CMS-Flow:Hydro Eqs

Governing Equation

The depth-averaged 2-D continuity equation may be written as

${\frac {\partial h}{\partial t}}+\nabla \cdot (h\mathbf {U} )=S$

(1)

where $h$ is the total water depth Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h=\zeta+\eta} , $\eta$ is the water surface elevation, $\zeta$ is the still water depth, and $\mathbf {U} =\left({{U}_{1}},{{U}_{2}}\right)$ is the depth-averaged current velocity, and $\nabla =\left({{\nabla }_{1}},{{\nabla }_{2}}\right)$ is the divergence operator.

The momentum equation can be written as

${\frac {\partial (h{{U}_{i}})}{\partial t}}+\nabla \cdot (h\mathbf {U} {{U}_{i}})-\mathbf {BU} =-gh{{\nabla }_{i}}\eta +\nabla \cdot \left({{\nu }_{t}}h\nabla {{U}_{i}}\right)+{\frac {1}{\rho }}\left({{\tau }_{wi}}+{{\tau }_{Si}}-{{\tau }_{bi}}\right)$

(2)

where $g$ is the gravitational constant, $\mathbf {B} =\left({\begin{matrix}0&{{f}_{c}}\\-{{f}_{c}}&0\\\end{matrix}}\right)$ where $f_{c}$ is the Coriolis parameter, is the eddy viscosity, is the wind stress, is the wave stresses, and is the combined wave-current mean bed shear stress.

for $i=1,2$ and $j=1,2$

Symbol	Description	Units
$t$	Time	sec
$h$	Total water depth $h=\zeta +\eta$	m
$\zeta$	Still water depth	m
$\eta$	Water surface elevation with respect to the still water elevation	m
$U_{j}$	Current velocity in the jth direction	m/sec
$S$	Sum of Precipitation and evaporation per unit area	m/sec
$g$	Gravitational constant	m/sec2
$\rho$	Water density	kg/m3
$p_{a}$	Atmospheric pressure	Pa
$\nu _{t}$	Turbulent eddy viscosity	m2/sec

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