Lord Rayleigh.

Lord Rayleigh, who was a hereditary peer, introduced the idea of presenting data in terms of non-dimensional plots. This turned out to have far reaching consequences throughout science and in the science of the movement of fluids has come to be extremely powerful. We must see how this came about.

 

The normal way of presenting results prior to Rayleigh, was to plot a family of graphs relating two variables with a third kept constant for each graph. In order to take into account a fourth variable a new family of graphs would be plotted. In some cases it was possible to group variables in ways that compressed the data and reduced the number of tables or graphs. These were called homologues. As most graphs are continuous and many are curves, the most common way of presenting results for use in calculation was to represent a power law presentation and of many other empirical expressions was well recognised and in 1888 Fourier[1] began the process which led to non-dimensional plotting by studying the characteristics of an equation relating physical quantities. In essence he said that if a physical system was seen to involve several physical quantities such as density, mass, viscosity, force, velocity, acceleration it should be possible to relate the relevant physical quantities by an expression of the form :      where , and  are physical quantities and  stands for "a mathematical function of…..". Fourier went on to note that all physical quantities can be expressed in terms of the more fundamental quantities of mass, length, time and temperature (although Fourier did not include mass because his paper was on heat transfer). In modern times engineers expect to work with systems of units that are consistent although this is really a relatively recent practice. We expect to find the units of Froude's constant by equating the indices of the units of mass, length and time on the two sides of the equation. This was new when Fourier proposed it. However Fourier took it one stage further and noted that for each fundamental dimension the sum of the indices as they appeared in the units was zero and that the equation could be rewritten in a new form where, if we use the modern presentation:-

                    , where  stands for a group of physical quantities which combine to have no dimension. Given the physical quantities that were relevant to a particular physical system the mathematically trained physicists of the day would not have had any difficulty in sorting out the best non-dimensional groups of variables.

 

These observations made little impact because they seemed not to have any application. Then in a paper published in 1899 Lord Rayleigh[2] made use of Fourier's work. The paper was concerned with the liquid dripping from the end of a vertical capillary tube with its lower end ground flat. During the formation of a drop the liquid is supported by the surface tension (ie the forces of attraction between the molecules at the surface of the liquid) that acts like a party balloon. As the amount of liquid supported at the end of the tube grows its shape changes and eventually it assumes a drop shape and a neck at which the surface tension acts to support the drop. When the weight of the liquid exceeds this force the neck contracts and a drop separates and falls.

 

Rayleigh set out the limitations as follows:

 

"....in most cases , viscosity may be neglected, the mass (  ) of a drop depends only on the density (  ) the surface tension (  ), the acceleration of gravity (  ) and the linear dimension of the tube (  ). In order to justify this assumption, the formation of the drop must be sufficiently slow, and certain restrictions must be imposed on the shape of the tube. For example, in the case of water delivered from a glass tube, which is cut off square and held vertically  will be the external radius; and it will be necessary to suppose that the ratio of the internal radius to  is constant, the cases of a ratio infinitely small, or infinitely near unity included. But if the fluid be mercury, the flat end remains unwetted, and the formation of the drop depends upon the internal diameter only."

 

Next Rayleigh considered dimensions and I quote:

 "The "dimensions" of the quantities on which  depends are:-

,  and  =

of which , a mass, is to be expressed as a function.

 

If we assume :

                                                  we have, considering in turn, length, time, and mass,

                                    ; ; and ;

so that

                                           ; ; ."

 

The straightforward substitution of these values would give:-

  , but this did not fit in with the way in which Rayleigh was tackling this problem. He had started from the observation that in some way the weight of the drop was related to the force exerted by the surface molecules at the neck of the liquid at the instant of separation. This, of course depends on the surface tension and the radius of the neck.

 

Rayleigh used the fact that if , we can also say that  or any other combination of the variables  and . Rayleigh continued and again I quote:

 "Accordingly                          . Since  is undetermined, all that we can conclude is that  is of the form:

                                                 , where  denotes an arbitrary function."

 

We can note that this expression can be rearranged to give:

                                                  , which is in accordance with Fourier's statement that any homogeneous expression between physical quantities can be rewritten in terms of non-dimensional groups. It should be observed that  is the ratio of the weight of the drop and the product of radius and surface tension, which, if multiplied by 2 , equals the weight of the forming drop. Rayleigh appears to have been convinced that, at the same value of , different, but geometrically similar systems, would share a flow pattern, that is, in Newton's words, be dynamically similar.

 

Rayleigh then experimented with water dropping from tubes having diameters up to 10 millimetres with various bores up to the thin walled case and plotted  against . This plot was a straight line up to the point where surface tension ceased to dominate the process of drop formation and Rayleigh deduced that, in any consistent system of units:

                                                                 .

 

Lord Rayleigh had shown that Fourier's ideas could be put to use in the science of fluid flow.

 

By any standards this six-page paper on a topic that had little practical application is quite remarkable. In it Rayleigh set out the complete structure for what is now known as the method of dimensions and provided a method of storing and retrieving experimental data in a new and very valuable way. Its value seems not to have been recognised by others but Rayleigh was sufficiently influential to ensure that it was explored more fully.

 

Rayleigh was the president of an Advisory Committee for Aeronautics formed in 1908 and reporting to the government. Its main work took place at the National Physical Laboratory. The work seems to have been totally out of step with the rate at which practical aeroplanes were being developed and in 1909 we find R.E.Froude experimenting to measure lift and drag of flat and curved square plates in a water tunnel. Rayleigh drew attention to the possibilities of making non-dimensional plots and of the likelihood of finding some relationship between the behaviour of geometrically similar systems. In the same report we find L.Bairstow and Harris Booth commenting on the possibility of re-appraising data from Eiffel and Lanchester on the resistance of square plates normal to a flow of air along the lines proposed by Rayleigh.

 

It is not surprising to find that the National Physical Laboratory set up two experimental programmes to look into Rayleigh's proposals. One was concerned with square flat plates set at an angle to the flow in an "air channel" or in modern parlance a wind tunnel. The other was to investigate the resistance to motion of fluids in pipes and it is this that interests us now.