Basics of Fluid Mechanics, 2014a

282 CHAPTER 9. DIMENSIONAL ANALYSIS
terms of the physical units. The first term L is the described by D with the units of
[L]. The time, [t], can be expressed by D/U. The mass, [M], can be expressed by
ρD 3 . Now the dimensionless groups can be constructed by looking at the remaining
physical parameters, ΔP , D and μ. The pressure difference, ΔP , has dimensions of
[M L −1 t −2 ] Therefore, ΔP M −1 Lt 2 is a dimensionless quantity and these values were
calculated just above this line. Thus, the first dimensionless group is
π 1 =
[M L −1 t −2 ]
{}}{
ΔP
[M −1 ]
{ }} {
1
ρD 3
The second dimensionless group (using D) is
[L]
{}}{
D
[t 2 ]
{}}{
D 2
U 2
unitless
{ }} {
= ΔP
ρU 2 (9.6)
π 2 =
[L] [L −1 ]
{}}{ {}}{
D l −1 = D L
(9.7)
The third dimensionless group (using μ dimension of [M L 1 t −1 ]) and therefore dimensionless
is
π 3 = μ
[M −1 ]
{ }} {
1
D 3 ρ
[L]
{}}{
D
[t]
{}}{
D
U = μ
DUρ
(9.8)
This analysis is not unique and there can be several other possibilities for selecting
dimensionless parameters which are “legitimately” correct for this approach.
There are roughly three categories of methods for obtaining the dimensionless
parameters. The first one solving it in one shot. This method is simple and useful for
a small number of parameters. Yet this method becomes complicated for large number
of parameters. The second method, some referred to as the building blocks method, is
described above. The third method is by using dimensional matrix which is used mostly
by mathematicians and is less useful for engineering purposes.
The second and third methods require to identification of the building blocks.
These building blocks are used to construct the dimensionless parameters. There are
several requirements on these building blocks which were discussed on page 281. The
main point that the building block unit has to contain at least the basic or fundamental
unit. This requirement is logical since it is a building block. The last method is mostly
used by mathematicians which leads and connects to linear algebra. The fact that this
method used is the hall mark that the material was written by mathematician. Here,
this material will be introduced for completeness sake with examples and several terms
associated with this technique.
9.2.2 Basic Units Blocks
In Thermo–Fluid science there are several basic physical quantities which summarized
in Table 9.1. In the table contains two additional physical/basic units that appear in