Unsteady Aerodynamics of Deformable Thin Airfoils

72 Pages·2009·0.43 MB·English

by  William Walker

Unsteady Aerodynamics of Deformable Thin Airfoils
William Paul Walker
Thesis submitted to the Faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
Master of Science
in
Aerospace Engineering
Mayuresh J. Patil, Chair
William J. Devenport
Robert A. Canfield
August 5, 2009
Blacksburg, Virginia
Keywords: Unsteady Aerodynamics, Deformable Airfoils
Copyright 2009, William Paul Walker
Unsteady Aerodynamics of Deformable Thin Airfoils
William Paul Walker
(ABSTRACT)
Unsteady aerodynamic theories are essential in the analysis of bird and insect flight.
The study of these types of locomotion is vital in the development of flapping wing aircraft.
This paper uses potential flow aerodynamics to extend the unsteady aerodynamic theory of
TheodorsenandGarrick(whichisrestrictedtorigidairfoilmotion)todeformablethinairfoils.
Frequency-domain lift, pitching moment and thrust expressions are derived for an airfoil
undergoing harmonic oscillations and deformation in the form of Chebychev polynomials.
The results are validated against the time-domain unsteady aerodynamic theory of Peters.
A case study is presented which analyzes several combinations of airfoil motion at different
phases and identifies various possibilities for thrust generation using a deformable airfoil.
Contents
1 Introduction 1
2 Background and Objectives 2
2.1 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Aerodynamic Theory 9
3.1 Lift and Pitching Moment Derivation . . . . . . . . . . . . . . . . . . . . . . 9
3.1.1 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 10
3.1.2 Noncirculatory Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1.3 Noncirculatory Pressure and Forces . . . . . . . . . . . . . . . . . . . 16
3.1.4 Circulatory Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.5 Theodorsen’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1.6 Total Lift and Pitching Moment Expressions . . . . . . . . . . . . . . 26
3.2 Thrust Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.1 Thrust due to Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.2 Thrust due to Leading Edge Suction . . . . . . . . . . . . . . . . . . 30
3.2.3 Total Thrust Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4 Verification of Aerodynamic Theory 35
4.1 Peters’ Unsteady Airloads Theory . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 Application of Peters’ Theory to Current Problem . . . . . . . . . . . . . . . 38
iii
5 Case Study 40
5.1 Lift and Pitching Moment Results . . . . . . . . . . . . . . . . . . . . . . . . 40
5.1.1 Reduction to Steady Thin Airfoil Theory . . . . . . . . . . . . . . . . 42
5.2 Thrust Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6 Conclusions and Future Work 55
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Bibliography 58
Appendix A 61
iv
List of Figures
3.1 Chebychev Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Joukowski mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Velocity due to generalized airfoil shape . . . . . . . . . . . . . . . . . . . . . 15
3.4 Vortex locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.5 Theodorsen’s function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.6 Fluid flow around leading edge . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.1 Phase of lift relative to airfoil motion . . . . . . . . . . . . . . . . . . . . . . 42
5.2 Phase of pitching moment relative to airfoil motion . . . . . . . . . . . . . . 43
5.3 Thrust coefficient due to the first 5 airfoil shapes . . . . . . . . . . . . . . . 46
5.4 Thrust coefficient due to h and α . . . . . . . . . . . . . . . . . . . . . . . . 47
5.5 Thrust coefficient due to h and κ . . . . . . . . . . . . . . . . . . . . . . . . 47
5.6 Thrust coefficient due to h and κ . . . . . . . . . . . . . . . . . . . . . . . . 48
2
5.7 Thrust coefficient due to h and κ . . . . . . . . . . . . . . . . . . . . . . . . 48
3
5.8 Thrust coefficient due to α and κ . . . . . . . . . . . . . . . . . . . . . . . . 49
5.9 Thrust coefficient due to α and κ . . . . . . . . . . . . . . . . . . . . . . . . 50
2
5.10 Thrust coefficient due to α and κ . . . . . . . . . . . . . . . . . . . . . . . . 51
3
5.11 Thrust coefficient due to κ and κ . . . . . . . . . . . . . . . . . . . . . . . . 51
2
5.12 Thrust coefficient due to κ and κ . . . . . . . . . . . . . . . . . . . . . . . . 52
3
5.13 Thrust coefficient due to κ and κ . . . . . . . . . . . . . . . . . . . . . . . 52
2 3
5.14 Thrust coefficient due to h, κ and κ (φ = −π/2) . . . . . . . . . . . . . . 53
2 3 κ2
5.15 Thrust coefficient due to h, κ and κ (φ = 0) . . . . . . . . . . . . . . . . 53
2 3 κ2
v
5.16 Thrust coefficient due to h, κ and κ (φ = π/2) . . . . . . . . . . . . . . . 54
2 3 κ2
5.17 Thrust coefficient due to h, κ and κ (φ = π) . . . . . . . . . . . . . . . . 54
2 3 κ2
A.1 Functions F and G as a function of 1/k . . . . . . . . . . . . . . . . . . . . . 63
vi
List of Tables
3.1 Velocity potential functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Lift force components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Pitching moment components . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Leading edge suction velocities . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.5 Thrust force due to h and α . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.6 Thrust force due to κ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.7 Thrust force due to κ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2
3.8 Thrust force due to κ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3
4.1 Lift force components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Pitching moment components . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.1 Lift and pitching moment amplitude and phase terms . . . . . . . . . . . . . 41
vii
Nomenclature
A airfoil shape coefficient
n
a joukowski mapping circle radius
b semichord
C(k) Theodorsen’s function
F(k) real component of Theodorsen’s Function
G(k) imaginary component of Theodorsen’s function
H (k) Hankel function
n
h plunge variable
¯
h plunge amplitude
i imaginary number
J (k) Bessel function of the first kind
n
k reduced frequency
L total lift
L circulatory lift
C
L noncirculatory lift
NC
M total Pitching moment
M circulatory Pitching moment
C
M noncirculatory Pitching moment
NC
p pressure
p total pressure
0
p free stream pressure
∞
viii
Q circulatory airfoil motion expression
S leading edge suction
t time
T total thrust
T Chebychev polynomials
n
T thrust due to pressure
L
T thrust due to leading edge suction
LES
u velocity in x direction
u(cid:48) disturbance velocity in x direction
v velocity in y direction
v(cid:48) disturbance velocity in y direction
X x location in unmapped plane
Y y location in unmapped plane
Y (k) Bessel function of the second kind
n
x x location in mapped plane
y y location in mapped plane
Γ point vortex circulation
α pitch variable
α¯ pitch amplitude
γ wake vorticity per unit length
w
(cid:15) source/sink strength
κ curvature variable
κ¯ curvature amplitude
κ higher Order shape terms
n
φ velocity potential
ϕ phase angle
ρ density
ω frequency
ix
Chapter 1
Introduction
MicroAirVehicles(MAVs)areagrowingareaofresearch. FlappingwingMAVsaresimilarto
birds and insects, and are highly maneuverable. The aerodynamics of flapping wing aircraft
is highly unsteady. Furthermore, during flight, bird and insect wings undergo significant
aeroelastic deformations in addition to the prescribed rigid-body kinematics. Understanding
thephysicsbehindpitching, plunginganddeformingairfoilswillhelpinthedesignofflapping
wing MAVs.
Analytical, frequency-domain, unsteady aerodynamics theory, such as Theodorsen [1]
and Garrick [2] theory, has proven quite useful in understanding aeroelastic stability and
thrust generation. However, Theodorsen and Garrick only modeled thin airfoils undergoing
rigid body motion. Extending this unsteady aerodynamics theory to deformable airfoils will
help in further understanding the unsteady aerodynamics and aeroelastic response of bird
and insect flight, and help in improving the design of MAVs.
Inviscid potential flow theory may not accurately model flows at low Reynolds num-
bers or flows which have significant viscous effects, but it will help in increasing funda-
mental understanding of the unsteady aerodynamic flow and aeroelasticity of flapping wing
MAVs. Furthermore, potential flow theory can serve as a check for verifying other more
complex/computational codes and methods used in design.
1