fluid_mechanics

360 Chapter 7 ■ Dimensional Analysis, Similitude, and Modeling
models, and essentially each problem must be considered on its own merits. The success of using
distorted models depends to a large extent on the skill and experience of the investigator responsible
for the design of the model and in the interpretation of experimental data obtained from the model.
Distorted models are widely used, and additional information can be found in the references at the
end of the chapter. References 14 and 15 contain detailed discussions of several practical examples
of distorted fluid flow and hydraulic models.
F l u i d s i n t h e N e w s
Old Man River in (large) miniature One of the world’s
largest scale models, a Mississippi River model, resides near
Jackson, Mississippi. It is a detailed, complex model that covers
many acres and replicates the 1,250,000 acre Mississippi
River basin. Built by the Army Corps of Engineers and used
from 1943 to 1973, today it has mostly gone to ruin. As with
many hydraulic models, this is a distorted model, with a horizontal
scale of 1 to 2000 and a vertical scale of 1 to 100. One
step along the model river corresponds to one mile along the
river. All essential river basin elements such as geological features,
levees, and railroad embankments were sculpted by hand
to match the actual contours. The main purpose of the model
was to predict floods. This was done by supplying specific
amounts of water at prescribed locations along the model and
then measuring the water depths up and down the model river.
Because of the length scale, there is a difference in the time
taken by the corresponding model and prototype events. Although
it takes days for the actual floodwaters to travel from
Sioux City, Iowa, to Omaha, Nebraska, it would take only minutes
for the simulated flow in the model.
7.9 Some Typical Model Studies
Models are used to investigate many different types of fluid mechanics problems, and it is difficult
to characterize in a general way all necessary similarity requirements, since each problem is
unique. We can, however, broadly classify many of the problems on the basis of the general nature
of the flow and subsequently develop some general characteristics of model designs in each
of these classifications. In the following sections we will consider models for the study of 112 flow
through closed conduits, 122 flow around immersed bodies, and 132 flow with a free surface. Turbomachine
models are considered in Chapter 12.
Geometric and
Reynolds number
similarity is usually
required for models
involving flow
through closed
conduits.
7.9.1 Flow through Closed Conduits
Common examples of this type of flow include pipe flow and flow through valves, fittings, and
metering devices. Although the conduit cross sections are often circular, they could have other
shapes as well and may contain expansions or contractions. Since there are no fluid interfaces
or free surfaces, the dominant forces are inertial and viscous so that the Reynolds number is an
important similarity parameter. For low Mach numbers 1Ma 6 0.32, compressibility effects are
usually negligible for both the flow of liquids or gases. For this class of problems, geometric
similarity between model and prototype must be maintained. Generally the geometric characteristics
can be described by a series of length terms, / 1 , / 2 , / 3 , . . . , / i , and /, where / is some particular
length dimension for the system. Such a series of length terms leads to a set of pi terms
of the form
ß i / i
/
where i 1, 2, . . . , and so on. In addition to the basic geometry of the system, the roughness of
the internal surface in contact with the fluid may be important. If the average height of surface
roughness elements is defined as e, then the pi term representing roughness will be e/. This parameter
indicates that for complete geometric similarity, surface roughness would also have to be
scaled. Note that this implies that for length scales less than 1, the model surfaces should be
smoother than those in the prototype since e m l / e. To further complicate matters, the pattern of
roughness elements in model and prototype would have to be similar. These are conditions that are
virtually impossible to satisfy exactly. Fortunately, in some problems the surface roughness plays