2.3. DIMENSIONAL ANALYSIS METHODOLOGY
37
In selecting fc-values for II3, it was imperative that k3 0 0 because force was the only
original variable not yet included in the complete set of dimensionless products.
Thus, we see that Reynolds number, Froude number, and Euler number constitute a complete set of dimensionless products describing flow problems in which the
aforementioned six variables are deemed important.
A final important step is to check each dimensionless product to assure that it is
indeed dimensionless. This is easily done by substituting the fundamental dimensions
for each of the original variables contained in the dimensionless product and showing
that they all cancel.
In Example 2.5, selection of the three exponents was contrived so that
the previously known dimensionless numbers were obtained. Obviously,
specifying other exponent values will result in a different complete set of
dimensionless products. However, any other selection of ^-values in Example 2.5 results in dimensionless numbers that are a combination of the
Reynolds, Froude, and Euler numbers.
Example 2.6. Fetch-Limited Wave Forecasting - Revisited
In Example 2.4 the five variables thought to be important for fetch-limited forecasting of significant wave height were listed, and a complete set of dimensionless
products was determined by inspection. This example obtains the same result by the
more formal method.
As noted previously, the variables of the wave forecasting problem can be arranged
into a dimensions matrix given as
g
Hs
U
h
X
L
1
1
1
1
1
T
-2
0
-1
0
0
M
0
0
0
0
0
It was also noted that (n — r) = (5 - 2) — 3; therefore, the complete set of dimensionless products contains three pi terms having the form
H = gkl Hsk2 Uk3 hk4 Xk*
The system of independent exponent equations is written directly from the matrix
as
(hi + à?2 + k3 -f- kt 4- k$) — 0
(-2jtj - k3) = 0
37
In selecting fc-values for II3, it was imperative that k3 0 0 because force was the only
original variable not yet included in the complete set of dimensionless products.
Thus, we see that Reynolds number, Froude number, and Euler number constitute a complete set of dimensionless products describing flow problems in which the
aforementioned six variables are deemed important.
A final important step is to check each dimensionless product to assure that it is
indeed dimensionless. This is easily done by substituting the fundamental dimensions
for each of the original variables contained in the dimensionless product and showing
that they all cancel.
In Example 2.5, selection of the three exponents was contrived so that
the previously known dimensionless numbers were obtained. Obviously,
specifying other exponent values will result in a different complete set of
dimensionless products. However, any other selection of ^-values in Example 2.5 results in dimensionless numbers that are a combination of the
Reynolds, Froude, and Euler numbers.
Example 2.6. Fetch-Limited Wave Forecasting - Revisited
In Example 2.4 the five variables thought to be important for fetch-limited forecasting of significant wave height were listed, and a complete set of dimensionless
products was determined by inspection. This example obtains the same result by the
more formal method.
As noted previously, the variables of the wave forecasting problem can be arranged
into a dimensions matrix given as
g
Hs
U
h
X
L
1
1
1
1
1
T
-2
0
-1
0
0
M
0
0
0
0
0
It was also noted that (n — r) = (5 - 2) — 3; therefore, the complete set of dimensionless products contains three pi terms having the form
H = gkl Hsk2 Uk3 hk4 Xk*
The system of independent exponent equations is written directly from the matrix
as
(hi + à?2 + k3 -f- kt 4- k$) — 0
(-2jtj - k3) = 0