CMS-Flow:Eddy Viscosity: Difference between revisions

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(Created page with "__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-relate...")
 
(Created page with __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-relate...)
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__NOTOC__
__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>
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}}
       {{Equation|math \nu_t  = \nu_0 + \nu_c + \nu_w /math  |2=1}}


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).
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).
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The  Falconer (1980) equation is  the method is the default method used in  the previous version of CMS,  known as M2D.
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
The first is the  Falconer (1980) equation  given by
       {{Equation|<math> \nu_c =  0.575c_b|U|h </math>|2=4}}
       {{Equation|math  \nu_c =  0.575c_b|U|h /math|2=4}}


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.
where  mathc_b/math is the bottom friction  coefficient,  mathU/math is the depth-averaged current  velocity, and  mathh/math is the total water depth.


===  Parabolic Model ===
===  Parabolic Model ===
The second option is the parabolic model  given by
The second option is the parabolic model  given by
       {{Equation|<math> \nu_c = c_0 u_{*} h  </math>|2=5}}
       {{Equation|math \nu_c = c_0 u_{*} h  /math|2=5}}


where  <math>c_0</math> is approximately equal to  <math>\kappa/6</math>.
where  mathc_0/math  is approximately equal to  math\kappa/6/math.


===  Subgrid Turbulence Model ===
===  Subgrid Turbulence Model ===
The third  option for calculating  <math>\nu_c</math> is the subgrid  turbulence model given by
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}}
       {{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
where  mathc_0/math and mathc_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. mathc_0/math is approximately  equal to 0.0667 (default) but may vary from 0.01-0.2.  mathc_{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}}
         {{Equation|math |\bar{S}| = \sqrt{2\bar{S}_{ij}\bar{S}_{ij}}
   = \sqrt{
   = \sqrt{
  2\biggl( \frac{ \partial U}{\partial  x} \biggr) ^2  +
  2\biggl( \frac{ \partial U}{\partial  x} \biggr) ^2  +
   2\biggl( \frac{ \partial V}{\partial  y} \biggr) ^2  +
   2\biggl( \frac{ \partial V}{\partial  y} \biggr) ^2  +
  \biggl(  \frac{ \partial U}{\partial y} +
  \biggl(  \frac{ \partial U}{\partial y} +
  \frac{ \partial V}{\partial x}  \biggr) ^2 } </math> |2=7}}
  \frac{ \partial V}{\partial x}  \biggr) ^2 } /math  |2=7}}


and
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}}
         {{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.
The  subgrid turbulence  model parameters may be changed in the advanced  cards  EDDY_VISCOSITY_BOTTOM, and EDDY_VISCOSITY_HORIZONTAL.
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==Wave-Related Eddy Viscosity  ==
==Wave-Related Eddy Viscosity  ==
The wave component of the eddy viscosity is calculated as
The wave component of the eddy viscosity is calculated as
       {{Equation|<math> \nu_w = \Lambda u_w H_s  </math>|2=2}}
       {{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.
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  mathu_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
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}}
       {{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
where mathH_s/math  is the significant wave height, mathT_p/math is the peak  wave period, mathk=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


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


----
----

Revision as of 20:05, 17 January 2011

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

  {{{1}}} (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

  U (4)

where mathc_b/math is the bottom friction coefficient, mathU/math is the depth-averaged current velocity, and mathh/math is the total water depth.

Parabolic Model

The second option is the parabolic model given by

  {{{1}}} (5)

where mathc_0/math is approximately equal to math\kappa/6/math.

Subgrid Turbulence Model

The third option for calculating math\nu_c/math is the subgrid turbulence model given by

  \bar{S} (6)

where mathc_0/math and mathc_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. mathc_0/math is approximately equal to 0.0667 (default) but may vary from 0.01-0.2. mathc_{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

  math (\bar{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  |2=7}}

and

  {{{1}}} (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

  {{{1}}} (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 mathu_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

  {{{1}}} (2)

where mathH_s/math is the significant wave height, mathT_p/math is the peak wave period, mathk=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


  {{{1}}} (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.

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