Lifting Line Theory
AOE 5104Advanced Aero- and Hydrodynamics
Dr. William Devenport andLeifur Thor Leifsson
2
Flow over a low aspect ratio wing
Bottom surface
Top surface
3
Lifting Line Theory• Applies to large aspect ratio unswept wings at small angle of attack.• Developed by Prandtl and Lanchester during the early 20th century. • Relevance
– Example of a phenomenological model in which elementary ideal flows are used to represent real viscous phenomena
– Basis of much of modern wing theory (e.g. helicopter rotor aerodynamic analysis, extends to vortex lattice method,)
– Basis of much of the qualitative understanding of induced drag and aspect ratio
1875-19531868-1946Γ h
Vπ4Γ
=
h
Biot Savart Law:Velocity produced by a semi-infinite segment of a vortex filament
4
Large Aspect Ratio? Unswept?
5
6
Physics of an Unswept Wing
Head, 1982, Rectangular Wing, 24o, Re=100000
Werle, 1974, NACA 0012, 12.5o, AR=4, Re=10000
Bippes, Clark Y, Rectangular Wing 9o, AR=2.4, Re=100000
7
Cessna Citation
Aircraft DataVelocity = 165 knots Wing Area = 29 m^2 Wing Span = 16 m
Mean Aerodynamic Chord = 2 m Weight = 8000 kg
Chord Reynolds Number = 1.18E7
8
Out
was
h
Physics of an Unswept Wingl, Γ
y
pu<pl pu≈ pl
Downwash
Inwash
s-s
Lift varies across span
Circulation is shed (Helmholz thm)
Vortical wake
Vortical wake induces downwash on wing…
…changing angle of attack just enough to produce variation of lift across span
9
Wing Nomenclature
l, Γ
y
Section A-A
αε
V∞
-w
l
ε
di
s-s
c
α=α (y) Geometric angle of attackε=ε (y) Downwash angle-w=-w (y) Downwash velocityc=c (y) ChordlengthS Planform areas Half spanb Spanc = S/b Average chordAR=b/c=b2/S Aspect ratioΛ Sweep angle of ¼ chord linel Lift per unit spandv Drag per unit span
Induced drag
bs
Area S
Total drag coeff
SVLCL 2
21
∞
=ρ
SVDC i
Di 221
∞
=ρ
Total lift coeff
A
A
10
LLT – The Wake ModelΓ
y1
ydy1
Strength of vortex shed at y1=
• Assume role up of wake unimportant• Assume wake remains in a plane parallel to the free stream• Model wake using single vortex sheet starting at the quarter chord
Downwash at y due to vortex shed at y1 )(4
)(1
11
yy
dydyd
ydw y
−
Γ−=−
π
Downwash at ydue to entire wake ∫
− −
Γ
=s
s
y
yy
dydyd
yw)(4
)(1
11
π
s-s
1
1
dydyd
y
Γ−
dΓ
11
LLT – The Section Model
αε
V∞
-w
l
ε
di
=Γ
==∞
∞
∞ cVV
cVlCl 2
212
21 ρ
ρρ
wccV πααπ +−=Γ ∞ )( 0
Sectional lift coefficient
• Assume flow over each section 2D and determined by downwash at ¼ chord, and thin airfoil theory
So,
Sectional forces Γ≈ ∞Vl ρ Γ−≈Γ≈ ∞ wVdv ρερ
∫−
∞ Γ≈s
s
dyVL ρ ∫−
Γ−≈s
si dywD ρ
∫−∞∞
Γ≈=s
sL dy
SVSVLC 2
221 ρ ∫
−∞∞
Γ≈=s
s
iD dyw
SVSVDC
i 2221
2ρ
Total Forcesintegrated over span
Total Coefficients
( )εααπ −− 02
12
The Monoplane EquationWake model
Section model∫− −
Γ
=s
s
y
yy
dydyd
yw)(4
)(1
11
π
wccV πααπ +−=Γ ∞ )( 0
l, Γ
ys-s
∫−
∞ −
Γ
+−=Γs
s
y
yy
dydyd
ccV1
1
01
4)( ααπ
0 π θ
θcos/ −=sy
Substitute for θ, and express Γ as a sine series in θ
∑∞
=∞=Γ
oddnn nAsU
,1)sin(4 θ
⎥⎦⎤
⎢⎣⎡ +=− ∑
∞
=
θπθθααπ sin4
)sin(sin)(4 ,1
0 scnnA
sc
oddnn The Monoplane Eqn.
13
Solving the Monoplane Eqn.: Glauert’s method
⎥⎦⎤
⎢⎣⎡ +=− ∑
∞
=
θπθθααπ sin4
)sin(sin)(4 ,1
0 scnnA
sc
oddnn
ys-s0 π θ
θcos/ −=sy
1. Decide on the number of terms Nneeded for the sine series for Γ
2. Select N points across the half span, evenly spaced in θ
3. At each point evaluate c, α, α0 and write down the corresponding monoplane equation
4. Solve the resulting N equations for the N unknown coefficient An
5. Evaluate Γ(y) and then the integrals for w(y), L and Di, and thus the aero coefficients
∑∞
=∞=Γ
oddnn nAsU
,1)sin(4 θ
∫− −
Γ
=s
s
y
yy
dydyd
yw)(4
)(1
11
π
∫−∞
Γ=s
sL dy
SVC 2
∫−∞
Γ=s
sD dyw
SVC
i 2
2
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Glauert’s Method Example
⎥⎦⎤
⎢⎣⎡ +=− ∑
∞
=
θπθθααπ sin4
)sin(sin)(4 ,1
0 scnnA
sc
oddnn
ys-s0 π θ
θcos/ −=sy
∑∞
=∞=Γ
oddnn nAsU
,1)sin(4 θ
-10 -8 -6 -4 -2 02
4
6
8
10
12
∫− −
Γ
=s
s
y
yy
dydyd
yw)(4
)(1
11
π
∫−∞
Γ=s
sL dy
SVC 2
∫−∞
Γ=s
sD dyw
SVC
i 2
2
15
Other results
∫− −
Γ
=s
s
y
yy
dydyd
yw)(4
)(1
11
π∫−∞
Γ=s
sL dy
SVC 2 ∫
−∞
Γ=s
sD dyw
SVC
i 2
2
∑∞
=∞=Γ
oddnn nAsU
,1)sin(4 θSubstituting
into
gives 1AARCL π=
)1(2
δπ
+=ARCC L
Di
∑∞
=
=oddn
n AAn,3
21)/(δ θ
θ
sin
)sin(,1∑∞
=
∞
−= oddnn nnA
Vw
• Lift increases with aspect ratio
• For planar wings at least lift goes linearly with angle of attack and lift curve slope increases with aspect ratio (to 2π at ∞) ?
• Drag decreases with aspect ratio and goes as the lift squared.
• Downwash tends to be largest at the wing tips ?
• Drag is minimum for a wing for which An=0 for n≥3 ?
So,
α
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The Elliptic WingSo the minimum drag occurs for a wing for which An=0 for n≥3. For this wing:
)sin(4 1 θsAU∞=Γ
1AVw
−=∞ AR
CC LDi π
2
=
)/arccos( sy−=θ14
22
1
=⎟⎠⎞
⎜⎝⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛ Γ⇒
∞ sy
sAVsince
andπααπ 10 )( AVV
c∞∞ −−
Γ=
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