Prandtl’s Lifting Line Introduction
Assumptions:
• 3-D steady potential flow (inviscid & irrotational)
• Incompressible
• High aspect ratio wing
• Low sweep
• Small crossflow (along span)
⇒ flow looks like 2-D flow locally with α adjusted
Outputs:
• Total lift and induced drag estimates
• Rolling moment
• Lift distribution along span
• Basic scaling of CDi with C L & geometry
2
CL2
⇒ dominated by A , where A = b
CDi =
S
π Ae
L

 q S
Di
=  2∞ 
q∞ S
πb Se
⇒ Di =
2
1  L
L2
=
2
π b q∞ e π q∞ e  b 
1  L
Di =
π q∞ e  b 
2
2
In steady level flight where L = W , we have the following options for reducing Di :
•
•
•
Raise q ∞ (i.e. raise cruise velocity) ⇐ Friction increases
W
Decrease span loading,
⇐ W & b coupled due to structures
b
Improve wing efficiency, e ⇐Can be difficult
Prandtl’s Lifting Line Introduction
Geometry & Basic Definitions for Lifting Line
y=-b/2
y=b/2
c(y)
y
Cl(αeff,y)
x
•
•
Chord is variable, c = c( y )
Angle of attack is variable, α = α ( y ) and is a sum of two pieces
α
( y) = α
N
N∞ + α g ( y )
local
α
•
freestream
α
N
geometric
twist
Effective angle of attack is modified by downwash from trailing vortices
α eff ( y ) = α ( y ) − α
i ( y)
N
induced
α
Note: αi > 0 for downwash
•
Local lift coefficient linear with α eff
Cl = ao ( y )[α eff ( y ) − α LO ( y )]
Fundamental Lifting Line Equation
Basic model results by:
•
Assuming Kutta-Joukowsky locally gives L ′ :
L′( y ) = ρ ∞V∞ Γ( y )
L′( y )
2 Γ( y )
⇒ Cl ( y ) =
=
q∞ c( y ) V∞ c( y )
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Prandtl’s Lifting Line Introduction
Distributing infinitely many horseshoe vortices along wing c -line to model
4
induced flow:
y
x
Γ(x)
Cl = ao ( y ) α eff ( y ) − α LO ( y )  =
2 Γ( y )
V∞ c( y )
= ao ( y ) [α ( y ) − α i ( y ) − α LO ( y )] =
α ( y) =
2 Γ( y )
V∞ c( y )
2 Γ( y )
+ α LO ( y ) + α i ( y )
ao ( y )V∞ c( y )
where α i ( y ) = −
1
w( y )
=
4πV∞
V∞
b
2
∫
−
b
2
dΓ
( yo )dyo
dy
y − yo
Note: only unknown is Γ( y ) !
We use a Fourier series to solve this.
N
b
Γ = 2bV∞ ∑ An sin nθ where y = − cos θ
2
n =1
Thus, the governing equation is:
α (θ ) =
N
N
4b
sin nθ
A
n
nAn
θ
+
α
θ
+
sin
(
)
∑
∑
n
LO
ao (θ )c(θ ) n =1
θ
sin
n =1
αi (θ )
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