Reynolds Number

k = r^wv^xa^yh^z

[M]⁰[L]⁰[T]⁰ = [M]^w[L]^-3w[L]^x[T]^-x[L]^y[M]^-z[L]^-z[T]^-z

(a) Equating powers of [M]; z = -w

(b) Equating powers of [L]; 3w = x+y-z

(c) Equating powers of [T]; x = -z

Substituting (a) into (c) gives x = w

Substituting for x and z into (b) gives y = w

So k = r^wv^wa^wh^-w. We can't specify w precisely, but since the combination is dimensionless, its value is irrelevant, and it is conventionally taken as 1. The threshold value of the combination, known as Reynolds' Number, at which there is a transition from laminar to turbulent flow, is about 2000.

R  = r v a / h (= 2000 at the laminar/turbulent threshold)