On the Relationship between Manning’s Roughness Coefficient and Relative Roughness

Manning’s roughness coefficient n is widely used in hydraulic engineering to estimate flow resistance, yet it lacks direct linkage to measurable physical parameters of flow and roughness geometry. In contrast, the Darcy–Weisbach friction factor f offers a physically grounded description of resistance through its direct link to surface roughness. This study provides a quantitative relationship between n and relative roughness ε/D by reconciling the Manning and Darcy–Weisbach formulations under fully rough, turbulent conditions. Using synthetic flow scenarios generated via numerical modeling, we examine how n systematically varies with ε/D in shallow open-channel flows. Results show that n increases monotonically with relative roughness, following trends consistent with the Moody diagram. This connection enables a more mechanistic interpretation of empirical resistance and provides a practical framework for estimating n from known roughness scales, bridging empirical and theoretical models in wall-bounded turbulent flow.