optimization of the aerodynamics of small-scale

193 Pages·2008·6.55 MB·English

OPTIMIZATION OF THE AERODYNAMICS OF
SMALL-SCALE FLAPPING AIRCRAFT IN HOVER
by
Sidney Lebental
Department of Mechanical Engineering
and Materials Science
Duke University
Date:
Approved:
Kenneth C. Hall, Supervisor
Donald B. Bliss
John Dolbow
Laurens E. Howle
Jonathan Protz
Dissertation submitted in partial fulfillment of the requirements
for the degree of Doctor of Philosophy in the
Department of Mechanical Engineering
and Materials Science in the
Graduate School of
Duke University
2008
ABSTRACT
OPTIMIZATION OF THE AERODYNAMICS OF
SMALL-SCALE FLAPPING AIRCRAFT IN HOVER
by
Sidney Lebental
Department of Mechanical Engineering
and Materials Science
Duke University
Date:
Approved:
Kenneth C. Hall, Supervisor
Donald B. Bliss
John Dolbow
Laurens E. Howle
Jonathan Protz
An abstract of a dissertation submitted in partial fulfillment of the
requirements for the degree of Doctor of Philosophy
in the Department of Mechanical Engineering
and Materials Science in the
Graduate School of
Duke University
2008
Copyright c 2008 by Sidney Lebental
(cid:13)
All rights reserved
Abstract
Flapping flight is one of the most widespread mean of transportation. It is a
complex unsteady aerodynamic problem that has been studied extensively in the
past century. Nevertheless, by its complex nature, flapping flight remains a chal-
lenging subject. With the development of micro air vehicles, researchers need new
computational methods to design these aircrafts efficiently.
In this dissertation, I will present three different methods of optimization for flap-
ping flight with an emphasis on hovering with each their advantages and drawbacks.
The first method was developed by Hall et al. It is an extremely fast and powerful
three-dimensional approach. However, the assumptions made to develop this theory
limit its use to lightly loaded wings. In addition, it only models the motion of the
trailing edge and not the actual motion of the wing.
In a second part, I will present a two-dimensional unsteady potential method.
It uses a freely convected wake which removes the lightly loaded restriction. This
method shows the existence of an optimal combination of plunging and pitching
motion. The motion is optimal in the sense that for a required force vector, the
aerodynamic power is minimal.
The last method incorporates the three-dimensional effects. These effects are
especially important for low aspect ratio wings. Thus, a three-dimensional unsteady
potential vortex method was developed. This method also exhibits the presence of
an optimal flapping/pitching motion. In addition, it agrees really well with the two
previous methods and with the actual kinematics of birds during hovering flapping
flight.
To conclude, some preliminary design tools for flapping wings in forward and
hovering flight are presented in this thesis.
iv
Contents
Abstract iv
List of Figures x
List of Tables xvii
Acknowledgements xx
I Introduction 1
I.1 General introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
I.2 Dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
I.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
I.4 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
II Literature Review 6
II.1 Dimensional analysis overview . . . . . . . . . . . . . . . . . . . . . . 6
II.2 Kinematics and aerodynamic forces production . . . . . . . . . . . . . 12
II.2.1 Biological study . . . . . . . . . . . . . . . . . . . . . . . . . . 12
a Forward flight case . . . . . . . . . . . . . . . . . . . . 12
b Hovering case . . . . . . . . . . . . . . . . . . . . . . 16
II.2.2 Experimental study . . . . . . . . . . . . . . . . . . . . . . . . 19
a Forward flight case . . . . . . . . . . . . . . . . . . . . 19
b Hovering case . . . . . . . . . . . . . . . . . . . . . . 19
II.2.3 Computational study . . . . . . . . . . . . . . . . . . . . . . . 22
a Analytical methods . . . . . . . . . . . . . . . . . . . 22
b Potential methods . . . . . . . . . . . . . . . . . . . . 23
v
b.1 Forward flight case . . . . . . . . . . . . . . . 23
b.2 Hovering case . . . . . . . . . . . . . . . . . . 25
c Full Navier-Stokes methods . . . . . . . . . . . . . . . 25
c.1 Forward flight case . . . . . . . . . . . . . . . 25
c.2 Hovering case . . . . . . . . . . . . . . . . . . 28
II.3 Power requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
III A Wake Approach to the Constrained Optimization Problem 36
III.1 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
III.2 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
III.2.1 Minimizing the induced power . . . . . . . . . . . . . . . . . . 38
III.2.2 Minimizing the viscous power . . . . . . . . . . . . . . . . . . 43
III.2.3 The hovering case . . . . . . . . . . . . . . . . . . . . . . . . . 48
III.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
III.3.1 Stroke angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
III.3.2 Flapping frequency . . . . . . . . . . . . . . . . . . . . . . . . 51
IV Two-Dimensional Potential Method 55
IV.1 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
IV.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
IV.3 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
IV.3.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
IV.3.2 Potential method . . . . . . . . . . . . . . . . . . . . . . . . . 59
a Inviscid formulation . . . . . . . . . . . . . . . . . . . 59
b Viscous formulation . . . . . . . . . . . . . . . . . . . 65
IV.3.3 Convergence study . . . . . . . . . . . . . . . . . . . . . . . . 66
vi
a Small amplitudes . . . . . . . . . . . . . . . . . . . . 66
a.1 Sudden change in pitch . . . . . . . . . . . . . 68
a.2 Plunging motion . . . . . . . . . . . . . . . . 69
a.3 Pitching motion . . . . . . . . . . . . . . . . . 71
b Wake patterns . . . . . . . . . . . . . . . . . . . . . . 74
c Large amplitude in hover . . . . . . . . . . . . . . . . 76
c.1 Number of cycles . . . . . . . . . . . . . . . . 76
c.2 Number of iterations with the freestream on . 80
c.3 Number of panels . . . . . . . . . . . . . . . . 80
c.4 Number of points per cycle . . . . . . . . . . 81
c.5 Freestream velocity . . . . . . . . . . . . . . . 82
c.6 Conclusion . . . . . . . . . . . . . . . . . . . 82
IV.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
IV.4.1 Inviscid case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
IV.4.2 Viscous model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 86
IV.4.3 Viscous model 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 90
V Three-Dimensional Potential Method 93
V.1 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
V.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
V.3 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
V.3.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
V.3.2 Potential method . . . . . . . . . . . . . . . . . . . . . . . . . 95
a Inviscid formulation . . . . . . . . . . . . . . . . . . . 95
a.1 Governing equations . . . . . . . . . . . . . . 95
vii
a.2 Laplace’s equation . . . . . . . . . . . . . . . 97
a.3 Poisson’s equation . . . . . . . . . . . . . . . 100
a.4 The Kutta condition . . . . . . . . . . . . . . 103
a.5 The wake description . . . . . . . . . . . . . . 103
b Viscous formulation . . . . . . . . . . . . . . . . . . . 109
V.3.3 Convergence study . . . . . . . . . . . . . . . . . . . . . . . . 110
a High aspect ratio with small amplitudes . . . . . . . . 110
a.1 Sudden change in pitch . . . . . . . . . . . . . 110
a.2 Plunging motion . . . . . . . . . . . . . . . . 112
a.3 Pitching motion . . . . . . . . . . . . . . . . . 118
b Finite aspect ratio wing . . . . . . . . . . . . . . . . . 120
b.1 Vortex rings versus vortons in the wake . . . . 120
b.2 Sudden change of pitch for finite aspect ratio
wings . . . . . . . . . . . . . . . . . . . . . . 123
c Convergence in hovering . . . . . . . . . . . . . . . . . 125
c.1 Convergence in time . . . . . . . . . . . . . . 125
c.2 Cut-off radius . . . . . . . . . . . . . . . . . . 128
c.3 ǫ in vortex line . . . . . . . . . . . . . . . . . 130
c.4 Freestream velocity . . . . . . . . . . . . . . . 132
c.5 Number of iterations with the freestream on . 134
c.6 Time step . . . . . . . . . . . . . . . . . . . . 136
c.7 Number of panels in the chordwise direction . 138
c.8 Number of panels in the spanwise direction . 140
c.9 Parameters used . . . . . . . . . . . . . . . . 142
viii
V.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
V.4.1 Inviscid case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
a Optimum motion . . . . . . . . . . . . . . . . . . . . 142
b Comparative study . . . . . . . . . . . . . . . . . . . 149
V.4.2 Viscous model 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 155
VI Conclusion and Recommendations 158
VI.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
VI.2 Further research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
A Flow relations 161
A.1 Potential flow identity . . . . . . . . . . . . . . . . . . . . . . . . . . 161
A.2 Actuator disk theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
B Mathematical methods 163
B.1 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Bibliography 164
Biography 170
ix
List of Figures
I.1 Data collected on various hovering flappers plotted as Π = f (Π ). . 3
2 6
II.1 The great flight diagram. Reprinted from Tennekes [1]. . . . . . . . 7
II.2 The relation between weight and wing loading represented in a pro-
portional diagram. When the weight increases by a factor of 100, the
wing loading increases by a factor of 5 and the forward speed by a
factor of more than 2. Reprinted from Tennekes [1]. . . . . . . . . . 9
II.3 Log-log plot of observed wingbeat frequency for various birds versus
the frequency parameter. Reprinted from Pennycuick [2]. . . . . . . 11
II.4 Log-logplotofobserved wingbeat wavelength forvariousbirdsversus
the frequency parameter. Reprinted from Pennycuick [2]. . . . . . . 11
II.5 Spanwise vorticity contour with superimposed velocity field for three
flight speeds (5,8,11m.s 1). Reprinted from Rosen [3]. . . . . . . . 14
−
II.6 Vortex loop formation by Drosophilia. Reprinted from Dickinson [4]. 16
II.7 Clapand Flingmechanism. Darkarrows show the induced velocities,
the light arrows the net force and the dark lines are the flow lines.
Reprinted from Sane [5]. . . . . . . . . . . . . . . . . . . . . . . . . 17
II.8 Schemaofthrustenhancement throughtheflingmechanism. Theleft
drawing is an airfoil in upward plunging motion, the right drawing
is an airfoil in upward plunging motion with the presence of the two
vortices due to the fling. Dark arrows show the induced velocities,
the light arrows the net force and the dark lines are the flow lines. . 18
II.9 Wing rotation mechanism. Reprinted from Dickinson [6]. . . . . . . 20
II.10 Development of vorticity during a stroke reversal for an axis of rota-
tion located respectively at 0.8, 0.5, 0.2 chord (A, B, C). Reprinted
from Dickinson [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
x