## 2 Theory and Literature Predictors for the Drag Coefficient
### Existing Predictors of the Drag Coefficient
[PERSON] and [PERSON] [61] introduced, under the hypothesis that the wall and bottom effects are negligible, the vegetation-related hydraulic radius, \(r_{v}\), which takes into account the density and the diameter of the vegetation, as \(r_{v}=(\pi/4)\big{(}\big{(}1-\phi_{veg}\big{)}/\phi_{veg}\big{)}D\). Thus, they defined the vegetation Reynolds number as \(Re_{v}=Ur_{v}/v\). Using the experimental data from several authors (random and staggered distributions and only two linear cases), [PERSON] and [PERSON] [61] showed that the drag coefficient decreases monotonically with an increase in \(Re_{v}\) and proposed the following equation:  
\[C_{D}=\frac{50}{Re_{v}^{0.43}}+0.7\bigg{[}1-exp\bigg{(}-\frac{Re_{v}}{15,000} \bigg{)}\bigg{]}. \tag{14}\]  
An estimate of the drag coefficient, using the direct measurements, was performed by [PERSON] et al. [62], [PERSON] et al. [11] and [PERSON] et al. [63], among others. In [PERSON] et al. [62], the cylinders were distributed on a grid with a staggered pattern forming angles of 45\({}^{\circ}\) with the flow direction and the action was calculated averaging the values obtained for 7 or 13 cylinders. In [PERSON] et al. [11], the angle formed with the flow direction was 30\({}^{\circ}\) and the action was determined on only one cylinder. Instead, [PERSON] et al. [63] used a linear arrangement of the cylinders and the action was calculated as the mean of that exerted on a cylinder group from 2 to 25. In the experiments of [PERSON] et al. [62], the drag coefficients differ significantly for the same stem Reynolds number, although the dependence is unclear. They provided three equations for the drag coefficient as a function of the areal density, depending on the flume bottom slope. In [PERSON] et al. [11], the density \(\phi_{veg}\) ranges between 0.0022 and 0.0885. For a subcritical flow, they obtained the following equation:  
\[C_{D}=1.53\big{[}1+0.45 ln\big{(}1+100\phi_{veg}\big{)}\big{]}Re_{D*}^{-3/50} \tag{15}\]  
in which \(C_{D}\) remarkably increases with \(\phi_{veg}\) and slightly varies with \(Re_{D*}\). [PERSON] et al. [63], on the basis of 70 tests with emerging stems in a linear arrangement, in uniform or quasi-uniform flow conditions, proposed an equation in which \(C_{D}\) is a function of the density \(\phi_{vcg}\) only:  
\[C_{D}=0.211 ln\big{(}100\phi_{vcg}\big{)}+0.784. \tag{16}\]  
The equation is valid for \(\phi_{vcg}\) from 0.003 to 0.05, \(i\) from 0.48% to 2.02% and \(Re_{D*}\) from 1000 to 10,000. The values obtained from Equation (16) are smaller than those obtained from Equation (15) for a given \(\phi_{vcg}\), owing to the two different cylinders arrangements (a triangular mesh in the case of [PERSON] et al. [11] and a square mesh in the case of [PERSON] et al. [63]). [PERSON] [64] assumed that the production of turbulence in the presence of vegetation is equal to the dissipation and that the turbulent intensity is equal to the drag forces. [PERSON] et al. [57] utilized genetic programming to develop a predictor of drag coefficients from the experimental data. They did not take into account the influence of the arrangement of the cylinders and expressed the drag coefficient as a function of \(Re_{D*}\), \(\phi_{vcg}\) and the blockage ratio \(\psi\). This latter, defined as \(\psi=D/L_{y}\), where \(L_{y}\) is the lateral distance between adjacent stems at the same streamwise location, reflects the blockage effect on a multi-cylinder array. [PERSON] et al. [57] employed 475 data points relative to random and staggered cylinder distributions. Among all the solutions provided by the software Eureqa, [PERSON] et al. [57] selected the final expression of \(C_{D}\) that could balance the complexity, accuracy and physical meaning to a great extent:  
\[C_{D}=\frac{189}{Re_{D*}}+0.82+\psi^{2}+6.02\phi_{vcg}. \tag{17}\]  
This expression has the same form as Equation (12) with the term \(\alpha_{1}\) function of the blockage factor and density. [PERSON] et al.