CMS-Flow:Subgrid Turbulence Model

Subgrid Turbulence Model

In CMS-Flow eddy viscosity is calculated as the sum of a base value $\nu _{0}$, the current-related eddy viscosity $\nu _{c}$ and the wave-related eddy viscosity $\nu _{w}$

     $\nu _{t}=\nu _{0}+\nu _{c}+\nu _{w}$  

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 $\nu _{c}$ is the subgrid turbulence model given by

     $\nu _{c}={\sqrt {(c_{0}u_{*})^{2}h+(c_{1}\Delta |S|)^{2}}}$

where $c_{0}$ and $c_{1}$ are empirical coefficients, and $\Delta$ is the average grid area. $c_{0}$ is approximately equal to 0.0667 (default) but may vary from 0.01-0.2. $c_{1}$ may vary from 0.1 to 0.5 and is set to a default value of 0.4. $|S|$ is equal to

     $|S|={\sqrt {{\biggl (}2{\frac {\partial U}{\partial x}}{\biggr )}^{2}+2{\biggl (}{\frac {\partial V}{\partial y}}{\biggr )}^{2}+{\biggl (}{\frac {\partial U}{\partial y}}+{\frac {\partial V}{\partial x}}{\biggr )}^{2}}}$

The wave component of the eddy viscosity is calculated as

     $\nu _{w}=\Lambda u_{w}H_{s}$

where $\Lambda$ is an empirical coefficient approximately equal to 0.5, $H_{s}$ is the significant wave height and $u_{w}$ is bottom orbital velocity based on the significant wave height. Outside of the surf zone the bottom orbital velocity is calculated as

     $u_{w}={\frac {\pi H_{s}}{T_{p}\sinh(kh)}}$

where $H_{s}$ is the significant wave height, $T_{p}$ is the peak wave period, $k=2\pi /L$ is the wave number. Inside the surf zone, the turbulence due to wave breaking is considered by increasing the bottom orbital velocity as

     $u_{w}={\frac {H_{s}}{2h}}{\sqrt {gh}}$

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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