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 $\nu _{w}$

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

There are two options to calculate $\nu _{c}$. The first is the Falconer (1980) equation given by

     $\nu _{c}=0.575C_{f}|U|h$

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

The second option is a subgrid turbulence model given by

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

where $c_{b}$ is an empirical coefficient approximately equal to 1/6 (default), $c_{h}$ is an empirical coefficient between 0.1-0.5 (default is 0.4), $\Delta A=\Delta x\Delta y$ is the local grid cell area, and $|S|$ is equal to

     $|S|={\sqrt {{\biggl (}{\frac {\partial U}{\partial x}}{\biggr )}^{2}+{\biggl (}{\frac {\partial V}{\partial y}}{\biggr )}^{2}+{\frac {1}{2}}{\biggl (}{\frac {\partial U}{\partial y}}+{\frac {\partial V}{\partial x}}{\biggr )}^{2}}}$

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 | SUBGRID-WU 

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

     EDDY_VISCOSITY_CONSTANT           1.0e-6    ![m^2/sec], kinematic viscosity, ~1.0e-6
     EDDY_VISCOSITY_BOTTOM             0.015     ![-], bottom shear coefficient, ~0.1667
     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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