Diffusion Wave Approximation to the Shallow Water Equations

In the previous section, Manning’s formula was used to estimate the bottom friction. If further constraints are assumed on the physics of the flow, a relation between barotropic pressure gradient and bottom friction is obtained from the diffusion wave form of the momentum equation. This relation is extremely useful due to its simplicity. However it must be noted that this relation can be applied only in a narrower scope than the more general momentum equation studied before. Under the conditions described in this section, the Diffusion Wave equation can be used in place of the momentum equation. It will be seen in subsequent sections that the corresponding model becomes a one equation model known as the Diffusion Wave Approximation of the Shallow Water equations (DSW).

Up to this point, we have described the hydraulics for momentum. From now on the discussion will gear towards the formulation and numerical methods of the solution. It will be convenient to denote the hydraulic radius and the face cross section areas as a function of the water surface elevation H, so R= R(H), A=A(H).

In shallow frictional and gravity controlled flow; unsteady, advection, turbulence and Coriolis terms of the momentum equation can be disregarded to arrive at a simplified version. Flow movement is driven by a barotropic pressure gradient balanced by bottom friction. Simplifying the momentum equation results in:

1)	$\begin{array}{l}\displaystyle \displaystyle \frac{g n^2}{R^{4/3}} \left\| \textbf{V} \right\| \textbf{V} = - g \nabla z_s - \frac{1}{\rho} \nabla p_a + \frac{\boldsymbol{\tau}_s}{\rho h}\end{array}$

where

$\begin{array}{l}\textbf{V}\end{array}$ : Velocity vector [L/T]
$\begin{array}{l}R\end{array}$ : Hydraulic radius [L]
$\begin{array}{l}z_s\end{array}$ : Water surface elevation elevation [L]
$\begin{array}{l}n\end{array}$ : Manning’s roughness coefficient [T/L^1/3]
$\begin{array}{l}\rho\end{array}$ : Water density [M/L³]
$\begin{array}{l}p_a\end{array}$ : Atmospheric pressure [M/L/T²]
$\begin{array}{l}\boldsymbol{\tau}_s\end{array}$ : Wind shear stress [M/L/T²]

Dividing both sides of the equation by the square root of their norm, the equation can be rearranged into the more classical form

2)	$\begin{array}{l}\displaystyle \displaystyle \textbf{V} = - \frac{R^{2/3}}{n} \frac{ \nabla z_s + \frac{1}{\rho g} \nabla p_a - \frac{\boldsymbol{\tau}_s}{\rho g h} }{\left\| \nabla z_s +\frac{1}{\rho g} \nabla p_a - \frac{\boldsymbol{\tau}_s}{\rho g h} \right\|^{1/2}}\end{array}$

When the velocity is determined by a balance between barotropic pressure gradient and bottom friction, the Diffusion Wave form of the Momentum, can be used in place of the full momentum equation, and the corresponding system of equations can in fact be simplified to a one equation model. Direct substitution of the Diffusion Wave approximation of the momentum equation in the mass conservation equation, yields the classical Diffusion-Wave Equation:

3)	$\begin{array}{l}\displaystyle \displaystyle \frac{\partial h}{\partial t} = \nabla \cdot (\beta \nabla z_s) + S + q\end{array}$

where:

$\begin{array}{l}\displaystyle \beta = \frac{h R^{2/3}}{n} \left| \nabla z_s + \frac{1}{\rho g} \nabla p_a - \frac{\boldsymbol{\tau}_s}{\rho g h} \right| ^{-1/2}\end{array}$
$\begin{array}{l}\displaystyle S = \nabla \cdot \left[ \beta \left( \frac{1}{\rho g} \nabla p_a - \frac{\boldsymbol{\tau}_s}{\rho g h} \right) \right]\end{array}$

Boundary Conditions

At any given time step, boundary conditions must be given at all the edges of the domain. Within HEC-RAS boundary conditions can be one of three different kinds:

Water surface elevation: The value of the water surface elevation $\begin{array}{l}z_s^{n+1} = z_{s,b}\end{array}$ is given at one of the boundary edges.
Normal Depth: The friction slope, $\begin{array}{l}S_{f,b}\end{array}$ , is specified and used to impose a flow boundary condition computed as $\begin{array}{l}Q= K S_{f,b}^{1/2}\end{array}$ .
Flow: The flow $\begin{array}{l}Q_b\end{array}$ that crosses the boundary is provided. In the continuity, this condition is implemented by direct substitution into the flow formula of the corresponding boundary faces.