Page 37 - Aerodynamics for Engineering Students

20 Aerodynamics for Engineering Students

In the above problem the frequency of eddies, n, will depend primarily on:

(i) the size of the cylinder, represented by its diameter, d
(ii) the speed of the stream, V
(iii) the density of the fluid, p
(iv) the kinematic viscosity of the fluid, u.
It should be noted that either p or u may be used to represent the viscosity of the
fluid.
The factors should also include the geometric shape of the body. Since the problem
here is concerned only with long circular cylinders with their axes perpendicular to
the stream, this factor will be common to all readings and may be ignored in this
analysis. It is also assumed that the speed is low compared to the speed of sound in
the fluid, so that compressibility (represented by the modulus of bulk elasticity) may
be ignored. Gravitational effects are also excluded.
Then

and, assuming that this function (. . .) may be put in the form

n = xCdaVbpeuf (1.33)

where Cis a constant and a, by e andfare some unknown indices; putting Eqn (1.33)
in dimensional form leads to

[T-l] = [La (LT -' )b (MLP3)" (L'T-' ) f] (1.34)

where each factor has been replaced by its dimensions. Now the dimensions of both
sides must be the same and therefore the indices of My L and T on the two sides of the
equation may be equated as follows:

Mass(M) O=e (1.35a)
Length (L) O=a+b-3e+2f (1.35b)
Time (T) -1 = -b-f (1.3%)

Here are three equations in four unknowns. One unknown must therefore be left
undetermined: f, the index of u, is selected for this role and the equations are solved
for a, b and e in terms off.
The solution is, therefore,

b=l-f ( 1.3 5d)
e=O (1.35e)
a=-l-f (1.35f)

Substituting these values in Eqn (1.33),

(1.36)