CMS-Flow:Hydro Eqs

Governing Equation

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

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

(1)

where $h$ is the total water depth $h=\zeta +\eta$, $\eta$ is the water surface elevation, $\zeta$ is the still water depth, and $\mathbf{𝐔}=(U_1,U_2)$ is the depth-averaged current velocity, and $\mathrm{\nabla }=({\mathrm{\nabla }}_1,{\mathrm{\nabla }}_2)$ is the divergence operator.

The momentum equation can be written as

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

(2)

where $g$ is the gravitational constant, $\mathbf{𝐁}=\left(\begin{array}{cc}0& f_c\\ -f_c& 0\\ \end{array}\right)$ where $f_c$ is the Coriolis parameter, ${\nu }_t$ is the eddy viscosity, ${\tau }_{wi}$ is the wind stress, ${\tau }_{Si}$ is the wave stresses, and ${\tau }_{bi}$ is the combined wave-current mean bed shear stress.

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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