fluid_mechanics

356 Chapter 7 ■ Dimensional Analysis, Similitude, and Modeling
Note that this model design requires not only geometric scaling, as specified by Eq. 7.13, but also
the correct scaling of the velocity in accordance with Eq. 7.14. This result is typical of most model
designs—there is more to the design than simply scaling the geometry!
With the foregoing similarity requirements satisfied, the prediction equation for the drag is
or
d
w 2 rV 2
d m
w 2 mr m V 2 m
d a w 2
b a r b a V 2
b d
w m r m
m V m
Similarity between
a model and a prototype
is achieved
by equating pi
terms.
V7.9 Environmental
models
Thus, a measured drag on the model, d m , must be multiplied by the ratio of the square of the plate
widths, the ratio of the fluid densities, and the ratio of the square of the velocities to obtain the
predicted value of the prototype drag, d.
Generally, as is illustrated in this example, to achieve similarity between model and prototype
behavior, all the corresponding pi terms must be equated between model and prototype.
Usually, one or more of these pi terms will involve ratios of important lengths 1such as wh in
the foregoing example2; that is, they are purely geometrical. Thus, when we equate the pi terms
involving length ratios, we are requiring that there be complete geometric similarity between
the model and prototype. This means that the model must be a scaled version of the prototype.
Geometric scaling may extend to the finest features of the system, such as surface roughness,
or small protuberances on a structure, since these kinds of geometric features may significantly
influence the flow. Any deviation from complete geometric similarity for a model must be
carefully considered. Sometimes complete geometric scaling may be difficult to achieve, particularly
when dealing with surface roughness, since roughness is difficult to characterize and
control.
Another group of typical pi terms 1such as the Reynolds number in the foregoing example2
involves force ratios as noted in Table 7.1. The equality of these pi terms requires the ratio
of like forces in model and prototype to be the same. Thus, for flows in which the Reynolds
numbers are equal, the ratio of viscous forces in model and prototype is equal to the ratio of inertia
forces. If other pi terms are involved, such as the Froude number or Weber number, a similar
conclusion can be drawn; that is, the equality of these pi terms requires the ratio of like
forces in model and prototype to be the same. Thus, when these types of pi terms are equal in
model and prototype, we have dynamic similarity between model and prototype. It follows that
with both geometric and dynamic similarity the streamline patterns will be the same and corresponding
velocity ratios 1V mV2 and acceleration ratios 1a ma2 are constant throughout the flow
field. Thus, kinematic similarity exists between model and prototype. To have complete similarity
between model and prototype, we must maintain geometric, kinematic, and dynamic similarity
between the two systems. This will automatically follow if all the important variables are included
in the dimensional analysis, and if all the similarity requirements based on the resulting
pi terms are satisfied.
F l u i d s i n t h e N e w s
Modeling parachutes in a water tunnel The first use of a parachute
with a free-fall jump from an aircraft occurred in 1914, although
parachute jumps from hot air balloons had occurred since the
late 1700s. In more modern times parachutes are commonly used by
the military, and for safety and sport. It is not surprising that there remains
interest in the design and characteristics of parachutes, and researchers
at the Worcester Polytechnic Institute have been studying
various aspects of the aerodynamics associated with parachutes. An
unusual part of their study is that they are using small-scale parachutes
tested in a water tunnel. The model parachutes are reduced in
size by a factor of 30 to 60 times. Various types of tests can be performed,
ranging from the study of the velocity fields in the wake of
the canopy with a steady free-stream velocity to the study of conditions
during rapid deployment of the canopy. According to the researchers,
the advantage of using water as the working fluid, rather
than air, is that the velocities and deployment dynamics are slower
than in the atmosphere, thus providing more time to collect detailed
experimental data. (See Problem 7.47.)