## 2 Theory and Literature Predictors for the Drag Coefficient  
### Overview and Basic Definition  
Very often, it is necessary to compute the free surface profile in an open channel for an assigned flow discharge. The main elements for the numerical calculation of the profile are summarized below, highlighting their peculiarities in the presence of emergent rigid vegetation. To this end, we refer to a GVF in a prismatic channel, where the energy grade line (EGL) slope can be locally evaluated using the equation for uniform flow with the relative coefficient of resistance and the local depth with an assumption that the flow is locally uniform. From the Bernoulli equation, the total head, \(H\), at any cross section can be determined as follows:  
\[H=z_{0}+h+\alpha\frac{V^{2}}{2g} \tag{1}\]  
where \(z_{0}\) is the bed elevation, \(h\) is the flow depth with respect to the channel bed, \(V\) is the mean flow velocity, \(\alpha\) the kinetic energy flux correction factor and \(g\) is the gravitational acceleration. Differentiating Equation (1) with respect to the coordinate in the flow direction \(x\), we obtain the following equation:  
\[\frac{dH}{dx}=-J=-i+\frac{dE}{dx} \tag{2}\]  
where \(J\) is the EGL slope, \(i\) is the bed slope (i.e., \(i=-dz_{0}/dx\)) and \(E\left(E=h+\alpha V^{2}/(2g)\right)\) is the specific energy (usually, \(\alpha=1\)). In the numerical computation of the GVF profiles, the local \(J\) can be calculated from Manning's equation using the local flow depth, assuming that the flow is locally uniform [37]:  
\[V=\frac{1}{n}R^{2/3}J^{1/2} \tag{3}\]  
where \(n\) is the Manning coefficient and \(R\) is the hydraulic radius. For an open-channel flow without vegetation, \(n\) can be estimated from the channel bed and bank roughness. However, the roughness estimate becomes complicated in presence of vegetation elements. Steady and locally uniform flow conditions require a local force balance between the flow driving the mechanism and the drag term. For a given length-scale \(dx\) along the streamwise direction, the flow driving mechanism is given by the component of the self-weight of the fluid along the flow direction, while the resistance is given by vegetation and the friction on the bottom and banks. Assuming that these latter can be neglected and considering a rectangular section, it is possible to obtain the following balance equation [1]:  
\[\gamma Bhdx\big{(}1-\phi_{veg}\big{)}J=BdxF_{D} \tag{4}\]  
where \(\gamma\) is the specific weight of the fluid, \(B\) is the cross-sectional width, \(\phi_{veg}\) is the areal concentration of vegetation and \(F_{D}\) is the drag force exerted on vegetation per unit of area. Simplifying the terms that appear in Equation (4), one can write:  
\[\gamma h\big{(}1-\phi_{veg}\big{)}J=F_{D}. \tag{5}\]  
If vegetation is simulated by rigid cylinders of diameter \(D\), the resistance due to vegetation per unit riverbed area, spatially averaged, is given by the following equation:  
\[F_{D}=\frac{1}{2}C_{D}mDh\rho U^{2} \tag{6}\]  
where \(C_{D}\) is the drag coefficient of the cylindrical vegetation, \(\rho\) is the fluid mass density and \(m\) is the number of vegetation stems per unit bed area, that is:  
\[\phi_{veg}=m\pi D^{2}/4 \tag{7}\]  
and \(U\) is the bulk velocity (or pore velocity), which is defined as follows:  
\[U=\frac{Q}{B\big{(}1-\phi_{veg}\big{)}h} \tag{8}\]  
where \(Q\) is the flow discharge. By replacing Equation (6) into Equation (5), one gets  
\[J=\frac{C_{D}mD}{\left(1-\phi_{veg}\right)}\frac{\mathcal{U}^{2}}{2g}. \tag{9}\]  
On the basis of the above equations, the central role assumed by the drag coefficient in the calculation of the free surface profile is noticeable. It should also be noted that the correct estimation of the drag coefficient is also important in the use of Reynolds Averaged Navier-Stokes (RANS)-based turbulence models [38; 39] applied to flow in vegetated channel.