Metodo_Theodorsen.pdf_Aerodynamics of oscillating airfoils

Aerodynamics of oscillating airfoils

1. \Aerodynamics of the oscillating airfoils"
University of Naples
November 4
th
, 2011.
Claudio Marongiu
1
[email protected]
1
CIRA - Italian Center for Aerospace Research
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th
2011 1 / 75
Aerodynamics of the oscillating airfoils

2. OUTLINE
Introduction
Theodorsen solution
CFD of the oscillating airfoils
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2011 2 / 75
Aerodynamics of the oscillating airfoils

3. References
1
Von Karman, T., and Burger J. M., 1935, \General Aerodynamic Theory. Perfect Fluids", Peter Smith Publisher, Inc.,
1976, pp. 280-310.
2
Theodorsen T., \General Theory of Aerodynamic Instability and the Mechanics of Flutter", National Advisory
Committee for Aeronautics, NACA Report. 496 (1935).
3
Bisplingho R. L., Ashley H. and Halfman R. L., \Aeroelasticity", Dover Publications, Inc., New York
4
Saman P. G. , \Vortex Dynamics", Cambridge University Press, 1992
5
McCroskey W. J., \The Phenomenon of Dynamic Stall", Lecture Notes presented at Von Karman Institute Lecture
Series on Unsteady Airloads and Aeroelasticity Problems in Separated and Transonic Flows, 9-13 March 1981.
6
Leishman J. G., (2000) \Principles of Helicopter Aerodynamics ". Cambridge University Press
Most of the material contained in these slides comes from my Ph.D. thesis made
between 2007 and 2010 and related publications in collaboration with Prof. Tognaccini,
(DIAS), University of Naples.
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Aerodynamics of the oscillating airfoils

4. Introduction
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Aerodynamics of the oscillating airfoils

5. Introduction
Helicopter
Turbomachinery
Manoeuvring Aircrafts
Wind Energy
Biological ows (Insect ight)
...
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Aerodynamics of the oscillating airfoils

6. Introduction
Examples: Helicopter blade motion.
Carr L. W., Progress in Analysis and Prediction of Dynamic
Stall, J. of Aircraft, Vol. 25, No. 1
By combining the rotation and the
advancing motion a time varying angle
of attack is seen by each blade section.
When the blade is in the advance
phase (0

180

) the local
angles of attack are lower (marked
compressible eects).
For (180

360

) the local
angles of attack increase and the
blades can stall (dynamic stall).
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Aerodynamics of the oscillating airfoils

7. Introduction
Examples: Helicopter blade motion.
Ex: main rotor of AW119, diameter = 10.83m, 400rpm,)Mt0:8
The dynamic stall causes signicant vibrations and torsional loads.
It is a strongly time dependent phenomenon.
In the 60's, it was discovered that the dynamic stall could be investigated similarly
on a two-dimensional airfoil under pitching conditions.
The combination of the asymptotic free stream and the airfoil motion produces a
change in the angle of attack.
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Aerodynamics of the oscillating airfoils

8. Introduction
Examples: Turbomachinery, rotor-stator interaction.
Rotor-stator interaction. Mach contours. Two-dimensional
simulations with ZEN at CIRA (2006), in collaboration with P.
L. Vitagliano. Sliding mesh technique.
The uid dynamics of turbomachinery
is another wide sector in which the
aerodynamics of oscillating airfoil is
extensively applied.
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Aerodynamics of the oscillating airfoils

9. Introduction
Examples: Wind EnergyU
¥
q= 0°
q= 45°
q= 90°
q= 135°
q= 180°
q= 225°
q= 270°
q= 315°
L
wR
L
wR
U
¥
wR
U
¥
wR
L
L
Similar phenomena occur on the
blades of the wind turbines.
The blades work in a wide range of
angles of attack.
The exact knowledge of the
aerodynamic loads can improve the
design and the structural life of the
plant in terms of fatigue limits.
Other requirements, such as the low
noise emission, must be respected in
order to reduce the environment
impact.
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2011 9 / 75
Aerodynamics of the oscillating airfoils

10. Introduction
Airfoil unsteady aerodynamics
The steady aerodynamics provides relations of kind
Cl=Cl(;Re1;M1)
In the unsteady aerodynamics, the dependency must be of kind
Cl=Cl(;_;;h;Re1;M1)
wherehis the vertical displacement.
Namely, the aerodynamic characteristics must include the dependency upon the
airfoil motion
The ow exhibits a memory of the past history.
Each ow phenomenon typical of the airfoil aerodynamics (transition, separation,
stall, buet, ... ) must be revisited under this perspective.
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Aerodynamics of the oscillating airfoils

11. Introduction
Airfoil unsteady aerodynamics. Dynamic Stall.
From Leishman (2000). Principles of Helicopter Aerodynamics.
The stage (1) is in correspondance of the static stall
angle of the airfoil. Under dynamic conditions the lift
curve continues to grow.
An extrapolation of the linear slope is observed up to
the points (2) or (3). The excess of lift is due to a
production of vorticity in the boundary layer.
At stage (2) the dynamic stall vortex is formed
changing the lift curve slope.
The moment coecient stalls between stages (2) and
(3).
The maximum peak of lift is achieved when the
dynamic stall vortex reaches the trailing edge.
During stages (3) and (4), the ow is separated.
The reattachment point advances at a velocity less than
U1. After, the ow acquires again a linear behaviour
(stage 5).
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Aerodynamics of the oscillating airfoils

12. Introduction
Airfoil unsteady aerodynamics. Dynamic Stall.
An important quantity is the aerodynamic damping
=
Z
Cmd
It represents the work done by the aerodynamic forces acting on the airfoil.
If >0 the uid receives energy from the airfoil (stable).
If <0 the uid transfers energy to the airfoil (unstable).
Geometrically,is the measure of the area enclosed by theCmcurve in the plane
Cm.αα
C
m
C
m
ζ

> 0ζ

< 0
Unstable Stable
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Aerodynamics of the oscillating airfoils

13. Introduction
Some concepts for ideal ows. Virtual or apparent mass
The aerodynamic force is related to thevirtualorapparent masscontribution,
dened as (see Saman):
I
B
=
Z
@B
ndS
is the velocity potential.
It is possible to show that:
F=
d
dt
I
B
IfMUis the momentum of the solid body andfthe resultant of the external
forces applied on the solid body, Newton's second law is:
d
dt
MU=F+f
Then we have:
d
dt
(MU+I
B
) =f
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Aerodynamics of the oscillating airfoils

14. Introduction
Some concepts for ideal ows. Virtual or apparent mass
The presence of thevirtual massI
B
alters the solid body inertia.
The termI
B
accounts for the eects of the uid surroundingB.
The external forcefapplied onBis balanced by the real mass of the body and the
uid virtual mass.
These eects appear in case of unsteady equilibrium only.
.avi
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Aerodynamics of the oscillating airfoils

15. Theodorsen solution.
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Aerodynamics of the oscillating airfoils

16. Theodorsen Solution
Hypothesis.
In 1935 Theodorsen obtained the unsteady ow solution for a thin
oscillating airfoil.
Inviscid ow
Incompressible ow
thin airfoil
small disturbances
Suppose the free stream velocity parallel to thexaxis and the wall normal
directed asz. We have
u=U1+u
0
(1)
u
0
;wV1 (2)
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Aerodynamics of the oscillating airfoils

17. Theodorsen Solution
Problem equations.
Suppose it is possible to introduce a potential perturbation
0
such that
@
0
@x
=u
0
;
@
0
@z
=w
The potential perturbation satises the equation of Laplace:
r
2

0
= 0 (3)
The linearized Bernoulli equation is also written as
pp1=U1u
0

@
0
@t
(4)
The problem is dened by setting the initial and boundary conditions.
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Aerodynamics of the oscillating airfoils

18. Theodorsen Solution
Road Map.Proof
Boundary
Condition
Conformal
Transformation
Circulatory
Part
Non
Circulatory
Part
Kutta
Condition
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Aerodynamics of the oscillating airfoils

19. Theodorsen Solution
Road Map.Proof
Boundary
Condition
Conformal
Transformation
Circulatory
Part
Non
Circulatory
Part
Kutta
Condition
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Aerodynamics of the oscillating airfoils

20. Theodorsen Solution
Part 1. Boundary conditions.
On the solid body, the wall normal component of the velocity must be specied.
The airfoil surface can be expressed in this form
Fu=zzu(x;z;t) = 0
FL=zzL(x;z;t) = 0
The boundary condition requires for the upper and lower sides that
DF
Dt
=
@z
@t
+u
@z
@x
+w= 0
Since of the hypothesis of small disturbances the upper and lower surfaces are
mathematically approximated as a plane surface atz= 0. Besides,
u
@z
@x
V1
@z
@x
Then we have the following condition
w
@z
@t
+U1
@z
@x
=wa(x;t) (5)
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Aerodynamics of the oscillating airfoils

21. Theodorsen Solution
Part 1. Boundary conditions.ba
α
h
x
z
b/2b/2
Scheme of the velocity components on the circumference in the complex plane
The plate oscillates atbafrom the half chord location.
The instantaneous position is:
za(x;t) =h(xba)
from which
wa(x;t) =_h_(xba)U1 (6)
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Aerodynamics of the oscillating airfoils

22. Theodorsen Solution
Road Map.Proof
Boundary
Condition
Conformal
Transformation
Circulatory
Part
Non
Circulatory
Part
Kutta
Condition
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Aerodynamics of the oscillating airfoils

23. Theodorsen Solution
Part 2. Conformal transformation.
The proof of Theodorsen's solution is achieved by transforming the at plate from
the physical planexzin a circle in the complex planeXZ.
Letc= 2bbe the chord of the airfoil. The conformal transformation is
x+i z=X+i Z+
b
2
4(X+i Z)
(7)
wherei=
p
1 is the imaginary unit.
The above relation transforms the plate of lengthcto a circle of radiusr=b=2.
In fact,X=rcosandZ=rsin, we have:
x+i z=r e
i
+
b
2
4re
i
= 2rcos
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Aerodynamics of the oscillating airfoils

24. Theodorsen Solution
Part 2. Conformal transformation.X
Z
XZ plane xz plane
r = b / 2
x
z
b b
!
q
r
q
!
u’
w
Conformal transformation of a circle in the planeXZto a at plate in the planexz.
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Aerodynamics of the oscillating airfoils

25. Theodorsen Solution
Part 2. Conformal transformation. Velocity.
Now, it is necessary to establish the transformation for the velocity components.
In the planeXZ, the velocity components are indicated withqXandqZ.
The complex velocity is obtained as
u
0
iw= (qxiqz)
d(X+i Z)
d(x+i z)
where:
d(x+i z)
d(X+i Z)
= 1
b
2
4(X+i Z)
2
(8)
Forr=b=2,

d(x+i z)
d(X+i Z)

r=b=2
= 2 sine
i
(9)
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Aerodynamics of the oscillating airfoils

26. Theodorsen Solution
Part 2. Conformal transformation. Velocity.
By calculatingqXandqZon the circler=b=2, we have:
u
0
i w=
qXi qZ
2 sin
e
i(=2)
=
1
2 sin

qXe
i(=2)
+qZe
i()

Since,e
i(=2)
= sinicos, ande
i()
=cosisin, we have:
u
0
i w=
1
2 sin

qXsinqZcosi(qXcos+qZsin)

The modules are:
ju
0
iwj=
p
u
02
+w
2
=
p
q
2
X
+q
2
Z
j2 sinj
=
p
q
2

+q
2
r
j2 sinj
(10)
whereqandqrare the radial and tangential components in theXZplane.
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Aerodynamics of the oscillating airfoils

27. Theodorsen Solution
Part 2. Conformal transformation. VelocityX
Z
XZ plane
xz plane
r = b / 2
x
z
−

bb
θ
q
r
q
θ
−
u’
w
α
α
q
Conformal transformations preserve the angle at which two lines intersect. Then,
jqj
jqj
=
ju
0
j
juj
) ju
0
j=
jqj
j2 sinj
andjwj=
jqrj
j2 sinj
According with the sign ofandxaxis, we deduct also that :
u
0
=
q
2 sin
andw=
qr
2 sin
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Aerodynamics of the oscillating airfoils

28. Theodorsen Solution
Part 2. Conformal transformation. Potential.
It is necessary to connect the velocity potentials between thexzandXZplanes in
such a way that
d(x;z) =d(X;Z)
For a path along the slit in thexzplane (the circle in theXZplane)
d(x;z) =u
0
dx d(X;Z) =
b
2
qd
The potential dierence between two points on the slit is
21=
Z
2
1
b
2
qd=
Z
x2
x1
u
0
dx (11)
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Aerodynamics of the oscillating airfoils

29. Theodorsen Solution
Road Map.Proof
Boundary
Condition
Conformal
Transformation
Circulatory
Part
Non
Circulatory
Part
Kutta
Condition
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2011 29 / 75
Aerodynamics of the oscillating airfoils

30. Theodorsen Solution
Part 3. Non circulatory contribution
Theodorsen achieved the solution by distributing sources (upper side) and sinks
(lower side) of equal strength.
In this way, the boundary conditionwa(x;t) was fullled.
The sources and sinks do not cancel each other on the plate surface.
The points outside the circle in the planeXZare mapped in the external eld
around the plate.
But, the points inside the circle are associated in the external eld of the plate as
well.
The whole planeXZcreates two overlapped sheets in the planexz(Riemann
surfaces).
We pass from a sheet to another only when we cross the plate from upper side to
the lower side and viceversa.
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Aerodynamics of the oscillating airfoils

31. Theodorsen Solution
Part 3. Non circulatory contribution
(a)xzplane (b)XZplane
Stream lines of the ow due to the distribution of sources and sinks. From Bisplingho et al., pg 256
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Aerodynamics of the oscillating airfoils

32. Theodorsen Solution
Part 3. Non circulatory contribution
LetH
+
be an innitesimal source sheet distributed on the upper side of the circle.
The potential function
0
in (x;z) induced byH
+
is given by:

0
(x;z;t) =
1
4
Z
b
b
H
+
(;t) ln

(x)
2
+z
2

d (12)
The result is related to the wall normal velocity as:
H
+
(x;t) = 2wa(x;t)
H
+
(;t) = 4wa(x;t) sin (14)
Similarly
H

(x;t) =2wa(x;t)
H

(;t) =4wa(x;t) sin (16)
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Aerodynamics of the oscillating airfoils

33. Theodorsen Solution
Part 3. Non circulatory contributionX
Z
XZ plane
P(r,
θ
)
Q
−
(r,−φ)
dq
+
dq
−
Q
+
(r,φ)
θ
φ
−

φ
dq
θ
Scheme of the velocity components on the circumference in the
complex plane
The velocity resulting from the
distribution of sources and sinks has
to be derived.
Consider two points,Q
+
(r; ) and
Q

(r; ) symmetrically located on
the circumference
The velocity in a pointP(r; )
induced byH
+
r d andH

r d is
built on the basis of geometrical
considerations.
Note that the induced velocity of the
source-sink sheet is such thatqr= 0,
otherwise the circle is a stream line.
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Aerodynamics of the oscillating airfoils

34. Theodorsen Solution
Part 3. Non circulatory contribution
The nal result is:
q(;t) =
2

Z

0
wa(x;t) sin
2

cos cos
d (17)
The potential function
0
is obtained:

0
U(;t)
0
(;t) =
b

Z


Z

0
wa(x;t) sin
2

cos cos
d d (18)
Because of the arbitrary time function in the denition of the potential
0
, it is
possible to put
0
(;t) = 0.
Besides, for the symmetry, we observe that the following relation subsists between
the lower and upper side potential:

0
L(;t) =
0
U(;t)
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Aerodynamics of the oscillating airfoils

35. Theodorsen Solution
Part 3. Non circulatory contribution
By taking into account thatx=bcos, the following integral gives the velocity
potential:

0
U(;t) =
b

(_h+U1)
Z


Z

0
sin
2

cos cos
d d+
b
2
_

Z


Z

0
sin
2
(cosa)
cos cos
d d (19)
After some algebra, the nal result is achieved in this form:

0
U(;t) =b(
_
h+U1) sin+b
2
_

1
2
cosa

(20)
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Aerodynamics of the oscillating airfoils

36. Theodorsen Solution
Part 3. Non circulatory contribution
By means of Bernoulli, the pressure relates to the potential function. Then
pUpL=

U1

@
0
U
@x

@
0
L
@x

+

@
0
U
@t

@
0
L
@t

(21)
By exploiting the symmetry properties of
0
, we have:
pUpL=2U1
@
0
U
@x
2
@
0
@t
=
2U1
bsin
@
0
U
@
2
@
0
@t
(22)
Thenon circulatorycontribution is:
LNC=
Z
b
b
(pUpL)dx=U1
h

0
U
i
b
b

Z
b
b
2
@
0
U
@t
dx (23)
Since there is no circulation,
0
(;t) is single valued. Because of

0
L(;t) =
0
U(;t), we have thatU(0;t) =L(0;t) = 0. Then
LNC= 2
@
@t
Z

0

0
Usind (24)
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Aerodynamics of the oscillating airfoils

37. Theodorsen Solution
Part 3. Non circulatory contribution
By means of Bernoulli, the pressure relates to the potential function. Then
LNC=b
2

h+U1_a b

(25)
We report also thenon circulatorypart of the aerodynamic moment:
MNC=b
2

U1
_h+bah+U
2
1b
2

1
8
+a
2



(26)
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Aerodynamics of the oscillating airfoils

38. Theodorsen Solution
Road Map.Proof
Boundary
Condition
Conformal
Transformation
Circulatory
Part
Non
Circulatory
Part
Kutta
Condition
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Aerodynamics of the oscillating airfoils

39. Theodorsen Solution
Part 4. Circulatory contribution
Thenon circolatorypart is not able to fulll Kutta's condition.
Theodorsen resolves the problem by superimposing a vorticity distribution on the
body surface (boundvorticity) and in the wake (freevorticity).
The technique of the image vortices are used.
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Aerodynamics of the oscillating airfoils

40. Theodorsen Solution
Part 4. Circulatory contributionX
Z
XZ plane
χ
q
−
q
+
P(r,θ)
b
2
/ 4χ
Γ
0
 − Γ
0
r
1
r
2
θ θ
1
θ
2
θ
1
−θ
θ
2
−θ
Boundvortex of intensity 0and its image of intensity0.
A vortex located at (;0) of intensity
0in the planeXZhas its image 0
in (b
2
=4;0).
The vortex pair respects the condition
of a stream line for the circumference.
It is possible to showqr= 0 on the
circle.
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Aerodynamics of the oscillating airfoils

41. Theodorsen Solution
Part 4. Circulatory contribution
The tangential componentqis
q=
0
2

r2cos(2)
r
2
2

r1cos(1)
r
2
1

It can also be observed that
r
2
2=
2
+

b
2

2
bcos;r
2
1=

b
2
4

2
+

b
2

2

b
3
4
cos;
and
r2cos(2) =
b
2
cos;r1cos(1) =
b
2

b
2
4
cos;
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Aerodynamics of the oscillating airfoils

42. Theodorsen Solution
Part 4. Circulatory contribution
By substituting the previous relations we have:
q=
0
b
"

2

b
2
4

2
+
b
2
4
bcos
#
(29)
The velocity potential is calculated

0
U(;t) =
Z


q
b
2
d=
=
0
2


2

b
2
4
Z


1

2
+
b
2
4
bcos
d (30)
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Aerodynamics of the oscillating airfoils

43. Theodorsen Solution
Part 4. Circulatory contribution
The result is

0
U(;t) =
0

tan
1

(
1
2
b)
(+
1
2
b)
r
1 + cos
1cos
!
(31)
By means of equation (31), we are able to compute the pressure distribution by
using Bernoulli.
Note that the time dependency appears through the variable(t) which indicated
the instantaneous position of the wake vortex.
The hypothesis that the vortex is shed at the free stream velocity is adopted.
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Aerodynamics of the oscillating airfoils

44. Theodorsen Solution
Part 4. Circulatory contribution
This assumption allows for the following transformation:
d
dt
=U1 (32)
whereis the vortex location in the planexzwhich corresponds to
=+
b
2
4
(33)
in the planeXZ.
Equation (33) can be cast as follows:
s
b
+b
=
(b=2)
+ (b=2)
(34)
In this way, equation (31) can be written as:

0
U(;t) =
0

tan
1
s
(b)(1 + cos)
(+b)(1cos)
(35)
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Aerodynamics of the oscillating airfoils

45. Theodorsen Solution
Part 4. Circulatory contribution
The lift produced by the pair of vortices of intensity 0is determined:
L0
=
Z

0
(pUpL)bsind=
U10
p

2
b
2
(36)
It can be noted that for! 1, (i.e.,t! 1) the lift tends to the value produced
by a single vortex of intensity 0.
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Aerodynamics of the oscillating airfoils

46. Theodorsen Solution
Part 4. Circulatory contribution
When we deal with a distribution of wake vorticity, the treatment must be referred
to an element of vorticity:
0=w(;t)d (37)
Now, the velocity is expressed by:
q=
Z
1
b
wd
b
"

2

b
2
4

2
+
b
2
4
bcos
#
(38)
The pressure dierence due to the complete system of wake vorticity is obtained by
pUpL=
U1
bsin
Z
1
b

+bcos
p

2
b
2
!
w(;t)d (39)
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Aerodynamics of the oscillating airfoils

47. Theodorsen Solution
Part 4. Circulatory contribution
By integrating from the trailing edge to innity we nd the complete eect of the
wake vorticity on the lift:
LC=U1
Z
1
b

p

2
b
2
w(;t)d (40)
Theodorsen indicates withQthe following integral:
Q=
1
2b
Z
1
b
s
+b
b
w(;t)d
Then, we can write the circulatory part of the lift as:
LC= 2b U1Q
Z
1
b

p

2
b
2
w(;t)d
Z
1
b
s
+b
b
w(;t)d
(41)
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Aerodynamics of the oscillating airfoils

48. Theodorsen Solution
Part 4. Circulatory contribution
By assuming simple harmonic oscillations in time the wake vorticityw(;t) takes
the form:
w(;t) = we
i!(t

U1
)
(42)
By dening the reduced frequencyk=!b=U1and

==b,
w(;t) = we
i!(tk

)
(43)
The ratio of the integrals can be manipulated as:
Z
1
b

p

2
b
2
w(;t)d
Z
1
b
s
+b
b
w(;t)d
=
Z
1
1


p

2
1
e
ik

d

Z
1
1
s


+ 1


1
e
ik

d

=C(k)
C(k) is a complex function of the reduced frequency only.
C(k) is said Theodorsen's function.
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Aerodynamics of the oscillating airfoils

49. Theodorsen Solution
Part 4. Circulatory contribution
The circulatory lift is
LC= 2b U1Q C(k) (45)
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Aerodynamics of the oscillating airfoils

50. Theodorsen Solution
Road Map.Proof
Boundary
Condition
Conformal
Transformation
Circulatory
Part
Non
Circulatory
Part
Kutta
Condition
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Aerodynamics of the oscillating airfoils

51. Theodorsen Solution
Part 5. Use of Kutta condition
Kutta's condition establishes the velocityqat the trailing edge (= 0) is zero.
By means of this further relation, the integral ratioQcan be computed.
The relations (17) and (29) provide the velocity at T.E.
q(= 0) =
2

Z

0
wa(x;t) sin
2

cos 1
d +
Z
1
b
wd
b
"

2

b
2
4

2
+
b
2
4
b
#
= 0
By taking into account the relation (34) betweenand, the Kutta condition can
be written as:
2

Z

0
wa(x;t) sin
2

cos 1
d +
1
b
Z
1
b
s
+b
b
w(;t)d= 0
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Aerodynamics of the oscillating airfoils

52. Theodorsen Solution
Part 5. Use of Kutta condition
Then,
q=
2

Z

0
wa(x;t) sin
2

cos 1
d 2Q= 0
By substituting the expression ofwa(x;t) in equation (6), we have:
Q=
_
h+U1+b

1
2
a

_ (46)
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2011 52 / 75
Aerodynamics of the oscillating airfoils

53. Theodorsen Solution
Final expression.
By collecting equations (25), (45) and (46) the Theodorsen solution is obtained:
L=b
2

h+U1_a b

+
+ 2b U1C(k)

_
h+U1+b

1
2
a

_

(47)
The expression of the aerodynamic moment is also reported:
M=b
2

bahU1b

1
2
a

_b
2

1
8
+a
2



+ 2U1b
2

a+
1
2

C(k)

_
h+U1+b

1
2
a

_

(48)
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2011 53 / 75
Aerodynamics of the oscillating airfoils

54. Theodorsen Solution
Considerations
In the circulatory part there is an equivalence between
_
handU1.
The body motion must produce small velocities according to the hypothesis of
small disturbances.
As a consequence, the reduced frequency is limited.
The eect of a mean steady angle of attack is taken into account by adding the
steady linear contribution.
Some special cases:
_
h=

h= 0;a=1=2;= e
i!t
;
The lift coecient is
Cl= 2[F(1 +ik) +G(ik)] e
i!t
+

i
k
2

e
i!t
whereF=Re(C) andG=Im(C).
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Aerodynamics of the oscillating airfoils

55. Theodorsen Solution
ConsiderationsF(k)
G(k)
0 0.25 0.5 0.75 1
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
k=0.01
k=0.2
k=1
F!1 ask!0
G!0 ask!0
F!0:5 ask! 1
G!0 ask! 1
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2011 55 / 75
Aerodynamics of the oscillating airfoils

56. Theodorsen Solution
Considerations
Approximated expression of Theodorsen's function.
F=
0:5005k
3
+ 0:51261k
2
+ 0:21040k+ 0:021573
k
3
+ 1:03538k
2
+ 0:25124k+ 0:02151
G=
0:00015k
3
+ 0:12240k
2
+ 0:32721k+ 0:001990
k
3
+ 2:48148k
2
+ 0:93453k+ 0:08932
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2011 56 / 75
Aerodynamics of the oscillating airfoils

57. Theodorsen Solution
Considerationsα
c
l
-10 -5 0 5 10
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
k=0.2
k=1.1
Various Theodorsen solutions.Cl
curves.
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2011 57 / 75
Aerodynamics of the oscillating airfoils

58. Theodorsen Solution
Considerationsα
c
l
-10 -5 0 5 10
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
k=0.2
k=1.1
Theodorsen solutions.Cl
curves.In-phase contribution.
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2011 58 / 75
Aerodynamics of the oscillating airfoils

59. Theodorsen Solution
Considerationsα
c
l
-10 -5 0 5 10
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
k=0.2
k=1.1
Theodorsen solutions.Clcurves.
Out-phase contribution.
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Aerodynamics of the oscillating airfoils

60. CFD of the oscillating airfoils.
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2011 60 / 75
Aerodynamics of the oscillating airfoils

61. CFD of the oscillating airfoils
Real ow around oscillating airfoils
Unsteadiness
Turbulence
Three-dimensional eects (even for airfoils)
Compressibility eects (even for low free stream Mach numbers)
Transition from laminar to turbulence
Vibration and structure deformation
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2011 61 / 75
Aerodynamics of the oscillating airfoils

62. CFD of the oscillating airfoils
Numerical methods.
The numerical methods can be listed
as the accuracy increases.
The numerical simulation around an
airfoil at ight Reynolds number:
(2080) Direct Navier Stokes
(2045) Large Eddy Simulation
(2000) Detached Eddy Simulation
(1995) Unsteady RANS
(1985) RANS
From Spalart 1999, 2000.
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2011 62 / 75
Aerodynamics of the oscillating airfoils

63. CFD of the oscillating airfoils
Numerical methods.
The progress in the prediction of the dynamic stall features is limited by
the following factors:
The numerical simulations are very time-consuming. Diculties in tuning the
parameters.
The experimental data often are not very accurate and detailed in order to make
close comparisons.
The experimental measures of the dynamic stall are made complex by the presence
of the mechanical devices for the oscillatory motion.
The pressure taps give reliable values only for the lift and not for the drag and
moment
Other techniques, PIV, LDV, provide more information but far from the solid wall
There no information about the presence of laminar separation bubble, transition
location ...
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2011 63 / 75
Aerodynamics of the oscillating airfoils

64. CFD of the oscillating airfoils
Numerical methods.
Some examples of dynamic stall with the CIRA CFD code, ZEN.
Geometry NACA0012
Re= 1:3510
5
2D Grid, 153600 cells.
!SST Menter.
No model for the transition.
Moving grid technique.
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Aerodynamics of the oscillating airfoils

65. CFD of the oscillating airfoils
Applications. Pre stallα
c
l
-10 -5 0 5 10
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Re= 1:3510
5
,M= 0:1.
= 7:5

sin(2kt),k= 0:05
Clcurve
Solid Line, unsteady RANS
Dashed Line, Theodorsen
.avi
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2011 65 / 75
Aerodynamics of the oscillating airfoils

66. CFD of the oscillating airfoils
Applications. Light stallα
c
l
-10 -5 0 5 10 15 20
-0.40
0.00
0.40
0.80
1.20
1.60
2.00
Re= 1:3510
5
,M= 0:1.
= 5

+ 10

sin(2kt),k= 0:05
Clcurve
Solid Line, unsteady RANS
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Aerodynamics of the oscillating airfoils

67. CFD of the oscillating airfoils
Applications. Light stallα
c
d
-8 -4 0 4 8 12 16 20
-0.10
0.00
0.10
0.20
0.30
0.40
0.50
0.60
Re= 1:3510
5
,M= 0:1.
= 5

+ 10

sin(2kt),k= 0:05
Cdcurve
Solid Line, unsteady RANS
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Aerodynamics of the oscillating airfoils

68. CFD of the oscillating airfoils
Applications. Light stallα
c
m
-10 -5 0 5 10 15 20
-0.50
-0.40
-0.30
-0.20
-0.10
0.00
0.10
0.20
Re= 1:3510
5
,M= 0:1.
= 5

+ 10

sin(2kt),k= 0:05
Cmcurve
Solid Line, unsteady RANS
.avi
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Aerodynamics of the oscillating airfoils

69. CFD of the oscillating airfoils
Applications. Deep stallα
c
l
0 5 10 15 20
0.00
0.25
0.50
0.75
1.00
1.25
1.50
Re= 1:3510
5
,M= 0:3.
= 10

+ 5

sin(2kt),k= 0:5
Clcurve
Solid Line, unsteady RANS
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Aerodynamics of the oscillating airfoils

70. CFD of the oscillating airfoils
Applications. Deep stallt
c
l
5 10 15 20 25
0
0.25
0.5
0.75
1
1.25
1.5
1.75
Re= 1:3510
5
,M= 0:3.
= 10

+ 5

sin(2kt),k= 0:5
Cltime curve
Solid Line, unsteady RANS
dashed, LES (from Nagarayan et al.
2006)
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Aerodynamics of the oscillating airfoils

71. CFD of the oscillating airfoils
Applications. Deep stallx/c
c
p
0 0.2 0.4 0.6 0.8 1
-4
-3
-2
-1
0
1
2
Re= 1:3510
5
,M= 0:3.
= 10

+ 5

sin(2kt),k= 0:5
CPdistribution on the airfoil.
Solid Line, unsteady RANS
, LES (from Nagarayan et al. 2006)
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Aerodynamics of the oscillating airfoils

72. CFD of the oscillating airfoils
Applications. Deep stallx/c
c
p
0 0.2 0.4 0.6 0.8 1
-4
-3
-2
-1
0
1
2
Re= 1:3510
5
,M= 0:3.
= 10

+ 5

sin(2kt),k= 0:5
CPdistribution on the airfoil.
Solid Line, unsteady RANS
, LES (from Nagarayan et al. 2006)
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Aerodynamics of the oscillating airfoils

73. CFD of the oscillating airfoils
Applications. Deep stallx/c
c
p
0 0.2 0.4 0.6 0.8 1
-4
-3
-2
-1
0
1
2
Re= 1:3510
5
,M= 0:3.
= 10

+ 5

sin(2kt),k= 0:5
CPdistribution on the airfoil.
Solid Line, unsteady RANS
, LES (from Nagarayan et al. 2006)
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Aerodynamics of the oscillating airfoils

74. CFD of the oscillating airfoils
Applications. Deep stallx/c
c
p
0 0.2 0.4 0.6 0.8 1
-4
-3
-2
-1
0
1
2
SST
LES
Re= 1:3510
5
,M= 0:3.
= 10

+ 5

sin(2kt),k= 0:5
CPdistribution on the airfoil.
Solid Line, unsteady RANS
, LES (from Nagarayan et al. 2006)
.avi
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2011 74 / 75
Aerodynamics of the oscillating airfoils

75. Conclusions
In case of ideal ow, the aerodynamic theories provide useful explanations of the
airfoil behavior.
When the angular amplitudes are wide enough (dynamic stall) there is no exact
theory able to predict the aerodynamics.
The analysis of the dynamic stall can be made only by CFD and by experimental
measurements.
Currently the dynamic stall is object of many industrial researches for its strong
technological impact.
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Aerodynamics of the oscillating airfoils