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One of the more difficult tasks for the inexperienced analyst is to decide Drawing up the list of independent variables

which independent variables to include in the dimensional analysis and which to exclude. It is important to include in the list of variables all quant- ities that influence the fundamental physics of the problem. On the other hand, there is no virtue in bringing in quantities that have no bearing on the situation. Here the discussion of the different kinds of dynamic similarity in Section 5.3 provides guidance on the variables that might require consider- ation in the field of fluid mechanics. In the context of dimensional analysis, the term independent variable sometimes needs to be interpreted broadly. This is because, in some experimental environments, it is not possible to vary particular quantities, despite the fact that they have a fundamental influence on an experiment. This is particularly so for problems in which gravity plays a fundamental role. Some students new to dimensional analysis omit g from the list of variables in a study where it in fact plays a fundamental role, simply because it is a physical quantity that cannot be varied in the labor- atory. In deciding whether or not g should be included amongst the list of independent variables, it is often helpful to consider whether the dependent variable would be affected if the experiment were performed on the surface of the moon, where g differs from the value on the surface of the earth.

Temperature is another quantity that causes difficulties, because physical properties, such as dynamic viscosity, vary significantly with temperature. Again, in fluid mechanics, it is sufficient to consider the question of dynamic similarity, to appreciate that the physical property, but not temperature, should be included in the list of variables. In problems of heat transmission, which are outside the scope of this book, temperature levels are important and must be included in the list of variables.

There are circumstances where, in specifying the list of independent vari- ables, a choice has to be made between a number of equally valid options. For example, when considering problems of internal flow, the flow rate can be represented by one of three variables, namely a representative mean velo- city u[LT−1], volumetric flow rate Q[L3T−1] or mass flow rate m[MT−1]. The case for any one of these can be argued but, by convention, velocity is generally used. However, there are circumstances when the use of either m or

176 Physical similarity and dimensional analysis

Q has the advantage of convenience. Another example arises when account

is taken of viscous effects, in both internal and external flows. Then, either dynamic viscosity_µ[ML−1T−1_{] or kinematic viscosity ν[L}2_T−1_{] can be used.}

Again by convention_{µ is usually chosen.}

The functional relation that emerges from a dimensional analysis is not

Alternative functional

forms unique. Depending upon the choice of repeating variables, the outcome can take a number of alternative, but equivalent, forms.

In the case discussed in Section 5.6.1, the functional relationship in dimensionless form was:

s ut = φ  at u  (5.10) However, we could have found the functional relation by dividing equation 5.6 throughout by at2to obtain

s at2 = ψ
u at  (5.11) where u/at = (at/u)−1 (5.12)

We note that the functional relation (5.11) can be obtained directly from (5.10) simply by multiplying both sides of (5.10) by the dimensionless group_(at/u)−1to yield

s ut  at u −1 = s at2 =  at u −1 φ  at u  = ψ
u at 

This case illustrates the important point that, even though the physical law underlying any phenomenon is unique (as exemplified here by eqn 5.6), the description of the phenomenon by means of a functional expression involving dimensionless groups does not lead to a single unique relation.

In the case of Example 5.1 three alternative functional forms emerged:

F_/(u2d2_{) = φ}2(Re) (5.13)

F_{/µud = φ3(Re)} (5.14)

F/µ2= φ4(Re) (5.15)

Define the non-dimensional groups

N1= F u2_d2; N3= F µud; N4= F_ µ2

Since Re= (ud/µ), and it is dimensionless, it follows by inspection that

N3= N1· Re and N4= N1· (Re)2

Alsoφ3(Re) = (Re) · φ2(Re) and φ4(Re) = (Re)2· φ2(Re).

The two cases just discussed illustrate the fact that, in functional relation- ships, a phenomenon can be expressed non-dimensionally in a variety of different but equivalent ways. The choice amongst the alternatives should be mainly governed by two criteria. The first of these is convenience.

Dimensional analysis 177

For example, the parameter F/µ2 is independent of d, and u, so experimental results in which either d, or u, are varied can be analysed in a very straightforward way by plotting F against d, or u. At the interpretation stage the grounds of convenience can again be invoked. Experience shows that F_/(u2_d2_{) 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 groupµ/(du) 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 NAand NBrelated by the expression NA= φ(NB). If

there is a variable common to NAand NB, then that variable can be elimin-

ated between NAand NBby creating a new dimensionless group NC. Writing NC= NA· (NB)δ, the value ofδ can be selected to eliminate the chosen vari-

able, resulting in the functional relation NC = ψ(NB). As an example of

this process, define NA = F/(u2d2) and NB = ud/µ, and assume that

we wish to eliminate the variable_{. Write N}C= (F/(u2d2)) · (ud/µ)δ. By

inspection_{δ = 1 and hence N}C= F/µud.

Example 5.2 The flow rate through differential-pressure flow-

metering 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  1 2g_h = Q E(π/4)d2  _ 2_p where E= {1 − (d/D)4}−1/2.

178 Physical similarity and dimensional analysis

The Reynolds number can be defined in terms of conditions in the pipe or at the throat, so that

ReD= u1

D

µ and Red=u2

d µ

Data are usually presented in the form

C= function(β, Re) or α = function (β, Re),

whereα = CE and β = d/D.

Whether Red or ReD is used is unimportant in the context of the

present arguments.

In order to evaluate the flow rate Q corresponding to a measured differential pressure or head, the value of C must first be established. But C is, in general, a function of Re, and Re itself depends upon the value of Q, since, from the continuity condition,

u1=

4Q

πD2 and u2=

4Q

πd2

Hence it is demonstrated that it is not possible to determine_{ directly} from this method of presentation of the data, whether in chart or equation form. In this situation, the use of a process of successive approximations to determine Q is often recommended, but it is rather cumbersome and is, in any event, unnecessary.

(b) A presentation which allows a direct approach to the calculation of the flow rate is to define a new non-dimensional parameter, based on the known valueh or p, rather than the unknown u1or u2.

Define N_pd= (p) 1/2_d µ = (2_gh)1/2_d µ and NpD= (p) 1/2_D µ = (2_gh)1/2_D µ

These new variables are related to Re by

Npd= Re_d √ 2_α and NpD= ReD √ 2_αβ2

A plot or correlating equation, with C as the dependent variable, and

β and Npd (or NpD) as independent variables, allows the flow rate

to be determined directly from measurements of differential pressure or head. The method is as follows: First, _{β and Npd} (or NpD) are

evaluated. Second, C is now determined. Finally, Q is calculated from the 2 equation Q= C(π/4)d 2_(2gh)1/2 (1 − β4₎1/2

The application of dynamic similarity 179

If, after performing a dimensional analysis, experimental data are plotted Scatter

in dimensionless form and they do not exhibit any obvious trends there are a number of possible explanations. First check that the test equipment and instrumentation are functioning correctly. Next check that the dimen- sional analysis has been processed correctly and that the derived quantities are indeed dimensionless. If these conditions are satisfied, the next stage is to check that the experiments are being implemented correctly. Each test should investigate the effect on the dependent dimensionless group of just one of the other dimensionless groups, whilst the remainder are all held constant. Without careful attention to detail, it is all too easy to overlook the fact that a dependent variable may appear in more than one dimensionless group, and so when its value is changed, it affects several dimensionless groups simultaneously. The cure here is either to test the effect of an alternative parameter, or to reformulate the dimen- sional analysis. Finally, if the scatter cannot be eliminated by action on these fronts, the explanation is probably that a factor that has a funda- mental influence on the experiment has been omitted from the original dimensional analysis. In this case it is necessary to start from square one again.