Dimensional analysis

1.4.1 Fundamental principles

The theory of dimensional homogeneity has additional uses to that described above. By predicting how one variable may depend on a number of others, it may be used to direct the course of an experiment or the analysis of experimental results. For example, when fluid flows past a circular cylinder the axis of which is perpendicular to the stream, eddies are formed behind the cylinder at a frequency that depends on a number of factors, such as the size of the cylinder, the speed of the stream, etc.

In an experiment to investigate the variation of eddy frequency the obvious procedure is to take several sizes of cylinder, place them in streams of various fluids at a number of different speeds and count the frequency of the eddies in each case. No matter how detailed, the results apply directly only to the cases tested, and it is necessary to find some pattern underlying the results. A theoretical guide is helpful in achieving this end, and it is in this direction that dimensional analysis is of use.

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, v.

It should be noted that either p от и 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

n = f(d, V, p, v)

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

n = Y^CdaVbpevf (1.33)

where C is a constant and a, b, e and f are some unknown indices; putting Eqn (1.33) in dimensional form leads to

[Г-1] = [La(LT-1)4(ML-3)e(L2T-1)^] (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where each factor has been replaced by its dimensions.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 M, L and T on the two sides of the equation may be equated as follows:

Mass (M) 0 = e (1.35a)

Length (L) 0 — a + b — 3e + 2f (1.35b)

Time (T) -1 = – b-f (1.35c)

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

b = -f e = 0

a = – l-f

Substituting these values in Eqn (1.33),

Rearranging Eqn (1.36), it becomes

(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/u, regardless of the actual values of the original variables. This means that if, for each observation, the values of nd/V and Vdjv 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

~ = [T-1 x L x (LT"1)-1] = [L°T°] = [1]

(b) Vd/v. The dimensions of this are given by

~ = [(LT_1) x L x (L2T-1)-1] = [1]

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/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.