# Modeling Open Channel Flows of a Viscous Fluid: Critical Transition and Apparent Bottom
## 1 Introduction  
The Shallow Water model (SWM) is largely used to describe the dynamics of stratified flows such as air and water in open channel flows, lakes and oceans, or atmospheric flows, where the air is decomposed into several layers [1, 2]. The SWM is derived from the Navier-Stokes model (NSM) in the limit of small fluid layer depth compared to the horizontal scale of the flow. Thus, the approximation is valid only when the boundary layer is slowly variable. The SWM fails to describe the flow over abrupt obstacles, which we will call \"macro-rugosities\" in contrast to wall rugosities that can be modeled by \"wall-laws\". Examples of turbulent flow over macro-rugosities include atmospheric boundary layer over an urban canopy or mountain, which often leads to Lee waves generation [3, 4, 5, 6]. The flows of lava [7], ice-sheets [8] or debris avalanches [9], instead, are examples of environmental flows in the laminar regime. The simplest model describing these kind of flows is the one derived in [10]. The problem of a two-layer flow over an abrupt topography has been studied in [11, 12, 13, 14, 15, 16], where the \"topography\" is normally represented by a semi-circular cylinder. The free surface inviscid flow over this geometry has been subjected to a large number of theoretical studies, such as [17, 18]. The fluid and semi-cylindrical obstacle interaction takes a variety of forms, depending on the normal (upstream) Froude number  
\[F_{n}=\frac{U_{n}}{\sqrt{gh_{n}\cos\theta}} \tag{1}\]  
and the blocking factor  
\[\alpha=\frac{H}{h_{n}} \tag{2}\]where \(H\) is the maximum obstacle height, \(h_{n}\) is the initial undisturbed layer depth, \(U_{n}\) is the velocity of the initial undisturbed fluid layer and \(\theta\) is the bottom slope. Assuming non-viscous flow and horizontal bottom, a theoretical curve for the transcritical transition has been derived in [4; 19] and is given by  
\[\alpha=1-\frac{3}{2}F_{n}^{2/3}+\frac{1}{2}F_{n}^{2}. \tag{3}\]  
This solution is shown in Figure 1 (solid lines). Under the lower branch, the flow is sub-critical in the entire domain while it remains super-critical above the upper branch. In the interval delimited by the two branches, different configurations of trans-critical flow can be observed. The phase diagram shown in Figure 1 had been first derived numerically in [4] and successively verified through experiments in other works, e.g., [20], for high Reynolds Number flows. Many different regimes have been found experimentally for two-layer flows [20; 21; 22; 23; 24; 25], three-layer flows [19], or flows with a continuous density gradient [26]. The two-layer turbulent flow over a semi-cylindrical obstacle has been also studied numerically using the Volume of Fluid Method (VOF) and the standard \(k-\epsilon\) turbulent model in [27; 28; 29], the CLEAR-VOF method and a large eddy simulation (LES) in [30], and an interface-tracking algorithm for computation of the free-surface with a moving grid in [31]. Most of the works on stratified flows consist of investigating the evolution of a free-surface when the bottom is known a priori. However, in many problems that are currently of interest, the position and the fluid velocity on the free-surface are given, but the shape and location of the solid bottom are unknown a priori. These free-surface problems are called indirect problems. The indirect problem can be solved using minimization techniques  
Figure 1: Diagram of the 16 numerical experiments for \(\alpha\in\{0.2,0.4,0.6,0.8\}\) and \(F_{n}\in\{0.25,0.75,1.25,1.75\}\). Theoretical critical curve \(\alpha=1-(3/2)\,F_{n}^{2/3}+(1/2)\,F_{n}^{2}\) in plain lines. Circles (\(\bigcirc\)): sub-critical regimes. Diamonds (\(\diamond\)): super-critical regimes. Squares (\(\square\)): trans-critical regimes. Up-triangles (\(\triangle\)). Down-triangles (\(\bigtriangledown\)). which are able to predict all the different shapes and positions of the bottom and solve iteratively the system with the boundary integral method.