34
CHAPTER 2. DIMENSIONAL ANALYSIS
Buckingham’s theorem, we should expect that the relationship between the variables
can be expressed by the function,
gH, _
( gh_ gX\
U2
\U2’ U2 J
By casting the relationship into dimensionless form, the number of independent variables has been reduced from 4 to 2 (gravity can be considered a constant in this
case). This greatly simplifies the task of examining field data from fetch-limited
regions. (Note that other dimensionless products could have been formed, such as
Hs/h or h/X, but experience with the data and physical reasoning has resulted in
the dimensionless parameters shown in the example.
2.3 Dimensional Analysis Methodology
2.3.1 Buckingham Pi Theorem
The systematic procedure for forming a complete set of dimensionless products from a given set of process variables begins with a determination of
how many dimensionless products can be formed. In all but a few cases the
following rule of thumb provides the correct number.
In a dimensionally homogeneous equation involving “n” variables, the number of dimensionless products that can be formed
from “n” variables is “n — r” where “r” is the number of fundamental dimensions encompassed by the variables.
This rule is known as the Buckingham Pi Theorem, so named because
Buckingham used the symbol II to represent the dimensionless products.
A dimensionally homogeneous equation involving n variables, such as
xl ~ f (æ2> æ3> æ4>•••> æn)
(2-5)
is an equation in which the dimensions of the variable on the left side of
the equality match the dimensions of any term that stands by itself on
the right side of the equality. The Buckingham pi theorem states that any
such equation can be rearranged into a new equation expressed in terms of
dimensionless products (pi terms), i.e.,
n1 = ^(n2, n3,...,nn_r)
(2.6)
I he required number of pi terms is less than the number of original variables by the amount r, where r is the number of fundamental dimensions
contained in the original variables. If the original n variables are independent, then the n — r dimensionless products will also be independent
CHAPTER 2. DIMENSIONAL ANALYSIS
Buckingham’s theorem, we should expect that the relationship between the variables
can be expressed by the function,
gH, _
( gh_ gX\
U2
\U2’ U2 J
By casting the relationship into dimensionless form, the number of independent variables has been reduced from 4 to 2 (gravity can be considered a constant in this
case). This greatly simplifies the task of examining field data from fetch-limited
regions. (Note that other dimensionless products could have been formed, such as
Hs/h or h/X, but experience with the data and physical reasoning has resulted in
the dimensionless parameters shown in the example.
2.3 Dimensional Analysis Methodology
2.3.1 Buckingham Pi Theorem
The systematic procedure for forming a complete set of dimensionless products from a given set of process variables begins with a determination of
how many dimensionless products can be formed. In all but a few cases the
following rule of thumb provides the correct number.
In a dimensionally homogeneous equation involving “n” variables, the number of dimensionless products that can be formed
from “n” variables is “n — r” where “r” is the number of fundamental dimensions encompassed by the variables.
This rule is known as the Buckingham Pi Theorem, so named because
Buckingham used the symbol II to represent the dimensionless products.
A dimensionally homogeneous equation involving n variables, such as
xl ~ f (æ2> æ3> æ4>•••> æn)
(2-5)
is an equation in which the dimensions of the variable on the left side of
the equality match the dimensions of any term that stands by itself on
the right side of the equality. The Buckingham pi theorem states that any
such equation can be rearranged into a new equation expressed in terms of
dimensionless products (pi terms), i.e.,
n1 = ^(n2, n3,...,nn_r)
(2.6)
I he required number of pi terms is less than the number of original variables by the amount r, where r is the number of fundamental dimensions
contained in the original variables. If the original n variables are independent, then the n — r dimensionless products will also be independent