## 2 Theory and Literature Predictors for the Drag Coefficient
### Existing Predictors of the Drag Coefficient
[65], in the evaluation of flow resistance models based on field experiments in a partly vegetated reclamation channel, considered \(C_{D}=1\). The above drag coefficient estimates are based on the experiments carried out under the conditions of uniform or quasi-uniform flow. More recently, the drag coefficient (or Manning coefficient) variation along the flow direction was also analyzed in the case of non-uniform steady flow. [PERSON] et al. [66] calculated the drag coefficient on the basis of approximately 70 experiments under the assumption of uniform and non-uniform steady flows. These authors stated that adopting the \(C_{D}\) values obtained under the conditions of uniform flow, when the flow is not uniform, leads to significant errors and, moreover, by excluding the smallest densities (\(\phi_{vcg}=0.3\%\)), the drag coefficient decreases with an increase in \(Re_{D}\). It should be noted that they referred to the average values collected in the field. [PERSON] et al. [1] analyzed the influence of vegetation on the non-uniform steady flow profile in the absence of a strong driving gradient. They observed eight M2 type profiles on a flume with \(i=0\%\), in vegetated section of lengths \(L\) between 0.52 and 0.67 m with densities between 1% and 41.9%. The flow rate was constant with \(Q=3.84\times 10^{-3}\) m\({}^{3}\) s\({}^{-1}\), while the upstream flow depths \(h_{0}\) were between 4.7 and 21 cm and the cylinder diameter \(D=1\) cm. The numerical simulation of the profile, carried out from upstream, resulted in an overestimation of the measured water depth when the drag coefficient was calculated with the equation for a single cylinder (Equation (11)). Instead, using the equation that takes into account the mutual influence of the cylinders (Equation (14)), the water depth was underestimated in the case of high vegetation density and overestimated for a low vegetation density. They, therefore, searched for an alternative expression to represent the drag coefficient. The measured water depth \(h\) as a function of the distance \(x\) from the beginning of the vegetated part of the channel was smoothed with the following mathematical function:  
\[h=c_{1}ln|x-c_{2}|+c_{3} \tag{18}\]where the constants \(c_{1}\), \(c_{2}\) and \(c_{3}\) were determined with the Matlab software (The MathWorks, Inc., Natick, Middlesex County, MA, USA). Having obtained the average flow profiles, [PERSON] et al. [1] computed the local drag coefficient \(C_{D}\) by means of Equation (8). They observed a non-monotonic pattern of the drag coefficient \(C_{D}\) versus the stem Reynolds number \(Re_{D*}\); indeed, the \(C_{D}\) increases from the inlet (low \(Re_{D*}\)), reaching a peak value, and then decreases toward the outlet (high \(Re_{D*}\)). Starting from Equation (2), [PERSON] et al. [1] proposed the following equation:  
\[C_{D-W}=\frac{2g\big{(}1-\phi_{vcg}\big{)}}{mD}[P^{*}-A^{*}] \tag{19}\]  
where \(P^{*}\) and \(A^{*}\) represent the advection and pressure components, respectively, and are functions of \(Re_{D*}\). Specifically, the two terms can be calculated as follows:  
\[P^{*}=S_{h}D^{2}v^{-2}Re_{D*}^{-2} \tag{20}\]  
and  
\[C_{D}=2\bigg{(}\frac{\alpha_{0}}{Re_{D*}}+\alpha_{1}\bigg{)} \tag{12}\]0  
where \(S_{h}\) is the free surface slope, which can be determined from  
\[C_{D}=2\bigg{(}\frac{\alpha_{0}}{Re_{D*}}+\alpha_{1}\bigg{)} \tag{12}\]1  
[PERSON] et al. [1] showed how \(P^{*}\) and \(A^{*}\) explain the non-monotonic behavior of \(C_{D}\) along \(x\) for the experiments with \(0.01\leq\phi_{vcg}\leq 0.419\) and also how, in the specific case of a constant flow discharge, the coefficients \(c_{1}\) and \(c_{1}/c_{2}\) depend on \(\phi_{vcg}\), while \(c_{3}\) depends on \(c_{1}\), \(c_{2}\) and the flow depth in the upstream cross section. In case of [PERSON]'s et al. [1] experiments, characterized by \(i=0\), Equation (19) introduced into Equation (2) allows the detected profiles to be reproduced in a quite correct way.