## 3 Results and Discussion
### Open Channel Flow  
The case of uniform open channel flow was carried out assuming a bed roughness \(z_{r}=0.03\) m. Employing Equations (32) and (33) it is possible to compute the water depth \(H\) and the friction velocity at the bottom \(u_{b}^{*}\) for a constant flow of 300 m\({}^{3}\)/s, respectively, to be equal to 2.428 m and 0.147 m/s. The conditions are those of a rough surface since the rough Reynolds number \(Re_{*}=u_{b}^{*}z_{r}/v=4420\) is much greater than 100 [37]. Figure 4 shows the comparison among the non dimensional vertical profiles of the longitudinal velocity, eddy viscosity and turbulent kinetic energy obtained with the available turbulence closures of Delft3D and ROMS. In particular, Delft3D adopts \(k-\epsilon\), \(k-L\) and algebraic closure models. Conversely, ROMS adopts \(k-\epsilon\), \(k-\omega\), \(k-kl\) and \(M\)Y25 turbulence models. Figure 4a,b show on a semi-log plot, the vertical profile of the longitudinal velocity made non dimensional with the bed friction velocity, together with the theoretical log law of Equation (34). Each profile obtained from the turbulence models is made non-dimensional either using the model bed shear stress outputs since \(u_{b}^{*}=\sqrt{\frac{\eta_{b}}{\rho_{o}}}\) or the theoretical \(u_{b}^{*}=0.147\) m/s. Such results are shown in panels (a) and (b) respectively. The choice of normalising the velocity profile with theoretical and output friction velocities is due to the fact that the curves should collapse over the rough wall law. In particular, comparing panels (a) and (b), it is possible to appreciate that the spread of the profiles is greater for high values of \(z/z_{r}\) adopting the output friction velocity. Results reported in Figure 4b show an increasing overlapping. Therefore, it is possible to argue that, although the profiles follow the expected slope in the semi-log plot, the output bottom stress from the model does not conform to the profiles obtained. In addition, our attention is drawn from the points at the bed since they do not follow the rough wall law. Such an issue is independent from the normalisation adopted since both panels show the same shortcomings. Boundary conditions at bottom do influence the bed friction velocity, and consequently, the velocity at the bottom layer. It is the evaluation of the bed friction velocity that must be modified adequately in order to produce a velocity at the bottom that is in agreement with the rough wall law. Another issue consists in the differences between the bed friction velocities generated from one turbulence model to another. As a result, whether we normalise with the theoretical bed friction velocity at the bottom or not, the curves tend to disperse around the wall law. This is the sign of possible inconsistencies between the various turbulence models adopted by Delft3D and ROMS. Figure 4c reports the vertical eddy viscosity normalised with theoretical friction velocity and the total depth for the several turbulence models adopted. The analytic solution was obtained assuming homogeneous turbulence, velocities \(V\) and \(W\) equal to zero and a logarithmic profile of velocity. As a result, the vertical eddy viscosity follows the well known parabolic profile that reads ([30]):  
Figure 4: Comparison of turbulence models used in Delft3D and ROMS for the open channel flow. (**a**) Velocity profile made non dimensional with bed friction velocity model outputs. (**b**) Velocity profile made non dimensional with theoretical bed friction velocity. (**c**) Vertical eddy viscosity \(v_{tv}\) and (**d**) turbulent kinetic energy \(k\). \[v_{t\text{\tiny D}}=\kappa u_{b}^{*}z\left(1-\frac{z}{H}\right). \tag{35}\]  
It is possible to see that there are high discrepancies in the central region, especially for the two-equation models, whereas the algebraic and the \(k-L\) model of Delft3D seem to show a better agreement with the expected one. Note that the best performance among the two equation models is achieved by the \(k-\omega\) formulation of ROMS.