Some of the erosion and deposition methods depend on the subgrid hydrodynamics. Depending on the transport potential formula applied, there may be one or more subgrid hydrodynamic variables which need to be estimated. These variables include subgrid current velocities, water depths, various types of shear stresses, and shear velocities:
| 1) |
h_{i}=\frac{\Omega _{i}}{a_{i}} |
where
h_{i} = subarea hydraulic depth [L]
\Omega _{i}=\sum _{j\cap i}h_{j}a_{j} = subarea water volume [L3]
The subcell current velocities are computed following Volp (2017) assuming a uniform over the cell friction slope is uniform within the cell. With this assumption and with the approximation of the hydraulic depth as the hydraulic radius for the friction coefficient, the following equation can be obtained
| 2) |
U_{i}=\frac{\Omega }{\sum _{j\in h>0}\frac{h_{j}^{5/3}}{n_{j}}a_{j}^{W}}\frac{h_{i}^{2/3}}{n_{i}}U |
The cell-averaged current velocity is defined as the volume average of the cell as
| 3) |
U=\frac{1}{\Omega }\sum _{i}h_{i}a_{i}^{W}U_{i} |
where
\Omega = cell volume [L3]
h_{i} = subcell water depths [L]
{a_{i}}^{W} = subcell wetted areas [L]
The constant friction slope assumption also leads to the following subarea shear stress formulation
| 4) |
S_{f}=\frac{\tau _{bi}}{\rho gh_{i}}=\frac{\tau _{b}}{\rho gh}\Rightarrow \tau _{bi}=\frac{h_{i}}{h}\tau _{b} |
The hydraulic radius is computed as
| 5) |
R_{h,i}=\left\{\begin{array} h_{i}\frac{a_{i}}{s_{i}+(h_{i+1}-h_{\mathrm{\ell }})\sqrt{a_{i}}}\,\,\,\mathrm{for}\,i=1,2,n-1\\ h_{i}\frac{a_{i}}{s_{i}}\,\,\,\mathrm{for}\,i=n \end{array}\right. |
where
h_{i} = subregion water depth [L]
η = water surface elevation [L]
z_{bi} = subregion bed elevation [L]
A^{W} = ϕ^{W}A = cell wetted area [L2]
{a_{i}}^{W} = {ϕ_{i}}^{W}a_i = subregion wetted area [L2]
R_{h, i} = subarea hydraulic radius [L]
n_{i} = subarea Manning’s roughness coefficient [L]
All other hydrodynamic variables are computed from the above variables.