Aerodynamics of Impulsive Insect Wing Design

8/12/2019 Aerodynamics of Impulsive Insect Wing Design

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Master of Science Thesis

The aerodynamics of an

impulsively-started insect wingAn experimental and numerical investigation

F.J. Venneman

June 26, 2009


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The aerodynamics of an

impulsively-started insect wingAn experimental and numerical investigation

Master of Science Thesis

For obtaining the degree of Master of Science in Applied Physics

at Delft University of Technology

F.J. Venneman

June 26, 2009

Faculty of Aerospace Engineering Faculty of Applied Sciences


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Delft University of Technology

Copyright c Aerospace Engineering & Applied Physics, Delft University of TechnologyAll rights reserved.


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DELFT UNIVERSITY OF TECHNOLOGY

DEPARTMENT OF AERODYNAMICS & APPLIED PHYSICS

The undersigned hereby certify that they have read and recommend to the Faculty of Ap-plied Sciences for acceptance the thesis entitled The aerodynamics of an impulsively-started insect wing by F.J. Venneman in fulfillment of the requirements for the degreeofMaster of Science.

Dated: June 26, 2009

Supervisors:Prof.dr. R.F. Mudde

Dr.eng. L.M. Portela

Dr.ir. M. Tummers

Prof.dr.ir.drs. H. Bijl

Dr.ir. B.W. van Oudheusden


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Preface

In the process of choosing a subject for my graduation project I knew for sure that I wantedsomething that I was really passionate for. I also wanted to do something that I could

basically explain on a birthday party. Many fields of applied physics seemed attractive, soit was a hard choice. Until a morning in the summer of 2008, under the shower, I realized Ihad to study the fluid dynamics of insects or birds. Flight of insects and specifically birdshas astonished me for years and fluid dynamics has always been the part of physics that Ifelt most confident with.

So I went shopping at various research groups (Multi-Scale Physics, the Laboratory forAero & Hydrodynamics and the Aerodynamics group) and ended in the position that theyall were part of my graduation project. At the start it was difficult to get a grip of theAerodynamics vocabulary, but once used to it, you realize all the terms relate to basicphysical transport phenomena. Besides the language, the challenge of graduation for mewas to keep focused on the insects while debugging my corrupt scripts. Luckily once in amonth a bug landed on one of my screens to tell me that he/she was the subject of mygraduation.

I would like to end this preface with some words of gratitude. First of all my girlfriend,Saskia, for understanding when I was in a dip and infinite support. Then I would like tothank, Christian Poelma for providing his piv data, Frank Bos for helping me out whenI got segmentation errors, Bas van Oudheusden for advice and reading my report overand over again, Rob Mudde for coaching me through my graduation project and of courseHester Bijl for her enthusiasm and keeping me on schedule.

Freek Venneman

Delft, June 15, 2009

MSc. Thesis F.J. Venneman


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vi Preface

F.J. Venneman MSc. Thesis


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Abstract

The design of Micro Air Vehicles (mavs) is currently an area of rapid growth. To improvemavs, a better understanding of bird and insect aerodynamics is very helpful. In this

study the time-dependent three-dimensional flow around an impulsively-started fruit-flywing is investigated at a Reynolds number of 256 and a constant angle of attack ( = 50).Both the forces on the wing and the flow-field structure are studied from an experimentaland a computational point of view.

On the experimental part, the work presented in this thesis is a first application ofa new tool to determine fluid dynamic forces in three dimensions. Experimentallyobtained velocity-fields around a dynamically-scaled robotic wing, with the previouslymentioned kinematics, are provided for the purpose of this study. Pressure-fields arededuced from these velocity-fields, by means of a new tool, called the planar Poisson

approach. The forces on the wing are obtained by the momentum approach, usingboth the velocity and the pressure-fields. These forces are in good agreement with theforces that were directly measured by a sensor mounted on the robotic wing. Excellentagreement is found in the steady part of the stroke, where the relative difference with themeasurements is within 13%.

For the computational element of this study, the Navier-Stokes equations aroundthe insect wing are solved using OpenFOAM as a framework. The incompressible flowis solved on a dynamically moving mesh, which deforms based on Radial Basis Functioninterpolation, which is a new mesh deformation tool. The forces obtained from the

computations are also in good agreement with the measurements, in the steady part ofthe stroke, the relative difference with the measurements in the steady part of the strokeis within 22%.



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Table of Contents

Preface v

Abstract vii

List of Figures xiii

List of Tables xv

1 Introduction 1

1.1 Goal and context of present study . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 MAVs and sustainability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Crop inspection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Disaster detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Environmental monitoring . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Brief background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Objectives and approach of present study . . . . . . . . . . . . . . . . . . . . 5

2 Flapping insect wing aerodynamics and problem definition 7

2.1 Physics of flapping insect wing aerodynamics . . . . . . . . . . . . . . . . . . 7

2.2 Experimental investigation of insect flight . . . . . . . . . . . . . . . . . . . . 10

2.3 Numerical investigation of insect flight . . . . . . . . . . . . . . . . . . . . . 11

2.3.1 2D Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.2 3D Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Comparison of experiments and computations. . . . . . . . . . . . . . . . . . 15

2.5 Problem definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16



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x Table of Contents

3 Experimental test case description 17

3.1 Experimental setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Kinematics of the wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3 PIV measurement technique . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4 Flowfield around the wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Determining forces on insect wings experimentally 23

4.1 The Blasius approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2 The momentum approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2.1 Scientific literature about the momentum approach. . . . . . . . . . . 26

4.2.2 Determining pressure from two dimensional cross-sections . . . . . . . 27Numerical scheme and boundary conditions . . . . . . . . . . . . . . . 28

Masking unreliable data regions . . . . . . . . . . . . . . . . . . . . . 29

Transforming velocity to a rotating frame of reference . . . . . . . . . 30

Details Poisson scheme . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2.3 Determining forces from two dimensional pressure-fields . . . . . . . . 31

4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3.1 Results Blasius approach. . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3.2 Results momentum approach . . . . . . . . . . . . . . . . . . . . . . 33

Results momentum approach: Pressure fields . . . . . . . . . . . . . . 33

Results momentum approach: Forces . . . . . . . . . . . . . . . . . . 35

4.4 Conclusion: Good estimation of forces on steady stroke . . . . . . . . . . . . 42

4.4.1 Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.4.2 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 Determining forces on insect wings numerically 45

5.1 Theory for a numerical study of insect wing aerodynamics . . . . . . . . . . . 45

5.2 2D numerical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.2.1 From three-dimensional to two-dimensional motion . . . . . . . . . . . 475.2.2 Building the grid in two dimensions . . . . . . . . . . . . . . . . . . . 48

5.2.3 Solving the system in two dimensions . . . . . . . . . . . . . . . . . . 48

5.2.4 Setup cases in two dimensions . . . . . . . . . . . . . . . . . . . . . . 50

Variation of grid density and Courant number. . . . . . . . . . . . . . 52

Validation impulsively-started wing . . . . . . . . . . . . . . . . . . . 52

Reynolds-Strouhal relation . . . . . . . . . . . . . . . . . . . . . . . . 53

5.2.5 Discussion of the two-dimensional results . . . . . . . . . . . . . . . . 54

Variation of grid density and Courant number. . . . . . . . . . . . . . 54



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Table of Contents xi

Validation impulsively-started wing . . . . . . . . . . . . . . . . . . . 54

Reynolds-Strouhal relation . . . . . . . . . . . . . . . . . . . . . . . . 545.3 3D numerical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.3.1 Building the grid in three dimensions . . . . . . . . . . . . . . . . . . 58

5.3.2 Solving the system in three dimensions . . . . . . . . . . . . . . . . . 60

5.3.3 Setup cases in three dimensions . . . . . . . . . . . . . . . . . . . . . 61

Variation of Courant number . . . . . . . . . . . . . . . . . . . . . . 62

Variation of grid density . . . . . . . . . . . . . . . . . . . . . . . . . 62

Variation of wing planform. . . . . . . . . . . . . . . . . . . . . . . . 62

Variation of the aspect ratio of the wing . . . . . . . . . . . . . . . . 63

Variation of kinematics of the start . . . . . . . . . . . . . . . . . . . 645.3.4 Discussion of the three-dimensional results . . . . . . . . . . . . . . . 64

Variation of Courant number . . . . . . . . . . . . . . . . . . . . . . 66

Variation of grid density . . . . . . . . . . . . . . . . . . . . . . . . . 67

Variation of wing planform. . . . . . . . . . . . . . . . . . . . . . . . 67

Variation of the aspect ratio of the wing . . . . . . . . . . . . . . . . 67

Variation of kinematics of the start . . . . . . . . . . . . . . . . . . . 67

5.3.5 Qualitative images of the flow . . . . . . . . . . . . . . . . . . . . . . 69

5.4 Conclusion: Good estimation of force-coefficients on steady stroke . . . . . . . 72

5.4.1 Two-dimensional simulations . . . . . . . . . . . . . . . . . . . . . . 725.4.2 Three-dimensional simulations . . . . . . . . . . . . . . . . . . . . . . 72

6 Conclusions and recommendations 75

6.1 Detailed conclusions experimental force determination . . . . . . . . . . . . . 76

Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776.2 Detailed conclusions numerical force determination . . . . . . . . . . . . . . . 77

Two-dimensional simulations . . . . . . . . . . . . . . . . . . . . . . 78Three-dimensional grid. . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.3 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Bibliography 80

A Derivations of approaches to determine force on an immersed body 85

A.1 Derivation of the momentum approach . . . . . . . . . . . . . . . . . . . . . 85

A.2 Derivation of the Blasius approach . . . . . . . . . . . . . . . . . . . . . . . 89

A.2.1 Velocity integrals in large bounded regions . . . . . . . . . . . . . . . 90

A.2.2 Force acting on a large body . . . . . . . . . . . . . . . . . . . . . . 92

A.2.3 Aerodynamic Force . . . . . . . . . . . . . . . . . . . . . . . . . . . 92



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xii Table of Contents

B Rotational motion 95

B.1 Transforming velocity to a rotating frame of reference . . . . . . . . . . . . . 95

B.2 Rotational frame of reference . . . . . . . . . . . . . . . . . . . . . . . . . . 97

B.3 Exponential fit for rotational motion . . . . . . . . . . . . . . . . . . . . . . 98

C Moving Grids 101

C.1 Radial Basis Function interpolation . . . . . . . . . . . . . . . . . . . . . . . 101

C.2 Space Conservation Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Mass conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Momentum conservation. . . . . . . . . . . . . . . . . . . . . . . . . 104

D Grid quality 105

D.1 Grid quality measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Aspect Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Non-orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Cell Skewness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

D.2 2D grid quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

D.3 3D grid quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

E OpenFOAM settings 111

E.1 Solution and algorithm control. . . . . . . . . . . . . . . . . . . . . . . . . . 111

E.2 2D OpenFOAM Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

E.3 3D OpenFOAM Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

F Fourier analysis of a force signal 115



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List of Figures

1.1 Three different types of mavs . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Schematic illustration of the leading-edge vortex on fruit-fly wings. . . . . . . 8

2.2 Inertial frame and rotating frame of reference . . . . . . . . . . . . . . . . . . 9

2.3 Smoke visualization of the flow around a tethered hawkmoth. . . . . . . . . . 11

2.4 Time courses of the vertical lift forces over a flapping cycle. . . . . . . . . . . 14

2.5 Comparison of time-history of computed lift with experiment. . . . . . . . . . 14

2.6 Lift and drag coefficients vs. sweeping angle for model wings . . . . . . . . . 15

2.7 Time-histories of model of plate finite elements . . . . . . . . . . . . . . . . . 16

3.1 Experimental setup and important angles.. . . . . . . . . . . . . . . . . . . . 18

3.2 Rotation, angular velocity and angular acceleration . . . . . . . . . . . . . . . 19

3.3 Top view of the experimental setup . . . . . . . . . . . . . . . . . . . . . . . 21

3.4 Isovorticity in six consecutive visualizations . . . . . . . . . . . . . . . . . . . 22

4.1 Control volume determining integral aerodynamic forces. . . . . . . . . . . . . 24

4.2 Forces at the beginning of the stroke of an impulsively-started wing . . . . . . 25

4.3 Force histories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.4 Simplified overview of the grid used to calculate the pressure-field.. . . . . . . 29

4.5 Control volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.6 Results for the Blasius approach . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.7 Timewise cross-sections of the pressure, t= 0.1 0.8 s . . . . . . . . . . . . 364.8 Timewise cross-sections of the pressure, t= 0.9 2.5 s . . . . . . . . . . . . 374.9 Planewise cross-sections of the pressure, 0.38R 0.66R . . . . . . . . . . . . 384.10 Planewise cross-sections of the pressure, 0.70R 0.98R . . . . . . . . . . . . 394.11 Contributions to the force of different terms Navier-Stokes equations . . . . . 40

4.12 Forces for different integration contours . . . . . . . . . . . . . . . . . . . . . 41

4.13 Result of the force determination on the impulsively-started wing. . . . . . . . 42



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xiv List of Figures

5.1 Most important aspects of the two-dimensional grid of46k cells . . . . . . . . 49

5.2 Boundary conditions of the two-dimensional case . . . . . . . . . . . . . . . . 515.3 Transient force and a phase forRe= 192,Comax= 1 and46k cells . . . . . . 52

5.4 CL andCD trajectories in time for 8 chord lengths of travel . . . . . . . . . . 55

5.5 Vorticity plots for the two-dimensional impulsively-started wing. . . . . . . . . 55

5.6 The force-coefficients plotted as a function of angle of attack. . . . . . . . . . 56

5.7 Vorticity plot and phase plot for flow around a wing at= 50 . . . . . . . . 57

5.8 Re St relation for a translating wing at = 50 . . . . . . . . . . . . . . . 585.9 Most important aspects of the grid of280k cells . . . . . . . . . . . . . . . . 59

5.10 Boundary conditions of the three-dimensional case . . . . . . . . . . . . . . . 61

5.11 Force-coefficients of the measurement . . . . . . . . . . . . . . . . . . . . . . 62

5.12 Planforms of the different wings used for the simulations. . . . . . . . . . . . 63

5.13 Rotation, angular velocity and angular acceleration of the alternative kinematics 65

5.14 Comparison of experimental and computational force-coefficients . . . . . . . 68

5.15 Iso-Q-surfaces of the flow around the cad-wing (530k mesh) . . . . . . . . . 70

5.16 Comparison of the pressure-fields . . . . . . . . . . . . . . . . . . . . . . . . 71

A.1 Control volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

A.2 Figure showing what forces act on what parts of the volume. . . . . . . . . . 93

B.1 Transforming velocity to a rotating frame of reference . . . . . . . . . . . . . 96

B.2 Robofly setup and angular velocity . . . . . . . . . . . . . . . . . . . . . . . 97

B.3 Angular velocity and angular velocity of the impulsively-started wing. . . . . . 99

C.1 A rectangular control volume whose size increases in time. . . . . . . . . . . . 104

D.1 Illustration to help explain the aspect ratio and non-orthogonality . . . . . . . 106

D.2 Skewness error on the face. . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

D.3 Figures indicating the quality of the two-dimensional grid of46k cells . . . . . 108

D.4 Histograms showing non-orthogonality and the skewness of 2D cells . . . . . . 109

F.1 Time-history and amplitude spectrum ofCL . . . . . . . . . . . . . . . . . . 115



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List of Tables

5.1 CD for a quiescent wing in a steady flow. Re= 192 . . . . . . . . . . . . . . 54

5.2 CL for a quiescent wing in a steady flow. Re= 192 . . . . . . . . . . . . . . 545.3 Aspect ratios for different wings . . . . . . . . . . . . . . . . . . . . . . . . . 64

A.1 Integration conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

B.1 Coefficients of exponential fit with95% confidence bands. . . . . . . . . . . . 98

C.1 Radial Basis Functions with global support . . . . . . . . . . . . . . . . . . . 102

C.2 Most important parameters for the RBF mesh deformation tool. . . . . . . . . 103

D.1 Indicators of the quality of the two dimensional mesh. . . . . . . . . . . . . . 107

D.2 Indicators of the quality of the 3D meshes at a maximum deformation. . . . . 110

E.1 Linear solvers and their tolerance used in the two-dimensional case . . . . . . 112

E.2 Numerical schemes used in the two-dimensional and three-dimensional case.. . 113

E.3 Linear solvers and their tolerance used in the three-dimensional case . . . . . . 113



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xvi List of Tables



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Chapter 1

Introduction

1.1 Goal and context of present study

The design of Micro Air Vehicles (mavs) is currently an area of rapid growth. The termmavgenerally refers to a new type of remotely controlled aircraft with a target dimensionof approximately 15cm. Development of insect-size aircraft is expected in the near future.Use in hazardous environment and enabling indoor flight are the driving factors of itsdevelopment. Ultimately, mavs should be able to fly independently, with sensors andflight control instrumentation on board to fly around obstacles. Three types of mavs are

under investigation. Airplane-like fixed wing models (Figure1.1(a)), helicopter-like rotarywing models (Figure1.1(b)) and bird- or insect- like ornithopter (flapping wing) models(Figure1.1(c)).

In this study we focus on the insect wing model. The physics of flapping insectwings has been studied extensively, in order to understand biological flight as well as toimprove mav design. Ellington et al. (1996) and Dickinson et al. (1999) showed that ininsect flight, exotic lift generating mechanisms, originating from vorticity generation, aredominating.The Delft University of Technology is an acknowledged player in the development ofmavs. Recently the DelFly (Figure 1.1(c)) and the Roboswift (in corporation withWageningen University) were developed with great success. Nevertheless, the physicalphenomena that produce the forces that keep these mavs aloft are still not understoodthoroughly. Both experiments and numerical simulations have been performed to seewhich parameters influence force production, but only incidentally the results werecompared quantitatively. Therefore the goal of this study is to determine forces ona specific test case (an impulsively-started revolving wing for which an experimentalflowfield database is provided) and compare the results of the experimental and numericalwork in a quantitative manner.



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2 Introduction

(a) Fixed wing model (b) Rotary wing model (c) Flapping insectwing model

Figure 1.1: Three different types of mavs. Sources: (a) AeroVironment Inc. (b) Interactive

Toy Concepts (c) DelFly team

1.2 MAVs and sustainability

In order to assess the sustainable character of mav development, a clear definition ofsustainability is needed. Multiple of these definitions are provided, but the best known isthe one stated by the Brundtland Commission, convened by the United Nations in 1983.It states:

Sustainable development is development that meets the needs of the presentwithout compromising the ability of future generations to meet their ownneeds.

With background knowledge about both mavs and sustainable development, it is possibleto show that the development of mavs supports sustainability. In the next paragraphs acloser look will be taken at the following areas of application of mavs:

Crop inspection

Disaster detection

Wildfire detection

Weather monitoring

Environmental monitoring

Crop inspection

mavs can be very useful for farmers. They can inspect crops and detect insect epidemicsin an early stage. Flying through the trees or the bushes, they can easily detect locustsor harm-causing beetles. As populations of these insects often grow exponentially it is



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1.2 MAVs and sustainability 3

advantageous to get rid of them in an early stage. This can be either by a very smallamount of insecticides or a predator that feeds on the harmful insects. So the mavs canboth minimize the damage as well as the amounts of anti-products. This would certainlyresult in higher yields of the fields. Besides flying above crops outside, they can alsoperform inspection in greenhouses (by using their unique ability to fly slowly or hoverinside buildings).

Disaster detection

mavs are extremely applicable in hazardous environments. This makes them the idealmachines to detect disasters and send an early warning to the authorities who can try to

minimize the results of the disaster. Two examples will be given below: wildfire detectionand weather monitoring.

Wildfire detection mavs can easily survey a forest with an infra-red camera onboard in order to detect heat. Not only by flying over the woods, but especially byflying through the woods (remember that they will be able to fly independently).This would permit it to detect wildfires in a very early stage. Once a fire is de-tected, they can immediately warn the local fire brigade which is able to extinguishthe fire when the fire is still on a small scale. This application is in line with the

Brundtland definition, because future generations can benefit from the forests, bothfor recreational purpose and storage of carbon dioxide.

Weather monitoring In general mavs can easily monitor the weather. Besidessensing and storing information, they can be used as an early warner in areas withhigh chance of sudden change in atmospheric conditions. For instance heavy rains(with risk of flooding) or tornados (with risk of damage on buildings or crops) can bedetected well in advance. Knowing that a sudden atmospheric change is coming up,people can protect there buildings or crops, in order to minimize disastrous effects.A lot of weather monitoring equipment already exists. However mavs will combine

their low costs, maneuverability and light-weight properties to be very competitivein this field of application.

Environmental monitoring

One of the fears is that in future wars, biological and chemical weapons will be used.Employing mavs to track down traces means that contaminated areas can be exploredwithout risking human lives. The minuscule size of the mav would allow it to be transportedquickly to areas to detect the presence of gases and test the safety of the environment forhuman operations.



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4 Introduction

1.3 Brief background

Flapping flight has inspired people for ages. Leonardo da Vinci was the first person whoconsidered to mimic biological wing motion, which led to the well-known Ornithopterdesign. In the 20th century people started developing analytical models as a firstapproximation of the forces that enable flight. It soon became clear that these classicalwing theory models were insufficient for the description of complex, flapping flight, as theydid not account for unsteady effects, so experiments were performed to visualize the flowaround wings. Once computers were able to solve the Navier-Stokes equation in a timeefficient manner, flow around wings could also be simulated numerically. In this section abrief history of the analytical, experimental and numerical work will be presented. For a

more detailed description about the experimental and numerical see sections2.2and2.3,respectively.

Analytical

It has proven to be almost impossible to capture flight performance of insects in ananalytical way. Up to today there exists no satisfactory equation that predicts flightperformance. The first well known attempt to relate forces with parameters like instan-taneous velocity, wing geometry and angle of attack was done by Weis-Fogh and Jensen(1956). In this study the quasi-steady theory was constructed, which greatly simplifiesthe time-dependent problem by converting it to a sequence of independent, steady-state

problems. Therefore it neglects wing motion and flow history and predicts unreliableforces. Sane and Dickinson (2002) tried to improve this theory by including somerotational effects but even then the results remain questionable.

Experimental

To fully understand what is happening in insect flight, many experiments have beenperformed in the past. For example hawkmoths were tethered and smoke was usedto visualize the flow pattern (Srygley and Thomas, 2002). Besides that Particle ImageVelocimetry (piv) was used to visualize the flow around a dynamically-scaled roboticwing that mimics insect flight (Dickinson et al., 1999). Although piv is becoming more

and more accurate, it is still very difficult to capture all the relevant aspects, notablysmall-scale flow phenomena as well as the flow very close to solid surfaces.

Numerical

Since computers become powerful enough to solve the Navier-Stokes equation on gridswith sufficient resolution, researchers have tried to simulate insect wing performance.Wang (2000b) for example investigated simple single translational motion with lowamplitude. Bos(2005) studied the influence of different wing kinematic models (includingrotational effects) of hovering insects. These two investigations were both performed intwo dimensions, in the last decade numerical simulations were also performed in threedimensions, which require a substantially greater computational effort.



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1.4 Objectives and approach of present study 5

1.4 Objectives and approach of present study

As already mentioned in the previous section, researchers have tried to study insect wingaerodynamics in various ways. Only a few have succeeded to compare experimental withnumerical work in a transparent manner. Therefore the objective of the present study isto compare forces deduced from experiments and computations on a specific kinematictest case, the impulsive start.

Experimentally obtained velocity-fields around this wing are provided by Dr. Ir.C. Poelma. He performed time-resolved, stereoscopic piv measurements around adynamically-scaled wing, known as the Robofly (Poelma et al.,2006).

Besides force acquisition from the force sensor mounted on the Robofly-wing, thepresent study applies a new tool to obtain forces, through a combination of the momen-tum approach and an algorithm to obtain pressure-fields from velocity-fields, called theplanar Poisson approach. This method deduces the pressure-fields from the velocity-fieldsby making use of the flow constitutive equations.

For the computational simulations, the kinematics of the Robofly-wing are mim-icked as accurately as possible. The Navier-Stokes equations are solved using OpenFOAMas a framework. The incompressible flow equations are solved on a dynamically deformingmesh. This mesh deforms based on Radial Basis Function interpolation, which is a new

mesh deformation tool developed at the Aerodynamics group of the Delft University ofTechnology.



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6 Introduction



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Chapter 2

Flapping insect wing aerodynamics and

problem definition

The dazzling physics of flow around bird and insect wings has inspired scientists for ages.In this thesis the flow around the wing of a common insect is studied, namely the fruit-fly (or Drosophila melanogaster). In this chapter a small overview of recent insect wingstudies will be provided. But first the basic physics of flapping wing aerodynamics will bediscussed in section2.1. Since we are especially interested in forces on insect wing in thisstudy, section2.2describes different methods to obtain forces from velocity-fields aroundinsect wings. This is followed by an overview of the computational work on insect wingaerodynamics, provided in section2.3. In section2.4some scientific articles are describedin which computations and experimental work are compared. In the last section (2.5) theproblem definition of this study is stated.

2.1 Physics of flapping insect wing aerodynamics

The flow around flapping fruit-fly wings can be considered incompressible since the Machnumber is extremely low;M 0.0015< 0.03 (Bos,2005). In physical transport phenomenais dealt with conservation of especially mass and momentum. The conservation of mass is

stated in the incompressible continuity equation:

u= 0 , (2.1)withuthe flow velocity. The conservation of momentum is represented by the incompress-ible Navier-Stokes equations (neglecting gravity):

u

t + (u ) u= p+ . (2.2)

With t the time, the fluid density, pthe static flow pressure and the stress tensor.The fact that insects are able to fly, highly depends on their ability to create vortices



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8 Flapping insect wing aerodynamics and problem definition

around their wings. The kinematics of insect wings is applied in such a way, that theymake use of these vortices in a very efficient way to create lift and thrust. The leading-edgevortex (or lev) is an excellent example of a lift creating vortex (a schematic illustration isprovided in Figure2.1). This vortex (with its main component in the spanwise direction)stays attached to the top of the wing during an upstroke or a downstroke.

Two dimensionless numbers are important in flapping insect wing aerodynamics. The

Figure 2.1: Schematic illustration of the leading-edge vortex on the wings of a Drosophilamelanogaster (top view). The shaded area represents the area swept by the wing.(Poelma et al., 2006)

first one is the Reynolds number, which can be interpreted as the ratio between the inertialforce and the viscous force. In an unsteady case where a body is excited it is also possibleto describe the Reynolds number as the relation between two time scales, the time-scale todissipate a vortex viscously and the time scale needed for convective transport. For flowaround airfoils the Reynolds number is defined as:

Re=cU

, (2.3)

wherec is the typical chord length,Uthe typical velocity andthe kinematic viscosity. Intwo dimensions the chord length, c, is evident and the velocity Uis the incoming velocity

(for a quiescent wing). In three dimensions the chord length c is the maximum chordlength over the wing and U is the velocity of the wing tip Utip. In scientific literature thetypical chord length and velocity are not consistently defined in this way, sometimes theaveragec and U are taken. Be aware of this in comparing results obtained in this thesiswith other results. In general, insect flight takes place in the intermediate Reynolds regime(10< Re


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2.1 Physics of flapping insect wing aerodynamics 9

where f is the frequency of the flow. In two-dimensional flow this is the vortex sheddingfrequency.The majority of flow problems are defined and solved in a fixed, or inertial, coordinatesystem. However, in this thesis, a case arises where we wish to use non-inertial coordinates,moving with a rotational system. Therefore we must modify the governing equations, inwhich the acceleration term is valid only if the absolute acceleration of the particles isrelative to the inertial coordinates. In Figure2.2the XY Zsystem is fixed in an inertialframe and the xyzsystem rotates relative to it with angular velocity .

Y

Z

X

y

x

z r

P

Figure 2.2: Inertial frame XY Zand rotating frame xyz. In the present study the rotatingframe is also rotating around the y-axis, just as depicted in this figure.

The expression for the absolute acceleration of pointPin Figure2.2is (Greenwood,1988):

2rPt2

=uP

t =

uP,rt

+

trP,r+

rP,r

+ 2 uP,r , (2.5)

where the subindexr refers to the coordinates in the rotational frame and is the angularvelocity. The first term on the right hand side of this equation is the acceleration relative

to the xyz frame (viewed by the rotating observer). The second and third term togetherrepresent the acceleration ofPrelative to the origin as viewed by the non-rotating observer.The second term represents the tangential acceleration and the third term represents thecentripetal acceleration (it points outward from P). The fourth term is known as theCoriolisterm, which is due to the changing direction in space of the velocity ofP relativeto the moving system.Once we substitute the absolute acceleration (2.5) to the Navier-Stokes equations (2.2) weobtain:

u

t + (u ) u= p +

tr +

r + 2 u . (2.6)



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Mind that the velocity u and displacement termr are now relative to the rotating frame

of reference (u

=u

P,r,r

=rP,r). Special notice should be taken of the term

t , which isoften neglected in scientific literature because the angular velocity is constant. Neverthelessit will turn out to be very important in the test case of this thesis. According to theconstitutive equation for an incompressible Newtonian fluid (i.e., =2u), equation(2.6) is equal to:

u

t + (u ) u

= p +2u

tr +

r

+ 2 u

. (2.7)

2.2 Experimental investigation of insect flight

To fully understand what is happening in insect flight, many experiments have been per-formed in the past. For example hawkmoths were tethered and smoke was used to visualizethe flow pattern (Ellington et al.,1996;Srygley and Thomas,2002) (see Figure2.3). Themost important feature considering flapping insect flight found from these studies, was theappearance of a leading-edge vortex (lev). Apart from direct observations of live animalspecies, many experiments have been carried out that simulate (insect) flight with me-chanical means using simplified configurations. Many researchers have tried to mimic thekinematics and planform of insect wings and measure the forces on these wings. In the nexttwo paragraphs, examples of measurements on insect wings in two and three dimensions

are shown.Dickinson and Gotz (1993) studied the aerodynamic forces on a two-dimensionalimpulsively-started wing. That study couples measurements of lift and drag on a two-dimensional flat plate with simultaneous flow visualizations. The Reynolds number, angleof attack, camber and roughness of the wing are varied in an independent manner. Themain conclusion is that the unsteady process of vortex generation at large angles of attackmay contribute to the production of aerodynamic forces in insect flight. Especially inter-esting in the scope of this thesis is that impulsive movement resulted in the production ofa leading-edge vortex that stayed attached to the wing the first 2 chord lengths of travel,resulting in 80% increase in lift compared to the performance measured 5 chord lengths

later.ByDickinson et al.(1999), the aerodynamic forces on a three-dimensionalDrosophilawingare studied. This is done by dynamically-scaled robotic wings, known as the Robofly (seefurther information in3.1. Three interacting mechanisms are distinguished: delayed stall,rotational circulation and wake capture. Delayed stall functions during the translationalportions of the stroke, when the wing sweeps through the air with a large angle of attack.It causes the formation of a leading-edge vortex that reduces pressure over the wing. Ro-tational circulation and wake capture generate aerodynamic forces during stroke reversals,when the wings rapidly rotate and change in direction. Rotational circulation causes liftwhen the insect rotates the angle of attack, increasing the speed on the top side of thewing relative to the bottom, creating a lower pressure on the top. Wake capture refers



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2.3 Numerical investigation of insect flight 11

Figure 2.3: Smoke visualization of the flow around a tethered hawkmoth late in the down-stroke (smoke injection at half the span of the wing, airspeed is 3.7 m s1). The flow

separates at the leading-edge and reattaches to the upper surface in the posterior half of thewing, enclosing a leading-edge vortex (lev) (Ellington et al.,1996).

to the principal that a wing benefits from the shed vorticity of the previous stroke. Theflow generated by one stroke can increase the effective fluid velocity at the start of thenext stroke and thereby increase force productions above that which could be explainedby translation alone.

2.3 Numerical investigation of insect flightSimulation of low Reynolds number flow around airfoil structures has been going on foryears. In this research we are focusing on insect wings. Because of the computational effortrequired for realistic (3D) simulations, most simulations were performed for a simplifiedgeometry in two dimensions (see section2.3.1), however, attempts for simulations in threedimensions were also performed in the last decade (see section2.3.2).

2.3.1 2D Simulation

In the two-dimensional case, flow around a cross-section of a single wing is simulated. Inrelevance to insect flight two different classes of wing simulations may be distinguished: aflapping rigid wing and a flexible non-flapping wing.

Fixed flexible wing

Insect wings have a high aspect ratio (large spanwise length with respect to thesurface area), which are easy to twist and turn. In order to simulate such a wing, it maybe argued that the wing behavior caused by interaction with the airflow (Fluid-StructureInteractions) must be analyzed. In this paragraph a study on flexible membrane wingairfoils is shown, performed by Gordnier (2008). A well-validated, robust Navier-Stokes



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solver is employed coupled with a membrane structural model suitable for the highlynonlinear structural response of the membrane. A low Reynolds number, Re = 2500,consistent with mavflight is chosen for the majority of the calculations. The most notableeffect of the membrane flexibility is the introduction of a mean camber to the membraneairfoil. A close coupling between unsteady vortex shedding and the dynamic structuralresponse is demonstrated. The dynamic motion of the membrane surface is also shown tosignificantly alter the unsteady flow over the membrane airfoil at high angles of attack.The coupling of this dynamic effect and the mean camber results in a delay in stall withenhanced lift and reduced drag for higher angles of attack. Exploratory computationsinvestigating the effects of angle of attack, membrane rigidity, membrane pretension andReynolds number on the membrane airfoil response are also presented.

Rigid flapping wingIn (Bos et al., 2008), the influence of the different wing kinematics models on theaerodynamic performance of a hovering insect is investigated by means of time-dependentNavier-Stokes simulations. With increasing complexity, a harmonic model, a Roboflymodel and two more realistic fruit-fly models are considered, all dynamically scaled atRe = 110. Details of the vortex dynamics, as well as the resulting lift and drag forces,were studied. The simulation results reveal that the fruit-fly wing kinematics result inforces that differ significantly from those resulting from the simplified wing kinematicmodels.In Lentink (2003) a flapping wing is modeled with a sinusoidal plunging airfoil. The

corresponding unsteady, incompressible Navier-Stokes equations are solved with respectto a body-fixed coordinate system. This non-inertial coordinate system introduces a bodyforce. The equations are solved with a validated and verified Navier-Stokes solver forReynolds number 150 and Strouhal number 0.25. Two airfoils with different shapes arecompared. It was found that the sub-critical airfoils outperform the supercritical airfoilsand that aft camber is highly important for good performance in insect flight.Wang(2000a) showed that a two-dimensional hovering motion is able to generate enoughlift to support a typical insect weight. That computation reveals a mechanism of creatinga downward dipole jet of counteracting vortices, which are from leading and trailing-edgevortices. The vortex dynamics further elucidates the role of the phase relation between

the wing translation and rotation in lift generation and explains why the instantaneousforces can reach a periodic state after only a few strokes.In (Wang, 2000b) a computational tool is devised to solve the Navier-Stokes equationsaround a moving wing, which mimics biological locomotion. The focus of the work isfrequency selection in forward flapping flight. Besides that time scales associated with theshedding of the trailing-edge vortex and leading-edge vortex, as well as the correspondingtime-dependent forces are investigated.



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2.3 Numerical investigation of insect flight 13

2.3.2 3D Simulation

Pure insect flight is inherently three-dimensional, so a pertinent question is the pertinenceof two-dimensional simulations and the impact of three-dimensional geometry on theflow effects. This section describes simulations of three-dimensional flapping insect flight.Since a 3D simulation (in contrast to a 2D simulation) also accounts for span wise flow,comparison with measurements is more relevant. The first two articles compare theirfindings with measured forces on a dynamically scaled robotic model (Fry et al., 2003,2005).In Aono et al. (2008) an integrative computational fluid dynamics study of near- andfar-field aerodynamics in insect hovering flight using a biology-inspired, dynamic flightsimulator is presented. This simulator, which has been built to encompass multiple

mechanisms and principles related to insect flight, is capable of flying an insect on thebasis of realistic wing-body morphologies and kinematics. The cfd study integratesnear- and far-field wake dynamics and shows the detailed three-dimensional near- andfar-field vortex flows: a horseshoe-shaped vortex (just as in Poelma et al. (2006)) isgenerated and wraps around the wing in the early down- and upstroke; subsequently,the horseshoe-shaped vortex grows into a doughnut-shaped vortex ring, with an intensejet-stream present in its core, forming the downwash; and eventually, the doughnut-shapedvortex rings of the wing pair break up into two circular vortex rings in the wake.The computed aerodynamic forces show reasonable agreement with experimental results(measured byFry et al.(2005)) in terms of both the mean force (vertical, horizontal and

sideslip forces) and the time course over one stroke cycle (lift and drag forces). A largeamount of lift force (approximately 62% of total lift force generated over a full wingbeatcycle) is generated during the upstroke, most likely due to the presence of intensive andstable, leading-edge vortices and wing tip vortices. See Figure2.4 for an example of theobtained forces and a comparison with measurements.

In Ramamurti and Sandberg (2007), three-dimensional unsteady computations ofthe flow past a fruit-fly under hovering and free flight conditions are computed. A finiteelement flow solver was employed to compute unsteady flow past a fruit-fly body. To carryout computations of the flow about oscillating geometries, the moving surface is coupled to

the volume grid. The volume grid in the proximity of the moving surface is then remeshedevery time step to eliminate badly distorted elements (Ramamurti and Sandberg, 2002).The kinematics of the wings and the body of the fruit-fly are prescribed from experimentalobservations. The computed unsteady lift and thrust forces are validated with experimen-tal results and are in good agreement (see Figure 2.5). The unsteady aerodynamic originof the time-varying yaw moment is identified. The differences in the kinematics betweenthe right and the left wings show that a subtle change in the stroke angle and deviationangle can result in the yaw moment for the turning maneuver. This investigation leads tothe conclusion that it is the forward force and a component of the lift force that combineto produce the turning moment while the side force alone produces the restoring torqueduring the maneuver.



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Figure 2.4: Time courses of the vertical lift forces over a flapping cycle. Blue, red and yellowlines represent the measurements of the upper (Exp-u), average (Exp-a) and lower (Exp-l)values obtained by Fry et al. (2005), respectively; the broken line is the computed result(Com-a). T is the dimensionless period of one flapping cycle. (Aono et al., 2008)

In Luo and Sun (2005) the effects of wing planform (shape and aspect ratio) on the

Figure 2.5: Comparison of time-history of lift (L) forces computed inRamamurti and Sandberg(2007) and experiments described in Fry et al. (2003). Gray andwhite bars indicate downstroke and upstroke, respectively.

aerodynamic force production of model insect wings in impulsively-started sweepingmotion at Reynolds number 200 at angle of attack of 40 are investigated. Wing-shapeand aspect ratio of ten representative insect wings are considered, amongst which thefruit-fly. A time-history of obtained force-coefficients is depicted in Figure2.6, in whichtypical peaks are shown followed by a constant force production for both lift and drag. It isconcluded that the variation in wing shape has only minor effects on the force-coefficients.This also counts for the aspect ratio. Increasing the aspect ratio, on one hand, increases



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2.4 Comparison of experiments and computations 15

the force due to reduction of three-dimensional flows; on the other hand, they willbe decreased due to shedding of part of the leading-edge-vortex. Luo and Sun (2005)conclude that these two effects approximately cancel each other, resulting in only minorchanges of the force-coefficients.

Figure 2.6: Lift and drag coefficients vs. sweeping angle for model wings with various shapes,but the same aspect ratio (= 40 ; Re 200). (Luo and Sun,2005)

2.4 Comparison of experiments and computations

In the previous section, research was described in which numerical modeling of an insectwing was performed. In most cases, the results were validated with experimental work((Fry et al., 2003, 2005), (Dickinson et al., 1999)). Some researchers also combinedexperiment and computation. Two articles describing a comparison of experiments andcomputation are described below.

Wang et al. (2004) compared 2D computational, 3D experimental and quasi-steadyforces in a hovering wing undergoing sinusoidal motion along a horizontal stroke plane forRe 100, unsteady effects of this motion are investigated. In all cases the drag compareswell, but the lift only compares for distinct kinds of rotation.The first conclusion of that investigation is a weak dependence of the stroke angle on theforce-coefficients. Secondly the forces are very sensitive to the phase between the strokeangle and the angle of attack. It was also found that the main difference between a 3Drevolving wing and a 2D translating wing is the absence of vortex shedding by a revolvingwing over a distance much longer than the typical stroke length of insects.Singh and Chopra(2008) validate a model of plate finite elements with 3D measurementson the Robofly-wing. The predicted vertical force was compared throughout one flapping



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cycle (see Figure2.7). As can be seen the two signals do not correspond very well, i.e. thethrust near the end of the downstroke (t

0.4) and upstroke (t

0.9) is overpredicted to

a large extent.

Figure 2.7: Time histories of the vertical force from Robofly experimental data and resultsof the model of plate finite elements. (Singh and Chopra,2008).

2.5 Problem definition

In this chapter the basics of the aerodynamics of insect flight have been shown. A majorinterest in studies on insect flight is to study the effect of kinematic modeling on thetime-histories of the force. Dynamically scaled robotic insect wings were used in orderto experimentally test this. Besides experimental work, a lot of numerical investigationsconcerning (flapping) insect wings have been performed.The Aerodynamics group at the Delft University of Technology has developed two toolsthat are very useful in the analysis of insect wing aerodynamics:

1. A procedure to obtain forces from velocity-field data, intended to be applied underexperimental conditions (see detailed information in section4.2,(Gurka et al.,1999)).

2. A mesh deformation tool based on Radial Basis Function interpolation, to efficientlycalculate flows around a moving body (see detailed information in Appendix C,(Boer et al.,2007))

These tools can be combined to put experimental and computational work on insect wingaerodynamics together in a quantitative comparison. A piv dataset of velocity-fieldsaround a dynamically scaled robotic wing (Robofly) was available from the research ofPoelma et al.(2006) carried out at CalTech Laboratories. The procedure to derive forcesfrom velocity will be applied to this dataset. These forces will then be validated with theresults of a numerical investigation using the same kinematics. A flow solver written withinthe open-source framework ofOpenFOAMwill be used for this purpose.



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Chapter 3

Experimental test case description

In this thesis the forces on insect wings are investigated experimentally and numerically.It was decided to emphasize on a specific motion, the impulsive start, for which an ex-perimental data base was available. These data were obtained from a dynamically-scaledrobotic wing moving in mineral oil. This robotic wing is better known as the Robofly,of which the experimental setup is described in section 3.1. The impulsive motion of theRoboflyis based on the kinematics derived from the real Drosophila Melanogaster (fruit-fly), as described in section3.2. Measurements of the velocity-field around this scaled wingwere performed with stereoscopic piv, this technique is explained in section 3.3 followed

by the resulting flow field, shown in section 3.4. As already mentioned, the original ex-periments were performed by Dr. ir. C. Poelma. For a more thorough explanation of theexperimental setup therefore seePoelma et al. (2006).

3.1 Experimental setup

As allready described in the introduction of this chapter, a dynamically-scaled roboticwing, known as the Robofly, has been used to mimic the wing motion of a fruit-fly. Thiswing was suspended in a large (11.53m3) rectangular tank. Using computer-controlledservo motors, the wing position and motion were accurately controlled with respect to fourdegrees of freedom. For this thesis only the angle of attack () and the rotation angle ()are important (see Figure3.1for the definition of these angles).The wing was cut in the planform shape of a Drosophilawing from a 2.25-mmthick acrylicsheet (see Figure3.1). The maximum chord length (c) of the wing is 10 cm. The distance(L) from the point-of-rotation (o) to the wing tip is 25 cm, but the first 7 cm were takenup by a gear box and a force sensor. This force sensor could measure forces perpendicularand parallel to the wing during the fluid velocity measurements. The mineral oil that wasused, had a density () of 880 kg m3 and a kinematic viscosity () of 115 106 m2 s1.After acceleration the wing moved at a typical angular velocity () of (3/8) rad s1,equivalent to a wing-tip velocity (Utip) of 0.29 m s

1. This leads to a maximum chord



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18 Experimental test case description

Figure 3.1: Picture with the experimental setup and the important angles in this study.Source: Christian Poelma

length based Reynolds number similar to that of a real Drosophila:

Re=cmaxUtip

= 256. (3.1)

3.2 Kinematics of the wing

In Figure3.2 the kinematics of the impulsively-started wing are shown. In Figure3.2(a)the elapsed rotation angle, , is plotted versus time, these data were provided by Dr. Ir.C. Poelma. This is followed by Figure3.2(b),in which the angular velocity versus time isplotted. The blue crosses indicate the results obtained from numerical differentiation of thedata in Figure3.2(a). Since the time resolution of the data is low ( 0.1 s, see for furtherinformation section3.3and Figure3.3), the derivative is not very accurate. Therefore anexponential fit was used for the angular velocity (red lines in 3.2(b)). The exponential fitis of a form prescribed byPoelma et al.(2006, eqn. 1):

(t) = max1 exp t

tc , (3.2)with the maximum angular velocity, max = 1.2 rad s

1 and the characteristic time tc =0.17 s (i.e., the wing reached 63% of its maximum velocity in the first 0 .17 s). The timederivative of this fit was used for the the angular acceleration (see Figure 3.2(c)). Thisangular acceleration was needed for later pressure and force calculation, as described insection4.2. An explanation how the fit was obtained is provided in AppendixB.3.



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3.2 Kinematics of the wing 19

time (s)

(rad)

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

(a) Rotation

time (s)

(rads

1)

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

(b) Angular velocity

time (s)

t

(rads

2)

0 0.5 1 1.5 2 2.5 3-1

0

1

2

3

4

5

6

7

8

(c) Angular acceleration

Figure 3.2: Rotation, angular velocity and angular acceleration of the impulsively-startedwing. Blue crosses indicate the (derivatives of the) data provided by Dr. Ir. C. Poelma, redlines indicate actually used values derived from the angular velocity (that was fitted, seeAppendixB.3).



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3.3 PIV measurement technique

Particle Image Velocimetry (piv) was used to obtain velocity-fields surrounding the roboticwing. pivis an optical method in which the fluid is seeded with tracer particles which aregenerally assumed to follow the flow dynamics. It is the motion of these seeding particlesthat is used to calculate velocity information of the flow being studied. Since this part ofthe thesis is about the post-processing of the velocity-fields, only a very brief overview ofthe measurement technique will be given here. For further information seePoelma et al.(2006). The local fluid velocity was measured by means of stereoscopic piv. Stereoscopicmeans that two cameras are used in order to extract the velocity information in threedimensions. A 25 25 cm2 field-of-view (fov) was recorded using two Imager Intensecameras (1, 376

1, 024 pixel resolution, LaVision).

Silver-coated hollow glass spheres with a mean diameter of 13 m were used as tracermaterial. Due to the high viscosity of the fluid, the particles could accurately follow allmotions of the fluid (their settling velocity was negligible 0.7 m s1).As a light source, an Nd:YAG laser (120 mJ pulse1 at 532nm) was used; the light sheetwas approximately 300 mm high and 2 mm thick. The flow was captured by means ofa phase-averaging approach at various phases and at different spanwise sections of thewing. This implies that the light sheet was kept on the same position and always parallelto the chord of the wing at the time of measurement. Image pairs were recorded with alaser pulse delay time of 5, 000 or 7, 500 s, depending on the flow pattern; the maximumtracer displacement was always kept below 8 pixels; a higher delay time led to an increased

error due to out-of-plane pair loss. For the processing, a three-pass cross-correlation waschosen (one pass at 64 64, two at 32 32 pixels) using a straightforward fft-basedcross-correlation algorithm (DaVis 7.0, LaVision GmbH).In some images reflections of the wing or the gear box were visible, which led to spuriousvectors due to parallax effects. These spurious vectors were very distinct, as they wereoften nearly an order of magnitude larger than the surrounding vectors. The reflectionswere removed by subtracting an averaged image using 32 recordings. Ideally, an imagewith only the reflections and no tracer particles should have been used for this. This wasnearly impossible, since every image also contains the light scattered by the tracer parti-cles. The averaged image contained the reflections, yet also weakened images of the tracer

(approximately 1/32 of their mean intensity). This meant that subtracting the averageimage reduces the image quality somewhat.Spurious vectors were detected in the post-processing step of the final ensemble of averagedvector fields, by means of a local median test; a threshold value of 1.5 times the local stan-dard deviation was chosen, and spurious vectors were replaced using bilinear interpolation.Due to the ensemble averaging, typically less than 2% of the vectors needed to be replacedin the ensemble result. For comparison, if individual image pairs were processed (i.e., noensemble averaging), less than 5% of the vectors needed to be replaced.The velocity-field around the impulsively-started wing was measured in 16 time steps.These time steps are depicted in Figure 3.3. At t = 1.5 s, the wing is shown with 18cross-sections (measurement planes), all 1 cm apart from each other. 2 other measurement



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3.4 Flowfield around the wing 21

Figure 3.3: Top view of the experimental setup. The 16time steps are shown with their times

(in seconds) at the outside of the semi-circle. These time/angle combinations correspondwith Figure 3.2(a). At t = 1.5 s, the wing is shown with 18 cross-sections (measurementplanes). The 2 other measurement planes are taken behind the wing tip.

planes were taken behind the wing tip. The total fovof one measurement plane is 33 24cm in size. There are 81 (Ni) datapoints in the x-direction and 60 (Nj) datapoints in they-direction, so the total field consists of 4860 datapoints in total per fov. The datapointsin the fovare 4.08 mmapart from each other.

3.4 Flowfield around the wing

Although our main focus is on the force development, some information on the flow fieldstructure is presented in this section to illustrate its character and evolution in time. InFigure 3.4, six consecutive panels are shown during the impulsive start. The view pointis from the center of rotation. Each panel shows isosurfaces of spanwise vorticity: blueand red represent clock-wise (z = 15 s1) and counter-clockwise motion (z = 10 s1),respectively. The vector in the first frame indicates the direction of the displacement ofthe wing. The outline of the wing is shown by the black line. Also indicated is a schematic

representation of the position in the impulsive start (red wing profile). In the first frameit is clearly shown that both a leading-edge vortex and a trailing-edge vortex develop. Apart of the trailing-edge vortex is shed in the third and fourth frame. In the stationairypart of the stroke (frame five and six) both a leading and a trailing-edge remain on thesurface of the wing.



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Figure 3.4: Composite of six consecutive visualizations (isovorticity in the spanwise direction)of an impulsively-started wing. The vector in the first frame indicates the direction of thewing. See the text in section3.4for a description and definition of the labels.



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Chapter 4

Determining forces on insect wings

experimentally

The previous chapter described how the velocity-field around the dynamically-scaledrobotic wing is obtained. In this chapter we are interested in obtaining the forces onthe wing. Besides measuring forces directly from force sensors mounted on the wing, it isvery challenging to indirectly obtain forces from the flow field information. A number ofmethods exist to obtain forces from velocity data, such as provided by piv. In scientificliterature, multiple approaches of evaluating forces from piv data are presented. Out of

these three approaches, two are applied in this thesis:

The Blasius approach. (section4.1,for results see section4.3.1) The momentum approach (section4.2, for results see section4.3.2)

The results of the tested approaches are shown in section 4.3. The conclusions of thischapter are stated in section4.4.These methods have in common that the force on an object is calculated using the mo-mentum approach. In this approach a fixed control volume V, boundary S and outwardpointing normal n is considered, enclosing an object as in Figure 4.1.



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24 Determining forces on insect wings experimentally

Figure 4.1: Control volume approach for determining integral aerodynamic forces in a two-dimensional flow configuration. The black body indicates a cross-section of a wing. The

dashed line is the integration path.

4.1 The Blasius approach

According to the Blasius theorem, instantaneous fluid dynamical forces can be related tothe time rate of change of the the first moment of vorticity in an infinite domain. InPoelma et al. (2006), the following straight forward and exact equation, derived by Wu(1981), is used to calculate the instantaneous force (an extensive derivation can be foundin AppendixA.2):

F (t) =

1

2

d

dt +

d

dt VS

udV . (4.1)

The first term in equation (4.1) accounts for the effect of the flow field under considerationon an immersed object. Whereis the first moment of vorticity and described by:

=

S

r dV , (4.2)

with , the vorticity andr the position vector enclosed in volume V.The second term in equation (4.1) accounts for the change in momentum of the displaced

fluid by the object (enclosed in volume VS, object velocity is u) itself. Since the wingis thin and acceleration is negligible for the case under consideration, the second term inequation (4.1) can be neglected in computing the total force.In (Poelma et al.,2006), force measurements, using strain gauges, were compared to forcepredictions extracted from piv data for the first time. The results, see Figure 4.2, cor-respond reasonably well for the initial stages of the impulsively started wing. However,in later stages of the stroke, when vorticity that leaves the field of view is not taken intoaccount anymore, the forces are seriously under predicted. Since we have obtained theraw velocity data fromPoelma et al.(2006), we could check this method. The results areshown in section4.3.1.

Birch and Dickinson(2003) also use the concept of the first moment of vorticity to predict



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4.1 The Blasius approach 25

Figure 4.2: Forces at the beginning of the stroke of an impulsively-started wing (total stroketakes 3 s). Continuous line and crosses represent the horizontal forces (drag), dashed lineand circles represent the vertical forces (lift). (Poelma et al.,2006)

forces. The same experimental setup as inPoelma et al.(2006) was used, but in that casethe forces are only compared relatively (see Figure 4.3, note that the units on the left axisare Nand on the right axis N m1).

Noca et al. (1999) has proposed another formulation that would permit to apply thevorticity concept on a finite domain, including the effect of vorticity convection across the

domain edges. The method proves to be successful for large normalized force coefficients( 2 3). On the other hand, the authors warn for use of large domains or smaller forcecoefficients, which may cause convergence problems. Birch and Dickinson(2003) also triedthis approach, but concluded that it was not robust for their case. Since the measurementsused in this study are obtained from the same experimental setup, this method would notbe an obvious choice.



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Figure 4.3: Predictions of equation (4.1) and measured forces give similar time histories atthe start of stroke one. The x-axis covers the first 16% of stroke one. Blue traces (left-hand y-axes) show measured lift and drag forces. Red circles (right-hand y-axes) plot valuesof sectional lift and drag calculated from Blasius approach calculation. Predictions beyondt = 0.15 were unreliable because starting vorticity moves out of the visualized frame. notethat the left and right axis have different units (Non the left axis and N m1 on the rightaxis), i.e. the total forces are compared with planar forces. (Birch and Dickinson,2003)

4.2 The momentum approachA more genereal approach to determine forces from velocity data is the momentum ap-proach. The scientific literature about this method is shortly described in section 4.2.1.This is followed by an extensive explanation how the momentum approach is used for thisthesis. Section4.2.2 shows how to obtain pressure-fields from velocity-fields and section4.2.3describes how ultimately the force is obtain from these pressure-fields. The results ofthis investigation are presented in section4.3.1.

4.2.1 Scientific literature about the momentum approach

The momentum approach, as discussed by Unal et al. (1997), relates the instantaneousvalue for the force experienced by the object to the flow variables as:

F (t) =

V

u

t dV

i

S

(u n) udS

ii

S

pndS

iii

+

S

ndS

iv

. (4.3)

withVthe control volume,Sits contour andnthe outward pointing normal (like in Figure4.1). Flow field properties are the velocity vector u, the pressure p, the density , and theviscous stress . Therefore, the principal contributions to the force are due to (i) time



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4.2 The momentum approach 27

rate of change of momentum within the control volume, (ii) the net momentum flux acrossthe boundaries of the control surfaces, (iii) the instantaneous pressure force acting on thecontrol surface, and (iv) the instantaneous shear force on the control surface. Assumingincompressible flow, the density is constant. The contribution of the viscous stress can beneglected when the control volume contour is taken sufficiently far away from the body. Adirect application of the control-volume formulation requires the velocity and accelerationdistribution inside the volume, and the pressure on the outer contour. However, determin-ing the pressure from the velocity-field is not a trivial task. First one obtains the pressuregradient (note: not the pressure itself) from the Navier-Stokes equations (2.2):

p=

du

dt + u u

+2u. (4.4)

From this equation it is clear that for incompressible flow, the instantaneous pressure gra-dient can be derived directly from the velocity information, after which the pressure itselfis obtained from spatial integration of the pressure gradient (using a Dirichlet condition,i.e. setting a reference value for the pressure). As can be seen in equation (4.3), only thepressure on the contour is required to obtain the integrated loads. However, the pressuregradient integration may be extended to the entire flow domain of interest, using eithersome sort of gradient-integration scheme or through solving the Poisson equation for thepressure (Gurka et al.(1999),Fujisawa et al.(2005),Oudheusden et al.(2007),Kat et al.(2008)).

4.2.2 Determining pressure from two dimensional cross-sections

In this thesis the pressure is obtained by calculating the pressure gradient from the Navier-Stokes equations for incompressible flow in a rotating frame of reference (repeating equation(2.6)):

u

t + (u ) u

= p+2u

tr +

r

+ 2 u

. (4.5)

In two dimensions the in-plane pressure gradient components look like this (see Kat et al.(2008) for the non-rotational terms and AppendixB.2for the rotational terms):

p

x =

u

t + u

u

x+ v

u

y+ w

u

z

+

2u

x2 +

2u

y2 +

2u

z2

tz 2x + 2w

, (4.6a)

p

y =

v

t + u

v

x+ v

v

y+ w

v

z

+

2v

x2 +

2v

y2+

2v

z2

. (4.6b)

These equations properly describe the pressure gradient in planar flow. There are norotational terms for the y-component of the pressure gradient, because the wing is ro-tating around this axis. Without rotational terms, these equations have been used byBaur and Kongeter(1999). In this study we used the planar Poisson approach, first stated



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Figure 4.4: Simplified overview of the grid used to calculate the pressure-field. The diagonalline represents the cross-section of the wing. As discussed in section 3.3there are 81 (Ni)datapoints in the x-direction and 60 (Nj) datapoints in the y-direction, so the total fieldconsists of4860 datapoints in total. The distance between two datapoints, h is 4.08 mm.

The Neumann conditions are directly obtained from the pressure gradients in (4.6). We setthe reference pressure (Dirichlet condition) in the bottom left to the pressure predicted bythe irrotational Bernoulli equation. This region was chosen since it complies best with the

Bernoulli restrictions (least time dependency, least vorticity). The value for the Dirichletpoint is therefore chosen as:

p= C 12

u2 +v2

, where C= 0 . (4.12)

Since there is only one point set as a Dirichlet condition, this point acts only as a referencepressure for the specific velocity-field at consideration, so there is no need to set it exactlyright.

Masking unreliable data regions

As the red datapoints in Figure4.4indicate, some regions of the grid are masked. Maskinga region makes sure that the velocity components of this region are not taken into accountfor the determination of the pressure-field. There are two specific regions chosen to bemasked:

1. The region immediately around the wingThe region immediately around the wing was masked because we are not sure of thereliability of the vectors in that region. This is related to the way the velocity wasdetermined there. The wing itself is only 2.25mmthick, but during cross correlation,64

64 interrogation windows where used (equal to approximately 1.5

1.5 cm),



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which definitely would overlap the wing. So for one interrogation window velocitycomponents from both sides of the wing where taken into account, which wouldcertainly lead to unreliable vectors. Therefore it was chosen to mask the vectorsless than 1.2 cm (about three vector distances) around the wing. This is a trade-offbetween not losing too much information and using reliable information.

2. The boundary of the fovSince the fovwas tilted a little in the yz-plane (Poelma et al., 2006, Fig. 5), somevectors at the boundary of the fovwhere not to be trusted. As well as for the regionaround the wing, a band of 1.2 cmwas masked.

Transforming velocity to a rotating frame of referenceThe cameras did not rotate with the wing, which means that the measured velocity com-ponents are measured with respect to a static reference frame. Nevertheless we do userotating terms in equation4.5,which is based on the definition of the coordinates systemattached to the wing. Therefore the velocity had to be transformed to a rotating frameof reference. This is very simple with the knowledge of (t) (see section3.2). The trans-formed velocities are obtained with the following relations for respectively the x, y and zcomponent of the velocity (see the derivation in AppendixB.1):

ur =u (t)zr, (4.13a)vr =v, (4.13b)

wr =w (t)xr. (4.13c)

Where symbols with subscript r are in the rotating frame of reference. The velocity in they-direction does not change since the wing is rotating around this axis.

Details Poisson scheme

In order to calculate the derivatives in equation 4.5the second order three-point rule wasused. At the boundary and next to the masked region a simple first order two point rule

was used.To speed up the process of convergence a first guess for the pressure-field was made usingthe Marching scheme. This scheme uses the local gradient to determine the pressure-field (by integrating the gradient over space). The scheme marches through the fieldcalculating the pressure at the points with the most neighbors. More about this can beread in (Turella,2008, page 24-25).After obtaining the first guess the Poisson scheme could start. For this iterative processa convergence criterion is needed. This convergence criterion requires a definition of theerror:

= 1

NiNj i=1,Ni j=1,Nj|pni,j pn1i,j | , (4.14)



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whereNiis the number of datapoints in the x-direction (81),Njis the number of datapointsin the y-direction (60) and n is the iteration parameter. The convergence criterion for was set to = 1 106. Fortan 90 was used to calculate the planar pressure-fields,each measurement plane would approximately take 10 seconds on a Pentium 4, 3.4 GHzprocessor. To calculate the pressure-field for all 20 planes and 16 time steps would thereforetake about one hour of computation time.A concern for the pressure calculation is the possible lack of resolution. The leading-edgevortex is only covered within 5 datapoints in width. This number is too low to get a goodreconstruction of the pressure drop. More important, it has a significant influence on thefar field as shown by preliminary calculations. The fact that lack of resolution in one placeinfluences the whole reconstructed pressure-field is is a true disadvantage of the Poissonscheme.

4.2.3 Determining forces from two dimensional pressure-fields

Having obtained the pressure-field from the velocity-field, it is fairly straight forward to

compute the force. As described in section 2.2, the momentum approach is used. Thederivation of this approach is in Appendix A.1,but a brief explanation will be presentedhere.In this thesis the momentum approach is based on the incompressible Navier Stokes equa-tions including the Coriolis and centripetal acceleration (2.6). Using a momentum approachthe force on the wing is written as the pressure and viscous force on the wing (see Figure4.5and equation (A.8)).

F(t) = EFGpndS+ EFG ndS

=

ABCI

(u n) u pn + ndS+

+

V

ut

r

2 u

trdV . (4.15)

Where V is the volume of the control volume with boundary S. The indices of theintegration path are depicted in Figure4.5. After some extensive rewriting (see Appendix



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Figure 4.5: Control volume. The black body indicates a cross-section of the wing. Thedashed line is the integration path

A.1), the planar lift and drag are calculated as follows:

F1(t) =

ABCI

u2dy

ABCI

uvdx

ABCI

2u

xdy+

+ ABCIu

y+

v

x dx+ ABCIpdy+ +

V

ut

+ 2x 2w t

xdV , (4.16a)

F2(t) =

ABCI

vudy

ABCI

v2dx+

ABCI

2v

ydx+

+

ABCI

u

y+

v

x

dy

ABCI

pdx+

+

V

vt

dV . (4.16b)

Where F1 is the drag force and F2 is the lift force. The resulting forces have units N m1,

because volumeV(see Figure4.5) is a two-dimensional volume. Calculating the force overeach cross-section and multiplying by their distance (1 cm, as described in section 3.3)gives the force with unit N.The major advantage of the momentum approach is that one can easily distinguish thecontribution of every physical aspect of the flow, this will be presented in section 4.3.2.It turned out that integrating close to the wing (like in Figure 4.5), resulted in morereliable results than taking the whole fovas a contour. This may be the result of lackresolution (the leading-edge vortex was typically only captured within 5 datapoints). Thislack probably even has a more significant influence on the far field than on the field near



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4.3 Results 33

the wing itself, since the Poisson algorithm propagates errors throughout the whole domain.The fact that relevant flow was masked, could also be a cause of unrealistic pressure-fields.The integration contour for the force determinations was taken 32 mm from the wing. Toget a measure of the uncertainty, the force on the wing was also determined for contoursin the range of 20 53mm, which will result in a range of possible forces that will be usedas an indication for the uncertainty of the deduced force.

4.3 Results

In this section, the results of the post-processing of the experimental data are shown.Section4.3.1shows the results for the Blasius approach, followed by section4.3.2in which

the results for the momentum approach are shown.

4.3.1 Results Blasius approach

The reproduced results of the force calculations using the Blasius approach are shown inFigure 4.6(a). Poelmas results are also shown in Figure 4.6(b) to see that both figuresare in good

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Aerodynamics of Impulsive Insect Wing Design

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