external flow example - Example Air enters a...

Example: Air enters a two-dimensional wind tunnel as shown. Because of the presence
of the boundary layer, fluid will be displaced away from the surface. In order to maintain
a constant velocity inside the tunnel, it is necessary to increase the cross-sectional size of
the tunnel.
(a) Determine the channel height, H(x), as a function of the distance measured from
the inlet of the tunnel, x. The tunnel velocity is 10 m/s and the tunnel inlet height
is 5 m. Assume the boundary layer has a profile u(y)=U
∞
sin( π y /2δ29.
(b) What is the momentum thickness θ (x) of the boundary layer flow.
(c) If a sphere has a diameter of 0.1 m is placed in the center of the wind tunnel, what
is the drag force exerted on the sphere. ρ
air
=1.2 kg/m
3
, ν =1.5x10
-5
m
2
/s.
C
D
U
∞
=10 m/s
5
m
x
H(x)
Sphere, 0.1
m diameter
C
D
=24/Re, when Re<1
Re=VD/ ν
0.5
400
300,000
C
D
=24/Re
0.646
,
when
1<Re<400

In order to maintain a constant velocity inside the tunnel, the tunnel wall has to be displaced
outward in order to accomodate the growth of the boundary layer.