CMS-Flow:Subgrid Turbulence Model: Difference between revisions

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== Eddy Viscosity  ==
== Subgrid Turbulence Model ==


In CMS-Flow eddy viscosity is calculated as the sum of the kinematic viscosity <math>\nu_{t0}</math>, the current-related eddy viscosity  <math>\nu_c</math> and the wave-related eddy viscosity  <math>\nu_w</math>  
In CMS-Flow eddy viscosity is calculated as <math> \nu_t = (1-\theta_m)\nu_{tc} + \theta_m \nu_m </math>  where <math>\theta_m</math> is weighting factor equal to <math>\theta_m = (H_s/h)^3 </math>  in which <math>H_s</math>  is the significant wave height and <math>\nu_{tc}</math> and <math>\nu_{tw}</math>  are the current- and wave-related eddy viscosity components respectively. The wave contribution is included using the equation of Kraus and Larson (1991)  <math> \nu_tw = \Lambda u_w h </math>, where <math>\Lambda</math> is an empirical coefficient (default is 0.5), and  <math>u_w</math> is the wave bottom orbital velocity. The current-related eddy viscosity is calculated as a function of the flow gradients, and the bottom shear stress


       <math> \nu_{t} = \nu_{t0} + \nu_c  + \nu_w  </math>
       <math> \nu_{tc} = \nu_{t0} + \sqrt{ (c_0 u_* h)^2 + (l_h^2 |S| )^2 } </math>


There are two options to calculate <math>\nu_c</math>. The first is the Falconer (1980) equation given by
where <math>\nu_{t0}</math> is a base value approximately equal to the dynamic viscosity, and <math>c_0</math> is an empirical coefficient, <math>l_h</math> is the subgrid mixing length and <math>|S|</math> is equal to


      <math> |S| = \sqrt{2 \biggl( \frac{ \partial U}{\partial x} \biggr) ^2 +  2 \biggl( \frac{ \partial V}{\partial y} \biggr) ^2 + \biggl( \frac{ \partial U}{\partial y} + \frac{ \partial V}{\partial x}  \biggr) ^2 } </math>


      <math> \nu_{c} = 0.575C_b|U|h </math>
The mixing length is calculated here as  
 
 
where <math>C_b</math> is the bottom friction coefficient, <math>U</math> is the depth-averaged current velocity, and <math>h</math> is the total water depth.
 
The second option is a subgrid turbulence model given by
 
      <math> \nu_{c} = \sqrt{ (c_0 u_* h)^2 + (c_{sm}^2 \Delta x \Delta y |S| )^2 } </math>
 
where <math>c_0</math> is an empirical coefficient approximately equal to 1/6, <math>c_{sm}</math> is an empirical coefficient between 0.25-0.5, <math>\Delta x</math> and <math>\Delta y</math>  are the grid dimensions in the x and y directions, and <math>|S|</math> is equal to
 
      <math> |S| = \sqrt{2 \biggl( \frac{ \partial U}{\partial x} \biggr) ^2 +  2 \biggl( \frac{ \partial V}{\partial y} \biggr) ^2 + \biggl( \frac{ \partial U}{\partial y} + \frac{ \partial V}{\partial x} \biggr) ^2 } </math>


The wave component of the eddy viscosity is calculated as
      <math>l_h = \kappa min( \sqrt{\Delta x \Delta y}, c_{sm} h) </math>


      <math> \nu_w = k(D/\rho)^{1/3}H_s </math>
where <math>c_{sm}</math> is an empirical coefficient (Smagorinsky coefficient).


where <math>k</math> is an empirical coefficient, <math> \rho </math> is the water density, and <math>D</math> is the total wave dissipation.
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'''References'''


LARSON, M.; HANSON, H., and KRAUS, N. C., 2003. Numerical modeling of beach topography change. Advances in Coastal Modeling, V.C. Lakhan (eds.), Elsevier Oceanography Series, 67, Amsterdam, The Netherlands, 337-365.


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[[CMS-Flow]]
[[CMS-Flow]]

Revision as of 18:40, 6 November 2009

Subgrid Turbulence Model

In CMS-Flow eddy viscosity is calculated as νt=(1−θm)νtc+θmνm where θm is weighting factor equal to θm=(Hs/h)3 in which Hs is the significant wave height and νtc and νtw are the current- and wave-related eddy viscosity components respectively. The wave contribution is included using the equation of Kraus and Larson (1991) νtw=Λuwh, where Λ is an empirical coefficient (default is 0.5), and uw is the wave bottom orbital velocity. The current-related eddy viscosity is calculated as a function of the flow gradients, and the bottom shear stress 

     νtc=νt0+(c0u*h)2+(lh2|S|)2

where νt0 is a base value approximately equal to the dynamic viscosity, and c0 is an empirical coefficient, lh is the subgrid mixing length and |S| is equal to 

     |S|=2(∂U∂x)2+2(∂V∂y)2+(∂U∂y+∂V∂x)2

The mixing length is calculated here as 

     lh=κmin(ΔxΔy,csmh)

where csm is an empirical coefficient (Smagorinsky coefficient).

References

LARSON, M.; HANSON, H., and KRAUS, N. C., 2003. Numerical modeling of beach topography change. Advances in Coastal Modeling, V.C. Lakhan (eds.), Elsevier Oceanography Series, 67, Amsterdam, The Netherlands, 337-365.

CMS-Flow

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