fluid_mechanics

7.3 Determination of Pi Terms 339
Step 1
Δp = f(D, r, m , V)
Step 2
Δp = FL 3 , ...
Step 3
k–r = 3
Step 4
D, V, r
Step 5
1 = Δp D a V b r c
Step 6
2 = mD a V b r c
Step 7
_____ Δp D
ρV 2 = F 0 L 0 T 0
Step 8
_____ Δp D
ρV 2
______ DΔp
ρV 2
m
DVρ
= ____
_____ ρVD
μ
The method of repeating
variables
can be most easily
carried out by following
a step-bystep
procedure.
Note that we end up with the correct number of pi terms as determined from Step 3.
At this point stop and check to make sure the pi terms are actually dimensionless 1Step 72.
We will check using both FLT and MLT dimensions. Thus,
or alternatively,
Finally 1Step 82, we can express the result of the dimensional analysis as
¢p / D
f˜ rV a m
2 DVr b
This result indicates that this problem can be studied in terms of these two pi terms, rather than
the original five variables we started with. The eight steps carried out to obtain this result are summarized
by the figure in the margin.
Dimensional analysis will not provide the form of the function f˜ . This can only be obtained
from a suitable set of experiments. If desired, the pi terms can be rearranged; that is, the reciprocal
of mDVr could be used, and of course the order in which we write the variables can be changed.
Thus, for example, could be expressed as
ß 2
ß 1 ¢p /D
rV 2
ß 2
ß 1 ¢p /D
rV 2
ß 2
ß 1
ß 2
and the relationship between and as
1FL 3 21L2
1FL 4 T 2 21LT 1 2 2 F 0 L 0 T 0
m
DVr 1FL 2 T 2
1L21LT 1 21FL 4 T 2 2 F 0 L 0 T 0
1ML2 T 2 21L2
1ML 3 21LT 1 2 2 M 0 L 0 T 0
m
DVr 1ML1 T 1 2
1L21LT 1 21ML 3 2 M 0 L 0 T 0
D ¢p /
rV 2
ß 2 rVD
m
f a rVD
m b
as shown by the figure in the margin.
This is the form we previously used in our initial discussion of this problem 1Eq. 7.22. The
dimensionless product rVDm is a very famous one in fluid mechanics—the Reynolds number.
This number has been briefly alluded to in Chapters 1 and 6 and will be further discussed in Section
7.6.
To summarize, the steps to be followed in performing a dimensional analysis using the method
of repeating variables are as follows:
Step 1 List all the variables that are involved in the problem.
Step 2 Express each of the variables in terms of basic dimensions.
Step 3 Determine the required number of pi terms.
Step 4 Select a number of repeating variables, where the number required is equal to the number
of reference dimensions 1usually the same as the number of basic dimensions2.
Step 5 Form a pi term by multiplying one of the nonrepeating variables by the product of
repeating variables each raised to an exponent that will make the combination
dimensionless.
Step 6 Repeat Step 5 for each of the remaining nonrepeating variables.
Step 7 Check all the resulting pi terms to make sure they are dimensionless and independent.
Step 8 Express the final form as a relationship among the pi terms and think about what it
means.