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| == Subgrid Turbulence Model == | | == Subgrid Turbulence Model == |
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| | In CMS-Flow eddy viscosity is calculated as <math> \nu_t = (1-\theta_m)\nu_{tc} + \theta_m \nu_m </math> where <math>\theta_m</math> is weighting factor equal to <math>\theta_m = (H_s/h)^3 </math> in which <math>H_s</math> is the significant wave height and <math>\nu_{tc}</math> and <math>\nu_{tw}</math> are the current- and wave-related eddy viscosity components respectively. The wave contribution is included using the equation of Kraus and Larson (1991) <math> \nu_tw = \Lambda u_w h </math>, where <math>\Lambda</math> is an empirical coefficient (default is 0.5), and <math>u_w</math> is the wave bottom orbital velocity. The current-related eddy viscosity is calculated as a function of the flow gradients, and the bottom shear stress |
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| In CMS-Flow eddy viscosity is calculated as the sum of the kinematic viscosity <math>\nu_{0}</math>, the current-related eddy viscosity <math>\nu_c</math> and the wave-related eddy viscosity <math>\nu_w</math>
| | <math> \nu_{tc} = \nu_{t0} + \sqrt{ (c_0 u_* h)^2 + (\sqrt{c_{sm} \Delta x \Delta y} |S| )^2 } </math> |
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| <math> \nu_{t} = \nu_0 + \nu_c + \nu_w </math>
| | where <math>\nu_{t0}</math> is a base value approximately equal to the dynamic viscosity, and <math>c_0</math> is an empirical coefficient and <math>l_h</math> is the subgrid mixing length. The mixing length is calculated here as |
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| There are two options to calculate <math>\nu_c</math>. The first is the Falconer (1980) equation given by
| | <math>l_h = c_{sm} \Delta x \Delta y </math> |
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| <math> \nu_c = 0.575C_f|U|h </math>
| | where <math>c_{sm}</math> is an empirical coefficient (Smagorinsky coefficient). |
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| where <math>C_f</math> is the bottom friction coefficient, <math>U</math> is the depth-averaged current velocity, and <math>h</math> is the total water depth. | |
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| The second option is a subgrid turbulence model given by
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| <math> \nu_{c} = c_b u_{*} h + c_h \Delta A |S| </math>
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| where <math>c_b</math> is an empirical coefficient approximately equal to 1/6 (default), <math>c_h</math> is an empirical coefficient between 0.1-0.5 (default is 0.4), <math>\Delta A = \Delta x \Delta y</math> is the local grid cell area, and <math>|S|</math> is equal to
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| <math> |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 } </math>
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| The wave component of the eddy viscosity is calculated as
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| <math> \nu_w = \Lambda u_w H_s </math>
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| where <math>\Lambda</math> is an empirical coefficient approximately equal to 0.5, <math> H_s </math> is the significant wave height and <math>u_w</math> is bottom orbital velocity based on the significant wave height. Outside of the surf zone the bottom orbital velocity is calculated as
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| <math> u_w = \frac{ \pi H_s}{T_p \sinh(kh) } </math>
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| where <math>H_s</math> is the significant wave height, <math>T_p</math> is the peak wave period, <math>k=2\pi/L</math> is the wave number. Inside the surf zone, the turbulence due to wave breaking is considered by increasing the bottom orbital velocity as
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| <math> u_w = \frac{ H_s}{2h}\sqrt{gh} </math>
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| The default turbulence model is the subgrid model, but may be changed with the advanced card
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| TURBULENCE_MODEL SUBGRID !FALCONER | PARABOLIC | SUBGRID | SUBGRID-WU
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| The turbulence model parameters may be changed in the advanced cards as
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| EDDY_VISCOSITY_CONSTANT 1.0e-6 ![m^2/sec], kinematic viscosity, ~1.0e-6
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| EDDY_VISCOSITY_BOTTOM 0.015 ![-], bottom shear coefficient, ~0.1667
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| EDDY_VISCOSITY_HORIZONTAL 0.2 ![-], smagorinsky coefficient, ~0.1-0.5
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| EDDY_VISCOSITY_WAVE 0.5 ![-], wave coefficient, ~0.25-0.5
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| ---- | | ---- |
Subgrid Turbulence Model
In CMS-Flow eddy viscosity is calculated as 
where 
is weighting factor equal to 
in which 
is the significant wave height and 
and 
are the current- and wave-related eddy viscosity components respectively. The wave contribution is included using the equation of Kraus and Larson (1991) 
, where 
is an empirical coefficient (default is 0.5), and 
is the wave bottom orbital velocity. The current-related eddy viscosity is calculated as a function of the flow gradients, and the bottom shear stress
where 
is a base value approximately equal to the dynamic viscosity, and 
is an empirical coefficient and 
is the subgrid mixing length. The mixing length is calculated here as
where 
is an empirical coefficient (Smagorinsky coefficient).
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.
CMS-Flow