CMS-Flow:Hydro Eqs

Governing Equations

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 $h=\zeta +\eta$, $\eta$ is the water surface elevation, $\zeta$ is the still water depth, $\mathbf {U} =\left({{U}_{1}},{{U}_{2}}\right)$ is the depth-averaged current velocity, $S$ is a source term due to precipitation and evaporation, 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, 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 \mathbf{B}=\left( \begin{matrix} 0 & {{f}_{c}} \\ -{{f}_{c}} & 0 \\ \end{matrix} \right) } where 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 f_{c}} is the Coriolis parameter, 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 \nu_t} is the eddy viscosity, 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 \tau_{wi} } is the wind stress, 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 \tau_{Si} } is the wave stresses, and 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 \tau_{bi} } is the combined wave-current mean bed shear stress.

Symbol	Description	Units
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 t }	Time	sec
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 }	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 }	m
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 \zeta }	Still water depth	m
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 \eta }	Water surface elevation with respect to the still water elevation	m
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 U_j }	Current velocity in the jth direction	m/sec
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 S }	Sum of Precipitation and evaporation per unit area	m/sec
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 g }	Gravitational constant	m/sec2
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 \rho }	Water density	kg/m3
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 p_a }	Atmospheric pressure	Pa
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 \nu_t }	Turbulent eddy viscosity	m2/sec

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