The first thing to accept is that, in order to use dimensional analysis you must have a good idea how the system that you want to investigate actually works. The method is reactive and not proactive. It is best explained by example.
The resistance to motion of a ship is the most recent example of fluid flow in this text. Let me use that as my example. The first thing to do is to write down all the physical quantities that are relevant to ship resistance. In order to do so we have to be clear how the resistance is actually created.
The resistance is due in part to wave-making and in part to friction between the hull and the water. The resistance caused by wave-making is dependent on the additional pressure created over the hull by the raising of waves above the normal level of the free surface. This additional pressure depends on the size and speed of the ship, on the density of water and on the acceleration due to gravity and it acts on some unspecified area. The skin drag will depend on the speed and the wetted area and on the density and the coefficient of viscosity of the water.
We can now easily list some of these quantities. They are speed , acceleration due to gravity , density and viscosity . This leaves a set of areas of the hull and the length of the hull and it is not obvious how these might be included. For the purposes of dimensional analysis it is supposed that the shape of the hull can be determined from one dimension just as archaeologists can draw a medieval ship from one dimension of the rudder. The dimension that is normally chosen is a typical length like the overall length .
Then we could say that the resistance to motion depends in some way on . You could write that where stands for “some function” that will have to be determined by experiment. The terms in the bracket are not multiplied together; they are just a list. They look to be very daunting because we now have to think of a way to set up the experiments. Experiments often throw up lots of experimental data and the experiment is wasted if the data is not stored in some way that makes it available for use in the future. Dimensional analysis can help us to group these variables in recognised ways and perhaps to go on to devise experiments that will lead to a satisfactory method of data storage.
I have pointed out that the function of this group of variables must be found by experiment and it is normal to record the function in a graph or family of graphs. Then a graph like, say, Froude’s graph for the frictional resistance of a board when towed edgewise is such a graph. There the graph can be represented by a polynomial expression to considerable advantage. But the graph of the resistance of the whole ship against speed cannot be represented by a single polynomial. However it is possible to represent it by lots of polynomials added together to give the correct shape. This fact permits us to make progress with our dimensional analysis by writing:-
In this series is just a term that is typical of the many terms in this series.
Now this equation is part of a description of a physical relationship between well-known quantities and it must be in consistent units so that, whatever the combination of the variables may be, they reduce in the end to those of force. Furthermore every physical quantity can be expressed in some consistent system of units but, more usefully to us, it can also be expressed in fundamental units of mass, length and time. I will give some examples :-
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Now the equation for the typical term can be expressed in these fundamental units. I have shown it in the first line of the table below. As the equation has every physical quantity given to some power, equations can be drawn up for the sums of these indices. We can have up to three equations, one each for mass, length and time but one might add other dimensions such as temperature for, say, heat transfer. I have put the three equations into a table because this is just a manipulation exercise that can go wrong very
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easily. At least in this tabular form it is easy to check the equations in their columns before you proceed. I use the same layout for longhand.
Now we can simplify.
We can now go back and re-write the typical term :-
This says that the resistance is equal to times a function of Reynolds number and the Froude number. The group is, in fact the product of a pressure and an area, the group being the pressure that is more recognisable as the stagnation pressure , and is the area. It looks quite exciting but if you ask whether this has improved our position relative to our knowledge of ship resistance it is hard to see what has changed.
Dimensional analysis has sorted out the variables but we are no nearer to designing an experiment to determine the function . However if you follow Froude’s argument we can see that we might write :-
and then and and, if you take the line that some espouse, that this separation is not justified, this change does not makes it any more legitimate. Froude chose an experimental plan that was correct in the event whatever objections there may be.
So we have a way to sort out relevant physical quantities but it is not a sure-fire method and must be used with care. Try doing a dimensional analysis for a sharp-edged orifice in the base of a cylindrical tank to see what I mean. This dimensional analysis is just another tool in the locker of those constructing an experimental science. It has its greatest value for pipes, as we have seen, for objects over which air flows steadily like aerofoils and objects that move underwater like torpedoes and hydrofoils.
Now there is a need for some examples to practice on.