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CHAPTER 2. DIMENSIONAL ANALYSIS
Important: When expressing functional relationships between the complete set of dimensionless products, remember to include those original
variables that were already dimensionless and left out of the analysis (in
the above example, the variable 0).
As mentioned, Buckingham’s pi theorem tells us, in nearly all cases, the
number independent dimensionless products that can be constructed from
a set of variables. The following example, taken from Langhaar (1951),
shows an exception to the rule and how to recognize when additional pi
terms must be formed to complete the set of dimensionless products.
Example 2.8. Wind Setup on a Lake
The height, S, of the setup that is caused by a steady wind blowing over a lake is
assumed to depend on the average depth of the lake (/i), the length of the lake (X)
parallel to the wind direction, specific weight of the water (7), and the shear stress
(r) of the wind on the water. The dimension matrix for these five variables is given
below
S
1
0
0
h
1
0
0
L
T
M
x
7
r
1
-2 -1
0
-2 -2
0
1
1
From Buckingham’s pi theorem the there should be 5—3 = 2 dimensional products
in the complete set, and they are assumed to have the form
n = Ski hk’ X * 3 yk4 rks
The equations for the exponents are written directly from the matrix as
(fci + &2 +
—
— ks) = 0
(-21=4-2fc5) = 0
(&4 + £5) = 0
Immediately we recognize that two of the equations are identical, and thus not
independent. So in this case r = 2, and n - r = 5 - 2 = 3. Therefore, we must solve
for three dimensionless products rather than two as originally anticipated.
Working through using the same procedure as in earlier examples culminates in
one set of dimensionless products is given by
which implies a relationship for setup given as
h
\7X' hJ