Hydromechanics 7.1 Dimensional Analysis, Basic Estimation, and Model Testing 129
Part A | 7.1
i. Eliminate length
Ä D
A
D
M
T 2 ;
Ä U
A
D
1
T
;
ŒŒA
3
D M ; ŒŒA D
M
T
:
We can see that the dimension of length has been
removed from the combined variables thus far and
that each step of the process will reduce the total number of variables by 1, until n m of them
remain. We will next repeat the process for the dimensions of M and T. Looking at the dimensions of
the combined variables we see that both U=A and
A
3 have simple dimensions of only one type of unit
each. Thus, we have a choice of using either set of
combined variables to remove one of the remaining
dimensions. Here we proceed next by eliminating
time.
ii. Eliminate time
Ä DA
U 2
D M ;
Ä A
2
U
D M ; ŒŒA
3
D M :
The only dimension remaining in the combined
variables is now mass. Remembering that we never
want to use the set of variables that contains the dependent variable, we must next choose which set of
combined variables we will use to eliminate mass.
This example illustrates one of the advantages of using the step-by-step elimination method over other
nondimensionalization approaches – it is possible
to examine the resulting combined variables at each
step of the process and if it becomes clear that any of
them could be reduced to one of the well-known dimensionless numbers, such as Reynolds number or
Froude number, we can elect to steer the elimination
process in that direction. Here it can be seen that if
the set A
3 is used to eliminate mass, the resulting
sets will be the inverse of the well-known Reynolds
number and a set similar to what is known as a drag
coefficient.
iii. Eliminate mass
˘ 1 D
D
U 2 A 2 ; ˘ 2 D
UA
:
Thus, we have identified the dimensionless ˘
groups (combined variables) which give a simpler
functional dependence (7.3) than the original (7.1).
This analysis only specifies that there is a functional dependence and which groups are likely to
be involved, not the actual function or form of the
dependence, which must typically be determined
from physical observation or experimental data.
Therefore, we can equally say that perhaps ˘ 1 D
.1=˘ 2 /, or that
C D D
D
1
2
U 2 A 2
D
 UA
Ã
D .Re/ ; (7.4)
where C D is the common form of the coefficient
of drag and Re is defined as the Reynolds number.
A plot of the relationship between C D and Re for
a sphere can be seen in Fig. 7.1.
The Exponent Method
If we can combine the variables involved in our expected functional dependence (7.1) as a product of
exponentials, the resulting equation must be dimensionally homogeneous
ŒD D ŒU
a
b
c A
d
:
(7.5)
Thus, solving for the exponential terms a; b; c; d which
make the relation (7.5) dimensionally homogeneous
should provide the nondimensional ˘ groups we seek.
Writing out the units in the terms of (7.5) gives
ML
T 2 D
 L
T
à a  M
L 3
à b  M
LT
à c
L
d
;
which results in having to solve the following system of
equations so that (7.5) is dimensionally homogeneous
L W 1 D a 3b c C d
M W 1 D b C c
T W W2 D a c :
(7.6)
This is an underdetermined system of equations. We can
solve for three of the variables in terms of a fourth. It is
convenient to choose that variable which appears most
often, which is c in this case. Solving the system of
equations (7.6), we find that a D 2 c, b D 1 c, and
d D 2 c. Re-writing (7.5) using these exponential values gives
D D U
2 A
2
Â
UA
à c
:
(7.7)
Thus, we can see that (7.7) is of the functional
form (7.4).
7.1.2 Physical Significance
of the Dimensionless ˘ Groups
˘ Groups can be interpreted as the ratio of typical values of two different, competing, physical effects. In
Part A | 7.1
i. Eliminate length
Ä D
A
D
M
T 2 ;
Ä U
A
D
1
T
;
ŒŒA
3
D M ; ŒŒA D
M
T
:
We can see that the dimension of length has been
removed from the combined variables thus far and
that each step of the process will reduce the total number of variables by 1, until n m of them
remain. We will next repeat the process for the dimensions of M and T. Looking at the dimensions of
the combined variables we see that both U=A and
A
3 have simple dimensions of only one type of unit
each. Thus, we have a choice of using either set of
combined variables to remove one of the remaining
dimensions. Here we proceed next by eliminating
time.
ii. Eliminate time
Ä DA
U 2
D M ;
Ä A
2
U
D M ; ŒŒA
3
D M :
The only dimension remaining in the combined
variables is now mass. Remembering that we never
want to use the set of variables that contains the dependent variable, we must next choose which set of
combined variables we will use to eliminate mass.
This example illustrates one of the advantages of using the step-by-step elimination method over other
nondimensionalization approaches – it is possible
to examine the resulting combined variables at each
step of the process and if it becomes clear that any of
them could be reduced to one of the well-known dimensionless numbers, such as Reynolds number or
Froude number, we can elect to steer the elimination
process in that direction. Here it can be seen that if
the set A
3 is used to eliminate mass, the resulting
sets will be the inverse of the well-known Reynolds
number and a set similar to what is known as a drag
coefficient.
iii. Eliminate mass
˘ 1 D
D
U 2 A 2 ; ˘ 2 D
UA
:
Thus, we have identified the dimensionless ˘
groups (combined variables) which give a simpler
functional dependence (7.3) than the original (7.1).
This analysis only specifies that there is a functional dependence and which groups are likely to
be involved, not the actual function or form of the
dependence, which must typically be determined
from physical observation or experimental data.
Therefore, we can equally say that perhaps ˘ 1 D
.1=˘ 2 /, or that
C D D
D
1
2
U 2 A 2
D
 UA
Ã
D .Re/ ; (7.4)
where C D is the common form of the coefficient
of drag and Re is defined as the Reynolds number.
A plot of the relationship between C D and Re for
a sphere can be seen in Fig. 7.1.
The Exponent Method
If we can combine the variables involved in our expected functional dependence (7.1) as a product of
exponentials, the resulting equation must be dimensionally homogeneous
ŒD D ŒU
a
b
c A
d
:
(7.5)
Thus, solving for the exponential terms a; b; c; d which
make the relation (7.5) dimensionally homogeneous
should provide the nondimensional ˘ groups we seek.
Writing out the units in the terms of (7.5) gives
ML
T 2 D
 L
T
à a  M
L 3
à b  M
LT
à c
L
d
;
which results in having to solve the following system of
equations so that (7.5) is dimensionally homogeneous
L W 1 D a 3b c C d
M W 1 D b C c
T W W2 D a c :
(7.6)
This is an underdetermined system of equations. We can
solve for three of the variables in terms of a fourth. It is
convenient to choose that variable which appears most
often, which is c in this case. Solving the system of
equations (7.6), we find that a D 2 c, b D 1 c, and
d D 2 c. Re-writing (7.5) using these exponential values gives
D D U
2 A
2
Â
UA
à c
:
(7.7)
Thus, we can see that (7.7) is of the functional
form (7.4).
7.1.2 Physical Significance
of the Dimensionless ˘ Groups
˘ Groups can be interpreted as the ratio of typical values of two different, competing, physical effects. In