# Modeling Open Channel Flows of a Viscous Fluid: Critical Transition and Apparent Bottom
## 5 Enhanced Parametrization for the SW Model  
### Analysis of the Parameters \(\lambda\) and \(\beta\)  
The results presented above show how the prediction of the simple SW model can differ from those of the complete NS one. The cause of this discrepancy must be sought in the approximation made to simplify the model. These approximations concern the terms \(\beta(t,x)\) and \(\lambda(t,x)\), considered equal to 1 in the SW model. Now, these parameters can actually be explicitly predicted from the full NSM simulations. In order to understand why the results of the two models differ, thus we report in Figures 7 and 8, respectively, the \(\beta\) and \(\lambda\) parameters as functions of the streamwise coordinate, for the different values of \(F_{n}\) and \(\alpha\). Each figure has four subplots in which four curves obtained at different blocking factors and same equilibrium Froude numbers are compared.  
The parameter \(\beta\), is the variance of the velocity profile plus 1, as defined in Equation (18). It is therefore equal to 1 when the variance is null, i.e., for a flat profile. A fluid in turbulent motion presents a logarithmic mean velocity profile whose variance is of order \(5\%\), so that approximating \(\beta\) equal to one is acceptable. However, when a viscous fluid is studied, this approximation is not suitable anymore. The velocity profile of a Newtonian fluid with a free boundary is a half parabola, whose variance is \(20\%\), leading to \(\beta=1.2\). This value is valid for the unperturbed flow but, in the proximity of the weir, the profile and consequently \(\beta\) can change.  
In Figure 7, the pattern of the factor \(\beta\) is shown. Far from the obstacle the fluid reaches the equilibrium state, and so \(\beta\) reaches the value 1.2 for the considered laminar flow. The distance at which the equilibrium is reached depends on both \(\alpha\) and \(F_{n}\). The smaller the \(\alpha\), the less \(\beta\) changes. This is obvious because having a smaller blocking factor is equivalent to having a smaller obstacle. On the other hand, the bigger the \(\alpha\), the more the velocity profile, and so \(\beta\), changes, with two pics prior and after the obstacle, the maximum being always located downstream the obstacle. This increase in \(\beta\) can be associated with the two circulation cells downstream and upstream the obstacle. Moreover, a minimum value of \(\beta\) is obtained at the top of the obstacle, with a value \(\beta\simeq 1\). At this position, the velocity profile is therefore close to a bulk flow, i.e., a flat velocity profile.  
The other parameter \(\lambda\), defined in Equation (19), is the ratio between the wall friction of the flow and the wall friction of the equilibrium state, which is equal to one if the bottom is smooth. \(\lambda\) is equal to 1 if the laminar flow reaches its equilibrium profile. Nevertheless,if the fluid flows over a rough bottom, the resulting motion can be complex, leading to vortices and secondary flows which reduce the similarity between the NS and SW model, that is only valid under the assumption of slowly variable bottom.  
In Figure 8, the pattern of \(\lambda\) is shown. As for \(\beta\), far upstream and downstream the obstacle, the equilibrium value, \(\lambda=1\) is reached. The non-dimensional dissipation parameter \(\lambda\) has, for all the considered cases, the following shape: it approaches 1 for \(x\rightarrow\pm\infty\), has two minima at the beginning and end points of the obstacle and a maximum slightly upstream the top of the obstacle.