fluid_mechanics

7.9 Some Typical Model Studies 365
E XAMPLE 7.7
Model Design Conditions and Predicted Prototype Performance
GIVEN The drag on the airplane shown in Fig. E7.7 cruising
at 240 mph in standard air is to be determined from tests on a 1:10
scale model placed in a pressurized wind tunnel. To minimize
compressibility effects, the air speed in the wind tunnel is also to
be 240 mph.
FIND Determine
(a) the required air pressure in the tunnel (assuming the same
air temperature for model and prototype) and
SOLUTION
(a) From Eq. 7.19 it follows that drag can be predicted from a
geometrically similar model if the Reynolds numbers in model
and prototype are the same. Thus,
For this example, V and / m/ 1 m V
10 so that
and therefore
r m V m / m
m m
r m
r m m
m
rV/
m
V /
V m
This result shows that the same fluid with r m r and m m m
cannot be used if Reynolds number similarity is to be maintained.
One possibility is to pressurize the wind tunnel to increase the
density of the air. We assume that an increase in pressure does not
significantly change the viscosity so that the required increase in
density is given by the relationship
r m
r 10
For an ideal gas, p rRT so that
p m
p r m
r
/ m
m m
m 1121102
r m
r 10 m m
m
(b) the drag on the prototype corresponding to a measured force
of 1 lb on the model.
V = 240 mph
F I G U R E E7.7
for constant temperature 1T T m 2. Therefore, the wind tunnel
would need to be pressurized so that
Since the prototype operates at standard atmospheric pressure,
the required pressure in the wind tunnel is 10 atmospheres or
(Ans)
COMMENT Thus, we see that a high pressure would be required
and this could not be achieved easily or inexpensively. However,
under these conditions, Reynolds similarity would be attained.
(b) The drag could be obtained from Eq. 7.19 so that
or
p m 10 114.7 psia2
147 psia
d
1
2 rV 2 / d m
2 1
2 r mVm/ 2 2 m
d r a V 2
b a / 2
b d
r m V m / m
m
a 1
10 b 1122 1102 2 d m
10d m
p m
p 10
Thus, for a drag of 1 lb on the model the corresponding drag on
the prototype is
d 10 lb
(Ans)
V7.16 Wind tunnel
train model
Fortunately, in many situations the flow characteristics are not strongly influenced by the
Reynolds number over the operating range of interest. In these cases we can avoid the rather stringent
similarity requirement of matching Reynolds numbers. To illustrate this point, consider the
variation in the drag coefficient with the Reynolds number for a smooth sphere of diameter d placed
in a uniform stream with approach velocity, V. Some typical data are shown in Fig. 7.7. We observe
that for Reynolds numbers between approximately 10 3 and 2 10 5 the drag coefficient is
relatively constant and does not strongly depend on the specific value of the Reynolds number.
Thus, exact Reynolds number similarity is not required in this range. For other geometric shapes
we would typically find that for high Reynolds numbers, inertial forces are dominant 1rather than
viscous forces2, and the drag is essentially independent of the Reynolds number.