fluid_mechanics

7.1 Dimensional Analysis 333
to study the phenomenon of interest under carefully controlled conditions. From these model studies,
empirical formulations can be developed, or specific predictions of one or more characteristics of
some other similar system can be made. To do this, it is necessary to establish the relationship between
the laboratory model and the “other” system. In the following sections, we find out how this
can be accomplished in a systematic manner.
F l u i d s i n t h e N e w s
Model study of New Orleans levee breach caused by
Hurricane Katrina Much of the devastation to New Orleans from
Hurricane Katrina in 2005 was a result of flood waters that surged
through a breach of the 17th Street Outfall Canal. To better understand
why this occurred and to determine what can be done to prevent
future occurrences, the U.S. Army Engineer Research and
Development Center Coastal and Hydraulics Laboratory is conducting
tests on a large (1:50 length scale) 15,000 square foot hydraulic
model that replicates 0.5 mile of the canal surrounding the
breach and more than a mile of the adjacent Lake Pontchartrain
front. The objective of the study is to obtain information regarding
the effect that waves had on the breaching of the canal and to investigate
the surging water currents within the canals. The waves
are generated by computer-controlled wave generators that can
produce waves of varying heights, periods, and directions similar to
the storm conditions that occurred during the hurricane. Data from
the study will be used to calibrate and validate information that
will be fed into various numerical model studies of the disaster.
7.1 Dimensional Analysis
It is important to
develop a meaningful
and systematic
way to perform an
experiment.
To illustrate a typical fluid mechanics problem in which experimentation is required, consider the
steady flow of an incompressible Newtonian fluid through a long, smooth-walled, horizontal, circular
pipe. An important characteristic of this system, which would be of interest to an engineer
designing a pipeline, is the pressure drop per unit length that develops along the pipe as a result
of friction. Although this would appear to be a relatively simple flow problem, it cannot generally
be solved analytically 1even with the aid of large computers2 without the use of experimental
data.
The first step in the planning of an experiment to study this problem would be to decide on
the factors, or variables, that will have an effect on the pressure drop per unit length,
¢p / 31lbft 2 2ft lbft 3 or Nm 3 4. We expect the list to include the pipe diameter, D, the fluid density,
r, fluid viscosity, m, and the mean velocity, V, at which the fluid is flowing through the pipe.
Thus, we can express this relationship as
¢p / f 1D, r, m, V2
which simply indicates mathematically that we expect the pressure drop per unit length to be some
function of the factors contained within the parentheses. At this point the nature of the function is
unknown and the objective of the experiments to be performed is to determine the nature of this
function.
To perform the experiments in a meaningful and systematic manner, it would be necessary
to change one of the variables, such as the velocity, while holding all others constant, and measure
the corresponding pressure drop. This series of tests would yield data that could be represented
graphically as is illustrated in Fig. 7.1a. It is to be noted that this plot would only be valid
for the specific pipe and for the specific fluid used in the tests; this certainly does not give us the
general formulation we are looking for. We could repeat the process by varying each of the other
variables in turn, as is illustrated in Figs. 7.1b, 7.1c, and 7.1d. This approach to determining the
functional relationship between the pressure drop and the various factors that influence it, although
logical in concept, is fraught with difficulties. Some of the experiments would be hard to carry
out—for example, to obtain the data illustrated in Fig. 7.1c it would be necessary to vary fluid density
while holding viscosity constant. How would you do this? Finally, once we obtained the various
curves shown in Figs. 7.1a, 7.1b, 7.1c, and 7.1d, how could we combine these data to obtain
the desired general functional relationship between ¢p / , D, r, m, and V which would be valid for
any similar pipe system?
Fortunately, there is a much simpler approach to this problem that will eliminate the difficulties
described above. In the following sections we will show that rather than working with the
(7.1)