SUPPLEMENTARY LOW SPEED AERODYNAMICS (DEN233)  Thin Boundary Lauer Equations 0 y V x U     0 1 2 2   y P y U x P y U V x U U              Momentum Integral Equation: 2 2 2 f e we e C U )H( xd Ud Uxd d     Thwaites Method xdu45.0 u x 0 5 e 6 e 2    xd Ud m e   2  09.0mseparation  0750.mstag  2 e w f u 2 C     xu Re ex  )m(S ue w    088.2 m14.0 0731.0 )m(H m107.0 m018.0 m402.122.0)m(S 0.0m0.1For       2m24.5m75.361.2)m(H 2m8.1m57.122.0)m(S 1.0m0For     Transition, Michel’s Method 4092 . trxtrx Re.)(Re   tre trx xU Re     eURe   Turbulent wall shear stress, you may assume this equation is valid for transitionxx  202028850 .xeW ReUρ.τ   Blasius Solution )(fUx)y,x( e   x U y e    y U    x V      δη)η(fη e δ e * U xν yd) U U (δ 0 0 1     δ ee yd) U U ( U U θ 0 1   δ η )η(f)η(f)η(f)η(f U xν e 0 2   * H  = ∂ ∂ = y U μτ x e Re fUρ 12  ν xU Re ex  ρ μ ν  f f  f  0 0 0 0.33206 0.2 0.00664 0.06641 0.3319 0.4 0.02656 0.13277 0.33147 0.6 0.05974 0.19894 0.33008 0.8 0.10611 0.26471 0.32739 1.0 0.16557 0.32979 0.32301 1.2 0.23795 0.39378 0.31659 1.4 0.32298 0.45627 0.30787 1.6 0.42032 0.51676 0.29667 1.8 0.52952 0.57477 0.28293 2.0 0.65003 0.62977 0.26675 2.2 0.78120 0.68132 0.24835 2.4 0.92230 0.72899 0.22809 2.6 1.07252 0.77246 0.20646 2.8 1.23099 0.81152 0.18401 3.0 1.39682 0.84605 0.16136 3.2 1.56911 0.87609 0.13913 3.4 1.74696 0.90177 0.11788 3.6 1.92954 0.92333 0.09809 3.8 2.11605 0.94112 0.08013 4.0 2.30576 0.95552 0.06424 4.2 2.49806 0.96696 0.05052 4.4 2.69238 0.97587 0.03897 4.6 2.88826 0.98269 0.02948 4.8 3.08534 0.98779 0.02187 5.0 3.28329 0.99155 0.01591 5.2 3.48189 0.99425 0.01134 5.4 3.68094 0.99616 0.00793 5.6 3.88031 0.99748 0.00543 5.8 4.07990 0.99838 0.00365 6.0 4.27964 0.99898 0.00240 6.2 4.47948 0.99937 0.00155 6.4 4.67938 0.99961 0.00098 6.6 4.87931 0.99977 0.00061                                              2 2 2 2 11 11 0 1 1              rr r rr )V( rr )rV( r )V( r )rV( r V, r V r V, r V r z r r r 2 2 2 2 2 0 yx y U x V y V x U y V, x U x V, y U z                                  Elementary Flows Uniform Flow   cosrUxU sinrUyU     Source/Sink rln π2 λ )yx(ln π4 λ φ θ π2 λ x y arctan π2 λ ψ 22 =+=.

1. SUPPLEMENTARY
LOW SPEED AERODYNAMICS (DEN233)
y
V
x
U
0
1

2. 2
2
y
P
y
U
x
P
y
U
V
x
U
U

3. 2
2
2
f
e
we
e
C

4. U
)H(
xd
Ud
Uxd
d
xdu45.0
u x
0
5
e
6
e
2

5. xd
Ud
m e
2
e
w
f
u
2
C
xu

6. ue
088.2
m14.0
0731.0
)m(H
m107.0
m018.0
m402.122.0)m(S
0.0m0.1For
2m24.5m75.361.2)m(H

7. 2m8.1m57.122.0)m(S
1.0m0For
4092 .
trxtrx
tre
trx
xU

8. valid for
202028850 .xeW ReUρ.τ
Solution
U
y e
y
U

9. x
V
e
δ
e
*
U

10. xν
yd)
U
U
(δ 0
0
δ
ee
yd)
U
U
(
U

11. U
θ
0
η
)η(f)η(f)η(f)η(f
U
xν
e
0

12. ∂
∂
=
y
U
μτ
x
e
Re
fUρ
ν
xU
ρ
μ

13. 0 0 0 0.33206
0.2 0.00664 0.06641 0.3319
0.4 0.02656 0.13277 0.33147
0.6 0.05974 0.19894 0.33008
0.8 0.10611 0.26471 0.32739
1.0 0.16557 0.32979 0.32301
1.2 0.23795 0.39378 0.31659
1.4 0.32298 0.45627 0.30787
1.6 0.42032 0.51676 0.29667
1.8 0.52952 0.57477 0.28293
2.0 0.65003 0.62977 0.26675
2.2 0.78120 0.68132 0.24835
2.4 0.92230 0.72899 0.22809
2.6 1.07252 0.77246 0.20646
2.8 1.23099 0.81152 0.18401
3.0 1.39682 0.84605 0.16136
3.2 1.56911 0.87609 0.13913
3.4 1.74696 0.90177 0.11788
3.6 1.92954 0.92333 0.09809

14. 3.8 2.11605 0.94112 0.08013
4.0 2.30576 0.95552 0.06424
4.2 2.49806 0.96696 0.05052
4.4 2.69238 0.97587 0.03897
4.6 2.88826 0.98269 0.02948
4.8 3.08534 0.98779 0.02187
5.0 3.28329 0.99155 0.01591
5.2 3.48189 0.99425 0.01134
5.4 3.68094 0.99616 0.00793
5.6 3.88031 0.99748 0.00543
5.8 4.07990 0.99838 0.00365
6.0 4.27964 0.99898 0.00240
6.2 4.47948 0.99937 0.00155
6.4 4.67938 0.99961 0.00098
6.6 4.87931 0.99977 0.00061

17. 2
2

18. 2
2 11
11
0
1
1

19. rr
r
rr
)V(
rr
)rV(
r

20. )V(
r
)rV(
r
V,
r
V
r
V,
r
V
r
z
r
r

21. r
2
2
2
2
2
0
yx
y
U
x
V
y

22. V
x
U
y
V,
x
U
x
V,
y
U
z

25. Elementary Flows
Uniform Flow
cosrUxU
sinrUyU

26. Source/Sink
rln
π2
λ
)yx(ln
π4
λ
φ
θ
π2
λ
x
y
arctan
π2

27. λ
ψ
22 =+=
==
Vortex
θ
π2
Γ
x
y
arctan
π2
Γ

28. φ
rln
π2
Γ
)yx(ln
π4
Γ
ψ 22
-=-=
=+=
Doublet
r
cos

29. yx
x
r
sin
yx
y

30. 22
22
22
22

31. Page 2 DEN233 (JUNE 2020)
For air: r=1.225 kg/m3 and n =1.46´10-5 m2/s
Question 1
Part a: Only one answer is most appropriate in each of the
following questions. For
each question write the number of your answer in your Answer
Booklet. Please note
that you must provide a clear reasoning and explanation why
you selected a
particular answer, otherwise you will not get any marks.

32. i) Induced drag on a wing is:
1) Zero;
2) a function of circulation;
3) always negative;
4) negative or positive depending on the direction of the lift
force.
ii) Which of the following statements is true?
1) The drag of a blunt object is less than the drag of an
equivalent streamlined
object at low Reynolds number;
2) The drag of a blunt object is always more than that of an
equivalent
3) streamlined object;
4) At high Reynolds numbers the drag of a blunt object is
always less than that
of an equivalent streamlined object;
5) None of the above.

33. iii) The transition of the laminar flows over wings can be
delayed by:
1) Applying adverse pressure gradient;
2) Reducing the surface roughness;
3) Heating the flow;
4) Increasing free stream turbulence.
iv) Which of the following statements is false?
1) Turbulent Boundary Layers are less resistive to adverse
pressure gradients;
2) Flow separation increases the form drag;
3) Reattachment of the boundary layer enhances the lift;
4) None of the above.
Question 1continues on the next page
or

34. DEN233 (JUNE 2020) Page 3
Question 1 continued
v) In an aircraft having laminar boundary layer over the wings
is preferable:
1) During the take-off;
2) During the landing;
3) During the cruise;
4) During the climb.
[2.0 marks each, 10.0 marks in total]
Part b:
1- In aeronautical applications explain what is meant by flow
control. Describe
passive and active control methods and provide an example for
an active control
method commonly used for the control of boundary layer
separation. Use a
sketch to illustrate and explain your answer.
[6 marks]

35. 2- The current aims of the European Commission in relation to
the Greening the Air
transport is to fund research in the areas related to the drag
reduction of the
aircraft by maintaining laminar flow as much as possible over
different parts of
the aircraft. Explain what the direct and indirect benefits of
laminar flows are and
which areas of an aircraft are targeted for laminar flows.
[6 marks]
Part c:
Provide a clear picture on the effects of the adverse pressure
gradient on the boundary
layer flows; in particular explain how adverse pressure gradient
affects the boundary
layer physical thickness and velocity distribution. Use sketches
with proper labelling to
illustrate your answer. During the cruise flight of a typical
civilian aircraft some parts of
the aircraft could be exposed to adverse pressure gradient.
Provide two examples and

36. use sketches to illustrate your answer.
[6 marks]
Part d:
A thin two-dimensional aerofoil with a chord of 2.0 m is
subjected to an external flow in
which is changing linearly with distance ( corresponds to the
leading edge of
the aerofoil) according to equation 1:
(Equation 1)
where and are two constants to be determined. Note that the
constant has a
unit of meter.
Question 1 continues on the next page
eU 0.0=x

37. )1(
A
xUU refe +=
refU A A
Page 4 DEN233 (JUNE 2020)
Question 1 continued
1) Surface flow visualisation and flow measurement indicate
that the boundary layer
on the one side of the aerofoil does separate at a point where the
local free
stream velocity and the local boundary layer momentum
thickness are 13.153 m/s
and 0.9359 mm respectively. Based on the measured data
determine .
and .
[8 marks]

38. 2) Determine the wall shear stress, boundary layer shape factor
and boundary layer
displacement thickness at point A ( m). Compare the boundary
layer
shape factor at point A with its corresponding value at the
separation point.
Clearly discuss the reason(s) behind the increase in shape factor
from point A to
the separation point. You should provide a sketch with proper
labelling to justify
and illustrate your answer. For this part you may assume m/s
and ;
[6 marks]
3) With clear explanations provide two flow control methods
(one active and one
passive) that can be used in order to delay the boundary layer
transition over the
wings. Use sketches in order to illustrate your answer
[4 marks]

39. 4) Clearly explain the effects of free stream turbulence,
favourable pressure gradient
and noise level on the boundary layer separation.
[4 marks]
Turn Over
refU
A sepx

40. 5.0=Ax
15=refU
0.10-=A
To
DEN233 (JUNE 2020) Page 5
Question 2
a) Provide a clear explanation of the physical meaning of the
boundary layer
displacement thickness. Explain how displacement thickness
could be used to
design the contour of ducts, for example the working section of
a wind tunnel.
Use sketch(s) with sufficient labelling to illustrate your answer.
[6 marks]

41. b) According to the thin boundary layer approximation, the
changes in pressure
across the boundary layer can be ignored. With sufficient
explanation provide
two examples in which this approximation could not be used.
[4 marks]
c) According to Prandtl the mean velocity for an incompressible
turbulent boundary
layer at zero pressure gradient can be approximated by a one-
seventh-power
law:
.
By using von-Karman integral equation, show that:
and .
You may assume .
[5 marks]
d) A simple wing at zero angle of attack is placed in a uniform

42. incompressible flow.
The drag of the wing has been measured for two extreme
conditions. One,
complete laminar flow over the wing and one complete
turbulent flow over the
wing. With sufficient and clear reasoning compare the power
requirement for the
turbulent case with the laminar case. Would you arrive at the
same answer if the
wing was exposed to adverse pressure gradient?
[5 marks]
e) With a clear physical argument explain the production of the
Reynolds shear and
normal stresses. Use sketches with proper details to illustrate
your answer.
[5 marks]
Turn over
7

43. 1
)(
dd
y
u
u
=
7/6Re162.0Re x»d 7/1Re
027.0
x
fC »
1/6/0.02≈ δf ReC
Page 6 DEN233 (JUNE 2020)

44. Question 3
A mountain ridge of maximum height m appears in a cross
section as the
upper half of a Rankine semi-infinite half-body. There is a cross
wind blowing in the
uphill direction with a uniform profile and free stream m/s. The
resulting two-
dimensional flow field, as shown in Figure Q-B2, can be
modelled as a superposition
of uniform flow along the x-axis and a source of strength
located in the origin of
the coordinate system.
Note: the ‘invisible’ lower part, mirrored along the x-axis, is
included in this
description.
Figure Q-B2
i) Give the stream function to describe the two-dimensional
flow.

45. Note: You should consider the complete semi-infinite half-body
including the invisible
lower part, mirrored along the x-axis;
[3 marks]
ii) Give expressions for the Cartesian velocity components and
V;
[3 marks]
Question 3 continues on the next page
100H =
5U =∞
λ
ψ
U

46. T
Y
DEN233 (JUNE 2020) Page 7
Question 3 continued
iii) Evaluate the strength of the source;
[5 marks]
iv) A sailplane as shown in Figure Q-B2, is flying at an altitude
of above the
source’s origin, i.e. at the coordinates ( , ), in descent flight
with a vertical
velocity of m/s.
Is the vertical velocity component created at the ridge sufficient
for the sailplane to
gain altitude? Clearly explain your result.

47. [6 marks]
v) What is the pressure coefficient at the ground station ( )
located at
coordinates ( shown in Figure Q-B2?
Note: Evaluate the value of the dividing streamline; secondly
find an equation the
dividing streamline, to evaluate the location of the ground
station. Finally use
Bernoulli’s equation to obtain pressure from velocity. In
answering part 5, you
must accompany your numerical calculation by physical
explanation, for example
how have you decided on the value of the dividing stream
function.
[8 marks]
End of the Examination Paper
λ
H2

48. 0 H2
-1.0=SailplaneV
V
PGC GP
)Y;X( GG
GX
omg