are shown for both the 12% thick and 2% thick airfoil sections.

CHAPTER 4. FLAPPING WING OPTIMIZATIONS 90

Table 4.12: Summary of planform e↵ects compared against the rectangular planform case.

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The thinner airfoil shows improved propulsive e ciency in both cases, with an 11.6% improvement in the linear twist case and a 8.7% improvement in the nonlinear twist case. In both cases the frequency is decreases while the flap angle and total twist increase. Examining the span-wise twist distribution plots in figure 4.10, the thinner wing has less aggressive twist towards the root, but more aggressive twist towards the tip in the non-linear twist case. The reduced frequency and the relative wing tip speed both decrease for the thinner case as well, and the reduced frequency remains within the range seen in the example from bird and insect flight.

The semi-elliptic plan form shapes lead to less-dramatic improvement in the propulsive e ciency, with a 2.8% improvement in the 0.5 tip-chord case and a 3.6% improvement in the 0.25 tip chord case. In these cases the frequency increases, the dihedral angle decreases and the total twist increases. Looking at the non-dimensional parameters, both the reduced frequency and relative wing tip speed both increase.

The potential mechanisms for the improvement seen from changes to the wing thickness and planform shape are most likely a result of changes to the dynamics of the flow, especially relating to boundary layer separation a the role of the leading and trailing edge vortices. The remainder of this chapter will deal with analysis of the flow physics of the the optimal flapping wing cases.

CHAPTER 4. FLAPPING WING OPTIMIZATIONS 91

4.4 Flow Physics The flow physics and specifically the dynamic behavior of the boundary layer and vortices near the wing and in the wake are of great interest in the study of flapping wings. During the flapping stroke the boundary layer may stay largely attached or may separate for some portion of the cycle. In cases where separation occurs, vortices tend to form at the leading and trailing edge and along the length of the span given su cient transfer of energy from the wing to the flow. The behavior of these vortices are thought to have a significant impact on performance in flapping flight. Simulations and experiments have shown that a stable leading-edge vortex (LEV) may be formed on the trailing surface of the wing during flapping flight and it is thought that the low-pressure region created by the LEV can enhance performance [33, 3]. Furthermore, evidence of stable LEV have been observed in several natural flyers [4, 3]. However, it is not yet clear how vortex interaction alters performance in flapping flight, nor is it clear what flight modes benefit from a stable LEV, nor what the degree of improvement might be. With this in mind, this section examines the dynamics of the boundary layer and vortex structures of the cases that optimize propulsive e ciency.





The vorticity isosurface plots in figures 4.11 and 4.12 provide some insight into the flow physics of the initial cases with varying twist complexity. Recall that these cases use a rectangular planform and a NACA0012 airfoil section. The twist-free case shows evidence of significant leading edge separation and vortex formation during the up and down strokes, especially towards the wing tips. In contrast, the leading edge vortex appears to be absent in the twisting cases, although there is evidence of instability or separation of the boundary layer around mid-chord during the middle portion of the up and down strokes on surface opposite the direction of motion. Furthermore, the severity and extent of this mid-chord separation appears to be less for the 2–1–0 when compared to the 1–1–0 case. These findings are similar to the findings in the 2D cases in that the addition of pitching/twisting improves e ciency by apparently reducing or eliminating leading edge separation, and that high propulsive e ciencies are achieved by motions that are at the limit of large-scale leading edge separation.

CHAPTER 4. FLAPPING WING OPTIMIZATIONS 92

Figure 4.14 shows plots of surface streamlines for each of the twist-complexity cases, one taken on the trailing face during the middle of the down-stroke and the other at the top of the stroke. The surface streamlines for the 0–1–0 case are highly non-uniform at both instances in the flapping cycle and illustrate the influence of the large-scale leading edge separation. In contrast, the 1–1–0 and 2–1–0 cases are much more uniform during the middle of the stroke. What is interesting to note, however, is that the surface streamlines point opposite the direction across the majority of the surface in both cases, indicating that the flow is, in fact, separated. Closer examination of the vorticity plots indicates that there is, in fact, a thin region of separated flow attached to the airfoil surface, but that this region neither grows nor rolls up to form a leading edge vortex during the middle portion of the stroke.

Towards the end of the stroke, this region can be seen to de-stabilize and form several transient vortices that begin near the leading edge and are shed down-stream. This instability can be seen in figure 4.11 in the plots from the top and bottom of the stroke. The streamlines in figure 4.14 show the e↵ects of these transient vortices as the flow becomes much less uniform under their influence. The streamlines at the top of the stroke are also dramatically di↵erent for the 1–1–0 and 2–1–0 cases despite being visually identical during the mid-stroke. This suggests that one e↵ect of the di↵ering twist distributions between the 1–1–0 and 2–1–0 cases is to influence the destabilization of the boundary layer at the top and bottom of the stroke.

CHAPTER 4. FLAPPING WING OPTIMIZATIONS 93

(a) 0–1–0 (b) 1–1–0 (c) 1–1–0 (no twist) (linear twist) (non-linear twist) Figure 4.11: Vorticity isosurface visualization of the flapping cycle for maximum propulsive e ciency. Color values are based on pressure. The left column depicts the case without twisting motion, the center column depicts the case with one twist control point, and the right column depicts the case with two twist control points.

CHAPTER 4. FLAPPING WING OPTIMIZATIONS 94

(a) 0–1–0 (b) 1–1–0 (c) 1–1–0 (no twist) (linear twist) (non-linear twist) Figure 4.12: Vorticity isosurface visualization of the flapping cycle for maximum propulsive e ciency. Color values are based on pressure. The left column depicts the case without twisting motion, the center column depicts the case with one twist control point, and the right column depicts the case with two twist control points.

CHAPTER 4. FLAPPING WING OPTIMIZATIONS 95

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0–1–0 1–1–0 2–1–0 Table 4.14: Visualization of instantaneous streamlines on the wing surface with varying twist. The left column depicts the wing at the middle of the upstroke (dihedral angle = 0 ), and the right column depicts the wing at the top of the stroke (dihedral angle = maximum).

CHAPTER 4. FLAPPING WING OPTIMIZATIONS 96

4.4.1 Thickness E↵ects This section examines the boundary layer and vortex dynamics for the 1–1–0 and 2–1–0 cases using a wing a 2% thickness NACA0002 airfoil section in comparison with the nominal 12% thickness NACA0012 section cases. The Q-criterion is used in this section and the next to examine the more-subtle di↵erences between the vortex structures [15]. Visualizing vortices using vorticity can sometimes be di cult because both shear layers and vortex cores have high values of vorticity. The Q-criterion is formulated to address this issue by di↵erentiating shear layers from vortex cores. The scalar Q is given by Sij Sij ) = (! 2 Q = (⌦ij ⌦ij 2Sij Sij ), (4.1)

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and ! is the standard vorticity magnitude. The symmetric component Sij represents the rate-of-strain and is large in magnitude in high-shear regions. The anti-symmetric component ⌦ is the rate-of-vorticity and is large in magnitude in circulating regions.

Q then represents the relative di↵erence between rate-of-strain and rate-of-vorticity.

Regions with Q 0 are increasingly shear-dominated, whereas regions with Q 0 are increasingly vortex-dominated.

Figures 4.13 and 4.

14 show span-wise Q contour slices for the 1–1–0 NACA0012 wing and the 1–1–0 and 2–1–0 NACA0002 wings. The contours display Q in the range 1-10 in all cases, illustrating the locations of vortex cores. Figure 4.13 shows the wings in the middle of the stroke, while figure 4.14 shows the wing at the top of the stroke. While the topology structure of the vortices is similar between cases, the intensity and size of the vortex cores are significantly smaller for the 2% thick wings.

In addition, the Q contour appear to be more uniform along the span in the 2–1–0 case when compared with the other cases. This is most noticeable along the leading

CHAPTER 4. FLAPPING WING OPTIMIZATIONS 97

edge and in the vortex cores at the top of the stroke. This indicates that the nonlinear distribution of twist serves, in part, to create a more uniform and consistent leading edge boundary layer and vortex structure.

Figures 4.15 and 4.

16 compare the surface stream lines for the NACA0012 and NACA0002 cases for the 1–1–0 and 2–1–0 cases. The streamlines at the mid-stroke point are consistent between all cases, the main di↵erence being that the saddle point near the root chord moves towards the trailing edge in the 2% thickness cases. In contrast, the streamlines at the top of the stroke di↵er greatly and show no clear trend with respect to thickness. Recall that the streamlines during this portion of the stroke are largely governed by the destabilization of the thin separated region present during the upstroke. The results here serve to further reinforce the high degree of sensitivity of this instability to the geometry and kinematics of the wing.

CHAPTER 4. FLAPPING WING OPTIMIZATIONS 98

(a) 1–1–0, NACA0012 (b) 1–1–0, NACA0002 (c) 2–1–0, NACA0002 Figure 4.13: Span-wise slices showing contours of Q-criterion at the middle of the down stroke. Depicted are the optimal cases for the 1–1–0 parameterization with 12% thickness airfoil, the 1–1–0 case with the 2% thickness airfoil and the 2–1–0 case with the 2% thickness airfoil.

(a) 1–1–0, NACA0012 (b) 1–1–0, NACA0002 (c) 2–1–0, NACA0002 Figure 4.14: Span-wise slices showing contours of Q-criterion at the top of the stroke.

Depicted are the optimal cases for the 1–1–0 parameterization with 12% thickness airfoil, the 1–1–0 case with the 2% thickness airfoil and the 2–1–0 case with the 2% thickness airfoil.



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