CMS-Flow:Hydro Eqs: Difference between revisions

From CIRPwiki
Jump to navigation Jump to search
Line 1: Line 1:
<big>
<big>
== Governing Equation ==
== Governing Equation ==
The depth-averaged 2-D continuity and momentum equations are given by
The depth-averaged 2-D continuity equation may be written as
 
{{Equation|<math> \frac{\partial h}{\partial t}+\nabla \cdot (h\mathbf{U})=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>
where <math>h</math> is the total water depth <math>h=\zeta+\eta</math>, <math>\eta</math> is the water surface elevation, <math>\zeta</math> is the still water depth, and <math> \mathbf{U}=\left( {{U}_{1}},{{U}_{2}} \right) </math> is the depth-averaged current velocity, and <math> \nabla =\left( {{\nabla }_{1}},{{\nabla }_{2}} \right) </math> is the divergence operator.
 
The momentum equation can be written as
{{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}}   
{{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}}   
 
where <math>g</math> is the gravitational constant, <math> \mathbf{B}=\left( \begin{matrix} 0 & {{f}_{c}}  \\  -{{f}_{c}} & 0  \\ \end{matrix} \right) </math> where <math>f_{c}</math> 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 <math> i=1,2 </math> and <math> j=1,2 </math>
for <math> i=1,2 </math> and <math> j=1,2 </math>

Revision as of 21:26, 31 March 2011

Governing Equation

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

  (1)

where is the total water depth , is the water surface elevation, is the still water depth, and is the depth-averaged current velocity, and is the divergence operator.

The momentum equation can be written as

  (2)

where is the gravitational constant, where 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 and

Symbol Description Units
Time sec
Total water depth m
Still water depth m
Water surface elevation with respect to the still water elevation m
Current velocity in the jth direction m/sec
Sum of Precipitation and evaporation per unit area m/sec
Gravitational constant m/sec2
Water density kg/m3
Atmospheric pressure Pa
Turbulent eddy viscosity m2/sec

Documentation Portal