Question

how to choose repeating variables in Buckingham's π theorem?

Answer 1

The repeating variables are any set of variables which, by themselves, cannot form a dimensionless group. Diameter, velocity, and height cannot be arranged in any way such that their dimensions would cancel, so they form a set of repeating variables.

For example, let's assume that we suspect that a fluid we're studying behaves as a function of several variables including a characteristic length, velocity, viscosity, density, surface tension, etc., and we want to see what dimensionless numbers we can make out of them. A possible choice of repeating variables would be length (ll), velocity (vv), and density (ρρ) (in MKS units they would be mm, msms, kgm3kgm3), because they cannot be combined in any way to make a dimensionless group. Now add surface tension (σσ, units of kgs2kgs2) and combine all four variables to make a dimensionless group like this:

ρlv2σρlv2σ

This is the Weber Number. Another possibility would be to use viscosity instead of surface tension:

ρlvμρlvμ

Which is of course the Reynolds Number. Ultimately, the choice of which combination to use out of all the possibilities comes down to whichever is more useful for the type of problem you're working on; Reynolds is good for turbulence and heat transfer, while Weber is more suitable for bubble and droplet formation.