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AE 416

APPLIED AERODYNAMICS

FALL 2014

HW#8

CARLOS ALBERTO NUNES DA SILVA JR.



1)

Analyze the following case:

AR = 4, 10, 16 (sweep)

λ = ct/co = 1

Λ = 0 degα = 10 deg

(a) What is the MAC and speed for each case to match the required Reynolds number?

The MAC is given by the following equation, where is the root chord.

22 1

3 1r

c c

(1.1)

The Re number is calculated based on the CMA and the freestream velocity as follows:

Re V c

(1.2)

Since Re number and (assumed sea level conditions) are fixed, we can solve the equation

1.2 for V and find the wing speed to match the Re.

To calculate lift curve slope for each wing we just divide the variation of CL by the

corresponding angle of attack change:

L

L

C C

(1.3)

The angle of attack in the simulations of this homework is 10 deg, and all the wings airfoil are

symmetrical ( = 0 = 0 ) so the lift curve slope is just the wing CL divided by .

To calculate the wing efficiency we just divide the CL by the 2D airfoil lift coefficient:

L

l

C e

C (1.4)

The for the airfoil e168 at 10 deg of angle of attack is equals to 1.12.

Since the aspect ratio and the tapered ratio are defined, we can set only the = 3.28 and

then the span and tip chord will be calculated; once all the wings have the same root and tip

chord (only for question 1) then the MAC and the wing velocity will not change with the AR,

as we can see in the figure below:



(b) Co-plot the spanwise lift coefficient distributions.

(c) Co-plot the downwash distributions.

(d) Co-plot the wing planforms.

0

0.2

0.4

0.6

0.8

1

1.2

-30 -20 -10 0 10 20 30

C

L

Wing Span [ft]

CL distributionsAR = 4AR = 10AR = 16

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

-30 -20 -10 0 10 20 30

D o w n w a s h [ f t / s ]

Wing Span [ft]

Downwash distributions AR = 4AR = 10AR = 16



(e) Tabulate the wing lift coefficients CL.

(f) Discuss the eff ect of aspect ratio on the Cl distribution.

The results shows that the wing CL is directly proportional to the AR. In the wings with high

aspect ratio the induced velocities due the tip vortices are smaller when compared with a low

AR wing therefore the induced angle of attack does not decrease too much, and then the CL is

similar to the 2D the airfoil CL. In the figure for CL distribution we can see that in the center

of the wing with aspect ratio of 16 the lift coefficient (1.04) is very similar to the airfoil CL

(1.09).

(g) Discuss the eff ect of aspect ratio on the downwash distribution.

The downwash distribution along the wing is inversely proportional with the distance of the

section to the wing tip vortices squared. In the simulations performed all the wings have the

same root and tip chord, so the wingspan changed in order to satisfy the aspect ratio desired

(the wingspan increased with the aspect ratio), therefore a high AR wing has the lower

downwash.

(h) Discuss the eff ect of aspect ratio on CL.

-6

-4

-2

0

2

4

6

8

10

12

14

-10 -5 0 5 10

Wing Span [ft]

Wing planform

AR = 4

AR = 10

AR = 16



Since the wing section CL increases with the AR the overall CL will present the same behavior.

If the aspect ratio becomes large the wing CL get closer to the airfoil 2D viscid lift coefficient.

(i) Calculate the lift curve slope of each wing and determine the wing efficiency e.

Show your work and discuss (as compared to each other and the elliptical case).

To calculate the lift curve slope we just have to use equation 1.3 with wing CLCL and

10 deg because the wing airfoil is symmetrical; the efficiency is calculated dividing the

wing CL by the airfoil Cl (which is approximately 2π). The trends of the CL results can be

graphically represent in the figure below:

(j) Based on your results, where do you expect stall to occur first?

The stall occur first at the middle of the wing. At the tips the induced velocity due the vortices

is large therefore the effective angle of attack is lower than the actual wing of attack; on the

other side in the center of the wing the induced angle is lower so the effective angle of attackis closer to the wing angle of attack so this region will stall first.

2. (15%) Analyze the following case:

AR = 10

λ = 0.2, 0.6, 1.0 (sweep)

Λ = 0 deg

α = 10 deg




(b)

Co-plot the spanwise lift coefficient distributions.



0

0.2

0.4

0.6

0.8

1

1.2

-20 -15 -10 -5 0 5 10 15 20

C L

Wing Span [ft]

CL distributions

lambda = .2lambda = .6lambda = 1

-50

-45-40

-35

-30

-25

-20

-15

-10

-5

0

-20 -15 -10 -5 0 5 10 15 20

D

o w n w a s h [ f t / s ]

Wing Span [ft]

Downwash distributionslambda = .2

lambda = .6

lambda = 1




(f) Discuss the eff ect of taper ratio on the CL distribution.

If we decrease the taper ratio (tip chord becomes lower than root chord) then the lowest

downwash is located at the wing tip, this implies in a lower induced angle of attack (the

effective angle of attack is closer to the wing angle of attack) therefore the is

effectively bigger in this region, which implies in a greater CL.

(g) Discuss the eff ect of taper ratio on the downwash distribution.

Due the circulation distribution in this wing the downwash is bigger in the middle of

the wing and lower in the wing tips.

(h) Discuss the eff ect of taper ratio on CL.

The wing with the taper ratio around .5 presents the maximum CL and is the wing that

presents a similar performance to the elliptical wing (the lowest induced drag planform)

(i)

Calculate the lift curve slope of each wing and determine the wing efficiency e.Show your work and discuss (as compared to each other and the elliptical case).

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6Wing Span [ft]

Wing planform

lambda = .2

lambda = .6

lambda = 1




If the wing is tapered, with the same airfoil and no twist angle, the stall occurs first at

the tip and then progress to the wing center.

3. (15%) Analyze the following case:

AR = 10

λ = 1

Λ = −30, 0, 30 deg (sweep)

α = 10 deg


(b)

Co-plot the spanwise lift coefficient distributions.

(c)

Co-plot the downwash distributions.

0

0.2

0.4

0.6

0.8

1

1.2

-20.00 -15.00 -10.00 -5.00 0.00 5.00 10.00 15.00 20.00

C L

Wing Span [ft]

CL distributions

gama [deg] = 30

gama [deg] = 0

gama [deg] = -30



(d)

Co-plot the wing planforms.


(f) Discuss the eff ect of sweep on the Cl distribution.

-45.00

-40.00

-35.00

-30.00

-25.00

-20.00

-15.00

-10.00

-5.00

0.00

-20.00 -15.00 -10.00 -5.00 0.00 5.00 10.00 15.00 20.00


Wing Span [ft]

Downwash distributions

gama [deg] = 30

gama [deg] = 0

gama [deg] = -30

-8

-6

-4

-2

0

2

4

-6 -4 -2 0 2 4 6

Wing Span [ft]

Wing planformgama [deg] = 30

gama [deg] = 0

gama [deg] = -30







0

0.2

0.4

0.6

0.8

1

1.2

-30.00 -20.00 -10.00 0.00 10.00 20.00 30.00

C L

Wing Span [ft]

CL distributions

lambda = 1

lambda = 1.5

lambda = 2

-45.00

-40.00

-35.00

-30.00

-25.00

-20.00

-15.00

-10.00

-5.00

0.00

5.00

-30.00 -20.00 -10.00 0.00 10.00 20.00 30.00


Wing Span [ft]

Downwash distributions

lambda = 1

lambda = 1.5

lambda = 2




(f) Discuss the eff ect of taper ratio on the Cl distribution.

Since we are using LLT method the sweep is not taken in account, so the results will

be the same if we simulate an unsweep wing. Moreover, if we increase the taper ratio

the wing section CL becomes greater in the central region of the wing than the tip.

(g) Discuss the eff ect of taper ratio on the downwash distribution.

When we increase the taper ratio the downwash tends to decrease until 0, when it

becomes an “upwash”, this means that the airfoil in the wing span center actually

experiences a bigger angle of attack than the wing. From the downwash distribution we

can conclude that the high tapered wing experiences less induced drag, but on the other

hand it cannot have good structural characteristics.

(h) Discuss the eff ect of taper ratio on CL.

Increase the tapper for values bigger than one decreases the wing CL, we can see that

from the CL distribution. The wing CL is proportional to the product of the section CL

by the its chord, and the region with large chord has lower lift coefficient and vice-

-8

-6

-4

-2

0

2

4

-10 -5 0 5 10

Wing Span [ft]

Wing planform

lambda = 1

lambda = 1.5

lambda = 2



versa; then increasing the taper ratio just maintain this kind of behavior (high lift

coefficient at the tip and lower in the root).

(i) Calculate the lift curve slope of each wing and determine the wing efficiency e.

Show your work and discuss (as compared to each other and the elliptical case).


The stall occurs first at the central region of the wing, because this part has higher

effective angle of attack.

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