Friction Losses in Turbulent Pipe-flow

Significant developments in the theory of turbulent flow in smooth and rough pipes are reviewed to establish a rational basis for the commonly accepted logarithmic laws for pipe friction. The Prandtl (1932)‡ smooth-pipe law, , where f is the friction coefficient in the formula , agrees with measured results on smooth pipes up to Reynolds numbers of at least 3 × 10⁶. With rough pipe walls and sufficiently high Reynolds numbers, viscosity (and hence Reynolds number) ceases to have any direct effect and the friction coefficient depends on wall roughness and pipe size only. Almost all practical cases of water flow in commercial pipes lie between these two extremes of completely smooth and fully rough conditions, where the friction coefficient varies with both Reynolds number and roughness.

Exponential flow formulae of the Manning type—V = ***Am₁xiy—can be rearranged into a more rational form f = B(Re)p(k/d)q*** relating f to Reynolds number and relative roughness for a given class of pipe carrying a fluid of given viscosity. A detailed study is made of published test data on wrought-iron and steel pipes which generally operate in the transition zone, and an exponential formula is derived which agrees with these results. This is found to be similar to that given by Blair for this class of pipe.

The relative merits of these exponential formulae, and of the Colebrook-White transition function, are discussed and it is concluded that, for most practical cases of water flow in pipes, the simple formulae are no less reliable.