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MAE 3241: AERODYNAMICS ANDFLIGHT MECHANICS
Introduction to Lifting Line Theory
April 11, 2011
Mechanical and Aerospace Engineering DepartmentFlorida Institute of Technology
D. R. Kirk
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NECESSARY TOOL
Return to vortex filament, which in general maybe curved
General treatment accomplished with Biot-Savart Law
34 r
rdl
dV
Electromechanical Analogy:Think of vortex filament as a wire carrying an electrical current I
The magnetic field strength, dB, induced at point P by segment dl is:
3
4 r
rdlIdB
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3
EXAMPLE APPLICATIONS
h
V
4
h
V
2
Case 1: Biot-Savart Law applied between
Case 2: Biot-Savart Law applied between fixed point A and 34 r
rdldV
Case 1 Case 2
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4
BIOT-SAVART LAW
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5
EXAMPLE APPLICATIONS
Case 1:
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HELMHOLTZS VORTEX THEOREMS
1. The strength of a vortex filament is constant along its length
2. A vortex filament cannot end in a fluid; it must extend to boundaries of fluid
(which can be ) or form a closed pathNote: Statement that vortex lines do not end in the fluid is kinematic, due to
definition of vorticity, w, (orxin Anderson) and totally general
We will use Helmholtzs vortex theorems for calculation of lift distribution which
will provide expressions for induced drag
L=L(y)=rV(y)
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CONSEQUENCE: ENGINE INLET VORTEX
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CHAPTER 4: AIRFOILEach is a vortex line
One each vortex line 1=constant
Strength can vary from line to line
Along airfoil, g=g(s)
Integrations done:
Leading edge to
Trailing edge
z/c
x/c
Side viewEntire airfoil has
14 7
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CHAPTER 5: WINGS
http://www.airliners.net/open.file?id=790618&size=L&sok=JURER%20%20%28ZNGPU%20%28nvepensg%2Cnveyvar%2Ccynpr%2Ccubgb_qngr%2Cpbhagel%2Cerznex%2Ccubgbtencure%2Crznvy%2Clrne%2Cert%2Cnvepensg_trarevp%2Cpa%2Cpbqr%29%20NTNVAFG%20%28%27%2B%22777%22%27%20VA%20OBBYRNA%20ZBQR%29%29%20%20beqre%20ol%20cubgb_vq%20QRFP&photo_nr=3417/23/2019 Introduction to Lifting Line theory
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PRANDTLS LIFTING LINE THEORY
Replace finite wing (span = b) with bound vortex filament extending from y = -b/2
to y = b/2 and origin located at center of bound vortex (center of wing)
Helmholtzs vorticity theorem: A vortex filament cannot end in a fluid
Filament continues as two free vorticies trailing from wing tips to infinity
This is called a Horseshoe Vortex
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PRANDTLS LIFTING LINE THEORY
Trailing vorticies induce velocity along bound vortex with both contributions in
downward direction (w is in negative z-direction)
22
2
4
24
24
4
yb
b
yw
yb
yb
yw
hV
Contribution from left trailing vortex
(trailing fromb/2)
Contribution from right trailing vortex
(trailing from b/2)
This has problems: It does not simulate downwash distribution of a real finite wing
Problem is that as y b/2, w
Physical basis for solution: Finite wing is not represented by uniform single bound
vortex filament, but rather has a distribution of(y)
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PRANDTLS LIFTING LINE THEORY
Represent wing by a large number of horseshoe vorticies, each with different
length of bound vortex, but with all bound vorticies coincident along a single line
This line is called the Lifting Line
Circulation, , varies along line of bound vorticies
Also have a series of trailing vorticies distributed over span
Strength of each trailing vortex = change in circulation along lifting line
Instead of=constant
We need a way to let =(y)
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PRANDTLS LIFTING LINE THEORY
Example shown here will use 3 horseshoe vorticies
d1
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PRANDTLS LIFTING LINE THEORY
d1
d2
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PRANDTLS LIFTING LINE THEORY
d1
d2d3
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PRANDTLS LIFTING LINE THEORY
Represent wing by a large number of horseshoe vorticies, each with differentlength of bound vortex, but with all bound vorticies coincident along a single line
This line is called the Lifting Line
Circulation, , varies along line of bound vorticies Also have a series of trailing vorticies distributed over span
Strength of each trailing vortex = change in circulation along lifting line
Example shown here uses 3 horseshoe vorticies
Consider infinite number of horseshoe vorticies superimposed on lifting line
d1
d2d3
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PRANDTLS LIFTING LINE THEORY
Infinite number of horseshoe vorticies superimposed along lifting line
Now have a continuous distribution such that = (y), at origin = 0
Trailing vorticies are now a continuous vortex sheet (parallel to V)
Total strength integrated across sheet of wing is zero
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PRANDTLS LIFTING LINE THEORY
Consider arbitrary location y0 along lifting line
Segment dx will induce velocity at y0 given by Biot-Savart law
Velocity dw at y0 induced by semi-infinite trailing vortex at y is:
Circulation at y is (y)
Change in circulation over dy is d = (d/dy)dy
Strength of trailing vortex at y = d along lifting line
yy
dydy
d
dw
04
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PRANDTLS LIFTING LINE THEORY
Total velocity w induced at y0 by entire trailing vortex sheet can be found by
integrating fromb/2 to b/2:
2
2 0
04
1b
b
dyyy
dy
d
yw
Equation gives value of
downwash at y0 due to
all trailing vorticies
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SUMMARY SO FAR
Weve done a lot of theory so far, what have we accomplished?
We have replaced a finite wing with a mathematical model
We did same thing with a 2-D airfoil
Mathematical model is called a Lifting Line
Circulation (y) varies continuously along lifting line
Obtained an expression for downwash, w, below the lifting line
We want is an expression so we can calculate (y) for finite wing (WHY?)
Calculate Lift, L (Kutta-Joukowski theorem)
Calculate CL
Calculate aeff
Calculate Induced Drag, CD,i (drag due to lift)
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FINITE WING DOWNWASH
Recall: Wing tip vortices induce a downward component of air velocity near wing
by dragging surrounding air with them
2
20
04
1 b
b
i dyyy
dy
d
Vy
a
ai
V
ywy
Vywy
i
i
0
0
010 tan
a
a
Equation for induced angle of attack
along finite wing in terms of(y)
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EFFECTIVE ANGLE OF ATTACK, aeff, EXPRESSION
0
0
0
00
0
0
00
2
00000
0
2
2
2
1
2
Leff
Leffl
l
l
LeffLeffl
effeff
ycV
y
yc
ycV
yc
yVcycVL
yyac
y
a
a
aa
rr
aaaa
aa aeffseen locally by airfoilRecall lift coefficient
expression (Ref, EQ: 4.60)
a0 = lift slope = 2
Definition of lift coefficient
and Kutta-Joukowski
Related both expressions
Solve foraeff
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COMBINE RESULTS FOR GOVERNING EQUATION
2
20
0
0
0
0
2
2
0
0
0
0
0
4
1
4
1
b
b
L
ieff
b
b
i
Leff
dyyy
dy
d
VycV
yy
dyyy
dy
d
Vy
ycV
y
a
a
aaa
a
a
a
Effective angle of attack
(from previous slide)
Induced angle of attack
(from two slides back)
Geometric angle of attack= Effective angle of attack+ Induced angle of attack
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PRANDTLS LIFTING LINE EQUATION
Fundamental Equation of Prandtls Lifting Line Theory
In Words: Geometric angle of attack is equal to sum of effective angle of
attack plus induced angle of attack
Mathematically: a = aeff+ ai
Only unknown is (y)
V, c, a, aL=0 are known for a finite wing of given design at a given a
Solution gives (y0), whereb/2 y0 b/2 along span
2
20
0
0
00
4
1b
b
L dyyy
dy
d
VycV
yy
a
a
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WHAT DO WE GET OUT OF THIS EQUATION?
1. Lift distribution
2. Total Lift and Lift Coefficient
3. Induced Drag
dyyy
SVSq
DC
dyyyVdyyyLD
LD
dyySVSq
LC
dyyVL
dyyLL
yVyL
b
b
ii
iD
i
b
b
i
b
b
i
iii
b
b
L
b
b
b
b
2
2
,
2
2
2
2
2
2
2
2
2
2
00
2
2
a
ara
a
r
r
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ELLIPTICAL LIFT DISTRIBUTION
For a wing with same airfoil shape across span and no twist, an elliptical
lift distribution is characteristic of an elliptical wing planform
AR
CC
ARC
LiD
Li
a
2
,
SPECIAL SOLUTION
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SPECIAL SOLUTION:
ELLIPTICAL LIFT DISTRIBUTION
Points to Note:
1. At origin (y=0) =0
2. Circulation varies elliptically with distance y along span
3. At wing tips (-b/2)=(b/2)=0
Circulation and Lift 0 at wing tips
2
0
2
0
21
21
b
yVyL
b
yy
r
SPECIAL SOLUTION
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SPECIAL SOLUTION:
ELLIPTICAL LIFT DISTRIBUTION
Elliptic distribution
Equation for downwash
Coordinate transformation q
See reference for integral
bVV
w
bw
db
w
db
dyb
y
dy
yy
b
y
y
byw
by
y
bdy
d
i
b
b
2
2
coscos
cos
2
sin2
;cos2
41
41
4
0
0
0
0 0
0
0
2
20
21
2
22
00
2
22
0
a
q
qqq
q
q
qqq
Downwash is constant over span for an elliptical lift distribution
Induced angle of attack is constant along span
Note: w and ai 0 as b
SPECIAL SOLUTION
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SPECIAL SOLUTION:
ELLIPTICAL LIFT DISTRIBUTION
AR
CC
dyySV
C
AR
C
S
b
AR
b
SC
bVdy
b
yVL
LiD
b
b
iiD
Li
Li
b
b
a
a
a
rr
2
,
2
2
,
2
2
0
2
2
21
2
2
0
2
4
41
CD,i is directly proportional to square of CL
Also called Drag due to Lift
We can develop a more
useful expression forai
Combine L definition for elliptic
profile with previous result forai
Define AR because it
occurs frequently
Useful expression forai
Calculate CD,i
SUMMARY TOTAL DRAG ON SUBSONIC WING
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SUMMARY: TOTAL DRAG ON SUBSONIC WING
eAR
CcSq
DcC
DDDDDDD
Lprofiled
iprofiledD
inducedprofile
inducedpressurefriction
2
,,
Also called drag due to lift
Profile Drag
Profile Drag coefficient
relatively constant withM at subsonic speeds
Look up
(Infinite Wing)
May be calculated from
Inviscid theory:
Lifting line theory
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SUMMARY
Induced drag is price you pay for generation of lift
CD,i proportional to CL2
Airplane on take-off or landing, induced drag major component
Significant at cruise (15-25% of total drag)
CD,i
inversely proportional to AR
Desire high AR to reduce induced drag
Compromise between structures and aerodynamics
AR important tool as designer (more control than span efficiency, e)
For an elliptic lift distribution, chord must vary elliptically along span
Wing planform is elliptical
Elliptical lift distribution gives good approximation for arbitrary finite wing
through use of span efficiency factor, e
WHAT IS NEXT?
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WHAT IS NEXT? Lots of theory in these slides Reinforce ideas with relevant examples
We have considered special case of elliptic lift distribution
Next step: develop expression for general lift distribution for arbitrary wing shape
How to calculate span efficiency factor, e
Further implications of AR and wing taper
Swept wings and delta wings
New A380:Wing is tapered and swept
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