CMS-Flow:Subgrid Turbulence Model

Subgrid Turbulence Model

In CMS-Flow eddy viscosity is calculated as the sum of the kinematic viscosity ${\nu }_0$, the current-related eddy viscosity ${\nu }_c$ and the wave-related eddy viscosity ${\nu }_w$

     ${\nu }_t={\nu }_0+{\nu }_c+{\nu }_w$  

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. 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.

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

The third option for calculating ${\nu }_c$ is the subgrid turbulence model given by

     ${\nu }_c=\sqrt{(c_0{u}_{*}{)}^{2}h+(c_1\mathrm{\Delta }|S|{)}^2}$

where $c_0$ and $c_1$ are empirical coefficients, and $\mathrm{\Delta }$ 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

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

The wave component of the eddy viscosity is calculated as

     ${\nu }_w=\mathrm{\Lambda }{u}_w{H}_s$

where $\mathrm{\Lambda }$ 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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