4. PREPARING FOR EXPERIMENTS
4.1 Introduction
Various methods of dimensional analysis were discussed in the last chapter. Application is fairly straightforward. Having written the fundamental equation describing the problem, the use of any method of dimensional analysis, except perhaps for the method of synthesis, is fairly mechanistic. Following the rules leads, with little thought, to the final dimensionless equation. The interesting part of preparing for experimental work lies in what is done before, and after, the application of dimensional analysis.
Before tackling the analysis, the researcher must write down the basic equation describing the physical behaviour of the phenomena to be studied. This is sometimes the most difficult part of the problem and the reader is encouraged to look again at Chapter 1 where the problem of specifying the relevant variables was discussed. Because of the importance of this step it is discussed again here.
Following the application of the dimensional analysis there are a number of questions to be considered before moving on to experiments. Is the arrangement of dimensional groups obtained the best arrangement for the research program envisaged ? Should compounding be used and, if so, to what end? Is it best to formulate the answer in terms of standard numbers or can novel arrangements be used? Is it desirable or possible to forecast the format that plotted results should take? Are there any particular rules for conducting experiments to ensure that the results obtained do, indeed, mean what they seem to mean? These questions are all dealt with in this Chapter. Also included in this chapter is a discussion of apparent anomalies ; phenomena which appear to contravene Buckingham’s law.
4.2 Choice of Relevant Variables
Chapter 1 provided guidance on how to write the initial starting equation using a simple geometric system as an example. Readers may wish to review the comments made in Chapter 1 before proceeding to the more complex situation described here.
Essentially, the starting point in any analysis is a functional equation which describes the physical phenomenon to be studied. Obviously then it is important to have some knowledge of that physical phenomenon. This may be obtained by thinking about the phenomenon and/or by reading about the phenomenon and/or by undertaking preliminary experiments to obtain some understanding.
Preliminary work then leads to the functional equation
A = φ ( B, C, D, E, ………. etc ) (4.1)
which implies that A depends on B, C, D, E - - - etc.
An alternative way to write this equation is
Φ ( A, B, C, D, E, ……..etc ) = 0 (4.2)
in which no assumptions are made regarding dependent or independent quantities but merely a stipulation that A, B, C, D, E¼. etc are related, in some way, to each other and to nothing else.
It is most important that the initial equation is complete (i.e. contains all relevant terms) but is not over specified (i.e. does not contain some terms which depend on only some of the others). For example, consider a rectangular box of length L, breadth B, depth D and diagonal (point to point) length P. If we are interested in the volume, V, then
V = φ ( L, B, D ) is correct (4.3)
V = φ ( L, D ) is incomplete (4.4)
and
V = φ (L, B, D , P ) is over specified (4.5)
because P depends on L, B and D
In many cases, there will be a number of separate, but related, dependent variables. For example, in considering the effect of a blast (or explosive) wave, Baker et al (1973) give the variables describing the blast source and the surrounding conditions as
w = total energy in blast sources (4.6 a)
r = size of source (4.6 b)
ri = shape of source (4.6 c)
p0 = ambient pressure ahead of blast front (or before blast occurs) (4.6 d)
a0 = sound velocity in ambient air (4.6 e)
Following the explosion a blast wave will radiate outwards from the source. The pressure Δp, at
the front of the wave will vary with distance of the front from the source R or with time t after the explosion. Researchers might also be interested in the density, ρ, of the gas at the front of the blast wave or perhaps in the particle velocity, u, at the front of the blast wave so that it would be correct to write
Δp or ρ or u = φ { w, r, ri, p0, ao, (R or t)} (4.7)
However, it would be quite incorrect to include all of these variables in an equation such as
Φ ( Δp, ρ, w, r, ri, po, ao, R, t ) = 0 (4.8)
Each of the variables in equation (4.8) is relevant to the problem of investigation conditions after the explosion. It is therefore acceptable to list them in a list of relevant variables. The problem with equation (4.8) is that it implies that any one of these variables is dependent on all the others. Equation (4.7) shows clearly that this is incorrect.
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Equation (4.7) deals with conditions at the front of the explosive wave as it radiates outwards from the source. Researchers might also be interested in conditions at one particular location, say X, located at a distance, Rx, from the source. Up to the time of the blast, and for some time thereafter, the pressure at X will be the ambient pressure. When the front of the blast wave arrives the pressure, px, will increase to a maximum and then, with the further passage of time, the pressure will drop again to the ambient pressure. Similar changes will be likely to occur to the density, ρx, and the particle velocity, ux, at the point X. In this case therefore variables defining the blast source and surrounding conditions must include the location of X and the time, t, in addition to the variables defined in equation (4.6). Conditions at X may therefore be given by
px or ux or ρx = φ ( w, r, ri, po, ao, Rx, t ) (4.9)
If we were only interested in the maximum pressure, pmax, at X, then the time at which this maximum pressure occurs, tmax, becomes a dependent variable because we cannot stipulate the time at which the maximum pressure will occur independently of the other blast conditions. Thus
pmax or umax or ρmax or tmax = φ ( w, r, ri, po, Rx ) (4.10)
Equations 4.6 to 4.10 show that it is usually not acceptable to simply list the relevant variables and then include them all in one functional equation. Variables which are dependent in one set of circumstances may be irrelevant in another and may even be independent in a third. Researchers must think carefully before deciding which variables to include and which to leave out. This is especially true when experiments are designed for more than one purpose. What is a dependent variable for one purpose may be independent for another.
In general, the meaning of the functional equation is clear and straightforward. If
A = φ ( B, C, D, E, ………..etc ) (4.11)
Then the value of A depends on the values of B, C, D, E …. etc, and on nothing else. Also the magnitude of each of the variables, B, C, D, E, …….etc must be capable of being specified independently of all the others. These two statements can be used to set up simple thought experiments. Using the variable B as an example researchers should ask themselves two simple questions. (1) "If I change the value of B, is this likely to affect the value of A ?" If the answer is "Yes" then B should be included in the functional equation provided it is truly independent. Independence is dealt with by the second question, (2), "Can I set the values for all of the variables on the right hand side of the functional equation except B ( i.e. values of C, D, E …… etc) and then choose a variety of values for B which are not in conflict with the values already set ( i.e. values already set for C, D, E …… etc). If the answer is "Yes" then B is, indeed, independent. If the answer is "No" then B depends to some extent on one or more of the other variables and should be excluded from the functional equation. Referring back to equation 4.5 it is clear that changes in the values of B, D, L and P will all affect the value of V. However it is also clear that it is not possible to specify B, D and L as 1.0, 2.0 and 3.0 mm respectively and at the same time specify P as 200 Km! Thus obviously P should be removed from the equation. Thinking of extreme values is helpful because the answer is then often quite clear. In more complex cases, such as the blast wave analysis referred to above, the answers may be less obvious and more careful thought is required.
4.3 Standard Numbers and Convenient Solutions
Various standard numbers (e.g. Froude, Reynolds, etc. see table 3.1) have been developed since the time of Buckingham and Rayleigh. Typically, these numbers provide a measure of the relative importance between two or more relevant force actions. For example, the Reynolds number determines the importance of viscous forces relative to inertial forces and the Froude number gives a measure of the relative importance of gravitational forces.
Many researchers consider that the end result of a dimensional analysis should, if possible, be couched in terms of these standard numbers (Woisin, 1992). However, while these numbers are important and useful (because they are easily recognised and because their numerical magnitude has some physical meaning) it is a mistake to be dogmatic (Sharp, 1993). Cases frequently occur in which the setting up of a solution in terms of standard numbers has retarded the determination of a convenient solution. Here we should define a convenient solution as one which allows us to easily determine the effect of the independent variables on the variable, or variables, of interest. Also, a convenient solution is one, which following experimentation to develop the functional relationship, allows the magnitude of the dependent variable to be calculated from known values if the independent variables. For example, if
Φ ( A, B, C, D ) = 0 (4.12)
is a dimensionally homogeneous equation, then one solution could be
Φ (A/B, C/B, D/B ) = 0 (4.13)
Now, if B is the variable of interest, then equation (4.13) is a highly inconvenient solution. A plot of experimental results based on equation (4.13), (i.e. values of A/B plotted against B/C for constant values of D/B) would tell us little about how variation in A, C and D affected B. It would also be impossible to obtain a direct solution for B knowing the values of A, C and D . The problem here is that the variable of interest, in this case B, appears more than once in the dimensionless equation. A convenient solution would be one in which the variable, or variables, of interest appear as infrequently as possible and preferably only once. Thus a better solution to equation (4.12) would be
Φ ( A/B, C/D, B/D ) = 0 (4.14)
Which would allow a direct solution for all variables except D.
The problem of relating frictional resistance to the flow in a pipeline was considered briefly at the end of section 3.2 and is a classic example of the problems of setting up a solution in terms of standard numbers. The basic problem was specified in equation (3.22) and is given again here for convenience.
V = φ ( d, k, Sg, μ, ρ ) (4.14)
The standard solution to this problem was given by equation (3.23) which is written in terms of a modified Froude number and a Reynolds number. Thus
Fs = φ ( R, d/k ) (4.15)
Where Fs = V2/Sgd is a form of Froude number with a modified gravitational acceleration, R = Vd/ν is the Reynolds number and d/k is the relative roughness of the pipe. The functional relationship is shown in the Moody diagram in Figure 3.1.
This arrangement is well known and has considerable significance as it divides the flow regime into regions of laminar, turbulent and transitional flow. However, the basic problem for engineers is to determine a pipe diameter rather than the state of the flow. Figure 3.1 and equation (4.15) are highly inconvenient for that purpose, because d appears three times in the arrangement of variables chosen for the Moody diagram. In the pipe flow problem there are really three variables of interest to engineers. These are, the pipe diameter, the discharge which can be carried by a particular pipe and the rate of head loss for a given pipe and discharge. Only the last can be determined directly from the Moody diagram.
When considered from the point of view of obtaining a convenient solution it is immediately obvious that equation (4.14) should be written in terms of discharge rather than velocity. However, velocity may be important in some cases (e.g. for deposition in sewers) so equation (4.14) could have been better written as
Q or V = φ (d, k, Sg, μ, ρ ) (4.16)
A convenient solution for practical purposes is now one in which Q, d and S (or Sg) appear only once in the final dimensionless arrangement. As indicated in section 3.2 this took many years to accomplish. The best arrangement is probably that of Ackers (1963) who used ν and k as repeating variables to obtain the arrangements given in equations (3.32) and (3.33).
Consider another classic fluid mechanics problem, the lock exchange flow (Barr, 1963). In lock exchange flow, a rectangular channel contains bodies of water of different densities, salt and fresh, but of the same depth, separated initially by a vertical barrier. When the barrier is suddenly removed, the denser water will flow under the less dense water and will form a saline wedge (see figure 4.1). If interest is in the distance, L, traveled in time, T, and in particular in the effects of viscous influence, then
Φ ( L, H, T, g', ν ) = 0 (4.17)
For the interests stated, it is important that any solution should contain the variables, L, T and ν as infrequently as possible. The arrangement which has been found to be particularly useful is
(4.18)