fluid_mechanics

7.11 Chapter Summary and Study Guide 373
advantage that the variables are known and the assumptions involved are clearly identified. In addition,
a physical interpretation of the various dimensionless groups can often be obtained.
7.11 Chapter Summary and Study Guide
similitude
dimensionless product
basic dimensions
pi term
Buckingham pi theorem
method of repeating
variables
model
modeling laws
prototype
prediction equation
model design conditions
similarity requirements
modeling laws
length scale
distorted model
true model
Many practical engineering problems involving fluid mechanics require experimental data for their
solution. Thus, laboratory studies and experimentation play a significant role in this field. It is important
to develop good procedures for the design of experiments so they can be efficiently completed
with as broad applicability as possible. To achieve this end the concept of similitude is often used in
which measurements made in the laboratory can be utilized for predicting the behavior of other similar
systems. In this chapter, dimensional analysis is used for designing such experiments, as an aid
for correlating experimental data, and as the basis for the design of physical models. As the name
implies, dimensional analysis is based on a consideration of the dimensions required to describe the
variables in a given problem. A discussion of the use of dimensions and the concept of dimensional
homogeneity (which forms the basis for dimensional analysis) was included in Chapter 1.
Essentially, dimensional analysis simplifies a given problem described by a certain set of variables
by reducing the number of variables that need to be considered. In addition to being fewer in
number, the new variables are dimensionless products of the original variables. Typically these new
dimensionless variables are much simpler to work with in performing the desired experiments. The
Buckingham pi theorem, which forms the theoretical basis for dimensional analysis, is introduced. This
theorem establishes the framework for reducing a given problem described in terms of a set of variables
to a new set of fewer dimensionless variables. A simple method, called the repeating variable
method, is described for actually forming the dimensionless variables (often called pi terms). Forming
dimensionless variables by inspection is also considered. It is shown how the use of dimensionless variables
can be of assistance in planning experiments and as an aid in correlating experimental data.
For problems in which there are a large number of variables, the use of physical models is
described. Models are used to make specific predictions from laboratory tests rather than formulating
a general relationship for the phenomenon of interest. The correct design of a model is
obviously imperative for the accurate predictions of other similar, but usually larger, systems. It
is shown how dimensional analysis can be used to establish a valid model design. An alternative
approach for establishing similarity requirements using governing equations (usually differential
equations) is presented.
The following checklist provides a study guide for this chapter. When your study of the
entire chapter and end-of-chapter exercies has been completed you should be able to
write out meanings of the terms listed here in the margin and understand each of the related
concepts. These terms are particularly important and are set in italic, bold, and color type
in the text.
use the Buckingham pi theorem to determine the number of independent dimensionless variables
needed for a given flow problem.
form a set of dimensionless variables using the method of repeating variables.
form a set of dimensionless variables by inspection.
use dimensionless variables as an aid in interpreting and correlating experimental data.
use dimensional analysis to establish a set of similarity requirements (and prediction equation)
for a model to be used to predict the behavior of another similar system (the prototype).
rewrite a given governing equation in a suitable nondimensional form and deduce similarity
requirements from the nondimensional form of the equation.
Some of the important equations in this chapter are:
Reynolds number
Re rV/
m
Froude number Fr V
1g/