fluid_mechanics

336 Chapter 7 ■ Dimensional Analysis, Similitude, and Modeling
the variables may be described by some combination of basic dimensions, such as MT 2 and L,
and in this case r would be equal to two rather than three. Although the use of the pi theorem
may appear to be a little mysterious and complicated, we will actually develop a simple, systematic
procedure for developing the pi terms for a given problem.
7.3 Determination of Pi Terms
A dimensional
analysis can be
performed using a
series of distinct
steps.
Several methods can be used to form the dimensionless products, or pi terms, that arise in a dimensional
analysis. Essentially we are looking for a method that will allow us to systematically form the pi
terms so that we are sure that they are dimensionless and independent, and that we have the right number.
The method we will describe in detail in this section is called the method of repeating variables.
It will be helpful to break the repeating variable method down into a series of distinct steps
that can be followed for any given problem. With a little practice you will be able to readily complete
a dimensional analysis for your problem.
Step 1 List all the variables that are involved in the problem. This step is the most difficult
one and it is, of course, vitally important that all pertinent variables be included. Otherwise
the dimensional analysis will not be correct! We are using the term “variable” to
include any quantity, including dimensional and nondimensional constants, which play a
role in the phenomenon under investigation. All such quantities should be included in
the list of “variables” to be considered for the dimensional analysis. The determination
of the variables must be accomplished by the experimenter’s knowledge of the problem
and the physical laws that govern the phenomenon. Typically the variables will include
those that are necessary to describe the geometry of the system 1such as a pipe diameter2,
to define any fluid properties 1such as a fluid viscosity2, and to indicate external
effects that influence the system 1such as a driving pressure drop per unit length2. These
general classes of variables are intended as broad categories that should be helpful in
identifying variables. It is likely, however, that there will be variables that do not fit easily
into one of these categories, and each problem needs to be carefully analyzed.
Since we wish to keep the number of variables to a minimum, so that we can minimize
the amount of laboratory work, it is important that all variables be independent. For
example, if in a certain problem the cross-sectional area of a pipe is an important variable,
either the area or the pipe diameter could be used, but not both, since they are obviously
not independent. Similarly, if both fluid density, r, and specific weight, g, are important
variables, we could list r and g, or r and g 1acceleration of gravity2, or g and g. However,
it would be incorrect to use all three since g rg; that is, r, g, and g are not independent.
Note that although g would normally be constant in a given experiment, that fact is irrelevant
as far as a dimensional analysis is concerned.
Step 2 Express each of the variables in terms of basic dimensions. For the typical fluid mechanics
problem the basic dimensions will be either M, L, and T or F, L, and T. Dimensionally
these two sets are related through Newton’s second law 1F ma2 so that F MLT 2 .
For example, r ML 3 or r FL 4 T 2 . Thus, either set can be used. The basic dimensions
for typical variables found in fluid mechanics problems are listed in Table 1.1 in Chapter 1.
Step 3 Determine the required number of pi terms. This can be accomplished by means of the
Buckingham pi theorem, which indicates that the number of pi terms is equal to k r,
where k is the number of variables in the problem 1which is determined from Step 12 and
r is the number of reference dimensions required to describe these variables 1which is determined
from Step 22. The reference dimensions usually correspond to the basic dimensions
and can be determined by an inspection of the dimensions of the variables obtained in Step
2. As previously noted, there may be occasions 1usually rare2 in which the basic dimensions
appear in combinations so that the number of reference dimensions is less than the
number of basic dimensions. This possibility is illustrated in Example 7.2.
Step 4 Select a number of repeating variables, where the number required is equal to the
number of reference dimensions. Essentially what we are doing here is selecting from
the original list of variables several of which can be combined with each of the remaining