Page 38 - Aerodynamics for Engineering Students

Basic concepts and definitions 21

Rearranging Eqn (1.36), it becomes

V Vd -f
n= c Cd (7) (1.37)
or, alternatively,

($) =g(?) (1.38)

where g represents some function which, as it includes the undetermined constant C
and index f, is unknown from the present analysis.
Although it may not appear so at first sight, Eqn (1.38) is extremely valuable, as it
shows that the values of nd/V should depend only on the corresponding value of
Vd/v, regardless of the actual values of the original variables. This means that if, for
each observation, the values of nd/V and Vd/v are calculated and plotted as a graph,
all the results should lie on a single curve, this curve representing the unknown
function g. A person wishing to estimate the eddy frequency for some given cylinder,
fluid and speed need only calculate the value of Vd/v, read from the curve the
corresponding value of nd/V and convert this to eddy frequency n. Thus the results
of the series of observations are now in a usable form.
Consider for a moment the two compound variables derived above:
(a) nd/V. The dimensions of this are given by
nd -1 -1
-= [T-I x L x (LT ) ] = [LOTo] = [l]
V
(b) Vd/v. The dimensions of this are given by

Vd
- [(LT-') x L x (L2T-')-'] = [l]
=
v
Thus the above analysis has collapsed the five original variables n, d, V, p and v
into two compound variables, both of which are non-dimensional. This has two
advantages: (i) that the values obtained for these two quantities are independent of
the consistent system of units used; and (ii) that the influence of four variables on a
fifth term can be shown on a single graph instead of an extensive range of graphs.
It can now be seen why the index f was left unresolved. The variables with indices
that were resolved appear in both dimensionless groups, although in the group nd/ V
the density p is to the power zero. These repeated variables have been combined in
turn with each of the other variables to form dimensionless groups.
There are certain problems, e.g. the frequency of vibration of a stretched string, in
which all the indices may be determined, leaving only the constant C undetermined.
It is, however, usual to have more indices than equations, requiring one index or
more to be left undetermined as above.
It must be noted that, while dimensional analysis will show which factors are not
relevant to a given problem, the method cannot indicate which relevant factors, if
any, have been left out. It is, therefore, advisable to include all factors likely to have
any bearing on a given problem, leaving out only those factors which, on a priori
considerations, can be shown to have little or no relevance.