Application of the Universal Velocity Profile to rough-wall pipe flow

The universal velocity profile (UVP), defined initially for a smooth pipe, is extended to pipe flow with a rough wall. Data from pipe experiments of fixed roughness fraction and varying Reynolds number are used to formulate a simple correction to the smooth-wall UVP using a function of the roughness Reynolds number. A good fit to the data is achieved using an additive velocity correction combined with a new roughness transition function that accounts for the delayed onset of roughness friction at low roughness Reynolds numbers. It is concluded that the UVP can represent rough-wall pipe flow with a very small error comparable to the smooth-wall case.