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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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HIGH FIDELITY OPTIMIZATION OF FLAPPING AIRFOILS

AND WINGS

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF AERONAUTICS &

ASTRONAUTICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Matthew Culbreth March 2013 © 2013 by Matthew Kohler Culbreth. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons AttributionNoncommercial 3.0 United States License.

http://creativecommons.org/licenses/by-nc/3.0/us/ This dissertation is online at: http://purl.stanford.edu/rb093bx6087 ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Antony Jameson, Primary Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Juan Alonso I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Thomas Pulliam Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives.

iii Abstract Flapping wings are interesting in many ways from both a scientific and an engineering perspective. They are also challenging to design and analyze due to the inherent complexity of both the kinematic motion of the wing and the resulting vortex-dominated fluid dynamics. One way to study flapping wings is to use an optimization approach to find kinematic motions that lead to e cient flight under various conditions. We use this approach, coupling high-fidelity 2D and 3D Navier-Stokes solvers with a gradient-based optimization algorithm.

Results are presented for several optimizations of 2D airfoils and 3D wings undergoing periodic, flapping-type motions. In 2D the pitching and plunging NACA0012 airfoil case is considered, with optimizations being carried out to maximize propulsive e ciency and also to minimize input power given a target thrust constraint. In 3D rectangular and semi-elliptic wings of varying thickness that are hinged at the root are considered. The motion of the 3D wing is parameterized by spline control points that allow span-wise variation of twist, dihedral and sweep, allowing complex wing motions and deformations with relatively few parameters. Propulsive e ciency is maximized for the 3D wing in a non-twisting case as well as using one, two and four of these span-wise twist control points, and a case with one each of twist, dihedral and sweep control points. Wing thickness and planform e↵ects are also considered, with optimizations being carried out for both 12% thick and 2% thick airfoil sections and using rectangular and semi-elliptic planform wings. The results of the optimizations lead to several conclusions, including that pitching and twisting can significantly improve the attainable propulsive e ciency, that twisting motions beyond a certain level of complexity o↵er no additional improvement in attainable propulsive e ciency, and iv that sweeping motions also do not increase attainable propulsive e ciency. Analysis of the flow physics of the optimal cases show that the high-performing cases operate in the absence of a persistent leading-edge vortex, but do display a stable, thin boundary region of recirculation during the middle portion of the stroke that destabilizes into shed vortices at the top and bottom of the stroke. The destabilization of this region is shown to be highly sensitive to variations in wing motion and geometry.

vAcknowledgement

I would first like to acknowledge my advisor Antony Jameson. We came to work together is somewhat of a round-about way (at least from my end), and throughout the process he proved to be a better mentor, advisor and friend than I could have hoped for. He has always been incredibly generous with his time and his substantial expertise and in allowing me the freedom to define and direct my research to towards my own interests. Having been able to work with him has certainly been one of the great highlights of my career.

I also owe great thanks to Juan Alonso and Tom Pulliam for agreeing to be members of my reading committee. I owe additional thanks to Tom for working with me and providing valuable insights throughout the development of my dissertation.

I would like to thank Robert MacCormack and Michael Saunders for serving on my oral defense committee. I would also like to further thank Michael Saunders and his colleague Walter Murray, both for helping me figure out how to use SNOPT, software they helped develop, as well as for providing numerous insights into the intricacies of numerical optimization.





I have thoroughly enjoyed the time I’ve spent with my colleagues in the Aerospace Computing Lab. They are: Andy, Charbel, Charlie, David, Edmond, Jen-Der, Josh, Kui, Lala, Manuel, Nawee, Patrice, Peter, Philip, Rui, Sachin, and Yves. Much fun was had in lab, at conferences and elsewhere in their company. I owe special thanks to Yves, my chief partner in crime. Not only did Yves agree to fix his various codes up to simulate my flapping wings (for a nominal fee), but has been a great collaborator during our investigations of low-cost task chair presence sensing, the practical performance limits of U-Haul trucks, and many other projects that probably vi shouldn’t be mentioned.

Last, but certainly not least, my family deserves enormous thanks for all of their support. My mother Pegi must have tirelessly scoured the calendars for every possible holiday she could use as an excuse to send me a card with some money and a note saying, ”have a treat!”, and my father Cully has always been a great companion on our many trips down to Monterey to check out endless quantities of rare and exotic cars every August, among the countless other ways my parents have helped me along the way. I would also like to thank my sister Lauren for always cheerfully introducing me as her brother the rocket scientist, letting me hang out with her in New York, and for generally being the coolest person I know.

This work is supported by the Department of the Army through the Army High Performance Computing Research Center, Cooperative Agreement No. W911NF-07and by the AFOSR under grant no. FA 9550-07-1-0195 from the Computational Math Program under the direction of Dr. Fariba Fahroo.

–  –  –

x

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)................................ 99

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)............................... 100

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)................... 103

xiList of Figures

2.1 Experimentally-obtained Eppler 387 polars for Re = 100, 000 and Re = 200, 000. Note the large discrepancy between di↵erent experiments at Re = 100, 000 [23]............................. 7

2.2 Reynolds number range for natural and man-made flyers [24]..... 8

2.3 Drop in section lift/drag ratio for transitional Reynolds numbers [21]. 8

2.4 Visualization of Mach number countour slices for a representative flapping wing test case. Contours visible for cases where M 0.3. The maximum Mach number encountered in this case is M = 0.42..... 9

2.5 Software block diagram.......................... 13

2.6 Boundary surface of a sample H-C mesh................ 22

2.7 An un-deformed mesh and the resulting mesh after a combination of twist and dihedral deformations are shown................ 24

2.8 2D Meshes used for optimization and analysis. The left image shows the boundaries and extent of the C-Mesh. The center figure shows a portion of the 1024 ⇥ 128 mesh used during function evaluations by the optimization algorithm. The left figure shows a portion of the 4096 ⇥ 512 DNS mesh used for validation, analysis and flow visualization. 27

2.9 3D mesh convergence with respect to lift (top) and drag (bottom), comparing 128 ⇥ 64 ⇥ 64, 256 ⇥ 64 ⇥ 64 and 384 ⇥ 96 ⇥ 96 meshes.. 28

2.10 2D quasi-periodic convergence of lift-versus-drag plots. Successive periods are compared, and the results indicate that a quasi-periodic state is reached for the second period..................... 31

–  –  –

3.1 2D Meshes used for optimization and analysis. The left image shows the boundaries and extent of the C-Mesh. The center figure shows a portion of the 1024 ⇥ 128 mesh used during function evaluations by the optimization algorithm. The left figure shows a portion of the 4096 ⇥ 512 DNS mesh used for validation, analysis and flow visualization. 50

3.2 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.............................. 50

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.

xiii

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...................... 52

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............. 52

3.6 Vorticity visualization of the flapping cycle for maximum propulsive e ciency. 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....... 54

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..................................... 55

3.8 Plots of the input power requirements resulting from thrust-constrained minimization............................... 60

3.9 Plots of the pitching and plunging motion parameters for each of the thrust-constrained optimization cases................... 61

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.............................. 62

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.............................. 62

–  –  –

xv

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....................... 79

4.3 Plots of the span-wise twist distribution for the linear and non-linear twist cases................................. 80

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.

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............... 83



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