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The Effect of Dynamic Span Wise Bending on the Forces of a The Effect of Dynamic Span Wise Bending on the Forces of a

Pitching Flat Plate Pitching Flat Plate

Manoj Prabakar Sargunaraj University of Central Florida

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STARS Citation STARS Citation Sargunaraj, Manoj Prabakar, "The Effect of Dynamic Span Wise Bending on the Forces of a Pitching Flat Plate" (2021). Electronic Theses and Dissertations, 2020-. 758. https://stars.library.ucf.edu/etd2020/758

THE EFFECT OF DYNAMIC SPANWISE BENDING ON THE FORCES

OF A PITCHING FLAT PLATE

by

MANOJ PRABAKAR SARGUNARAJ

B.S. Aero-Mechanical Engineering, Aeronautical Society of India, 2014

MS (by research) Aerospace Engineering, Indian Institute of Technology, Madras. 2018

A thesis submitted in partial fulfillment of the requirements

for the degree of Master of Science in the Department of Mechanical and Aerospace Engineering

in the College of Engineering and Computer Science at the University of Central Florida

Orlando, Florida

Summer Term

2021

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© 2021 MANOJ PRABAKAR SARGUNARAJ

iii

ABSTRACT

A novel experiment has been conducted to investigate the effect of dynamic spanwise bending

on the formation of a dynamic stall vortex on a rapidly pitching flat plate. Experiments have

been performed in a towing tank at different nondimensional pitch rates (Kp) in the range of

0.4112 < Kp < 0.8225, and four maximum pitch angles (30◦, 45◦, 60◦and 90◦) at a Reynolds

number of 12,000. Synchronized direct force measurements and particle image velocimetry

(PIV) are used to characterize the effect of bending on the unsteady forces and the flow field.

An unsteady analytical model based on the bending and pitching kinematics is used to model

the lift force histories. It is found that a spanwise bending of a pitching wing alleviated the

unsteady lift forces. However, the main contribution was found to come from the non-

circulatory forces. The circulation and pressure analysis of the pitching wing revealed little or

no sensitivity to the wing bending motions examined.

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Dedicated to the memory of my stillborn baby, Aadhira

v

ACKNOWLEDGMENTS

I express my sincere thanks to the thesis Advisor, Dr. Samik Bhattacharya, and thesis

committee members, Prof. Jayanta Kapat and Dr. Erik Fernandez for taking the time to review

my progress and their useful suggestions. I would also like to give my sincere thanks to Prof.

Gou, the Program Coordinator, and Prof. Peles, the Chair of the Department of Mechanical and

Aerospace Engineering, and all the faculty members for their timely help. Also, I would like

to thank Dr.Mansy and Dr. Sagnik for allowing me to work as a Graduate teaching assistant.

I am thankful to all the office and technical personnel of the Mechanical and Aerospace

Engineering Department, especially Gabriela, Amanda, Lynn, Patty, Kamryn for their efforts

and help whenever approached, without exception. I am grateful to UCF Graduate office

personnel in the academic section and the thesis editors for their guidance and help with all the

thesis work.

I am immensely grateful to all the scholars and students whose inputs and suggestions helped

me on various levels. I am thankful for the help provided by Dibya raj Adhikari, Carlos E Soto,

Kamlesh, Taha while making the experimental setup. Also, I thank my friends whom, I have

unintentionally forgotten here, but who surely deserve as many thanks as the individuals I have

mentioned. I hope they will accept my apologies. for their love, patience, and understanding.

I am immensely grateful to my parents for their unfailing support through, before and beyond,

the entire course of my MS and life. Without all the love and effort from all above said people,

this work would not have been possible. Above all, I would like to thank my wife Nandhini

Raju for her love, encouragement. and Profound understanding, without her support this thesis

could not have been completed.

vi

TABLE OF CONTENTS

LIST OF FIGURES .................................................................................................................. ix

LIST OF TABLES ................................................................................................................... xii

NOMENCLATURE ................................................................................................................ xii

Greek Symbols ................................................................................................................ xiv

Subscripts and Superscripts, ........................................................................................... xiv

CHAPTER 1 INTRODUCTION ......................................................................................... 1

1.1 Background ................................................................................................................ 1

1.2 Literature Review of a Morphing Wing Aerodynamics ............................................ 3

1.2.1 Chordwise-flexible wing structures ....................................................................... 5

1.3 Spanwise-Flexible Structures..................................................................................... 8

1.4 Motivation and Objectives of The Present Study .................................................... 11

1.5 Outline of the Thesis ................................................................................................ 12

CHAPTER 2 EXPERIMENTAL SETUP .......................................................................... 14

2.1 Experimental Facility ............................................................................................... 14

2.2 Facility ..................................................................................................................... 14

2.2.1 Driving mechanism .............................................................................................. 15

vii

2.2.2 Wing Construction ............................................................................................... 16

2.2.3 Force measurement .............................................................................................. 17

2.2.4 PIV measurements ............................................................................................... 18

2.2.5 Wing profile measurements ................................................................................. 19

2.3 Experimental Procedure ........................................................................................... 20

2.4 Flow Field Analysis Vortex Dynamics .................................................................... 22

CHAPTER 3 ONE DIMENSIONAL UN STEADY THEDORSEN MODEL

DEVELOPMENT .................................................................................................................... 23

3.1 Theodorsen's Unsteady Classical Theory ................................................................ 23

3.1.1 Assumptions for classical unsteady flow theory .................................................. 23

3.1.2 Problem statement ................................................................................................ 23

3.1.3 Non-circulatory terms .......................................................................................... 25

3.1.4 Circulatory terms ................................................................................................. 28

3.2 Current Analytical Procedure .................................................................................. 30

CHAPTER 4 RESULTS AND DISCUSSION .................................................................. 33

4.1 An Unsteady pitching flat plates, ............................................................................. 33

4.2 Force Measurements ................................................................................................ 34

viii

4.3 PIV Measurements ................................................................................................... 36

4.4 Total Circulation Temporal Evolution ..................................................................... 39

4.5 Discussion And Outlook .......................................................................................... 41

CHAPTER 5 CONCLUSION ............................................................................................ 45

APPENDIX: INSTURUMENTATIONS ................................................................................ 46

REFERENCES ........................................................................................................................ 48

ix

LIST OF FIGURES

Figure 1 (A) Falcon Bird Wing Morphing While Preying (B) Major Forces And Moments

Acting On A Bird ............................................................................................................... 2

Figure 2 (A)Vortices Signature Of The Starling’s Bird Wake With Different Beat Frequencies

(A To D) (Ben-Gida Et Al., 2013) (B) Difference In Induced Flow Between Flapping And

Gliding Flights Of The Swifts Bird (Henningsson, Hedenström And Bomphrey, 2014)

(Reproduced Under Creative Commons Attribution License ) ........................................ 4

Figure 3 (A) Chord-Wise Wing Morphing (B) Spanwise Wing Morphing. ............................. 5

Figure 4 (A) Overall Experimental Setup That Includes The Motion Mechanisms And Dpiv

System (B) Details Of The Mechanical Model................................................................ 14

Figure 5 Experimental Setup ................................................................................................... 15

Figure 6 Schematic Of The Spanwise Flexible Wing Attachment .......................................... 16

Figure 7 Sample Force Results Of 5 Run Average Input Data And The Filtered Output Data Of

A Pitching And Pitching-Bending Case (0-90 Deg-0 Deg). ............................................ 17

Figure 8 Experimental Setup With Laser Stand ...................................................................... 18

Figure 9, A, Wing Schematic, 2 B, Bending Part (8 Black Dots From The Bottom Of Bending

Parts, A Tip Is Considered As 9th Point) At Time = 0 Sec (The Coordinates Of The Nine

Points (Pt2, To Pt9) Were Tracked At Different Instants Of Time To Obtain Their

Trajectories. ..................................................................................................................... 19

Figure 10, Bending Profile With Forward Bending Rate 1 Sec (Dimensions Centimeters) ... 20

Figure 11 Experimental Setup Showing The Flow Field Illuminated By Two Lasers ............ 22

Figure 12 Airfoil In Real Flow (Considering All Unsteady Motion) ...................................... 23

x

Figure 14 Conformal Mapping (Non-Circulatory Terms) Of Cylindrical Co-Ordinates Along

With Source And Sink Distribution ................................................................................ 26

Figure 15 Small Deflections (ℇ) In Cylindrical Co-Ordinate .................................................. 26

Figure 16 Vortex Placed Away From Cylinder (Circulatory Terms Considered) ................... 28

Figure 17 Wing Pitch Motion And Wing Bend During One Cycle (0- 2 Seconds) ................ 29

Figure 18 Pitch-Bent Wing Schematic And Nomenclature ..................................................... 30

Figure 19 Effect Of Angle Of Attack On A Lift And Drag Force Of A Pitching Wing. ........ 33

Figure 20 Comparison Of Lift Force And Drag Force Of The Pure Pitching Case (Black)

Compared Against A Pitch-Bend Case (Grey). Green Shading Shows Where The Pitching

Wing Force Is Higher Than The Pitch-Bent Case, While Yellow Shows When The Pitch-

Bent Case Is More Than The Pitch Case. ........................................................................ 35

Figure 21. Snapshots Results Of The Spanwise Vorticity Fields Related To Several Reduced

Pitch Rate F∗ At Re = 12000 The Solid Black Rectangle Denotes The Flat Plate Cross

Section (A) F∗ = 0.4112, F = 0.5 Hz, Αmax = 30 O (B) F∗ = 0.6119, F = 0.5 Hz Αmax = 45

O (C) F∗ = 0.8225, F = 0.5 Hz Αmax= 60 O The Vorticity Is Scaled By C/U∞, Clockwise

(Red) And Negative, Anticlockwise (Blue) ..................................................................... 37

Figure 22 Comparison Of The Pressure Distribution F∗ = 0.8225 At Re = 12000 The Solid

Black Rectangle Denotes The Flat Plate Cross-Section (A) 50% Span,

αmax= 60 O (B) 70% Span, αmax= 60 O (C) 86% Span, αmax= 60 O . ................................. 38

Figure 23 Historical Development Of The Total Circulation For Various Motion Profiles. .. 39

Figure 24 (Color Online) (A–C) Snapshots Results Of The Spanwise Vorticity Fields Related

To Several Reduced Pitch Rate F∗ = 0.8225 At Re = 12000 The Solid Black Rectangle

Denotes The Flat Plate Cross-Section (A) 50% Span, αmax= 60o (B) 70% Span, αmax=

xi

60 O (C) 86% Span, αmax= 60 O. The Vorticity Is Scaled By C/U∞, Clockwise (Red) And

Negative, Anticlockwise (Blue) ....................................................................................... 41

Figure 25 Time History Of Non-Circulatory Mass Forces ...................................................... 43

Figure 26 Model Instrumentation ............................................................................................ 47

xii

LIST OF TABLES

Table 1 Summary of the Parameter Space ............................................................................................................ 21

xiii

NOMENCLATURE

c = chord

CA = force coefficient in the y-direction

CL = Lift coefficient

Cp = pressure coefficient (P- Po/ 2𝜌𝜌 U∞2)

CN = force coefficient in the x-direction

f = Airfoil oscillation pitch–bend frequency

FN = Normal Force acting on the wing

FA = Axial Force acting on the wing

h = bend position as a function of time

K = Reduced frequency(𝞹𝞹fc/U)

Kp = Reduced pitch rate (K αmax/2)

P = static pressure of the fluid

P0 = Total pressure of the fluid

Re = Reynolds number

t* = Convective time (tU∞ /c)

xiv

U∞ = free-stream velocity

Ub = bending velocity, (dh/dt)

Ueff = effective velocity

xp = pitch pivot location

Greek Symbols

Α = geometric angle of attack, degrees

αeff = Effective angle of attack, α +tan−1 (Ub/U∞)

αmax = Maximum angle of attack

ωz = spanwise vorticity

θbend = bending angle.

α = Pitch rate, radians per second

𝜌𝜌 = Density of the fluid

Subscripts and Superscripts,

L = Lift Force

N = Normal Force

1

CHAPTER 1 INTRODUCTION

1.1 Background

The observation of bird flight encouraged the human desire to fly, eventually evolving an

aircraft. In recent times a study of bioinspired wings attracted special attention among the

aerodynamics research community due to its potential application ranging from

Micro Air Vehicles (MAV), Unmanned Aerial Vehicles (UAV), or small manmade aircrafts

mimicking the birds. Until now, Flapping wing manmade commercial aircraft are sparse in the

market due to their design difficulties and lack of understanding of morphing wing

aerodynamics (Sofla et al., 2010)(Li et al., 2018a). So, it is necessary to understand the flow

physics around the bird flight which provides access to low-cost unconventional technologies

(Pennycuick, 1968; G R Spedding, 2003; G. R. Spedding, 2003). The common motivation of

all earlier research to morph the wing is to increase aerodynamics performance fuel efficiency,

maneuverability, and to add stability to an aircraft (Bergou et al., 2015; Hedenstrom and

Johansson, 2015). The pitching wing and flapping wings are used in lab conditions to simulate

the flow dynamics around the birds. But only the pitching or flapping-wing without morphing

does not resemble the complete ability of the bird's wing. For example, birds could fly through

turbulent flow effectively by the dynamic motion of the wing. Similarly, birds can morph their

wing according to their various need such as prey chasing as shown in Figure 1a. It is important

to find the effects of wing morphing motion on major forces acting on the bird as shown in

Figure 1b. It helps the aerodynamics community to harness the effect of morphing wings into

real aircraft. Many investigations have been undertaken to find bird flow dynamics using the

rigid wing pitching and flapping wing motion. Still, the effect of spanwise dynamic morphing

in a pitching wing motion has persisted mysterious despite some effort done using spanwise

2

flexible wings(Liu and Bose, 1997; Heathcote, Wang, and Gursul, 2008; Yang et al., 2012;

Wong and Rival, 2017; Tyler Scofield, Kun Jia, 2019).

Figure 1 (a) Falcon bird wing morphing while preying (b) major forces and moments

acting on a bird

The pitching motion of the wing is investigated for many decades(Corke and Thomas,

2015)(Adhikari et al., 2020). Still, the vortex dynamics of the pitching wings are more popular

in unsteady aerodynamics because of their complexity. It is found that during pitching the tip

effects are confined initially near the wingtips but begin to strongly affect the leading-edge

vortex (LEV) as the motion of the plate proceeds(Jantzen et al., 2014). Especially, the LEV of

a stalled wing is significant for rapidly maneuvering flyers as the bound vortex is of negligible

strength(Pitt Ford and Babinsky, 2013). Also, earlier researchers have stated that the presence

of spanwise flow from the root to the tip on the suction side of the flying insect’s wing LEV

stability. It is confirmed that vortex development near the tip region shows dissimilarities

x

Drag

Side force

Lift

roll

Weight

Pitch

Yaw

y

z

Centre of gravity

Aerodynamic Centre

(b)(a)

3

between the flat-rigid and deformable bent plates of low aspect ratio. Wing deformation

resulted in a reduced drag coefficient in deformable bent plates compared to the rigid flat

plate(Gharib, 2011). A recent research investigation also identified that bird uses bending of

the wing which delays the breakdown of leading-edge vortex near the wingtip, responsible for

augmenting the aerodynamic force-production (Nakata and Liu, 2012).

Even though many studies used animal locomotion to extract the effects of a natural flexible

wing on fluid dynamics, not many conceptual research findings are available explaining the

effect of dynamics bending contributes to unsteady aerodynamics and energetics in pitching

wing. The main concern of this study is about vortex creation, transportation, and their

organization with the dynamics of the morphing wing; the interplay between the deformable

wing variables (bending ratio, bending rate) and vortex formation on the lift and drag created

on the wing along its length.

It is learned from the literature that factors affecting the dynamic stall vortex stability of an

unsteady pitching wing are the Reynolds number (Re), mean angle of attack, the reduced Pitch

rate (Kp). Also, the steady and unsteady aerodynamic forces depend on these salient factors

mentioned later. at Re ≈ 12000, angle of incidence θ = 30,45,60,90o and reduced pitch rate in

the range 0.4112 < Kp < 0.8225. In a steady cursing aircraft, slow changes may be enough to

vary the performance of the wing, fast changes may be required to control rapidly maneuvering

aircraft. It is also important to study the rate at which the bending (Tbend) of the wing occurs.

1.2 Literature Review of a Morphing Wing Aerodynamics

Despite many advances in the field of aerodynamics, there are still many abilities of the bird’s

wing that are not adapted into real flight such as morphing of a wing. It is also hypothesized

4

that wing morphing helps the birds to improve their aerodynamics performance and flight

stability and control. Many researchers have conducted some excellent experiments on the real

bird to understand the bird flight. Hadar Ben-Gida et al(Ben-Gida et al., 2013) conducted wind

tunnel experiments to study unsteady flow features around the free-flying starling. Particle

image velocimetry is used to study the wake evolution of flapping wings. Henningsson et

al(Henningsson, Hedenström, and Bomphrey, 2014) conducted series of experiments to study

the similarities between the gliding flight (straight wing) and flapping wing (morphing wing)

of a swift bird as shown in Figure 3. It is found that the flapping wing has the highest span

efficiency and lift production compared to the gliding flight. It is hypothesized that the

morphing wing is more efficient while generating lift during flapping flight.

Figure 2 (A)Vortices signature of the starling’s bird wake with different beat frequencies

(a to d) (Ben-Gida et al., 2013) (B) Difference in induced flow between flapping and gliding

flights of the swifts bird (Henningsson, Hedenström and Bomphrey, 2014) (Reproduced

under Creative Commons Attribution License )

Similarly, numerous research works have been conducted to adapt Bio-inspired shape

morphing technology in the field of aeronautics. The morphing technology is mainly studied

5

through three main approaches in the research labs such as structural, material, multi-

disciplinary approach, etc., But in this section, a brief review is given to the literature related

to structural morphing of chordwise and spanwise morphing wing concepts as shown in Figure

3. Therefore, the morphing concepts other than the chordwise and spanwise morphing such as

chord length change, wingspan resizing, span sweeping angle are not covered in this literature

review. Many excellent review papers of the morphing concepts can be referred to, to learn

more about morphing wings (Sofla et al., 2010; Barbarino et al., 2011; Li et al., 2018b, 2018a).

Figure 3 (a) Chord-wise Wing morphing (b) Spanwise Wing morphing.

1.2.1 Chordwise-flexible wing structures

Zhu et al numerical investigated a flapping foil with chordwise and span-wise flexibility. The

foil ins were investigated with two fluids medium air and water. It is found that the chord-wise

flexibility increases the thrust and propulsive efficiency in the first and second scenarios. But

6

the span-wise flexibility increases the performance in the first scenario not in the second

scenario.

Eldredge et al(Eldredge, Toomey and Medina, 2010) conducted a numerical simulation of a

chord-wise flexible wing with a wide variety of stiffness and motion parameters. The benefits

of the chord-wise flexible are seen in the power requirements and the aerodynamic performance

improvement for the specific maneuvers compared to the rigid wings. It is found that the

interaction of flexible wings with the leading-edge vortex formation plays a major role in the

observed difference between rigid and flexible wings.

Bluman et al(Bluman, Sridhar and Kang, 2018) investigated the effect of chordwise wing

flexibility of an insect wing. For the investigation, two-dimensional Navier stokes equations

are used with the tight fluid-structure interaction scheme. It is found that the chord-wise flexible

wing gives more stability to a passively flexible wing. It is hypothesized that insect is using

their passive flexibility of the wing to stabilize the flight.

Kang et al(Kang and Shyy, 2013) reported a scaling law to find the insect wing chord-wise

flexibility on the aerodynamics of the flight. For the investigation, Navier stokes equations

coupled with the structural dynamics are used to investigate the insect wing in the hovering

motion. It is observed that a chordwise flexible wing enhances a lift force by streamlining its

shape to mitigate the degrading wing wake interaction that happens using the rigid wing.

Cheng et al(Cheng and Lan, 2015) numerically simulated the chord-wise flexible wing to study

the aerodynamics effects. It is observed that the chord-wise flexible wings generally reducing

the lift performance of the wing except for few cases such as delayed pitch rotation. Also, the

wing with smaller flexibility gives marginally higher efficiency compared to its counterpart.

7

The aerodynamics performance of the flexible wing is not sensitive to phase difference and

stroke angle of attack.

Suzuki et al(Suzuki, Aoki and Yoshino, 2019) numerically investigated a butterfly model to

study the effect of chord-wise flexibility using an immersed boundary-lattice Boltzmann

method. They found that the lift forces and thrust forces of a wing increased with the range of

spring stiffness. The reason for the improvement is observed to be flexible producing lift and

thrust force during the stroke. Also, the simulation of the model wing goes off balance with the

pitching motion.

Beals et al(Beals and Jones, 2015) experimented with the rigid rectangle wing and chord-wise

flexible wing of aspect ratio 2 undergoing a rotational motion. Force measurements and dye

flow visualization are done to understand the flow physics of a rotating wing. Also, the lift

produced by both the wings compared with the thin airfoil theory lift. It is shown that the wing's

passive flexibility played a major role in lift production at a high angle of attack by increasing

the camber section of the wing. It is suggested that the wing chordwise flexibility will be

desirable in an unsteady environment.

Dai et al(Dai, Luo and Doyle, 2012) Numerically simulated an elastic wing using the viscous

incompressible solver-based immersed boundary method. The results show that passive and

rate of flexibility majorly affect the aerodynamics of the wing. It is also shown that when the

ratio between flapping frequency and the natural frequency of the wing ≤ is 0.3, the wing

flexibility enhances the lift and aerodynamic efficiency of the wing.

8

1.3 Spanwise-Flexible Structures

Combes et al(Combes and Daniel, 2003) experimental investigated Manduca wings to find the

influences of aerodynamics and inertial forces of the wing bending. The comparison was made

between the experiments wing bending in normal air and helium air during flapping. It is found

that fluid dynamics forces have minimal effect on the bending of the wing. Further finite

element analysis suggested that the differences of the flapping wing at air versus helium major

due to the fluid damping.

Kang et al(Kang et al., 2011) studied the effect of chordwise and spanwise flexibility on the

Propulsive efficiency of the flapping wings are investigated using the order of magnitude and

energy analysis. It is explained that the major parameter effects the propulsive efficiency are

the density ratio, Strouhal number (St) and flapping frequency(k). It is found that maximum

propulsive is found in the near-resonant flapping frequency and the optimum propulsive

efficiency is found near the half of the natural frequency of the wing. The obtained scaling will

aid the design of micro air vehicle design and the performance analysis.

Daegyoum Kim et al(Gharib, 2011) experimentally investigated the low aspect ratio translating

rectangular plates with three different tips of passive deformation, rigid, curved tips. it is found

that the plate with a rigid tip experiences less drag compared to the plated with a rigid tip. It is

hypothesized that the vortex development near the tip of the flexible tip is the reason for the

reduction in drag.

Liang Zhao et al(Zhao, Huang and Deng, 2010) studied aerodynamics effects of trailing edge

flexibility of the wings. it is found that flapping wing at a high angle of attack generated sable

9

leading-edge vortices compared to the rigid wing. It improves the aerodynamics performance

of the flexible flapping wing with a flexible trailing edge.

Deepa et al(Kodali et al., 2017) presented two coupled fluid-structure interaction models of the

flapping wing with spanwise flexibility. The model is validated with experimental data with

flexibility. The effort has been made to access the aeroelastic performance of goose and

passerine wings. It is shown that the optimal performance is found near the actual aspect ratio

of the wing and near the structural resonance of the wing.

Feshalami et al(Feshalami, Djavareshkian and Zaree, 2018) investigated the aerodynamic

performance of the flapping wing with span-wise flexibility. It is found that the aerodynamics

performance of the flapping wing the span-wise bending increased the performance of the

wing. It also increases the thrust and power loading of the wing.

Zhu et al(Zhu, 2007) numerically investigated a flapping foil with chordwise and span-wise

flexibility. The foil ins were investigated with two fluids medium air and water. It is found that

the chord-wise flexibility increases the thrust and propulsive efficiency in the first and second

scenarios. But the span-wise flexibility increases the performance in the first scenario not in

the second scenario.

Gamble et al(Gamble, Pankonien and Inman, 2017) developed a nonlinear lifting model to

optimize spanwise trail edge morphing of the wing to recover from the dynamic stall. It is also

shown that morphing is an adapter to nonlinear aerodynamics and reduces the drag penalty

while operating off the design. These findings enable the ability to use morphing wings in the

future design of aircraft.

10

Nakata et al(Nakata and Liu, 2012) performed a fluid-structure interaction analysis on a

hovering hawkmoth, Manduca. It is found that wing flexibility helps with an increase in the

aerodynamic performance of the wings. It is also observed that dynamic wing bending and

twisting combination improve the aerodynamic force production. Further studies have been

conducted to observe the effect of wing stiffness on wing structures. It is observed that the

wing structure can be optimized further in terms of wing strength, wing circulation ability, and

stability and control.

WenQing et al(Yang et al., 2012) conducted a numerical simulation of a wing with span-wise

and chord-wise flexibility. The simulation results show that the large-scale span-wise

flexibility decreases the lift, and the small-scale flexibility of the span increases the lift. But

the chord-wise flexibility of the wing at a certain range increases the aerodynamic performance

of the wing. Also, the combination of spanwise and chord-wise flexibility is more effective to

increase aerodynamics performance.

Wong et al(Jain, Wong and Rival, 2015) studied the effect of spanwise flexibility of an

impulsively started plated at a low Reynolds number. A vortex growth model is derived and

validated for a rigid and flexible flat plate in a towing motion. The model shows the phase

delay lift between rigid and flexible plates. The Particle image velocimetry experiments also

show the delay in vortex growth in various spanwise locations due to lower shear layer

velocities towards the tip. It is concluded that spanwise flexibility results in vortex convention

along the span and constant circulation distribution. But, the circulation distribution results

show a uniform spanwise distribution.

Heathcote et al(Heathcote, Wang and Gursul, 2008) conducted water experiments to study the

effect of spanwise flexibility of the ing for the range of Strouhal number. The PIV results show

11

that the spanwise flexibility increases the thrust and power coefficient. But a further increase

in spanwise bending results in a detrimental effect of a decrease in thrust and power coefficient.

It is concluded that spanwise flexibility may result in aerodynamic efficiency benefits.

Wong et al(Wong and Rival, 2015) hypothesized that ta LEV stability of a flapping wing can

be achieved by moderating the LEV growth by distributing the vorticity in the span-wise

direction. It is also hypothesized that the reduced frequency and profile wing sweep are the

most critical parameters that affect the LEV stability and growth. The hypothesis was later

proved using the PIV and Particle Tracking Velocimetry (PTV) experiments.

Wong et al(Wong and Rival, 2017) investigated the effect of span-wise bending using combine

pitching and flapping motion on a two-dimensional NACA0012 profile. A 4D-PTV technique

is used to find the vorticity transport and circulation budget of the wing. The results show that

the tip leading kinematics increases LEV size and strength and tip lagging kinematics results

in a decrease in LEV size and circulation growth.

Li et al(Li et al., 2021) numerically simulated pitching wing with three different trailing edge

flexibility stiffness. It is found that passive flexibility of the plate redistributing the pressure

gradient thugs improves the aerodynamic efficiency. It shows that moderately flexible plated

helps transfer the moment to the fluid thus increasing the thrust.

1.4 Motivation and Objectives of The Present Study

A recent statistical study monitoring the bending of the various natural animals. It is found that

animals are bending their propulsor’s span at about two-third of the total span. Also, it is

hypothesized that the bending of an animal’s body and bird wings reason for its excellent

propulsive ability(Lucas et al., 2014). Inspired by these benefits, current research is

12

conducted to predict and evaluate the role of spanwise bending while studying bird flight and

stability.

In the present study flow physics of the dynamic pitch-bend wing will be studied and reveal

the unsteady aerodynamics force production mechanisms of it, by synchronized PIV, unsteady

force measurements, and by applying classical unsteady aerodynamics theory. The author

presents an experimental result of the deformable pitching wing in cursing flight under a

different set of deformation patterns. The experimental results will give more insights to

estimate. The steps followed by the present study.

(1) the effects of bending wing on the flow topology of a pitching wing at a different angle

of attack. The detailed PIV experiments are conducted at three different planes with a

focus on circulatory force production.

(2) measured unsteady forces to examine aerodynamic effects of wing bending during

pitching motion at different pitch rates (Kp)

(3) detail of the flow field around the pitching and bending wing by computing the vortex

structure, circulation, and PIV-based pressure loading on it.

(4) . provide detailed experimental data required for future computational and theoretical

analysis. and to (2) The unsteady lift force is measured and compared with the

generalized unsteady model for the pitching-bending wing

1.5 Outline of the Thesis

Chapter -1 Introduction and literature review

include the Introduction and the motivation behind the study, literature survey, problem

statement.

13

Chapter-2 Experimental setup

explains Experimental details of the towing tank, automation of the setup, PIV, and the force

sensor setup.

chapter -3, 1D model development

Explain Theodorsen's theory and the modification of theory for the pitch-bent wing.

chapter -4 Results

it shows the experimental results (circulation, force, vortex dynamics) of the pitching wing

and pitch-bent wing for four different reduced frequencies.

chapter -5 Conclusion

contribution of the research and future work suggestions

14

CHAPTER 2 EXPERIMENTAL SETUP

The unsteady flow over pitching and bending flat plate is conducted using experimental

approaches such as Particle Image Velocimetry and force sensor in a towing tank. In this

section, the experimental facilities and procedures for these approaches are explained including

the water tank and the geometry of the flat plate, and all the stepper and servo motors used to

pitch and bend the plate are described.

2.1 Experimental Facility

Figure 4 (a) overall experimental setup that includes the motion mechanisms and DPIV

system (b) Details of the mechanical model.

2.2 Facility

Experiments were performed at the Centre for Advanced Turbomachinery and Energy

Research (CATER) laboratory at the University of Central Florida. Experiments are conducted

at an open tank 0.9 m x 0.45 m in cross-section and flow speed range of 0–0.2 m/s. The water

Traverse

Plunging motor

Pitching motor

Force sensor

Servo motor

Wing

a b

15

tank is raised from the bottom and firmly supported by a frame made from 80/20 aluminum

beams. The experimental test system has three main systems, namely the pitching and

morphing motion control system, Force Measurement System, and DPIV system as shown in

Figure 4.

Figure 5 Experimental setup

2.2.1 Driving mechanism

The driving mechanism of the facility is shown in Figure 4. The turning movement of the

Panasonic technologies AC servo motor was converted into a linear movement with a linear

driver to move the traverse. Two separate servo motors operated by microcontroller Arduino

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Uno modulates the morphing of the wing, while the pitching is controlled by a separated

Zaber technologies RSW 60A stepper motor rotates/pitches the wing through a shaft, and the

motor movement is controlled by the Zaber X-MCB2 controller. One linear motor is used to

move the wing with the desired velocity horizontally, with a second rotary motor operating

vertically, resulting in the autonomous control of pitch or rotation about the quarter chord

point.

2.2.2 Wing Construction

Figure 6 schematic of the spanwise flexible wing attachment

In this research, a flat plate of 300 mm span, 100 mm chord, and 5 mm thickness are used. The

plate is additively manufactured (3D Printed) using nylon plastic material. A present rod was

inserted into the flat plate as shown in Figure 6. The morphing of the plate is achieved by

rotating the pre-bent rod using the servos. Also, the plate can be kept flat by adjusting the

rotation of the rod. The flat plate wing arrangement is considered to have a physical aspect

ratio of 3. The flat plate is made rigid in chordwise directions, and it has a rounded leading

edge and sharp trading edge to ensure the Kutta condition. The experienced chord-wise

17

deflection in all experiments is negligible at all reduced frequencies. Morphing wing design

can be accomplished by turning any section of the wing or the whole wing. It leads to variable

bending of the wing. In this study, bending of the wing is achieved by rotating the servo rods

located within the wing by servos. The bending rate is controlled by changing the speed of

rotation of the servos.

Figure 7 Sample force results of 5 run average input data and the filtered output data of a pitching and pitching-bending case (0-90 deg-0 deg).

2.2.3 Force measurement

An ATI Mini40 Multi-Axis Force and Torque sensor is used to measure direct force and torque

measurements in the x y, z directions. The position of the force sensor is shown in Figure 4 b.

The force (Fx, Fy) measured in the force sensor in the and y-direction equal to lift and drag

force of the wing. The force transducers' results are measured by LABVIEW with a data rate

of 10 kHz. The force transducers are calibrated by adding the known precision weights. And

the measured forces with 95% confidence intervals lie within 2%. The filtering was performed

using a MATLAB-based 1D digital filter to smoothen the noisy data. The sample force trace

between the unfiltered 5 run average data (green and Red) is plotted along with the filtered data

18

(purple and blue) in Figure 7. The filtering is removed the noisy fluctuations from the unfiltered

data.

Figure 8 Experimental setup with laser stand

2.2.4 PIV measurements

The flow field is illuminated by a continuous-wave green laser (Dragon laser) with a power of

2 W and a wavelength of 532 nm. The output from the laser head consists of a beam with a

diameter of 2 mm which is expanded into a thin sheet of 1 mm thickness using a cylindrical

and a spherical lens as shown in. Seeding was provided with 10μm hollow glass particles with

a stokes number of approximately 10-6. The particle follows stream lines closely if the stoke

number is less than 0.1(Raffel et al., 2017). The stokes number less than 0.1 is a measure of

the tracing ability of the particle in PIV. In other words, the particle response time is faster than

the smallest time scale of the flow field. The illuminated flow field is captured by a PCO high-

speed camera at 400 Hz. Image pairs are captures focusing on the same plane and an

19

unassembled average is formed to calculate the velocity field. The captured image pairs are

processed using PIV lab software.

2.2.5 Wing profile measurements

Figure 9, A, wing schematic, 2 B, bending part (8 black dots from the bottom of bending

parts, a tip is considered as 9th point) at time = 0 sec (The coordinates of the nine points

(pt2, to pt9) were tracked at different instants of time to obtain their trajectories.

Only bending part dimensions are given in all Figure 9. The straight part of the wing (22cm)

should be added to all the bent profile values. Figure 9 is the wing schematic for reference.

The bending of the experiments happens at the suction surface/upper surface of the plate.

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Figure 10, bending profile with forward bending rate 1 sec (dimensions centimeters)

2.3 Experimental Procedure

The computer-controlled traverse accelerated linearly from rest to its constant speed. All the

experiments are conducted at traverse velocity and acceleration 100m/s and 100 m/s2

respectively. The pitching motor and servos are triggered once the traverse reached a constant

velocity of 100 m/s. All the experiments are conducted in 2seconds. No phase lag is given

between the pitching motor and traverse. It means the pitching motor and bending servos reach

a maximum(0-1s) and minimum(1s-0s) amplitude at the same time. The wing follows the

bending profile given in the previous section in the forward section. Also, the wing follows the

same bending profile during the reverse motion. The experiments are conducted between 5

min intervals to let turbulence levels decay. The parameter space of all the experiments is given

in Table 1. below.

21

Table 1 Summary of the parameter space

Variable Values

Plate chord (c) 100 mm

Free stream speed (U) 0.1 m/s

Reynolds number (Re) 12000

Aspect ratio (AR) 3

The angle of attack Range 30°,45°,60°,90o

Convection time

(t*= tU/c)

2

Bending Ratio (Bending part Length /Total

Length)

0.733

Flexion angle 160

The images are acquired by the camera kept perpendicular to the imaging plane as shown in

Figure 7. To conduct the phase-locked experiments the camera, linear traverse, pitching motor

and Arduino controlled servo, the force sensor is synchronized using a voltage trigger signal

that was programmed by Arduino Uno microcontroller. The camera is connected to the traverse

and travels along the measurement plane. The flow field is illuminated by the two continuous-

wave green lasers as shown in Figure 10. The pitching angle of attack of the flat plate is

monitored by the Pitching motor encoder. The velocity of the traverse is monitored by the

sensor based on the Ultra Sonic waves. The flat plate at which the traverse reaches the constant

velocity is monitored by the Ultrasonic sensor. When the wing reaches the constant velocity

the trigger signal is sent to the pitching motor, servos connected to the bent rod, camera, and

the force sensor.

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Figure 11 Experimental setup showing the flow field illuminated by two lasers

2.4 Flow Field Analysis Vortex Dynamics

The method used here to calculate to find the total circulation is adapted from literature(Carr,

Devoria and Ringuette, 2015). In this method, the total circulation is determined by

constraining the flow field data within the rectangle region aligned above the chord of the wing.

The vorticity field data below the wing and ahead of the wing is not used due to the shadow

formed by the laser sheet. The rectangle domain uses all positive vorticity in a region near the

leading edge and any positive vorticity near the plate. The circulation due to the Trailing edge

is neglected since it would contribute to the incorrect total circulation measurements(Carr,

Devoria and Ringuette, 2015).

23

CHAPTER 3 ONE DIMENSIONAL UN STEADY THEDORSEN

MODEL DEVELOPMENT

3.1 Theodorsen's Unsteady Classical Theory

In this section steps required to get the unsteady lift forces of pitching bending wing using an

inviscid potential flow theory are elaborated.

3.1.1 Assumptions for classical unsteady flow theory

• One dimensional Inviscid Potential flow was considered along with harmonic

functions.

• No penetration boundary conditions on the surface of the airfoil.

• Kutta condition was applied (for circulatory terms)

• Conservation of circulation

3.1.2 Problem statement

Figure 12 airfoil in real flow (considering all unsteady motion)

24

Consider a two-dimensional aero foil as shown in Figure 11 the moving with a velocity U

in a flow field. In a real flight, the wing may undergo various unsteady motions such as flutter.

Theodorsen tried to predict the aerodynamic effects of this structural oscillation using

analytical calculations. The major objective of the theory is to understand the aerodynamics

reactions during the flutter motion. In the theory, the unsteady motions are considered as

pitching, plunging, and oscillating motions as shown in Figure 13. To predict the lift generation

Theodorsen's unsteady classical theory is derived here by considering an aerofoil oscillation

about an axis.

The lift generation due to the unsteady flow is dived into two different segments as

circulatory and non-circulatory. The complex variable approach is used in this theory to derive

the unsteady lift. In this theory, a cylinder with a radius of b/2 is considered. For simplification,

the source and sink with unknown strength are placed above the bottom of the cylinder as

shown in Figure 13. A conformal mapping approach is used to transform the cylinder with the

source and sink terms into an aerofoil of minimum thickness as shown in Figure 13.

By following assumptions, the below equations were derived. Initially, unsteady

Bernoulli's equation is used to derive the unsteady potential term.

𝛁𝛁𝟐𝟐(𝝋𝝋) = 𝟎𝟎

∇2(𝜓𝜓) = 0

𝐹𝐹 = (𝜑𝜑) + 𝑖𝑖(𝜓𝜓)

∇2𝐹𝐹 = 0

On the plate at 𝑦𝑦 = 0, 𝑟𝑟 = 𝑏𝑏/2

25

𝑥𝑥 = 𝑏𝑏 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐; 𝑢𝑢 = − 𝑉𝑉𝜃𝜃2 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠

; 𝑣𝑣 = 𝑣𝑣𝑟𝑟2𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠

;

Unsteady Bernoulli’s equation

𝑷𝑷 + �𝟏𝟏𝟐𝟐� 𝝆𝝆𝒖𝒖𝟐𝟐 + 𝝆𝝆𝝋𝝋𝒕𝒕 = 𝑪𝑪(𝒕𝒕)---------- 3-1

C(t) here is constant which was independent of space and depends on time.

Linearized, 𝑷𝑷 − 𝑷𝑷∞ = −𝝆𝝆 [𝑼𝑼∞𝝋𝝋𝒙𝒙 + 𝝋𝝋𝒕𝒕] ---------- 3-2

From equation 2, it can be derived lift as

𝑳𝑳 = 𝟐𝟐𝝆𝝆 �∫ 𝝋𝝋𝒕𝒕,𝒖𝒖𝒅𝒅𝒙𝒙 + 𝑼𝑼∞(𝝋𝝋𝑻𝑻𝑻𝑻 − 𝝋𝝋𝑳𝑳𝑻𝑻)𝒃𝒃−𝒃𝒃 �---------- 3-3

By considering the first term in equation3, 𝜑𝜑𝑡𝑡,𝑢𝑢- is the upper surface and required this value

to calculate L

𝝋𝝋𝒙𝒙−𝝋𝝋𝑳𝑳𝑻𝑻 = ∫ 𝝏𝝏𝝋𝝋𝝏𝝏𝒙𝒙

𝒙𝒙𝑳𝑳𝑻𝑻 (𝒅𝒅𝒙𝒙)---------- 3-4

Equation 4 can be further transformed n cylindrical co-ordinate by using conformal mapping.

From fig 2, it can be observed that,

𝝋𝝋𝒄𝒄𝒄𝒄𝒄𝒄(𝜽𝜽) = −𝒃𝒃𝟐𝟐 ∫ 𝒗𝒗𝜽𝜽(ϛ)𝝅𝝅

𝟎𝟎 𝒅𝒅ϛ---------- 3-5

So, now Vθ is also required.

By using all these 5 equations non-circulatory and circulatory terms for the lift can be

calculated.

3.1.3 Non-circulatory terms

Theodorsen frequency response model usually considers the source and sink pair over the

cylindrical or airfoil coordinates with unknown strength “q”. It is shown in Figure 13. By

considering the fluctuations due to aerodynamic flutter small deflections were considered and

mentioned in Figure 14.

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Figure 14 Conformal mapping (non-circulatory terms) of cylindrical co-ordinates along

with source and sink distribution

Figure 15 small deflections (ℇ) in cylindrical co-ordinate

From Figure 13, Figure 14, it can be observed that source and sink were the singularities used

for this study. To move further with So complex potential functions, super positioning the

two flow field functions can be used.

𝑭𝑭 =�𝒒𝒒(ϛ)𝒃𝒃

𝟐𝟐 𝒅𝒅ϛ�

𝟐𝟐𝝅𝝅�𝒄𝒄𝒏𝒏 �𝝃𝝃 − 𝒃𝒃

𝟐𝟐𝒆𝒆𝒊𝒊ϛ� − 𝒄𝒄𝒏𝒏 �𝝃𝝃 − 𝒃𝒃

𝟐𝟐𝒆𝒆𝒊𝒊ϛ��---------- 3-6

Here q was considered as source/sink strength per unit length.

Differentiating equation 6 with ξ,

Ḟ =𝒒𝒒𝒃𝒃

𝟐𝟐

𝟒𝟒 𝒅𝒅ϛ

𝟐𝟐𝝅𝝅𝒆𝒆𝒊𝒊𝜽𝜽 �(𝟐𝟐𝒊𝒊𝟐𝟐𝒊𝒊𝒏𝒏ϛ)(𝑹𝑹−𝒊𝒊𝒊𝒊)

𝑹𝑹𝟐𝟐+𝒊𝒊𝟐𝟐�---------- 3-7

27

Equation 7 can be further simplified as real and imaginary parts. From equation 7, real parts

were considered as radial velocity components, and imaginary parts were considered as

tangential components of velocity. Based on this, Vθ can be derived as,

(𝒗𝒗𝜽𝜽) = −𝟏𝟏𝟐𝟐𝝅𝝅 ∫

𝒒𝒒(ϛ)𝟐𝟐𝒊𝒊𝒏𝒏ϛ 𝒅𝒅ϛ𝒄𝒄𝒄𝒄𝟐𝟐𝜽𝜽−𝒄𝒄𝒄𝒄𝟐𝟐ϛ

𝝅𝝅𝟎𝟎 ---------- 3-8

Similarly, Vr from equation 7 can be derived as,

(𝒗𝒗𝒓𝒓) =𝒒𝒒(Ɵ)𝟐𝟐

𝒗𝒗𝒓𝒓(𝒓𝒓,𝜽𝜽) = ∫(𝒒𝒒 𝟐𝟐𝒊𝒊𝒏𝒏𝒔𝒔𝒅𝒅𝒔𝒔)�𝒃𝒃/𝟒𝟒)𝟐𝟐�

𝝅𝝅�𝑹𝑹𝟐𝟐+𝒊𝒊𝟐𝟐�𝝅𝝅𝟎𝟎 ----- 3-9

No penetration boundary conditions were applied.

𝑣𝑣𝑝𝑝𝑝𝑝𝑝𝑝𝑡𝑡𝑝𝑝 = 𝑣𝑣𝑚𝑚𝑚𝑚𝑡𝑡𝑠𝑠𝑚𝑚𝑠𝑠

𝑉𝑉𝑟𝑟,𝑐𝑐𝑐𝑐𝑝𝑝𝑠𝑠𝑠𝑠𝑐𝑐𝑝𝑝𝑟𝑟 = 𝑣𝑣𝑟𝑟,𝑚𝑚𝑚𝑚𝑡𝑡𝑠𝑠𝑚𝑚𝑠𝑠

𝑣𝑣𝑝𝑝𝑝𝑝𝑝𝑝𝑡𝑡𝑝𝑝 =𝑣𝑣𝑟𝑟,𝑐𝑐𝑐𝑐𝑝𝑝𝑠𝑠𝑠𝑠𝑐𝑐𝑝𝑝𝑟𝑟

2𝑐𝑐𝑖𝑖𝑠𝑠𝑐𝑐

By considering fig 3, 𝑣𝑣𝑚𝑚𝑚𝑚𝑡𝑡𝑠𝑠𝑚𝑚𝑠𝑠 can be derived as below

𝑣𝑣𝑝𝑝𝑝𝑝𝑝𝑝𝑡𝑡𝑝𝑝 = −U sin𝛼𝛼 −Ἠ cos𝛼𝛼 − 𝛼𝛼[𝑋𝑋 − 𝑎𝑎𝑏𝑏]

sin ∝ = ∝; cos ∝= 1(Because α here is infinitesimal)

𝑣𝑣𝑝𝑝𝑝𝑝𝑝𝑝𝑡𝑡𝑝𝑝 = �−𝑈𝑈 ∝ −Ἠ−∝ (𝑋𝑋 − 𝑎𝑎𝑏𝑏)�

28

Equations 8 and 9 were further integrated and small-angle considerations were included.

Derived V(Ɵ) and V(r) values were substituted in equation 3 to find the lift force. Also,

𝜑𝜑𝐿𝐿𝐿𝐿 = 𝜑𝜑𝐿𝐿𝐿𝐿 = 𝟎𝟎. After substitutions and consideration equation 3 can be,

𝑳𝑳𝑵𝑵𝑪𝑪 = 𝝅𝝅𝝆𝝆𝒃𝒃𝟐𝟐(𝑼𝑼ἀ + Ἠ− ἄ𝒂𝒂𝒃𝒃)----- 3-10

The term 𝜋𝜋𝜌𝜌𝑏𝑏2 is added mass or virtual mass. Added mass is the mass of fluid elements that

came along with a solid surface due to its motion in the medium. There was a required force

that needed to accelerate surround flow to move along with the airfoil. The remaining terms

in the above equation were acceleration terms. 𝐿𝐿𝑁𝑁𝑁𝑁 is non-circulatory terms of lift.

3.1.4 Circulatory terms

Figure 16 vortex placed away from cylinder (circulatory terms considered)

To derive circulatory terms, the Vortex was placed far away from the cylinder. One more

vortex was placed away from origin with distance d. Like non-circulatory terms, applied

complex variable approach for circulatory terms too. The function can be a superposition of

vortices. Differentiating this superposition equation gave Vr and VƟ values.

(𝒗𝒗𝜽𝜽) = −𝟏𝟏𝝅𝝅𝒃𝒃 ∫ 𝜸𝜸(𝒛𝒛)∞

𝒃𝒃 �𝒛𝒛+𝒃𝒃𝒛𝒛−𝒃𝒃

𝒅𝒅𝒛𝒛----- 3-11

Here, the strength of vortex 𝛾𝛾(𝑧𝑧) was unknown, to find out this, the Kutta condition was

applied. It resulted as below

29

∫ 𝜸𝜸(𝒛𝒛)∞𝒃𝒃 �𝒛𝒛+𝒃𝒃

𝒛𝒛−𝒃𝒃𝒅𝒅𝒛𝒛 = 𝟐𝟐𝝅𝝅𝑼𝑼𝒃𝒃𝜶𝜶𝟑𝟑/𝟒𝟒 = 𝜞𝜞𝒒𝒒𝟐𝟐----- 3-12

Equation 12 is the integral equation for wake circulation. By giving VƟ value in equation 5,

ϕ(Ɵ) can be derived and it can be helpful to derive lift

𝑳𝑳 = 𝝆𝝆𝑼𝑼𝜞𝜞(𝟏𝟏 + 𝒛𝒛−�𝒛𝒛𝟐𝟐−𝒃𝒃𝟐𝟐

�𝒛𝒛𝟐𝟐−𝒃𝒃𝟐𝟐)----- 3-13

Equation 13 was known as a final form of the Theodorsen lift generation equation and

included steady ( 𝜌𝜌𝑈𝑈𝜌𝜌 )and unsteady terms in it. Equation 13 can be further simplified by

Hankel(𝐻𝐻𝑠𝑠) and Bessel functions(𝐾𝐾𝑠𝑠). This final form of the Theodorsen equation can be

applied to any pitching, plunging case by changing the input information. These functions

can be further derived in the numerical model.

Figure 17 Wing pitch motion and wing bend during one cycle (0- 2 seconds)

30

3.2 Current Analytical Procedure

A simple model is required to understand the individual contributions of the morphing portion,

center of mass, and wing shape and actuation patterns to the performance of the whole wing. In

this section, an effort has been made to check the ability of the classical unsteady model to

predict the aerodynamic performance capabilities of the pitching and bending flat plate. It will

aid to explain the mechanism by which lift is created. These underlying physical mechanisms

are explained by a potential flow model, followed by a simplified pitch-bend

coupling term to predict the lift generation of airfoil when it undergoes unsteady motion as

shown in Figure 17.

Figure 18 Pitch-bent wing schematic and Nomenclature

Unsteady aerodynamic forces and moments acting on the harmonically oscillating

wing section undergo unsteady motions of multiple numbers of degrees of freedom is

discussed in detail in the literature (Theodorsen, 1935; Y.C.Fung, 1993; RL Bisplinghoff, H

Ashley, 1996) and will not be discussed in detail here. However, for clarity in

the subsequent section, it is worthwhile to briefly describe the system under consideration and

define the nomenclature used in the analysis as shown in Figure 17.

31

In the present analysis, a wing is considered as two different sections in contrary to the

Theodorsen, one section (lp) undergoes only the sinusoidal pitch motion, and the other section

(lpb) undergoes sinusoidal pitch and bend forcing about a pivot point (ab) located at the center

of the chord, therefore, ab = 0 for the current case. Figure 17b, c& d is a schematic of the 2D

airfoil displacements in the vertical (h) and angular (α) motion, which has amplitudes of hm and

αm respectively. The complex unsteady lift equation per unit span from the classical

Theodorsen theory is expressed in terms of circulatory lift (LC) and non-circulatory lift (LNC),

which are given by

𝑳𝑳 = −𝞺𝞺𝒃𝒃𝟐𝟐�𝑼𝑼𝞹𝞹��𝜶 + 𝞹𝞹��𝒉 − 𝞹𝞹𝒂𝒂𝒃𝒃��𝜶��������������������𝐿𝐿𝑁𝑁𝑁𝑁

−𝟐𝟐𝞹𝞹𝞺𝞺𝑼𝑼𝒃𝒃𝑪𝑪(𝒌𝒌) �𝑼𝑼𝜶𝜶 + ��𝒉 + 𝒃𝒃 �𝟏𝟏𝟐𝟐− 𝒂𝒂� ��𝜶����������������������������

𝐿𝐿𝑁𝑁

3-14

The lift equation (L) can be reduced for the current case of pure bending and pure pitch, that

is

𝑳𝑳𝒃𝒃𝒆𝒆𝒏𝒏𝒅𝒅 = −𝝅𝝅𝝆𝝆𝒃𝒃𝟐𝟐��𝒉𝒄𝒄𝒑𝒑𝒃𝒃���������𝐿𝐿𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏−𝑁𝑁𝑁𝑁

− 𝟐𝟐𝝅𝝅𝝆𝝆𝑼𝑼𝒃𝒃ℎ𝑪𝑪(𝒌𝒌)𝒄𝒄𝒑𝒑𝒃𝒃�������������𝐿𝐿𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏−𝑁𝑁

3-15

𝑳𝑳𝒑𝒑𝒊𝒊𝒕𝒕𝒄𝒄𝒉𝒉 = −𝞹𝞹𝞺𝞺𝒃𝒃𝟐𝟐𝒄𝒄𝑼𝑼��𝜶�������𝐿𝐿𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝ℎ−𝑁𝑁𝑁𝑁

−𝟐𝟐𝞹𝞹𝞺𝞺𝑼𝑼𝒃𝒃𝒄𝒄𝑪𝑪(𝒌𝒌) �𝑼𝑼𝜶𝜶 + 𝒃𝒃 �𝟏𝟏𝟐𝟐− 𝒂𝒂� ��𝜶��������������������������

𝐿𝐿𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝ℎ−𝑁𝑁

3-16

𝑳𝑳𝒑𝒑𝒊𝒊𝒕𝒕𝒄𝒄𝒉𝒉−𝒃𝒃𝒆𝒆𝒏𝒏𝒅𝒅 = 𝑳𝑳𝒃𝒃𝒆𝒆𝒏𝒏𝒅𝒅 + 𝑳𝑳𝒑𝒑𝒊𝒊𝒕𝒕𝒄𝒄𝒉𝒉 3-17

32

The Lift force ( 𝐿𝐿𝑏𝑏𝑝𝑝𝑠𝑠𝑐𝑐) in a pure bending wing consists of two terms, the first term is the

inertial term and the second term is the induced angle of attack, and the second term lift due

to the induced angle attack modified by the based on the Theodorsen function C (k) = F+iG

defined in terms of Hankel functions HV (2) = Jv – iYv where Jv and Yv are Bessel first and

second kind functions respectively, and expressed as

𝐶𝐶(𝑘𝑘) ∶= 𝐻𝐻1

(2)(𝑘𝑘)𝐻𝐻1

(2)(𝑘𝑘) + 𝑖𝑖𝐻𝐻0(2)(𝑘𝑘)

and k ∶= 𝞹𝞹fc/U is the reduced frequency.

Similarly, the lift force (𝑳𝑳𝒑𝒑𝒊𝒊𝒕𝒕𝒄𝒄𝒉𝒉) in a pure pitching wing consists of an inertial term

proportional to the angular acceleration (��𝜶) involving the product 𝑈𝑈��𝜶 expressed as the

‘Magnus’ term equivalent to the second Fourier harmonic for quasi-steady theory(Stevens

and Babinsky, 2017). And the second term includes circulatory lift due to the change of angle

of attack and rate of change of angle of attack modified by the Theodorsen deficiency

function C (k).

33

CHAPTER 4 RESULTS AND DISCUSSION

4.1 An Unsteady pitching flat plates,

This section the results of the pitching thin flat plate net lift and aerodynamic force history at

the Re= 11882 and 0.131 < K < 0.3925. The flat plate is rotated at the different reduced

frequencies from the angle of attack 0 deg to AOA (30,45,60,90deg) ends at 0 deg. Lift and

drag forces are measured in the force senor are projections of the flat plate wing-normal force

in the respective directions.

Figure 19 Effect of angle of attack on a Lift and Drag force of a pitching wing.

In all the cases the total cycle time (t* = 2) of the pitching wing was kept constant. So, the pitch

rate increases or decreases according to the maximum angle of attack achieved in total cycle

time. The effect of the angle of attack along with the reduced pitch rate is immediately visible

in Figure 18. At a lower reduced pitch rate, K = 0.131 with the maximum AOA of 30o, the

increase in a lift is slow compared to a rapid rise in lift the higher angle of attack cases of 90

deg at K = 0.3925. At a higher reduced pitch rate, K = 0.3925 with the maximum AOA of 90o,

one can observe a decrease in a lift after the rapid rise in lift force earlier compared to a lower

reduced pitch rate, lift curve is increasing monotonically with the increase in the angle of attack

34

(AOA) until it reaches peak lift angle of attack. It is observed from Figure 18 peak lift saturates

early in the case of higher K = 0.1927 around ≈ 15o. But in the case of lower reduced pitch

rate K = 0.1308, the peak lift saturates at higher AOA around ≈ 20o. The decrease in

aerodynamics force is qualitative evidence of detachment of leading-edge vortices from the flat

plate. It is evident from the comparable lift and drags force trends in the results at each

respective reduced pitch rate. It suggests that dominant aerodynamics force is exerted in a plate

in normal directions. Similar trends are observed at different velocities and reduced pitch rates.

It is apparent from the results that, the lift and drag forces do not follow the same while rating

forward and reverse. It is clear evidence of the existing pitching rate-dependent hysteresis as

shown in Figure 18. With the higher reduced pitch rate, the difference in lift and drag force

values in the forward and reverse directions is small compared to the large difference in the

higher reduced pitch rate.

4.2 Force Measurements

Initially, the effect of dynamic bending on a pitching flat plate, moving at a constant velocity

of 0.1 m/s, is quantified using dynamic force measurements. The tests have been performed

for sinusoidal pitch-up to three different maximum geometric AoAs (αmax) of 30°, 45°, and 60°,

followed by an immediate pitch-down motion to 0o (0°- αmax - 0o) during the forward motion.

For the pitch-bent case, the flat plate was dynamically bent along with the pitching motion

simultaneously without any phase difference (Figure 17). The measured lift and drag

coefficient curves for the pitch-only and pitch-bent configurations are presented in Figure 19.

It is observed that with increasing pitch rate, the lift coefficient increases during the upstroke

(0°-30°, 0°-45°, 0°-60°). This is expected due to a higher rate of change of momentum which

has also been reported by Grunnland et al. (2012). The rapid increase in the lift coefficient is

35

accompanied by a steep drop in the downstroke part of the motion. It is to be noted that the

upstroke motion is completed at t* = 1, and the downstroke starts soon after that. Similar lift

augmentation was not observed during the downstroke part (30°-0°). We will show later that

the origin of this rapid increase in lift coefficient in both the pitch-only and pitch-bend case is

non-circulatory in nature.

During the pitch-bend case, the lift coefficient values were observed to reduce during the

upstroke part of the motion, while they increased during the downstroke part, compared to the

pitch-only case. This difference is negligible in the lowest pitch rate case (0-30-0). However,

the difference becomes prominent with the increasing pitch rate. This observation alludes to

the excellent force manipulation capabilities of morphing during an unsteady maneuver. We

Figure 20 Comparison of lift force and drag force of the pure pitching case (black) compared against a pitch-bend case (grey). Green shading shows where the pitching wing force is higher than the pitch-bent case, while yellow shows when the pitch-bent case is more than the pitch case.

36

surmise that by varying the bending ratio or flexion ratio and the amplitude of bending, it is

possible to reduce the lift excursions to a further extent. Our present morphing mechanism

did not permit us to try higher flexion amplitudes, but a higher bending amplitude (or flexion

angle) during the upstroke will result in a more decrease in the lift peak.

4.3 PIV Measurements

In this section, vorticity fields are calculated from the velocity fields measured by PIV. The

vorticity dynamics of the pitching and translation are majorly influenced by the pitch rate

(Kp)(Granlund, Ol and Bernal, 2013; Onoue and Breuer, 2016; Chen, Kolomenskiy and Liu,

2017; Eldredge and Jones, 2018). Similar dependency is observed in the current PIV

Experiments as shown in Figure 23, during convection time, t* = 0.00–1.50 s. From top to

bottom, the rows represent three different instants in the pitching cycle: t* = 0.5, 1, and 1.5

respectively. At the initial stage of pitching

At t* = 0.5, the initial stage of pitching, the emergence of the Trailing Edge Vortex (TEV)

(blue) is caused by the initial acceleration of the wing. At this stage between the time t* = 0-1

s, the emergence of the Leading-Edge Vortex (LEV)rapidly grows near the leading edge with

the increase in α. The LEV (red contours) is attached to the leading edge of the wing until the

pitch-up wing increase in α ceases. In the meantime, the TEV is convected away downstream

from the Trailing edge and further new vortices are produced. After the pitch-up motion at t*

= 1 s, circulation addition through the shear layer to LEV is stopped. At t* = 1–2 s the pitch

down motion of the wing starts. As the flow evolves the reattachment point of the flow field

moves downstream and the LEV separates and convects downstream. Around t*= 1.5 s LEV

pinch off from the upper surface of the wing as the vortex convects downstream. The secondary

vortex formation with the opposite sign to the LEV is seen between the LEV and the plate.

37

This flow field remained the same for all the cases discussed. Similar observations are made in

the previous investigations at high pitch rates.(Ol and Bernal, 2009; Baik et al., 2012; Buchner

et al., 2012). The changes in the vortex structure are expected between the pitch and pitch-bend

case. Meanwhile, the vortex size and position comparison pitch-bend case closely follow the

pure pitching case, including the Tailing edge vortex formation at t* = 0.5 s.

Figure 21. Snapshots results of the spanwise vorticity fields related to several reduced

pitch rate f∗ at Re = 12000 The solid black rectangle denotes the flat plate cross section

(a) f∗ = 0.4112, f = 0.5 Hz, αmax = 30 o (b) f∗ = 0.6119, f = 0.5 Hz αmax = 45 o (c) f∗ = 0.8225,

f = 0.5 Hz αmax= 60 o The vorticity is scaled by c/U∞, clockwise (red) and negative,

anticlockwise (blue)

To demonstrate this effect, the normalized pressure field is shown in Figure 21 for pure pitching

and pitch-bend motion airfoils at different time intervals. Current non-intrusive PIV-based

pressure analysis is the same method shown in Dabiri et al.(Dabiri et al., 2014) was used to

38

observe the pressure loads on the wing. The pressure field derived from the velocity data

captured important flow features near the wing. In particular, the field shows a low-pressure

region in the place of LEV and a high-pressure region where the fluid is accelerating from the

wing. The pressure field at the various span location from the tip to the mid-span shows the

variation in the pressure values. And the upward displacement of the Pitch-bend wing shows

a minimal effect on the wing pressure loading compared to the pressure loading of the pitching

wing.

Figure 22 comparison of the pressure distribution f∗ = 0.8225 at Re = 12000 The solid

black rectangle denotes the flat plate cross-section (a) 50% span, αmax= 60 o (b) 70%

span, αmax= 60 o (c) 86% span, αmax= 60 o .

39

4.4 Total Circulation Temporal Evolution

Figure 23 Historical development of the Total circulation for various motion profiles.

40

To determine the strength of the vortex above the wing, the spanwise circulation of the wing is

calculated for various pitch and pitch-bend kinematics. The experiments are conducted over

the various pitch rates in the range of 0.4112 < Kp < 0.8225, amplitudes αmax = 30,45,60 deg

at Re = 12000. In this section, several total circulation histories are compared for the pitch and

pitch-bend cases as shown in Figure 22. The method used here to calculate to find the total

circulation is adapted from literature(Carr, Devoria and Ringuette, 2015). In this method, the

total circulation is determined by constraining the flow field data within the rectangle region

aligned above the chord of the wing. The vorticity field data below the wing and ahead of the

wing is not used due to the shadow formed by the laser sheet. The rectangle domain uses all

positive vorticity in a region near the leading edge and any positive vorticity near the plate.

The circulation due to the Trailing edge is neglected since it would contribute to the incorrect

total circulation measurements(Carr, Devoria and Ringuette, 2015).

Total circulation values below and ahead of the plate are not captured due to a laser shadow

and the camera lines-of-sight being blocked by the plate. Here t*= 0 is a time instant where the

plate starts the pitching from idle at α = 0. It is observed for all the cases discussed here; the

total circulation of the wing increases monotonically with the increase in the angle of attack

until the LEV separates from the plate. Subsequently, the circulation decreases as the LEV

convects downstream during the pitch down motion. The effect of reduced pitch rate with the

increase in amplitude is seen in the circulation values. The circulation trend is expected since

the feeding shear layer velocity/tip velocity of the wing increases with the increase in pitch

rate. The main objective of this study is to address the effect of bending on the pitching wing

circulation growth. But manipulation of the tip velocity introduced by the bending motion on

the vortex development is not observed in all the cases. This highlights that introduced

manipulation on the pitching wing total circulation growth of the wing is insignificant.

41

Figure 24 (Color online) (a–c) Snapshots results of the spanwise vorticity fields related

to several reduced pitch rate f∗ = 0.8225 at Re = 12000 The solid black rectangle denotes

the flat plate cross-section (a) 50% span, αmax= 60o (b) 70% span, αmax= 60 o (c) 86%

span, αmax= 60 o. The vorticity is scaled by c/U∞, clockwise (red) and negative,

anticlockwise (blue)

4.5 Discussion And Outlook

The performance results of the dynamic bending on the pitching wing have been discussed

42

until now. In this section, an effort has been made to understand the load alleviation observed

due to the morphing wing. In the experimentally measured unsteady aerodynamic force results

shown clearly show the difference in loads acting on the wing. It is well known that the total

aerodynamics force acting on the wing is a combination of circulatory and Non-circulatory

effects. To test the significance of the non-circulatory force due to the bending wing on the

force results. Lift due to bending wing contribution from a pitching wing isolated to a single

parameter to understand the performance of the morphing wing as calculated as ∆𝐿𝐿𝑏𝑏𝑝𝑝𝑠𝑠𝑐𝑐 =

𝐿𝐿𝑝𝑝𝑠𝑠𝑡𝑡𝑐𝑐ℎ − 𝐿𝐿𝑝𝑝𝑠𝑠𝑡𝑡𝑐𝑐ℎ−𝑏𝑏𝑝𝑝𝑠𝑠𝑐𝑐. The difference is shown as the shades as the isolated Lift force due to

the bending wing (∆𝐿𝐿𝑏𝑏𝑝𝑝𝑠𝑠𝑐𝑐) compared with the Noncirculatory force of the bending wing

(𝐿𝐿𝑏𝑏𝑝𝑝𝑠𝑠𝑐𝑐−𝑁𝑁𝑁𝑁) obtained by the analytical model. An analytical model based on the Theodorsen

theory is used to calculate non-circulatory forces. This model has shown an ability to estimate

the non-circulatory forces even in separated flow such as surge and stall regions

recently(Corkery, Babinsky and Graham, 2019)(Bull et al., 2020). It was observed that the

force from the model compares well with the experimentally measured lift forces. It is noted

that the non-circulatory forces contributed to the positive lift in the first part of the pitching

wing (0-1 sec) and it gives a positive lift in the next part (1-2 sec) of the pitching wing. It is

understood that discrepancy between the model non-circulatory values and the experimental

results. But the model considers only non-circulatory effects to show the isolated effects. In the

previous section, it is seen that the sensitivity of the circulatory effects on the morphing wing

in terms of the circulation values around the wing and the vortex structure is less or nil. Also,

the effectiveness of the 2D model predicting the experimental values is less compared to the

3D model(Liu and Sun, 2018).

43

Figure 25 Time history of non-circulatory mass forces

From the above results, it is evident that the non-circulatory forces are the major contributors

to the performance of the morphing wing similar to the aerodynamic force production

mechanisms of a flapping wing(Liu and Sun, 2018; Guvernyuk et al., 2020; Liu, Du and Sun,

44

2020). The effects of non-circulatory forces could not be eliminated while finding the total

aerodynamics forces of the morphing wing.

45

CHAPTER 5 CONCLUSION

In this study, the force production mechanisms around pitching and the bending wing are

experimentally investigated for pitch up and pitch down motion kinematics over a range of

maximum angle of attack and reduced frequency. Based on the research study concluding

remarks are mentioned below.

• It is found that pitch-bent wing results in less lift force during the pitch-up motion

and higher lift forces in the pitch-down motion.

• PIV experiments are conducted to explore the observed force difference with the

morphing wing compared to the non-morphing wing. The PIV results indicate that

there are very few changes in flow field topology. Further vortex circulation

analyses are conducted to observe the difference between morphing and non-

morphing wing. It is observed that the total circulation growth of the morphing

wing is like the non-morphing wing.

• Finally, an analytical model based on the Theodorsen theory is used to calculate

non-circulatory forces. It is found that non-circulatory forces play a greater role in

the load alleviation in a morphing wing. It should be noted that while finding a

force acting on a morphing wing; It is important not to neglect the added Mass

Contribution.

46

APPENDIX: INSTURUMENTATIONS

47

Figure 26 Model instrumentation

A proximity sensor is used to detect and control the motion of the wing.

Arduino Uno micro controller is used for triggering the remaining instruments i.e., Pitching

motor, PCO camera, Servo motor for the bending motion, ATI mini 40 force sensor.

48

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