fluid_mechanics

342 Chapter 7 ■ Dimensional Analysis, Similitude, and Modeling
External Effects. This terminology is used to denote any variable that produces, or tends
to produce, a change in the system. For example, in structural mechanics, forces 1either concentrated
or distributed2 applied to a system tend to change its geometry, and such forces would need
to be considered as pertinent variables. For fluid mechanics, variables in this class would be related
to pressures, velocities, or gravity.
The above general classes of variables are intended as broad categories that should be helpful
in identifying variables. It is likely, however, that there will be important variables that do not
fit easily into one of the above categories and each problem needs to be carefully analyzed.
Since we wish to keep the number of variables to a minimum, it is important that all variables
are independent. For example, if in a given problem we know that the moment of inertia
of the area of a circular plate is an important variable, we could list either the moment of inertia
or the plate diameter as the pertinent variable. However, it would be unnecessary to include
both moment of inertia and diameter, assuming that the diameter enters the problem only
through the moment of inertia. In more general terms, if we have a problem in which the variables
are
f 1 p, q, r, . . . , u, v, w, . . .2 0
and it is known that there is an additional relationship among some of the variables, for example,
q f 1 1u, v, w, . . .2
(7.4)
then q is not required and can be omitted. Conversely, if it is known that the only way the variables
u, v, w, . . . enter the problem is through the relationship expressed by Eq. 7.4, then the
variables u, v, w, . . . can be replaced by the single variable q, therefore reducing the number of
variables.
In summary, the following points should be considered in the selection of variables:
1. Clearly define the problem. What is the main variable of interest 1the dependent variable2?
2. Consider the basic laws that govern the phenomenon. Even a crude theory that describes the
essential aspects of the system may be helpful.
3. Start the variable selection process by grouping the variables into three broad classes: geometry,
material properties, and external effects.
4. Consider other variables that may not fall into one of the above categories. For example, time
will be an important variable if any of the variables are time dependent.
5. Be sure to include all quantities that enter the problem even though some of them may be
held constant 1e.g., the acceleration of gravity, g2. For a dimensional analysis it is the dimensions
of the quantities that are important—not specific values!
6. Make sure that all variables are independent. Look for relationships among subsets of the
variables.
(7.3)
Typically, in fluid
mechanics, the required
number of
reference dimensions
is three, but
in some problems
only one or two are
required.
7.4.2 Determination of Reference Dimensions
For any given problem it is obviously desirable to reduce the number of pi terms to a minimum
and, therefore, we wish to reduce the number of variables to a minimum; that is, we certainly do
not want to include extraneous variables. It is also important to know how many reference dimensions
are required to describe the variables. As we have seen in the preceding examples, F, L, and
T appear to be a convenient set of basic dimensions for characterizing fluid-mechanical quantities.
There is, however, really nothing “fundamental” about this set, and as previously noted M, L, and
T would also be suitable. Actually any set of measurable quantities could be used as basic dimensions
provided that the selected combination can be used to describe all secondary quantities. However,
the use of FLT or MLT as basic dimensions is the simplest, and these dimensions can be used
to describe fluid-mechanical phenomena. Of course, in some problems only one or two of these
are required. In addition, we occasionally find that the number of reference dimensions needed to
describe all variables is smaller than the number of basic dimensions. This point is illustrated in
Example 7.2. Interesting discussions, both practical and philosophical, relative to the concept of
basic dimensions can be found in the books by Huntley 1Ref. 42 and by Isaacson and Isaacson
1Ref. 122.