Unsteady Aerodynamics

165 Pages·2012·1.31 MB·English

by  William Walker

Optimization of Harmonically Deforming Thin Airfoils and
Membrane Wings for Optimum Thrust and Efficiency
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
Doctor of Philosophy
in
Aerospace Engineering
Mayuresh J. Patil, Chair
Robert A. Canfield
William J. Devenport
William H. Mason
May 2012
Blacksburg, Virginia
Keywords: Unsteady Aerodynamics, Deformable Airfoils, Membrane Aeroelasticity,
Aeroelastic Tailoring, Optimization
Copyright 2012, William Paul Walker
Optimization of Harmonically Deforming Thin Airfoils and Membrane
Wings for Optimum Thrust and Efficiency
William Paul Walker
(ABSTRACT)
This dissertation presents both analytical and numerical approaches to optimizing
thrust and thrust efficiency of harmonically deforming thin airfoils and membrane wings. A
frequency domain unsteady aerodynamic theory for deformable thin airfoils, with Chebychev
polynomials as the basis functions is presented. Stroke-averaged thrust and thrust efficiency
expressions are presented in a quadratic matrix form. The motion and deformation of the
airfoil is optimized for maximum thrust and efficiency. Pareto fronts are generated showing
optimum deformation conditions (magnitude and phase) for various reduced frequencies and
constraints. It is shown that prescribing the airfoil to deform in a linear combination of basis
functions with optimal magnitude and phase results in a larger thrust as compared to rigid
plunging, especially at low reduced frequencies. It is further shown that the problem can
be constrained significantly such that thrust is due entirely to pressure with no leading edge
suction, and associated leading edge separation.
The complete aeroelastic system for a membrane wing is also optimized. The aero-
dynamic theory for deformable thin airfoils is used as the forcing in a membrane vibration
problem. Due to the nature of the two dimensional theory, the membrane vibration problem
is reduced to two dimensions via the Galerkin method and nondimensionalized such that
the only terms are nondimesional tension, mass ratio and reduced frequency. The maximum
thrust for the membrane wing is calculated by optimizing the tension in the membrane so
that the the aeroelastic deformation due to wing motion leads to optimal thrust and/or effi-
ciency. A function which describes the optimal variation of spanwise tension along the chord
is calculated. It is shown that one can always find a range of membrane tension for which
the flexible membrane wings performs better than the rigid wing. These results can be used
in preliminary flapping wing MAV design.
Contents
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Literature Survey 5
2.1 Unsteady Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Aerodynamic Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Aeroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Aeroelastic Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Aerodynamic Theory 18
3.1 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Lift and Pitching Moment Derivation . . . . . . . . . . . . . . . . . . . . . . 21
3.2.1 Noncirculatory Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.2 Circulatory Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.3 Complete Lift and Pitching Moment Expressions . . . . . . . . . . . 29
3.2.4 Complete Pressure Expression . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Thrust Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4.1 Lift and Pitching Moment . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4.2 Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
iii
4 Aerodynamic Optimization 49
4.1 Energy and Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2.1 Magnitude Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.2 Efficiency Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.3 Leading Edge Suction Constraint . . . . . . . . . . . . . . . . . . . . 57
5 Aerodynamic Optimization Results 62
5.1 Thrust Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2 Thrust and Efficiency Optimization . . . . . . . . . . . . . . . . . . . . . . . 66
5.3 Airfoil Shape and Pressure Distribution for Maximum Thrust . . . . . . . . 76
6 Membrane Wing Aeroelasticity 82
6.1 General Equations for Membrane Vibration . . . . . . . . . . . . . . . . . . 82
6.2 Reduction of Membrane Vibration Theory to Two Dimensions . . . . . . . . 84
6.3 Solution to Membrane Vibration Problem via the Assumed Modes Method . 88
6.4 Orthogonal Polynomials for Membrane Vibration Problem . . . . . . . . . . 90
6.5 Generalized Force Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7 Analytic Sensitivity Analysis 93
7.1 Sensitivity of Stroke-Averaged Thrust with Respect to the Design Variables . 93
7.2 Sensitivity of Normalized Stroke-Averaged Thrust with Respect to the Design
Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.3 Sensitivity of Various Constraints with Respect to the Design Variables . . . 99
7.3.1 Sensitivity of Chordwise and Spanwise Tension Constraints with Re-
spect to the Design Variables . . . . . . . . . . . . . . . . . . . . . . 99
7.3.2 SensitivityofDivergenceConstraintwithRespecttotheDesignVariables105
7.3.3 Sensitivity of the Efficiency Constraint with Respect to the Design
Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
iv
8 Aeroelastic Tailoring Results 108
8.1 Aeroelastic Tailoring for Maximum Thrust of a Plunging Membrane Wing . 109
8.1.1 Analysis and Optimization of a One Degree of Freedom Membrane Wing109
8.1.2 Analysis and Optimization of a Three Degree of Freedom Membrane
Wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8.1.3 Investigation of Exponential Increase in Thrust . . . . . . . . . . . . 120
8.2 Aeroelastic Tailoring for Maximum Normalized Thrust of a Pitching and
Plunging Membrane Wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
8.3 AeroelasticTailoringforMaximumNormalizedThrustandPropulsiveEfficiency130
8.4 Optimal Design for Various Cases . . . . . . . . . . . . . . . . . . . . . . . . 136
9 Conclusions and Future Work 143
9.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
9.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Bibliography 147
v
List of Figures
3.1 Airfoil motion/deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Chebychev Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Vortex locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Lift magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.5 Pitching moment magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.6 Lift phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.7 Pitching moment phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.8 Thrust coefficient due to the first 5 deformation shapes . . . . . . . . . . . . 39
3.9 Sensitivity of thrust coefficient due to h calculated at h = 1 . . . . . . . . . 40
1 0
3.10 Sensitivity of thrust coefficient due to h calculated at h = 1 . . . . . . . . . 40
2 0
3.11 Sensitivity of thrust coefficient due to h calculated at h = 1 . . . . . . . . . 41
3 0
3.12 Sensitivity of thrust coefficient due to h calculated at h = 1 . . . . . . . . . 41
4 0
5.1 Maximum thrust with magnitude constraint . . . . . . . . . . . . . . . . . . 64
5.2 Maximum thrust with magnitude and leading edge suction constraint . . . . 65
5.3 Efficiency of maximum thrust with magnitude constraint . . . . . . . . . . . 65
5.4 Efficiency of maximum thrust with magnitude and leading edge suction con-
straint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.5 Pareto Front for k = 0.01 (Magnitude constraint only) . . . . . . . . . . . . 67
5.6 Pareto Front for k = 0.25 (Magnitude constraint only) . . . . . . . . . . . . 68
5.7 Pareto Front for k = 1 (Magnitude constraint only) . . . . . . . . . . . . . . 68
5.8 Pareto Front for k = 4 (Magnitude constraint only) . . . . . . . . . . . . . . 69
vi
5.9 Pareto Front for k = 0.01 (Magnitude and leading edge suction constraint) . 70
5.10 Pareto Front for k = 0.25 (Magnitude and leading edge suction constraint) . 70
5.11 Pareto Front for k = 1 (Magnitude and leading edge suction constraint) . . . 71
5.12 Pareto Front for k = 4 (Magnitude and leading edge suction constraint) . . . 71
5.13 Stroke-averaged work for T = T at k = 4 . . . . . . . . . . . . . . . . . . . 72
h0
5.14 Thrust for η = 75% (Magnitude constraint) . . . . . . . . . . . . . . . . . . 73
5.15 Thrust for η = 75% (Magnitude and leading edge suction constraint) . . . . 74
5.16 Efficiency for T = T /2 (Magnitude Constraint) . . . . . . . . . . . . . . . . 74
h0
5.17 Efficiency for T = T /2 (Magnitude and leading edge suction constraint) . . 75
h0
5.18 Airfoil Shape for (k = 0.1) (Magnitude constraint) . . . . . . . . . . . . . . . 76
5.19 Pressure Distribution for (k = 0.1) (Magnitude constraint) . . . . . . . . . . 77
5.20 Airfoil Shape for (k = 0.1) (Magnitude and leading edge suction constraint) . 77
5.21 Pressure Distribution for (k = 0.1) (Magnitude and leading edge suction con-
straint) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.22 Airfoil Shape for (k = 4) (Magnitude constraint) . . . . . . . . . . . . . . . . 79
5.23 Pressure Distribution for (k = 4) (Magnitude constraint) . . . . . . . . . . . 80
5.24 Airfoil Shape for (k = 4) (Magnitude and leading edge suction constraint) . . 80
5.25 Pressure Distribution for (k = 4) (Magnitude and leading edge suction con-
straint) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.1 Membrane wing in air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
8.1 Effect of stiffness on average thrust for k = 0.25 (1 degree of freedom) . . . . 111
8.2 Effect of stiffness on average thrust for σ = 5 (1 degree of freedom) . . . . . 111
8.3 Maximum thrust as a function of reduced frequency (1 degree of freedom) . . 112
8.4 Stiffness for maximum thrust (1 degree of freedom) . . . . . . . . . . . . . . 112
8.5 Effect of stiffness on average thrust for k = 0.5 and σ = 5 (3 degrees of freedom)115
8.6 Effect of stiffness on average thrust for k = 0.25 and σ = 1 (3 degrees of freedom)115
8.7 Maximum thrust for a 3 degree of freedom system using a single design variable116
vii
8.8 Difference in results between the 3 degree of freedom system and 1 degree of
freedom system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.9 Improvement of maximum thrust with 2 design variables for a 3 DOF system 118
8.10 Improvement of maximum thrust with 3 design variables for a 3 degree of
freedom system for σ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8.11 Spanwise tension for maximum thrust of a 3 design variable, 3 degree of free-
dom system for σ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
8.12 Magnitudeofdeformationrequiredformaximumthrustwith3designvariables
for σ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
8.13 Maximum thrust normalized by the deformation with 3 design variables for
σ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
8.14 Determinate of the aeroelastic system with 3 design variables for σ = 1 . . . 123
8.15 Maximum Thrust (σ = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
8.16 Maximum Thrust (σ = 15) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
8.17 Maximum Thrust (σ = 40) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8.18 Maximum Thrust (σ = 150) . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
8.19 Maximum Thrust (σ = 40) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.20 Pareto front (k = 0.2 and σ = 40) . . . . . . . . . . . . . . . . . . . . . . . . 131
8.21 Pareto front (k = 0.3 and σ = 40) . . . . . . . . . . . . . . . . . . . . . . . . 132
8.22 Pareto front (k = 0.4 and σ = 40) . . . . . . . . . . . . . . . . . . . . . . . . 133
8.23 Maximum thrust at 60% Efficiency (σ = 40) . . . . . . . . . . . . . . . . . . 134
8.24 Maximum thrust at 70% Efficiency (σ = 40) . . . . . . . . . . . . . . . . . . 135
8.25 Spanwise tension for maximum thrust σ = 40 . . . . . . . . . . . . . . . . . 137
8.26 Spanwise tension for maximum thrust at 60% efficiency σ = 40 . . . . . . . . 138
8.27 Spanwise tension for maximum thrust at 70% efficiency σ = 40 . . . . . . . . 139
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List of Tables
3.1 Velocity potential functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Lift force components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Pitching moment components . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 Leading edge suction velocities . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5 Lift and pitching moment amplitude and phase terms . . . . . . . . . . . . . 35
8.1 Design vector for different numbers of design variables . . . . . . . . . . . . . 124
8.2 Magnitude of pitching for optimal design . . . . . . . . . . . . . . . . . . . . 138
8.3 Phase of pitching in degrees for optimal design . . . . . . . . . . . . . . . . . 139
8.4 Magnitude of pitching for 3 variable optimal design . . . . . . . . . . . . . . 140
8.5 Phase of pitching in degrees for 3 variable optimal design . . . . . . . . . . . 140
8.6 Nondimensional tension (spanwise and chordwise) for 3 variable optimal design140
8.7 Magnitude of pitching in degrees for 4 variable optimal design . . . . . . . . 141
8.8 Phase of pitching in degrees for 4 variable optimal design . . . . . . . . . . . 141
8.9 Nondimensional chordwise tension for 4 variable optimal design . . . . . . . 142
8.10 Nondimensional spanwise tension for 4 variable optimal design . . . . . . . . 142
ix
Nomenclature
a rigid body plunging of membrane
0
a rigid body pitching of membrane
1
b semichord
C thrust coefficient
T
C(k) Theodorsen’s function
F(k) real component of Theodorsen’s Function
G(k) imaginary component of Theodorsen’s function
h real airfoil shape coefficient
i
H(k) Hankel function
(cid:104) (cid:105)
H thrust Hessian
Tavg
(cid:104) (cid:105)
H structural work Hessian
Savg
i imaginary number
J(k) Bessel function of first kind
K stiffness matrix
k reduced frequency
L lift
M pitching moment (or mass matrix)
p pressure
q nondimensional chordwise tension
x
q nondimensional spanwise tension
y
S leading edge suction
x