Mechanics of Fluids

For example, the parameter Fϱ/µ 2 is independent of d, and u, so
experimental results in which either d, oru, are varied can be analysed in a
very straightforward way by plotting F against d, oru. At the interpretation
stage the grounds of convenience can again be invoked. Experience shows
that F/(ϱu 2 d 2 ) varies only slowly with Re at large values of Re, and so the
functional form (5.13) should be used at high Reynolds numbers. At low
values of Re, the parameter F/µud is constant or varies only slightly with
Re, and so the form (5.14) is the obvious choice at low Reynolds numbers.
The second consideration is to use what are regarded as traditional forms of
dimensionless groups. For example, the dimensionless group µ/(duϱ) often
arises from the process of dimensional analysis. This should be recognised
as the reciprocal of Reynolds number, by which it should be replaced.
We conclude this Section by noting that it is sometimes desirable to
reformulate a dimensional analysis to eliminate a specific variable from a
dimensionless group, often the one containing the dependent variable. This
can be done by creating an alternative dimensionless group. Consider the
dimensionless groups NA and NB related by the expression NA = φ(NB). If
there is a variable common to NA and NB, then that variable can be eliminated
between NA and NB by creating a new dimensionless group NC. Writing
NC = NA · (NB) δ , the value of δ can be selected to eliminate the chosen variable,
resulting in the functional relation NC = ψ(NB). As an example of
this process, define NA = F/(ϱu 2 d 2 ) and NB = ϱud/µ, and assume that
we wish to eliminate the variable ϱ. Write NC = (F/(ϱu 2 d 2 )) · (ϱud/µ) δ .By
inspection δ = 1 and hence NC = F/µud.
Example 5.2 The flow rate through differential-pressure flowmetering
devices, such as venturi and orifice-plate meters, can be
calculated when the value of the discharge coefficient is known.
Information on the discharge coefficient is conventionally presented
as a function of the Reynolds number.
(a) Shown that this method of presenting the data is inconvenient for
many purposes.
(b) By using different dimensionless parameters, show that these
difficulties can be avoided.
Solution
Define the mean velocities in the pipe and at the throat by u1 and
u2, and the upstream pipe diameter and throat diameter by D and d,
respectively.
(a) Reference to Chapter 3 shows that the discharge coefficient, C,
is defined as
C =
Q
E(π/4)d2 �
where E ={1 − (d/D) 4 } −1/2 .
1
2g�h =
Q
E(π/4)d2 �
ϱ
2�p
Dimensional analysis 177