Asymptotic similarity of turbulence structures in smooth- and rough-walled pipes

Work recently reported by the authors (Perry & Abell 1975) on smooth-walled pipe flow showed support for the Townsend (1976) structural similarity principle as regards viscosity not being directly relevant in controlling the mean relative motions and the energy-containing turbulent motions. The work also supported a universal spectral behaviour in the wall region of the flow. In many hypotheses for rough-walled pipe flow, surface roughness, like viscosity, enters the problem only via external boundary conditions. Data obtained in a rough pipe are reported here and on first appearance the results seem to contradict the Townsend hypothesis and to threaten the very foundation upon which many similarity laws for rough-walled flows are based. However, on closer examination of the spectrum scaling of smooth-walled pipe flow the low and high wavenumber energy not necessarily associated with the universal similarity range can be accounted for. The broad-band longitudinal turbulence results for a rough-walled pipe can then be predicted from the smooth-wall scaling. The conclusion is that, despite the apparent anomalies, the turbulence structure in a rough pipe appears to follow the same scaling laws as for a smooth pipe, given a sufficient length of flow development in both cases. The deduced functional forms are consistent with Townsend's (1976) attached-eddy hypothesis.