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 where is weighting factor equal to in which is the significant wave height and and are the current- and wave-related eddy viscosity components respectively. The wave contribution is included using the equation of Kraus and Larson (1991) , where is an empirical coefficient (default is 0.5), and 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

where is a base value approximately equal to the dynamic viscosity, and is an empirical coefficient, is the subgrid mixing length and is equal to

The mixing length is calculated here as

where 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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