A Simple Dimensional Method for Hydraulic Problems

Every physical magnitude in a problem has a unique dynamic number. The physical magnitude is expressed dimensionally in terms of the measure-system components. This expression is divided by the physical magnitude itself. The quotient is the dynamic number. In hydraulic problems, the measure-system components are usually ρ (density), L (length), V (velocity), M (mass), L, T (time) dimensions are transformed into ρ, L, V dimensions through the identitites M ≡ ρL³ and T ≡ L/V. Thus the dynamic number for viscosity is formed as follows: μ (viscosity) has the dimensions M/LT, which become ρ L V. Dividing by μ gives the dynamic number N_μ = ρ L V/_μ = R. Application of this simple system, to all of the physical magnitudes involved, yields a set of dynamic number N₀, N₁, N₂, N₃....... It is shown that any useful physical equation may be reduced to the form N₀ = f (N₁, N₂, N₃.......), the nature of the function f being revealed only by physical analysis of the problem. When the appropriate dynamic numbers are substituted in this expression, the result is identical with that obtained by the more lengthy conventional methods of dimensional analysis such as the use of the π-theorem.