Aerodynamics AERO 464 - Winter 2022
Lecture 8: Thin-airfoil theory
Instructor: ©Dr. Mojtaba Kheiri
Department of Mechanical, Industrial and Aerospace Engineering
Concordia University
March 15, 2022
Montréal, Canada
Thin-airfoil theory Symmetric airfoils Cambered airfoils
Classical thin airfoil theory
Approach: a vortex sheet is placed along the camber line of the airfoil,
and the goal is to calculate the strength of the vortex sheet, γ(s), such
that the camber line becomes a streamline of the flow and such that the
Kutta condition is satisfied at the trailing edge, i.e. γ(TE) = 0. Once we
have found the appropriate γ(s), the total circulation Γ around the airfoil
is found by integrating γ(s) from the L.E. to T.E. Knowing Γ, the lift is
then calculated via the Kutta-Joukowski theorem.
Fig. 1 Vortex sheet placed on the camber line
Ref: Anderson Jr, J.D., 2017. Fundamentals of Aerodynamics. Sixth Ed., McGraw-Hill Education.
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Thin-airfoil theory Symmetric airfoils Cambered airfoils
Classical thin airfoil theory
Approach: a vortex sheet is placed along the chord line of the airfoil,
and the goal is to calculate the strength of the vortex sheet, γ(x ), such
that the camber line becomes a streamline of the flow and such that the
Kutta condition is satisfied at the trailing edge, i.e. γ(c) = 0. Once we
have found the appropriate γ(x ), the total circulation Γ around the airfoil
is found by integrating γ(x ) from the L.E. to T.E. Knowing Γ, the lift is
then calculated via the Kutta-Joukowski theorem.
Fig. 2 Vortex sheet placed on the chord line
Ref: Anderson Jr, J.D., 2017. Fundamentals of Aerodynamics. Sixth Ed., McGraw-Hill Education.
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Thin-airfoil theory Symmetric airfoils Cambered airfoils
Thin-airfoil theory implementation: V∞,n
For the camber line to be a streamline,
the component of flow velocity normal to
the camber line must be zero at all points
along the camber line: V∞,n + w ′ (s) =
0, where V∞,n is the component of the
freestream velocity normal to the camber
line; w ′ (s) is the component of velocity
normal to the camber line induced by the Fig. 3 Finding the component of freestream velocity
normal to camber line (Anderson 2017)
vortex sheet
Referring to above figure, V∞,n is obtained as
h
dz i
V∞,n = V∞ sin α + arctan(− ) ,
dx
which for a thin airfoil and small α reduces to
dz V∞,n = V∞ α −
.
dx
Ref: Anderson Jr, J.D., 2017. Fundamentals of Aerodynamics. Sixth Ed., McGraw-Hill Education.
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Thin-airfoil theory Symmetric airfoils Cambered airfoils
Thin-airfoil theory implementation: w ′ (s)
It is consistent with thin airfoil theory
to make the approximation that w ′ (s) ≈
w (x )
The velocity induced at point x by the
elemental vortex of the strength γdξ located at point ξ is written as
γ(ξ)dξ
.
dw = −
2π(x − ξ)
Fig. 4 Finding the induced velocity at the chord line
(Anderson 2017)
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The total induced velocity at point x , that is induced by all the elemental
vortices along the chord line, is obtained by integrating equation (3) from the
leading edge (ξ = 0) to the trailing edge (ξ = c) as
Z c
γ(ξ)dξ
w (x ) = −
.
(4)
2π(x
− ξ)
0
Ref: Anderson Jr, J.D., 2017. Fundamentals of Aerodynamics. Sixth Ed., McGraw-Hill Education.
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Thin-airfoil theory Symmetric airfoils Cambered airfoils
The fundamental equation of thin airfoil theory
The fundamental equation of thin airfoil theory is obtained by applying
the condition that the flow velocity normal to the chord line must be zero
at every point along the chord line:
1
2π
Z c
γ(ξ)dξ
0
x −ξ
subject to the Kutta condition, γ(c) = 0.
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= V∞ α −
dz ,
dx
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Thin-airfoil theory Symmetric airfoils Cambered airfoils
Thin-airfoil theory: symmetric airfoils
For symmetric airfoils: dz/dx = 0.
c
We define a new variable θ: ξ = (1 − cos θ), where θ varies between 0
2
and π
c
At point x , θ = θ0 ⇒ x = (1 − cos θ0 )
2
1 R π γ(θ) sin θ dθ
The thin-airfoil theory becomes:
= V∞ α
2π 0 cos θ − cos θ0
The closed-form solution to the above equation is written as
γ(θ) = 2αV∞
1 + cos θ
.
sin θ
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It can easily be shown that the Kutta condition is satisfied, i.e. γ(π) = 0
The Rtotal circulationR around the airfoil is obtained from
c
π
Γ = 0 γ(ξ)dξ = c2 0 γ(θ) sin θdθ = παcv∞
2
From the Kutta-Joukowski theorem, we find L′ = ρ∞ V∞ Γ = παcρ∞ V∞
Ref: Anderson Jr, J.D., 2017. Fundamentals of Aerodynamics. Sixth Ed., McGraw-Hill Education.
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Thin-airfoil theory Symmetric airfoils Cambered airfoils
Thin-airfoil theory: symmetric airfoils (Cont.)
The lift coefficient
is obtained as:
L′
cl = 1
= 2πα
2
2 ρ∞ V∞ c × 1
The lift coefficient is linearly
proportional to angle of attack –
this is supported by experimental
results; see Fig. 6
The theoretical lift slope is 2π
rad−1 , which is 0.11 degree−1
Fig. 5 Comparison between theory and experiment for
the lift and moment coefficients for NACA 0012 airfoil
(Anderson 2017)
The total moment about the leading
edge (per unit
Rc
R c span) due to entire vortex
′
sheet is written as: MLE
= − 0 ξdL′ = −ρ∞ V∞ 0 ξγ(ξ) dξ; the pitching
moment about L.E. becomes: cm,le = −cl /4; the pitching moment about the
quarter-chord point is obtained from: cm,c/4 = cm,le + c4l = 0
Theoretically, the quarter-chord point is both the center of pressure and
the aerodynamic center for a symmetric airfoil.
Ref: Anderson Jr, J.D., 2017. Fundamentals of Aerodynamics. Sixth Ed., McGraw-Hill Education.
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Thin-airfoil theory Symmetric airfoils Cambered airfoils
Thin-airfoil theory: cambered airfoils
In this case, the expression of the vortex sheet strength becomes more complex:
which results in
∞
1 + cos θ X
γ(θ) = 2V∞ A0
+
An sin nθ ,
sin θ
n=1
(7)
∞
X
dz
= (α − A0 ) +
An cos nθ0 .
dx
n=1
(8)
From equations for a Fourier cosine series, we can obtain:
Z
1 π dz
A0 = α −
dθ0 ,
π
dx
Z π 0
2
dz
An =
cos nθ0 dθ0 ,
π 0 dx
where dz/dx is a function of θ0 .
Ref: Anderson Jr, J.D., 2017. Fundamentals of Aerodynamics. Sixth Ed., McGraw-Hill Education.
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Thin-airfoil theory Symmetric airfoils Cambered airfoils
Thin-airfoil theory: cambered airfoils (Cont.)
The total circulation due to entire vortex sheet from the leading edge to the
trailing edge is
Z c
Z
c π
π γ(θ) sin θ dθ = cV∞ πA0 + A1 .
Γ=
γ(ξ)dξ =
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2 0
2
0
From the above equation and using the Kutta-Joukowski theorem, L′ is
obtained as:
π 2
L′ = ρ∞ V∞ Γ = ρ∞ V∞
c πA0 + A1 .
2
The lift coefficient cl is obtained as
Z
i
h
1 π dz
(cos θ0 − 1) dθ0 .
cl = 2π α +
π 0 dx
The pitching moment coefficient about the leading edge becomes:
hc
i
π
l
cm,le = −
+ (A1 − A2 ) .
4
4
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Thin-airfoil theory Symmetric airfoils Cambered airfoils
Thin-airfoil theory: cambered airfoils (Cont.)
The pitching moment coefficient about the quarter-chord point is simply
obtained from the fact that cm,c/4 = (cl /4) + cm,le :
cm,c/4 =
π
(A2 − A1 ).
4
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Unlike the symmetric airfoil, where cm,c/4 = 0, the above equation shows that
cm,c/4 is finite for a cambered airfoil, and thus, the quarter chord is not the
center of pressure for a cambered airfoil.
However, since A1 and A2 are independent of the angle of attack, cm,c/4 is
independent of α, and thus, the quarter-chord point is the theoretical location
of the aerodynamic center for a cambered airfoil.
The location of the center of pressure can be obtained as
xcp = −
i
′
MLE
cm,le c
ch
π
=
−
=
1
+
(A
−
A
)
.
1
2
L′
cL
4
cl
Ref: Anderson Jr, J.D., 2017. Fundamentals of Aerodynamics. Sixth Ed., McGraw-Hill Education.
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