Face Hydraulic Properties

The hydrodynamic model requires three hydraulic property tables at faces:

Vertical wetted face area
Hydraulic radius
Manning's roughness coefficient

The vertical face area A_k is simply the integration of the cumulative horizontal wetted width W_k as:

1)	$\begin{array}{l}A_{k}=\begin{cases}{l} 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,\,k=1\\ A_{k-1}+(\eta _{k}-\eta _{k-1})W_{k-1}\,\,\,\mathrm{for}\,\,k>1 \end{cases}\right.\end{array}$

In which η_k is the face water surface elevation, and W_k is the face width. The curve A_k is piece-wise linear and W_k is piece-wise constant. Similarly the face widths may be obtained from the cell areas as

2)	$\begin{array}{l}W_{k}=\begin{cases} \frac{A_{k+1}-A_{k}}{\eta _{k+1}-\eta _{k}}\,\,\,\,\mathrm{for}\,k< M\\ W_{f}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,k=M \end{cases}\right.\end{array}$

where W_f is the maximum face width which is the same as the face length and M is the number of faces. It is noted that the W₁ is the minimum face width even when the face is completely dry.

The face conveyance is updated using the Single Channel Method and the Manning’s roughness coefficient as

3)	$\begin{array}{l}\displaystyle K=\frac{AR^{2/3}}{n}=\frac{A^{5/3}}{nP^{2/3}}\end{array}$

where

$\begin{array}{l}n\end{array}$ : Manning’s roughness coefficient [T/L^1/3]

$\begin{array}{l}A\end{array}$ : wetted area [L²]

$\begin{array}{l}R\end{array}$ : hydraulic radius [L]

$\begin{array}{l}P\end{array}$ : wetted perimeter [L]

The subface shape factor is utilized to compute the wetted perimeter and during the simulation as the face bed elevations are updated. Initially, the wetted perimeter P_i is computed from a high-resolution terrain model. This allows the model to take into account all of the details of the terrain. As the subface bed elevations are updated, new estimates of the wetted perimeter are necessary. This is done by assuming that the wetted perimeter is of the form:

4)	$\begin{array}{l}P_{k}=\begin{cases}{l} 0\,\,\,\,\,\mathrm{for}\,\,k=1\\ P_{k-1}+K_{k}\sqrt{W_{k-1}^{2}+\left(\eta _{k}-\eta _{k-1}\right)^{2}}\,\,\,\,\mathrm{for}\,k>1 \end{cases}\right.\end{array}$

where

$\begin{array}{l}P_{k}\end{array}$ : cumulative wetted perimeter corresponding to η_k [L]

$\begin{array}{l}K_{k}\end{array}$ : shape factor for subsegment k [-]

$\begin{array}{l}W_{k}\end{array}$ : effective top width corresponding to η_k [L]

$\begin{array}{l}{η_{k }}\end{array}$ : elevation of subsegment k [L]

The shape factor K_k is computed with the above equation and the initial given wetted perimeter and is assumed to remain constant throughout the simulation.

Once an estimate of the wetted perimeter is obtained, the hydraulic radius is simply

5)	$\begin{array}{l}\displaystyle R_{h,k}=\frac{A_{k}}{P_{k}}\end{array}$

where

$\begin{array}{l}R_{h,k}\end{array}$ : face hydraulic radius corresponding to η_k [L]

$\begin{array}{l}A_{k}\end{array}$ : vertical face area corresponding to η_k [L²]

$\begin{array}{l}P_{k}\end{array}$ : face wetted perimeter corresponding to η_k [L]