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8/12/2019 FIT Intro to LLT
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8/12/2019 FIT Intro to LLT
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2
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, (or xin 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)=rVG(y)
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3
CONSEQUENCE: ENGINE INLET VORTEX
8/12/2019 FIT Intro to LLT
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4
CHAPTER 4: AIRFOILEach is a vortex line
One each vortex line G1=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 G
G1G4 G7
8/12/2019 FIT Intro to LLT
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5
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=3418/12/2019 FIT Intro to LLT
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6
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
8/12/2019 FIT Intro to LLT
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7
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
G
G
G
G
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 boundvortex filament, but rather has a distribution of G(y)
8/12/2019 FIT Intro to LLT
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8
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, G, 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 G=constant
We need a way to let G=G(y)
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PRANDTLS LIFTING LINE THEORY
Example shown here will use 3 horseshoe vorticies
dG1
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PRANDTLS LIFTING LINE THEORY
dG1
dG2
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PRANDTLS LIFTING LINE THEORY
dG1
dG2dG3
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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, G, 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
dG1
dG2dG3
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PRANDTLS LIFTING LINE THEORY
Infinite number of horseshoe vorticies superimposed along lifting line
Now have a continuous distribution such that G = G(y), at origin G = G0
Trailing vorticies are now a continuous vortex sheet (parallel to V)
Total strength integrated across sheet of wing is zero
8/12/2019 FIT Intro to LLT
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PRANDTLS LIFTING LINE THEORY
Consider arbitrary location y0along lifting line
Segment dxwill induce velocity at y0given by Biot-Savart law
Velocity dw at y0induced by semi-infinite trailing vortex at y is:
Circulation at y is G(y)
Change in circulation over dy is dG= (dG/dy)dy
Strength of trailing vortex at y = dGalong lifting line
yy
dydy
d
dw
G
04
8/12/2019 FIT Intro to LLT
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PRANDTLS LIFTING LINE THEORY
Total velocity w induced at y0by entire trailing vortex sheet can be found by
integrating fromb/2 to b/2:
G
2
2 0
04
1b
b
dyyy
dy
d
yw
Equation gives value of
downwash at y0due 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 G(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 G(y) for finite wing (WHY?)
Calculate Lift, L (Kutta-Joukowski theorem)
Calculate CL
Calculate aeff
Calculate Induced Drag, CD,i(drag due to lift)
8/12/2019 FIT Intro to LLT
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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
G
2
20
04
1b
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 G(y)
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EFFECTIVE ANGLE OF ATTACK, eff, EXPRESSION
0
0
0
00
0
0
00
2
00000
0
2
2
2
1
2
G
G
G
Leff
Leffl
l
l
LeffLeffl
effeff
ycV
y
yc
ycV
yc
yVcycVL
yyac
y
a
a
aa
rr
aaaa
aa aeff seen 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 for aeff
8/12/2019 FIT Intro to LLT
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COMBINE RESULTS FOR GOVERNING EQUATION
G
G
G
G
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
8/12/2019 FIT Intro to LLT
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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 G(y)
V, c, a, aL=0are known for a finite wing of given design at a given a
Solution gives G(y0), whereb/2 y0 b/2 along span
G
G
2
20
0
0
00
4
1b
b
L dyyy
dy
d
VycV
yy
a
a
8/12/2019 FIT Intro to LLT
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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
b
ii
iD
i
b
b
i
b
b
i
iii
b
b
L
b
b
b
b
G
G
G
G
G
2
2
,
2
2
2
2
2
2
2
2
2
2
00
2
2
a
ara
a
r
r
8/12/2019 FIT Intro to LLT
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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
http://images.google.com/imgres?imgurl=http://www.stelzriede.com/ms/photos/planes/v1spit.jpg&imgrefurl=http://www.stelzriede.com/ms/html/sub/marshwvw.htm&h=452&w=401&sz=19&tbnid=vudGzyWC8gMJ:&tbnh=124&tbnw=110&start=2&prev=/images%3Fq%3Dbritish%2Bspitfire%26hl%3Den%26lr%3D8/12/2019 FIT Intro to LLT
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SPECIAL SOLUTION:
ELLIPTICAL LIFT DISTRIBUTION
Points to Note:
1. At origin (y=0) G=G0
2. Circulation varies elliptically with distance y along span
3. At wing tips G(-b/2)=G(b/2)=0
Circulation and Lift 0 at wing tips
2
0
2
0
21
21
G
GG
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
G
G
G
G
G
G
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
00
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
S C SO O
8/12/2019 FIT Intro to LLT
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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
G
G
G
CD,iis directly proportional to square of CL
Also called Drag due to Lift
We can develop a more
useful expression for ai
Combine L definition for elliptic
profile with previous result for ai
Define AR because it
occurs frequently
Useful expression for ai
Calculate CD,i
8/12/2019 FIT Intro to LLT
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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 withMat subsonic speeds
Look up
(Infinite Wing)
May be calculated from
Inviscid theory:
Lifting line theory
8/12/2019 FIT Intro to LLT
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SUMMARY
Induced drag is price you pay for generation of lift
CD,iproportional 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