fluid_mechanics

346 Chapter 7 ■ Dimensional Analysis, Similitude, and Modeling
Finally, to make the combination dimensionless we multiply by D so that
Thus,
Next, we will form the second pi term by selecting the variable that was not used in ß 1 ,
which in this case is m. We simply combine m with the other variables to make the combination
dimensionless 1but do not use ¢p / in ß 2 , since we want the dependent variable to appear only in
ß 1 2. For example, divide m by r 1to eliminate F2, then by V 1to eliminate T2, and finally by D 1to
eliminate L2. Thus,
and, therefore,
a ¢p /
rV 2b D a 1 b 1L2 L0 1cancels
L
ß 2
m
rVD 1FL 2 T 2
1FL 4 T 2 21LT 1 21L2 F 0 L 0 T 0
¢p / D
rV 2
ß 1 ¢p /D
rV 2
f a m
rVD b
L2
which is, of course, the same result we obtained by using the method of repeating variables.
An additional concern, when one is forming pi terms by inspection, is to make certain that
they are all independent. In the pipe flow example, ß 2 contains m, which does not appear in ß 1 ,
and therefore these two pi terms are obviously independent. In a more general case a pi term would
not be independent of the others in a given problem if it can be formed by some combination of
the others. For example, if ß 2 can be formed by a combination of say ß 3 , ß 4 , and ß 5 such as
ß 2 ß2 3 ß 4
ß 5
then ß 2 is not an independent pi term. We can ensure that each pi term is independent of those
preceding it by incorporating a new variable in each pi term.
Although forming pi terms by inspection is essentially equivalent to the repeating variable
method, it is less structured. With a little practice the pi terms can be readily formed by inspection,
and this method offers an alternative to more formal procedures.
7.6 Common Dimensionless Groups in Fluid Mechanics
A useful physical
interpretation can
often be given to dimensionless
groups.
At the top of Table 7.1 is a list of variables that commonly arise in fluid mechanics problems.
The list is obviously not exhaustive but does indicate a broad range of variables likely to be found
in a typical problem. Fortunately, not all of these variables would be encountered in all problems.
However, when combinations of these variables are present, it is standard practice to combine
them into some of the common dimensionless groups 1pi terms2 given in Table 7.1. These
combinations appear so frequently that special names are associated with them, as indicated in
the table.
It is also often possible to provide a physical interpretation to the dimensionless groups which
can be helpful in assessing their influence in a particular application. For example, the Froude number
is an index of the ratio of the force due to the acceleration of a fluid particle to the force due
to gravity 1weight2. This can be demonstrated by considering a fluid particle moving along a streamline
1Fig. 7.32. The magnitude of the component of inertia force F I along the streamline can be expressed
as F I a s m, where a s is the magnitude of the acceleration along the streamline for a particle
having a mass m. From our study of particle motion along a curved path 1see Section 3.12 we
know that
a s dV s
dt
V s dV s
ds