## 2 Theory and Literature Predictors for the Drag Coefficient
### Existing Predictors of the Drag Coefficient  
In the case of a single cylinder, the drag coefficient is a function of the Reynolds number, \(Re_{D}=VD/\
u\), where \(\
u\) is the kinematic viscosity of the fluid. [PERSON] [51] proposed to estimate \(C_{D}\) as follows:  
\[ \begin{array}{ll}C_{D}=3.07\ Re_{D}^{-0.168}&\text{ for }Re_{D}<800\\ C_{D}=1.0&\text{ for }800\leq Re_{D}<8000\\ C_{D}=1.2&\text{ for }8000\leq Re_{D}<10^{5}.\end{array}  \tag{10}\]  
Moreover, [PERSON] [52] suggested using the following equation:  
\[C_{D}=11 Re_{D}^{-0.75}+0.9\bigg{[}1-exp\bigg{(}-\frac{1000}{Re_{D}}\bigg{)} \bigg{]}+1.2\bigg{[}1-exp\Bigg{(}-\bigg{(}\frac{Re_{D}}{4500}\bigg{)}^{0.7} \Bigg{)}\bigg{]}. \tag{11}\]  
When there is a set of cylinders, a mutual influence rises among them, owing to the strong interaction between the wakes and also between the cylinders and the wakes, especially when the vegetation density is high. In this way, the drag coefficient can be significantly different from unity. A number of studies were carried out in this regard. The methods used to estimate the drag coefficient were based on theoretical analysis, on indirect or direct measurement, on the link with turbulence, on using computational fluid mechanics and on genetic programming. Some of the proposed formulas are recalled below and reference is made to [PERSON] et al. [53], [PERSON] et al. [54], [PERSON] et al. [55; 56] and [PERSON] et al. [57] for a more extensive overview. [PERSON] and [PERSON] [58] analyzed the effects of different vegetation distributions on the resistance to flow and sediment transport. For the description of the flow field downstream of the cylinders that schematize the vegetation, they used the laws introduced by [PERSON] [59]. Considering \(C_{D}=1.2\) for a single cylinder, they determined the average asymptotic \(C_{D}\) associated with a number of cylinders, parallelly and alternately distributed, and at different mutual distances. As an example, with a distance among the cylinders equaling 10 times their diameter, and a channel slope of 0.002, they obtained an average \(C_{D}\) value of 1.1 in the case of alternate distribution and 0.75 for a parallel-cylinder distribution. This is one of the few studies in which it was analyzed how the drag coefficient varies with the flow direction. The results obtained by [PERSON] and [PERSON] [58] showed that the flow velocity is remarkably influenced by the cylinder number and the way they are spatially distributed. In particular, for a constant flow rate condition, the flow depth increases with the cylinders' number and depends on their spatial distribution. The flow depth increases with an increase in vegetation density, while the mean flow velocity and the shear stress at the walls decrease. Moreover, for constant flow rate and vegetation density, the flow depth is remarkably higher in case of alternate distribution with respect to the parallel one. Later, [PERSON] and [PERSON] [60] proposed the following equation:  
\[C_{D}=2\bigg{(}\frac{\alpha_{0}}{Re_{D*}}+\alpha_{1}\bigg{)} \tag{12}\]  
where \(\alpha_{0}\) is the contribution relative to the viscous forces on the cylinder surface, \(\alpha_{1}\) is the contribution from the inertial forces deriving by the pressure drop downstream of the cylinders, and \(Re_{D*}=UD/v\). They found that \(\alpha_{1}\) varies linearly with the vegetation density, whereas \(\alpha_{0}\) is independent of the cylinder array characteristics for \(\phi_{veg}>0.15\). [PERSON] et al. [54], considering the literature data with square, staggered and random arrangements of the rigid cylinders, suggested the following equation:  
\[C_{D}=2\bigg{(}\frac{6475D+32}{Re_{D*}}+17D+3.2\phi_{veg}+0.5\bigg{)} \tag{13}\]  
where the coefficients of the \(D\) terms have units of m\({}^{-1}\) to preserve the non-dimensionality.