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Transcript of 1 MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS Introduction to Lifting Line Theory April 11, 2011...
1
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
2
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
3
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
4
BIOT-SAVART LAW
5
EXAMPLE APPLICATIONS
Case 1:
6
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)
7
CONSEQUENCE: ENGINE INLET VORTEX
8
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
10
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’
11
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)
12
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)
13
PRANDTL’S LIFTING LINE THEORY
• Example shown here will use 3 horseshoe vorticies
d1
14
PRANDTL’S LIFTING LINE THEORY
d1
d2
15
PRANDTL’S LIFTING LINE THEORY
d1
d2
d3
16
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
18
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
19
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
20
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)
21
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)
22
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
24
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
25
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
32
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