CMS-Flow:Subgrid Turbulence Model: Difference between revisions

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== Subgrid Turbulence Model ==
== Subgrid Turbulence Model ==


In CMS-Flow eddy viscosity is calculated as


      <math> \nu_t = \nu_0 + \nu_c + \nu_w </math>
In CMS-Flow eddy viscosity is calculated as the sum of the kinematic viscosity <math>\nu_{0}</math>, the current-related eddy viscosity  <math>\nu_c</math> and the wave-related eddy viscosity  <math>\nu_w</math>


where <math>\nu_c</math> is the kinematic viscosity, and <math>\nu_c</math> and <math>\nu_w</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_{t} = \nu_0 + \nu_c + \nu_w </math>


      <math> \nu_w = \Lambda u_w h </math>
There are two options to calculate  <math>\nu_c</math>. The first is the Falconer (1980) equation given by


where  <math>\Lambda</math> is an empirical coefficient (default is 0.5), and  <math>u_w</math> is the wave bottom orbital velocity and <math>h</math> is the water depth. The current-related eddy viscosity is calculated as a function of the flow gradients, and the bottom shear stress
      <math> \nu_c = 0.575C_f|U|h </math>


      <math> \nu_{c} = \nu_{0} + \sqrt{ (c_0 u_* h)^2 + (c_{sm}\Delta |S| )^2 } </math>
where <math>C_f</math> is the bottom friction coefficient, <math>U</math> is the depth-averaged current velocity, and <math>h</math> is the total water depth.


where  <math>\nu_{0}</math> is a base value approximately equal to the kinematic viscosity, <math>c_0</math> is an empirical coefficient, <math>c_{sm}</math> is an empirical coefficient (Smagorinsky coefficient), <math> \Delta </math> is the local cell area, and <math>|S|</math> is equal to  
The second option is a subgrid turbulence model given by
 
      <math> \nu_{c} = c_b u_{*} h + c_h \Delta A |S|  </math>
 
where <math>c_b</math> is an empirical coefficient approximately equal to 1/6 (default), <math>c_h</math> is an empirical coefficient between 0.1-0.5 (default is 0.4), <math>\Delta A = \Delta x \Delta y</math> is the local grid cell area, and <math>|S|</math> is equal to


       <math> |S| = \sqrt{ \biggl( \frac{ \partial U}{\partial x} \biggr) ^2 +  \biggl( \frac{ \partial V}{\partial y} \biggr) ^2 + \frac{1}{2} \biggl( \frac{ \partial U}{\partial y} + \frac{ \partial V}{\partial x}  \biggr) ^2 } </math>
       <math> |S| = \sqrt{ \biggl( \frac{ \partial U}{\partial x} \biggr) ^2 +  \biggl( \frac{ \partial V}{\partial y} \biggr) ^2 + \frac{1}{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> \nu_w = \Lambda u_w H_s  </math>
where <math>\Lambda</math> is an empirical coefficient approximately equal to 0.5, <math> H_s </math> is the significant wave height and <math>u_w</math> is bottom orbital velocity based on the significant wave height. Outside of the surf zone the bottom orbital velocity is calculated as
      <math> u_w = \frac{ \pi H_s}{T_p \sinh(kh) } </math>
where <math>H_s</math> is the significant wave height, <math>T_p</math> is the peak wave period, <math>k=2\pi/L</math> is the wave number. Inside the surf zone, the turbulence due to wave breaking is considered by increasing the bottom orbital velocity as
      <math> u_w = \frac{ H_s}{2h}\sqrt{gh} </math>
The default turbulence model is the subgrid model, but may be changed with the advanced card
      TURBULENCE_MODEL                  SUBGRID  !FALCONER | PARABOLIC | SUBGRID | SUBGRID-WU
The turbulence model parameters may be changed in the advanced cards as
      EDDY_VISCOSITY_CONSTANT          1.0e-6    ![m^2/sec], kinematic viscosity, ~1.0e-6
      EDDY_VISCOSITY_BOTTOM            0.015    ![-], bottom shear coefficient, ~0.1667
      EDDY_VISCOSITY_HORIZONTAL        0.2      ![-], smagorinsky coefficient, ~0.1-0.5
      EDDY_VISCOSITY_WAVE              0.5      ![-], wave coefficient, ~0.25-0.5


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Revision as of 18:35, 5 May 2010

Subgrid Turbulence Model

In CMS-Flow eddy viscosity is calculated as the sum of the kinematic viscosity , the current-related eddy viscosity and the wave-related eddy viscosity

There are two options to calculate . The first is the Falconer (1980) equation given by

where is the bottom friction coefficient, is the depth-averaged current velocity, and is the total water depth.

The second option is a subgrid turbulence model given by

where is an empirical coefficient approximately equal to 1/6 (default), is an empirical coefficient between 0.1-0.5 (default is 0.4), is the local grid cell area, and is equal to

The wave component of the eddy viscosity is calculated as

where is an empirical coefficient approximately equal to 0.5, is the significant wave height and is bottom orbital velocity based on the significant wave height. Outside of the surf zone the bottom orbital velocity is calculated as

where is the significant wave height, is the peak wave period, is the wave number. Inside the surf zone, the turbulence due to wave breaking is considered by increasing the bottom orbital velocity as

The default turbulence model is the subgrid model, but may be changed with the advanced card 

     TURBULENCE_MODEL                  SUBGRID   !FALCONER | PARABOLIC | SUBGRID | SUBGRID-WU

The turbulence model parameters may be changed in the advanced cards as 

     EDDY_VISCOSITY_CONSTANT           1.0e-6    ![m^2/sec], kinematic viscosity, ~1.0e-6
     EDDY_VISCOSITY_BOTTOM             0.015     ![-], bottom shear coefficient, ~0.1667
     EDDY_VISCOSITY_HORIZONTAL         0.2       ![-], smagorinsky coefficient, ~0.1-0.5
     EDDY_VISCOSITY_WAVE               0.5       ![-], wave coefficient, ~0.25-0.5

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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