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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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Figures 3.18 and 3.19 show the vorticity contour plots for the ↵0 = 5 and ↵0 = 10 cases. These cases in general show wake vortex patterns that are similar to the ↵0 = 0 cases, with a relatively large leading edge vortex being formed and then detached at the top and bottom of the stroke as well as the continuous shedding of small vortices from the trailing edge. The flow in these cases is di↵erent for the up and down strokes as a result of the angle of attack of the incoming flow. The most prominent consequence of the angle of attack is that the leading edge vortex formed on the down-stroke is much stronger. The leading edge vortex also appears to form later in the down stroke, growing quickly during the period of fast pitching-up. In contrast, the leading edge vortex is much smaller or, as is particularly seen in the ↵0 = 10 cases, not present during the upstroke.

CHAPTER 3. PITCHING & PLUNGING AIRFOIL OPTIMIZATIONS 69

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Figure 3.18: Vorticity visualization of the flapping cycle for constrained power minimization cases with ↵0 = 5.

CHAPTER 3. PITCHING & PLUNGING AIRFOIL OPTIMIZATIONS 70

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Figure 3.19: Vorticity visualization of the flapping cycle for constrained power minimization cases with ↵0 = 5.

Chapter 4 Flapping Wing Optimizations This section contains results for the maximization of propulsive e ciency for the 3D flapping wing case. Several combinations of degrees of freedom are considered. In all cases the motion is parameterized using a single control point for root dihedral angle. Cases with between one and four equally-spaced span-wise twist control points as well as the case with one twist and one sweep control point are considered. The goal with these optimizations is to determine the trade-o↵ between increasing the complexity of motion by adding parameters versus the potential gain in achievable propulsive e ciency. In addition, wing geometry variations are considered. These include performing optimizations with a thinner airfoil section (2% vs 12%), as well as di↵ering wing planform shapes. The goal with these optimizations is to identify general trends resulting in variation of these static geometric parameters. The resulting optimal cases are also analyzed using the resulting flow-field data from the flow solver to identify fluid-dynamic phenomenon related to the cases that maximize propulsive e ciency.

The wing configurations used in all cases have a span of 8 and root-chord of 1, and use a symmetric NACA airfoil section. The domain is discretized using an H-C mesh with 256 ⇥ 64 ⇥ 64 cells and employs a symmetry plane, as can be seen in figure

4.1. A mach number of 0.2 and a Reynolds number of 2000 based on the chord length are used throughout. Flow solutions are advanced through two flapping cycles during the optimization process and then verified over five cycles. The averaged force and

CHAPTER 4. FLAPPING WING OPTIMIZATIONS 72

power coe cients are integrated over the final flapping cycle. Flow solutions are run in parallel, typically using 256 cores, requiring approximately 1-2 hours of wall-time per flapping cycle.

In the remainder of this chapter an abbreviated notation to describe the di↵erent permutations of the kinematic parameters is used. The notation encodes the number of degrees of freedom as # twist–# dihedral –# sweep, so for example a case with 3 twisting degrees of freedom, 2 dihedral degree of freedom and 1 sweep degree of freedom would be denoted “3–2–1”.

CHAPTER 4. FLAPPING WING OPTIMIZATIONS 73

Figure 4.1: The 256 ⇥ 64 ⇥ 64 H-C mesh and rectangular wing geometry used for all 3D optimization cases

CHAPTER 4. FLAPPING WING OPTIMIZATIONS 74

4.1 Periodic Twisting and Dihedral Motion We first consider four cases with a single dihedral degree of freedom and zero, one, two and four twisting degrees of freedom. In all cases the wing has a rectangular planform and a NACA0012 airfoil section. These cases are analogous to the 2D pitching and plunging cases as each span-wise airfoil section is undergoing a pitching and plunging motion. The span-wise dihedral angle is constant as a result of having a single dihedral control point. The pure dihedral (0–1–0) case is analogous to pure plunging. The 1–1–0 case has a linearly-varying span-wise twist distribution, starting from zero degrees at the root. The 2–1–0 and 4–1–0 cases allow increasingly complex non-linear variation in the twist distribution, including non-monotonic distributions such as the example case show in figure 2.15. The variable assignments used for the four cases considered are shown in tables 4.1, 4.2, 4.3, and 4.4.

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The 0–1–0 case was run first and allowed to converge to an optimum. The 1–1–0 case was then initialized with the optimal parameters from the 0–1–0 cases and then allowed to reach a new optimal value. The 2–1–0 case was then run using the optimal parameters from the 1–1–0 case, and so on.

Table 4.5 contains a summary of the motion parameters and the resulting propulsive e ciency for the optimal cases.





In the 4–1–0 case the optimization algorithm failed to improve on the results from the 2–1–0 case, converging to essentially the same propulsive e ciency value as the 2–1–0 case. The 4–1–0 case was initialized from several other starting points but in all cases it failed to produce a further improvement in propulsive e ciency. This suggests that the 4–1–0 case provides essentially no advantage over the 2–1–0 case despite the increase in potential complexity of twist motion. It is also possible that the 4–1–0 optimization space is more multimodal as a result of the increased number of parameters, causing the optimization to become stuck in local optimizers of lesser propulsive e ciency. As with the 2D cases, it is, unfortunately, too computationally expensive at present to employ optimization algorithms that are better-suited to multi-modal problems. Further analysis of the 4–1–0 case is not included due to this lack of improvement.

The data shown in 4.5 show that the addition of wing twist produces a significant improvement in the modified propulsive e ciency, increasing the achievable value from around 10% to almost 47% by simply allowing linear twist and at the cost of two extra parameters. Adding two more parameters to allow non-linear twisting further improves the achievable e ciency to nearly 50%. A significant decrease in frequency and increase in dihedral amplitude is observed with the addition of twisting when comparing the 0–1–0 and 1–1–0 cases, while there is a relatively minor change in the overall parameter values between the 1–1–0 and the 2–1–0 cases. All phase angles, which have been omitted for brevity, are within the range 90 ± 10.

Table 4.6 shows several non-dimensional parameters derived from the data in table 4.

5. The non-dimensional parameters taken from several bird and insect species are displayed again in table 4.7 for comparison. Looking specifically at the reduced frequency, the values from the optimal cases are, with the exception of the nontwisting case, very similar to the values for the bird and insect species, which range

CHAPTER 4. FLAPPING WING OPTIMIZATIONS 77

in value from 0.17 to 0.35. Both the Strouhal number and the ratio Utip /U1, which relate to the relative amount of forward speed versus wing tip speed, indicate that the wing tip speed is much higher than the forward speed. This is to be expected for a near-hovering type condition.

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The flapping wing motions are visualized in figure 4.2, with surface coloring corresponding to pressure values. Figure 4.2 further illustrates the significant di↵erence in the dihedral amplitude between the twist-free (0–1–0) case and the twisting (1– 1–0) cases, as well as the relatively minor visible di↵erence between the 1–1–0 and the 2–1–0 cases. Figure 4.3 shows the span-wise twist distributions for for the 1–1– 0 and 2–1–0 cases. The the 2–1–0 case displays more aggressive twist towards the root chord, reaching approximately 78% of the final tip twist angle of 58.4 between the wing root and the mid-span control point, while relatively little additional twist occurs between mid-span and the wing tip.

CHAPTER 4. FLAPPING WING OPTIMIZATIONS 79

(a) 0–1–0 (b) 1–1–0 (c) 2–1–0 (no twist) (linear twist) (non-linear twist) Figure 4.2: Vorticity isosurface visualization of the flapping cycle for maximum propulsive e ciency. Color values are based on pressure. The top row depicts the case without twisting motion, the middle row depicts the case with one twist control point, and the bottom row depicts the case with two twist control points.

CHAPTER 4. FLAPPING WING OPTIMIZATIONS 80

Figure 4.3: Plots of the span-wise twist distribution for the linear and non-linear twist cases.

CHAPTER 4. FLAPPING WING OPTIMIZATIONS 81

The polar plot in figure 4.4 show trends that are similar to the 2D e ciency maximization case. The polar for the twist-free case has a large CL range and a low-magnitude but largely negative (thrust producing) CD range resulting in low e ciency. The twisting cases show a reduced CL and increased thrust, both contributing to the increase in e ciency. Between the 1–1–0 and 2–1–0 cases the maximum thrust coe cient increases from approximately 0.8 to 1.2 without a significant change in the CL range. A noted di↵erence between the 2D and 3D cases is that, while in the 2D case the CL range increases for the pitching and plunging case, in 3D the CL range decreases for the twisting cases. It is not clear if this is a result of the fundamental di↵erence between 2D and 3D flows, or if it an indication of potential room for further optimization of the 2D results.

CHAPTER 4. FLAPPING WING OPTIMIZATIONS 82

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Figure 4.4: CL versus CD polars for optimal 3D cases.

The vertical axis represent lift and the horizontal axis represents drag. The left plot shows the polar for the non-twisting case, the center plot shows the polar for the single control point case and the right plot shows the polar for the 2 control point case.

CHAPTER 4. FLAPPING WING OPTIMIZATIONS 83

4.1.1 Optimization Convergence Figures 4.5, 4.5, 4.5, and 4.5 show the sequence of objective values computed by SNOPT during each of the optimizations. Note that each optimization begins with a period of pseudo-random function calls to establish an initial basis for the optimization. The small number of parameters and a good initial parameter set resulted in relatively few function calls for the 0–1–0 and 1–1–0 cases. The 2–1–0 case took many more function calls due to a relatively bad initial estimate, however the plot indicates steady improvement from the optimization algorithm. The 4–1–0 case, which was initialized from the 2–1–0 optimum, required fewer function calls despite the increase in number of parameters due to the fact that the optimal parameter set, which did not improve on the 2–1–0 case, was very similar to the optimal set from the 2–1–0 case.

Figure 4.5: History of function evaluations conducted by SNOPT during the optimization of the 0–1–0 (2 parameter) 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. Note that this case was started very near the optimal set of parameters and thus converged with relatively few (25) function evaluations.

CHAPTER 4. FLAPPING WING OPTIMIZATIONS 84

Figure 4.6: History of function evaluations conducted by SNOPT during the optimization of the 1–1–0 (4 parameter) 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. Note that this case too was started very near the optimal set of parameters and thus converged with relatively few (74) function evaluations.

Figure 4.7: History of function evaluations conducted by SNOPT during the optimization of the 2–1–0 (6 parameter) 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. Note that this case was started with a set of parameters relatively far from the optimal set and thus required relatively many (375) function evaluations.

CHAPTER 4. FLAPPING WING OPTIMIZATIONS 85

Figure 4.8: History of function evaluations conducted by SNOPT during the optimization of the 4–1–0 (10 parameter) 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. This case reached a parameter set satisfying the optimality conditions but the propulsive e ciency value attained did not exceed that of the 2–1–0 case.

CHAPTER 4. FLAPPING WING OPTIMIZATIONS 86

4.2 Adding Sweeping Motion This section presents the results from the maximization of propulsive e ciency for a case with twisting, dihedral and sweeping degrees of freedom, specifically the 1–1–1 case with a single control point for each degree of freedom. This optimization was initialized using the optimal parameters from the 1–1–0 case. The variable assignment for this problem is show in table 4.8.

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Table 4.9 shows the resulting parameters and modified propulsive e ciency for the 1–1–1 case.



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