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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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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............... 84

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........... 84 xvi

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..... 85

4.9 Non-rectangular planform shapes considered with tip chords of 50% and 25% of the root chord......................... 88

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............................ 89

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.............. 93

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.............. 94

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........ 98

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........ 98 xvii

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......................... 102

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 semielliptic planforms. Note that the vortex structure is again topologically similar across planforms......................... 102

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Background

1.1 Introduction Flapping wings have been the dominant mechanism of flight the the last 350 million years, and it is only within the last one hundred or so years that craft have been propelled and held aloft by anything other than flapping wings. Flapping wings are not well suited to flight at the scales required for human transport and as a result much of the scientific and engineering e↵orts to understand the aerodynamics of atmospheric flight have focused on large, fixed wings with a separate engine to provide forward thrust. Massive e↵orts in the century since the first manned flight have led to a thorough understanding of the aerodynamics of fixed wings across a vast range of flight speeds. With the maturation of fixed-wing aerodynamics has come a new interest in the relatively little-explored problem of understanding flapping-wing flight.

Research interest in the aerodynamics of flapping wings has increased significantly within the last decade. Flapping wing flight has become a significant engineering research direction, with the miniaturization of electronic components and increases in battery energy density allowing for the creation of a class of small-scale unmanned flying vehicles generally referred to as micro air vehicles (MAVs). Flapping wings are of particular interest for this application both because of the numerous, highlysuccessful examples of flapping wing flight found in nature and because flapping wings CHAPTER 1. BACKGROUND 2 provide an attractive flight system that simultaneously packages mechanisms for lift, thrust and control. Investigation of flapping wing flight is also of significant importance to the fundamental science of aerodynamics. The development of aircraft over the last century has provided a significant and mature body of understanding of the physics of steady-state fluid dynamics across a wide range of flight speeds. This work generally has little application, however, to the complex and unsteady flapping wing flight regime. Fortunately, high-fidelity computational tools and associated powerful computational hardware now allow for accurate simulation of flapping wings, making it possible to begin to understand the physics of this type of flight. These highdelity simulation tools have also opened new possibilities for research into the fluid dynamics of bird and insect flight within the biological sciences. These tools now allow detailed examination in areas such as the relationship between wing kinematics and the resulting forces such as lift and thrust, as well as the specific flow physics associated with the numerous examples of animal flight mechanisms.





From a design stand-point, flapping wing flight presents a significant increase in the potential degrees of freedom that need to be addressed. The standard wing geometry parameters from fixed-wing aircraft such as airfoil section shape, planform shape, aspect ratio, dihedral angle, sweep angle, and twist all have the potential to become time-dependent functions. A key objective for flapping wing design is thus to reduce this infinite-dimensional design space to one that consists of a manageable finite set of parameters that specify the motion and deformation of the wing in time.

The aim of this thesis is to perform a series of studies at a specific small-scale flight condition with the goal being both to identify a compact set of parameters that can e↵ectively prescribe flapping motions and find the specific values of these parameters that lead to e cient flight performance. This is accomplished by using a numerical optimization process over varying types and complexities of kinematic and geometric parameter sets. 2D and 3D viscous flow solvers have been developed that are specifically tailored to simulate oscillating airfoils and flapping wings. These codes are coupled to a gradient-based numerical optimization algorithm. The optimization process is then used in conjunction with parameterizations of varying complexity to both identify optimal flapping motions and also investigate the trade-o↵ between CHAPTER 1. BACKGROUND 3 the complexity of the motion and the attainable performance. In this thesis we specifically address the optimization of propulsive e ciency in the 2D and 3D cases.

The 2D investigations focus on the pitching and plunging airfoil case, while the 3D investigations focus on a flapping and twisting wing that is hinged at the wing root and allowed to move with varying combinations of twist, dihedral and sweep. Wing thickness and planform e↵ects are also investigated in the 3D case. Results from the various parameter combinations are compared both to understand how each degree of freedom a↵ects attainable e ciency, and to gain insight into the associated flow physics.

1.2 Background The study of flapping wings dates back to the beginnings of the study of aerodynamics itself, with numerous examples of attempts to construct flapping wing vehicles that predate the first powered aircraft. The development of the theory of flapping wings through the twentieth century was somewhat of a footnote in the evolution of aerodynamic theory, which was largely driven by understanding subsonic, transonic and supersonic fixed-wing flight. Early examples of flapping wing theory come from the work of Theodorsen [35] and Garrick[5], who each developed analytic models for airfoils undergoing small-amplitude oscillations in 2D potential flow, work that was largely motivated by attempts to understand aeroelastic flutter. Lighthill [20] and Weis-Fogh [37], among others, performed pioneering research on the aerodynamics of biological fliers including the elucidation of some of the complex aerodynamic mechanisms used by these creatures.

Interest in flapping flight increased significantly towards the end of the twentieth century as steady aerodynamic theory reached maturity and improvements in algorithms for computational fluid dynamics (CFD) and increasing computing power made higher-fidelity simulations of flapping wings possible. Early numerical simulations of flapping wings were carried out using unsteady vortex-lattice and panel methods [22, 17]. Current computational capabilities allow for high-fidelity, 3D, unsteady Navier-Stokes simulations of flapping wings. A number of detailed CFD studies CHAPTER 1. BACKGROUND 4 have been completed, including work by Jones, et. al. [18, 16], Shyy et. al. [32], Persson et. al. [28], and Ou et. al. [27].

There has also been long-standing research interest in optimization of flapping flight, based partially on the assumption that natural fliers operate in a manner that is in some way optimal. R.T. Jones [19] developed an expression for the optimal lift distribution along the wing during flapping motion by minimizing induced drag for a given wing bending moment in potential flow. Hall and Hall [9] compute the optimal span-wise circulation distribution on a thrusting and lifting wing using a 1D integral solution in the small amplitude case and a vortex-lattice code in the large amplitude case. Hamdaoui et. al [10] use a multi-objective evolutionary algorithm coupled with analytic flapping wing models to optimize various flight metrics. Ito [11] couples a vortex-lattice model with a hybrid optimization method that combines a genetic algorithm with a sequential quadratic programming algorithm. Strang [34] utilizes a vortex-lattice code in conjunction with the gradient-based optimization to investigate the optimal flapping gait of a Pterosaur wing. Milano and Gharib [25] couple a genetic algorithm to an experimental apparatus with a two degree of freedom flapping rectangular plate to maximize average lift force. Tuncer and Kaya [36] use gradient-based optimization coupled with a 2D overset-grid Navier-Stokes solver to maximize thrust and propulsive e ciency of a pitching and plunging airfoil. Willis et. al. [38] use a multi-fidelity approach to optimize flapping wing performance metrics. In their work, flapping motions are generated using optimal wake vorticity distributions generated by a wake only method, which are then further refined using a panel method and finally verified using a Discontinuous Galerkin-based 3D NavierStokes solver.

In this work we propose to couple a 3D Navier-Stokes solver with a gradient-based optimization in order create a framework for investigating various combinations of flapping motions and optimization objectives. The details of this framework and several optimization results are presented in the remainder of this document.

Chapter 2 Methodology This section presents the details of the tools used to obtain the optimal flapping wing motions and the decisions that led to their design.

2.1 Physical Aspects of the Problem Flapping wing flight in nature occurs over chord Reynolds number ranges between roughly 200, 000 for large birds-of-prey to roughly 100 for the smallest insects, as can be seen in figure 2.2. There is a broad variation in the behavior of the flow across wings across this Reynolds number range. At the upper end, on the order of Re = 200, 000, flow over a fixed wing tends to be fully attached and largely turbulent, with transition occurring around 10% 20% of the chord. These flows tend to be relatively easy to measure experimentally and simulate numerically given suitable boundary layer treatment.

As the chord Reynolds number drops below Re ⇡ 100, 000 laminar transition becomes delayed to extent that laminar separation can occur relatively easily. Flow separation induces a transition to turbulence that can lead to reattachment of the flow, creating a laminar separation bubble. The interaction between laminar separation, turbulence transition and flow reattachment is a highly interdependent and essentially chaotic phenomenon. This Reynolds number range is fundamentally unpredictable. Experimental measurements of quantities such as lift and drag in this

CHAPTER 2. METHODOLOGY 6

Reynolds number range show hysteresis as well as significant sensitivity to parameters such as free-stream turbulence levels and surface roughness on the test model, as can be seen in the experimental airfoil drag polars shown in figure 2.1. Numerical simulations in this Reynolds number range like-wise show strong sensitivity to aspects such as numerical formulations, mesh quality and treatment of turbulence.

Below Re ⇡ 10, 000 the flow is dominated by viscous e↵ects and lacks su cient momentum to transition to turbulence. Flows in this regime are once again relatively easy to measure experimentally and can be e↵ectively simulated using CFD with the added benefit that no turbulence model is required.



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