DYNAMICS AND CONTROL OF A FLAPPING-WING AIRCRAFT

NGUYEN DUC VINH (B.Eng, Ho Chi Minh University of Technology)

A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE 2005

Acknowledgements

The author wishes to express sincere appreciation of the assistance and professional

suggestions given by my supervisor, Assoc. Prof Dr. Lim Kah Bin, Research Fellow

Dr. Hu Yu and PhD Student Mr. Tay Wee Beng.

The author would also like to thank laboratory officers Mrs. Ooi-Toh Chew Hoey,

Miss Tshin Oi Meng, Ms Hamidah Bte Jasman, Mr. Sakthiyavan Kyppusamy and Mr.

Yee Choon Seng in Control and Mechantronics Lab 1, Assoc. Prof Lim Tee Tai and

laboratory officer Mr. Looi Siew Wah in Fluid Dynamics Lab for providing excellent

facilities to carry out the project.

Lastly, the author would like to thank his family members and friends who have given

him many useful suggestions and moral support.

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Table of Contents

Summary I

List of Figures IV

List of Tables VII

List of Symbols VIII

Chapter 1 Introduction 1

Chapter 2 Literature Review 4

Chapter 3 Unsteady Vortex-lattice Method 12 3.1 Theory and numerical approach 13

3.1.1 The basic formulation 13

3.1.2 The numerical solution 14

3.1.2.1 Select singular elements 15

3.1.2.2 Discretization and grid generation 18

3.1.2.3 Defining the kinematics of the wing 18

3.1.2.4 The wake shedding procedure 20

3.1.2.5 The influence coefficients 21

3.1.2.6 The linear set of equations of Newman boundary condition 23

3.1.2.7 Pressure, velocity and load computations 23

3.1.2.8 Wake roll-up computation 24

3.2 UVLM program 25

3.2.1 Code implementation 25

3.2.2 Features and limitations 27

Chapter 4 A pressure-based segregated finite volume solver for modelling flows in moving and deforming zones 30

4.1 Introduction 30

4.2 Grid conservation law 31

4.3 Dynamic mesh conservation equations 31

4.4 Dynamic mesh update methods 33

4.4.1 Spring-based smoothing 33

4.4.2 Local remeshing method 35

4.5 Solid-body kinematics 36

iii

4.6 General procedure for dynamic mesh solution in FLUENT 38

Chapter 5 Investigations of two-dimensional flows 40 5.1 Introduction 40

5.2 Description of wing-motion 40

5.3 Problem setup 42

5.3.1 UVLM simulations 43

5.3.2 FLUENT simulations 43

5.4 Results and discussion 45

5.4.1 Comparison between unsteady vortex-lattice methods 46

5.4.2 Comparison between inviscid and viscous solutions 47

5.5 Conclusion 49

Chapter 6 Investigations of three-dimensional flows 50 6.1 Introduction 50

6.2 Description of wing-motion 50

6.3 Solution procedure 52

6.4 Results 52

6.4.1 Effect of wing camber 52

6.4.2 Effect of flapping amplitude 54

6.4.3 Effect of wing pitch angle 54

6.4.4 Effect of flapping frequency 56

6.5 Phenomena in flow field 58

6.6 Discussion and conclusion 60

Chapter 7 Experimental Studies 64 7.1 Testing model 65

7.2 Wing pitch controller 66

7.3 Force balance 67

7.3.1 Characteristics of a force measurement system 67

7.3.2 The design and development of a 2-axis force measurement system 69

7.3.2.1 Hardware 69

7.3.2.2 Software 73

7.3.3 Calibration 74

7.3.3.1 Calibration set-up 74

7.3.3.2 Calibration procedures 75

7.3.3.3 Calibration results 75

7.4 Wind tunnel facilities 77

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7.5 Experiment set-up 78

7.6 Testing procedure 79

7.7 Results and discussion 79

7.7.1 Profile of instantaneous force 79

7.7.2 Effect of pitch angle on lift force 81

7.7.3 Lift force versus reduced frequency and Reynolds number 82

7.7.4 Measured and computed force data 83

7.8 Summary of experiment studies 84

Chapter 8 Conclusion and Recommendation 87 8.1 Conclusion 87

8.2 Recommendation 89

8.2.1 Computational studies 89

8.2.2 Experimental studies 90

References 91

Appendix A Two dimensional investigations 95 A.1 User-defined function for 2D pure plunging motion in FLUENT 95

A.2 Investigation for 2D airfoil in pitching and plunging motion using FLUENT 95

A.3 Investigation for 2D airfoil flapping in Ground effect 95

Appendix B Three dimensional investigations 98 B.1 User-defined function for 3D flapping motion in FLUENT 98

Appendix C Experimental Studies 99 C.1 Specifications of the Basic Stamp 2 Microcontroller 99

C.2 Controlling Servos 99

C.3 Load cell specifications 100

C.4 Specifications and circuit diagram of the amplifier 100

C.5 Block diagram of the computer-based data acquisition module 101

C.6 Detail-drawings of the force balance 104

v

Summary

Micro Air Vehicles are defined as flying devices which have maximum dimension of

less than 15 cm. The demand for developing such small vehicles is increasing not only

in civil operation but also in military mission. These vehicles are useful to operate in

confined space as well as for surveillance or reconnaissance because of their small

size. An abundant evidence of such small flying devices can be seen in nature (birds,

insects…) and most of them utilize flapping-wing for their flight. This is a sign which

shows that flapping-wing has greater advantage for the aerodynamics of small flying

objects in comparison with the conventional fixed-wing.

The availability of flapping wing flight for Micro Air Vehicle (MAV) applications has

been studied at the National University of Singapore for several years. Several projects

in developing micro flapping-wing aircraft have been carried out. The first endeavour

was to build an electric-power ornithopter (EPO) which could be airborne for several

seconds. In this first project [1], thrust measurement for a membrane wing in stationary

flow was done to understand the affect of wing shape and flapping parameters on the

thrust generation. The best flight time in the total of nine designs from this project was

around 7 seconds. At the same time, a computer simulation for a rigid flapping wing

by Kamakoti et al. [2] based on the strip theory model of DeLaurier [3] was developed.

The code was successfully used to study the aerodynamic performance of a flying

Pterosaur (based on published parameters) and some other wing platforms. However,

as strip theory is only appropriate for design of large-scale flapping wing, this code

cannot be used for our micro air vehicle design. Based on the experience gained from

I

the sustained flight of the membrane wing from a previous project, a controllable EPO

(with span less than 30 cm) was designed and built, it could achieve a sustained flight

of 5 minutes. This effort then was mostly focused on the optimization of the previous

version of the ornithopter. Many improvements have been made since the previous

prototypes. Two new types of wing were introduced, spring membrane wing and

cambered wing, and were partially investigated. Membrane wing was still mainly used

in this project. Several flapping mechanisms were investigated and optimum flapping

mechanism was chosen based on intensive torque analysis. Light material as well as

small-scale light-weight electric devices were carefully chosen. The final design of a

controllable EPO from this Master project [4] weights around 41gram and can sustain

flight for around 8 minutes.

This project is a continuation of the above projects. Although membrane wing has

been successfully used to make the ornithopter sustaining flight, but its aerodynamic

performance is not yet studied. Thus, this lack of knowledge hinders the design

process. In addition, the parameters for this membrane wing affecting flight are

difficult to study. The need to investigate another kind of wing for flapping flight is

therefore necessary and the thin rigid cambered wing in our opinion is the best choice.

The aerodynamic performance of this wing can be explored both numerically and

experimentally. The numerical evaluation of this wing was carried using panel method

and pressure-based finite volume solver. Both inviscid and viscous solutions were

investigated. Experimental assessment was also carried out in a low-speed wind tunnel.

The numerical simulations proved that thin rigid cambered wing can be used as an

alternative to the lift/propulsion generation for flapping wing applications at high

Reynolds number. A good control of wing pitch angle during the flapping cycle and

II

flapping frequency can significantly enhance the lift and thrust performance of the

wing. A 30cm-prototype has been built for the wind tunnel experiment to study the

aerodynamic performance of this wing at MAV Reynolds Number regime. The

instantaneous lift and thrust of the model are measured using a 2-axis force balance

system. The investigation of the thin rigid cambered wing both numerically and

experimentally is believed to provide a good step to boost the design of flapping-wing

micro air vehicles in the future.

III

List of Figures

Fig. 2.1: Wing section aerodynamic forces (left) and aerodynamic model for flapping wing flight by Delaurier.................................................................................................. 7

Fig. 3.1: Schematic flow chart for the numerical solution of unsteady wing problems14

Fig. 3.2: Nomenclature for the vortex ring elements. This picture is taken from Katz and Plotkin [22]............................................................................................................. 15

Fig. 3.3: Vortex ring element model for a thin-lifting surface. This picture is taken from Katz and Plotkin [22]. .......................................................................................... 15

Fig. 3.4: Arrangement of vortex ring elements in a rectangular array. This picture is taken from Katz and Plotkin [22].................................................................................. 17

Fig. 3.5: Inertial and body coordinates used to describe the motion of the body. ........ 19

Fig. 3.6: Wake shedding from the unsteady lifting surface during the first time step (upper) and during the second time step (lower). This picture is taken from Katz and Plotkin [22]. .................................................................................................................. 20

Fig. 3.7: Flow chart of the UVLM code ....................................................................... 25

Fig. 3.8: GUI of the UVLM code ................................................................................. 28

Fig. 3.9: An example of pop-up dialog of computed force........................................... 28

Fig. 3.10: Wake after a moth wing (top) and rolling wing (bottom). ........................... 28

Fig. 4.1: Spring-based connections between any two mesh nodes. .............................. 34

Fig. 4.2: Compression of cylinder before (left) and after (right) spring-based smoothing...................................................................................................................... 35

Fig. 4.3: Expansion of mesh around a cylinder before (left) and after (right) local remeshing. ..................................................................................................................... 36

Fig. 4.4: Solution procedure in FLUENT ..................................................................... 37

Fig. 4.5: Gambit GUI (top) and FLUENT GUI (bottom) ............................................. 38

Fig. 5.1: Profile of airfoil path in pure plunging motion .............................................. 41

Fig. 5.2: Three-dimensional rectangular wing with half-cosine and full-cosine distribution along wing span and wing-chord respectively in CMARC....................... 42

Fig. 5.3: Wake after NACA0014 airfoil in pure plunging motion................................ 43

Fig. 5.4: FLUENT 2D Navier-Stokes grid. The picture on the left is the initial grid consisting of deforming zone and non-deforming zone separated by grid interface (solid lines). The picture on the right shows the deformation of grid-cells near the airfoil boundary after one cycle. ................................................................................... 44

Fig. 5.5 : Installation of test model in wind tunnel by K.D. Jones et al........................ 46

Fig. 5.6 : Unsteady lift coefficient, 0.4κ = ................................................................. 47

IV

Fig. 5.7: Unsteady lift and thrust coefficient between 2D FLUENT (Inviscid) and panel method at 0.2κ = ............................................................................................... 47

Fig. 5.8: Mean lift (left) and thrust (right) coefficient between inviscid and viscous solutions at 0.2κ = ...................................................................................................... 48

Fig. 6.1: Three-dimensional flapping wing model........................................................ 51

Fig. 6.2: Net angle of attack of the three dimensional flapping wing at 00 30θ = and

00γ = ............................................................................................................................ 51

Fig. 6.3: Wake behind three-dimensional rigid cambered wing in one flapping cycle.52

Fig. 6.4: Mean camber lines of various airfoils in percentage of wing chord .............. 53

Fig. 6.5: Unsteady lift and thrust coefficient at various cambers ................................. 53

Fig. 6.6: Mean lift and thrust coefficient versus flapping amplitude............................ 54

Fig. 6.7: Unsteady lift and thrust coefficient at various wing pitch angles. ................. 54

Fig. 6.8: Improvement of thrust in the downstroke (left) and reduction of drag in the upstroke (right) with respect to zero pitch angle. ......................................................... 55

Fig. 6.9: Improvement of thrust and reduction of lift at various coupled pitch angles with respect to zero coupled pitch angle....................................................................... 55

Fig. 6.10: Illustration of coupled wing pitch angles in flapping flight ......................... 56

Fig. 6.11: Unsteady lift and thrust coefficient for coupled wing pitch angle of 020 at various flapping frequency ........................................................................................... 57

Fig. 6.12: Mean lift and thrust coefficient at various flapping frequency .................... 58

Fig. 6.13: Lift and thrust coefficient between panel method and FLUENT ................. 59

Fig. 6.14: Low pressure regions on wing surface in downstroke (left) and upstroke (right) ............................................................................................................................ 59

Fig. 6.15: Leading vortex in the mid downstroke computed by FLUENT (left) and experiment (right) ......................................................................................................... 59

Fig. 6.16: Net angle of attack at wing root (top) and wing tip (bottom) for various wing pitch angles .......................................................................................................... 62

Fig. 7.1: Real model for wind tunnel experiment ......................................................... 65

Fig. 7.2: CAD model and plot of flapping angle .......................................................... 66

Fig. 7.3: BasicStamp 2 microcontroller (left) and set-up of servos on the carbon rod (right) ............................................................................................................................ 66

Fig. 7.4: Typical output characteristics of a force measurement system...................... 68

Fig. 7.5: CAD model of the force balance .................................................................... 70

Fig. 7.6: Real force balance .......................................................................................... 71

Fig. 7.7: LCL Omega 454g load cell ............................................................................ 71

Fig. 7.8: RS Strain-gage amplifier ................................................................................ 72

Fig. 7.9: National Instruments DAQCard-1200............................................................ 73

Fig. 7.10: GUI of the computer-based data acquisition module ................................... 73

V

Fig. 7.11: Set-up for the calibration process ................................................................. 74

Fig. 7.12: Relation of voltage output and applied force on lift direction (top) , drag direction (bottom) ......................................................................................................... 76

Fig. 7.13: Wind tunnel facilities ................................................................................... 78

Fig. 7.14: Set-up force balance and test model inside the wind tunnel ........................ 78

Fig. 7.15: Breakdown of force components: unsteady lift, inertial loading and total force in Newton............................................................................................................. 80

Fig. 7.16: Experimental mean lift versus pitch angles.................................................. 81

Fig. 7.17: Experimental lift versus reduced frequency and Reynolds number at zero pitch angle. .................................................................................................................... 82

Fig. 7.18: Measured and computed unsteady lift in Newton for top (3.34m/s and 4.42Hz), bottom (6.92m/s and 2.79Hz) ........................................................................ 83

Fig A.1 : Vortex shedding after two-dimensional airfoil in combined pitching and plunging motion computed by 2D panel method (top) and FLUENT (bottom)........... 96

Fig A.2 : Vortex shedding in the wake of a NACA0015 flapping in ground effect computed by two-dimensional panel code (top) and FLUENT (bottom)..................... 97

Fig C.1 : Dimensions of the 454g Omega load cell .................................................... 100

Fig C.2: Basic circuit for printed circuit board RS stock no. 435-692 (gain approx. 1000) ........................................................................................................................... 101

Fig C.3: Block diagram of the computer-based DAQ module (section 1) ................. 102

Fig C.4: Block diagram of computer-based DAQ module (section 2) ....................... 103

VI

List of Tables

Table 5.1: Simulation parameters for two-dimensional plunging motion. ................... 42

Table 5.2: Simulation parameters for 2D unsteady panel method................................ 42

Table 5.3: Setting-parameters for updating mesh of two-dimensional FLUENT problem. ........................................................................................................................ 45

Table 7.1: Basic parameters of the test model .............................................................. 65

Table 7.2: Range of experiment parameters ................................................................. 79

Table A.1: Parameters for NACA0012 airfoil in combined pitching and plunging motion ........................................................................................................................... 96

Table A.2: Parameters for NACA0015 airfoil flapping in ground effect ..................... 96

VII

List of Symbols

AR wing aspect ratio

c wing chord length

LC lift coefficient

TC thrust coefficient

f flapping frequency

h plunging amplitude

κ reduced frequency . .f cUπ

∞

Re chord Reynolds number .U cv∞

t time

τ non-dimensional time .t Uc∞

T period of one flapping cycle

U∞ free stream speed

v kinematic viscosity

0θ flapping amplitude

γ pitch angle

q∆ velocity induced at an arbitrary point by a typical vortex segment with constant

circulation

VIII

Chapter 1 Introduction

The objective of this project is to investigate the aerodynamic performance of a thin-

rigid cambered wing in flapping-wing motion at high and low Reynolds numbers. This

investigation is helpful in the development of flapping wing micro air vehicle

applications.

Recently, there is a growing recognized need for miniature flight vehicles with multi-

functional capabilities such as Micro Air Vehicles (MAVs) with span less than 15 cm.

And the interest for developing such MAVs using mechanical flapping wing flight is

motivated by the notion that flapping wing at small-scales may offer some unique

aerodynamic advantages over a fixed wing solution. The evidence for such notion can

be easily seen in nature with the presence of flying vertebrates such as birds and

insects.

In aerodynamic aspect, our knowledge of how to design a conventional fixed wing

aircraft with moderate aspect ratio is fully understood. However, applying such

knowledge to the design of MAVs or smaller-scale aircraft is not a right approach. The

reason is that the aerodynamic performance of stationary airfoil drops at low Reynolds

number of order 104-105, which is the Reynolds regime of MAVs. Moreover, the large

difference between the flow characteristics around the conventional fixed wing and the

flapping wing also makes a big gap in applying the conventional aerodynamics to

explain the aerodynamics of flapping wing flight. The movement of the wing in

1

flapping motion makes the flow around the flapping flyers highly unsteady in

comparison with relatively steady flow around the fixed wing.

A first step in the effort of resolving the aerodynamic performance around a rigid

flapping wing has been made by two final year projects [5,6]. In these projects, a

Matlab-code application based on the strip theory of DeLaurier [3] was successfully

developed. The theory assumes that flow past through wing with sufficiently high

aspect ratio is attached-chordwise and that each section’s aerodynamics is determined

by its local angle of attack as influenced by wake effects. The resulting sectional

aerodynamic forces can then be resolved and integrated to give lift and thrust

behaviour. This approach is sufficiently accurate to be successfully applied to the

design of large ornithopters.

The current project aims to resolve the aerodynamic performance of a thin rigid

cambered wing for large-scale as well as small-scale flapping-wing aircraft. Lift and

thrust behaviour of the wing has been investigated both numerically and

experimentally. The numerical simulation is performed using three-dimensional

unsteady vortex-lattice method and pressure-based finite volume solver. Both inviscid

and viscous unsteady flows were modelled. Effects of flight parameters and wing

geometry on lift/thrust performance were carefully investigated. New concept of

improving lift/thrust performance for rigid cambered wing was developed. The

assessment of the numerical investigation is then carried out by experiments in a low

speed wind tunnel.

The content of this thesis is summarized as follows:

• Chapter 2: Background and summary of previous work

2

• Chapter 3: Basic theory and numerical implementation of the unsteady vortex-

lattice method.

• Chapter 4: Basic theory and problem setup for modelling flows in deforming and

moving zones using finite volume solver FLUENT.

• Chapter 5: Investigation of a two-dimensional airfoil in pure plunging motion to

validate the unsteady vortex-lattice method (UVLM) and FLUENT solver.

• Chapter 6: Investigation of a three-dimensional rectangular rigid cambered wing in

flapping flight at high Reynolds number. Effects of wing camber, flapping

amplitude, wing pitch angle and flapping frequency were studied. New concept of

improving lift/thrust performance of the rigid cambered wing in flapping flight was

developed here.

• Chapter 7: Experimental study for a 30-cm rigid cambered rectangular wing in

flapping flight at low Reynolds number in a wind tunnel. This chapter includes the

design and building of the test model as well as a 2-axis force balance for

measuring lift and thrust of the test model.

• Chapter 8: Conclusion and recommendation.

3

Chapter 2 Literature Review

The objective of this project is to investigate the lift and thrust generation of a large-

scale as well as a small-scale rigid cambered wing in flapping flight at high and low

Reynolds number.

The development of flying machines so far can be divided into two major groups. The

first group includes those which use separate systems for lift and thrust generation.

This group consists of well-developed large-scale fixed wing aircrafts and several

rotary-wing helicopters. This concept of flying machines [7] is first introduced by Sir

George Cayley in 1799. Since then, flying machines that applied this concept have

been successfully built using a systematically engineering approach. On the other

hand, the development of the second group utilizing combined lift and thrust

generation system is still in its early stage. This group consists of small-scale flying

machines at the size of birds and insects. The interest in these vehicles is renewed by

the recent need to develop micro air vehicles with maximum span of 15cm to operate

in confined space as well as for surveillance and for reconnaissance purposes. At a first

glance, mechanical flapping wing (bird-like or insect-like) has been chosen as a good

alternative as these small-scale aircraft can make use of knowledge from the

understanding of the super aerodynamic performance of birds and insects in nature.

Major researches in flapping wing flight in the past two decades can be classified into

two categories: researches of biologists or zoologists and researches of aerospace

4

engineers and designers. Many investigations have been performed. In our reviews,

only measurements, observations and flow visualization from animal-flight studies

which are appropriate to the design of flapping wing MAVs are presented. Numerical

simulations as well as experiments from aerospace engineers are reviewed in the way

that we can acquire a systematic knowledge on the development of flapping wing

MAVs so far. A three-dimensional simulation and experiment for a rigid cambered

rectangular flapping wing is also presented in this project.

The primary motivation for studying animal-flight is to explain the physics of a

creature that is known to fly. One thing clear from these animal-flight investigations is

that the entire creature uses two specific mechanisms to overcome the small-scales

aerodynamic limitations of their wings: flexibility and flapping. The coupling between

flexible wing and aerodynamic forces is combined in the way that the aeroelastic

deformations of the wing improve the aerodynamic performance. Rayner [8] studied

the thrust and drag in steady and level flapping flight. According to him, birds and bats

uses two patterns of wing movements to maximize their thrust generation: the vortex

gait and continuous vortex gait. In the vortex gait, characteristic of slow-flight, the

wake is composed of ring vortices, where in the continuous gaits, characteristic of

long-winged birds; the wake is a pair of sub-linear trailing vortices that approaches the

line vortices of fixed wing. From the calculations based on his continuous vortex wake

model, Rayner presented his formulae for predicting the flight parameters of MAVs

based on its size and design. Maximum speed range V and mechanical power at

that speed can be expressed as

mr mrP

0.190( / ) 12.49V m s M= (2.1)

1.161( ) 14.95mr

mrP W M=

5

in term of total mass M (kg) alone, or

(2.2) 275.0818.1590.121.27)( SBMWPmr

mr−=

095.0553.0413.000.10)/( SBMsmV −−=

in term of total mass, wingspan B(m) and wing platform area S(m2). The continuous

wake model also showed that the mechanical power against speed curve for a bird in

steady level flight follows the U-shape as usually seen in the low-speed fixed wing

aircraft; and the maximum speed V should be interpreted as a parameter

characterizing this power curve (not necessarily the speed at which a bird or an MAV

should fly). Wing shape and drag reduction techniques used by flapping birds are also

investigated in Rayner’s research. Many adaptations which are believed to improve the

aerodynamic performance of birds have been clearly explained. However, as suggested

by Rayner as well as other biologists, it is slavishly impractical to copy such

aerodynamic features or wing motion of birds or insects. Instead the determination of

which aerodynamic features as well as which movements of insects and birds are truly

necessary for flapping MAVs is the key for the successful development of flapping

wing vehicles.

mr

In contrast with researches of biologists, researches of most aerospace engineers or

designers usually consist of comprehensive analysis with rigorous experiments over a

wide range of parameters. Three main numerical methods have been successfully used

to study the aerodynamics of flapping wing flight by researchers at several universities.

These methods include modified strip theory, panel method and Euler/Navier-Stokes

based method.

6

The modified strip theory was first proposed by DeLaurier (1993) [3] in his study of

average lift and drag of a Pterosaur replica. Assuming attached flow and no spanwise

cross flow, the wing can be divided into a number of thin strips. Lift and thrust of each

strip can be calculated independently and then summed up to give the average lift and

drag of the whole wing. By considering the wing as a set of airfoils, this kinetic model

can be easily used to account for effect of dynamic twisting, bending and chordwise

variation along the span of the wing in flapping flight. Kamakoti et al. [2] has

enhanced the faithfulness of the model by developing a procedure which is capable of

predicting the domain of flight (to sustain flight) for different flight parameters as well

as the maximum power efficiency when certain conditions prevail. Their optimization

shows that the propulsive efficiency achieves optimum solely depending on dynamic

twist and neither flapping angle nor flapping amplitude. This conclusion emphasizes

the fact that birds utilize the flexibility of the wing to improve its aerodynamic

performance. Although the modified strip theory is quite easy to implemented, but the

assumption of attached flow may not be appropriate to apply for small-scale flapping

wing where lift and thrust are dominated by vortex shedding.

The panel method has shown its advantages to study the aerodynamic performance of

flapping wing flight. This method has been extensively used by Jones et al. [9,10 ] to

study the propulsion of plunging and pitching airfoils over the last decade. Many

Fig. 2.1: Wing section aerodynamic forces (left) and aerodynamic model for flapping wing flight

by Delaurier.

7

numerical simulations as well as experiments in wind and water tunnel have been

carried out for airfoil undergoing pure plunging, pure pitching and combined plunging

- pitching motion. The numerical simulations are performed using an unsteady, two-

dimensional potential-flow code with a nonlinear deforming wake model. Results from

their simulations show good agreement with linearize theory of Garrick [11] and some

validated Euler/Navier-Stokes solvers at high Reynolds number. Thrust coefficient and

flapping efficiency are meticulously computed for several configurations including

single and combined airfoils. The opposed-plunge configuration, which consists of two

airfoil plunges symmetrically through a horizontal plane (the so-called ground effect),

has been chosen as the propulsion generation for their MAV design owing to its high

thrust as well as high efficiency. The final MAV design using this propulsion

generation configuration has 27cm span, 18cm length and weight of about 13.4g with

two channel control and enough battery capacity for 15-20 minutes flight. The most

remarkable advantage of this MAV is the ability to sustain flight at relatively low

speed of 2m/s to 5m/s and virtually stall-proof. The success of this MAV is the result

of extensive systematic simulations and wind-tunnel experiments. Moreover, the

agreement of their two-dimensional panel code with Garrick’s theory now provides a

good prediction tool for aerodynamic performance of their future MAV design.

The Euler solution for flapping wing flight is investigated by Neef and Hummel [12].

In their studies, flow field around a three-dimensional rigid rectangular wing

undergoing flapping and twisting motion at high Reynolds number is computed.

FLOWer, the code which was used in their simulation, is developed at the Institute of

Design Aerodynamics of DLR Braunschweig for solving the three-dimensional Euler

equation as well as the Reynolds-averaged Navier-Stokes equation in integral form on

structure meshes. For solving time-dependent flapping wing, a geometric conservative

8

law [13] is embedded into FLOWer code to correct the change of local cell volumes

made by the moving boundaries. Grids at some given time-steps are generated and an

interpolation scheme is used to compute grids for all remaining time-steps. Attached-

flow condition is required. The method is first validated by studying flow-field of two-

dimensional plunging and pitching airfoil. A good agreement of this study leads to the

expansion of the method for solving the three-dimensional case which involves a

rectangular NACA 0012 of aspect ratio of 8 in O-O topology grid. Flapping and

twisting vary linearly along the span with constant phase shift of 900. The effect of

flapping and twisting in three-dimensional case is similar to that of plunging and

pitching in two-dimensional case. However, as shown in this study, the thrust output

and efficiency in three-dimensional case are greatly dominated by unsteady trailing

vortices at the wing tip, especially at large mean angle of attack. Therefore, thrust

efficiency in three-dimensional is considerably reduced compared to in two-

dimensional situation. Navier-Stokes solution is their future focus in order to find the

optimal motion parameters for the thrust generation in three-dimensional case. Their

final aim is to provide a tool for evaluating the effect of arbitrary wing motions on the

performance of flapping wing propulsion for attached-flow conditions.

Beside the three prevailing methods above, there are other approaches used by several

researchers for the study of the performance of micro air vehicles both numerically and

experimentally. Unsteady lifting-line theory has been used to study the low-frequency

flapping by Willmott [14], Phlips et al. [15] , Ahmadi and Widnall [16]. Unsteady

quasi-steady vortex-lattice method for predicting the flapping efficiency of various

planforms and flapping motions has been used by Lan [17]. However, as indicated by

Hall and Hall [18], none of these above analyses addresses the issue of minimum

power flapping flight. This issue is important to flapping flight because the sectional

9

coefficients of lift and drag vary significantly over a flapping cycle. Hall and Hall [18]

proposed an approach, the so-called vortex-lattice –based variation method, in which

desired thrust and lift for a given flapping wing can be obtained in such a way that no

portion of the wing can operate in stall condition. The optimal frequency for this

optimal condition can be found. The method has been used by them to compute the

minimum power loss circulation distributions for flapping wings at both high and low

Reynolds numbers. The study shows that efficiency of flapping flight at low Reynolds

number is not significantly different from that of propeller-driven aircraft. More

research should be done to clarify this conclusion.

Going along with the investigations in aerodynamics, some flapping-wing MAVs have

been successfully launched. Most of these MAVs, or the so-called ornithopters, uses

membrane wing. The wing is made of thin Mylar firm or Japanese tissue paper glued

to carbon rods. The advantage of this membrane wing is light weight. A radio control

MAV using this membrane wing can sustain flight by using super light weight motors,

servos and transmitters/receivers. Such designs are widely found among ornithopter

hobbyists. An improvement in the design of membrane wing is introduced by Nick

Pornsin-Sirirak et al. [19,20]. In their improvement, carbon rods are replaced by

titanium-alloy metal for wing structures and thin Mylar* film is replaced by parylene-

C†, the so-called Micro Electro-Mechanical System (MEMS) technology. This new

MEMS wing technology together with photolithography technology can be used to

fabricate some complicated wing structures such as dragonfly, butterfly and beetle

wing. Three prototypes called “Micro bat” using this MEMS wing have been built. The

* Mylar is a trade name of DuPoint Teijin Films of Hopewell, VA for biaxially-oriented polyethylene terephthalate polyester film used for high tensile strength, chemical and dimensional stability, transparency and electrical insulation. † The term “parylene”- a contraction of poly (para-xylene) – denotes a family of vapour-deposited polyemers. In parylene-C (the most common form of parylene), a chlorine atom is substituted for one of the hydrogen atom on the benzene ring of each para-xylene moiety.

10

final battery-powered with radio control system flapping-wing MAV [21] weights

around 11.5g and can fly for approximately 6 minutes 17 seconds. The successful

launch of this micro bat shows a promise for future development of MAVs using

MEMS technology. However, the difficulty in modelling the aerodynamic

performance of the membrane wing in-general has limited its interest among

aerodynamicists and aerospace engineers.

The development of flapping wing MAVs over the past two decades has progressed

gradually. A general theoretical framework for flapping wing flight has not yet been

established. Most of the aerodynamic investigations are limited to two-dimensional.

Membrane wing is very difficult to study numerically. The three-dimensional rigid

cambered wing for MAVs application is introduced to overcome these gaps. The

aerodynamic performance of this kind of wing was studied by using both numerical

simulations as well as experiments in this project.

11

Chapter 3 Unsteady Vortex-lattice Method

In this chapter, the basic theory of the unsteady vortex-lattice method, presented in the

book by Katz and Plotkin [22], is summarized here to give reader some basic

background of the theory.

The vortex lattice method (VLM) is a classical boundary element method which relies

on developing a distribution of source and doublet “singularities” on wing and body

surfaces and with doublet “singularities” to represent the wake. Unsteady vortex-lattice

method (UVLM) is a transformation from the steady VLM to treat problems in

unsteady flows. To date, the VLM has been used, almost exclusively, to analyze the

aerodynamic forces on aircraft. The VLM is based on potential theory which assumes

non-viscous flow. The advantages of this method are that it accommodates the

detailing of the trailing wake, and can include dynamics effects in a distributed

manner. Moreover, it is also capable of taking into consideration the flexibility and

interference effects. Such advantages render the UVLM more useful than other

methods such as momentum, blade elements, hybrid momentum or lifting-line

methods when analyzing the aerodynamic forces on flapping wings [23]. This method

has been proven to be a good representation for modelling flows at Reynolds number

in the order of 10 or above. It was also used in this project for modelling flapping

wing flight at both high (10 or above) and low (10 or below) Reynolds number.

5

6 4

12

This chapter is divided into two parts. The first part presents the basic theory of a

lifting-surface solution for unsteady flows. The procedure to transform the steady-state

VLM to treat for unsteady flows is clearly presented in “Low Speed Aerodynamics,

from the theory to Panel Methods” by Katz and Plotkin [22]. The second part provides

the descriptions of the UVLM code developed by our team (Dr Hu Yu, PhD Student

Mr. Tay Wee Beng and M. Eng student Mr. Nguyen Duc Vinh) in this project. This

code has been intensively used in this project to investigate the flapping wing flight of

a three-dimensional rigid cambered wing.

3.1 Theory and numerical approach

3.1.1 The basic formulation

The governing equation of the unsteady vortex-lattice method is the Laplace equation

with several boundary conditions.

As singularity elements are distributed on the wing surface, the solution is to find the

strength of these singularity elements subject to several boundary conditions. For thin

wings, the Newman boundary condition on the wing surface is used. The condition

states that the normal velocity induced by all the singularity elements on the wing

surface shall be zero. In unsteady case, this normal velocity must include the velocity

induced by the movement of the wing and therefore the Newman boundary condition

can be written as follows:

( ) 00 .relV v r n∇Φ− − −Ω× =r rr r r

r

(3.1)

where ∇Φ is the velocity induced by all singularity elements on the wing surface

0V is the velocity of the body frame in the inertial frame

13

relvr

r

rr

is the velocity of the wing in the body fixed frame

Ω is the angular velocity of the body fixed frame in the inertial frame

is the position vector of the body fixed frame in the inertial frame

If the wing is divided into a number of small panels, then Eqn(3.1) must be for all

distributed panel on the wing surface. Note that Eqn(3.1) is computed in the body fixed

frame.

3.1.2 The numerical solution

End

Definition of Geometry

Flight Path Information

Calculation of Influent Coefficients

Calculation of Momentary RHS Vector

Solution of Matrix

Wake roll-up

Calculation of Velocity Components, Pressure, Load, etc.

ti = ti-1 + ∆t

Fig. 3.1: Schematic flow chart for the numerical solution of unsteady wing problems

14

The flow chart for the numerical solution of the unsteady wing problem is presented in

Fig. 3.1. Transforming from steady case, two new blocks are added to treat the

unsteady problems: flight path information and wake roll-up. Detail solution of each

block is presented in the subsequent paragraph.

3.1.2.1 Select singular elements

For three dimensional thin lifting surface problems, the vortex ring elements are used.

A picture with nomenclature for some typical vortex ring elements is shown in Fig.

Fig. 3.2: Nomenclature for the vortex ring elements. This picture is taken from Katz and

Plotkin [22].

Fig. 3.3: Vortex ring element model for a thin-lifting surface. This picture is taken from Katz and Plotkin [22].

15

3.2. The ring is in the form of a rectangle with constant vortex distributed on its four

edges. The main advantage of this element is the simple programming effort that it

requires (although its computational efficiency can be further improved). Additionally,

the exact boundary conditions will be satisfied on the actual wing surface, which can

have camber and various planform shapes.

The vortex ring elements chosen above are based on the vortex line solution of the

incompressible continuity equation. The boundary condition that must be satisfied by

the solution is the zero normal flow across the thin wing’s solid surface which is

derived in Eqn (3.1) for unsteady case.

To solve the lifting-surface problem, the wing is divided into panels containing vortex

ring singularities as shown in Fig. 3.3. The leading segment of the vortex ring is placed

on the panel quarter-chord line and the collocation point is at the center of the three

quarter-chord line. The normal vector nr is defined at this point too. A positive

circulation is defined here according to the right-hand side rule as shown in the Fig.

3.3.

Γ

∆The velocity induced at an arbitrary point q ( ), ,P x y z

dl

by a typical vortex segment

with constant circulation Γ can be computed based on the Biot-Savart’s law as

follows:

34q

rπdl rΓ ×

∆ = (3.2)

The numerical computation of this induced velocity should be grouped into a

subroutine called:

16

( ) ( ), , , , , , , , , , ,u v w VORTXL x y z x y z x y z 1 1 1 2 2 2= Γ

,i j

( , ,P x y z i j

( )

(3.3)

As the wing is divided into panels containing vortex ring elements as shown in Fig.

3.3, from the numerical point of view these vortex ring elements can be stored in

rectangular patches with indices as shown in Fig. 3.4. The induced velocity at an

arbitrary point by a typical vortex ring at location can be computed by

applying the vortex line routine in Eqn(3.3) to the rings’ four segments:

) ( , )

( )( ) ( )( )

1 1 1 , , , , 1 , 1 , 1 ,

2 2 2 , 1 , 1 , 1 1, 1 1, 1 1, 1 ,

3 3 3 1, 1 1, 1 1, 1 1, 1, 1

, , , , , , , , , , ,

, , , , , , , , , , ,

, , , , , , , , , ,

i j i j i j i j i j i j i j

i j i j i j i j i j i j i j

i j i j i j i j i j i

u v w VORTXL x y z x y z x y z

u v w VORTXL x y z x y z x y z

u v w VORTXL x y z x y z x y z

+ + +

+ + + + + + + + +

+ + + + + + + + +

= Γ

= Γ

= ( )( ) ( ), , ,

, ,

4 4 4 1, 1, 1, ,

,

, , , , , , , , , , ,i j i j i j

j i j

i j i j i j i ju v w VORTXL x y z x y z x y z+ + +

Γ

= Γ

(3.4)

The velocity induced by the four vortex segments is then:

( ) ( ) ( )u v w u v w u v w u v w u v w= + + +

( )

1 1 1 2 2 2 3 3 3 4 4 4( , , ) ( , , ) , , , , , , (3.5)

It is convenient to include those computations in Eqn (3.5) a subroutine such that:

, , ( , , , , , )u v w VORING x y z i j= Γ (3.6)

Fig. 3.4: Arrangement of vortex ring elements in a rectangular array. This picture is taken from Katz and Plotkin [22].

17

The use of this subroutine can considerably shorten the programming effort; however,

for the inner vortex rings the velocity induced by the vortex segment is computed

twice. For the sake of simplicity this subroutine is also used in the UVLM code

developed by us, but more advanced programming can easily correct this loss of

computational effort.

3.1.2.2 Discretization and grid generation

The wing surface is divided into panels as shown in Fig. 3.2. To reduce the

computational effort, only wing semispan should be modelled and the mirror-image

method will be used to account for the other semispan. The leading edge of the vortex

ring is placed on the panel’s quarter chord and the collocation point is placed at the

center of the three quarter chord line. The normal vector n is defined at this collocation

point. A positive is defined here as the right-hand rotation about an axis. For

pressure distribution calculations, the local circulation is needed, which for the leading

edge panel is equal to but for all the elements behind it is equal to the

difference . In the case of increased surface curvature, the above-described

vortex rings will not be placed exactly on the wing surface and a finer grid needs to be

used or the wing surface can be redefined accordingly. By placing the leading edge

segment of the vortex ring at the quarter-chord line of the panel the two dimensional

Kutta condition is satisfied along the chord

Γ

Γ

−Γ −Γ

i

1i i

3.1.2.3 Defining the kinematics of the wing

18

Consider an inertial frame X,Y,Z which is stationary and a body frame x,y,z which

moves to left of the page as shown in Fig. 3.5. The flight path of the origin and

orientation of the x ,y ,z system is assumed to be known and is prescribed as:

( ) ( ) ( ) X X t Y Y t Z Z t= = =0 0 0 0 0 (3.7)

( ) ( ) ( ) = =t t tφ φ θ θ ψ ψ= (3.8)

( ) ( ) ( ), ,V t W t , ,U t in the The time-dependent kinematic velocity components x y z

frame due to the translation velocity and rotation of the body frame of reference can be

computed as follows:

3

3

3

t

t

t

U U qz ryV V ry pzW W py qx

tη

⎡ ⎤⎢ ⎥− +⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥= + − +⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ∂⎣ ⎦ ⎣ ⎦ − + −⎢ ⎥

∂⎣ ⎦

(3.9)

where

[ ]3 3 3, ,U V W T

( )

is the translational velocity components observed in the body fixed frame

( ) ( ) ( ) ( ), , , ,p q r t t tφ θ ψ= && & is the angular speed of the body fixed frame with respect

x

y

z

X

Y

Z

Stationary inertial frame of reference

Body-fixed frame of reference

Starting vortex

Fig. 3.5: Inertial and body coordinates used to describe the motion of the body.

19

to inertial frame

η is the function that defines the geometry of the wing surface and ( ), ,x y tη η=

( )

The velocity with reference to the body fixed frame can be calculated by

taking a transformation between the two coordinates and may be computed as follow:

3 3 3, ,U V W

03

3 0

3 0

1 0 0 cos 0 sin cos sin 00 cos sin 0 1 0 sin cos 00 sin cos sin 0 cos 0 0 1

t t t t

t t t t

t t t t

XUV YW Z

θ θ ψ ψφ φ ψ ψφ φ θ θ

−−⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢= − ⎢⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢− −⎣ ⎦ ⎣ ⎦ ⎣⎣ ⎦ ⎣ ⎦

&

&

⎡ ⎤⎤⎥ − ⎥⎥⎥⎦

&

( ), ,

(3.10)

t t tφ θ ψ

, ,

are Euler angles of the body fixed frame around three main principal axes

X Y Z of the inertial frame. To define such a three-dimensional transformation

uniquely, the order of Euler sequence must be specified.

3.1.2.4 The wake shedding procedure

The wake vortex ring corner points will be created at each time step. At the first time

step, only two corner points of the wake are created. These two points are coincident

with the trailing segment of the trailing edge vortex ring and play the role of the

starting vortex. Therefore, during the first time step, no wake vortex ring exists. In the

Fig. 3.6: Wake shedding from the unsteady lifting surface during the first time step (upper) and during the second time step (lower). This picture is taken from Katz and Plotkin [22].

20

second time step, the wing trailing points of the trailing edge vortex ring advances and

free wake vortex rings can be created using the two new aft points of the trailing edge

vortex ring and the two points created in the previous time step. This shedding

procedure is repeated at each time step and a set of new trailing edge wake vortex rings

is created. An example of the wake created in the first and second time step is shown

in Fig. 3.6.

The strength of the most recently shed wake vortex ring is set to be equal to the

strength of the shedding vortex in the previous time step:

.tw T Et t−∆ (3.11) Γ = Γ

Once the wake is shed, its strength is unchanged and the wake carries no load and

moves with the local velocity. The above procedure is called wake roll up.

3.1.2.5 The influence coefficients

The influence coefficient is the normal velocity induced at the collocation point of the

ith vortex panel by the other vortex panels with unit strength circulation. According to

the Newman boundary condition derived in Eqn(3.1), the normal components of the

singularity-induced velocity plus the normal components of wing-wake motion

induced must be zero. This boundary condition can be expressed in term of influence

coefficients as follows:

[ ] 1 1 2 2 ... ... , , . 0i i ik k im im t w t w t w iia a a a U u V v W w nΓ + Γ + + Γ + + Γ + + + + =

r

nr

(3.12)

where

i is the normal vector of the ith vortex panel at its collocation point

21

kΓ

(

is the vortex strength of panel k

), ,t t tU V W

(

is the velocity induced by the movement of the wing

), ,u v w

a

un

w w w is the velocity induced by the wake vortex rings

ik is the normal velocity at collocation point i induced by unit vortex ring of panel k.

This coefficient can be calculated by:

,

.ik i

i k

a vw

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

r (3.13)

and can be calculated using the subroutine given in Eqn (3.5). The above

influence coefficients are calculated in the body fixed frame.

( ), ,u v w,i k

( ), ,U V W

( ), ,u v w

m MxN

In Eqn(3.12), t t t are the time-dependent kinematics velocity components due

to the motion of the wing and can be calculated using Eqn(3.9), are the

velocity components induced by the wake vortex rings and can be computed from the

strength of wake vortex rings in the previous time step;

w w w

= . By shifting the

known value to the right hand side, this equation (Eqn (3.14)) can allow us to solve for

the unknown vortex strengths of the present time step:

( ) inΓ + Γ + + Γ + + Γ = − + + +r

1 1 2 2 ... ... , , .i i ik k im m t w t w t wa a a a U u V v W w (3.14)

22

3.1.2.6 The linear set of equations of Newman boundary condition

Since the Newman boundary condition must be satisfied on the whole wing surface,

Eqn (3.14) is valid for every collocation point and a set of equations can be obtained as

follows:

[ ][ ]

[ ]

11111 12 1

221 22 2 2 2

1 2

, , ...., , ....

... ... ... ... ... ........ , , .

t w t w t wM

t w t w t wM

M M MN M t w t w t w MM

U u V v W w na a aU u V v W w na a a

a a a U u V v W w n

Γ⎛ ⎞⎡ ⎤ ⎜ ⎟⎜ ⎟⎢ ⎥ − + + +Γ ⎜ ⎟⎜ ⎟⎢ ⎥ = ⎜ ⎟⎜ ⎟⎢ ⎥ ⎜ ⎟⎜ ⎟⎢ ⎥ ⎜ ⎟Γ − + + +⎣ ⎦ ⎝ ⎠ ⎝ ⎠

r

r

⎛ ⎞− + + +r

(3.15)

Eqn (3.15) can be re-written in the form below:

[ ] [ ] [ ]1 1MxN Mx MxA RHSΓ = (3.16)

Unknown vortex strength of all the vortices then can be solved by multiplying the

inverse of the influence coefficient matrix [ ]MxNA to the two sides of Eqn(3.16):

[ ] [ ] [ ]1

11Mx MxNA RHSΓ =

Mx

− (3.17)

3.1.2.7 Pressure, velocity and load computations

Once all the vortex strengths are known, other flow field parameters can be calculated.

Pressure distribution over the wing surface can be computed using the unsteady form

of Bernoulli’s equation. The pressure difference across a vortex panel can be derived

from this unsteady form as follows:

( ), , 1, 1, ,

, ,, ,

. . . .t w t w

i j i ji j i j i jij t w i t w j

i j i jt w t wi j i j

U u U up V v V v

c bW w W w

ρ τ τ −−

+ +⎡ ⎤ ⎡ ⎤Γ −Γ⎜ ⎟Γ −Γ ∂Γ⎢ ⎥ ⎢ ⎥∆ = + + + +⎜ ⎟⎢ ⎥ ⎢ ⎥∆ ∆⎜ ⎟⎢ ⎥ ⎢ ⎥+ +⎣ ⎦ ⎣ ⎦⎝ ⎠

r r

t

⎛ ⎞

∂ (3.18)

23

where are the chord length and spanwise length of the vortex panel located

at point ( i ) in the array of panels;

,i j i j,,c b∆ ∆

, j

iτr is the chordwise tangential vector of vortex panel at its collocation point

jτr is the spanwise tangential vector of vortex panel at its collocation point

When the pressure difference is known, the force acting on the wing surface due to this

pressure difference can be calculated by:

( ) ,n, ,.i j i ji j

F p S∆ = − ∆ ∆r r (3.19)

The total force acting on the whole wing surface can be obtained by summing the force

on each panel. The force acting on each panel are located at the panel collocation

point.

3.1.2.8 Wake roll-up computation

Since the vortex wake is force-free, each vortex must move with the local stream

velocity. The local velocity is a result of the velocity components induced by the wing

and wake, and is usually measured in the inertial frame of reference ( , , )X Y Z

)

, at each

vortex ring corner point. To obtain the vortex wake roll up, at each time step the

induced velocity ( is calculated and then the vortex elements are moved by: , ,l

u v w

( ) ( ), , , , l

x y z u v w t∆ ∆ ∆ = ∆ (3.20)

The velocity induced at the wake vortex point is a combination of the wing and the

wake influence and can be obtained by using the same influence routine (Eqn(3.6)):

24

( )

( )

1

1

, , ( , , , , , )

, , , , ,w

l l l klkN

l l l wkk

u v w VORING x y z i j

VORING x y z i j

=

=

M N×

= Γ

+ Γ

∑

∑ (3.21)

MxN Nfor wing panels and wake panels. w

In the case of strong wake rollup, the size of the wake vortex ring can increase (or be

stretched) and if a vortex line segment length increases its strength must be reduced.

For the method presented above, it is assumed that this stretching is small and

therefore it is not accounted for.

3.2 UVLM program

3.2.1 Code implementation

Preprocessor Module

Computation Module

Postprocessor Module

Input

Kinematics parameters,wing geometerical parameters

The grid parameters,The transformed kinematic parametersThe velocity due to wing motion and wake

The circulation (¦£i ) distribution,Tangential Speed Distribution due to the wake and motion over the wing

Output

The distributed forces andtotal forces

The wake rollup parameters Iterate over time

The data flow The process pipeline Fig. 3.7: Flow chart of the UVLM code

25

The implementation of our ULVM code using the unsteady vortex-lattice algorithm

above is depicted in Fig. 3.7. To facilitate the coding process, the code is divided into

three main modules: pre-processor module, computation model and postprocessor

module. The role of each module is clearly shown in the figure. The discretization and

grid generation are computed in the pre-processor module. Influence coefficients and

vortex strengths of all lattices are computed in the computational module. Pressure

distribution and other flight parameters are computed in the postprocessor module. The

graphic user interface of the program along with an example of pop-up dialog of

computed force is shown in Fig. 3.8 and Fig. 3.9 respectively. Computed forces from

the program can be exported in Excel format file for further analysis.

The ULVM code has been developed based on the sample code provided in the

textbook of Katz and Plotkin [22]. This sample code was written to solve for

rectangular planforms in heaving and pitching oscillations with ground effect. We have

modified the code to solve for cambered wing in any arbitrary motion. Details of this

modification are given as follows:

Geometry model

Along the spanwise direction, the wing surface is divided into some major panels.

These panels are modelled using four corner points, root cambered line and rear

cambered line. Each major panel is then divided into small minor panels based on

the given number of rows and columns. The discretization procedure described

above allows to model wing with camber and spanwise chord variation.

Movement of the wing

26

The movement of the wing is a prescribed motion. This prescribed motion is a

combination of two separated movement: movement of the wing frame in the body

frame and movement of the body frame in the inertial frame. Prior to the execution

of the UVLM code, these motion parameters below should be manually calculated

and provided as input data for the code:

- The origin and Euler angles of the wing frame versus body frame in each

time steps.

- The origin and Euler angles of the body frame versus inertial frame in each

time steps.

Using the kinematic model above, any prescribed motion can be used for UVLM

code. This feature allows to use UVLM code not only for flapping wing problems

but also for other complex wing movement.

Graphic user interface and vortex visualization

To facilitate the user input, a friendly graphic user interface is provided. Sample input

data files are also provided to instruct users how to prepare proper input for the

program. Vortex visualization is added. Computed forces is presented in the chart and

can be exported to Excel format file for verification purposes.

3.2.2 Features and limitations

Assuming attached flows, the program can be used to model unsteady flows around

rigid wing problems. The wing is modelled using its mean camber line surface (zero

thickness). Some problems that can be modelled in the program are:

• Wing with chord variation along the spanwise direction

27

28

Fig. 3.8: GUI of the UVLM code

Fig. 3.9: An example of pop-up dialog of computed force

Fig. 3.10: Wake after a moth wing (top) and rolling wing (bottom).

• Wing movement at any prescribed motion. Pictures of wake after a moth wing and

rolling wing are presented in Fig. 3.10.

• Multi-wing modelling as shown in top picture of Fig. 3.10.

Some features that haven’t been incorporated in the program include:

• Wing flexibility ( wing twisting)

• Interference between wing-wing and wing-fuselage, and

• Viscous effect

29

Chapter 4 A pressure-based segregated finite volume solver for modelling flows in moving and deforming zones

4.1 Introduction

The unsteady vortex-lattice method presented in Chapter 3 is suitable for modelling

inviscid (non-viscous) flow. Although VLM has been shown to be a good method to

solve for flow at Reynolds number of 10 or above, its applicability in low Reynolds

regime of order 10 or below is still a question. When the viscous force becomes

dominant at the low Reynolds number (flight regime of small birds, insects and

MAVs), the question on the validity of non-viscous assumption must be answered

before applying them in micro air vehicle applications. A pressure-based finite volume

method solver, FLUENT, has been used for this purpose. This solver provides a mean

of comparison with the UVLM. The dynamic mesh model in this solver can be used to

model flows where the shape of domain changes with time due to the motion of the

domain boundaries. The motion can be a prescribed motion (e.g. we can specify the

linear and angular velocities about the center of gravity of a solid body with time) or

unprescribed motion where the subsequent motion is determined based on the solution

at the current time step (e.g. the linear and angular velocities are calculated from the

force balance on a solid body). When a starting mesh and description of any moving

zones in the model are provided, the update of volume mesh at the subsequent time

step will be handled automatically by the solver based on the new positions of the

boundaries. The unsteady Navier-Stokes equations will be solved for the pressure

5

4

30

distribution. Geometric conservation law is used to update flow-field parameters from

the current time step to the next time step. Similarly, just like any finite volume or

finite element method, this solver requires more computational effort than the unsteady

vortex-lattice method. Therefore only some selected cases are solved in this project

using the solver.

This chapter provides the basic theory that has been used by FLUENT to model for

flows in moving and deforming zones based on the user’s guide [24] of Fluent Inc.

This theory includes the basic grid conservation law (use to update flow field

parameters between time steps), dynamic mesh conservation equation, dynamic mesh

update methods (for remeshing and mesh-updating) and solid-body kinematics (for

defining solid-body motions). A brief description on setting-up a dynamic mesh

solution in FLUENT environment is also provided.

4.2 Grid conservation law

The grid conservation law is derived from the conservation of mass by setting

1ρ = 0andν = . It can be written as follows:

.AV AdV w dA

dt=∫ ∫

d

n 1nt

(3.22)

In words, it can be understood that the change in volume of each control volume

between two time instants, t and + , must be equal to the volume swept by the cell

boundary during that time interval 1n nt t t+∆ = − .

4.3 Dynamic mesh conservation equations

The integral form of the conservation equation of a general scalar,φ , on an arbitrary

control volume, V, whose boundary is moving can be written as:

31

( ). .gV V V VdV u u dA dA S dV

dt φρφ ρφ φ∂ ∂

+ − = Γ∇ +∫ ∫ ∫ ∫d r rr r (3.23)

where ρ is the fluid density

is the flow velocity vector ur

gur

Γ

S

is the grid velocity of the moving mesh

is the diffusion coefficient

φ is the source term of φ

is the control volume V

Vand ∂ is the boundary of the control volume V.

The time derivative term in Eqn(3.23) can be rewritten using the first order backward

difference scheme, as

( ) ( )1n n+

1n

V

V Vd dVdt t

ρφ ρφρφ

−=

∆∫ (3.24)

where n and n+1 denote the respective quantities at the current and the next time level.

The n+1 time level volume V + is computed from

1n nV Vdt

+ dV t= + ∆ (3.25)

where dVdt is the time derivative of the control volume. In order to satisfy the grid

conservation law, dVdt is computed from

32

. .fn

g gj jVj

dV u dA u Adt ∂

= = ∑∫r rr r (3.26)

where fn is the number of faces on the control volume and jAr

is the j face area

vector. The dot product .gj ju on each control volume face is calculated from Ar

. jgj ju A

tVδ

=∆

r

V

r (3.27)

where jδ is the volume swept out by the control volume through the control volume

face j over time step . t∆

4.4 Dynamic mesh update methods

Three mesh motion methods are available in FLUENT to update the volume mesh in

the deforming regions subject to the motion defined at the boundaries:

• Spring-based smoothing

• Dynamic layering

• Local remeshing

Two methods which are used in the modelling of flapping wing are spring-based

smoothing and local remeshing.

4.4.1 Spring-based smoothing

In the spring-based smoothing method, the edges between any two mesh nodes are

idealized as a network of interconnected springs as shown in Fig. 4.1. The spring

stiffness of the edge between node i and node j is defined as follows:

33

1ij

i j

kx x

=−

r r (3.28)

where and jix xr r and j

)in

i

are the position vectors of nodes i respectively. Using Hook’s

law, the force on a mesh node can be written as:

(1

i ij jj

F k x x=

= ∆ −∆∑r r r (3.29)

where ix∆r and jx∆ are the displacements of node i and its neighbour r j , is the

number of neighbour nodes connected to node i . At equilibrium, the net force on a

node due to all the springs connected to the node must be zero. This condition results

in an iterative equation as shown in Eqn (3.30):

in

1i

ij jjn

i n

ijj

k xx

k

+

in

∆∆ =

∑

∑

r

r (3.30)

Since displacements are known at the boundary nodes (after boundary nodes have been

updated), Eqn.(3.30) is solved for Jacobi sweep on all interior nodes. At convergence,

the positions are updated such that:

1n n n convergedi ix x x+ −= + ∆r r r (3.31) i

By default, spring-based smoothing method is used

only on triangular and tetrahedral cell zones. For non-

triangular and non-tetrahedral cell zones, spring-based

smoothing method can be used but subject to Fig. 4.1: Spring-based connections

between any two mesh nodes.

34

several conditions such that:

• The boundary of the cell zones moves predominantly in one direction.

• The motion is predominantly normal to the boundary zone.

An example of meshes before and after spring-based smoothing is shown in Fig. 4.2.

Fig. 4.2: Compression of cylinder before (left) and after (right) spring-based smoothing.

4.4.2 Local remeshing method

On zones with triangular or tetrahedral mesh, the spring-based smoothing method is

normally used. When the boundary displacement is large compared to the local cell

sizes, the cell quality can deteriorate or the cells can become degenerate. This will

invalidate the mesh (e.g. result in negative cell volume) and consequently, will lead to

divergence solutions at the next time step.

To solve this problem, FLUENT will mark cell that violates the skewness or size

criteria and locally remeshes the marked cells; otherwise, the mesh is locally updated

with the new cells (with solution interpolated from the old cells).

FLUENT evaluates each cell and marks it for remeshing if it meets one or more of the

following criteria:

• It is smaller than a specified minimum size

• It is larger than a specified maximum size

35

• It has a skewness which is greater than a specified maximum skewness value.

The local remeshing method is used only in cell zones that contain tetrahedral or

triangular cells. Pictures of mesh around a cylinder before and after applying local

remeshing method are shown in Fig. 4.3.

Fig. 4.3: Expansion of mesh around a cylinder before (left) and after (right) local remeshing.

4.5 Solid-body kinematics

FLUENT uses solid-body kinematics if the motion is prescribed based on the position

and orientation of the center of gravity of a moving object. This is applicable to both

cell and face zones.

The motion of the solid-body can be specified by the linear and angular velocities of

the center of gravity. FLUENT allows the velocities to be specified either as profiles or

user-defined functions (UDFs). FLUENT assumes that the motion is specified in the

inertial frame.

If the motion is specified using a profiles, the components of the velocities must be

described using the following profiles fields:

• Linear velocity ( ), ,x y zv v v as a function of time

• Angular velocity as a function of time. ( , ,x y zw w w )

36

In addition to the motion description, the starting location of the center gravity and

orientation of the solid body must be specified. FLUENT automatically updates the

center of gravity position and orientation at every time step such that:

1 .n n n

1 . .cg cg cg

n n ncg cg cg

x x v t+ = + ∆r r r

G tθ θ+ = + Ω ∆r r r (3.32)

where cgxr and cgθr

are the position and orientation of the center of gravity, and cgvr cgΩr

are the linear and angular velocities of the center of gravity, and G is the

transformation matrix that defines the choice of θr

. By default, θr

, ,

is the vector

describing the three Euler angles around the three principal axes X Y Z of the inertial

frame, thus is taken to be an identity matrix. If G θr

is chosen to be a different set of

Euler angles, the solid-body motion must be specified using a user-define motion

where the appropriate form of can be defined. G

Fig. 4.4: Solution procedure in FLUENT

37

4.6 General procedure for dynamic mesh solution in FLUENT

The general procedure for setting up a dynamic mesh problem in FLUENT is

illustrated in Fig. 4.4. Mesh generation is performed by GAMBIT, a pre-processor

module of FLUENT. Mesh can be generated from solid models created inside

GAMBIT or imported from other CAD software. In this project, the three-dimensional

wing is drawn in SolidWorks® and imported into GAMBIT using parasolid format.

When the mesh has been successfully imported into FLUENT and other solution

parameters have been defined, before iteration starts, preview of the mesh motion must

Fig. 4.5: Gambit GUI (top) and FLUENT GUI (bottom)

38

be done first to ensure that the solver can handle successfully the mesh updating

process at all time steps.

39

Chapter 5 Investigations of two-dimensional flows

5.1 Introduction

The theory and numerical solutions related to the two numerical methods which were

used for the numerical computation in this project have been presented in Chapter 3

and Chapter 4. This chapter presents the computation of a two-dimensional airfoil

undergoing pure plunging motion using these two numerical methods. The case has

been investigated by Jones et al. [25] using two-dimensional panel method and

Euler/Navier-Stokes solver. It was again computed in this project in order to validate

the accuracy of the UVLM and FLUENT solvers before extending them to treat three-

dimensional cases.

5.2 Description of wing-motion

In this two-dimensional computation, a NACA0014 airfoil which undergoes pure

plunging motion (shown in Fig. 5.1) is investigated. The plunging motion is defined

such that

( ) ( )cosz t h κτ= (4.1)

where h is the plunging amplitude, τ is the non-dimensional time and is the reduced

frequency. The reduced frequency is the non-dimensional form of the plunging

(oscillating) frequency, which is defined by:

κ

40

fcUπκ

∞

= (4.2)

Fig. 5.1: Profile of airfoil path in pure plunging motion

Non-dimensional time is written in the form such that

tUc

τ ∞= (4.3)

A plunge amplitude of is chosen for all cases. The parameter-space for the

investigation follows strictly the work of Jones at al. [

0.4h c=

25] for the purpose of

comparing results. These parameters include reduced frequency, wing aspect ratio and

mean angle of attack.

Numerical results in the above-mentioned paper were evaluated using a broad range of

numerical codes: 2-D panel method, three-dimensional panel method (CMARC), two-

dimensional unsteady compressible Euler/Navier Stokes solver (NSTRANS) and the

finite volume compressible Euler/Navier-Stokes solver (FLOWer). CMARC is a PC-

based version of PMARC (Panel Method Ames Research Center), a low-order 3D

panel code from Aerologic. CMARC has some advantages over our UVLM as follows:

• CMARC allows us to model wing with upper and lower surface so it can be

used with thick wing. UVLM models the wing using it mean camber surface

and that’s why it can only be used with thin wing (thickness <12% ). c

• Panels in CMARC are distributed with half-cosine along the span, with the

tightest spacing at the tip and full-cosine spacing in the chord-wise direction.

41

This feature gives a better result on the effect of wing-tip vortex on

aerodynamic forces. A picture of three-dimensional wing in CMARC is shown

in Fig. 5.2. Currently our UVLM assumes uniform distribution along chordwise

direction as well as spanwise direction. The number of panels along spanwise is

twice as that of chordwise.

Fig. 5.2: Three-dimensional rectangular wing with half-cosine and full-cosine distribution along

wing span and wing-chord respectively in CMARC

In order to compare the results between CMRAC and our UVLM, the results from

CMARC as well as from above numerical codes were extracted directly from the

paper.

5.3 Problem setup

The Reynolds number and reduced frequency are used to match results between our

simulations with published results. This matching defines the simulation parameters

Table 5.1: Simulation parameters for two-dimensional plunging motion. Non-dimensional values in the paper

Chord length c(m) Stream velocity U∞ (m/s)

Plunging freq f(Hz)

6Re 10 , 0.4κ= = 1 15.6 1.986 6Re 10 , 0.2κ= = 1 15.6 0.993

Table 5.2: Simulation parameters for 2D unsteady panel method. No Motion configuration Wing aspect ratio Simulation 1 6Re 10 , 0.4κ= = 4AR = Simulation 2 6Re 10 , 0.4κ= = 8AR = Simulation 3 6Re 10 , 0.4κ= = 20AR = Simulation 4 6Re 10 , 0.4κ= = 100AR =

42

used for ULVM and FLUENT as shown in Table 5.1. In this table, chord length of the

airfoil is fixed at1 . The free stream velocity is calculated from Reynolds number

and plunging frequency is calculated from the reduced frequency as follows:

( )m

Re. .v U, 2 .

U fc c

κπ

∞ ∞∞ = = (4.4)

where v is the kinematic viscosity at standard atmosphere condition. 51.56 10−= ×

4,8, 20

∞

5.3.1 UVLM simulations

A finite three-dimensional flat plate was used to model the two-dimensional airfoil. A

total of 722 panels were used to model the plate, which is 1.5 times of the total panels

used in CMARC. Three cycles are required for each simulation in order to diminish the

transient. Each cycle has 50 time steps. Four simulations at four aspect ratio

and 100 were computed on a Pentium IV-2.9GHz desktop and requires around 1 hour

to complete three cycles. Wake after the airfoil after one completed cycle can be seen

in Fig. 5.3. The parameters of each simulation are presented in Table 5.2.

Fig. 5.3: Wake after NACA0014 airfoil in pure plunging motion

5.3.2 FLUENT simulations

43

A two-dimensional grid around an NACA0014 airfoil is generated in GAMBIT. The

grid is divided into two parts: a non-deforming grid and a deforming grid. The

deforming grid consists of cells which are adjacent to the boundary of the airfoil and

deformed due to the movement of the airfoil. The non-deforming grid consists of cells

which aren’t affected by the movement of the airfoil boundary. These two grids are

connected by using grid-interface. Pictures of these grids are shown in Fig. 5.4.

The movement of the airfoil is defined using the user-defined function (UDF). This

UDF defines the plunging velocity of the quarter chord point of the airfoil and can be

found in Appendix A.1. This plunging velocity is derived from the plunging motion in

Eqn(4.1) as follows:

( ) ( ) ( )sinv z hτ τ κ κτ= = −& (4.5)

Spring-based smoothing and local-remeshing with their parameters presented in Table

5.3 are used for mesh-updating. A total of 100 time-steps is needed for each cycle.

This number of time step is sufficient to prevent the negative mesh volume which can

occur in the mesh-updating process if the movement of the airfoil boundary is greater

than the minimum cell height of adjacent cell zone. Two cycles are performed for each

Fig. 5.4: FLUENT 2D Navier-Stokes grid. The picture on the left is the initial grid consisting of

deforming zone and non-deforming zone separated by grid interface (solid lines). The picture on the right shows the deformation of grid-cells near the airfoil boundary after one cycle.

44

Table 5.3: Setting-parameters for updating mesh of two-dimensional FLUENT problem. Options Parameters ValueSpring-based Smoothing Spring Constant Factor 0.4 Boundary Node Relaxation 0.7 Convergence Tolerance 0.001 Number of iterations 20Local Remeshing Minimum Cell Volume 1e-13 Maximum Cell Volume 0.0016 Maximum Cell Skewness 0.2 Size Remesh Interval 30 Size Function Resolution 3 Size Function Variation 3.8327 Size Function Rate 0.3

of the simulations. Both inviscid and viscous solutions are computed.

5.4 Results and discussion

Some comparisons have been made between our results and previous solution obtained

by Jones et al. [25]. The first comparison is performed between the results obtained

using our UVLM code and CMARC. The purpose is to determine the accuracy of our

UVLM code with respect to this commercial code. The second comparison is made

between solutions obtained using 2D FLUENT (both inviscid and viscous solutions)

with solutions computed by 2D panel method. The aim is to find the difference

between the panel method and the finite volume method as well as between the

inviscid and viscous solutions in flapping flight.

Experiments for the above-mentioned two dimensional airfoil were also carried by the

group of Jones in their low speed wind tunnel. Shown in Fig. 5.5 is the installation of

their test model in the wind tunnel. For this test model, it is very difficult to isolate the

drag which is created by the wing and the drag created by the rest of the model during

the experiments. Therefore, in order to facilitate comparisons with theory and various

numerical methods, in the original work of Jones et al [25] steady-drag was already

removed from all the experiments as well as numerical results. Thus, to compare our

45

results with their published results, steady-drag is also removed from our results. The

steady drag is defined as the drag created by the non-moving wing at its present mean

angle of attack. This steady drag is calculated by running simulation of the non-

moving wing at their appropriate mean angle of attack in each solver prior to starting

the unsteady motion.

Fig. 5.5 : Installation of test model in wind tunnel by K.D. Jones et al.

As the steady drag is calculated, the mean thrust coefficient is then computed by:

steadyT D DC C C= − +

4,8,100

(4.6)

5.4.1 Comparison between unsteady vortex-lattice methods

In Fig. 5.6, results for inviscid three-dimensional are compared. The results shown in

the figure are lift coefficient for values of aspect ratio of . There’s a good

agreement between the results obtained using our UVLM and CMARC. The unsteady

lift coefficient in the three-dimensional UVLM as well as CMARC is close to the two-

dimensional solution with increasing wing aspect ratio. As thickness distribution is not

accounted for in ULVM code, thrust prediction could not be compared because UVLM

failed to predict the drag for a flat-plate (zero thrust).

46

-1-0.8-0.6-0.4-0.2

00.20.40.60.8

1

0.00 0.20 0.40 0.60 0.80 1.00

t/T

CL

3DPANEL-CMARC3D PANEL UVLM2D PANEL

AR=4AR=8AR=100

Fig. 5.6 : Unsteady lift coefficient, 0.4κ =

-0.6-0.4-0.2

00.20.40.60.8

1

0 0.2 0.4 0.6 0.8 1

t/T

CL

2D PANEL2D FLUENT(Inviscid)

α=4α=2α=0

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t/T

CT

2D PANEL2D FLUENT (Inviscid)

α=4α=2α=0

Fig. 5.7: Unsteady lift and thrust coefficient between 2D FLUENT (Inviscid) and panel

method at 0.2κ =

5.4.2 Comparison between inviscid and viscous solutions

In this section, two comparisons are made to investigate the use of FLUENT in both

inviscid and viscous computation. The first comparison shown in Fig. 5.7 is to

illustrate the use of FLUENT in inviscid computation. This comparison is made

47

between the published results obtained using 2D panel method and those obtained

using 2D FLUENT in inviscid computation. And the second comparison shown in Fig.

5.8 is for the purpose of illustrating the use of FLUENT in viscous computation.

Fig. 5.7 is the results for two-dimensional inviscid solutions obtained using 2D panel

method and 2D Fluent in inviscid. A good agreement can be seen in the lift direction,

especially at zero mean angle of attack. At higher mean angle of attack, unsteady lift in

the downstroke of 2D FLUENT falls below that obtaining by 2D panel method while

the lift in the upstroke is equal or a bit higher than the results obtained by 2D panel.

The same trend can be found in the thrust prediction shown in Fig. 5.7.

Shown in Fig. 5.8 is a comparison between the published results obtained using 2D

panel method with those obtained using Fluent in inviscid and in full viscous. In this

figure, the mean lift and mean thrust coefficient of one cycle is compared. The highest

prediction of the above coefficients belongs to 2D panel method and the lowest

prediction belongs to FLUENT in full viscous. Prediction of FLUENT in inviscid falls

in the middle of the prediction of 2D panel method and prediction of full viscous

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 1 2 3 4AOA

Mea

n C

L

5

Mean lift PANELMean lift FLUENT (Viscous)Mean lift FLUENT (Inviscid)

0

0.01

0.02

0.03

0.04

0.05

0 1 2 3 4 5AOA

Mea

n C

T

Mean Thrust PANELMean Thrust FLUENT (Viscous)Mean Thrust FLUENT (Inviscid)

Fig. 5.8: Mean lift (left) and thrust (right) coefficient between inviscid and viscous solutions at 0.2κ =

48

FLUENT. As the angle of attack increases, the difference in prediction of mean lift and

mean thrust coefficients using these above methods slightly increase.

5.5 Conclusion

The comparison between UVLM and CMARC proved that the numerical

implementation of UVLM is good and it can be used for further study. For FLUENT,

investigations in both inviscid and viscous solution has been done. However, in other

to have a fully evaluation of its applicability for flapping wing problems, more

investigations at finer mesh should be required. In this thesis, due to time constraints,

this is not investigated fully. In the mean time, FLUENT can be used as a tool to study

the effect of viscous and other phenomena which cannot be captured in the panel

method.

49

Chapter 6 Investigations of three-dimensional flows

6.1 Introduction

The two-dimensional investigation using UVLM and FLUENT in Chapter 5 proved

that these codes can be used to further investigate flows in three-dimensional cases.

The investigation in this chapter is limited to a high Reynolds numbers of 10 . As

UVLM can provide the results without high computation effort, it is mainly used in

this three-dimensional investigation to study effects of some flapping parameters such

as reduced frequency, flapping amplitude, and wing pitch angle. The flow field at

some selected cases were performed in FLUENT to capture phenomenon such as

leading-edge vortex. The application of panel methods and FLUENT for solving

flapping flight at low Reynolds number of 10 will be evaluated based on the

comparison between computed and measured force data which will be presented in

Chapter 7.

5

4

4

d c=

6.2 Description of wing-motion

The model used in this three-dimensional investigation is a rectangular wing with

aspect ratio of undergoing flapping and pitching motion. An isometric view of the

model is shown in Fig. 6.1. For the purpose of installing the wing pitch controller, the

wing root is put far away from the flapping axis at a distance . The pitching

motion is defined as the rotation angle of the wing at its quarter-chord axis. This

pitching angle is denoted by γ . The flapping is a simple harmonic motion with equal

time for downstroke and upstroke, which is defined by:

50

Fig. 6.1: Three-dimensional flapping wing model.

( )cos 0θ θ κτ= (5.1)

where 0θ is the flapping amplitude and κ is the reduced frequency.

The net angle of attack is a combination of the pitch angle γ and a time-dependent

angle induced by the flapping motion which is calculated by:

( ) 1tan y

Uα τ −

∞

⎛ ⎞= ⎜

⎝ ⎠

v⎟ (5.2)

where is the velocity induced at position y along the wing span. A plot of this angle

of attack is shown in Fig. 6.2. As the wing experiences through a complete sinusoidal

flapping motion, the net angle of attack also follows a sinusoidal curve with magnitude

yv

-30

-20

-10

0

10

20

30

0 0.2 0.4 0.6 0.8

deg

1

Net angle of attackat w ing tip

Net angle of attackat half span

t/T

Fig. 6.2: Net angle of attack of the three dimensional flapping wing at and 00 30θ = 00γ =

51

increasing from wing root to wing tip. The figure also shows that flapping wing flight

is dominated by large instantaneous angle of attack.

6.3 Solution procedure

Simulations were performed on a wide range of parameters. These parameters include

wing camber, wing pitch angle, flapping frequency and flapping amplitude. Most of

the simulations were performed at Reynolds number of , reduced frequency

and flapping angle of unless otherwise stated when studying effect of

flapping frequency or flapping amplitude. The effect of wing camber is computed first

and the best camber will be used for the subsequence simulation. A picture of wake

after the wing for a completed cycle in shown in Fig. 6.3.

57.27 10×

0.24κ = 030±

Fig. 6.3: Wake behind three-dimensional rigid cambered wing in one flapping cycle.

6.4 Results

6.4.1 Effect of wing camber

Four different wings with different level of camber shown in Fig. 6.4 (in percentage of

wing chord %c) are chosen to study the effect of camber in flapping flight.

52

Fig. 6.4: Mean camber lines of various airfoils in percentage of wing chord

Variation of unsteady lift and thrust coefficients with respect to wing cambers is

shown in Fig. 6.5. The lift coefficient increases with the increasing mean camber. Plate

generates zero thrust. For camber wing, thrust is generated in the downstroke and drag

is produced in the upstroke. As thrust and drag increase as camber increases, the

thrust/drag ratio is used in this project to choose the best wing among the four above-

mentioned wings. The highest ratio is found in the case of the wing using S1091

airfoil. As a result, this wing is selected for subsequent simulations to study the effect

of other parameters. We can notice in Fig. 6.5 that thrust in the downstroke begins to

drop when the wing camber reaches 6.1%, which is around the mean camber of the

wing using S1091 airfoil.

-0.4-0.3-0.2-0.1

00.10.20.3

0 0.2 0.4 0.6 0.8 1

t/T

CT

Fig. 6.5: Unsteady lift and thrust coefficient* at various cambers

3 Plate

2 NACA2412

1 S1091C L SARATOV

0 0 1 0.2 0.4 0.6 0.8-1

-2 t/T

53

6.4.2 Effect of flapping amplitude

Simulations in this section are performed for the wing with S1091 at zero wing pitch

angle for both downstroke and upstroke; flapping frequency was at 1Hz. Mean lift and

thrust coefficient are shown in Fig. 6.6. Mean lift and mean thrust increase versus

flapping amplitude. The figure shows that positive thrust is generated at flapping

amplitude greater than . 030

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 10 20 30 40 5Flapping amplitude (deg)

Coe

ffici

ent

0

Mean thrust coeff icient

Mean lif t coeff icient

Fig. 6.6: Mean lift and thrust coefficient versus flapping amplitude

6.4.3 Effect of wing pitch angle

The variation of lift and thrust coefficient with respect to wing pitch angle are shown

in Fig. 6.7. More thrust is generated when the wing pitches down in the downstroke

-4

-2

0

2

4

6

0 0.2 0.4 0.6 0.8 1t/T

CL

-1.5

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1t/T

CT

0 deg5 deg-5 deg

Fig. 6.7: Unsteady lift and thrust coefficient at various wing pitch angles.

54

Fig. 6.8: Improvement of thrust in the downstroke (left) and reduction of drag in the upstroke

(right) with respect to zero pitch angle.

050

100

150200250300

350400

0 5 10 15 20 25 30Coupled pitch angle (deg)

(%) Thrust improvement

Lif t reduction

Fig. 6.9: Improvement of thrust and reduction of lift at various coupled pitch angles with respect

to zero coupled pitch angle.

and less drag is produced when the wing pitches up in the upstroke. During the

downstroke, a pitch angle of order of -50 gives a thrust which is 40% greater than that

of a zero pitch angle. In the upstroke, a pitch angle of order of +50 gives a drag which

is 45% less than that of a zero pitch angle. The trend of improving thrust in the

downstroke and reducing drag in the upstroke are clearly shown out in Fig. 6.8. The

right curve in this figure denotes the amount of drag that can be reduced (drag

reduction) with respect to the drag at zero pitch angle during the upstroke. The left

curve in the figure denotes the amount of thrust that can be increased (thrust

improvement) with respect to the thrust at zero pitch angle in the downstroke. An

55

optimum pitch angle exists in the thrust improvement as well as drag reduction. In the

downstroke, the left curve in Fig. 6.8 shows that thrust can be increased up to 80% at

the optimum pitch angle of -170 with respect to at zero pitch angle. In the upstroke, the

right curve in this figure also shows that drag can be reduced to nearly 100% at the

optimum pitch angle of +170 with respect to that at zero pitch angle (this means that

drag can be reduced to zero in the upstroke). Therefore, by turning the wing to the

optimum pitch angle in each stroke, thrust can be generated without any power loss by

drag. Assuming that the wing pitches down in the downstroke and pitches up in the

upstroke at the same angle (i.e. pitch down to 05− and pitch up to+ ), the

improvement of the thrust in comparison with the loss in lift is shown in Fig. 6.9. The

figure shows that this configuration can increase the thrust to reach 3.5 times of that at

zero-pitch angle configuration while maintaining the loss in lift at around 25% of the

lift at zero-pitch angle configuration (i.e. wing pitch angle of the wing maintains at

zero during both downstroke and upstroke). Again, the configuration shows a very

good compromise between lift and thrust generation. As a result of this, the so-called

coupled wing pitch angles configuration is used in the study of flapping frequency. An

illustration of the above configuration is shown in Fig. 6.10.

05

6.4.4 Effect of flapping frequency

Fig. 6.10: Illustration of coupled wing pitch angles in flapping flight

56

The study of the effect of flapping frequency on the lift and thrust coefficients was

performed based on the couple wing pitch angles configuration found in section 6.4.3.

All the simulations in this study were performed at the coupled wing pitch angles of

(i.e. the wing pitches down in the downstroke and pitches up in the

upstroke respectively).

020 020− 020+

Fig. 6.11 shows the variation of instantaneous lift and thrust coefficients with respect

to flapping frequency. This figure shows that at high flapping frequency, lift and thrust

of the rigid cambered wing with coupled wing pitch angles can be greatly improved.

At high flapping frequency, a portion of thrust is also generated in the first half of the

upstroke. The mean lift and thrust coefficients versus flapping frequency is shown in

Fig. 6.12. At flapping frequency of 5Hz, the mean lift coefficient is 20 times while the

mean thrust coefficient is nearly 60 times of those at flapping frequency of 1Hz. This

-60

-40

-20

0

20

40

60

80

0 0.2 0.4 0.6 0.8 1

t/T

CL

-35-30-25-20-15-10

-505

10

0 0.2 0.4 0.6 0.8 1

t/T

CT 1 Hz

2 Hz

3 Hz

4 Hz

5 Hz

Fig. 6.11: Unsteady lift and thrust coefficient for coupled wing pitch angle of at various

flapping frequency

020

57

Fig. 6.12: Mean lift and thrust coefficient at various flapping frequency

-70 -60 -50 -40 -30 -20 -10

0 10 20 30

1 2 3 4 5

Average DragAverage Lift

Hz

presents a great promise for the application of rigid cambered wing with coupled pitch

angles in flapping wing flight.

6.5 Phenomena in flow field

The study of flow field around the three-dimensional rigid cambered wing in flapping

flight was performed using the unsteady three-dimensional solver FLUENT. The

investigation was also carried out for the coupled wing pitch angles of 200. Shown in

Fig. 6.13 is a breakdown of instantaneous lift and thrust coefficients between non-

viscous solution of UVLM and viscous solution of FLUENT solver at the flapping

frequency of 1Hz. In the downstroke, FLUENT predicts a larger lift and smaller thrust

than UVLM. In the upstroke, UVLM gives a larger lift force and smaller thrust force

than FLUENT solver. The difference of lift prediction in the downstroke and upstroke

between UVLM and FLUENT can be explained based on an aerodynamic feature, the

leading edge vortex (LEV), which is not captured in panel method. The pictures in

Fig. 6.14 ( printed in colour mode) denote the pressure distribution on the wing surface

at its mid downstroke and mid upstroke. When the wing flaps down, this LEV (which

was shown in Fig. 6.15) appear on the upper surface of the wing, near the leading edge

58

-4

-2

0

2

4

6

8

0 0.2 0.4 0.6 0.8 1Lift

coe

ffci

ent

UVLM 1Hz FLUENT3d 1Hz

-2

-1

0

1

2

3

4

0 0.2 0.4 0.6 0.8 1Dra

g co

effc

ient

Fig. 6.13: Lift and thrust coefficient between panel method and FLUENT

Fig. 6.14: Low pressure regions on wing surface in downstroke (left) and upstroke (right)

Fig. 6.15: Leading vortex in the mid downstroke computed by FLUENT (left) and experiment

(right)

t/T

CT

t/T

CL

59

and results in a low pressure region at the leading edge. The present of this low

pressure region (the blue region) enhances the lift generation in the downstroke, which

gives greater lift prediction than vortex-lattice method. The same LEV occurs in the

upstroke but in lower surface, which reduces the lift generation in the upstroke. An

experiment [26] of such leading-edge vortex in mid downstroke of a taper wing in

flapping motion was found as shown in the right picture of Fig. 6.15. For the thrust

prediction, there is no immediate and satisfactory explanation for the difference, but

we suspect that this difference could be due to viscosity. In this thesis, due to time

constraints, this is not investigated fully.

6.6 Discussion and conclusion

The effects of wing camber, flapping amplitude, wing pitch angle and flapping

frequency in flapping flight of a rigid rectangular wing at high Reynolds number is

summarized in this section. Although in flapping flight we are more concerned is with

low Reynolds number, but all simulations in this chapter are conducted at high

Reynolds number first. The reason is that our UVLM and FLUENT haven not been

proven for solving flapping-flight at low Reynolds number. Vortex shedding and

detached flow which are dominant at flapping-flight at low Reynolds number has not

been accounted for in our UVLM. Therefore, all simulations carried out using the

above solvers have been investigated at high Reynolds number with two purposes. The

first purpose is to provide us a general understanding in the aerodynamics of the rigid

cambered wing in flapping flight. The second one is to provide us some basic

parameters to set up our wind tunnel experiments for the rigid cambered wing at low

Reynolds number.

60

The lift/thrust performance of rigid cambered wing in flapping flight at high Reynolds

number can be summarized as follows:

• Unsteady lift increases with respect to the increase in wing camber. Generally,

thrust is generated in the downstroke and drag is generated in the upstroke.

Maximum thrust/drag ratio is obtained at the mean wing camber around 6%c.

• Both mean lift and thrust increase with increasing flapping amplitude. However,

flapping amplitude of 300 or above is required for positive thrust.

• For maximize thrust performance, an appropriate change in the pitch angle in the

downstroke and upstroke is required. A negative pitch angle in the downstroke is

required to maximize the thrust and positive pitch angle in the upstroke is required

to minimize the drag. Using the so-called couple pitch angle configuration above,

thrust can be improved to reach 3.5 times of that at zero-pitch angle configuration

while the loss in lift is only 25% of the lift at zero pitch angle configuration (wing

pitch angle maintains at zero during both strokes). Optimum pitch angles exist for

each stroke and they are equal in magnitude.

• At the coupled pitch angles of 200 ( i.e. the wing pitches down to -200 in the

downstroke and pitches up to +200 in the upstroke), mean lift coefficient increases

up to 20 times and mean thrust coefficient increases up to 60 times when flapping

frequency was adjusted from 1Hz to 5Hz.

To explain the principle behind these effects, more investigations in flow field should

be done. However, the profiles of the net angle of attack below can provide a hint for

this principal. Fig. 6.16 shows several profiles of the net angle of attack at the wing

root and the wing tip of the rigid cambered wing above. The maximum angle of attack

at the wing root is relatively small than that at the wing tip. At zero pitch angle, the

61

maximum angle of attack at the wing tip can be increased up to 500. This high angle of

attack may cause dynamic stall and produce larger drag at some portions of the wing

and therefore reduce the lift and thrust generation of the wing. As shown in the figure,

a constant pitch angle only reduces this high angle of attack at one stroke and increases

this high angle of attack in the other. And therefore, lift and thrust can not be improved

much. The coupled pitch angle is a combination of both and moderate angle of attack

can be obtained at both downstroke and upstroke. At the optimum pitch angle, the

maximum angle of attack along the wing span is around . This maximum angle of

attack may ensure no portion of the wing fails to generate lift/thrust and that’s why the

maximum lift and thrust coefficient is reached at this optimum angle. This explanation

is an assumption and more research should be done to verify it.

030±

-30

-20

-10

0

10

20

30

0 0.2 0.4 0.6 0.8 1deg

-70-60-50-40-30-20-10

010203040506070

0 0.2 0.4 0.6 0.8 1

Non-dimensional time

deg

Zero deg (Constant pitch) -10 deg (Constant pitch)

10 deg (Constant pitch) 17 deg (Optimum coupled pitch)

Fig. 6.16: Net angle of attack at wing root (top) and wing tip (bottom) for various wing pitch angles

62

In conclusion, the coupled pitch angle for lift and thrust generation of a rigid cambered

wing presents a promise in its application for flapping wing flight. However, the above

investigation is performed at relative high Reynolds number; the same investigation

should be performed at the Reynolds regime of order of 10 before this configuration

can be applied to the application of micro air vehicles. The experiment in Chapter 7 is

performed for this purpose.

4

63

Chapter 7 Experimental Studies

The three-dimensional investigation of flows at Reynolds number of in

Chapter 6 shows that the rigid cambered wing can be used as an alternative for lift and

thrust generation in flapping wing flight at high Reynolds number. A dramatic

improvement in lift and thrust coefficient can be obtained based on the combination

effect of coupled pitch angles and flapping frequency. However, the observation

cannot be readily applied to micro air vehicles flying at low Reynolds number.

Performance of the wing at the high Reynolds number can change at low Reynolds

number. Although UVLM has been proven to be an adequate tool to understand flow

around a rigid cambered wing at high Reynolds number but a previous work by Ames

et al. [28] proved that it may not be appropriate for studying flapping wing flight at

low Reynolds number. Therefore, in order to evaluate the application of the rigid

cambered wing for micro air vehicles, both simulation and experiment have been

carried out at the MAV Reynolds number regime in this project. The experiment plays

an important role in this project. It is a tool for assessing not only the performance of

the rigid cambered wing in flapping flight based on actual flight condition but also the

accuracy of our ULVM code in solving unsteady flows at low Reynolds number.

57.27 10×

The experiments in this chapter were performed for a 30cm span flapping model at

Reynolds number of the order of 104. The model is constructed using balsa wood.

Some controllers are added to the model to control the pitch angle and flapping

frequency. Force data were recorded using a 2-axis force balance system. The force

64

data were used to verify the numerical results computed by UVLM at the same

Reynolds number.

7.1 Testing model

The basic parameters of the test model are shown in Table 7.1 and a picture of the

actual model is shown in Fig. 7.1. The airfoil and flapping amplitude are similar to the

three-dimensional case in Chapter 6. The wing is made of balsa wood with a carbon

tube placed at the quarter-chord as the spar. A carbon rod was put inside the carbon

tube with one end connected to the flapping mechanism and the other end glued with a

secured-bolt to secure the wing during flapping. The flapping mechanism was

borrowed from previous project [4]. A small modification is made to adjust the

flapping angle to the desired value. The whole flapping mechanism is redrawn in

SolidWorks and motion simulation is done to find out the appropriate length of the

crank-shaft for the desired value of flapping angle. The CAD model of the mechanism

along with a plot of its flapping angle is shown in Fig. 7.2.

Table 7.1: Basic parameters of the test model Wing span of each wing 10 cmWing chord 5 cmWing root to flapping axis 5 cmAirfoil section S1091Aspect ratio 4Flapping amplitude 30± deg

Fig. 7.1: Real model for wind tunnel experiment

65

Fig. 7.2: CAD model and plot of flapping angle

7.2 Wing pitch controller

The pitch angle around the main spar at the quarter-chord of the wing is controlled

using a servo and two servo-arms placed in parallel as shown in Fig. 7.2. A BS2-IC

microcontroller from Parallax. Inc shown in Fig. 7.3 ( with its specifications in

Appendix C.1) is used to command the servo to rotate to the desired angle precisely.

Each servo has three wires: +5 volts, ground and signal. To make the servo move, we

connect two power wires to an appropriate power supply and connect the signal wire to

an output pin of the microcontroller. Before commanding the servo to move to the

desired angle, an experiment was done to figure out the relation between the pulse

width and the rotation angle of the servo in use. Two positive-going pulses, 1

milliseconds and 2 milliseconds long; were sent to the servo. The differential angle

between the two rotations commanded by these two signals determines the maximum

Fig. 7.3: BasicStamp 2 microcontroller (left) and set-up of servos on the carbon rod (right)

66

deflection angle of the servo. From this experiment, we can determine the pulse length

to command the servo to rotate 1 deg. For the servo used in this experiment, this value

is9 / degsµ . With this value in hand, commanding the servo rotating to the desired

angle can be easily done using a small code provided in Appendix C.2.

The microcontroller above can be programmed for automatically changing the pitch

angle in each stroke with the help of two limit switches. One limit switch is placed at

the top-death end and the other is placed at the bottom-death end as shown in the right

picture of Fig. 7.3. A signal will be created each time the wing reaches its dead-end

and this signal is fed back to the microcontroller and a signal will be sent to the servos

to rotate the wing to the desired pitch angle. This configuration can be used to study

the effect of coupled pitch angle as mentioned in section 6.4.3. However, each servo

takes an amount of time to rotate to the desired angle, usually 0.09-0.12s/600, this is

the reason why the configuration can only be used at low flapping frequency around 1-

2Hz.

7.3 Force balance

7.3.1 Characteristics of a force measurement system

Force measurement systems can involve a number of different physical principles but

their performance can be described by a number of common characteristics and terms,

and the behaviour of a system or transducer can be expressed graphically as a response

curve - by plotting the indicated output value (e.g. voltage) from the system against the

force applied to it. An idealized response curve is shown in Fig. 7.4 - where the force

applied increases from zero to the rated capacity of the force measurement system and

then back again to zero. The deviation of the response curve from a straight line is

magnified in the figure for the purposes of clarity.

67

Fig. 7.4: Typical output characteristics of a force measurement system

Characterizing the performance of a force measuring system is commonly based on

calculating such a best-fit least-squares line and stating the measurement errors with

respect to it.

Vertical deviation from this line is referred to as non-linearity and generally the largest

value is given in the specifications of a system.

The difference in readings between the increasing and decreasing forces at any given

rated force is defined as hysteresis. The largest value of hysteresis is usually at the

mid-range of the system.

Sometimes non-linearity and hysteresis are combined into a single figure - usually by

drawing two lines parallel to the best-fit line such that they enclose the increasing and

decreasing force curves as shown. The maximum difference (in terms of output) is

then halved and referred to as the ±combined error.

Any difference between the indicated value of force and the true value is known as an

error of measurement. Such errors are usually expressed as either a percentage of the

68

force applied at that particular point on the characteristic or as a percentage of the

maximum force.

The rated capacity is the maximum force that a force transducer is designed to

measure.

Full-scale output, also known as span or rated output, is the output at the rated

capacity minus the output at zero applied force.

Sensitivity is defined as the full-scale output divided by the rated capacity of a given

transducer/load cell.

7.3.2 The design and development of a 2-axis force measurement system

The design and development of the force balance used in this project is based on the

model of a force balance that was used to measure drag of a micro air vehicle by Luke

[27]. This force balance model is found to be appropriate for the purpose of measuring

lift and thrust of our flapping wing model presented in section 7.1. The design and

development of this force balance consists of two parts: hardware and software. The

hardware includes the design and testing of the frame, load cells, amplifier and data

acquisition system. The software part includes the development of a program in

LabView to record the signals from the hardware, convert it to force data, plot them in

graph and save them to the disk for further analysis. After finishing hardware and

software parts, a calibration process is performed and then the force balance can be

used in the experiment.

7.3.2.1 Hardware

7.3.2.1.1 Aluminium structure

69

The structure of the force balance was designed in SolidWorks® based on the principal

of the force model found in a drag measurement application [27] for a fixed-wing

micro air vehicle. The CAD model of the force balance can be seen in Fig. 7.5. The

force balance consists of three main platforms: base, lift platform and drag platform.

The detail-drawings of these platforms can be found in Appendix C.6. As shown in the

figure, the lift platform is fixed to the base using two shims and the drag platform is

then fixed to the lift platform by other two shims. The shims added to the lift platform

Fig. 7.5: CAD model of the force balance

70

allow the lift platform to move upward and downward in the lift direction and prevent

it to move in the drag direction. The shims added to the drag platform allow the drag

platform to move forward and backward in the drag direction and prevent it to move in

the lift direction. Therefore, with this configuration, the force measured in each

direction will be independent to each other.

The sensitivity of the force balance depends on the bending of shims and that’s why it

depends on the shim material as well as thickness. Two types of shim have been used:

a plastic shim and copper shim. The plastic shim has a thickness of 1mm and the

copper shim has a thickness of 0.3mm. The plastic shim performs not as good as the

copper shim. Using the copper shim, the sensitivity of the force balance is around

50mN. Structure optimization can be done to find appropriate shim thickness for the

desired sensitivity. A picture of the fabricated force balance is shown in Fig. 7.6.

Fig. 7.6: Real force balance

7.3.2.1.2 Load cell

Fig. 7.7: LCL Omega 454g load cell

Two thin beam Omega LCL-454g load cells are

used for the force balance. As shown in their

specifications in Appendix-C.3, they have a rated

capacity of 454g and combined error of 0.25% of

full scale output.

71

7.3.2.1.3 Amplifier

Signal from load cell is very small; usually 1-2mV/V, and therefore it should be

amplified before feeding back to data acquisition system. Each load cell in this force

balance is amplified using the RS strain-gauge amplifier shown in Fig. 7.8. The

amplifier is operated by a dual supply ± 15 DC power. The bridge excitation voltage

can be adjusted by the variable resistor VR2 and zero force adjustment can be

controlled by turning the variable resistor VR1. Excitation voltage for each load cell is

connected through BS±

input±

terminals; output from the load cell is connected through

terminals. If lead wire from the load cell to the amplifier is more than 10

metre, the compensation wire from the load cell should be connected to the C-terminal

on the PCB board. The specification and circuit diagram of the amplifier are shown in

Appendix C.4.

Fig. 7.8: RS Strain-gage amplifier

7.3.2.1.4 Data acquisition card

The DAQCard-1200 PCMCIA from National Instrument shown in Fig. 7.9 is used to

convert the analog signal from the amplifier to digital signal to process in the

computer. The card is equipped with a 12-bit ADC and 8 analog inputs/2 analog

outputs. The maximum sampling rate is 100kS/s. Two analog input channels are used

to read voltage signals from the amplifiers. Signals from the two amplifiers are

72

connected to the appropriate terminals of the DAQ card using differential mode. For

each channel, a100 resistor is wired from the negative input terminal to analog

input ground reference of the DAQCard to stabilize the reading of the DAQ.

KΩ

Fig. 7.9: National Instruments DAQCard-1200

7.3.2.2 Software

7.3.2.2.1 Computer-based data acquisition module

Signals fed to the analog input channels of the DAQCard are read and processed using

a LabView®, a graphic programming software for measurement instruments from

National Instruments. The GUI of the program written for this force balance is shown

in Fig. 7.10. The acquisition starts by choosing the DAQ device number, appropriate

Fig. 7.10: GUI of the computer-based data acquisition module

73

analog input channel for lift load cell and drag load cell, the lowest limit and highest

limit voltage of each channel, then triggering the Acquire-switch. The instantaneous lift

and drag will be displayed on the graph and in digital display box next to the graph.

The lift and drag data can be fed to the table by triggering the Table-switch. To save

the data in the table, trigger the Hold-switch on before triggering off the Acquire-

switch and trigger the Save-switch. The block diagram of the program can be found in

Appendix C.4.

7.3.3 Calibration

7.3.3.1 Calibration set-up

A system of strings and pulleys is used to calibrate the force balance. A picture of the

set-up for the calibration process is shown in Fig. 7.11. A load is applied to the

negative lift direction (the direction of gravity) by placing a mass hanging from a

string, whose other end is tied directly to the strut of the force balance. For drag

direction and thrust directions, a load is applied to one end of a string, which is draped

over a pulley and attached directly to the strut. To reduce hysteresis caused by

calibration system, the string should be strong enough in order to avoid stretch under

Fig. 7.11: Set-up for the calibration process

74

the applied load.

7.3.3.2 Calibration procedures

A total of 4 calibration procedures were performed. The first procedure begins by

increasing load in the lift direction and then unloading in the reverse order. No load is

applied on the drag direction during this procedure. The same procedure is applied to

the drag direction. These two first procedures will determine the relationship between

voltage output and applied force in each direction. After finishing the first two

procedures, a load of 100g is applied in the drag direction and the third procedure

starts by increasing load in the lift direction then unload following the reverse order.

The same procedure is then applied to the drag direction. The purpose of the two final

procedures is to check for mutual effect between lift and drag directions.

7.3.3.3 Calibration results

Results from the first calibration shows that the voltage output and applied force is

linear for both lift and drag direction. Plots of this relation in lift and drag direction are

shown in Fig. 7.12.

From the calibration data, linear regression is carried out for each data set to find out

the best-fit straight line between voltage and applied force for each channel. For lift

channel, this relationship is expressed by:

1278.8 5117.3y x= + (7.1)

where y is the applied force and x is the output voltage. The regression on lift channel

data gives value of R-square equal to 1, showing that the regression correlates very

well with the calibration data and no hysteresis exists on the lift channel.

75

For drag direction, the relationship is as follows (see Fig. 7.12):

2162.5 8651.1y x= − − (7.2)

Lift direction (-450 to 0g)

y = 1278.8x + 5117.3

R2 = 1

-500

-400

-300

-200

-100

0-4.4 -4.3 -4.2 -4.1 -4

Voltage(V)

Wei

ght(

g)

Load Unload Linear (Load) Linear (Unload)

Drag direction (0 to 450g)

y = -2162.5x - 8651.1R2 = 0.9998

R2 = 0.9994

0

100

200

300

400

500

600

-4.25 -4.2 -4.15 -4.1 -4.05 -4Voltage(V)

Wei

ght(g

)

Load Unload Linear (Load) Linear (Unload)

Fig. 7.12: Relation of voltage output and applied force on lift direction (top) , drag direction (bottom)

Regression on drag channel gives R-square of 0.9998 in the load direction and 0.9994

in the unload direction. This value shows that small hysteresis occurs on drag channel.

However, this hysteresis is not caused by the force balance itself but is caused by

76

calibration system (string and pulley do not get back to its initial position after

unloading).

7.4 Wind tunnel facilities

The experiment was done in the wind tunnel in the Fluid Mechanics Lab. This wind

tunnel is powered by a 60HP motor with a test section of 2 1m m× . The wind speed

ranges from 3-10m/s. Shown in Fig. 7.13 are wind tunnel test section, inlet, wind-

speed controller and manometer. The inlet has a contraction ratio of 4:1 with

rectangular grid at the end of the inlet to straighten the flow before entering the test

section. The length of the test section is around 5m with side-windows. The

manometer provides the pressure difference inside and outside the wind tunnel. This

pressure difference is calculated using the reading from the water column height of the

manometer as follows:

( ) m mp g Zρ ρ∆ = − ∆ (7.3)

where mρ is the density of the fluid inside the manometer

ρ is the density of the fluid inside the wind tunnel (air)

Z∆ is the height of the fluid column inside the manometer m

Using the pressure difference above, the wind speed inside the wind tunnel is then

calculated by:

2 pUρ∞∆

= (7.4)

77

Fig. 7.13: Wind tunnel facilities

7.5 Experiment set-up

The force balance was hung up to the wind tunnel as shown in Fig. 7.14. The picture in

the top-left corner shows the force balance’s bases tighten to the bottom surface of the

Fig. 7.14: Set-up force balance and test model inside the wind tunnel

78

test section using four screws. The strut protrudes into the wind tunnel and shielded by

an airfoil-shape shroud as seen in the top-right corner picture. The purpose of the

shroud is to prevent the effect of surrounding air which can cause undesired reading on

the force balance. The test model is tied to the strut and two supports using fibre wires

as shown in the bottom left picture. Radio receiver, transmitter and speed controller are

used to control the flapping frequency. The picture in the bottom-right corner is the

power supplies, oscilloscope and computer for recording the force data.

7.6 Testing procedure

The experiment is divided into three sections. Three parameters that can be changed in

this experiment are the flapping frequency, wind speed and pitch angle. Only constant

pitch angle (the same pitch angle for both downstroke and upstroke) was tested in this

experiment. In each section, two parameters are held constant and the other is changed.

The minimum and maximum values of these parameters are shown in Table 7.2. Based

on the dimensions of the test model and range of these parameters, the Reynolds

number and reduced frequency are calculated and also are presented in Table 7.2. This

range of Reynolds number is in the absolutely low Reynolds regime. Not many

previous experiments have been done for flapping flight at this regime. The reduced

frequency is held constant in the range of small birds.

7.7 Results and discussion

7.7.1 Profile of instantaneous force

Table 7.2: Range of experiment parameters Parameter Unit Minimum MaximumFlapping frequency Hz 1 5Wind speed m/s 3 8Pitch angle Deg -30 30Re 40.8 10× 42.2 10×Reduced frequency 0.04 0.5

79

The results presented below are limited to the lift direction. Due to limitation of the

force balance, the force in the thrust direction couldn’t be captured. In the lift direction,

at high flapping frequency, the inertial load will be dominant and that’s why a dynamic

model is used to subtract the force generated by inertial load on the reading of the

force balance. This dynamic model can be found in a similar experiment for a

rectangular plate made by Ames et al. [28] and is defined as follows:

( ) ( )( )2sin cosL F md θ θ θ θ= − +&& &

L

F

m

d

(7.5)

where is the aerodynamic force ( lift )

is the total force recorded by the force balance

is the total mass including the wing and two servos

is the distance from the center of gravity of the whole wing to the flapping axis

andθ is the flapping angle.

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1

Total forceLift force

Inertial loading

Fig. 7.15: Breakdown of force components: unsteady lift, inertial loading and total force in Newton.

80

Fig. 7.15 shows a breakdown of force components mentioned in Eqn(7.5) at flapping

frequency of 4.42Hz and free stream air velocity of 3.34m/s. The figure shows that the

inertial loading reaches its highest value at the beginning of the downstroke and

reaches its lowest value at middle of each stroke. The instantaneous lift force contains

fluctuations. This type of fluctuation force hints at some types of vortices shedding of

the wing which are not clearly understood at present.

7.7.2 Effect of pitch angle on lift force

The effect of the pitch angles on the unsteady lift is shown in Fig. 7.16. The figure

shows results of two configurations: lowest reduced frequency versus highest

Reynolds number and highest reduced frequency versus lowest Reynolds number. At

the Reynolds number of 1.9 and reduced frequency of 410× 0.2k < , the trend of mean

lift versus pitch angle is quite clear, the unsteady mean lift increases when the pitch

angles increases and begins to drop at around 200. At Reynolds of and reduced

frequency of , the trend is quite unexplainable. Mean lift is around zero at zero

pitch angle, increases with positive pitch angles and still increases even when the pitch

9400

0.4k >

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

-40 -30 -20 -10 0 10 20 30 40

Pitch angle (deg)

Lift

(N)

k = [0.13-0.18], Re=1.94x10e4

k = [0.4-0.5], Re=9.35x10e3

Fig. 7.16: Experimental mean lift versus pitch angles.

81

angle reaches up to 300. It is then suddenly increases when the wing was turned to

negative pitch angle from zero pitch and constantly maintained when wing pitch angle

is down to -300. At pitch angle from[ 1 , it seems that the mean lift in

unchanged versus reduced frequency and Reynolds number.

0 00 , 30 ]− −

0.45N

4×

410×

The range of mean lift versus pitch angle shown in Fig. 7.16 is from zero to .

This amount of lift can lift up a maximum weight of 45gram, which is quite enough

for micro air vehicle applications.

7.7.3 Lift force versus reduced frequency and Reynolds number

Fig. 7.17 is a bubble plot of mean lift versus reduced frequency and Reynolds number.

In the vertical direction which shows the effect of reduced frequency, mean lift

increases gradually. In the horizontal direction which shows the effect of Reynolds

number, mean lift increases rapidly. When Reynolds number changes from 1 1

to1.5 , mean lift increases about 2.2 times. Fig. 7.17 is plotted to show the

combination effect of flapping frequency and free stream speed. As shown in the

0

0.11679

0.11968

0.13698

0.205530.21525

0.272050.09692

0

0.1

0.2

0.3

0.4

0.5

0.6

0.000 0.500 1.000 1.500 2.000 2.500

Reynolds number

Red

uced

freq

uenc

y (k

)

410×

Fig. 7.17: Experimental lift versus reduced frequency and Reynolds number at zero pitch angle.

82

figure, along the diagonal (e.g. both flapping frequency and wind speed increase) the

degree of increase of mean lift is the greatest.

7.7.4 Measured and computed force data

Shown in Fig. 7.18 is the measured and computed force data for two combinations in

the experiment: Case I (lowest reduced frequency of order of 0.04 with highest

Reynolds number of order of 2.2 ) and Case II (highest reduced frequency of

order of 0.5 with lowest Reynolds number of order of 0.8 ). The computed data

were obtained using panel code. Around 722 panels are used to model the wing

geometry and 300 time steps are required for two flapping cycles. Case I shows that

the panel method underpredicts the force variation. Averaging over one flapping cycle,

410×

410×

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1

ExperimentSimulation

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1

Measured

Computed

Fig. 7.18: Measured and computed unsteady lift in Newton for top (3.34m/s and 4.42Hz), bottom

(6.92m/s and 2.79Hz)

83

the measured force is about 10 times greater than the computed force. In Case II,

which is computed at highest Reynolds number ( 2.2 ), it shows a better agreement

than Case I. The computed average lift is 3 times smaller than the measured data. The

two cases suggest that panel method is not a good representation for simulating

flapping flight at this extreme low Reynolds number. The same conclusion was found

by Ames et al. [

4×10

28] in their experiment for a flat plate at similar wing chord and

Reynolds number. According to Ames, at the limit of high reduced frequency and low

Reynolds number, the flow field is dominated by large instantaneous angle of attack,

separated flows, smeared distributed regions of vorticity and shear layers with discrete

vertical structures, none of which can be expected to be captured in the panel code.

In the above section, a comparison between computed force obtained using UVLM

code and measured forced obtained from experiment has been made to exploit the

applicability of UVLM code for flapping wing problems at low Reynolds number. The

above comparison suggested that UVLM may not be appropriate to use for flapping

wing problems at this low Reynolds regime. In this case, the applicability of FLUENT

at the above Reynolds regime should be investigated. However, it was only during the

later stage of our works we had tried to use FLUENT in our experiment. And since

FLUENT requires a very high computational efforts and time-consuming for a three-

dimensional computation, we did not have adequate time to fully exploit its

applicability at low Reynolds number.

7.8 Summary of experiment studies

The experiment for a rectangular rigid cambered wing has been carried out and lift

performance of a 30-cm rigid cambered wing at Reynolds range of

84

3 4[9.35 10 ,1.94 10 ]× × has been studied. Due to the limitation of the force balance, the

thrust performance of this wing can not be measured. The experiment shows that:

• The instantaneous lift force fluctuates rather extensively. These fluctuations

indicate some kind of vortex shedding which is not clearly understood at present.

• At positive pitch angle, mean lift increases with the increase in wing pitch angle.

For negative pitch angle around , mean lift seems to remain constant

versus changes in reduced frequency and Reynolds number. The lift force versus

pitch angle ranges from zero to 0.45N, which can lift up a maximum weight of

45gram. This lift force is quite enough for micro air vehicle application.

0 0[ 10 , 30 ]− −

• At zero pitch angle, mean lift gradually increases with the increase in reduced

frequency but rapidly increases versus the increase in Reynolds number. However,

the degree of increasing mean lift is the highest along the diagonal, which is a

combination of increasing flapping frequency and wind speed.

The comparison between measured and computed lift force shows that UVLM under-

predicts the variation of the lift force at low Reynolds number. At highest Reynolds

number studied in this experiment, the maximum amplitude of the computed force is

1/3 of that in the experiment. At lowest Reynolds number, this ratio is even much

higher. This finding agrees with the finding of Ames et al. [28] in their study about the

aerodynamics of a flat plate in flapping flight at similar Reynolds condition. These

findings suggest that UVLM or vortex-lattice method may not be appropriate to model

flapping flight at low Reynolds number. However, UVLM was first developed due to

the fact that we did not have any available tool to simulate the 3D flapping wing,

moreover, it was found to be able to provide quick results and require less

computational efforts. And as a result of the above conclusion, the obvious question is

85

whether FLUENT can be used to model flapping wing problems at this low Reynolds

regime. However, it was only during the later stages of our works reported in this

thesis that we had tried to use FLUENT in our experiment. Thus, we did not have

adequate time to fully exploit its applicability at low Reynolds number.

86

Chapter 8 Conclusion and Recommendation

8.1 Conclusion

The objective of this project is to investigate the aerodynamic performance of a thin

rigid cambered wing in flapping flight at high and low Reynolds numbers. The

investigation shows that thin rigid cambered wing can be a good alternative for

lift/thrust generation in flapping flight at high and low Reynolds numbers.

At high Reynolds number, intensive numerical computations performed using UVLM

and FLUENT lead to some conclusions on the lift/thrust generation of a thin rigid

cambered wing in flapping flight as follows:

• The maximum camber for maximize thrust performance of the camber wing is

around 6%c.

• Lift and thrust of a thin rigid cambered wing increase when flapping amplitude

increases. For positive thrust generation, flapping amplitude of 300 or above is

required.

• For a rigid cambered wing, thrust is usually generated in the downstroke and drag

is usually generated in the upstroke. A negative pitch angle in the downstroke will

increase the thrust and a positive pitch angle in the upstroke will decrease the drag.

Optimum pitch angle for maximize aerodynamic performance in each stroke exists

and they are equal in magnitude. By turning the wing to the negative optimal pitch

87

angle in the downstroke and turning it back to the positive optimal pitch angle in

the upstroke, thrust performance of the wing in one flapping cycle can improved up

to reach 3.5time of that at zero-pitch angle configuration while the loss in lift is

only 25% of the lift at zero-pitch angle configuration. The configuration in which

the wing is turned to its appropriate pitch angle in each stroke for maximizing the

thrust performance is called “coupled wing pitch angle”.

• The lift and thrust performance of the coupled wing pitch angle” can be further

improved by increasing flapping frequency. At the coupled wing pitch angle of

200, a flapping frequency of 5Hz can give the mean lift coefficient 20 times greater

and mean thrust coefficient around 60 times greater than those at flapping

frequency of 1Hz.

With those features presented above about the lift/thrust performance of the rigid

cambered wing, it is believed that the thin rigid cambered wing accompanied by the

coupled wing pitch angle can be an alternative for lift/thrust generation in flapping

wing flight at high Reynolds number.

At low Reynolds number, an experiment for measuring the lift of a 30-cm span

flapping model in a low speed wind tunnel showed that:

• The profile of the instantaneous lift force contains a lot of fluctuations which is

believed to be caused by some vortex shedding that is not understood at present.

• In the range of constant positive pitch angle, the increase in mean lift versus pitch

angle follows a polynomial curve (as shown in Fig. 7.16 in Section 7.7.2). At pitch

angle in the range of , the mean lift seems to remain at constant

regardless of reduced frequency and Reynolds number.

0 010 , 30⎡− −⎣ ⎤⎦

88

• At zero pitch angle, mean lift gradually increases versus the increase reduced

frequency and drops rapidly with reducing Reynolds number. Along the diagonal

which combines a slightly increase in reduced frequency and Reynolds number, the

degree of increasing mean lift is the highest.

• The mean lift of the 30-cm model in the experiment ranges from 0-0.45N. This

mean lift is quite enough for micro air vehicle applications. However, more

experiment on both lift and thrust must be done to further investigate the

applicability of the rigid cambered wing in sustaining flight.

• UVLM or vortex-lattice method under-predicts the variation of lift force at low

Reynolds number. At Reynolds number of , the ratio of maximum

amplitude between computed force and measured force is about 1/3. At lower

Reynolds number, this ratio is even higher. A reason for such poor performance of

UVLM, as stated by Ames et al. [2 ], is that UVLM has not been developed to

model some typical phenomena which are dominant in flapping wing flight at low

Reynolds number such as large instantaneous angle of attack, separated flows and

vortex shedding in the flow field.

41.94 10×

8

8.2 Recommendation

8.2.1 Computational studies

UVLM and FLUENT have been proven to be adequate tools for numerical

computation of flapping wing flight at high Reynolds number. But at low Reynolds

number, experiment showed that UVLM is not a good tool because it hasn’t been

developed to model some dominant phenomena in the flow field of flapping wing

flight. FLUENT has not been investigated at this Reynolds number because it requires

intensive computational effort and is rather time-consuming. Therefore, the future

development of this project should focus on how to model separated flows and vortex

89

shedding for UVLM. However, such development turns out to be a very difficult task

because none of effort for modelling such phenomena for vortex-lattice method exists.

This is the reason why consideration should also be made on Euler/Navier-Stoke

solution. Although it is quite time-consuming but further investigation using FLUENT

is also a good choice.

8.2.2 Experimental studies

Lift performance of a rigid cambered wing at low Reynolds number has been

investigated. Due to the limitation of the force balance, thrust performance of this wing

has not yet been considered. To study the thrust performance of the rigid cambered

wing at low Reynolds number, some suggestions below should be considered:

• The thrust generated by the testing model is so small that it went out of the

working range of the force balance. The dimension of the wing in this experiment

is 5cm and the area is . A slightly bigger wing should be considered but at

the same time respect the maximum size of a Micro Air Vehicle.

20.01m

• The sensitivity of the force balance is around 5g, it means that it can only detect a

minimum force of 50mN. As stated in section 7.3.2.1.1, this sensitivity of the force

balance can be further improved by using thinner shims or other material for the

shims. The present shims are made of copper with thickness of around 0.3mm.

Some studies in the choice of better materials will enhance the force sensor

sensitivity.

90

References

[1] H.P. Ng, “Dynamics and Control of Flapping Wing Aircraft”, B.Eng Thesis,

Department of Mechanical Engineering, National University of Singapore, 2001/2003.

[2] R. Kamakoti, M. Berg, D. Ljungqvist and W. Shyy, “A Computational Study for

Biological Flapping Wing Flight”, Transactions of the Aeronautical and Astronautical

Society of the Republic of China, Vol.32, No.4, pp.265-279 (2000).

[3] J.D. DeLaurier, “An aerodynamic model for flapping wing flight”, Aeronautical

Journal, Vol. 97, April 1993, pp. 125-130.

[4] W.B. Tay, “Dynamics and Control of Flapping Wing Aircraft”, M.Eng Thesis,

Department of Mechanical Engineering, National University of Singapore, 2002/2004.

[5] L.F. Ang, “Computer simulation of Flapping Wing Flight”, B.Eng Thesis,

Department of Mechanical Engineering, National University of Singapore, 2001/2002.

[6] J.H. Ling, Adeline, “Modelling and Simulation of a Flapping Wing aircraft”,

B.Eng Thesis, Department of Mechanical Engineering, National University of

Singapore, 2002/2003.

[7] T.J. Mueller and J.D. DeLaurier, “An Overview of Micro Air Vehicle

Aerodynamics”, Conference on Fixed, Flapping and Rotary Wing Aerodynamics at

Very Low Reynolds Number, edited by T.J. Mueller, Vol. 195, 2001, pp. 1-10.

91

[8] J.M.V. Rayner, “Thrust and Drag in Flying Birds: Applications to BirdLike Micro

Air Vehicles”, Conference on Fixed Flapping and Rotary Wing Aerodynamics at Low

Reynolds Number, edited by T.J. Mueller, Vol. 195, 2001, pp. 217-230.

[9] K.D. Jones, T.C. Lund, and M.F. Platzer, “Experimental and Computational

Investigation of Flapping Wing Propulsion for Micro Air Vehicles”, Conference on

Fixed Flapping and Rotary Wing Aerodynamics at Low Reynolds Number, edited by

Thomas J.Mueller, Vol. 195, 2001, pp. 307-339.

[10] K.D. Jones, C.J. Bradshaw, J. Papadopoulos and M.F. Platzer, “Development and

Flight Testing of Flapping-Wing Propelled Micro Air Vehicles”, AIAA 2003-6549.

[11] I.E. Garrick, “Propulsion of a Flapping and Oscillating Airfoil”, NACA Rept.

567, 1936.

[12] M.F. Neef and D. Hummel, “Euler Solutions for a Finite-Span Flapping Wing”,

Conference on Fixed Flapping and Rotary Wing Aerodynamics at Low Reynolds

Number, edited by T.J. Mueller, Vol. 195, 2001, pp. 429-451.

[13] R. Kamakoti and W. Shyy, “Evaluation of geometric conservation law using

pressured-based fluid solver and moving grid technique”, International Journal of

Numerical Methods for Heat & Fluid flow, Vol. 14 No.7, 2004, pp. 849-863.

[14] P. Wilmott, “Unsteady Lifting Line Theory by the Method of Matched

Asymptotic Expansions”, Journal of Fluid Mechanics, Vol. 186, Jan. 1998, pp. 303-

320.

[15] P.J. Phlips, R.A. East and N.H. Pratt, “Un steady Lifting Line Theory of Flapping

Wings with Application to the Forward Flight of Birds”, Journal of Fluid Mechanics,

Vol. 112, Nov 1981, pp. 97-125.

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[16] A.R. Ahmadi and S.E. Widnall, “Unsteady Lifting Line Theory as a Singular-

Perturbation Problem”, Journal of Fluid Mechanics, Vol. 153, April 1985, pp. 59-81.

[17] C.E. Lan, “The Unsteady Quasi-Vortex-Lattice Method with Applications to

Animal Propulsion”, Journal of Fluid Mechanics, Vol. 93, No. 4, 1979, pp. 747-765.

[18] K.C. Hall and S.R. Hall, “A Rational Engineering Analysis of Efficiency of

Flapping Flight”, Conference on Fixed, Flapping and Rotary Wing Aerodynamics at

Low Reynolds Number, edited by Thomas J.Mueuller, Vol. 195, 2001, pp. 249-274.

[19] T. Nick Pornsin-Sirirak, Y.C. Tai, H. Nassef and C.M. Ho, "Titanium-Alloy

MEMS Wing Technology for A Micro Aerial Vehicle Application," Sensors and

Actuators, A: Physical, vol. 89, Issue 1-2, Mar 20, 2001, pp. 95-103.

[20] T. Nick Pornsin-Sirirak, S.W. Lee, H. Nassef, J. Grasmeyer, Y.C. Tai, C.M. Ho

and M. Keennon, "MEMS Wing Technology for A Battery-Powered Ornithopter," The

13th IEEE International Conference on Micro Electro Mechanical Systems (MEMS

'00), Miyazaki, Japan, Jan 23-27, 2000, pp. 799-804.

[21] N.P. Siriak and M. Liger, Research Abstract in MEMS technology for Micro Air

Vehicle application – Microbat Project Update. Retrieved from Caltech

Micromachining Lab’s website http://touch.caltech.edu/research/bat/bat.html.

[22] J. Katz and A. Plotkin, “Low Speed Aerodynamics, From Wing Theory to Panel

Methods”, ISBN 0-07-050446-6.

[23] M.J.C. Smith, P.J. Wilkin and M.H. Williams, “The advantages of an unsteady

panel method in modelling the aerodynamic forces on rigid flapping wings”, The

Journal of Experimental Biology 199 (1996).

93

[24 ] Fluent Inc., “Fluent 6.1 User’s Guide”, Fluent Inc. January 28th 2003. National

University of Singapore, Supercomputing and Visualization Unit http://e-

svu.nus.edu.sg (accessed January 19th 2005).

[25] K.D. Jones, B.M. Castro, O. Mahmoud, S.J. Pollard, M.F. Platzer, M.F. Neef, K.

Gonet and D. Hummel, “A collaborative numerical and experimental investigation of

flapping-wing propulsion”AIAA-2002-0706.

[26] K.S. Yeo, T.T. Lim and K.B Lua, “Aerodynamics of Micro Flyers”. Retrieved

Feb 2005, from National University of Singapore, Faculty of Engineering’s Website:

http://www.eng.nus.edu.sg/EResnews/0502/rd/rd_11.html.

[27] M. Luke, “Predicting drag polars for micro air vehicles”, Thesis submitted for

Department of Mechanical Engineering, Brigham Young University, December 2003.

[28] R. Ames, O. Wong and N. Komerath, “On the Flowfield and Forces Generated by

a Flapping Rectangular Wing at Low Reynolds Number”, Fixed and Flapping Wing

Aerodynamics for Micro Air Vehicle Applications, edited by Thomas J.Muller,

Volume 195, p.p 287-305.

94

Appendix A

Two dimensional investigations

A.1 User-defined function for 2D pure plunging motion in FLUENT

#include "udf.h" /* this function defines velocity of center of gravity for pure plunging motion*/ /* Plunging motion equation is z(t)=h*sin(2*pi*f*t) */ DEFINE_CG_MOTION(plunging_motion, dt, vel, omega, time, dtime) Thread *t; face_t *f; /*reset velocities */ NV_S(vel,=,0.0); NV_S(omega,=,0.0); if (!Data_Valid_P()) return; /* Get the thread pointer for which this motion is defined */ /* t=DT_THREAD(dt); */ /* vel[1] is the vertical plunging velocity */ /* These velocity below is for h=0.4c and f=0.993Hz */ vel[1] = 2.495*cos(6.2392*time);

A.2 Investigation for 2D airfoil in pitching and plunging motion using FLUENT

The investigation is performed for a NACA0012 undergoing sinusoidal pitching and

plunging motion. The basic parameters of these motions are presented in Table A.1. A

similar in vortex shedding in the wake of the airfoil computed by two-dimensional

panel code and FLUENT can be found in Fig A.1.

A.3 Investigation for 2D airfoil flapping in Ground effect

The investigation is performed for two NACA0015 airfoils undergoing sinusoidal

pitching and plunging motion symmetrically through a ground surface. The basic

parameters of the investigation are presented in Table A.2. A similar in vortex

95

shedding in the wake of the airfoil computed using panel code and FLUENT can be

found in Fig A.2.

Fig A.1 : Vortex shedding after two-dimensional airfoil in combined pitching and plunging motion

computed by 2D panel method (top) and FLUENT (bottom).

Table A.1: Parameters for NACA0012 airfoil in combined pitching and plunging motion Pitching amplitude 010± Plunging amplitude 0.2c Reduced frequency 4 Phase 090

Table A.2: Parameters for NACA0015 airfoil flapping in ground effect Pitching amplitude 010± Plunging amplitude 0.2c Mean altitude 0.7c Reduced frequency 4 Phase 0180

96

Fig A.2 : Vortex shedding in the wake of a NACA0015 flapping in ground effect computed by two-

dimensional panel code (top) and FLUENT (bottom)

97

Appendix B

Three dimensional investigations

B.1 User-defined function for 3D flapping motion in FLUENT

#include "udf.h" DEFINE_CG_MOTION(flapping, dt, vel, omega, time, dtime) #if !RP_NODE Thread *t; face_t *f; #endif NV_S(vel,=,0.0); NV_S(omega,=,0.0); if (!Data_Valid_P()) return; /*flapping frequency is 1Hz */ omega[0]=0.5236*2*3.1416*sin(2*3.1416*time);

98

Appendix C

Experimental Studies

C.1 Specifications of the Basic Stamp 2 Microcontroller

Processor Speed 20 MHz Program

Execution Speed~4,000

instructions/sec.

RAM Size 32 Bytes (6 I/O, 26 Variable)

EEPROM (Program) Size

2K Bytes, ~500 instructions

I/O Pins 16 +2 Dedicated Serial

Voltage Requirements 5 - 15 vdc

Current Draw at 5V

3 mA Run / 50 µA Sleep

PBASIC Commands 42

Size 1.2"x0.6"x0.4"

C.2 Controlling Servos

The following code is written using the Basic Stamp Editor v2.1

' $STAMP BS2 ' $PBASIC 2.5 n_pulse VAR Word pitch_angle VAR Word i VAR Byte sign VAR Bit ' bit to hold the sign 'Enter desired pitch angle here pitch_angle = 10 ' value is in degree sign = pitch_angle.BIT15 ' determine result sign n_pulse = ABS pitch_angle * 9 / 2 ' Compute the pulse width IF (sign = 1) THEN n_pulse = - n_pulse ' correct sign if negative DEBUG SDEC? pitch_angle DEBUG SDEC ? n_pulse 'Begin to command servos servo: PULSOUT 0, 750 + n_pulse PULSOUT 1,750 - n_pulse PAUSE 20 GOTO servo

99

C.3 Load cell specifications

Excitation: 5 Vdc, 12 volts max. Rated Output: 2 mV/V ±20% (to minimize ±20% tolerance, end-user must calibrate with a known weight) Zero Balance: +0.3 mV/V Combined Error: 0.25% full scale Operating Temperature: -65 to 200°F (-54 to 93°C) Compensated Temp.: 20 to 120°F (-7 to 49°C) Temperature Effects: Zero Balance 0.02% FS/°F; Output 0.02%/°F Resistance: (Input and output) 1200 Ω ±300 Ω Insulation Resistance: 1000 @ 50 Vdc Seal: Urethane coated Safe Overload: 150% FS Full Scale Deflection: 0.010 to 0.050 in Lead Wire: 9 in shielded PVC four conductor 30 AWG Material: 301 SS (beryllium copper 1⁄4 and 1⁄2 lb units)

Fig C.1 : Dimensions of the 454g Omega load cell

C.4 Specifications and circuit diagram of the amplifier

Specification (At 25ºC ambient and ±12V supply unless otherwise stated.) Supply voltage _____________________ ±2 to ±20Vdc Input offset voltage ____________________200µV max. Input offset voltage/temperature _______0.5µV/oC max. Input offset voltage/supply ______________ 3µV/V max. Input offset voltage/time___________0.3µV/month max. Input impedance ______________________ >5Mmin. Input noise voltage __________________0.9µVp.p max.

100

Band width (unity gain)_____________________450kHz Output current______________________________ 5mA Output voltage span _____________________ ±(Vs-2)V Closed loop gain (adjustable) ___________ 3 to 60,000 Open loop gain __________________________ >120dB Common mode rejection ratio ______________>120dB Bridge supply voltage/temperature ________ 20µV/oC Maximum bridge supply current _____________ 12mA Power dissipation __________________________ 0.5W Warm up time _____________________________5 mins Operating temperature range _______ -25oC to + 85oC Component values ( Fig. C.2) R1 100k R7 47R C2, C5 10n (typ.) R2100R R810R C3, C4 10µ (tant.) R3100k* R91k0 T1BD 135 R4 68R* R10680R T2BD 136 R510R R11680R T3BC 108 R6100R(typ.) C1, C6, C7 100n (typ.) D1, D2 1N827

Fig C.2: Basic circuit for printed circuit board RS stock no. 435-692 (gain approx. 1000)

C.5 Block diagram of the computer-based data acquisition module

101

102Fig C.3: Block diagram of the computer-based DAQ module (section 1)

103Fig C.4: Block diagram of computer-based DAQ module (section 2)

C.6 Detail-drawings of the force balance

104

105

105

106

106

107

107

108

108

109

109

110

110

111

111

112

112

113

113

114

114