Most kinds of fluid flow that are important in natural environments do not lend themselves to analytical solutions, even when no sediment is moved, so experiment and observation are a valuable way to learn something about them. I have expatiated upon dimensionless variables and their use in expressing experimental results because this sort of analysis, usually called dimensional analysis, is so useful in dealing with problems of fluid flow and sediment movement. Dimensional analysis is a way of getting some useful information about a problem when you cannot obtain an analytical solution and may not even know anything about the form of the solution, but you have some ideas about important physical effects or variables. You will encounter many examples of its use in later chapters.
Suppose that you are dealing with a fluid-flow problem that can be simplified somehow, perhaps in geometry or in time variability, to be manageable but still representative. Use your experience and physical intuition to identify the important variables. Form a set of dimensionless variables by which the observational results can be expressed. This represents the most efficient means of dealing with experimental data, and it usually makes it possible to get some idea of the ranges in which certain physical effects are important or unimportant. Do not worry too much about guessing wrong about important variables; the example of flow past a sphere shows how you can find out and change course.
The number of dimensionless variables equivalent to a given set of original variables is given by the Pi theorem, also called Buckingham’s theorem. By the Pi theorem, the number of dimensionless variables corresponding to a number \(n\) of original variables that describe some physical problem is equal to \(n -m\), where \(m\) is the number of dimensions by which the problem must be expressed. If you want to go back to the original source of the proofs (the theorem was not proved in the foregoing material, just demonstrated), see Buckingham (1914, 1915).