Temp

This is a page for setting up pages before loading them.

${\frac {\partial (hU_{i})}{\partial t}}+{\frac {\partial (hU_{i}U_{j})}{\partial x_{j}}}-\epsilon _{ij3}f_{c}U_{j}h=-gh{\frac {\partial \eta }{\partial x_{i}}}-{\frac {h}{\rho _{0}}}{\frac {\partial p_{a}}{\partial x_{i}}}+{\frac {\partial }{\partial x_{j}}}{\biggl (}\nu _{t}h{\frac {\partial U_{i}}{\partial x_{j}}}{\biggr )}+{\frac {\tau _{i}}{\rho }}$

(2)

The discretized momentum equations are Failed to parse (syntax error): {\displaystyle \frac{\partial }{\partial t} \int_{A} h U_i dA + \oint_{F} \biggl{ \frac{ \partial }{\partial x_j} \big[ (h U_i U_j ) - \nu_t h \frac{\partial U_i }{\partial x_j} \big) - \frac{h}{\rho_0} \frac{\partial p_a }{\partial x_i} - \epsilon_{ij3} f_c U_j h = - g h \frac{\partial \eta }{\partial x_i} + \frac{\tau_i }{\rho} }

$U_{i,P}^{n+1}={\frac {1}{a_{i,P}}}{\biggl (}\sum _{k=1}a_{i,k}U_{i,k}^{n+1}+S_{i}{\biggr )}-{\frac {h_{P}}{a_{i,P}}}\sum _{k=1}n_{ik}\Delta s_{k}p_{k}^{n+1}$

The continuity equation is discretized as

$h^{n+1}-\mathbf {S}$

where the subscript $k$ indicates the cell face, $p=g\eta$ with $\eta$ being the water surface elevation, $n_{ik}$ is equal to the dot product of the velocity unit vector and the cell face unit vector.

The coefficient $a_{i,P}$ is equal to $a_{i,P}=\sum a_{i,k}+a_{P}^{0}$

The continuity equation is discretized as $p_{P}^{n+1}=p_{P}^{n}-g{\frac {\Delta t}{\Delta A_{P}}}\sum _{k=1}n_{k}F_{k}^{n+1}$

where $n_{k}$ is the dot product of the cell face unit vector and

The depth-averaged 2-D continuity and momentum equations are given by

${\frac {\partial h}{\partial t}}+{\frac {\partial (hU_{j})}{\partial x_{j}}}=S$

(1)

for $j=1,2$