# Modeling Open Channel Flows of a Viscous Fluid: Critical Transition and Apparent Bottom
## 6 Conclusions  
We have compared the steady solutions of a two phase flow on a titled bottom with a semi-circular obstacle for both the 2D Navier-Stokes (NS) equations and the Shallow Water (SW) equations. For constant values of the geometrical Reynolds number \(R_{g}=26\) and the lower fluid viscosity \(\
u\), different values of the angle \(\theta\) and the flux discharge \(q\) have to be chosen in order to explore 16 values of the couple \((F_{n},\alpha)\) while keeping the normal Reynolds number \(R_{n}\) small enough to ensure that the flow remains laminar. The numerical simulations have shown that the sharpness of the obstacle is responsible for the existence of hydraulic jumps and trans-critical regimes for the SW model that were not observed for the NS model. This rapid variation of the bottom is also responsible for departure of the velocity profile from the parabolic shape, that exists far upstream and downstream the obstacle, which has been quantified due to the departure of \(\beta(x)\) and \(\lambda(x)\), respectively, the velocity variance and dissipation parameters, from their asymptotic values. The physical reason behind the differences between the SW and NS results is the static pressure modeling. In the SW model, the pressure is considered o be hydrostatic, as in Equation (15). This approximation is reasonable, however around a rough topography the static pressure departs from hydrostatic values. Because in first approximation, the total pressure, i.e., static plus dynamic, is constant, this significantly affects the fluid kinetic energy around the obstacle. This is reflected in both the changes in \(\beta(x)\) and \(\lambda(x)\). In particular, the dissipation increases sharply upstream the obstacle, while the kinetic energy increases downstream it. The increase dissipation around the obstacle and the increased velocity gradient, i.e., velocity variance, both prevent the occurrence of hydraulic jumps in the NS model. We have shown that the SW model could give better prediction of stratified flows over macro-rugosities by replacing the topography with a smoother, apparent bottom. The determination of the apparent bottom is based on an exact match between the free surface height of both the NS and SW equations, which constitutes the main advantage of this approach, since NS is a more reliable model than SW, as well as its novelty. However, in an attempt to reproduce the Stokes waves, for a certain range of Froude numbers and blocking factors, this procedure adds further bumps on the bottom. This makes the shape of the apparent bottom more complex than the original topography, which is an undesirable feature. The main conclusions of this work are quantified as follows:  
* The SW and the NS model give different results in the presence of a sharp topography;
* The two models differs most for supercritical conditions, i.e., \(F>1\);
* The local velocity profile variance \(\beta(x)\) and the local dissipation coefficient \(\lambda(x)\) have been deemed responsible for the differences between the two models;
* The apparent bottom obtained from Equation (28) is able to reproduce free-surface profile;
* However, this procedure is highly sensitive to any ripples at the interface and the apparent bottom shape is very complex, especially in subcritical (\(F<1\)) flows. Further studies are required to remove the additional rugosities by imposing artificial constraints on the regularity of the apparent bottom. This would lead to a systematic determination of an apparent bottom for different types of \"macro-rugosities\". Furthermore, this study is limited to a two-dimensional, highly viscous flow. Future investigation should deal with more realistic, three-dimensional turbulent flows. [PERSON] was responsible for developing the FORTRAN SW code and running the NS Simulations with OpenFOAM. [PERSON] and [PERSON] supervised the work and advised on the definition of apparent bottom. All the authors contributed equally in drafting the paper. All authors have read and agreed to the published version of the manuscript.