lab work: lifting profiles
TRANSCRIPT
LAB WORK: LIFTING PROFILES
Mater SMA Hydrodynamics 06/06/2014
Manas PHATAK
1
Objectives:
To generate the lifting profiles of different types of the wings: rectangular, semi-elliptic and triangular.
To code the Prandtl method and then compare the numerical results with the theory and the
wings performances to different aspect ratios (λ) and angles of attack (α).
INTRODUCTION
Three types of plane wings have been considered: rectangular, triangular and semi-elliptic (on the
codes, the variables linked to each type of wing contain the letter t, r or e at the end, for instance Cxr
for the drag coefficient on the rectangular wing). These wings are defined by a chord l(y) and the
span 2L. For a given calculation the same span and surfaces (linked to the aspect ratio) has been
taken for the three types of wings in order to compare the performances. The fluid used has been air
( ) and a shape coefficient of has been used. We have taken aspect ratios higher than 4 and angles of
attack lower than 15° in order to not reach the critical angle and have stall. With aspect ratios higher
than 4, and angle for the triangular wing higher than 30° is guaranteed. Coding of the Prandtl equation and the solution to the problem
To obtain the forces that act on the wings we use the lifting line theory. To do so, first we calculate the circulation along the wing with the Prandtl equation (Eq. 1).
( ) ( ) ( ) ( ( ) ∫ ( ) ) Eq. 1
( )
To obtain the circulation on each position y of the span we discretize the span in N points of
discretization { } and N-1 elements between –L and +L. We calculate the circulation on
each point on the span according to Eq. 2. We suppose that ( ) ( ) and so
. We take control points { } located at the center of each element, in order
to avoid singular values due to a division by zero in Eq.2.
( ∑ ) Eq. 2
2
We can see on the equation above that we need the chord on each point of discretization, . So we
implement the shapes on the code, defining the chord on each discretization point (an example is
shown on Fig. 1).
2
rectangular
1.8 triangular
1.6
semi-elliptic
1.4
(ch
ord
) 1.2
1
l(y)
0.8
0.6
0.4
0.2
0
-5 -4 -3 -2 -1 0 1 2 3 4 5
y (position on the wing)
Fig. 1 Shape of the 3 types of wing for a span of 10 m and λ=10 (the scale is not the same on both axis)
From the equation above (Eq. 2) we see that to get the value of the unknown we need to know the
values of the circulation in all the points which we don’t know. Therefore we have to construct a
linear system derived from the equation above to get the vector of circulations along the wing. This
linear system is defined by matrix A and vector B of independent terms, which are obtained as stated
in Eq. 3 to Eq.6. Once the system is solved by inversing the matrix A and multiplying it by the vector
B, we obtain the circulation in the N points of discretization (see Fig. 2). When the aspect ratio
increases the profile of the circulation along the span it gets closer to the profile of the wing. We can
see on Fig. 2 that the profile of the circulation explicitly indicates which of the three wings each one
is.
Eq. 3
(
) Eq. 4
(
) Eq. 5
Eq. 6
3
4
rectangular
3.5 triangular
semi-elliptic
3
(cir
cu
latio
n)
2.5
2
(y
)
1.5
1
0.5
0
-5 -4 -3 -2 -1 0 1 2 3 4 5
y (position on the wing)
Fig. 2 Circulation along the wing for a span of 10 m, λ=10 and α=10°
Once we have the circulation along the span, we have to calculate the induced velocity given by Eq.7.
The induced velocity is produced by the tip vortices and it changes the effective angle of attack, and
so it is needed to calculate the coefficients, in particular drag coefficient. The induced velocity is also
used to calculate the circulation, added to the angle of attack, what means changing the effective
angle of attack. However to calculate the circulation it has been substituted by its expression directly
(Eq.1).
( ) ∫ ( )
Eq. 7
( )
Eq.7 can be discretized as well along the span as done with the circulation (Eq. 8). Notice that we can solve directly the induced velocity on each point of discretization without having to solve a system,
since we know all the unknowns except unknown wi, which is isolated.
∑ Eq. 8
( )
On Fig. 3 and Fig.4 the induced velocity along the wing is plotted for an span of 10 m, λ=10 and
α=10°.The two plots are the same, on Fig. 4 the induced velocity for the last points on the two edges
of the span are not presented in order to see better the profile of the induced velocity. We can
observe that except the point on the edges of the span the induced velocity is negative.
4
7
rectangular
6 triangular
semi-elliptic
5
velo
city)
4
(in
du
ce
d
3
w(y
)
2
1
0
-1
-5 -4 -3 -2 -1 0 1 2 3 4 5
y (position on the wing)
Fig.3 Induced velocity w(y) for a span of 10 m, λ=10 and α=10°
0.6
rectangular
0.4
triangular
semi-elliptic
velo
city) 0.2
0
(in
du
ce
d
-0.2
w(y
)
-0.4
-0.6
-0.8
-5 -4 -3 -2 -1 0 1 2 3 4 5
y (position on the wing)
Fig.4 Induced velocity w(y) for a span of 10 m, λ=10 and α=10°
5
Once we know the circulation and the induced velocity along the span we can obtain the drag and lift coefficients (Eq.9 and Eq. 10).
∫ ( ) Eq. 9
∫ ( ) ( ) Eq. 10
The formulas of the coefficients above can be discretized as follows (Eq.11 and Eq. 12), where dy=2L/N:
∑ ( ) ∑
Eq. 11
∑ ( ) ∑ Eq. 12
For the pitch moment we calculate it taking into account that the aerodynamic center is located at ¼
chord from the leading edge. We calculate the moment produced by the lift and the drag respect an
arbitrary location along the chord which has been fixed as half chord (point O on the figure below),
according to the theory of the course. (Eq. 13).
∫ ( ) ( ) ( ( ) ) Eq. 13
6
We discretise eq. 13 into eq. 14. Where is the mean chord, which is obtained from
∑ ( ) ( ) ( ( ) ) Eq. 14 We finally compute the drag, the lift and the pitch moment with the drag, lift and moment coefficients, the surface, the density and the velocity.
Eq. 15
Eq. 16 The main program we have created on Matlab asks the user the velocity, the span, the angle of
attack and the aspect ratio and it calculates the circulation, the lift, the drag and the pitch moment
for each wing. The program is wing.m
We have calculated the roll moment and the yaw moment and their coefficients. It was not
necessary because they are zero as the wings are symmetrical. However we have included them on
the main program wing.m to see if there are zero as expected and it was the case.
In order to complete the different tasks asked on this practice, three more programs has been created, one per task (convergence.m, theory.m and forces.m). Three functions that support these programs have also been coded, which are convwing(N,L,lamda,alfa,U), ellipticwing( N,L,lamda,alfa,U) and wingforces(N,L,lamda,alfa,U).
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RESULTS
Analysis of convergence
The program linked to this part is convergence.m which calls de function convwing(N,L,lamda,alfa,U). For the three types of wings we have checked the convergence of the results on the circulation and
the forces (drag and lift), according to our discretization (number of points along the span). To verify
the convergence of the results on the circulation we have taken three control points which are
located at –L/2, 0 and L/2.
To do so, we have calculated the relative error and plotted it against the number of points of
discretization. A program has been created which solves the problem for different amounts of N
discretization points (convergence.m). The relative error has been calculated with the value on the
step in question and the value of the following step (Eq. 17). Notice that the error is in tan per one
and not tan per cent.
Eq. 17
Fig. 5 and Fig.6 represent the relative error for the drag and lift forces. The difference between both
is that in Fig. 5 the X axis is in logarithm scale. The Y axis is always presented on logarithm scale with
base ten). Fig. 7 and Fig. 8 show the relative error on the circulation.
Analysing the graphs of convergence, one can see that the solution converges. It seems as well that
N=100 is a good choice, since a remarkable change of tendency appears at a value of N=100. The
plots has been obtained for different ranges of spans, aspect ratios and angles of attack, and it
always present the same phenomena at N=100.
8
log(E
r) R
ela
tive e
rror
in the
forc
es
0
lift rectangular
-1 lift triangular
lift semi-elliptic
drag rectangular
-2 drag triangualar
drag semi-elliptic
-3
-4
-5
-6
-7 0.5 1 1.5 2 2.5 3
log(N) (number of discretization points)
Fig. 5 Relative error on the drag and lift forces 0
lift rectangular
-1
lift triangular
lift semi-elliptic
forc
es drag rectangular
-2 drag triangualar
drag semi-elliptic
in t
he
-3
err
or
Re
lative
-4
log
(Er)
-5
-6
-70 100 200 300 400 500 600 700
N (number of discretization points)
Fig. 6 Relative error on the drag and lift forces
9
log(E
r) R
ela
tive e
rror
in t
he
circulq
tion
0
-L/2 rectangular
-L/2 triangular
-L/2 semi-elliptic
-1 0 rectangular
0 triangualar
0 semi-elliptic
L/2 rectangular
-2 L/2 triangualar
L/2 semi-elliptic
-3
-4
-5
-6 0.5 1 1.5 2 2.5 3
log(N) (number of discretization points)
Fig. 7 Relative error on the circulation
log(E
r) R
ela
tive
err
or
in the
circula
tion
0
-L/2 rectangular
-L/2 triangular
-1 -L/2 semi-elliptic
0 rectangular
0 triangualar
0 semi-elliptic
-2 L/2 rectangular
L/2 triangualar
L/2 semi-elliptic
-3
-4
-5
-60
100 200 300 400 500 600 700
N (number of discretization points)
Fig. 8 Relative error on the circulation
10
Comparison with the theory
The program linked to this part is theory.m which calls the function ellipticwing(N,L,lamda,alfa,U)
In the case of the elliptic wing we have checked that the numerical results are in agreement with the theory. The formulas for the elliptic wing for different aspect ratios ( ) are:
Eq. 18
Eq. 19 To compare the first formula (Eq. 18), the problem has been solved for different spans and velocities,
a range of aspect ratio from 5 to 250 and a range of angle of attack from 0° to 15°. Executing the
program theory.m (entering the value of the velocity and the span), the relative error between the
results and the theory is stored on the matrix Cxerror, whose columns indicate the aspect ratio and
whose rows indicate the angle of attack. In the case of a span of 10 m and a velocity of 5m/s, all the
components of the matrix are around of 0,25. That means that for the ranges of aspect ratio
between 5 and 250 and angles of attack from 0° to 15° the error is less than 1% in comparison with
the theory, and as well for different spans and velocities.
A graph is also presented for an aspect ratio of 5, 10 and 100 (Fig.9a), where for N=400 points of
discretization the numerical and theoretical results has been plotted . On this graph the theory and
the numerical lines are superimposed due to the small error between them. If we decrease the
number of points the numerical and theoretical curves are not as close (Fig.9b). This graph (Fig. 9a)
can also be compared with the curve obtained by Comolet, 1994 that can be found on the slides of
the course (Fig. 10). We can see that the curves are of similar shape and increasing the aspect ratio
the curve goes more vertically. The most difference between them is that the curves obtained by
Comolet takes into account the friction and therefore they don’t pass over the center of coordinates,
while on Fig. 9a the curves pass through the center of coordinates as we don’t take into account the
frictional drag on our numerical solution.
11
1.8
theory
1.6 α=15° numerical
1.4
1.2
1 α=10°
C z
0.8
0.6 α=5°
0.4
0.2
0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
Cx
Fig. 9a Relation between drag and lift coefficient for different aspect ratios and N=400
1.8
theory
1.6 numerical
1.4
1.2
1
C z
0.8
0.6
0.4
0.2
0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Cx Fig. 9b Relation between drag and lift coefficient for different aspect ratios and N=40
12
Fig. 10 Polar curve of a wing versus aspect ratio (Colomet, 1994)
To see what happens varying only the angle of attack and not the aspect ratio, we have solved the
problem with different aspect ratio and a fixed angle of attack. Almost no difference between theory
(Eq. 18) and numerical solution is observed (Fig.11). The lift increases when increasing the aspect
ratio and the drag decreases when increasing the aspect ratio. This leads us to the conclusion that
the higher the aspect ratio the best the lift in comparison with the drag. This will be better explained
after, when talking about the lift-to-drag ratio.
1.15
theory
1.1 λ =250 numerical
1.05
1 λ =30
Cz
0.95
0.9 λ =20
0.85
0.8
λ =5
0.75 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
Cx
Fig. 11 Relation between drag and lift coefficient for different aspect ratio and α=10°
13
For the Eq. 19 which links the variation of the lift coefficient respect to α with the aspect ratio, a good fit has been observed as well between the theory and the numerical results.
On Fig.12 the lift coefficient against the angle of attack is presented for different aspect ratios. This
plot is interesting as it gives information of how the lift depends on the two parameters. From this
plot one can see that the lift increases linearly with the angle of attack, and by increasing the aspect
ratio the lift increases as well.
1.8
1.6
1.4
1.2
1
C z =5
0.8 =6
=7
=8
0.6 =9
=10
=12
0.4 =15
=20
=30
0.2 =50
=100
0 =250
0 5 10 15
Fig. 12 Variation of lift coefficient with angle of attack α
Finally, we have calculated the slopes of the lines of Fig. 12 to compare them with Eq. 19. On Fig.13
the relative error in percentage is presented for different aspect ratio. We can see that the error with
the theory is lower than 0,09 % on the worst case, and it decreases quickly by increasing the aspect
ratio.
14
0.09
0.08
0.07
(%)
dC
z/d
0.06
0.05
err
or
rela
tive
0.04
0.03
0.02
0.010 50 100 150 200 250
Fig. 13 Relative error on dCz/dα between the theory and numerical results
15
Performances
The program linked to this part is forces.m which calls the function wingforces(N,L,lamda,alfa,U)
For the three types of wings we have compared the performances, for different angle of attack and
aspect ratio. To compare the performances we compare the drag, lift and moment coefficients as
they are non-dimensional.
First we will present two graphs of the lift and drag coefficients on the rectangular wing for angle of
attack ranging from 1° to 15° and aspect ratio from 5 to 30 (Fig.14 and Fig.15). For the lift, we see
that it increases linearly with the angle of attack, and it increases but not linearly with the aspect
ratio. Therefore to have more lift we can increase the aspect ratio or the angle of attack or both. In
reference to the drag, which we want to be as low as possible, we see that it increases when
increasing the angle of attack and it decreases when increasing the aspect ratio. Therefore if we want
a low drag we have to increase the aspect ratio and decrease the angle of attack.
Lift co
effic
ien
t fo
r re
cta
ng
ula
r w
ing
2
1.5
1
0.5
0
15
10 30
25
angle of attack 5 20
15 aspect ratio
0 10
5
Fig. 14 Lift on the rectangular wing
16
Dra
g c
oe
ffic
ien
t fo
r re
cta
ngu
lar
win
g
0.1
0.08 0.06
0.04
0.02
0
15
10 30
25
angle of attack 5
20
15
0
10 aspect ratio
5
Fig. 15 Drag on the rectangular wing
On Fig. 16 and Fig. 17 we observe that the lift increases with the angle of attack and with the aspect
ratio. With the aspect ratio it increases a lot when we are at low aspect ratios. For the angle of
attack, the relation is linear, so the lift is directly proportional to the angle of attack. From these
graphs we can also see that the best performances are for the semi-elliptic wing. The lift is always
higher for the semi-elliptical wing given that the surface, velocity, aspect ratio, and angle of attack
are the same for all three wings. The lift on the triangular wing is higher than the lift on the rectangular wing for mid and high aspect ratios (λ>10), and a bit higher on the rectangular wing for low aspect ratios (λ<10).
In what concerns to the drag, it increases parabolicaly with the angle of attack. This is clear since by
increasing the angle of attack the component of the drag due to the pressure is higher and it
becomes more and more important. The drag decreases with the aspect ratio, it is inversely
proportional to the aspect ratio.
The drag is almost always higher for the triangular wing, except when the aspect ratio is higher than
60. In that case, the drag is higher for the rectangular wing. The drag is almost always lower for the
semi-elliptic wing (fig.18), except for very low aspect ratios where it is much close to the other two
wings.
To organize the thoughts, the lift is higher on the semi-elliptic wing and the drag is lower on the semi-elliptic wing. Therefore the performances are better for the semi-elliptic wing.
17
1.1
1.05
1
Lift
co
eff
icie
nt 0.95
0.9
0.85
0.8
rectangular
0.75 triangular
semi-elliptic
0.7 0 10 20 30 40 50 60 70 80 90 100
aspect ratio
Fig. 16 Lift for different aspect ratio and α=10°
1.4
1.2
1
co
eff
icie
nt
0.8
Lift 0.6
0.4
0.2 rectangular
triangular
semi-elliptic
0 0 5 10 15
angle of attack
Fig. 17 Lift for different angle of attack and λ=10
18
0.04
rectangular
0.035 triangular
semi-elliptic
0.03
Dra
g c
oe
ffic
ien
t 0.025
0.02
0.015
0.01
0.005
0 0 10 20 30 40 50 60 70 80 90 100
aspect ratio
Fig. 18 Drag for different aspect ratios and α=10°
0.07
0.06
0.05
co
eff
icie
nt
0.04
Dra
g
0.03
0.02
0.01 rectangular
triangular
semi-elliptic
0 0 5 10 15
angle of attack
Fig. 19 Drag for different angle of attack and λ=10
19
Finally to better understand the performances of the wings, the lift-to-drag ratio has been obtained
for different values of the parameters, and presented versus the angle of attack (Fig. 20) and versus
the aspect ratio (Fig. 21).
From both figures we can deduce that the best wing is the semi-elliptic wing, as the lift-to-drag
coefficient is always higher than on the other two wings. This is in agreement with what we have told
before. We can also see that the lift-to-drag ratio is even higher for the semi-elliptic wing in
comparison with the other two wings when the aspect ratio increases (Fig. 21). The ratio is almost
the same for the rectangular and triangular wings, for low aspect ratio a bit higher on the rectangular
wing, and for high aspect ratio a bit higher on the triangular wing.
We can also see that the lift-to-ratio is directly proportional to the aspect ratio (Fig. 21) and inversely
proportional to the angle of attack (Fig. 20). Therefore the relative difference between lift and drag is
higher for low angles of attack and for high aspect ratios.
We have said before that by increasing the angle of attack we increase the performances (lift),
however we have to take into account that we increase as well the drag and it becomes more and
more important as it is reflexed on the decrease of the lift-to-drag ratio (fig. 20). We also have to take
into account that we can arrive to the critical angle and have stall, in that case the lift drops and the
drag increases.
350
rectangular
300
triangular
semi-elliptic
ratio 250
-to
-dra
g
200
Lif
t
150
Cz/C
x
100
50
0 0 5 10 15
angle of attack
Fig. 20 Lift-to-drag ratio for different angle of attack and λ=10
20
300
250
-to
-dra
g r
atio
200
150
Lift
Cz/C
x
100
50 rectangular
triangular
semi-elliptic
0 0 10 20 30 40 50 60 70 80 90 100
aspect ratio
Fig. 21 Lift-to-drag ratio for different aspect ratios and α=10°
We have analysed the pitch moment as well for the three wings and for different angles of attack and
aspect ratio. From Fig. 22 and Fig. 23 we obtain that the pitch moment is always much higher on the
triangular wing and the lowest value of pitch moment is always on the rectangular wing.
We want the pitch moment as low as possible since the pitch moment destabilizes the wing. Therefore, the rectangular wing is the best on this aspect, while the triangular wing is the worst.
In Fig.22 we can see that the pitch moment seems to be linear with the angle of attack, this is due to
the fact that the pitch moment is mostly due to lift and we have seen before (Fig. 17) that the lift
varies linearly with the angle of attack. In Fig.23 we see how the pitch moment increases fast in the range of low aspect ratios when increasing the aspect ratio and it increases slowly when the aspect ratio is high.
21
0.45
0.4
0.35
co
eff
icie
nt
0.3
0.25
mo
me
nt
0.2
Pitch
0.15
0.1
rectangular
0.05 triangular
semi-elliptic
0 0 5 10 15
angle of attack
Fig. 22 Pitch moment coefficient for different angle of attack and λ=10
0.36
0.34
0.32
co
eff
icie
nt
0.3
0.28
mo
me
nt
0.26
Pitch
0.24
0.22
rectangular
0.2 triangular
semi-elliptic
0.18 0 10 20 30 40 50 60 70 80 90 100
aspect ratio
Fig. 23 Pitch moment coefficient for different aspect ratios and α=10°
22
To have some idea on the order of magnitude some results are presented for different angles of attack, velocities and aspect ratios. The span is 2L=10 m.
CASE 1 CASE 2 CASE 3
V=5ms-1
, λ=5, V=30ms-1
, λ=10, V=100ms-1
, λ=100, S=20 m
2 ,α=5 S=10 m
2, α=10 S=1m
2 , α=14
Rect Tri Semi Rect Tri Semi Rect Tri Semi
Fx (drag) [N] 2,81 2,92 2,93 144,78 155,56 143,6 60,35 57,16 43,31
Fy (lift) [N] 112,87 111,25 117,46 4754 4736 4934 8912 8968 9030
MM (Pitch moment) 56,10 69,04 63,10 1164 1477 1306 215,8 287,4 236,4
[Nm]
Table 1 Results for different values of parameters
On table 1 we can see that the velocity has a great effect on the lift and drag forces. This is because
the forces are proportional to the square of the velocity. In the third case, even if the surface is ten
times smaller than in the second case, the lift is double. This is because the velocity is almost three
times higher than the second case, but also because the aspect ratio and the angle of attack are
higher. However the drag is lower than the drag of the second case, due to the fact that by increasing
the aspect ratio, the drag diminishes (see Fig. 16 and Fig. 18, where lift coefficient increases and drag
coefficient decreases when increasing the aspect ratio).
The lift is always higher on the semi-elliptical wing (in agreement with Fig. 16 and Fig.17), and it is verified in the three cases.
On the first case the drag force is higher for the semi-elliptic wing, because we have a low aspect
ratio. If we increase a bit the aspect ratio the lift force is higher on the triangular wing and lower on
the semi-elliptic wing (in accordance with Fig. 18). Finally, when the aspect ratio is very high (third
case) the drag force is higher on the rectangular wing and lower on the semi-elliptic wing (again in
agreement with Fig. 18).
On the three cases, the pitch moment is higher on the triangular wing and lower on the rectangular wing, which is in agreement with what we have discuss before (fig. 22 and fig. 23).
On three cases the best option is the semi-elliptic wing, as the lift is the highest and the drag the
lowest (except the first case which the drag is almost the same for the three wings). The pitch
moment is not as good as on the rectangular wing but not as worse as on the triangular wing.
23
CONCLUSIONS
We have seen that for the same given parameters (area, aspect ratio, angle of attack and velocity)
the best performances are on the semi-elliptic wing. This is due to the fact that the lift is higher and
the drag is lower. In fact this behaviour has been checked with the lift-to-drag ratio, which for the
semi-elliptic wing is higher. However, the semi-elliptic wing does not present the lowest pitch
moment. The rectangular wing is better in reference to the pitch moment and the triangular wing the
worst. Between the rectangular wing and the triangular wing, we have seen that for low aspect ratio (λ<10-
15) the performances are better for the rectangular wing, a higher lift and lower drag and pitch
moment (higher lift-to-drag ratio). On the other side, for high aspect ratio (λ>10-15) the
performances are better for the triangular wing, higher lift and lower drag, although higher pitch
moment. After doing an analysis on the parameters that influences the drag, the lift and the pitch moment, we have seen:
The lift increases with the angle of attack and the aspect ratio The drag increases with the angle of attack and decreases with the aspect ratio The pitch moment increases with the angle of attack and the aspect ratio.
From these observations we can conclude that the best performances occur for high aspect ratios,
since the lift increases and the drag is reduced. If our objective is to have a higher lift, increasing the
angle of attack is a solution, but we have to be careful because the drag increases as well, and so we
will need more energy (propulsion) to defeat the drag. We have to be careful as well with increasing
to much the angle of attack to prevent from stall.
BIBLIOGRAPHY
Slides of the cours ‘Basics of hydrodynamics’ in master SMA of Ecole Centrale de Nantes
Comolet R., Mécanique expérimentale des fluides, tome II, 4ème
édition, ed. Masson, 1994.
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