fluid_mechanics

364 Chapter 7 ■ Dimensional Analysis, Similitude, and Modeling
It is clear from Eq. 7.19 that geometric similarity
For flow around
bodies, drag is often
the dependent
variable of interest.
as well as Reynolds number similarity
must be maintained. If these conditions are met, then
or
/ im
/ i
/ m /
r m V m / m
m m
e m
/ m
e /
rV/
m
d m
d
1
2rV 2 / 2 1
2r m V 2 2
m / m
d r a V 2
b a / 2
b d
r m V m / m
m
Measurements of model drag, d m , can then be used to predict the corresponding drag, d, on the
prototype from this relationship.
As was discussed in the previous section, one of the common difficulties with models is related
to the Reynolds number similarity requirement which establishes the model velocity as
or
V m m m
m
V m n m
n
r
r m
/
/ m
V
/
/ m
V
(7.20)
(7.21)
where n m n is the ratio of kinematic viscosities. If the same fluid is used for model and prototype
so that n m n, then
V m / / m
V
V7.14 Model airplane
test in water
V7.15 Large scale
wind tunnel
and, therefore, the required model velocity will be higher than the prototype velocity for // m
greater than 1. Since this ratio is often relatively large, the required value of V m may be large. For
1
example, for a 10 length scale, and a prototype velocity of 50 mph, the required model velocity is
500 mph. This is a value that is unreasonably high to achieve with liquids, and for gas flows this
would be in the range where compressibility would be important in the model 1but not in the
prototype2.
As an alternative, we see from Eq. 7.21 that V m could be reduced by using a different fluid
in the model such that n m n 6 1. For example, the ratio of the kinematic viscosity of water to that
1
of air is approximately 10, so that if the prototype fluid were air, tests might be run on the model
using water. This would reduce the required model velocity, but it still may be difficult to achieve
the necessary velocity in a suitable test facility, such as a water tunnel.
Another possibility for wind tunnel tests would be to increase the air pressure in the tunnel
so that r m 7 r, thus reducing the required model velocity as specified by Eq. 7.20. Fluid viscosity
is not strongly influenced by pressure. Although pressurized tunnels have been used, they are
obviously more complicated and expensive.
The required model velocity can also be reduced if the length scale is modest; that is, the
model is relatively large. For wind tunnel testing, this requires a large test section which greatly
increases the cost of the facility. However, large wind tunnels suitable for testing very large models
1or prototypes2 are in use. One such tunnel, located at the NASA Ames Research Center, Moffett
Field, California, has a test section that is 40 ft by 80 ft and can accommodate test speeds to
345 mph. Such a large and expensive test facility is obviously not feasible for university or industrial
laboratories, so most model testing has to be accomplished with relatively small models.