CMS-Flow:Subgrid Turbulence Model: Difference between revisions

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== Subgrid Turbulence Model ==
__NOTOC__
In CMS-Flow eddy viscosity is calculated as the sum of a base value  <math>\nu_{0}</math>, the current-related eddy viscosity  <math>\nu_c</math> and the wave-related eddy viscosity  <math>\nu_w</math>
      {{Equation|<math> \nu_t  = \nu_0 + \nu_c + \nu_w </math> |2=1}}


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
The base value for the eddy viscosity is approximately equal to the kinematic eddy viscosity can be changed using the advanced cards (Click [[http://cirp.usace.army.mil/wiki/CMS-Flow_Eddy_Viscosity here]] for further details).
==Current-Related Eddy Viscosity Component==
There are four options for the current-related eddy viscosity: FALCONER, PARABOLIC, SUBGRID, and MIXING-LENGTH. The default turbulence model is the subgrid model, but may be changed with the advanced card  TURBULENCE_MODEL.


      <math> \nu_{tc} = \nu_{t0} + \sqrt{ (c_0 u_* h)^2 + (\sqrt{c_{sm} \Delta x \Delta y} |S| )^2 } </math>
=== Falconer Equation ===
The Falconer (1980) equation is  the method is the default method used in the previous version of CMS,  known as M2D.
The first is the Falconer (1980) equation  given by
      {{Equation|<math> \nu_c = 0.575c_b|U|h </math>|2=4}}


where <math>\nu_{t0}</math> is a base value approximately equal to the dynamic viscosity, and <math>c_0</math>  is an empirical coefficient and <math>l_h</math> is the subgrid mixing length. 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.


       <math>l_h = c_{sm} \Delta x \Delta y </math>  
=== Parabolic Model ===
The second option is the parabolic model given by
       {{Equation|<math> \nu_c = c_0 u_{*} </math>|2=5}}


where <math>c_{sm}</math>  is an empirical coefficient (Smagorinsky coefficient).
where <math>c_0</math> is approximately equal to <math>\kappa/6</math>.


----
=== Subgrid Turbulence Model ===
'''References'''
The third  option for calculating <math>\nu_c</math> is the subgrid  turbulence model given by
      {{Equation|<math>  \nu_{c} = \sqrt{ (c_0 u_{*} h)^2  + (c_1 \Delta |\bar{S}|)^2}  </math>|2=6}}
 
where  <math>c_0</math> and <math>c_1</math> are  empirical coefficients related the turbulence produced by the bed and  horizontal velocity gradients, and <math>\Delta</math> is  the average grid area. <math>c_0</math> is approximately  equal to 0.0667 (default) but may vary from 0.01-0.2.  <math>c_{1}</math> is equal to approximately the square of  the Smagorinsky coefficient and may vary from 0.1 to 0.5 (default is  0.4). <math>|\bar{S}|</math> is equal to
      {{Equation|<math> |\bar{S}| = \sqrt{2\bar{S}_{ij}\bar{S}_{ij}}
= \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>  |2=7}}
 
and
      {{Equation|<math> \bar{S}_{ij} = \frac{1}{2} \biggl( \frac{  \partial U_i} { \partial x_j} +\frac{ \partial U_j} { \partial x_i}  \biggr) </math> |2=8}}
 
The  subgrid turbulence model parameters may be changed in the advanced  cards EDDY_VISCOSITY_BOTTOM, and EDDY_VISCOSITY_HORIZONTAL.
 
=== Mixing Length Model ===
 
==Wave-Related Eddy Viscosity ==
The wave component of the eddy viscosity is calculated as
      {{Equation|<math> \nu_w = \Lambda u_w H_s  </math>|2=2}}
 
where <math>\Lambda</math> is an empirical coefficient with a default value of 0.5 but may vary between 0.25 and 1.0. <math> H_s </math> is the significant wave height and <math>u_w</math> is bottom orbital velocity based on the significant wave height. <math>\Lambda</math> may be changed using the advanced card EDDY_VISCOSITY_WAVE.
 
Outside of the surf zone the bottom orbital velocity is calculated as
      {{Equation|<math> u_w = \frac{ \pi H_s}{T_p \sinh(kh) } </math>|2=2}}
 
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


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.
      {{Equation|<math> u_w = \frac{ H_s}{2h}\sqrt{gh} </math>|2=3}}


----
----
== 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]]
----
[[CMS#Documentation_Portal | Documentation Portal]]

Latest revision as of 21:56, 8 September 2011

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

  (1)

The base value for the eddy viscosity is approximately equal to the kinematic eddy viscosity can be changed using the advanced cards (Click [here] for further details).

Current-Related Eddy Viscosity Component

There are four options for the current-related eddy viscosity: FALCONER, PARABOLIC, SUBGRID, and MIXING-LENGTH. The default turbulence model is the subgrid model, but may be changed with the advanced card TURBULENCE_MODEL.

Falconer Equation

The Falconer (1980) equation is the method is the default method used in the previous version of CMS, known as M2D. The first is the Falconer (1980) equation given by

  (4)

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

Parabolic Model

The second option is the parabolic model given by

  (5)

where is approximately equal to .

Subgrid Turbulence Model

The third option for calculating is the subgrid turbulence model given by

  (6)

where and are empirical coefficients related the turbulence produced by the bed and horizontal velocity gradients, and is the average grid area. is approximately equal to 0.0667 (default) but may vary from 0.01-0.2. is equal to approximately the square of the Smagorinsky coefficient and may vary from 0.1 to 0.5 (default is 0.4). is equal to

  (7)

and

  (8)

The subgrid turbulence model parameters may be changed in the advanced cards EDDY_VISCOSITY_BOTTOM, and EDDY_VISCOSITY_HORIZONTAL.

Mixing Length Model

Wave-Related Eddy Viscosity

The wave component of the eddy viscosity is calculated as

  (2)

where is an empirical coefficient with a default value of 0.5 but may vary between 0.25 and 1.0. is the significant wave height and is bottom orbital velocity based on the significant wave height. may be changed using the advanced card EDDY_VISCOSITY_WAVE.

Outside of the surf zone the bottom orbital velocity is calculated as

  (2)

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


  (3)

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.

Documentation Portal