If you have worked through the examples above or even if you just read the questions you must have realised that dimensional analysis did not change the world and make everything in fluid flow simple. Look at question 1 on the orifice tank. The outcome of the dimensional analysis is scarcely recognisable when compared with the rational expression that we normally use. In truth if the rational expression had not existed we would have had to create it from the result of the dimensional analysis in order to make progress. I do not think that the result for ship resistance has much to commend it over Froude’s method. In fact when you look at the uses that have been found for the Froude number they look as though they were “forced” uses and did not come naturally from some analysis or experimental plan.
The fact is that the important fields of fluid flow like pipes, channels, sewerage, ships, lubrication and hydraulic machines had all been sorted out quite satisfactorily before dimensional analysis came along. All this experience was not going to be replaced by new methods based on non-dimensional groups unless there was some very significant advantage. The great fields of activity that had yet to develop when dimensional analysis appeared was aerodynamics and model testing and, as luck would have it, they are the ones that are most suited to the use of non-dimensional numbers.
We now have non-dimensional groups in profusion all over science and it is hard to escape the thought that some have been devised to ensure that someone’s name will survive into the future. In my view one should always start to think about something that is new to you by deciding how it works. Dimensional analysis will not tell you nor will mathematics. When I start, I follow a golden rule: “look for the forces”. It always works where momentum and the like just hides the real modus operandi.
However dimensional analysis should not be ignored. It is always there and might fit in with what you are doing. But do not fall for the idea that only mathematics is acceptable and that somehow data that has been gathered by experience and experiment and then systematised for storage and retrieval is second best waiting only for something better to be devised. It is almost certainly correct and have stood the test of time and is available in engineering handbooks now and not some time in the future.
The juxtaposition of dimensional analysis and rational expressions leads to examination questions that may or may not have some value to the practising engineer. I give some here with their solutions.