Manning's Formula and the Strickler Scaling Derived from a Co-spectral Budget Model

Manning's empirical formula in conjunction with Strickler's scaling is widely used to predict the bulk velocity (V) from the hydraulic radius (Rh), the roughness size (r), and the slope of the energy grade line (S) in uniform channel flows at high bulk Reynolds numbers. Despite their importance in science and engineering, both Manning's and Strickler's formulations have awaited decades before finding a theoretical explanation. This explanation was provided using phenomenological arguments by Gioia and Bombardelli (2002) and is hereafter labeled as GB02. The main finding was a link between the Strickler and the Kolmogorov scaling exponents, the latter pertaining to velocity fluctuations in the inertial sub-range of the turbulence spectrum that is presumed to be universal. In this work, the GB02 analysis is revisited showing that GB02 employed some ad-hoc scaling assumptions for the turbulent kinetic energy dissipation rate and, although implicitly, for the mean velocity gradient adjacent to the roughness elements. The similarity constants arising from the GB02 scaling assumptions were presumed to be independent of r/Rh inconsistent with known flow properties in the near-wall region of turbulent wall flows. Because of the dependency of these similarity constants on r/Rh, GB02 requires the validity of the Strickler scaling to cancel the dependency of these constants on r/Rh so as to arrive at the Strickler scaling and thus Manning's formula. Here, the GB02 approach is corroborated using a co-spectral budget (CSB) model for the wall shear stress at the interface between the roughness sublayer and the log-region. Assuming a simplified shape for the spectrum of the vertical velocity (Eww), the proposed CSB model allows Manning's formula to be derived. To substantiate this approach, numerical solutions to the CSB over the entire flow depth using different spectral shapes for Eww are carried out across a wide range of r/Rh. The results support the simplifying hypotheses used to derive Manning's equation. While none of the investigated spectral shapes allows the recovery of the Strickler scaling, the numerical solutions of the CSB reproduce the Nikuradse data in the fully rough regime thereby confirming that the Strickler's scaling represents only an approximate fit for the friction factor for granular roughness.