Analysis of Steady and Unsteady Flows past
Fixed and Oscillating Airfoils
Mohammed Abdo
Department of Mechanical Engineering
McGiIl University
Montreai, Quebec, Canada
March, 2000
A nesis submitted to the Faculty of Graduate Studies and Research in partiai
fulnllment of the requirements for the degree of Master of Engineering.
Q Mohammed Abdo, 2000
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Abstract
This thesis presents a method based on the vetocity singdarities for the analysis
of oscillating airfoils. The method of velocity singularities has k e n ongindly developed
by Mateescu for the analysis of the steady flows past airfoik. Tbis method makes use of
special singulanties associated to the Ieading edge and ridges that duectiy represent the
cornplex perturbation velocity.
Closed form sulutions were derived for the pressure distributions and for the
aerodynamic forces and moments acting on the oscillating airfoils with or without
oscillating ailerons.
The solutions obtained with the velocity singularity method Cor steady flows past
airfoils were f o n d to be in very good agreement with the exact solutions obtained by
conformal transformation in the case of thin Joukowski airfoiis. as well as with the
previous solutions for the case of flexible-membrane airfoils obtained by Mateescu &
Newman, NieIsen and Thwaites.
For unsteady flows. this method has been validated for airfoils executing
oscillatory motions in translation and pitching rotation and for airfoils with ailerons
executing oscillatory rotations. The pressure distributions and the aerodynamic forces and
moments obtained with the present method were found to be in excellent agreement with
the previous solutions obtained by Theodorsen and Poste1 and Leppert.
The method has then been extended to the unsteady flows past airfoiIs executing
flexural harmonic osciIIations. Closed form soIutions were also daived for the pressure
distributions and the aerodpamic forces and moments acting on an airfoil with an aileron
executing flexural osdations. No comparisons were presented for the cases of flelcural
oscillations since there are no previous resuIts known.
The present approach dispIayed an excelient accuracy and efficiency in aU
problems smdied.
Cette thèse présente une méthode d'analyse basée sur les singularités de vitesse
pour les profïis aérodynamiques ayant des mouvements osciiiatok. La méthode des
singularités de vitesse a été développée originalement par Mateescu pour l'analyse
d'écoulements stationnaires autour des profils aérodynamiques. Cette methode utilise des
singularités spéciales asociées au bord d'attaque et aux arêtes qui représentent directement
la vitesse complexe de perturbation .
Des solutions explicites ont été établies pour la distribution de pression et pour les
forces et les moments aérodynamiques agissant sur les profils aérodynamiques en
mouvements oscillatoirç. avec ou sans aileron en mouvement oscillatoire.
Les solutions obtenues avec la méthode des singularités de vitesse pour les
écoulements stationnaires autour des profils aérodynamiques sont en excellent accord
avec la solution exacte obtenue par transformation conformale pour Ie cas des profils
aérodynamiques de Joukowski, ainsi que avec les solutions antérieures obtenues par
Mateescu & Newman, Nielsen and Thwaites pour les profils aérodynamiques flexibles.
Pour les ecodements instatiomaires, cette méthode a été validée pour les profils
aérodynamiques exécutant des mouvements oscillatoires en translation et en rotation,
ainsi que pour des profils munies d'un aileron oscillant. Les distributions de pression et
les forces et moments aérodynamiques obtenus par la présente méthode ont été trouvées
en excellent accord avec les solutions obtenues dans ces cas par lleodorsen et par Postel
et Leppert.
La méthode a été puis utilisée pour étudier les écoulements instatioMaires autour
des pronls aérodynamiques exécutant des oscillations harmoniques en flexion. Des
formules explicites ont été aussi établies pour la distribution de pression et pour les forces
et moments aérodynamiques agissant sur un profil aérodynamique muni d'un aileron
exécutant des oscilIatioas flexuraIes. Aucune comparaison n'est pas présentée pour le cas
des oscillations en flexion parce qu'il n'y a pas des résultats précédent connus.
La présente méthode a démontré une excellente précision et efficacité dans tous
Ies problémes étudiés.
Acknowledgements
1 wodd k e to thank my thesis supervisor. Professor Dan Mateescy for his
guidance, knowledge, and dedication throughout the course of this work.
I am gratefùi to my parents for their support and encouragement.
I would iike also to th& al1 my &ends at McGill University.
Contents
Page
Abstract.. ..................................................*. .............................
1
Résumé.. ..................................................................................
II
..
.
Acknowledgements.. ...................................................................
*.
Table of contents.. ..........................,...........................- - .--.
..........
iv
List of figures. .............,.............................................................
vii
List of tables......................................................... ... .................
x
1 Introduction
1
2 Problem formulation and review of Theodorsen theory
5
2.1 ProbIem formulation..
........-................................................
111
5
2.2 Previous studies for unsteady flows past oscillating airfoils:
Theodorsen's method. .................. .......
..............................
2.2-1 Motion without circulation.. ,..,.........,...........................
2.2.2 Motion with circulation around a flat plate. .......................2.2.3 Complete motion. ......................................... .............
+.
3
Method of velocity singuiarities for steady flows
1O
12
14
16
19
3.1 Method of velocity singuiarities for steady flows pst airfoils.. .......
19
3.2 How solutions for continuous1y cambered rïgid airfoils................
25
3.3 Flow solutions for flexible-membrane airfoib. ..........................
5O
3.4 Flow solutions for jet-flapped d o i I s,.,....-.......-.....................
35
4 Method of velocity singularities for nnsteady flows
4.1 Method of velocity singuiarities for unsteady flows past airfoiIs.. .....
39
39
Page
4 2 Prototype problem...............................................................................
44
4.3 The complete oscillating airfoi1 problem..............................................
53
4.3.1
The reduced circulation around the airfoil.........................
53
4.3.2
The reduced pressure coefficient....................................
54
4.3.3
The reduced lift and pitching moment coefficients. ..............
55
4.4 Method of velocity singularities for unsteady flows past airfoils with
. .
oscrllatuig ailerons.. .............................................................................
56
4.4.1
Prototype problem for the case of an oscillating aileron............
57
4.4.2
The complete problem for the airfoil with an oscillating aileron...
64
4.5 Unsteady flow solutions for airfoils executing oscillatory translations
and rotations....................................................................................
4.5.1
Boundary conditions on the oscillating airfoil and aileron........
73
4.5.2
Umteady flow solution past an airfoil in oscillatory translation..
74
4.5.3
Unsteady flow solution past an airfoi1 in oscillatory rotation ....
77
4.5.4
Unsteady flow solution past an airfoil executing flexural oscillations. 79
4.6 Unsteady flow past an airfoi1 with an oscillating aileron ................
4.7 Unsteady flow past an airfoil with an aileron executing
. .
f l e d oscillations.............................................................
5
Results and discussion
5-1 Case of airfoils executing osciIIatory translations ......................
5.2 Case of airfoils executing oscillatory rotations.................... ............
..
5.3 Case of airfoils executing flexurai oscillations.................................
5.4 Case of an airfoil with an aileron executing pitching oscillations.......
5.5 Case of an airfoil with an aiieron executing îlexuraI oscillations.........
6
73
Conclusions
Bibliography
Appendices
A General integrals
A.1 The integrai F (z).....................................................................
A.2 The integral 3 (x) .................................................................
A.3 Recurrence formulas for the integrai Ik........................................
A.4 Recurrence formulas for the integrai
Q, .......................................
A 3 Recurrence formulas for the integrai Fk.......................................
A.6 Integrds related to I, ............................................................
A.7 Recurrence formula for the integrai 1, ........................................
B Theodorsen's formulas
B .1 Fonnulas reIated to Theodorsen's results......................................
.......................................
B.2 Theodorsen's function..............
.
.
.
C
Special complex functions
C.1 The cornplex function H.(qs)..................................................
C.2 The complex function A, (z.,a)
..................................................
C.3 Summary of the behavior of the special complex Functions.......................
D
Derivation of the unsteady pressure coefficient
D.1 Unsteady pressure coefficient for the airfoil...............................................
D.2 Unsteady pressure ccefficient for the airfoil with an oscillaring aileron.....
List of figures
page
2.1 Geometry of an airfoi1 at an incidence a in uniform flow of velocity LI, .....
2.2 Geometry of a flat plate airfoi1executing translation and pitching rotation
. .
oscillations.. ......................................................................
2.3 Decomposition of the unsteady flow aromd an oscillating airfoil into: (a) The
steady flow past a fixed airfoii,and (b) The unsteady motion past an oscillahg
flat plate....................................................................................................
2.4 Effect of a jump in the vertical veIocity on a tIat plate in the motion
without circuIation.. .............................................................
2.5 Motion with circulation around a flat plate due to a vortex placed
downstream.......................................................................
2.6 Flat plate geometry as presented in Theodorsen's solution.................
3.1 Geometry of a thin flapped airfoil: a) in the physicd complex plane;
b) in the complex plane defined by the conformai transformation.. .......
3.2 The jump of the irnaginary component of the complex velocity w (s)...
3.3 Comparison between the present solution (-) and Glauert's solution (O,A,n)
with various number of terms N for a flapped airfoil
(a=0,B =O.lrad.s/c=0.6) atthreechordwise locations...............
3.4 Geometry of a parabolic thin airfo.li
......................................
3.5a Comparison between the present soIution and the exact solution for
a circula arc airfoil at a = 5' and E = 0.02 .................................
3 3 Comparison between the present soiution and the exact solution for
a circuiararc airfoil at a = 2' and E = 0.01 .................................
3.6 Geometry of a flexible membrane airfoil......................................
3.7 Geometry of a jet-flapped airfoil.. ..............................................
4.1 Rat plate airfoil with a boundary condition e(x,r) ..................................
4.2 Geometry of a flat plate airfoil indicating the kvortex distribution......
4.3 Free vortices shedding at the trading edge of the airfoi1............................
4.4 Representation of the velocity jumps in the boundary conditioas of the
prototype problem of an oscillating airfoil................................................
4.5 Geometry of an osciIlating aileron................................................
4.6 Velocity jumps representations in the boundiuy conditions on an airfoi1
. .
with an oscrllatmg aileron...........................................................................
4.7 Geometry of an airfoil and aileron under unsteady osciilarions...............
4.8 Geometry of an airfoi1execdng verticai osciliatory translation h (f ).......
4.9 Geomeq of an airfoil executing osciffatory rotation 0 (r ) ...................
4.10 Geometry of flexurai oscillations on an airfoi1.................................
4.1 1 Geometry of an airfoi1 and aileron executing oscillatory ratauon P (1) .....
5.1 Real and imaginary parts of the pressure ciifference for an airfoil executing
osciilatory transiation...............................................................
5.2 Real and imaginary parts of the lift coefficient for an airfoil executing
osciliatory translation................................................................
5.3 Red and imaginary parts of the pitching moment coefficient for an airfoil
executing oscillatory translation. .................................................
5.4 Typical variations in t h e of the lift and pitching moment coefficients for
......
an airfoil executing oscillatory translation ......................
........
5.5 Real and imaginary parts of the pressure difference for an airfoil executing
oscillatory rotation., ................................................................
5.6 Red and irnagiaary parts of the lifi coefficient For an airfoil executing
oscilIatory rotation...................................................................
5.7 Real and imaginary p m of the pitching moment coefficient for an airfoil
executing osciliatory rotation ......................................................
5.8 Typicai variations in t h e of the lift and pitching moment coefficients for
au airfoii executing oscilIatory rotation......................................
5.9 Real and imaginarv parts of the pressure difference for an airfoi1 executing
parabolic flexirral oscillations.....................................................
Page
Real and imagulary parts of the Lift coefficient for an airfoil executing
parabotic flexural oscillations .........................................- .......
Real and imaginary parts of the pitching moment coefficient for an airfoil
executing parabolic f l e x d osciilations.. ..................................
Typical variations in t h e of the lifl and pitching moment coefficients For
an airfoil executing parabolic flexural oscillations...........
.
.
.
.......
Red and imaginary parts of the pressure ciifference coefficient for an
airfoil with an aileron (s, I c = 0.75) executing oscillatory rotations.......
Real and imaginary parts of the reduced üft coefficient for an airfoil with
an aileron (s, 1c = 0.70)executing oscillatory rotations............................
Red and imaginary parts of the reduced pitching moment coefficient for an
airfoil with an aileron (s, 1c = 0.70) executing oscillatory rotations.........
Real and imaginary parts of the pressure difference coefficient for an
airfoil with an aileron (s, / c = 0.60) executing parabolic flexural
. .
osciilations............................................................................................
Red and imaghary parts of the reduced lift coefficient for an airfoil with an
aileron (s, I c = 0.60) executing parabolic flexurd oscillations...............
IO9
Real and imaginary parts of the reduced pitching moment coefficient for an
airfoil with an aiIeron (s,1c = 0.60) executing parabolic flexural
. .
osclIIations..............................................................
110
Real and imaginary parts of the pressure difference coefficient for an
airfoil with an aileron (s,l e = 0.75) executing parabotic f l e d
.
.
osciiianons............................................................................................
111
Real and imaginary parts of the reduced lift coefficient for an airfoil with an
aileron (s, 1c = 0.75) executing parabolic fiexurai oscillations...............
Real and imaginary parts of the reduced pitching moment coefficient for an
airfoi1 with an aileron (s,1c = 0.75) executing parabo tic flexinal
112
Page
Real and irnagiaary parts of the pressure dserence coefficient for an
airfoii with an aileron (s,1c = 0.80) executing parabolic flexural
- .
osctlla~ons
............................................................................................
114
Red and imaginary parts of the reduced lift coefncient for an airfoi1 with an
aileron (s, / c = 0.80) executing parabolic flexurai osciIIations...............
115
Red and imaginary parts of the reduced pitching moment coefficient for an
airfoi1 with an aileron (s, 1c = 0.80) executing parabolic flexurai
. .
oscillations............................................................................................
116
List of tables
3. I Cornparison between the present solution for flexible-membrane airfoils
and the previous results of Nielsen and Thwaites............................
B. 1 Theodorsen8sfunction
c (k) = F + iG ........................................
34
135
.................
133
C.2 General behavior of a special compIex singularity A, (r,a).................
143
C. 1 Gened behavior of a speciai compIex singularity H, (2,s)
Chapter 1
Introduction
The analysis of unsteady flows past oscillating airfoils and wings is very important
for the aeronautical applications. The problem of predicting the aerodynamic
characteristics of such airfoils is among the first ones that have been studied in the
development of the aeronautical sciences. This anaiysis has proved to be of considerable
practical importance. Studies of unsteady airfoil flows require predicting the unsteady
aerodynamic loads acting on thin lifting surfaces. PotentialIy beneficial effects of
unsteadiness has also been given some attention. such as controlled periodic vortex
generation, irnprove the performance of turbo-machinery, helicopter rotors. and wind
turbines by controlling the unsteady forces in some optimum way. Most of these studies
concern either periodic motion of an airfoil in a uniform Stream or periodic fluctuations in
the approaching flow. There is a need to develop efficient methods of analysis that c m be
used in conjunction with tbe structurai research to study the dynamics and stabiiity of the
airfoil and wing structure subjected to the unsteady aerodynamic forces.
The anaiysis of thin airfoi1 theory has been a abject under research and development
for many years. Since it is in general ciifficuit to develop an exact solution for the ideal
flow past an airfoi1 of arbitrary shape, approxirnate methods have been deveIoped to
solve tbis problem. Among the pioneering works are the ones of Glauert [6,I4]and
Birnbaum, Glauert's method approximates the airfoil by its camber line and by modehg
this camber line as a vortex sheet. An integral equation resuited for the distribution of the
vortex sheet by iinearinng the boundary conditions and transferring these boundar~
conditions fiom the d a c e of the airfoi1 to the chordline. GIauert solved the integral
equation by means of Fourier series. This approach is the basis of thin airfoil theories.
The fact that the boundary conditions are in general non-linear rnakes the probiem of
fincihg an exact solution for the motion of an ideal flow very complicated. Few soiutions
are available for a limited number of geometries only. For instance, Joukowski [21,14]
airfoils, which can be directiy obtained fiom the conforma1 transformation of a circle.
However, these airfoils are of Limiteci use in practice because of tfieir predetermined
shapes, but they are essentid in assessing the accuracy of the approxhate methods.
The method of velocity singularities has been fkst htroduced for supersonic flow
past wings and wing-fuselage systems, Mateescu [Il]. Mateescu E12.131 has also
intmduced a new method for the solution of the steady flow past thin airfoils in subsonic
fiows. The method of velocity singdarities deveioped by Mateescu, makes use of special
singdarities at the airfoil Ieading edge and ridges dong the airfoil. This method. does not
diredy use the velocity potential but instead operates directly on the veiocity fieid by
considering the singular behavior of the flow at geometncaIly important eiements of the
airfoil. In addition, this soiution satisfies the Kutta condition at the trading edge. This
method is characterized by a simple and direct approach. leading to closed-form soIutions
in ail cases when the airfbil contour is specified. ALthough the initiai method was
developed as a Iinw theory, a non-linear development is dso possible [12]. This method
has been extended to analyze the problems of flexible airfoits and jet-tlapped airfoiis, in
which the shape of the jet sheet or the flexible membrane depends on the pressure
difference a m s s them, see Mateescu & Newman [13], Nielsen [22] and Thwaites 1271.
Studies of unsteady-airfoil flows have been motivated mostfy by efforts to prevent or
reduce such undesirable effects as fiutter and vibrations. Among the best known and most
enlightewing analysis of this class of problems are those by Theodorsen [26,15] and Von
K h h & Sears [28], who considered a thin flat pIate and a t d i n g Bat wake of vortices
in incornpressrile Buid. Theodorsen developed the velocity potentials due to the unsteady
flow around the airfoil. The aerodynamic forces and moments on an oscillaihg akfoil are
then determined on this basis.
Kiissner and Schwarz [5,9,25] obtained a solution of the Lift and moment for a .
airfoii osciüating with arbitrary mode, which depends on a Fowier series expansion-
A more specinc solution for the unsteady lift and moment for an airfoii was
introduced by Biot [SI. The solution is S i t e d to the special cases of an airfoii executing
vertical-translation and rotationai oscillations.
Many important features of the unsteady airfoil behavior were desmied by
McCroskey [18,19]. A number of scientists such as Dietz, Schwarz, Kemp, and Basu
[2,4,8,25] covered various aspects of the unsteady flow past airfoils. in these
investigations, the primary objective was to obtain the lift forces and moments; therefore,
computations were not made to show typicai pressure distributions. Theodorsen [26]
denved the forces and moments for the harmonicaily oscillating airfoil without achially
calculating the pressure distributions. Postel and Leppert [24] have caiculated pressures
acting on a thin airfoii with harmonic pIunging and pitching oscillations of srnail
amplitude in incompressible flow. These previous soIutions are restricted to the special
cases of oscillatory translation and rotation of an aidoil. However, the previous formulas
of the aerodynamic forces are complicated and are obtained for speciai cases of the
airfoils oscillating as rigid bodies without considering the flexural oscillations.
In the present thesis. a new analytical method of solution for the analysis of the
unsteady flow past airfoils and ailerons. based on velocity singularities, is developed.
Closed form formdas of the aerodynamic forces dong with the pressures acting on a thin
aii.foil oscillating with various harmonic motions are determined. The formulas obmined
are of a simple and efficient form and the mathematicai treatrnent provides an effective
approach for the general case of oscillations, including the flexurai oscillations.
The first objective of the present work is to obtain the solutions of the unsteady flow
past oscillating airfoils, which codd efficiently avoid the mathematicai dficulties
encountered in the classical theories. The second objective of the present snidy is to
develop a comprehensive approach for the analysis of the unsteady flow past oscillating
airfoils.
Chapter 2 is devoted to the problem formulation of the steady and unsteady flows
past fixed or osciilating aUrfoiIs. The previous theory developed by Theodorsen [263 is
ais0 presented
in Chapter 3, the method of velocity singdarities for steady flows past airfoils is
discussed. This method of solution is applied for various problems of specifïc airfoils.
The method of velocity singularities is first validated by comparison with the exact
solutions obtained by conformal transformation [13,I4], and then with the d t s
obtained by Nielsen [Eland Thwaites [271 for flexible-membrane airf~ils,
in Chapter 4, a new method for the analysis of the unsteady flow past oscillating
airfoits using velocity singdarities is presented. The method is extended to the analysis of
the unsteady flow past airfoils with oscillating ailerons, The aerodynamic forces and the
pressure distribution on the oscillating airfoik and ailerons are then determined.
The solutions of the flow past airfoils executing more general osciIIations such as,
flexural oscillations are aiso determined.
Chapter 5 presents numerical results for various cases of oscillating airfoils obtained
with the method of solution developed in Chapter 4. First, the present solutions are
compared for validation with the results obtained by Theodorsen [26] and Postel &
Leppert [24] for the case of oscillations in transIation and pitching rotation. Then, the
present solutions obtained for airfoils with ailerons executing flemiraI oscilIations are
presented.
The Iast chapter is devoted to the conclusions and recommendations for future
further studies.
Chapter 2
Problem formulation and review of Theodorsen
theory
2.1 Problem formulation
Consider the flow around an airfoil having an angle of attack a with respect to the free
stream velocity U,as shown in Figure 2.1.
Fig 2.1 Geometry of an airfoil at an incidence a in uniform flow of velocity U, .
The airfoil could be d o n a r y or can execute various types of osci1Iations. ï h e
most common types of oscillauons are the linear transIation and the pitching rotation
defined, respectively, as,
h(r)= ho c ~ s ( o > t + ~ , ) ,
(2-1)
e(r)= 8, cos(air +ql,).
(2-2)
where, o and r are the fiequency of osciilations and the time, respectively. In linear
translation osciiiations, ho and cp, are the amplitude and the phase angle. SimiIariy, 0,
and cp, are the amplitude and the phase angle of the pitching rotation oscillations.
The above equations can also be expressed in complex fom as,
h(t) = ~ e { ehm t } .
e(t)= ~ e ( P
é
~},
where the complex amplitudes iand
6 are d e h e d by.
h = b etqb,
(2-5)
ê=e,
(2-6)
el.",
For simplification, in the foiiowing, the reai part symbol, ~ e ).(will be omitted and thus
equations (2-3), (2-4) are expressed in the form,
iet'*,
0 (t ) = 6 elm'.
h(t)=
(2-7)
(2-8)
However, this will imply that the real part will have to be taken fiom the final complex
solution (in fact, this is a cornmon practice for this type of hamonic oscillation
problems).
The velocity poteutid equation for unsteady potential flows defined by the fluid velocity
and velocity components is,
where,
4 represents the velocity potential and V is the fluid velocity, are defined as.
-
V=Ve,
-
V = (u--casa + u ) 4 + (LI.
-sina+v)-j .
(3-10)
(2- 11)
The Bernoulli-Lagrange equation for barotropic incompressibIe fluids (for which the
density is a function of pressure only) is,
The pressure coefficient equation in second order approximation is,
It is important to note that in the case of incompressible unsteady potentiai flows, the
velocity potential equation and the pressure coefficient equation are,
C, =-,P-P*
1
-pu:
9
[n
Cartesian coordinates, the perturbation velocity components
(11 ,v)
and the
perturbation velocity potential ( 9 )equations are,
Boundary conditions
The boundary condition on an airfoi1 executing vertical translation and pitching rotation
oscillations is determined in the assumption of small amplitude oscillations. Consider a
body executing a srnail oscillations, the equation of the body surface is expressed
generally as.
(2-19)
f (xTyT-i,t)= O .
The bolmdary condition on the body surface is given by,
af
-+(U,
+q).vf =o.
at
where, U, is the uniform Stream veIocity and
q is the disturbance velocity.
Consider a flat plate airfoi1 in oscillation as showa in Figure 2.2.
Fig 2.2 Geometry of a flat plate airfoil executing translation and pitching rotation
oscillations.
The equation of the body surface of flat plate airfoi1 is.
(2-2 1 )
y=h(t)-xtanû.
In the assumption of srnaIl amplitude oscillations ( tane t 0 ), the equation of the body
surface f (x,y, t ), is expressed as,
f ( ~ , ~=
, ty+x-8(t)-h(t)
)
=0.
The boundacy condition on the oscillating plate is denved tkom,
af
-+(ü=
++7f
at
=o.
where,U, is the uniform stream velocity and q is the disturbance vel.ocity. The
equations of which, are expressed as,
-
U, = iCr,,
q=iu+jw,
in which, Vf is derived from,
a f and Vf in equation (2-23) are expressed as,
The values of dr
vf = i 0 ( t ) + j 1.
The vertical translation and pitching rotation osciilations are given by,
A
h(t)= e'm'.
e (t)= é elm1,
af
ïhe value of the time derivative - is expressed as,
ar
df
-=ie[è
x - h ] elY.
at
Hencc, the boundary condition on the oscillating plate is,
w ( W l ,du =
[ -u. è + i a > ( k - é x ) ] e r m .
which can be expressed as,
W(X,~.~)(,,
=[ii.(x.y)
IMNe
eial.
where the reduced verticai disturbance veIocity W(x.
,
=
- i + i o (h - è x ) .
is expressed as.
2.2 Previous studies for unsteady flows past oscillating airfoils:
Theodonen's theory
The unsteady flow past an airfoi1 osciIlaîhg acound a mean position defined by
the mean incidence a ,can be decomposed into the steady flow past the 6xed airfoi1 at the
mean incidences and the unsteady motion pst an oscillating flat plate in the assumption
of small perturbations, Figure 2.3.
Fig 2.3 Decornposition of the unsteady flow around an oscillating airfoi1 h o : (a) The
steady flow past a Gxed airfoil, and (b) The unsteady motion past an oscillating flat plate.
For the unsteady flow past an osciIlating plate. ïheodorsen [15361 has developed
a method to determine the aerodynamic forces on an oscillating airfoil. The theory was
based on the potential flow and the Kutta condition.
In this method,
the perturbation
veIocity potential,~,around the osciliating airfoil is decomposed into two parts:(i) the
perturbation potential p, , correspondhg to the motion without circulation arotmd the
airfoil, ad,(iï) the potential p, ,corresponding to the motion with circulation due to the
shedding vortices.
The equation of the perturbation potentiai for the uzlsteady incompressible flow is given
by,
The pressure coefficient equation is given by,
ep
Introdwing the reduced potential @ (r.y ) and the reduced pressure ~~effieient (*.Y )
in the fom,
g,(x, y,[) = @ ( . L Y ) ~ ' ~ .
(2-37)
y, r ) = ê, (x. y)el".
(2-38)
C,
(.Y,
the above equations become,
The general form of the boundary condition on the oscitlating airfoil is given by.
where.
For the considered case of phmging and pitching osciliations, the boundary conditions are
given by equaîion (2-34).
In this manner, the unsteady aerodynamics problem has been reduced to the study of the
reduced motion dehed by the reduced potential.
23.1 Motion without circulation
Consider the flow without circulation around a flat plate, the effect of the vertical
velocity jump is replaced by a source (+A Q)and sïnk (- A Q) system situated on the upper
and lower sides of the plate as show in Figure 2.4. The relation between the velocity
jump and the strength of the source ( A Q)is given by.
(2-44)
A Q = 2 b W(X,~ ) d r,,
Fig 2.4 Effect of a jump in the verticai velocity on a flat plate in the motion without
circulation.
Joukowski's conformal transformation is used to transform the flat pIate in the plane
z = x + iy into a circle in the plane
C = 5 + iq .The transformation is defmed by,
The complex potential of the source and sink is given by,
where,
The velocity perturbation potential for the flow without circulation is denoted by,
where,
The axial perturbation velocity is given by,
a
d u , (x,x,,r)=-[dq
b-ik
,,
d
(x,x,.t)]=~(.r,,r)-
The corresponding pressure coefficient is expressai by.
The pressure difference across the plate is given by,
&-v:
cr-.rl)JL7a
(2-51)
2.2.2 Motion with circulation around a flat plate
The motion with circdation around a flat ptate is studied by considering an elementary
shedding vortex A
situated downstream at z = x,.
Joukowski's conformai
d o n n a t i o n is used to transforrn the flat plate into a circle and hence, a vortex of
opposite sense bas to be introduced inside the circle at the symmetricd point in order to
render the circle, the streamliae property, as shown in Figure 2.5.
AI-
Fig 2.5 Motion with circulation around a flat pIate due to a vortex placed downstream.
The cornplex potential is given by,
The perturbation velocity potentid is,
The axial perturbation vebcity is,
dldt
In the unsteady case. at any instant of time t. a fhe vortex - d t
is shedding at the
trailing edge of the profile. Since this free vortex is moving downstream with the velocity
Urnof the uniform stream,the intensity of the free vortex distribution y /, along the axis
of the plate, is expressed as,
A fiee vortex Located at the instant t at the distance b .r,. was shedding at the trailing edge
at a previous instant of time, and hence,
The reduced frequency k and the harmonic oscillations are given by,
where, yi is a constant that will be determined by imposing the Kutta condition at the
trailing edge of the profile. The free vortex distriiution is related to the perturbation
potential by,
.
~ r = ~ , ( x b&
~,t)
(2-61)
Since the fiee vortex is moving towards downstream with the velocity of the Eke stream.
the pressure coeficient corresponding to the pureiy circulatory motion produced by the
perturbation velocity potential after considering the effect of ail the fiee vortices situated
in the wake o f the oscillating airfoiI, is given by,
d A p, = p C', y,'
elk
e'"'
2
-
1
"
Ie
-
233 Complete motion
By aciding the potential comsponding to the motion without circulation amund the flat
plate, due to the w
d veiocity jump
w (x, ,t) ,and the potential correspondmg to the
pure circulatory motion, producd by the shedding &ee vortices in the wake of the airfoil,
one obtains the total perturbation velocity potential for the complete motion around the
oscilIating plate or airfoil.
Replacing the unknown constant y: by the equation,
and irnposing the Kutta condition at the trailing edse of the profile, one obtains the
unknown constant y, .The axial disturbance velocity is given by,
By imposing the Kutra condition at the rraiiing edge (x = i), one obtains,
[d u (x, x,
.
(2-65)
t}Lmi= finite ,
where ~ g ? (k)
) ,H/')(k) represent the Hankel hctions of second kind of orders
zero and one [7,20,29]. See Appendices A and 6 for deîails, The pressure difference
across the plate combines a quasi-steady term, inertia force tenu and the effect of the fiee
vortices in the wake of the airfoil. The equation of the pressure equation is given by,
2
~ p G . i j =a
1+x
1
/=-
1+XI dr,
IV(x, , t ) 1-x, x-x,
where Theodorsen's function ~ ( k,is) definecl as,
where Hi') (k),H!')(k) represent the Hankel fiinctions of second kind of orden zero and
one [7,20,29].See Appendices A and B for detaik
The function L (x,x, ) is given by,
Theodorsen's results are presented Far the following oscillatory motions of the airfoil
(Figure 2.6),
Fig 2.6 Flat plate geometry as presented in Theodorsen's solution.
The oscillations of the airfoiI are denoted as,
h(t)= h-elmt,
a(t)=&.e'"',
~(=
t )fi-da',
The boundary conditions on the airfoil-aileron combination are dehed as,
w ( x , r ) = -aum-8-b(x-a)a
for x E (- 1. c, ) ,
(7-73)
~ ( x , r ) = - a ~-h-b(x-a)a-b(x-eI)(i-pum
.
for xx(cl -1).
(2-74)
The lift force is given by.
The pitching moment equation is,
The hinge moment equation is given by,
A list of the terms that appear in the above formulas is given in Appenditc B.
Chapter 3
Method of velocity singnlarities for steady flows
3.1 Method of velocity singularities for steady flows past airfoils
Prototype Problem
Consider a very thin airfoi1 extending on the red axis in the complex plane z = x+i y
fiom z = O to = c with a sudden change of slope, due to a single ridge, at z = s Figure
3.1. A typical configuration of a flapped, utherwise uncambered airfoil Figure 3. l a is fmt
considered. This gives the basic suigdarities at the kading edge L(X=O) and the ridge
R ( x = s ) , where the airfoil slope suddenly changes by an angle
P.
Fig 3.1 Geometry of a thin flapped airfoil: a) in the physicd cornplex pIane; b) in the
compIex plane defined by the conformai transformation.
The boundary conditions are,
U(T-a)=vo,
U(s-a-f3)=v0 +AV.
for O c x c s
for s < x < c
where a is the angle of attack, r is the leading-edge slope with respect to the cor4
v, = U (s - a) represents the normal-ta-chord perturbation velocity v at the Ieading edge
and A v = -BU denotes the change in v at the ridge. In the complex plane z = x +i y,
the conjugate complex perturbation velocity is defmed as,
w(z)=u -i(v - v o ) .
(3-2)
The boundary conditions (3-1) appiied on the airfoil chord are, thereby, sirnplified to,
O
1me[w(z)13r={-Av
for O < x < s
fors<x<c'
and, due to the antisymrnetric nature of the perturbation flow past the airfoil in the
complex plane,
,.
REAL[w(z)]
=O
forx<O, x > c .
(3-4
The local behavior of the perturbation velocities at the leading and trailing edges are.
respectively,
w q
:-,
+
(where A, is independent of z)
where the füst condition represents the leading-edge singuiarity for a flat plate at r = O ,
and the second is the Kutta condition at the traiIing edge (z = c).
The complex velocity w ( z ) shodd aIso contain a logarithmic singularity in order to
provide the jump A v at the ndge (z = s):
Iadeed, considering z - s =r, ele in the vicinity of the ridge R (r = s) Figure 3 2, one can
successively write,
Fig 3.1 The jump of the imaginary component of the complex veiocity w ( r ) .
The expression of w ( z ) c m be easily determined in an auxiliary phne j = j
+ iq
Figure 3.1 b, defined by the conformal transformation of the Schwarz-Christoffel type.
-1
5' =
.
(PZ)
(3-8)
which transforms the airfoil (the x m i s between O and c ) into the whole of the real m i s
in the & plane. and the rest of the x axis into the imaginary mis. The velocity
components do not change under conforma1 üansformation in contrast to the usual
complex potentid theory, the botmdary conditions becorne,
Tbe leading-edge singularity becomes a doublet singuIarity I / < at the origin, and the
ndge singularity becomes in this plane t(AVl n)ùi(~
i a),and, thus.
where the constant A has to be determiaed.
In the physicd complex plane r ,the solution becomes,
and the constant A cornes from the requirement that u = v = O at z + -a,. resulting in.
On the upper surface of the airfoil (y = O. z = x). the chordwise perturbation velocity
14
= REAL w(z) may,therefore. be expressed as,
c-x
2
--AVG(C,S.X),
X
X
where the singular ridge contribution ~ ( c , s . xis) defined as.
1O
for x < O and.r>c
The pressure ciifference across the airfoi1 in dimensionless form AC, = 4ulU, where
_+
u are the chordwise perturbation veiocity components on the upper and Iower surfaces
of the airfoil, can now be expressed as,
where v, =(r-a)(/and AV =-PU.
The Iift coefficient and the pitching moment coefficient about the leadiing edge are
obtained by integration of the pressure coeficient over the aidoil:
[ jr JE)].
CL=2n(a-r)+4~cos-' - +
3;2
4
Ill
(1-j) .
A cornparison between the present closed-fom solution and Glauert's [6,13,14] thin
aufoi1 theory based on a modified Fourier expansion with N terms is given in Figure
3.3. One can notice that the results obtained with Gduert's theory using an inmashg
number of ternis, in the Fourier expansion osciiIate about the present closed-form
solution or tend asymptotically, but very stowiy, toward the present solution.
Fig 3.3 Cornparison between the present solution (-) and Glauert's solution (0d.Û) with
various number of terms rV for a flapped airfoi1 (a= O. P = 0.1 rad. s / c = 0.6) at three
chordwise locations.
3.2 Flow solutions for continuously cambered rigid airfoils
Consider a thin continuously cambered airfoii detined by the camber slope.
hV(x)=a
+v,(X)l~,
(3-19)
where v, is the normal-to-chord perturbation velocity v at the solid boundary, and
X = x l c is a nondiensionai coordinate. Using the two typicai singular contributions
for the leadig edge and ridge, the solution for any camberline shape cm be obtained by
superimposing infinitesirnally flapped airfoiis equation (3-16) in the form.
4
ACp (X)
=--{U
[
1FY;
v,, (O)+?
@)COS-I
JSds
'=O
where X = x / c a n d S = s / c .
Considering a polynomiai representation for the camberline slope (which was aiso
adopted in the design of the NACA 5 digit airfoil series),
the pressure difference across the airfoil is obtained in nondimensiond form as,
where,
The lift and pitching moment coefficients are,
-
Parabolic thin airfoil
Consider a parabolic thin airfoil (Figure 3 -4) defined by,
Fig 3.4 Geometry of a parabolic thin airfoiI.
The derivative of the airfoii contour is expressed as.
From equation (3-X ),
ho = 4 &
.
h, = - 8 ~ and h, = O for k 2 2 .
The expression of the pressure difference coefficient is given by.
The Iift and pitchuig moment coefficients are expressed, respectively, as,
C, = 2 x ( a + 2 ~ ) .
(3-30)
The general expression of the geometry of a circdar arc a o i l is given by,
where,
The derivative of the a o i l is expanded in Taylor series and is expressed by,
where,
Y=
.r (c - x)
cl
Bo and B, =
64e2
(1-4~'p'
From equation (3-21), the 9 terms are expressed by,
3
ho = A , ,hl = - A , ( 2 + ~ ) , h = A l ( f 4 + g ~ ~ ) , h ,
16
(3-36)
25
+-B.'
16
(
h, = -B,'
64
35
h8 = Al
(%;
),
128
(3-37)
(3-38)
h,, = -A,(:B;),
hk =O* k 2 10.
(3-39)
where.
(3-40)
A numericd cornparison made for a circular arc airfoil (with 2 and 9 teras) with a
camber ratio
E = 0.0 1
and
E = 0.02
(at the angles of incidence a = 2" and a = 5" )
indicated that the present solution was in a very good agreement with the exact conformai
transformationsolution, as shown in Figures 3.5% 3.56.
---
.
.
-Conformai
transformation
i
-e-
Present method with 2 terms
+Present method with 9 tems
-.
--
Fig 3.5a Cornparison between the present solution and the exact solution for a circular arc
airfoil at a = 5" and E = 0.02.
Fig 3.5b Cornparison between the present solution and the exact solution for a circular
arc airfoil at a = 2" and E = 0.0 1.
3.3 Flow solutions for flexiblemembrane airfoils
When the airfoil is a flexible, impenrious, and nonstretchable membrane (or a twodimensionai sail, as shown in Figure 3.6), its shape is unknown but it has to satis@ the
normal equilibrium equation between the tension and the local pressure diierence across
the membrane:
The latter approximation for the radius of curvature R is for srnaIl slopes. The tension T.
per unit span, can be considered constant since the skin biction is zero in theory and very
srnail in practice.
Fig 3.6 Geometry of a flexiile membrane airfoil.
inmducing the nondimensiod coefiient AC, = ?A p l (p~U' ) and the coefficient
Cr = ZT I ( ~ U ' C )where
,
c is the chord length, equation (3-41) becomes.
AC,
=-cc,h9(x),
(3-42)
where the nondimensionai pressure ciifference is AC, = 424/ U .
The aerodynarnic boundary condition on ttte airfoil is expressed in terms of the normal
perturbation velocity component v in the form,
v = Y* (x) s [-ac ~ ' ( x ) ] u ,
(3-43)
Nielsen [13,22] treated this problem by using Glauert's [6,13,14] approach based on a
modiied Fourier expansion for the circulation y = 2U AC, in the form,
where the constants A,, and il, are related to the cosine Fowier expansion of the airfoii
slope, h'(x) ,in terms of the variable 0 = cos-' (2xi c - 1) in the form.
On this basis, equation (3-42)was rewritten as,
which had to be used to determine the coefficients A, and A, that define the membrane
shape. However, at the Ieading edge. 0 = x , equation (3-46) leads to A, = 0. which
corresponds to the ideal incidence for which the Kutta condition is also satisfied at the
leading edge. At any other incidence, the constant do should not be zero. and then
equation (3-46)is not satisfied at the leading edge. Nielsen [1332]avoided this particular
difficulty by expanding each term of equation (3-46), including 1 and cos 0 . as sine
Fourier series, and equating the coefficients of sinke. The present method does not
contain the above ambiguity. By differentiating equation (3-43), the membrane
equilibrium equation (3-42) may be recast oniy in terms of u and v as.
4 u(x)
cc, u
h"(x)=v; (x)/u = ---.
For any shape of the camberline, equation (3-20) shows that u contains the factor
J(I-X)IX = ,,/W.
which satides the Kutta condition at
X-"'
= f singuiarity
i
at the leading edge.
X = 1 and has the
Hence, according to equation (3-47),
hW(x)and v; ( x ) both have to contain the same singularity factor
J(1as u .
The foiiowing expansion for v, (x) satisfïes d of these requirements:
where, to satise the Kutta condition at the h.ailing edge (X = 1) in equation (3-47),
The lead temi of equation (3-48) contains [(1/2) - 2 X ] aah although not essential. leads
to a more compact expression
for v, (x). The antisyrnmetrical chordwise velocity
component u on the flexible membrane corresponding to the above expansion of v, (x)
is obtained fiom equation (3-20) in the fom,
(3-5 1)
where gk-, is defined by equation (3-33).
The camber shape of the flexible membrane can be obtained by integrating equation (349), in the form.
where.
There are (n+ 2) coefficients dehing the membrane camber defined by the above
equations, a and bk (k = 0.1 J..., n), which have to be deterrnined using the condition (350) and the membrane equilibrium equation (3-47). At the leading edge (X =O),
equation (337), which involves u and v; ,reduces to.
Other (n -1) equations can be obtained satisfjing equation (3-47) at other convenient
locations X, (i = 1,2,..........n - 1) dong the chord,
The final equation is obtained fiom (3-52) by requiring that the chord lies dong the x-
mis, Le., h(1) = h(0),
ELiminating the coefficient a . using equation (3-56). the system of equations reduces to
the form,
This can be solved for the coefficients (b, lu), which are independent of a. provided
that the matrix [D,] is not singular, as, for exampie, when C, = 1.727. which
corresponds to the case of a flexible membrane at an ideal incidence, a = 0 .
At this ideal incidence. a = O and, hence, the pressure distribution. as well as the
membrane curvature, do not contain the leading edge singularity x'"' .
The excess Length, defined as E = Il c - 1,is given by the relation,
and tiie lift coefficient of the flexible membrane airfoil is expressed, using equation
A numerical comparison (Table 3.1) of the resuits obtained for the overall parameters
al C,,a 1JÉ,and Cr between
the present method and Nielsen [l3,Z] and Thwaites
[13,27] methods, indicated that the results are generally in fair agreement, although there
are detailed ciifferences. The present method is in good agreement for Cr> 8 with
Thwaites method. The comparison with Nielsen's method shows good agreement for
C, < 8 .The resuits are presented in Table 3-1.
Table 3.1 Comparison between the present soiution for flexible-membrane airfoils and
the previous results of Nielsen and Thwaites.
-T
n
Present rnethod
Nielsen
Thwaites
Present method
Nielsen
Thwaites
Present method
Nielsen
ïhwaites
Present method
Nielsen
Thwaites
15
Present method
Nieisen
Thwaites
1 O0
Present method
Nielsen
Thwaites
400
Present method
Nielsen
Thwaites
a/&
a / C,
0.322
28.250
3.4 Flow solutions for jet-flapped airfoils
Consider a thin cambered airfoil of chord length c, providesi with a thin jet flap inched
at an angle P ,with respect to the chord, at the trailing edge Figure 3.7.
Fig 3.7 Geometry of a jet-fl apped airfoil.
The jet is assurned to be a thin sheet and have a small dope, and to have a constant
momentum J per unit length of slot. The sheet curvature II R 2 r W ( x,) where y, = e (.Y)
is the jet-flap shape, is related to the pressure difference across the jet sheet by the
momentum equation normal to the jet:
A ~ J =I R =J r R ( x ) ,
(3-60)
Using the diimensionless jet mornentum coefficient C, = 2 J / (pu'c), this equation can
also be expressed in terms of u and v as,
(41c J ) u ( x )= cvP(x).
(3-61)
The aerodynamic boundary condition (343) has to be satisfied on the airfoil. where h (x)
is specified, and furthemore, the same condition has to be satisfied on the jet sheet itseif:
.
v = v, ( x ) = [-a + e r ( x ) ] u
where the jet flap shape, y, = e(x), has to be detemined.
(342)
At the trailing edge (x = c), the jet slope with respect to the chord is specified as
B,
which le& to the additional condition,
er(c)= -tan9 = -p ,
(3-63)
v(c)=-~sin(ct+p)/cosp = - ( ~ + P ) u .
(3-64)
or
Due to the pressure difference across it, the jet sheet is deflected upwards and.
eventuaiiy, at large distance I behind the airfoil, becomes flat and parallel to the free
ream direction. This cm be expressed by,
er(f)= en([)= 0 ,
(3-65)
or
v(l)= vr(l)= 0'
[also u(1) = O].
(3-66)
where 1 tends. theoreticaily, to infinity. However, the jet curvature ptactically tends to
zero at a finite distance (1 - c), measured from the trailing edge. which depends on the jet
coefficient CJ ,the jet angle B ,and the incidence a.and, in general, has the sarne order
of magnitude as the chord length c .
In this situation, the jet-flapped airfoil can be considered as a fictitious rigid airfoil of an
overdl chord length I ,with the Kutta condition u(1) = O satisfied at x = 1.
Considering the nondimensional coordinates X = x / c and L = l l c , the following
expansion of the normal-toîhord velocity component on the jet sheet vJ
the preceding considerations
and, by inteption,
where L = l l c , a n d
(x), satisfies
The last tenu of equation (3-67) accouuts for traiiing-edge singularity appearing in u
when the jet is not tangent to the camberline, Le., when h'(1)
The boundary condition regardhg the jet dope
P
#
tan B = -Q .
at the M i n g edge c m be expressed
in the fom,
At the other end of the jet sheet (X = L), the boundary conditions can be expressed as.
However, the tems (3-71) & (3-72) may be omitted in order to avoid having to mmy
terms in expansion (3-68) since the jet dope at X = L is at least in order of magnitude
smaller than the slope at the üailing edge
(since L is chosen to be larger than 2).
The velocity v, (s)should correspond to two different variations of the camberline dope
on the d o i l itself. v, (s),and on the jet sheet, vJ (s).i.e.,
where S represents the new nondimensiond coordinates with respect to
C(S
= s / c) .For
convenience, a potynorniai expansion is considered on the jet sheet,
The chordwise perturbation veIocity u on the airfoil and the jet is obtained as,
where G (L, 1. X) has the expression (3-1 5).
The expressions of I, and J , are defined by,
-
The constant a is given as,
.
The (n - 1) unknown coefficients e, (k = 21,.......n ) can be determined by satisfying
equation (3-6 1 ) at (n - 1) convenient locations X I , situated between X = 1 and X = L
The Iifi coefficient is obtained by integrating the pressure ciifference, in the tom,
CL= 4a L cos-' E - 4 [
hk[21k+,cos-' i
F + ~k +dl = i
/=O
I,
)+
.
Chapter 4
Method of velocity singularities for unsteady flows
4.1 Method of veiocity singularities for unsteady flows past airfoils
Consider the general flat plate airfoil given in Figure 4.1,
Fig 4. I Flat plate airfoil with a boundary condition e (x,t ) .
The vertical displacement of the airfoi1 d a c e is defined by.
y = e(x.t),
(4-1
where eCr.t ) is given by
e(x,r)= i ( ~ ) e ' .~ '
The equation of the body surfacef (x,
(4-2)
is hence
f ( ~ ~ ~ * -t e) ( =x . r ) = ~ ,
(4-3)
The boimdary condition on the surface of the aidoil is defined by
where u and v are the perturbation veImities in the x and y directions, respectively.
By performing the denvatives and inserting them into the boundary equation, one obtains
the perturbation velocity v in terms of e(x,t),
The boundary condition on the airfoil cm be expressed as
v =v(x),
where.
The velocity
v ( x ) can be also expressed by
v ( x ) = P (x)el*'.
where,
[n the srnall perturbation assumption-
tl
II,
is smaller than the unity and hence c m be
negIected in the above equations. The general fonn of the velocity boundary condition is
given by
where the coefficients b, can be determined for each specific case from the boundary
conditions on the oscillating airfoil. The compIex perturbation veIocity is given by
w (z)=u(x,y)-
jv(x,y),
(4- I2)
and cm be expressed in the form.
w (z)= w (z)eiW.
(4-13)
w(z)= Û ( X , ~ ) - j v ( ~ , ~ ) ,
(4- 14)
where,
Note that,
W-15)
j = C .
The boundary conditions on the osciIIating airfoil can now be expressed in the compIex
form, (whm M G ,[ ] represents the imaginary part of the comsponduig expression).
MG,
k(r)i., = Y(x) .
for O < x c c
(416'
To derive the boundary conditions on the wake of the airfoil, consider a flat plate airfoil
.
-
with a chord c placed on the x axis as shown in Figure 4.2,
Fig 4.2 Geornetry of a flat plate airfoil indicating the fiee vortex distribution.
The circulation ~ ( c , t )around the airfoi1 is expressed as a function of the reduced
circulation f (c),
r(~,t)=i(c)e?
(4- 17)
where o is the frequency and r is ihe time.
dldt
At any Uzstant of time t a fiee vortex -dt
is shedding at ihe trailing edge of the profiIe
as shown in Figure 4.3.
A
Fig 4.3 Free vortices shedhg at the trailing edge of the airfoil.
The shedding (trailmg edge) fiee vortex is moving downstream with the uniform stream
velocity U, . This means that during the tirne interval dtT the shedding free vortex will
move dong the distance d x, given by,
Hence, the corresponding distribution of the intensity of the shedding free vortices, in the
proximity of the trailing edge is given by,
The intensity of the fiee vortex dimibution y,(c,r) at the trailing edge of the airfoil is
expressed as,
where U-is the fke strearn velocity.
The intensity of the ûee vortex distribution y,(c.t) can be expressed in terms of the
,
reduced intensity of the fiee vortex j (c). as,
v,(c,t)=j,(c)elm'.
(4-2 1 )
m e r taking the derivative of equation (4-1 7), the intensity of the free vortex distribution
at the trailing edge of the airfoi1 is given by,
The reduced intensity of free vortex disaibution is given by,
The intensity of fke vortex distriiution, at a distance o behind the trailing edge of the
airfoil, is calculated considering the fact that a fiee vortex at a distance behind the traiiing
edge moves with the fiee Stream veIocity U,and was initiated at a previous tirne interval
At
(time Iag) fiom the trailing edge of the airfoiI.
The time lag is expressed as,
The intensity of the fke vortex disûibution at a distance a behind the trailing edge of the
airfoi1 y ,
(qt)
is denved as,
The reduced frequency h , is expressed as,
A=-. O C
u m
The reduced free vortex distribution at o is given by,
The reduced k e vortex distribution at G. is related to the reduced velocity Û(a)by the
relation,
Y,(O)= 28(~).
(4-2 8)
and hence,
1
1A
Ù(cr)= - f , (cr) = -i--î(c)e
2
2c
-,.(;-,]
The boundary condition on the wake of the aufoil is hence,
REAL, [w(r)l,. = CG).
for x > c
where, REAL, [ ] represents the ieai part of the correspondhg expression.
4.2 Prototype problem
Consider a flat plate airfoi1 with the jumps of velocities due to the bound vortices on the
airfoil are represented by a constant bo and a variable
6Y. The jump of velocity due to
the fiee vortices behind the trailing edge of the airfoil is represented as &as
shown in
Fig 4.4 Representation of the velocity jumps in the boundary conditions of the prototype
problem of an oscilIating airfoil.
Complex singularity functions
Special singuiarities are used to detennine the complex perturbation veiocity
W(Z)
(rather than the cornplex potential) in the airfoi1 plane (see Figure 4.4). The
velocity singularities are expressed by.
At the leading edge. r = O
At the velocity jump on the airfoiI, s 5 r Ic
At the velocityjump due to the k e vortices outside the airfoil, c < z S cr
The generd behavior and the derivations of the above special complex singdarities are
given in Appendix C.
The elementaiy complex velocity &*(.z)
is eqressed in t e m of the velocity jurnps,
due to the bound and fiee voaices, and singuiarities related to the complex variable
z = x + j .The expression of ihe cornplex velocity
The velocity
(t)is given by
c(a) is given by
The velocity jump SÛ due to an eiementary k e vortex behind the trailing edge is
defined by
where
(0) is defined by equation (4-29).
From equations (4-34) and (4-35),
Taking the effects of al1 the fke vortices fiom the trailing edge c
complex velocity &(z)
ir exp~ssedby,
to
infinity a. the
By taking into account equation (4-30, equation (4-37) can be recast as,
where,
-
0,
F(z)= ~ i m (j e
6
,
-1
I
-('Y-c)
cos-' $""dc
o(o- L)
1.
The complex fùnction F (2) is denved in Appendix A and is given by,
To detemine the constant &. we note that at z +m. the perturbation velocity vanishes.
&(a)= O .
The complex velocity at infinity is given by,
where.
The real and imaginary parts of the cornptex velocity at inflliity are equal to zero.
14-42)
Imposing the condition,
R E A L ~ [ ~ ~ (O.~ ) J =
one obtains for the indeterminate value of the constant E, ,
The above conclusion can also be obtained fiom the Riemann-Lebesque lemma on
Fourier Integrais 15,331, when the theory of distribution is used (seaiso Fung [SI).
Hence,
Imposing the condition.
one obtains,
By inserthg M into &(r) and noting the fact that E. = O. the equation (4-38) of the
complex velocity becomes.
Determination of the constant &(c)
The reduced circulation around the airfoü
k(c)for the prototype problem, is determhed
by imposing the fact that the reduced bound vortex distribution on the airfoii is related to
the complex veiocity by the folIowing expression,
f,(x)=
ZREAL, b ( z ) L x = 2Ü(x),
(4-52)
The reduced circulation at any position x on the airfoil f(x) is related to the reduced
bound vortex distribution by,
The expression of
6i(x) is given by,
To determine 6i(c), the condition of x = c is introduced in equation (4-54).
The derivation of the various integrais in equation (4-55) is given in Appendk A. and the
final formulas of the integrais are obtained as,
where Hi2)(Al 2) and Hi'' ( ~ 1 2 are
) the Hankei's Uitegrals of second khd of orders of
one and zero [7,20,29], respectively, and
introducing the above integrais into equation (4-59, one obtains,
where,
The ~ d u c e dpressure coefficient 6ep(+)
for the prototype oscillating airfoi1 probiem is
expressed in ternis of the reduced velocity potentiai
s C, (x) -- -+i~smb,vi+w],
u, u n
&
7
The equation of the reduced pressure coefficient se, (x) is then expmsed as.
where  is the reduced fÎequeucy,
The expression of the reduced velocity on the airfoil 6 û is given by,
a û = REAL 1~ (i)l*x
,
where the r e m REAL ,O represents the red part of the cornplex fiincrion.
By t a h g the reai part of the complex veloeity 6~ ( 2 ) the
. expression of Sû is given by,
M e r inserting the expressions of 66 and
di(*) into the expression of
the reduced
pressure coefficient 66,(x), one obtaios,
3
*
t]
--si.p(x)=(-)[-ao
2
C
--wcos-l
K
~(.-.t
Details of the caiculations are shown in Appendix D.
The above equation can be represented d e r mathematical arrangements ast
where ~ ( h / 2 is) Theodorsen's function [1526]. The d u e s of the real and imaginary
parts of the Theodorsen h c t i o n are given in Appendix B.
Determination of b t L
The reduced lift coefficient 6 t L for the protome problern is related to the reduced
pressure dBerence coefficient 6[
~ t(x) ,] by the expression,
- =-la[
1'
sc,
AC,(X) ] a = - - 7 JsC,(x)m.
Co
0
The relation between the reduced pressure coefficient 6kp(x) and the reduced pressure
difference coefficient 6[
cf[
AC, (x) ] is expressed as,
AC,(X) ]= -2 sf ,(.t),
(4-69)
M e r evaluating the related integrais (see Appendix A), the expression of the reduced lifi
coefficient is given by,
Determination of se,,,
The reduced pitching moment coefficient
&Cmfor the prototype pmblem is defined by,
The detailed integral derivations related to bCm are given in Appendix A. and the
expression of the reduced pitching moment coefficient 6
where
c (h/ 2 ) is Theodoaen's [I 5 J q function.
Cmis obtained as.
4.3 The complete oscillating aidoil probiem
Let us consider the boundary conditions on the oscillating airfoil in the general fonn.
The jump of velocity 6Y on the airfoi1cm be expressed as,
and hence,
where the coefficients 6, are deterrnined for each specific case fiom the boundary
conditions on the oscillating airfoi1 as it is shown for example in section 4.5.2. The
complete motion is obtained by inserting the eiementary velocity jurnp defined by (4-74)
into the equations of the prototype problem and then integrating dong the chord of the
airfoil, as shown in the following:
where,
The expression of the reduced pressure coefficient is given by,
The reduced iift coefficient is expressed as,
The reduced pitching moment coefficient is d e h e d by,
4.3.1 The reduced circulation around the airfoil
The expression of the total reduced circulation mund the airfoil
f' (c) is given by.
The detailed derivations of the integrais are given in Appendix A.
The resdtam expression of the totai reduced circulation ?(c) is given by,
where,
2k-1
fk =c It-l
2k
K
3 7
where, Io = R II= C-2 I2 = gRC*.
The detailed derivation of the recurrence formula of 1, is given in Appendix A.
4.3.2 The reduced pressure coefficient
The expression of the total miuced pressure coefficient k , ( x ) is determined by
introduchg the expression of the velocity jimip 6Y hto the expression of 6C, (x). The
integral form of the equation is given as,
The detaiIed derivations of the integnls related to equation (4-84) are given in Appendix
A. The resuitant equation of the reduced pressure eoeficient
ep(x) is expressed as
Note that the reduced pressure ciifference coefficient hep(x), is given by the relation.
h e p ( x ) = - Cp(x),
The pressure difference coeEcient for the airfoil is expressed by,
AC, (x, r ) = ~e
[aCp(.r)e1&1 .
(4-86)
4.3.3 The reduced lift and pitching moment coefficients
By inserthg the expression of the velocity jump S f into the reduced lift coefficient
Se,, equation
(4-70), and by taking the integrals amund the airfoil, the general
is given by,
expression for the total reduced lift coefficient IL
The detailed derivations of the integrais within equation (4-88), are given in Appendiu A.
The resuitant expression of the total reduced lift coefficient CL is given by.
where c(U2)is Theodonen's fùnction , A is the reduced Frequency and 1, is a
recurrence expression given by equation (4-83) rewritten beIow as,
2k-1
~t
3
c 1,-, where, Io = x Il = c2k
2 I2 =
lk =-
7
Tc-
The total reduced pitching moment coefficient
kmis derived in the same way and is
given by, (see Appendix A for detailed integrai calculations),
" k
+ zk-I- bk+I
k
(4-90)
2i
2 k + 1 1 (2k+1)(2k+3)
I* [ - : ( l - T C ( A i 2 ) ) + 12
[ ~ +S~A ) -+k + 2 12 ( k + 2 ) ( k + 3 )
4.4 Method of velocity singuiarities for unsteady flows past airfoils with
oscillating ailerons
The boundary condition on the a i ~ o iw
lith an oscillating aileron is given by the veiocity
Y ( x ) in the form
The boundary condition on the airfoi1 at the position of the aileron (x = s, ) is given in
ternis of the velocity jump
Y (s, ) that is given by,
where the coefficients p, are determineci fiom each specific case fiom the boundary
conditions on the airfoil with an oscilIating aileron.
4.4.1 Prototype problem for the case of an oscillating aileron
Consider the unsteady flow past an airfoil with an oscillating aileron, as shown in Figure
4.5.
Fig 4.5 Geometry of an oscillating aileron.
The various velocity jurnps on the airfoil are represented in Figure 4.6,
Fig 4.6 Velocity jumps representations in the boundary conditions on an airfoil with an
osciiiating aileron.
Complex singuiarity functions
Special singuiarities are used to determine the complex perturbation velocity
w ( z ) (rather than the complex potential) in the airfoil and aileron plane(see Figure 4.6).
The velocity singularities are expressed by,
At the leading edge, z = O
At the velocity jump on the airfoil, s Iz 5 c
NI(z.s)= cosh-'
id;";,
At the velocity jump due to the fke vortices outside the aidoil, c < z 2 o
The elernentary cornpiex velocity &e(z) is expressed in ternis of the velocity jumps.
due to the bound and free vortices, and singularities related to the complex variable
r = x + iy .The expression of the complex v e l o c i ~d@< (i)is given by
The velocity
O(=)
is given by,
ï h e velocity jump 6Lj due to the fiee vortices behind the traiiing edge is expressed by
(see dso the anaiysis in section 4.1),
The velocity jump dI for the sinoil with an osciiiating aileron is given by
From equation (4-98),
Taking the effects of al1 the Cree vortices fiom the trailing edge c to infinity a, the
complex velocity &(z)
is expressed by,
(4- 102)
By taking into account equation (4-10 1). equation (4-1 02) can be recast as.
where.
The compIex function F (2) is derived in Appendii A and is given by,
To determine of the constant $4, we note that at ,- + oo the perturbation velocity
vanishes.
The cornplex velocity at infinity is given by,
where as shown in equation (447),
The reai and imaginary parts of the cornpiex velocity. at inhnjty, are equai to zero.
From a simila trdysis to that shown in section 4.2, one obtains,
By Uiserting d4 into 6W ( z ) ,the equation of the complex velocity becomes,
Detemination of &(c) for the prototype problem of the airfoü with an oscülriting
aileron
The cuculation for the prototype aileron problem around the airfoi1 is given by,
By taking the reai part of the complex velocity and performing the integration around the
airfoil, one obtains,
(4-1 13)
The derivations of the above intepis are given in Appendix A.
The resdtant equation of the reduced circuiation for the airfoil with an oscillating aileron
is given by,
where.
where H/?)
(hl 2) and
'AH
( ~ 1 2 are
) the Hankel's integrals of second kind of orden of
one and zero [7,20,29], respectively.
Determinition of 6 t p
(x)
The reduced pressure coefficient 6 t p ( x )for the prototype aileron pmblem is expressed
in terms of the reduced velocity potential S6 as,
where,
1
+]a;
d - j 2 s û < t r =2 - 6 i ( ~ ) .
O
(4-1 17)
2 0
n i e equation of the reduced pressure coefficient 6Cp(x) is then expressed as.
where h is the reduced Frequency,
The expression of the reduced velocity on the airfoil Sû is given by.
s ù = REAL [6W( r ) L x .
(4-I 19)
By taking the real part of the cornplex velocity 6 ~ ( 1 )the
. expression of 6 û is given by.
The expression of the partiai reduced circdation at any position x is given by,
The above integrals are evaiuated in Appendix A.
By inserting the above equations into the expression of the reduced pressure coefficient,
one obtains,
+
(- -),/=(I
1
c-x
C
X
(-A.
1
E]
-AP(~,
?r
)[,/(.-.T)FI
-
Detaiied caiculations are s h o w in Appendix D.
The reduced lifi coeficient 6CL For the prototype aileron problem is related to the
1
reduced pressure difference coefficient 6[ AC, ( x ) by the expression.
After evduating the related integrals (see Appendk A). the expression of the reduced lifi
coefficient is given by,
The reduced pitching moment coefficient 6 t mfor the prototype aileron problem is
evaluated by
The detailed integrai derivations dateci to d e r nare given in Appendix A. the expression
of the reduced pitching moment coefficient S Cmis given by.
where
c (A / 2) is Theodorsen's [I 5,261 function.
4.4.2 The complete problem for the airfoil with an oscillating aileron
In the equations derived in the previous section, the velocity jurnp
Fom equations (4-92) and (4-100) in the form,
for x e@,s,)
foi X E(s,,C) .
The reduced circulation is given in the gened fom as.
6Y can be obtained
By using the above expression and performing the integration fiom x = s, to the trailing
edge of the airfoil x = c ,the reduced circuiation can be expressed as,
Details of the integrais calculations are shown in Appendix A.
M e r performing the integration in equation (4-139), the expression of the reduced
circulation for the aidoil with an osciiiating aileron f (c) is given by,
where the recurrence formula 1, for the airfoil with an oscillating aileron is given by.
where,
The reduced pressure roefiident tp(x) for the a s o i l with an aiieron
The generai expression of the reduced pressure coefficient for the airfoil with an aiieron
is obtained by substituthg the expression of the vetocity jump
6Y
into the equation
obtained for the reduced pressure coefficient of the prototype aileron problem,6Cp(x).
The expression is given in general fom as,
By integrating fiom
x = s, to the trailing edge of the airfoil x = c one obtains the
reduced pressure coefficient 6,(x) in the general integral fom,
K
El-
The resuitant expression of the redud pressure coefficient e,,(x) for the airfoü with an
osciliating aileron is given by,
--C,(x)=
2
-
-
- --v(s,)ms-l
(if)[
g r )-
where,
C-X
o xs 1
p
c
+~ ( x ) .
The recurrence expression I, is given by equations (4-13 1) and (4-132),
where,
The values of the coefficients(Bk, k = 0 -P n)can be detemined fiom the specific
boundary conditions on the airfoil with an oscillating aileroa.
The reduced lift coefficient
eLfor the airfoü with an d e r o i
The generd expression for the reduced lift coefficient
equation of' the velocity jump
CLis obtained by inserting the
6Y into the expression obtained for 66, and perforrning
the integration from the aileron position x = s, to the traihg edge of the airfoilx = c .
The expression is given by.
The rendtant expression of the reduced lift coefficient
EL for the airfoil with an
oscillating aileron is given in an integral form by,
Detailed cdculations of the integrah are given in Appendix A.
The expression of the reduced lifi coefficient
is then given in the form
e, for the airfbil with an oscillating aiieron
The same analyticai steps w d to obtain the reduced lift coefficient
CL for the airfoi1
with an oscillating aileron are repeated for the reduced pitching moment coefficient
em.
The generai expression is defined by,
(4-149)
The resultant expression of the reduced pitctiing moment coefficient for the airfoil with
an aileron is expressed in an integrai form as
-
[-
V(S,)COS
C
t.1
iSk-l
cos-'
id.]{^(^)^^
24 c
i L4C ( A l 2 )
'1
(4- 150)
Detailed caiculations of the integds are given in Appendix A.
The expression of the reduced pitching moment coefficient for the airfoi1 with an
oscillating aileron is then expressed as,
The recurrence expression I , is given by equations (4-13 1) and (4-133),
where,
The values of the coeficients(~ for k >O) cm be determined fiom the specinc
boundary conditions on the airfoil with an oscillating aileron.
4.5 Unsteady flow solutions for airfoils executing oscülatory translations
and rotations
4.5.1 Boundary conditions on the oscillating airfoi1 and aileron
Consider an airfoil executing vertical oscilIatory translation h(f) and oscillatory p i t c b g
rotation 0 ( t ) ,while the aileron executes oscillatory rotational motion ~ ( t )as, sbown in
Figure 4.7.
Fig 4 7 Geornetry of an airfoi1 and aileron under unsteady oscillations.
The verticai oscüiatory translation h (t), the pitching osciliatory rotation 0 (r) and the
oscillatory rotation of the aileron P (r), are denoted by (as shom in Chapter 2),
h(f)= Le'",
(4- 154)
e(t)= 6 e l m ,
(4-1 55)
p(r)=Be4",
(4-156)
The correspondhg boundary conditions are,
durn
ê+im ( i - ê x ) ) ~
-urnê + i m ( ~ - ê x ) - i m j ( x - s , ) - ~ ))el"
~
for X E ( 0 , ~ ~ )
for x E (s,,c)
The vertical disturbance velocity Y ( q t ) cm be expressed as.
.
v(.r,r) = I (x)edu
where
?( x ) is the reduced vertical disturbance velocity given by,
- ~ j + i w( h - é x ) )
for x E (O, si )
(4- 159)
- ~ ~ ê + i c( io- ê x ) - i m é ~ - s , ) - ~ ~ f i ) for I E ( S ~ , C )
4.5.2 Unsteady flow solution past an airfoil in oscillatory translation
Consider an airfoil executing vertical oscillatory translation h ( t ) , as shown in Figure 4.8.
Fig 4.8 Geometry of an airfoil executhg vertical oscillatory translation h (t).
The vertical oscillatory cransIation h(t) is expressed as,
h ( t ) = ~ e J m,'
(4- 160)
where o and t are the kquency and time, tespectively.
The boundary condition on the osciilating airfoil is given fiom equation (4-1 59) by,
P(x)=ioh,
(4-1 6 1)
From the method of velocity singuIarity for unsteady flows, the polynomial
representation of the vertical vdocity on the airfoil is given by,
By comparing the two vertka1 velocities on the airfoil, one concludes the values of the
constants b,,
bo=iuh
and b,=O, f o r k z l .
(4- 163)
Reduced pressure difference coefficient for oscillatory translation
The expression of the reduced pressure coefficient C,(x) for an airfoil executing
oscillatory translational motion. &er substituting the values of the constants b, in the
general expression of the reduced pressure coefficient equation (4-85), is given by,
--u, C- p ( ~ ) = ( - : ]
3
where
,/a
-/y
[bo Io]+ 1 c - x
it
bo {(L-C(U2))I0- I ~ ) .
.
6, = i w h and Io = A .
The general expression of the reduced pressure coefficient
2,(x) for an airfoil executing
oscillatory translation afler arrangements, is given by.
The reduced pressure difference coefficient A ~ , ( x is
) given by,
The pressure diifference coefficient AC, (x) is expmsed as,
Redaced lift and pitching moment coefficients for oscillatory translation
The reduce lift coefficient is calculated in the same manner as the reduced pressure
coefficient. From equatioo (4-89)
CLis giveo by,
The expression of the Iifi L for an airfoi1 executing oscillatory translation c m be
calculated by,
where p is the density. The general expression of the lift is given by,
The reduced pitching moment coefficient
km
for an allfoil executing oscillatory
translation is given fiom equation (4-9 1) as,
The pitching moment coefficient C m is expressed as.
The pitching moment around the teading edge of the airfoil is expressed as,
where the reduced ûequency is given by,
4.53 Unsteady flow solution past an airfoil in oscillatory rotation
Consider a thin airfoi1 executing oscillatory rotation 8 (r), as shown in Figure 4.9.
Fig 4.9 Geometry of an airfoil executing oscillatory rotation 0 (t ) .
The oscillatory rotation 0 (r), is expressed as,
e(t)=ééW
.
(4- 178)
The boundary condition on the oscillating airfoil in pure rotation 0(r) is expressed (fiorn
equation (4- 159)) as,
Y (x)= -u,é-iméx.
(4- 179)
By cornparing the terms in the polynomial representation of the vertical velocity
@en by,
one obtains,
b,=-U,8
-
, b,=-id
and b,=O f o r k l î .
v (x)
Reduced pressure coefficient for oscillatory rotation
By inserting the constants (obtained from the boundary condition on the airfoii) into the
general expression of the reduced pressure coefficient for the airfoil, equation (4-85), one
obtains the reduced pressure coefficient e,(x) for an airfoil executing osciilatory
rotation,
(4- 182)
where the pressure coefficient C, (XJ) is given by,
.
C, (x, r ) = kP(x)eJW'
The pressure difference coefficient AC, (XJ) =
(4- 183)
~ eexp(ia>
, r ) is given by,
AC,(X,~)=-2~,(x,t).
(4- 184)
~e~(x) = -2 êp(x) .
(4- 185)
Reduced lift and pitching moment coefficients for oscillatory rotation
By inserting the values of the constants b, into the general equations of the reduced lift
and moment coefficients. equations (489) and (4-9 1) respectively, one obtains,
The generai expressions of the lifl L and pitching moment M around the leading edge of
the airfoil are given, respectively, as,
4.5.4
Unsteady flow solution past an airfoil executing flexural
oscillations
Consider an airfoi1 executing f l e d oscilIation of a parabolic type, as shown in Figure
4.10.
-5
Fig 4.10 Geometry of flexural oscillations on an airfoil.
The parabolic flexural oscillation y(.r,t) is assumed in the fom,
where
E,
is a constant.
From equations (4-1) and (4-2) one obtains,
By substituting in equation (4-IO), the equation of the reduced vertical velocity Y(x) for
an airfoi1 under flexural osciIIations is given by,
By cornparhg the terms in the polynomid representation of the vertical velocity on the
airfoil, one obtains,
bo=O , b , = Z U , ~ ~ l c, b t = i o ~ o l cand b , = O f o r k L 3 .
(4-193)
Reduced pressure coefficient for paraboüc flexnral oscüIations
By substituthg the above constants into the generaI expression of the reduced pressure
coefficient (equation (4-85)), one obtains the expression of the reduced pressure
coefficient C,(x) for an aidoil executing parabolic Bexurai oscilIations,
where the pressure coefficient c,(x,~) is given by,
C, (.Y, t ) = èp(x)éa'
.
The pressure difference coefficient AC, (x,r) is given by,
AC,(X.I)= - 2 ~ , ( . r . r ) ,
Reduced lift and pitcbing moment coefficients for parabolic fiexural oscillations
By inserting the vaiues of the coefficients b, into the general equations of the reduced 1 3
and moment coefficients, equations (4-89) and (4-91) respectively, one obtains afier
mathematicai simplifications,
where C (A 12 ) is Theodorsen's hction and h. is the reduced frequency.
4.6 Uasteady flow past an airfoii with an cwcülating aileron
Consider an airfoi1 having an aileron that is executing oscillatory rotation P (r), as shown
in Figure 4.1 1.
W
C
-
Fig 4.1 1 Geometry of an airfoi1 and deron executing oscillatory rotation P ($).
The aileron oscillatory rotation B ( t ) is given by,
The boundary condition is representeà in terms of the reduced vertical velocity 3(x).
which in the case of aiLeron oscillations is given by,
The bomidary condition on the airfoil with an osciIIating aileron is given by,
P
for x E (o,s,)
From the above conditions one obtains the vaiues of the boundary constants,
/ 3 , = - U , ~ + i a i ~ s , , fl,=-irub and
In this case, the value of C' (si) is,
B,=O f o r k > 2 -
(4-202)
The resultant expression of the pressure ciifference coefficient for an airfoil with an
aiieron executing pitching osciiiations is given by,
2
-
2 c-x
--vt&t.d--/-c,
7r
K
where.
<,(x) = ( x - S , ) H (x).
where,
H (x)=
X
The resultant expression of the lift coefficient for an airfoi1 with an aileron executùig
pitcbg oscillations is given by,
The resultant expression of the pitching moment coefficient for an airfoil with an aileron
executing pitching oscillations is given by,
where,
4.7 Unsteady flow past an airfoil with an aileron executing flexural
oscillations
Consider an airfoil having an aileron under parabolic flexurai osciIlations y (x,t). The
parabolic flexural oscilIation y (x,r) on the aileron is assumed in the form,
where
K~
is a constant.
From equation (4-2), one obtains,
From the above anaiysis and h m equation (4-IO), the equation of the reduced vertical
velocity
(x) for an airfoil with an aileron executing t l e d osciIIations is given by.
where,
(0
for x E (O,$,)
By comparing the ternis in the polynornial representation of the boundary conditions on
the airfoil, one obtains,
and
Bk= O
for k 2 3 .
In this case, the value of f (s,) is,
i(s,) = o.
The resuitant expression of the pressure ciifference coefficient for an airfoil with an
aileron executing parabolic flexurai oscillations is given by,
w here,
where.
The resuitant expression of the lif&coefficient for an airfoil with an aileron executing
parabolic f l e d oscillations is given by,
where,
The resuitant expression of the pitching moment coeficient for an airfoil with an deron
executing parabolic flemrai oscilIations is given by,
where,
Y,
=z,,
where,
Il = ,/(.-.,b,+c
cos'
g,
3
Il = - J - ( + - c - I , , 4,
Chapter 5
Results and discussion
The present rnethod has been validated by comparison with the results obtained by
ïheodorsen [15,26] and Postel and Leppert [24] for the case of an airfoi1 executing
oscillatory translation and rotation and for the case of an airfoil with an aiferon executing
oscillatory rotations.
M e r validation, the present method has been used to obtain solutions for airfoils
executing f l e d oscillations and for airfoils with ailerons executing f l e x d
osciliations.
No cornparisons were presented for the case of flexud oscillations since there are no
previous results known.
For the sake of cornparison, the following coefficients have been introduced:
which is the reduced oscillatory pressure difference coefncient, and
K, = REAL(K; elY ).
(5-3)
which are the reduced oscilIatory Iifi coefficient and the oscillatory Lift coefficient.
respectively, and
which are the reduced oscillatory pitching moment coefficient and the oscillatory
pitching moment coefficient,respectively.
5.1 Case of airfods executing oscillatory translations
The numericd results were obtained for various values of the reduced frequency R. ,such
as il = 0.48,0.60,0.68. The resdts are presented in Figure 5.1 and in Figures 5.2,5.3, for
the reai and imaginary parts of the reduced oscillatory pressure difference coefficient
(- cl(2b))AK,, where ho = h ,represents the complex amplitude of oscillations, and for
the reai and imaginary parts of the reduced oscillatory lift and pitching moment
coefficients, ( 2 c l ( d , , ) ) ~ I and ( 4 c l ( n f i , ) ) ~ brespectiveiy.
,
The corresponding typical
variations in t h e of the oscillatory lifk and pitching moment coefficients have been
calculated for several values of R ,such as il = 0.4!3,0.68, as shown in Figure 5.4.
The r e d t s of the real and imaginary parts of the pressure difference coefficient are in
very good agreement with the previous r e d t s obtained by Postel & Leppert [24]. The
results of the Iifi and pitching moment coefficients obtained by the present method are in
excellent agreement with Theodorsen's solution [1536].
/f
.-
o , A ,Postel
~ & Leppert
Fig 5.1 Real and Maginary parts of the pressure ciifference for an airfoil executing
oscillatory translation.
-Present sohîbn
Fig 5.2 Real and imaginary parts of the Lift coefficient for an airfoi1 executing oscillatory
a~lll~lution.
- Presents o W n
-Present soiution
-
O
0.2
0.4
Theodorsen's solution
-
0.6
0.8
-
1
1.2
Fig 5.3 Real and imaginary parts of the pitching moment coefficient for an airfoil
executing oscillatory transintion.
Fig 5.4 Typical variations in tirne of the lift and pitching moment coefficients for an
aidoil exemrnig osciZlatory translation.
5.2 Case of airfoils executing oscülatory rotations
The numerical results were obtained for various values of the reduced fiequency
Â
,such
as R = 0.48,0.60,0.68. The results are presented in Figure 5.5 and in Figures 5.6, 5.7 for
the real and imaginary parts of the reduced oscillatory pressure difierence coefficient
(116.
where 8, = 6 represents the complex amplitude of oscillations, and for the
real and imaginary parts of the reduced osdatory lift and pitching moment coefficients,
(- 4 /(RB, ) KI
and (- 8 /(do
))K:, respectively. The corresponding typical variations in
time of the oscillatory lift and pitching moment coefficients have been calcuiated for
various values of A ,such as A = 0.48.0.68, as shown in Figure 5.8.
The results of the real and imaginary parts of the pressure difference coefficient are in
very good agreement with the previous results obtained by Postel & Leppert [24]. The
results of the lifl and pitching moment coefficients obtained by the present method are in
excellent agreement with Theodorsen's soiution [15,26],
Fig 5.5 Real and imaginary parts of the pressure ciifference for an airfoil erenrring
osciIZatory rotation.
-Present solution
9\
O
0.2
-
0.4
0.6
0.8
1
Theodorsen's solution -
0
1.4
1.2
1.6
1.8
2
REDUCED FREQUENCY, A/2
-
O
0.3
0.6
0.9
1.2
1.5
1.8
o
-
ïheodorsen's solution
. .---
2.1
2.4
2.7
3
Fig 5.6 Real and imaginary parts of the lift coefncient for an airfoil executing oscillarory
rotation.
-
-Present solution
Theodorsen's solution -
o
O
0.5
1
1.5
2
2.5
3
3.5
4
REDUCED FREQUENCY, h/2
-Present sotution
O
Theodorsen's solution
--
-
Fig 5.7 Real and imaglliary parts of the pitching moment coefficient for an airfoi1
executing oscillatory rotation.
Fig 5.8 Typical variations in t h e of the Lift and pitcbing moment coefficients for an
airfoi1 executing oscillato~rotation.
5.3 Case of airfoils executing tiexural oscillations
The numerical r e d t s were obtained for various values of the reduced fiequencyR ,such
as 1 = O.48,0.60,0.68. The results are presented in Figure 5.9 and in Figures 5-10, 5.1 1
for the real and imaginary parts of the reduced oscillatory pressure difference coefficient
(-2146, )AlCi,
and for the reai and imaginary parts of the reduced oscillatory Mt and
pitching moment coefficients. (8 /(3mO))K; and (8 /(3m,, ))Ki , respectively. The
corresponding typical variations in time of the oscillatory Lifl and pitching moment
coefficients have been calculated for various values of R , such as
À
=0.48.0.68 as
shown in Figure 5-12. No cornparisons were presented for the case of flexurai oscillations
since there are no previous results known.
Fig 5.9 Real and imaginary parts of the pressure ciifference for an airfoi1 execuring
parabolic frexurut osciiZa~ons.
k-----2
3
4
-
+Present Method
5
-
6
EtEDUCED FREQUENCY, A
+Present method
4
6
8
REDUCED FREQUENCY, 1
Fig 5.10 Red and imaghary parts of the U
t coefficient for an airfoi1 execuringparabolic
fremal oscilIatiom.
--
+Present solution
-
1.5
--
2
4
6
REDUCED FREQUENCY, A
Fig 5.11 Real and imaginary parts of the pitching moment coefficient for an airfoi1
executingparabolicjlexural osciilahons.
8
6
4
A
s 2
x'
5
E
ri
0
" -2
P,
-4
-6
-8
Fig 5.12 TypicaI variations in time of the lift and pitching moment coefficients for an
airfoil executingparabolic frsnrral oscillations.
102
5.4 Case of an airfoü with an aileron executing pitching oscillations
The numerical resuits were obtained for various values of the reduced fiequency A ,such
as A = O.48,0.60,0.68 .The results are presented in Figure 5.13 and in Figures 5.14, 5.1 5
for the real and imaginary parts of the reduced oscillatory pressure difference coefficient
0 1 p,,
where
8, = represents the cornplex amplitude of osciiiations, and for the
real and imaginary parts of the reduced oscillatory iifi and pitching moment coefficients,
(- 41 &)Kl and (-81 4 ) ~respectively.
:.
The results of the real and imaginary parts of the pressure difference coefficient for an
airCoil with an aileron (s, 1c = 0.75), were found to be in very good agreement with the
previous results obtained by Postel & Leppert [24]. The results of the Lift and pitching
moment coefficients obtained by the present method for an airfoil with an aileron Located
at (s,/ c = 0.70) were in excellent agreement with Theodorsen7ssolution [1536].
Postel& Leppert
Fig 5.13 Real and imaginary parts of the pressure ciifference coefficient for an airfoil
wirh an aileron (s, / c = 0.75) exniring oscillatory rotations.
I
-Present solution
-\--
O
Theodorsen's solution -
REDUCED FREQUENCY, W2
-Present solution
O
Theodorsen's solution
-. --
2
4
6
8
REDUCED FREQUENCY, Al2
Fig 5.14 Real and imaginary parts of the reduced lift coefficient for an airfoi1 with an
aileron (s,I c = 0.70) executing oscillatory rotationr.
--
0
-r ---m
-Present solution
-O Theodorsen'ssolution
-
REDUCED FREQUENCY, h/2
I
L
Theodorsen's solution
O
-
Fig 5.15 Red and imaginary parts of the reduced pitching moment coefficient for an
airfoil with an aileron (s,/ c = 0.70) execziring oscillatory rotations.
5.5 Case of an airfoil witb an aileron executing flexural oscillations
The numerical results were obtained for various values of the reduced frequency A ,such
as A = 0.48,0.6010.68. The results are presented in Figures 5.16, 5.19, 5.22 and in
Figures 5.17, 5.18, 5.20, 5.2 1, 5.23,5.24, for the reai and irnagùiary parts of the reduced
oscillatory pressure ciifference coefficient (-cl
Mi, and for the real and imaginary
K~
parts of the reduced oscillatory lift and pitching moment coefficients, ( 4 ~ 1 %)K; and
(8c1K ~ ) K : ,respectively. The results were performed for various aileron positions such
as s,1c = 0.60, 0.75, 0.80. No comparisons were presented for the case of flexural
oscillations since there are no previotts d t s known.
Fig 5-16 Red and baginary parts of the pressure difference coefficient for an airfoi1
with an aileron (s, / E = 0.60) executingpmabolicjrextaaI oscillations.
t
+Present solution
2
4
6
REDUCED FREQCIENCY, A
--'
+
Present solution
-
- --
--
2
4
6
REDUCED FREQUENCY, h
Fig 5-17 Real and imaginary parts of the reduced iift coefficient for an airfoi1 with an
aileron (s,/ c = 0.60) execuingpmab~licflexuraloscillations.
\
'
+Present solution
REDUCED FREQmNCY, 7L
i
P
-
-
+-Present solution
-
Fig 5.18 Red and imaginary parts of the reduced pitcbg moment coefficient for an
airfoi1 with an aiteran (s,/ c = 0.60)executingparuboIic~~aI
osciIIatio~ts
Fig 5.19 Real and imaginary parts of the pressure ciifference coefficient for an airf'oil
with an aileron (s,/ c = 0.75) executingparabolicjrexural oscillatiom.
IF-
-
-
- --
-
--
+Present solution
+fresent solution
---
-
- -
Fig 5.20 Real and imaginw-parts ofthe reduced lift coefficient for an airfoil with an
aileron (s,1 c = 0.75) executingparabolicflexu~uloscillatinm.
REDUCED FREQUENCY, h
+Present method
4
6
8
10
REDUCED FREQUENCY, h
Fig 5.21 Red and imaginary parts of the reduced pitching marnent coefficient for an
airfoil with an aileron (s, / c = 0.75) executingparab~licfrexural
oscilIations
Fig 5.22 Real and haginary parts of the pressure ciifference coefficient for an airfoil
wiîh an aileron (s, / c = 0.80) executingparabolic~exura1oscillations.
+-Present solution
--
-
--
6
4
8
REDUCED FREQUENCY,h
-
-
+Present solution
-
4
6
-
-
8
REDUCED FREQUENCY, h
Fig 533 Real and imaginary parts of the reduced Si coefficientfor an airfoii with an
aileron (s, 1c = 0.80) executingpmabolicflexuralosciIIations.
-4 Present solution
- -
REDUCED FREQUENCY, h
Pb--p
4
+Present solution
6
-
8
REDUCED FREQUENCY, A
Fig 5.24 Real and imaginary parts ofthe reduced pitching moment coefficientfor an
ahfoil with an aileron (s, 1c = 0.80) execuringparabolicftexural osciltatiom.
Chapter 6
Conclusions
In this thesis, the steady and unsteady flow past fixed or oscillating airfoils has
been analyzed. This work presents a new method based on velocity singularities for the
analysis of the unsteady flows past oscillating airfoils,
The method OF velocity singularities developed by Mateescu [Il, l3,14,16] For
steady flows past airfoils. has been vaiidated for the cases of ngid and flexible airfoils, in
comparison with the previous solutions based on conformal transformations [1431], or
obtained by Nielsen and Thwaites [1322,27]. A very good agreement has been obtained
between the present solutions based on the velocity singularity method for steady flows
and the previous results.
Closed fom solutions were obtained for the pressure distribution and the
aerodynamic forces acting on airfoils executing various hamonic oscillations. Closed
form formulas were also derived for the pressure distribution and the aerodynamic forces
acting on an airfoil with an aileron executing harmonic oscillations.
The method of velocity singuides for the unsteady flow past airfoils and
ailerons executing harmonic oscillations, developed in this thesis, has proven to lead to
accurate solutions, computationally efficient, in al1 studied problems. The solution of the
unsteady flows obtained by the present method of velocity singularity is relatively
simple, which avoid the mathematicai difficulties encomtered in the classicai theories.
The present method has been fkit vdidated for airfoils executing osciliatory translation
and pitching rotation and for aidoils with aiierons executing pitching oscillations, in
comparison with the previous redts obtained by Theodorsen [15,26] and Poste1 and
Leppert [24]. An exceiient agreement bas been obtained between the present velocity
singularity solutions for unsteady flows and the previous results.
The method has then been extended to the unsteady flows past airfoils executing flexurd
harmonic oscillations. Closed form solutions were also derived for the pressure
distributions and the aerodynamic forces acting on an airîoil with an aileron executing
flexurai oscillations. No cornparisons were presented for the case of flexurai oscilIatioas
since there are no previous results known.
In al1 problems treated in this thesis, the rnethod of velocity singularities for
steady and unsteady flows past tïxed or oscillating airfoils has proven to be accurate,
efficient and stable. The mathematicd treatment also provides a convenient arrangement
permitting a smooth treatment of ail cases of oscilIations.
As a suggestion for future work, the present method c m be extended to the non-
ünear analysis of the unsteady flow past the oscillating airfoils. Also the method can be
extended to the case when the amplitude of oscillations is decaying exponentially in t h e
instead of being constant. as it is ofien the case in practice.
Bibliography
1- Abbott, H., and Von Doenhoff, AE.,Theory of wing sections., Dover, New York,
1959.
2- Basu, B.C., and Hancock, G.J.,'The Unsteady Motion of a Two-Dimensionai Airfoi1
in Incompressible Inviscid Flow," J o m d of FIWd Mechanics, Vol. 87,1978, pp.
159-78.
3- Carafoli E., Mateescu D., and Nastase A., Wing Theory in Supersonic Flow.,
Pergarnon Press, 1969, pp. 1-591.
4- Dietz, "The Airforces on a Harmonicdly Osciiiating Self Deforming Plate,"
Luftfahrtforschung, Vol. 16, No. 2,1939, p.84.
5- Fung, Y.C.,An Introduction to the Theory of Aeroelasticity, Dover, New York, 1993,
pp. 381462.
6- Glauert, H., The Elements of Aerofoil and Airscrew Theory,, Cambridge University
Press, Second edition.. 1926.
7- Gray, A., and Mathews, G.B., "A Treatise on Bessel Functions and Their
Applications to physics," Dover, Second edition, pp. 20-35.
8- Kemp, N.H.,and Homicz, G., "Approximate Unsteady Thin Airfoil Theory for
Subsonic Flow," AiAA Journal, Vol. 14.1976, pp. 1083-89.
9- Kussner, H.G., "Nonstationary Theory of Airfoils of Finite Thickness in
incompressible Flow," AGARD Manual on Aeroelasticity, Part II, Ch.8, 1960,
Neuilly-sur-Seine, Fmce.
10- Mateescu, D., " A Hyprid Pane1 method for Aerofoil Aerodynamics," B o u n d q
Elements XII, Vol. 2, Applications in Fluid Mechanics and Field problems, edited by
M. Tanaka, C. A.. Brebia. and T.Honma, Computational Mechanics PubIications,
Southampton, England, UK, and Springer-Verlag, Berlin, 1990, pp.3-14.
11- Mateescu, D., "Wùig and Conical Body of Arbitrary Cross-Section in Supersonic
Flow," Journal of Aircraft, VoI.24, No,4,1987, pp.239-247.
12- Mateescu, D., Nadeau, Y., ' A nonlinear a d y t i c d solution for airfoüs in irrotationd
flow," Proceedings of the third International congress of Fluid Mechanics, Cairo,
Egypt, Vol. IV, January 1990, pp. 14214432,
13- Mateescu, D., and Newman,B.G.,"Analysis of Flexible-Membrane and Jet-FIapped
Ahfoils Using Velocity Singularilies," Journal of Aircraft, Vo1.28, No.11, 1991,
pp.789-795.
14- Mateescu, D., "Subsonic Aerodynamics," Lecture Notes, McGill University, 1998.
15- Mateescu, D., "Unsteady Aerodynamics," Lecture Notes, McGill University, 1999.
16- Mateescu, D.,
"
High-Speed Aerodynamics," Lecture Notes, McGilI University,
1999.
17- Mateescu, D., "Private Communications," McGill Uuiversity, Montreal, Canada,
1999.
18- McCroskey, W.J., "Unsteady Airfoils," Ann. Rev. Fluid Mech. 1982.14: 285311.
19- McCroskey, W.J., "Inviscid Flow Field of an Unsteady Airfoit," N A A Journal,
Vol. 11,1973, pp.1130-37.
"Bessel Functions For Engineers," M o r d at the Clarendon Press,
20- McLachian. N.W.,
1934, pp. 61-70.
21- Milne-Thomson, LM.,Theoretical Aerodynamics.. 4~edition, Dover, New York.
1966, pp. 136- 145.
22- Nielsen, J.N., 'Thtory of Flexible Aerodynamic Surfaces." Journal of Applied
Mechanics, Vol. 30, No. 9,1963, pp. 435-442.
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269378.
24- Postel, E.E.. and Leppert, E.L., Jr. 1948, 'Theoretical Pressure Distribution for a thin
Airfoil Oscillating in IncompressiiIe flow," Journd of Aeronauticd Sciences. 15:
486-92.
25- Sch-
"Calcdation of the Pressure Distriiution of a Wig Harmonicaiiy
Oscillating in Two-Dimensional Flow," Lufüahrtforschung, Vol. 17, Nos. 11,12,
1940, p. 379.
26- Theodorsen, Theodore, "Generd Theory of Aerodynamic instability and the
Mechanism of FIutter," N.A.C.A. T.R No. 496,1935.
27- Thwaites, B., "The Aerodynarnic Theory of safis: Two-DimensionaI Sails,"
Proceedings of the Royal Society, series A, Vol. 261,1961, pp. 402-422.
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Journal of Aeromutical Sciences. 5: 379-90.
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edition, 1966, pp. 160-183.
Appendix A: General integrals
A.1 The integral F ( z )
Consider the integrai F (z) given by,
-
F (r)= lirn [ le
.'
c(Cr
-Z )
y
By taking the derivative, one obtains,
Integratiag by parts yields.
A 3 The integral J (x)
Consider the integrai 3 ( x ) defrned by ,
By integrating J (x) fiom x = O to x .one obtains,
The second integrai is evaiuated by,
One obtains thus two known integrals and the overaii integrai is given by,
By substituting the second integral into the integrai of ~(x),one concludes,
(A4
The value of the definite integrai fiom x = O to x = c is given by,
R
IJ~)&=iim[ l e
2
C
-&-c)
O
g
A
do]-Ciirn[
0.-
2 cm-
-1-(O-C)
[e
c
du].
(A-9)
C
By substituting O = - (<+ 1) into the f ~ t integrai
e
appearing in the integral of ~ ( x ) one
.
2
obtains,
(A- 1O)
Note that Hankel's integrals are defined by,
(A- 12)
where H ~ ' ) ( x11) aod ~ / " ( 1 12) are Hankel's nuictions of second kind of zero and fust
By substituting the kuown integrals of Hankel into the above integral, one obtains,
(A- 13)
An important integral to note is that related to Fourier integrais given by (after integrating
by parts),
(A- 14)
where as s h o w in (4-46),
(A- 15)
which can be obtained fiom the Riemann-Lebesgue Iemma on Fourier integrals 15,231,
when the theory of distribution is used, Hence, the integrai defuied in (A-9) becomes.
(A- 16)
A 3 Recurrence formulas for the integral Ik
Consider the recunence integral defined by,
Note that sk can be written by,
(A- 18)
By substituting the vaiue of sk into the recurrence integd, one concludes,
(A- I 9)
The second integrai can be then given by,
Substituting the above integral into the dennition of the recurrence integrai, one obtains,
By arrauging the t e m , one concludes,
The value of the inte& at k = O ( Io) is given by,
-
1
I,, = jJmh=-2cos-'
S C-S
i:
-
The value of the integral at k = 1 ( 1,) is given by,
The recurrence integral 1, with limits is given by,
a) ï h e integrai 1s'-'
-4
ch is related to the recurreace integral as follows.
O
b) The integrid jsk (/,-
dr is related to the recurrence integral as follows,
O
where,
2k+1
cz (2k+1)(2k+3)
c lk and Ik,2 = 4 ( k t l ) ( k + 2 ) Ik
Ik+l
= 2 (k+ 1)
The resultant formuIa is given by,
The recurrence integral Ik is evduated h m x = s, to x = c and is given by,
C
2, Ik = 9
sk
C-S
/:
Io = 2 cos-' -
1
'
where,
The resultant formda of the integral is given by,
The foIIowing integrals are related to tbe recurrence integrai and are evaluated in the
same manner as above,
where,
A.4 Recurrence formulas for the integral
Q,
-
Consider the integrai Q, dehed by,
By taking the derivative of the cosh-'(
Hi
= c0sh-l RI
.
} function, one obtains.
R,
By integating by parts, one concludes,
By raking xk+' = s:"
k
- (s, -x)
xn s
:"
the above recumnce formula wiii be.
n-O
where fiom section A.5,
where, Ïo= -2 cos-'
The following integrals are special cases of the recmnce integral QI,
By taking the bits of the above integral fiom x = O to
.
x =c one obtains,
no= j c /a(~
--)-9
~° 9 ~ =2 G2 JO.-,s ) ( j ~ ) + ~
0
C X
C(S X)
The real part of the above integrai is given by,
The resultant formula is eqressed as.
The real part of the above integrai is given by,
1=
~ e a l k , f(s
+:)dm-
By substituting into the recmnce integrai Q,,one concludes,
The resdtant formula is given by,
The real part of the integai is evaluated as,
The foliovuhg integrals are reIated to the recurrence integraI Q, and are given as,
A S Recurrence formulas for the integral F,
Consider the recurrence integrai F, defined by,
TT
By integrating by parts, one concludes,
where,
-JC - .rjx
d H,
-=
d s, ? (s,- r)J&TJ '
k
By taking s:'
= xk+' + (s,
- x ) I s , " xk"
R-O
Frorn section AS,
one obtains.
.
By substituthg into the recurrence formula of Fk one concludes,
The real part of the integrai is given by,
Note that,
2n-1
In =?
c I,,
it
where, Io = n , I, = C F .
-
The following is a special case of the above recurrence integral with Iimits and is
concludeci as,
The following is an integral that is related to the recurrence forsnula and is given by.
The recurrence integrai Fk is evaiuated for the case of the limits fiom x = s, to x = c
and is given by,
where,
j:
Io = 2 cos-' -
.;/,
I, =3(.-k+c
cos-'
A special case of the above recunence formula is given below as,
A.6 Integrals related to 1,
The following are integrals related to the recurrence FormuIa I, ,
1) jsk-' cos-'
g*
O
By performing the derivative, one obtaùis,
Uitegrating by parts,
where,
2k-1
2k
1, =-
4-t
where, Io = n , I, =c.;
x
-
The fouowing intepi is a special case of the above recurrence formuia,
The redtant expression is &en by,
where,
Io = 2 cos-'
:J
A.7 Recurrence formula for the integral Ir
Consider the recurrence integral I, defined by,
where (fiom section A.4),
The following integrais are related to the recunence integral I r .
C-X
1) / X q T . =
C-X
[Xr
J(C-")Xk
=CI'
The resultmt expression is given by,
where,
IO = X , II =c-
II
3
and Iz
3 c'
=-X.
8
-LI
The foiiowing integrals are related to the recurreace integrai 1, at various values of r,
Appendix B: Theodorsen's formulas
B.1 Formulas related to Theodorsen's results
*,
,
=
-1 -Jgc:)+
(1
clcos-'
+
cl,
3
0-9)
(B-10)
(B-11)
(B-12)
(B-13)
(B-14)
(B-15)
(B-16)
(B-17)
q, = 41-c,' +cos-'c , .
2-,' =cos-l c,(1-2C,)+Jq(2-cI).
T,, = J 3 ( 2 + c I
1
1
T,, = ~ + - y c , .
Note that,
v=I/*.
)-EOS-I
cl@cl + I),
B.2 Theodorsen's hnction
The function ~ ( k ) i called
s
Theodorsen's function, The exact expression of it is given by,
where @(k)
and H12)(k)are Hankel functions [7,20,29]
of second kind (zero and £ïrst
order, respectively). The standard notations for the reai and imaginary parts of ~ ( k ) a r eF
and G, which are tabulated in Table B. 1.
Table 8.1 Theodorsen's function ~ ( k=)F t iG .
Appendix C: Special complex functions
C.1 The complex function H , (LS)
Consider the folIowing complex function,
Hl (2,s)= cosh-' R, ,
where,
The derivative of the complex function is given by,
where,
The derivative is given by.
The complex firnction can be expressed by,
where,
The derivative of the complex hction is given by,
where,
(C-13)
(C- 14)
The derivative is given by,
From the above decivations one concludes,
(C- 16)
Note thak
cl,
H,= h ( ~+ ,, / q ) = l n ( % + j ~ T ) = i z j c o s - l R, =kjK1Note that,
El =cos-' K.
(C- 18)
(C-19)
By comparing the derivatives, one conchdes,
For z = x < s < c ,
The complex function in this domain is expressed by.
H,= cosh-l R, = I ~ ( R , + 1/- ,* 1 R
- )= +jcos-' RI = +jH,.
Note that,
By comparing the derivatives, one concludes,
Note thatfor s < z = x < c .
The following are special cases of the complex function HI(z.s),
One concludes.
Tt
H,(z,~)= j-
2'
for any z ,
H,( a v )= O,
C.2 The complex funetion Al (;;c)
Consider the compiex function A, (z, a) given by.
A, (z, a)= cos-'
.
R,
where,
The derivative o f the above cornpiex function is given by,
where.
The derivative is given by,
For a > z = x > c ,
The cornplex function in this range is expressed as,
Note that,
The derivative of the above complex function is given by,
where,
The derivative is given by.
By comparing the derivatives in this range, one concludes,
For z = x > ~ > c , o n e o b t a i n s ,
The complex iünction is expressed by.
The above conclusion is supported by comparing the derivatives,
For z = x > o > c ,one obtains,
For ~ = ~ < O < c ~ u , o n e o b t a i n s ,
The complex fùnction is eqressed by.
The derivative of the complex function in this range is given by.
By comparing the derivatives, one conc[udes,
The following are speciai cases of the complex lünction A, (z, D),
A, (2, a)= cos-'
c(0-f)'
The finai expression of the complex function for this special case is given by,
cos-'
i
C-z
c
C3 Summary of the behavior of the special complex fonctions
Below is a summary of the behavior of the special complex functions, represented in
Tables C. 1 and C.2.
Table C.1 Generai behavior of a speciai complex shgularity Hl (ZJ).
Part
J'"
C-I
R,= C(S - I)
Table C.2 Generai behavior o f a specid complex singularity Al (z,o).
A, = cos-' RI
1;
The two compiex singulanties at speciai positions almg the airfoil bebave as,
cos-'
ic:=
-
Appendix D
Derivation of the unsteady pressure coefficient
D.1 Unsteady pressure coefficient for the airfoii
The equation of the reduced pressure coefficient 6ep(x) is expressed as,
where k is the reduced frequency.
The expression of the reduced velocity on the airfoil Sû is given by, (See chapter 4),
By taking the reai part of the cornplex velocity 6 ~ ( z ) the
. expression of 6 û is given by.
The expression of k(x)is expressed as,
By evaluating the integrals appearing in equation @4),see Appendix A. one obtains,
The expression of &(c) is Men by
where,
The integral J (x) is defmed by,
The above integrai cm be expressed as.
By performing the second integrai, one obtains,
(D-10)
The integral of the above equation is given by,
By performing the integrals in the above equation, one obtains.
(D- 12)
By inserting equations (D-3) and @-5) into the expression of the pressure coefiient
se, (1)(equation (D-i)), one obtains,
(D-I 3)
Let us coosider y(.r) as.
(D-14)
By inserthg the t e m that appear in equation (D-14).
one obtains,
Alter arrangement, one Obtains,
=
G 2
irJ
-i:J
C-X
-e
x
'-A
@)(A / 2).
(D16)
Note that,
ix c-x
+n,/+l-ctli2))
i
By inserting equation @-17) into the expression of the pressure coefficient, one obtains.
(D-18)
The above equation can be represented after mathematical arrangements as,
(D-19)
where C(J. 12) is Theodorsen7sfunction, The values of Theodorsen tùnction are given in
Appendix B.
D.2 Unsteady pressure coefficient for the airfoil with an osciiiating
aileron
The equation of the reduced pressure coefficient 6ep(n)ir expressed as,
The expression of the reduced velocity on the airfoil 6 î1 is given by,
6 û = REAL,
[m~(z)l-~,
The expression of 6û is given by,
The expression of the reduced circulation for the prototype problem at any position x is
given by,
The resultant of the above equation is given by,
The above integrals are evaluated in Appendix A.
The resultant equation of the reduced circulation for the airfoir with an oscillating aileron
is given by,
where,
where H,(')(A/?) and
'AH
(hl2) are the Hankel's integrals of second kind of orden of
one and zero, respectively. By inserthg the above equations into the expression of the
pressure coefficient and performing a similar analysis to section D.1.one obtains.