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 }

(1)

Base Value 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.

Wave-Related Eddy Viscosity Component

The wave component of the eddy viscosity is calculated as

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

(2)

where $\Lambda$ is an empirical coefficient with a default value of 0.5 but may vary between 0.25 and 1.0. $H_{s}$ is the significant wave height and $u_{w}$ is bottom orbital velocity based on the significant wave height. $\Lambda$ may be changed using the advanced card

     EDDY_VISCOSITY_WAVE               0.5       ![-], wave coefficient

Outside of the surf zone the bottom orbital velocity is calculated as

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

(2)

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

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

(3)

Current-Related Eddy Viscosity Component

There are three options for the current-related eddy viscosity: FALCONER, PARABOLIC, and SUBGRID. The default turbulence model is the subgrid model, but may be changed with the advanced card

     TURBULENCE_MODEL                  SUBGRID   !FALCONER | PARABOLIC | SUBGRID

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

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

(4)

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

Parabolic Model

The second option is the parabolic model given by

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

(5)

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

     EDDY_VISCOSITY_BOTTOM             0.0667     ![-], bottom shear coefficient

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_{*} h)^2 + (c_1 \Delta |\bar{S}|)^2} }

(6)

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 related the turbulence produced by the bed and horizontal velocity gradients, 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} is equal to approximately the square of the Smagorinsky coefficient and may vary from 0.1 to 0.5 (default is 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 |\bar{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 |\bar{S}| = \sqrt{2\bar{S}_{ij}\bar{S}_{ij}} = \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 } }

(7)

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 \bar{S}_{ij} = \frac{1}{2} \biggl( \frac{ \partial U_i} { \partial x_j} +\frac{ \partial U_i} { \partial x_j} \biggr) }

(8)

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

     EDDY_VISCOSITY_BOTTOM             0.0667     ![-], bottom shear coefficient
     EDDY_VISCOSITY_HORIZONTAL         0.4       ![-], horizontal shear 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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