fluid_mechanics

354 Chapter 7 ■ Dimensional Analysis, Similitude, and Modeling
Π 1
k – r = 3
Π 3 = C 1 (constant)
Π 3 = C 2
Π 3 = C 3
Π 3 = C 4
Π
F I G U R E 7.5 The graphical
presentation of data for problems
involving three pi terms.
way. It may also be possible to determine a suitable empirical equation relating the three pi terms.
However, as the number of pi terms continues to increase, corresponding to an increase in the general
complexity of the problem of interest, both the graphical presentation and the determination
of a suitable empirical equation become intractable. For these more complicated problems, it is often
more feasible to use models to predict specific characteristics of the system rather than to try
to develop general correlations.
7.8 Modeling and Similitude
V
V m
Prototype
Model
V7.8 Model
airplane
m
Models are widely used in fluid mechanics. Major engineering projects involving structures, aircraft,
ships, rivers, harbors, dams, air and water pollution, and so on, frequently involve the use of
models. Although the term “model” is used in many different contexts, the “engineering model”
generally conforms to the following definition. A model is a representation of a physical system
that may be used to predict the behavior of the system in some desired respect. The physical system
for which the predictions are to be made is called the prototype. Although mathematical or
computer models may also conform to this definition, our interest will be in physical models, that
is, models that resemble the prototype but are generally of a different size, may involve different
fluids, and often operate under different conditions 1pressures, velocities, etc.2. As shown by the
figure in the margin, usually a model is smaller than the prototype. Therefore, it is more easily
handled in the laboratory and less expensive to construct and operate than a large prototype (it
should be noted that variables or pi terms without a subscript will refer to the prototype, whereas
the subscript m will be used to designate the model variables or pi terms). Occasionally, if the prototype
is very small, it may be advantageous to have a model that is larger than the prototype so
that it can be more easily studied. For example, large models have been used to study the motion
of red blood cells, which are approximately 8 mm in diameter. With the successful development of
a valid model, it is possible to predict the behavior of the prototype under a certain set of conditions.
We may also wish to examine a priori the effect of possible design changes that are proposed
for a hydraulic structure or fluid-flow system. There is, of course, an inherent danger in the
use of models in that predictions can be made that are in error and the error not detected until the
prototype is found not to perform as predicted. It is, therefore, imperative that the model be properly
designed and tested and that the results be interpreted correctly. In the following sections we
will develop the procedures for designing models so that the model and prototype will behave in
a similar fashion.
7.8.1 Theory of Models
The theory of models can be readily developed by using the principles of dimensional analysis. It
has been shown that any given problem can be described in terms of a set of pi terms as
ß 1 f1ß 2 , ß 3 , . . . , ß n 2
(7.7)
In formulating this relationship, only a knowledge of the general nature of the physical phenomenon,
and the variables involved, is required. Specific values for variables 1size of components, fluid
properties, and so on2 are not needed to perform the dimensional analysis. Thus, Eq. 7.7 applies