CMS-Flow:Subgrid Turbulence Model

Subgrid Turbulence Model

In CMS-Flow eddy viscosity is calculated as the sum of a base value Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \nu_{0}} , the current-related eddy viscosity Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \nu_c} and the wave-related eddy viscosity Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \nu_w}

     Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle  \nu_t = \nu_0 + \nu_c + \nu_w }
  

Base Eddy Viscosity

The base value for the eddy viscosity is approximately equal to the kinematic eddy viscosity can be changed using the advanced card

     EDDY_VISCOSITY_CONSTANT           1.0e-6    ![m^2/sec], kinematic viscosity, ~1.0e-6

There are three options for calculating the current-related eddy viscosity.

Current-Related Eddy Viscosity

1. 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

     $\nu _{c}=0.575c_{b}|U|h$

where $c_{b}$ is the bottom friction coefficient, $U$ is the depth-averaged current velocity, and $h$ is the total water depth.

2. Parabolic Model

The second option is the parabolic model given by

     $\nu _{c}=c_{0}u_{*}h$

where $c_{0}$ is approximately equal to $\kappa /6$ and may be changed using the advanced card

     EDDY_VISCOSITY_BOTTOM             0.015     ![-], bottom shear coefficient, ~0.1667

2. Subgrid Turbulence Model

The third option for calculating Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \nu_c} is the subgrid turbulence model given by

     Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle  \nu_{c} = \sqrt{ (c_0 u_{*})^2 h + (c_1 \Delta |S|)^2}  }

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c_0} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c_1} are empirical coefficients, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta} is the average grid area. Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c_0} is approximately equal to 0.0667 (default) but may vary from 0.01-0.2. Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c_1} may vary from 0.1 to 0.5 and is set to a default value of 0.4. Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |S|} is equal to

     Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle  |S| = \sqrt{ \biggl( 2\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 } }

The wave component of the eddy viscosity is calculated as

     Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle  \nu_w = \Lambda u_w H_s  }

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Lambda} is an empirical coefficient approximately equal to 0.5, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H_s } is the significant wave height and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_w} is bottom orbital velocity based on the significant wave height. Outside of the surf zone the bottom orbital velocity is calculated as

     Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle  u_w = \frac{ \pi H_s}{T_p \sinh(kh) } }

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H_s} is the significant wave height, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T_p} is the peak wave period, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k=2\pi/L} is the wave number. Inside the surf zone, the turbulence due to wave breaking is considered by increasing the bottom orbital velocity as

     Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle  u_w = \frac{ H_s}{2h}\sqrt{gh} }

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

     TURBULENCE_MODEL                  SUBGRID   !FALCONER | PARABOLIC | SUBGRID

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

     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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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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