CMS-Flow:Hydro Eqs: Difference between revisions

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Revision as of 21:21, 31 March 2011

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

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

(1)

for $j=1,2$

{\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)

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

@@ Line 3: / Line 3: @@
 The depth-averaged 2-D continuity and momentum equations are given by
-{{Equation|<math> \frac{\partial h  }{\partial t} + \frac{\partial (h U_j )}{\partial x_j} = S </math>|2=1}}
+{{Equation|<math> \frac{\partial h}{\partial t}+\nabla \cdot (h\mathbf{U})=S </math>|2=1}}
 for  <math>  j=1,2  </math>
-{{Equation| <math> \frac{\partial ( h U_i ) }{\partial t} + \frac{\partial (h U_i U_j )}{\partial x_j}
+{{Equation| <math> \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) </math>|2=2}}
-- \epsilon_{ij3} f_c U_j h = - g h \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}
- </math>|2=2}}
 for <math> i=1,2 </math> and <math> j=1,2 </math>