## 2 Material and Methods
### Description of the Test Cases
#### 2.2.1 Open Channel Flow  
The uniform open channel flow is a free-surface channel flow characterised by a constant slope and a constant discharge. The vertical velocity profile is the well-known logarithmic profile [33]. Figure 1 shows a sketch of the free-surface channel flow we modelled. The bottom of the channel is depicted in red while the surface in blue. In steady conditions these surfaces should be parallel to each other. The profiles of velocity, tangential stresses and turbulent eddy viscosity are shown as a function of the vertical axis \(z\).  
We define a non-dimensional conductance coefficient \(C\) as the ratio between the depth-averaged velocity \(\overline{U}\) and the bed friction velocity \(u_{b}^{*}\), which can be related to the flow depth and the sediment diameter \(d_{s}\) through the Keulegan [33] equation for fully rough turbulent flows:  
\[C=\frac{\overline{U}}{u_{b}^{*}}=\frac{1}{\kappa}\log\left(\frac{11.09R_{i}}{ 2.5d_{s}}\right), \tag{32}\]  
where \(R_{i}\) is the hydraulic radius, i.e., \(R_{i}=\Omega/(W+2H)\), \(W\) being the width of the channel and \(H\) the water depth, and \(d_{s}\) is the average diameter of the sediments. In this expression \(2.5d_{s}\) is considered as the Nikuradse roughness length scale \(z_{r}\). The choice of this expression is justified by the fact that it derives directly from the rough wall law and it is adjusted for rectangular riverbeds. The expected water depth can be evaluated thanks to the uniformity of the flow by solving iteratively the following equation:  
\[Q=\Omega C\sqrt{gR_{i}i_{f}}, \tag{33}\]  
where \(Q\) is the flow, \(\Omega\) the cross section of the channel, \(g\) the acceleration of gravity and \(i_{f}\) the slope.  
Figure 1: Sketch of uniform open-channel flow modelled with Delft3D and ROMS. Velocity, tangential stresses and vertical eddy viscosity are plotted in order to show the expected theoretical trends; i.e., a logarithmic, linear and parabolic profiles, respectively.  
We simulate such a flow over a 1000 m long and 50 m wide channel where a constant discharge of 300 m\({}^{3}\)/s flows. The grid is made of constant square cells of 5 m sides. The velocity profile and all the results are taken at the centre of the channel in order to discard possible effects of the boundaries. The channel presents a constant slope of 0.1%. Delft3D presents two open boundaries conditions of total discharge at both ends of the channel. The total discharge is modelled in order to increase linearly from a zero value to 300 m\({}^{3}\)/s in one hour. After that, steady conditions of flows are reached. Analogously for ROMS, conditions of a constant current parallel to the bed are chosen since conditions of total discharge are not provided as a default option in the code. The current velocity is chosen as 2.47 m/s in order to allow a constant flow of 300 m\({}^{3}\)/s in steady conditions and to carry out a fruitful comparison with Delft3D. The simulation parameters are summarised in Table 2.  
Results of the benchmark are analysed in Section 3 and compared with the theoretical solution expected, which is the usual logarithmic profile:  
\[\frac{U}{u_{b}^{*}}=\frac{1}{\kappa}\log\left(\frac{z}{z_{r}}\right)+8.5. \tag{34}\]  
The velocity \(U\) goes to zero when \(z=\
icefrac{{z_{r}}}{{30}}\). Consequently, tangential stresses and vertical eddy viscosity were expected to have a linear and a parabolic profile, respectively [34].