On turbulent friction in straight ducts with complex cross-section: the wall law and the hydraulic diameter

We develop predictive formulae for frictional resistance in ducts with complex cross-sectional shape based on the use of the log law and neglect of wall shear stress non-uniformities. The traditional hydraulic diameter naturally emerges from the analysis as the controlling length scale for common duct shapes such as triangles and regular polygons. The analysis also suggests that a new effective diameter should be used in more general cases, yielding corrections of a few percent to friction estimates based on the traditional hydraulic diameter. Fair, but consistent, predictive improvement is shown for duct geometries of practical relevance, including rectangular and annular ducts, and circular rod bundles.