2.3. DIMENSIONAL ANALYSIS METHODOLOGY
43
The three pi terms found in this example can be shown to be independent simply
by demonstrating that one pi term cannot be formed by some combination of the
other two pi terms, i.e.,
n3 / (ni) **
(n2)b
or
zxx1
zsy ( r \b
\h) * \h)
)
Inspection confirms that no values for the exponents, a and b, exist that would
make the pi term on the left side in the above equation a combination of the two pi
terms on the right side.
2.3.3 Dimensionless Products and Experimental Data
One of the most important uses of dimensional analysis is to aid in correlation of experimental or field-measured data. Dimensional analysis only
provides a method for obtaining convenient groupings of important variables under the assumption that a relationship exists among the new dimensionless variables. Determination of any possible relationship requires
data obtained from experiments or measured in the field (Munson, et al.
1990). Historically, fluid mechanics has relied heavily on empirical results,
so it is not surprising that dimensional analysis is considered by some to
be an important tool in coastal engineering.
The difficulty of establishing an empirical relationship between pi terms
obtained through dimensional analysis depends largely on the number of
pi terms that have to be considered and the difficulty of obtaining reliable
measurements (Munson, et al. 1990). Obviously, as the number of pi terms
increases, much more data must be obtained to quantify any relationship
between variables over a sufficient range of values.
In rare instances, the difference between the number of variables and the
number of fundamental dimensions encompassed by the variables (n — r)
will equal unity. If this occurs, all the variables can be grouped into a single
pi term which must be equal to a constant, i.e.,
Hi = C
(2.8)
This is the one time that dimensional analysis reveals the specific form
of the relationship between process variables. The value of the constant
must be determined by theoretical considerations, experiments, or field
measurements. Although in principle only one experiment is needed to
determine “C”, it is wise to make multiple determinations to establish a
reliable value and to assess variability of the constant. Significant variation
in the constant that cannot be ascribed to measurement or experiment error
43
The three pi terms found in this example can be shown to be independent simply
by demonstrating that one pi term cannot be formed by some combination of the
other two pi terms, i.e.,
n3 / (ni) **
(n2)b
or
zxx1
zsy ( r \b
\h) * \h)
)
Inspection confirms that no values for the exponents, a and b, exist that would
make the pi term on the left side in the above equation a combination of the two pi
terms on the right side.
2.3.3 Dimensionless Products and Experimental Data
One of the most important uses of dimensional analysis is to aid in correlation of experimental or field-measured data. Dimensional analysis only
provides a method for obtaining convenient groupings of important variables under the assumption that a relationship exists among the new dimensionless variables. Determination of any possible relationship requires
data obtained from experiments or measured in the field (Munson, et al.
1990). Historically, fluid mechanics has relied heavily on empirical results,
so it is not surprising that dimensional analysis is considered by some to
be an important tool in coastal engineering.
The difficulty of establishing an empirical relationship between pi terms
obtained through dimensional analysis depends largely on the number of
pi terms that have to be considered and the difficulty of obtaining reliable
measurements (Munson, et al. 1990). Obviously, as the number of pi terms
increases, much more data must be obtained to quantify any relationship
between variables over a sufficient range of values.
In rare instances, the difference between the number of variables and the
number of fundamental dimensions encompassed by the variables (n — r)
will equal unity. If this occurs, all the variables can be grouped into a single
pi term which must be equal to a constant, i.e.,
Hi = C
(2.8)
This is the one time that dimensional analysis reveals the specific form
of the relationship between process variables. The value of the constant
must be determined by theoretical considerations, experiments, or field
measurements. Although in principle only one experiment is needed to
determine “C”, it is wise to make multiple determinations to establish a
reliable value and to assess variability of the constant. Significant variation
in the constant that cannot be ascribed to measurement or experiment error