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12% 2% Table 4.15: Visualization of instantaneous streamlines on the wing surface comparing the 1–1–0 case with two wing thicknesses. 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).


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12% 2% Table 4.16: Visualization of instantaneous streamlines on the wing surface comparing the 2–1–0 case in two wing thicknesses. 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).


4.2 Planform E↵ects This section deals with the analysis of the flow physics of the semi-elliptic planforms in comparison with the nominal rectangular planform. Recall that these cases all use a NACA0012 airfoil section and are parameterized with one twist and one dihedral control point. The smaller tip chords should, in theory, lead to a weaker wingtip vortex than a rectangular planform. A weaker wingtip vortex will, in turn, have less influence on span-wise vortex structures, and thus may aid in stabilizing a leading edge vortex. The following figures examine this possibility.

Figures 4.15 and 4.

16 show span-wise Q contour slices for the rectangular and semi-elliptic wings. The contours display Q in the range 1-10 in all cases, illustrating the locations of vortex cores. The topology of the vortex structures is again very similar between the di↵erent planform shapes. The middle of the stroke shows a vortex core towards the trailing edge and detaching near the mid-span point in all cases. The top of the stoke shows a triple vortex struction along the span in all cases, with one large aft core and two smaller cores nearer the trailing edge. In the rectangular case the vortex structure appears to destabilize towards the wing tip, whereas the semi-elliptic planforms maintain a more consistent core diameter and Q value. This suggests that the strength of the wing-tip vortex can, in fact, have a destabilizing e↵ect on span-wise vortex structures.

Figure 4.17 show surface streamline plots for the rectangular and semi-elliptic planforms at the mid-stroke and top of stroke points.

The streamlines at mid-stroke show the familiar pattern of relatively uniform reversed flow, though the saddle point is nearer the trailing edge for the semi-elliptic cases. At the top of the stroke the streamlines are relatively consistent between the two semi-elliptic case, displaying an attracting line of span-wise flow that starts around mid-chord near the wing root and moves to the trailing edge at two-thirds to three-quarters span. This observation is in contrast to the previous cases that all show significant variation in the surface streamlines at the top of the stroke. This observation lends further evidence to the stabilizing e↵ect of a reduced tip-chord on span-wise vortex structures.


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Figure 4.15: Span-wise slices showing contours of Q-criterion at the middle of the down stroke.

Depicted are the optimal cases using the rectangular and semi-elliptic planforms. Note that the vortex structure is topologically similar across planforms.

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Figure 4.16: Span-wise slices showing contours of Q-criterion at the top of the up stroke.

Depicted are the optimal cases using the rectangular and semi-elliptic planforms. Note that the vortex structure is again topologically similar across planforms.


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100% 50% 25% Table 4.17: Visualization of instantaneous streamlines on the wing surface. 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 5 Conclusions The results obtained from this research lead to a number of conclusions about efcient flapping-wing flight. They also raise a number new questions and lead to several new hypotheses. The first conclusion is that the objective function spaces examined, namely the maximization of propulsive e ciency in 2D and 3D and the thrust-constrained minimization of propulsive e ciency in 2D, are su ciently continuous and smooth in the region of optima to allow reliable gradient evaluations and thus enable the use of gradient-based optimization algorithms. There are no indications that the maximization of propulsive e ciency problem is problematically multi-modal since consistent optima were found from multiple starting points. The constrained power minimization problems, however, do show indications of either multi-modality or a poorly conditioned objective functions space since several cases showed dependency on initial conditions.

In terms of conclusions relating to design of flapping wing systems, the results of this research show that adding span-wise twist to a purely dihedral flapping motion leads to a significant improvement in the achievable maximum propulsive e ciency.

The greatest improvement comes from switching from no twist to simple linear spanwise twist, while a smaller improvement is realized by utilizing a two-parameter nonlinear twist distribution. Increasing the complexity beyond the two parameter case does not yield a significant improvement. The same conclusion can be draw from the 2D analog of flapping wing, the pitching and plunging airfoil. In this case the


addition of pitching motion yields a dramatic improvement in attainable propulsive e ciency over plunging motion alone. Returning to the flapping wing case, addition of sweeping motion as a degree of freedom does not yield an improvement in achievable propulsive e ciency. Taken together, these results show that propulsive e ciencies on the order of 50% can be achieved using a relatively compact parameterization (six to ten parameters) that includes periodic dihedral and twisting motions. Reducing the wing thickness from 12% to 2% in both the linear and non-linear twisting cases yields an improvement in propulsive e ciency of roughly 10%. Finally, changing the wing planform shape from rectangular to semi-elliptic profiles tip-chords yields a slight improvement in propulsive e ciency on the order of a few percent.

The results from this research indicate that the thrust-constrained power minimization problem is more di cult and subtle. Two main design conclusions can be drawn from the results. First, there is a definitive and seemingly linear correlation between target thrust coe cient and the minimum required power to achieve it. Second, increasing the angle of attack with a fixed target thrust coe cient results in an increase in the minimum power required to achieve that thrust. There are no clear trends with respect to the parameters of frequency, pitch and plunge amplitudes and phase di↵erence. This indicates either that the problem is relatively insensitive to these parameters, or that the objective function space is multimodal.

Post-processing of the solver field data leads to several conclusion about the fluid dynamics of the optimal cases. The formation or suppression of the leading-edge vortex appears to be a significant factor in the cases that maximize propulsive e ciency both in 2D and 3D. A leading edge vortex is clearly evident in the relatively ine cient twist/pitch free cases. The leading edge vortex forms as a result of flow separation induced by high local angles of attack as the wing moves perpendicular to the free-stream flow. In the pitching and twisting cases the airfoil/wing the local angle of attack is reduced as the airfoil/wing rotates into the direction of motion.

The reduction in local angle of attack has the e↵ect of suppressing the formation of the leading-edge vortex in the optimal propulsive e ciency cases. Thus, improved propulsive e ciency is correlated with the absence of a leading edge vortex. Close examination of the flow field around both the wing and the airfoil show that there is


a thin, separated and recirculating region attached to the surface during a large portion of the up and down stroke. This region is present in all twisting cases regardless of wing thickness and planform shape.The top and bottom of the stroke show much greater sensitivity to the twist distribution and wing geometry, with the fluid behavior for this portion of the stroke being governed by the destabilization and roll-up of the thin, recirculating region. All rectangular planform cases show significant di↵erences in surface streamlines during this portion of the stroke. The two semi-elliptic planforms are much more similar in comparison, indicating that one e↵ect of the smaller wing tip is to stabilize the span-wise vortex structures at the top and bottom of the stroke, possibly as a result of reducing the influence from the wing-tip vortex.

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[8] Philip E Gill, Walter Murray, and Michael A Saunders. Snopt: An sqp algorithm for large-scale constrained optimization. SIAM journal on optimization, 12(4):979–1006, 2002.

[9] Kenneth C. Hall and Steven R. Hall. Minimum induced power requirements for flapping flight. Journal of Fluid Mechanics, 323:285–315, 8 1996.

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[11] K ITO and S SUZUKI. Optimization of flapping wing motion. In Proceedings of Aircraft Symposium, volume 39, pages 160–163, 2001.

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[13] Antony Jameson. The construction of discretely conservative finite volume schemes that also globally conserve energy or entropy. Journal of Scientific Computing, 34(2):152–187, 2008.

[14] Antony Jameson. Formulation of kinetic energy preserving conservative schemes for gas dynamics and direct numerical simulation of one-dimensional viscous compressible flow in a shock tube using entropy and kinetic energy preserving schemes. Journal of Scientific Computing, 34(2):188–208, 2008.

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[18] KD Jones and MF Platzer. Numerical computation of flapping-wing propulsion and power extraction. AIAA paper, 97:0826, 1997.

[19] R. T. Jones. Wing flapping with minimum energy. NASA Technical Memorandum, 81:174, 1980.

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[21] P. B. S. Lissaman. Low-Reynolds-Number Airfoils. Annual Review of Fluid Mechanics, 15:223–239, 1983.

[22] Hao Liu and K Kawachi. A numerical study of insect flight. Journal of Computational Physics, 146(1):124–156, 1998.

[23] C.A. Lyon, A.P. Broeren, P. Gigure, A. Gopalarathnam,, and M.S. Selig. Summary of Low-Speed Airfoil Data, volume 3. SoarTech Publications, 1998.

[24] John H McMasters and Michael L Henderson. Low-speed single-element airfoil synthesis. 1979., pages 1–31, 1979.

[25] Michele Milano and Morteza Gharib. Uncovering the physics of flapping flat plates with artificial evolution. Journal of Fluid Mechanics, 534(1):403–409, 2005.

[26] Siva K Nadarajah and Antony Jameson. Optimum shape design for unsteady flows with time-accurate continuous and discrete adjoint methods. AIAA journal, 45(7):1478, 2007.

[27] K. Ou, P. Castonguay, and A. Jameson. 3d flapping wing simulation with high order spectral di↵erence method on deformable mesh. In 49th AIAA Aerospace Sciences Meeting, Orlando, Florida, 2011.

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[29] David L Raney and Eric C Slominski. Mechanization and control concepts for biologically inspired micro air vehicles. Journal of aircraft, 41(6):1257–1265, 2004.

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[33] Wei SHYY and HAO LIU. Flapping wings and aerodynamic lift: The role of leading-edge vortices. AIAA journal, 45(12):2817–2819, 2007.

[34] Karl Axel Strang. E cient flapping flight of pterosaurs. Stanford University, 2009.

[35] Theodore Theodorsen. General theory of aerodynamic instability and the mechanism of flutter, naca report 496, 1935. This paper is also included in A Modern View of Theodore Theodorsenpublished by AIAA in, pages 2–21, 1992.

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