# Modelling Regime Changes of Dunes to Upper-Stage Plane Bed in Flumes and in Rivers
## 2 Dune Model
### Flow Model  
In general the flow is forced by the difference in water level along the domain. While the water depth at the start and end of domain are the same due to the periodic boundary conditions, the water level differs because the domain is sloped. The average bed level is taken as zero and has a slope (this average bed slope is an input parameter for the model). By solving the flow equations with a certain average water depth, a discharge is found. The average water depth is adjusted until this discharge matches the discharge given as input.  
The flow in the model of [PERSON] et al. (2009) [21] is described by the steady two-dimensional shallow water equations in a vertical plane (2-DV), assuming hydrostatic pressure conditions. For small Froude numbers and small vertical flow acceleration, the momentum equation in the vertical direction reduces to the hydrostatic pressure condition, and the time variations in the horizontal momentum equation can be dropped. The governing model equations that result are shown in Equations (1) and (2).  
\[u\frac{\partial u}{\partial x}+w\frac{\partial u}{\partial z}=-g\frac{ \partial\zeta}{\partial x}+A_{v}\frac{\partial^{2}u}{\partial z^{2}}+g\dot{i} \tag{1}\]  
\[\frac{\partial u}{\partial x}+\frac{\partial w}{\partial z}=0 \tag{2}\]  
The velocities in the x and z directions are \(u\) and \(w\), respectively. The water surface elevation is denoted by \(\zeta\), \(i\) is the average channel slope, \(g\) is the acceleration due to gravity, and \(A_{v}\) denotes the constant vertical eddy viscosity. Note that a steady flow model is used to compute unsteady flow of a flood wave. This is a reasonable approach, because the length scale of a dune is small, so the remaining terms are large compared to the time derivatives. The computational domain is shown in Figure 2.  
In Figure 2 the symbol \(\lambda\) denotes the dune length (in m), \(h\) is the domain-averaged water depth (in m), and \(z_{b}\) is the bed level relative to the x-axis (in m). The flow is forced in the domain because the x-axis is actually at a slope \(i\) with regard to the real horizontal plane, creating a water level difference along the domain.  
Figure 2: The computational domain.  
The boundary conditions are defined at the water surface (\(z=h+\zeta\)) and at the bed (\(z=z_{b}\)). The boundary conditions at the water surface, Equation (3) represents no flow through the surface, and Equation (4) means no shear stress at the surface (so wind stress is assumed to be negligible). The kinematic boundary condition at the bed, Equation (5) yields that there is no flow through the bed.  
\[u\left.\frac{\partial\zeta}{\partial x}\right|_{z=h+\zeta}=w \tag{3}\]  
\[\left.\frac{\partial u}{\partial z}\right|_{z=h+\zeta}=0 \tag{4}\]  
\[u\left.\frac{\partial z_{b}}{\partial x}\right|_{z=z_{b}}=w \tag{5}\]  
In order to close the turbulence model, a time- and depth-independent eddy viscosity is assumed. In order to represent the bed shear stress correctly for a constant eddy viscosity, a partial slip condition at the bed, Equation (6) is necessary.  
\[\tau_{b}=A_{v}\left.\frac{\partial u}{\partial z}\right|_{z=z_{b}}=Su_{b} \tag{6}\]  
In Equation (6), \(\tau_{b}\) (m\({}^{2}\)/s\({}^{2}\)) represents the volumetric bed shear stress (i.e., without the density), \(u_{b}\) (m/s) is the flow velocity along the bed, and the resistance parameter \(S\) (m/s) controls the resistance at the bed. For more details about the model equations and numerical solution procedure, reference is made to [PERSON] et al. (2009) [21] and [PERSON] et al. (2012) [35]. For more details, the reader is referred to [PERSON] et al. (2016) [26].