Derivation of Canopy Resistance in Turbulent Flow from First-Order Closure Models

Quantification of roughness effects on free surface flows is unquestionably necessary when describing water and material transport within ecosystems. The conventional hydrodynamic resistance formula empirically shows that the Darcy&ndash;Weisbach friction factor f~(r/h<inf>w</inf>)1/3 describes the energy loss of flowing water caused by small-scale roughness elements characterized by size r (<<h<inf>w</inf>), where h<inf>w</inf> is the water depth. When the roughness obstacle size becomes large (but <h<inf>w</inf>) as may be encountered in flow within canopies covering wetlands or river ecosystem, the f becomes far more complicated. The presence of a canopy introduces additional length scales above and beyond r/h<inf>w</inf> such as canopy height h<inf>v</inf>, arrangement density m, frontal element width D, and an adjustment length scale that varies with the canopy drag coefficient C<inf>d</inf>. Linking those length scales to the friction factor f frames the scope of this work. By adopting a scaling analysis on the mean momentum equation and closing the turbulent stress with a first-order closure model, the mean velocity profile, its depth-integrated value defining the bulk velocity, as well as f can be determined. The work here showed that f varies with two dimensionless groups that depend on the canopy submergence depth and a canopy length scale. The relation between f and these two length scales was quantified using first-order closure models for a wide range of canopy and depth configurations that span much of the published experiments. Evaluation through experiments suggests that the proposed model can be imminently employed in eco-hydrology or eco-hydraulics when using the De Saint-Venant equations.