Exam TEP4160, 2013
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Norwegian University of Science and Technology
NTNU
Faculty of Engineering Sciences and Technology
Dept. of Energy and Process Technology
Contact persons during the exam:
Per-Åge Krogstad, ph.: 93710
ENGLISH VERSION
Exam TEP 4160 Aerodynamics
Date: 4. June 2013
Time: kl. 09:00-13:00
Allowed material: B)
- Approved calculator with empty memory.
- All handwritten or printed material.
Problem 1
We are interested in the flow past an airfoil as shown in Figure 1. The airfoil is flying at a
velocity Uinf and is operated with a blown flap as indicated. This produces a strong, coupled
wake as sketched in the figure. The wake behind the flap is the smallest and has a width of
2a in the z-direction, perpendicular to the flow. Similarly, there is a wake behind the main
airfoil which has a width of 2b. Let us for simplicity assume that the velocity in the two
wakes varies linearly with z and that the depth of the wake is the same in both cases, giving
a minimum velocity Uinf /2.
As shown in the figure, the wake flow may be split into 5 velocity regions. If z is measured
upwards from where the two wakes intersect, the velocity distribution in the flap wake may
be written as
• Region 1: u = −Uinf (z/2a) for −2a ≤ z < −a
• Region 2: u = Uinf (1 + z/2a) for −a ≤ z < 0
while the velocity in the wing wake is
• Region 3: u = Uinf (1 − z/b) for 0 ≤ z < b/2
• Region 4: u = Uinf /2 for b/2 ≤ z < 3b/2
• Region 3: u = −Uinf (1 − z/b) for 3b/2 ≤ z < 2b
a) In the figure we have indicated two streamlines that enclose the airfoil and the wake. Far
upstream they originate from two positions that are separated by a distance ∆z = 2h in the
z direction. Find an expression that shows how h is related to a and b.
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Uinf/2
Uinf
Uinf
Exam TEP4160, 2013
5
4
2h
3
2
1
2b
3b/2 z
b/2
-a
-2a
Figure 1: Airfoil with deflected blown flap
b) From the momentum lost in the flow as it passes the airfoil, the drag may be computed if
you assume that the pressure along the two streamlines is constant. Find an expression for
the drag coefficient,
2D
CD =
2 c ,
ρUinf
as function of a/c and b/c. Here D is the calculated drag and c is the airfoil chord length.
If the wake behind the wing is four times as wide as the wake behind the flap (b = 4a), how
large fraction of the total drag comes from the wing and how much comes from the flap?
Problem 2
A small turboprop airplane has the following specifications:
• Wing span: b = 14.5m
• Wing area: A = 27m2
• Max take-off weight: m = 4050kg
• Cruise speed: Vcruise = 390km/h at altitude h = 10.000m
a) Assume that the wing is without sweep, twist or dihedral, and that the taper ratio=1.
Find the aspect ratio AR and the wing chord c.
The pilot is flying at 10.000m of altitude at cruise speed, in steady horizontal flight.
What is the lift coefficient CLcruise that the wing needs to produce, if you assume standard
atmosphere conditions?
b) The wing produces CL =0.1604 at zero angle of attack (α=0) and the lift-curve slope is
CL,α =0.0802. What is the angle of attack for zero lift, α0L ? The lift is linearly dependent on
the angle of attack for a range between α=-6 and 10 degrees. Sketch the CL curve in the linear
range, comparing the curve for the wing and the theoretical curve for the two-dimensional
airfoil. What is the efficiency factor e = for the wing?
Exam TEP4160, 2013
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c) Assume that the flight conditions in question a) apply. What is the wing angle of attack
αcruise needed to achieve them? What is the lift-induced drag coefficient, if the lift coefficient
at minimum drag is CLmin =0?
d) The pilot rolls the plane to 45 degrees banking and pulls the stick so that the plane
performs a coordinated turn at constant altitude. What angle of attack must the wing have
now in order to continue the turn at the same speed? What is the turn radius? What is the
ratio between the power required for the turn flight to the need at straight, level flight, if you
assume that all drag is caused by the drag generated by lift?
Problem 3
We want to design a thin airfoil with a small amount of camber. The camber line is approximated by two straight lines, given by
z ∗ = ax∗ for 0 ≤ x∗ < 0.2 , z ∗ =
a
(1 − x∗ ) for 0.2 ≤ x∗ ≤ 1 ,
4
(1)
where z ∗ = z/c and x∗ = x/c are the dimensionless coordinates perpendicular to and along
the camber line. a is an amplitude parameter.
a) We want the airfoil to have 2% camber. What is then the value to be used for a?
In order to make estimates of the airfoil performance we do a standard analysis for thin
cambered airfoil, building on the vorticity distribution for a symmetric airfoil. This is done
by adding an infinite Fourier series. In its general form, the Fourier series contains both sine
and cosine terms. Why are the cosine terms dropped in the thin airfoil analysis?
b) Determine A0 , A1 and A2 in the vorticity distribution.
c) Determine the lift and quarter chord moment coefficients, CL and CM (x∗ = 0.25).
d) What is the angle of attack for zero lift, α (CL = 0)? What is the lift coefficient at zero
angle of attack, CL (α = 0)?
What is the meaning of the aerodynamic centre and the centre of pressure of an airfoil. What
is the difference?
The centre of pressure, x∗cp , for a cambered airfoil is off-set from the aerodynamic centre,
which is located at x∗ac = 0.25. Where is x∗cp for this airfoil at zero angle of attack? And at
zero lift?