CMS-Flow:Subgrid Turbulence Model

Subgrid Turbulence Model

In CMS-Flow 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_t = (1-\theta_m)\nu_c + \theta_m \nu_w } 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 \theta_m} is weighting factor equal to $\theta _{m}=(H_{s}/h)^{3}$ in which $H_{s}$ is the significant wave height and $\nu _{c}$ and $\nu _{w}$ are the current- and wave-related eddy viscosity components respectively. The wave contribution is included using the equation of Kraus and Larson (1991) $\nu _{w}=\Lambda u_{w}h$, where $\Lambda$ is an empirical coefficient (default is 0.5), and $u_{w}$ is the wave bottom orbital velocity and $h$ is the water depth. The current-related eddy viscosity is calculated as a function of the flow gradients, and the bottom shear stress

     $\nu _{tc}=\nu _{t0}+{\sqrt {(c_{0}u_{*}h)^{2}+(l_{h}^{2}|S|)^{2}}}$

where $\nu _{t0}$ is a base value approximately equal to the dynamic viscosity, and $c_{0}$ is an empirical coefficient, $l_{h}$ is the subgrid mixing length and $|S|$ 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{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 } }

The mixing length is calculated here 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 l_h = \kappa \textrm{min}( \sqrt{\Delta x \Delta y}, c_{sm} h )  }
 

where $c_{sm}$ is an empirical coefficient (Smagorinsky coefficient).

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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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CMS-Flow