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7/31/2019 AERO3630-Lec_Lifting Line_Finite Wing Theory
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
•You are now starting to learn how to
determine lift on a wing (3D)
Start from where you left off in Thin AirfoilTheory
Imagine the wing is made up of infinitesimal
airfoil sections spanning a distance ‘b’
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
Make the same Assumptions:
Ideal Flow (Potential Flow)
Incompressible flow
Further assumptions: •Low angle of incidence
•Negligible airfoil thickness
Using thin airfoil theory obtain the total Circulation for each
section so that the circulation varies in the y direction ,)( y
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
In the thin airfoil theory, i.e., for infinite wing
span, the camber line was replaced by a string
of line vortices of infinitesimal strengths
Now in a similar manner we are looking for a
vortex strength distribution produced by the
flow field around a 3D lifting body
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
Now a Wing can be replaced by a system of
vortex systems:
•Starting Vortex
•Free Trailing Vortex
•Bound Vortex
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
The starting vortex is created when
the wing starts its motion or when air starts to flowover a wing.
The free trailing vortex is also created during themotion
Hence, starting and trailing vortices are physical
entities and can even be seen if conditions are right
The difficulty lies with bound vortex.
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
All we will consider is that:
The bound vortex system is a hypotheticalarrangement of vortices which replaces the
real wing in every way except that of
thickness.
It is largely concerned with developing the
bound vortex system which simulatesaccurately or at least a little distance away, allthe properties, effects, disturbances, force
systems etc., due to the real airfoil.
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
WARNING!
Do not interpret the representation of the
wing by a bound vortex system to be a
rigorous model. What it does is to allow a
relationship to be established between the:
•physical load distribution for the wing
•trailing vortex system
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
Prandtl hypothesised that:
Each section of the wing acts as though it is an
isolated 2D section provided the span wise
flow is not great.
Let us now see how the vortex system that
replaces the wing looks like in the next slide
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
Now further assumptions:
We can always argue that after a long time thestarting vortex system will be pushed away at such a
great distance from the wing that its effects would
have been mitigated by viscosity and other factors
and become negligible
Let us see how this looks like in the next slide
Note: Pay attention to the lifting line and the natureof the Circulation distribution on this line
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
We are now going to apply some logical argument
based on practical observations to justify the natureof circulation in the span wise direction
The flow is going to behave :•more like a 2D flow at the centre of the wing , i.e.,
Circulation is maximum at the middle or mid-span.
•and less 2D or become more 3D as it moves
towards the wing tip
•Since the flow has to break away at wing tip, the
Circulation will be zero at wing tip
)( y
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
Then:
The Maximum is at mid point, i.e., at y=0
And at the two ends , i.e., at the tips:
and
)(0 y
0)( ss
0)( ss
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
Now since this is finite wing, according to Newton’s Laws
every action has an equal and equal and opposite reaction.
This reaction is essentially the measure of the induced drag or
the downwash created.
Using Biot-Savart law, the flow velocity induced or associated
with a 3D vortex filament element can be expressed as:
Or
r
y ywi
4
)()(
)(2
1)(
2
1)(
0 y ydy
yd ydwi
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
If the origin is taken at the centre of the bound
vortex, then the velocity at any point y along thebound vortex induced by the trailing semi infinite
vortices is:
Or
)2 / (4)(
)2 / (4)()(
yb y
yb y ywi
22)2 / (4
)()(
yb
y ywi
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
Let us now see how the effects of the trailing vortices
can be superimposed on the lifting line graphically.
We will see that this looks like a horse shoe vortex
system in the next slide
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
Now we are in a position to work out the lift andinduced drag on a finite wing
Obtain the circulation for each wing section (airfoil).
Place the total circulation on the quarter chord
length of each airfoil section on the wing.
The locus of these points is the ‘Lifting Line’
Let us see how this looks like in the next slide
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
Once the circulation distribution and the induced velocity
distribution are known or defined, By putting the co-ordinate system at the centre of the wing:
the total lift and the drag for the wing can be obtained in
the following manner
Where s and –s are the semi span or b/2 and –b/2
s
s
dy yU y L )()(
s
s
ii dy y yw y D )()()(
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
CONSIDR AN EXAMPLE:
The lift and induced drag for:Elliptic Lift Distribution
So that:
And:
2
1)(
s
y y o
s
s
dys
yU y L
2
0 1)(
s
s
i dys
y yw y D
2
01 1)()(
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
Similar to thin airfoil theory, it is better to work in angles:
Where , the location of any point on the lifting line is given by
coss y
dysU y L
s
s
sin)cos1()(2
0
0
2
0
04
sin)( U bd sU y L
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
The lift co-efficient then becomes:
Where S is the wing area
02
S U
bC L
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
Before working out the induced drag let us first work out
the total downwash velocity:
So that:
dy y y ys
y
sw
s
s
i
)(40
22
0
dy y y ys
y
sdy
d s
s
)( 0
222
0
dy y y ys
y
y y ys
y y
sw
s
s
i
)
)()
)((
40
22
0
0
22
00
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
Integrating between -s and s with respect to y:
Note: The value is constant across the span
ss
wi4
)0(4
00
s
wi
4
0
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
Now:
Where, for elliptic distribution:
giving:
Or:
s
s
ii dys
y
yw D
2
0 1)(
s
wi
4
0
2
0
8
i D
0
2
00 sincos1
4d s
s Di
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
Now:
Where S is the area of the wing
Recalling, aspect ratio is given by: b2/S
S U
C i D
20
4
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
Recalling, aspect ratio to be given by: b2/S
This is the expression for minimum induced drag
2
22
2
0 2
44
b
S U C
S U S U C L Di
)(
2
2
2
AR
C
b
S C C L L Di
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
Some other points of note:
The resultant induced velocity at a point is, in general in thedownward direction, and is called ‘downwash’, where
Or
i
i
U
w
U
w y y
i )(tan1
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
The downwash has the effect of ‘tilting’ the undisturbed air,
so the effective angle of attack of the aerodynamic centre
(i.e., the quarter chord) is
Where is the effective angle of attack (3D)
is the downwash angle or induced angle
is the angle of attack (2D)
ie
e
i
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
The downwash has the effect of ‘tilting’ the undisturbed air,
so the effective angle of attack of the aerodynamic centre
(i.e., the quarter chord) is
ie
e
i
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
The lift curve slope relations are:
)(1 2
2
3
ARa
a
d
dC a
D
D L
D
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AERO3630 Lecture on : Techniques for General Circulation Distribution
Consider a span wise circulation distribution than can berepresented by a Fourier sine series consisting of n terms
Then:
Note, here too applies:
and
coss
y
]sin[4)(
1
n BsU n
n
]sin[4)()(1
2 n BU sU Ln
n
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
But:
And
Replacing by
)(2
cU
C L
)( 03 e D L aC
LC )(2 cU
)()]()([)(
)(20
U aU ac e
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
But
Then:
dy y y
dy
d
wU
s
s
)(4
1
0
sin
sin1
n
n nnBU w
)]()([
)(
)(20
aU
ac e
sin
sin1
n
n nnBU
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
Since:
Substituting and rearranging:
Let
And rearranging:
This is the Mono-plane equation
]sin[4)(1
n BsU n
n
n Bcas
n
n
e
sin81 )]()( 0 a
sin
sin1
n
n nn B
)]()([sin 0 a
s
cae
8
)sin(sin1
nnnBn
n
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
224 U s
d n Bn sinsin0 L
If we consider the symmetrical loading distributions, only odd terms of the
series need considering:
......]5sin3sin]sin[4)( 531 B B BsU
L22
4
U s
0
1
4
2sin
2
B
224 U s
03 1
)1sin(
1
)1sin(
n
n
n
n Bn
n
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
Now:
And:
Giving:
But
Giving
24
2sin
2 01
B
1,01
)1sin(
1
)1sin(
03
n
n
n
n
n Bn
n
1
22)
2
1)(4( BU s L
S U C L L )21(
2
)(1 AR BC L
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
Similarly for Induced drag:
But
Thus:
s
s
ii dy y yw D )()(
0
22
4 U s Di d n B
n
n sin1
0
nnBn
n sin1
d n Bn
n sin1
n
nnB1
2
2
)( ARC i D
n
nnB1
2
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
Now recall:
Giving
But
Hence:
Or
)(1 AR BC L
)(1
AR
C B L
)( ARC i
D n
nnB1
2
)(
2
AR
C C L
Di
nn
B
Bn
1
2
1
)(
)(
2
AR
C C L
Di .......)]
753(1[
2
1
2
7
2
1
2
5
2
1
2
3 B
B
B
B
B
B
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AERO3630 Lecture on : Lifting Line/Finite Wing Theory
Remember: If we consider the symmetrical loading distributions, only odd terms of
the series need considering:
So that
Or
where:
And [can you guess why??]
Then:
)(
2
AR
C C L
Di .......)]
753(1[
2
1
2
7
2
1
2
5
2
1
2
3 B
B
B
B
B
B
)(
2
ARC C L
Di )1(
0
.......)753
(2
1
2
7
2
1
2
5
2
1
2
3
B
B
B
B
B
B