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