1 mae 3241: aerodynamics and flight mechanics introduction to lifting line theory april 11, 2011...
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MAE 3241: AERODYNAMICS ANDFLIGHT MECHANICS
Introduction to Lifting Line Theory
April 11, 2011
Mechanical and Aerospace Engineering Department
Florida 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
rdldV
Electromechanical Analogy:Think of vortex filament as a wire carrying an electrical current IThe magnetic field strength, dB, induced at point P by segment dl is:
34 r
rdlIdB
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EXAMPLE APPLICATIONS
hV
4
hV
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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BIOT-SAVART LAW
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EXAMPLE APPLICATIONS
Case 1:
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HELMHOLTZ’S VORTEX THEOREMS1. 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 path
Note: Statement that “vortex lines do not end in the fluid” is kinematic, due to definition of vorticity, , (or in Anderson) and totally general
• We will use Helmholtz’s vortex theorems for calculation of lift distribution which will provide expressions for induced drag
L’=L’(y)=∞V∞(y)
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CONSEQUENCE: ENGINE INLET VORTEX
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CHAPTER 4: AIRFOILEach is a vortex lineOne each vortex line =constantStrength can vary from line to lineAlong airfoil, =(s)
Integrations done:Leading edge toTrailing edge
z/c
x/c
Side viewEntire airfoil has
14 7
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PRANDTL’S 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)
• Helmholtz’s 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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PRANDTL’S LIFTING LINE THEORY• Trailing vorticies induce velocity along bound vortex with both contributions in
downward direction (w is in negative z-direction)
2
2
2
4
24
24
4
yb
byw
yb
yb
yw
hV
Contribution from left trailing vortex(trailing from –b/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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PRANDTL’S 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 =constantWe need a way to let =(y)
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PRANDTL’S LIFTING LINE THEORY
• Example shown here will use 3 horseshoe vorticies
d1
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PRANDTL’S LIFTING LINE THEORY
d1
d2
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PRANDTL’S LIFTING LINE THEORY
d1
d2
d3
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PRANDTL’S 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
• Example shown here uses 3 horseshoe vorticies→ Consider infinite number of horseshoe vorticies superimposed on lifting line
d1
d2
d3
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PRANDTL’S LIFTING LINE THEORY
• Infinite number of horseshoe vorticies superimposed along lifting line
– Now have a continuous distribution such that = (y), at origin =
• Trailing vorticies are now a continuous vortex sheet (parallel to V∞)
– Total strength integrated across sheet of wing is zero
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PRANDTL’S 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
dydyd
dw
04
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PRANDTL’S LIFTING LINE THEORY
• Total velocity w induced at y0 by entire trailing vortex sheet can be found by integrating from –b/2 to b/2:
2
20
0 4
1b
b
dyyy
dyd
yw
Equation gives value ofdownwash at y0 due toall trailing vorticies
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SUMMARY SO FAR• We’ve 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 eff
– 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
0 4
1b
bi dy
yy
dyd
Vy
i
V
ywy
V
ywy
i
i
00
010 tan
Equation for induced angle of attackalong finite wing in terms of (y)
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EFFECTIVE ANGLE OF ATTACK, eff, EXPRESSION
0
0
0
00
0
0
002
00000
0
2
22
1
2
Leff
Leffl
l
l
LeffLeffl
effeff
ycV
y
yc
ycV
yc
yVcycVL
yyac
y
eff seen locally by airfoilRecall lift coefficientexpression (Ref, EQ: 4.60)a0 = lift slope = 2
Definition of lift coefficient and Kutta-Joukowski
Related both expressions
Solve for eff
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COMBINE RESULTS FOR GOVERNING EQUATION
2
20
00
00
2
20
0
00
0
4
1
4
1
b
bL
ieff
b
bi
Leff
dyyy
dyd
VycV
yy
dyyy
dyd
Vy
ycV
y
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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PRANDTL’S LIFTING LINE EQUATION
• Fundamental Equation of Prandtl’s Lifting Line Theory
– In Words: Geometric angle of attack is equal to sum of effective angle of attack plus induced angle of attack
– Mathematically: = eff + i
• Only unknown is (y)
– V∞, c, , L=0 are known for a finite wing of given design at a given a
– Solution gives (y0), where –b/2 ≤ y0 ≤ b/2 along span
2
20
00
00 4
1b
bL dy
yy
dyd
VycV
yy
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WHAT DO WE GET OUT OF THIS EQUATION?
1. Lift distribution
2. Total Lift and Lift Coefficient
3. Induced Drag
dyyySVSq
DC
dyyyVdyyyLD
LD
dyySVSq
LC
dyyVL
dyyLL
yVyL
b
bi
iiD
i
b
bi
b
bi
iii
b
bL
b
b
b
b
2
2
,
2
2
2
2
2
2
2
2
2
2
00
2
2
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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
AR
C
LiD
Li
2
,
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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
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SPECIAL SOLUTION:ELLIPTICAL LIFT DISTRIBUTION
Elliptic distribution
Equation for downwash
Coordinate transformation →
See reference for integral
bVV
wb
w
db
w
db
dyb
y
dy
yyby
y
byw
by
y
bdy
d
i
b
b
2
2
coscos
cos
2
sin2
;cos2
41
41
4
0
00
0 0
00
2
20
21
2
22
00
2
220
Downwash is constant over span for an elliptical lift distribution
Induced angle of attack is constant along spanNote: w and i → 0 as b → ∞
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SPECIAL SOLUTION:ELLIPTICAL LIFT DISTRIBUTION
AR
CC
dyySV
C
AR
CS
bAR
b
SC
bVdy
b
yVL
LiD
b
b
iiD
Li
Li
b
b
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 moreuseful expression for i
Combine L definition for elliptic profile with previous result for i
Define AR because itoccurs frequently
Useful expression for i
Calculate CD,i
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SUMMARY: TOTAL DRAG ON SUBSONIC WING
eAR
Cc
Sq
DcC
DDD
DDDD
Lprofiled
iprofiledD
inducedprofile
inducedpressurefriction
2
,,
Also called drag due to lift
Profile DragProfile Drag coefficient relatively constant with M∞ at subsonic speeds
Look up(Infinite Wing)
May be calculated fromInviscid 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
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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