fluid_mechanics

7.4 Some Additional Comments about Dimensional Analysis 341
If these do not check, go back to the original list of variables and
make sure you have the correct dimensions for each of the variables
and then check the algebra you used to obtain the exponents
a, b, and c.
Finally, we can express the results of the dimensional analysis
in the form
d
w 2 V 2 r f˜ a h w , m
wVr b
(Ans)
Since at this stage in the analysis the nature of the function f˜ is
unknown, we could rearrange the pi terms if we so desire. For
example, we could express the final result in the form
d
w 2 rV f aw 2 h , rVw
m b
(Ans)
which would be more conventional, since the ratio of the plate
width to height, wh, is called the aspect ratio, and rVwm is the
Reynolds number.
COMMENT To proceed, it would be necessary to perform a
set of experiments to determine the nature of the function f, as
discussed in Section 7.7.
7.4 Some Additional Comments about Dimensional Analysis
The preceding section provides a systematic approach for performing a dimensional analysis. Other
methods could be used, although we think the method of repeating variables is the easiest for the
beginning student to use. Pi terms can also be formed by inspection, as is discussed in Section 7.5.
Regardless of the specific method used for the dimensional analysis, there are certain aspects of
this important engineering tool that must seem a little baffling and mysterious to the student 1and
sometimes to the experienced investigator as well2. In this section we will attempt to elaborate on
some of the more subtle points that, based on our experience, can prove to be puzzling to students.
It is often helpful to
classify variables
into three groups—
geometry, material
properties, and external
effects.
7.4.1 Selection of Variables
One of the most important, and difficult, steps in applying dimensional analysis to any given problem
is the selection of the variables that are involved. As noted previously, for convenience we will
use the term variable to indicate any quantity involved, including dimensional and nondimensional
constants. There is no simple procedure whereby the variables can be easily identified. Generally,
one must rely on a good understanding of the phenomenon involved and the governing physical
laws. If extraneous variables are included, then too many pi terms appear in the final solution, and
it may be difficult, time consuming, and expensive to eliminate these experimentally. If important
variables are omitted, then an incorrect result will be obtained; and again, this may prove to be
costly and difficult to ascertain. It is, therefore, imperative that sufficient time and attention be
given to this first step in which the variables are determined.
Most engineering problems involve certain simplifying assumptions that have an influence on
the variables to be considered. Usually we wish to keep the problem as simple as possible, perhaps
even if some accuracy is sacrificed. A suitable balance between simplicity and accuracy is a desirable
goal. How “accurate” the solution must be depends on the objective of the study; that is, we may be
only concerned with general trends and, therefore, some variables that are thought to have only a minor
influence in the problem may be neglected for simplicity.
For most engineering problems 1including areas outside of fluid mechanics2, pertinent variables
can be classified into three general groups—geometry, material properties, and external effects.
Geometry. The geometric characteristics can usually be described by a series of lengths
and angles. In most problems the geometry of the system plays an important role, and a sufficient
number of geometric variables must be included to describe the system. These variables can usually
be readily identified.
Material Properties. Since the response of a system to applied external effects such as
forces, pressures, and changes in temperature is dependent on the nature of the materials involved
in the system, the material properties that relate the external effects and the responses must be included
as variables. For example, for Newtonian fluids the viscosity of the fluid is the property
that relates the applied forces to the rates of deformation of the fluid. As the material behavior becomes
more complex, such as would be true for non-Newtonian fluids, the determination of material
properties becomes difficult, and this class of variables can be troublesome to identify.