CMS-Flow:Subgrid Turbulence Model: Difference between revisions
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In CMS-Flow eddy viscosity is calculated as the sum of a base value <math>\nu_{0}</math>, the current-related eddy viscosity <math>\nu_c</math> and the wave-related eddy viscosity <math>\nu_w</math> | In CMS-Flow eddy viscosity is calculated as the sum of a base value <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_t = \nu_0 + \nu_c + \nu_w </math> | {{Equation|<math> \nu_t = \nu_0 + \nu_c + \nu_w </math> |2=1}} | ||
===Base Value Eddy Viscosity=== | ===Base Value Eddy Viscosity=== | ||
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The wave component of the eddy viscosity is calculated as | The wave component of the eddy viscosity is calculated as | ||
<math> \nu_w = \Lambda u_w H_s </math> | {{Equation|<math> \nu_w = \Lambda u_w H_s </math>|2=2}} | ||
where <math>\Lambda</math> is an empirical coefficient with a default value of 0.5 but may vary between 0.25 and 1.0. <math> H_s </math> is the significant wave height and <math>u_w</math> is bottom orbital velocity based on the significant wave height. <math>\Lambda</math> may be changed using the advanced card | where <math>\Lambda</math> is an empirical coefficient with a default value of 0.5 but may vary between 0.25 and 1.0. <math> H_s </math> is the significant wave height and <math>u_w</math> is bottom orbital velocity based on the significant wave height. <math>\Lambda</math> may be changed using the advanced card | ||
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Outside of the surf zone the bottom orbital velocity is calculated as | Outside of the surf zone the bottom orbital velocity is calculated as | ||
<math> u_w = \frac{ \pi H_s}{T_p \sinh(kh) } </math> | {{Equation|<math> u_w = \frac{ \pi H_s}{T_p \sinh(kh) } </math>|2=2}} | ||
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 | 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 | ||
<math> u_w = \frac{ H_s}{2h}\sqrt{gh} </math> | {{Equation|<math> u_w = \frac{ H_s}{2h}\sqrt{gh} </math>|2=3}} | ||
===Current-Related Eddy Viscosity Component=== | ===Current-Related Eddy Viscosity Component=== | ||
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The first is the Falconer (1980) equation given by | The first is the Falconer (1980) equation given by | ||
<math> \nu_c = 0.575c_b|U|h </math> | {{Equation|<math> \nu_c = 0.575c_b|U|h </math>|2=4}} | ||
where <math>c_b</math> is the bottom friction coefficient, <math>U</math> is the depth-averaged current velocity, and <math>h</math> is the total water depth. | where <math>c_b</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 the parabolic model given by | The second option is the parabolic model given by | ||
<math> \nu_c = c_0 u_{*} h </math> | {{Equation|<math> \nu_c = c_0 u_{*} h </math>|2=5}} | ||
where <math>c_0</math> is approximately equal to <math>\kappa/6</math> and may be changed using the advanced card | where <math>c_0</math> is approximately equal to <math>\kappa/6</math> and may be changed using the advanced card | ||
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The third option for calculating <math>\nu_c</math> is the subgrid turbulence model given by | The third option for calculating <math>\nu_c</math> is the subgrid turbulence model given by | ||
<math> \nu_{c} = \sqrt{ (c_0 u_{*} h)^2 + (c_1 \Delta |\bar{S}|)^2} </math> | {{Equation|<math> \nu_{c} = \sqrt{ (c_0 u_{*} h)^2 + (c_1 \Delta |\bar{S}|)^2} </math>|2=6}} | ||
where <math>c_0</math> and <math>c_1</math> are empirical coefficients related the turbulence produced by the bed and horizontal velocity gradients, and <math>\Delta</math> is the average grid area. <math>c_0</math> is approximately equal to 0.0667 (default) but may vary from 0.01-0.2. <math>c_1</math> is equal to approximately the square of the Smagorinsky coefficient and may vary from 0.1 to 0.5 (default is 0.4). <math>|\bar{S}|</math> is equal to | where <math>c_0</math> and <math>c_1</math> are empirical coefficients related the turbulence produced by the bed and horizontal velocity gradients, and <math>\Delta</math> is the average grid area. <math>c_0</math> is approximately equal to 0.0667 (default) but may vary from 0.01-0.2. <math>c_1</math> is equal to approximately the square of the Smagorinsky coefficient and may vary from 0.1 to 0.5 (default is 0.4). <math>|\bar{S}|</math> is equal to | ||
<math> |\bar{S}| = \sqrt{2\bar{S}_{ij}\bar{S}_{ij}} | {{Equation|<math> |\bar{S}| = \sqrt{2\bar{S}_{ij}\bar{S}_{ij}} | ||
= \sqrt{ | = \sqrt{ | ||
2\biggl( \frac{ \partial U}{\partial x} \biggr) ^2 + | 2\biggl( \frac{ \partial U}{\partial x} \biggr) ^2 + | ||
2\biggl( \frac{ \partial V}{\partial y} \biggr) ^2 + | 2\biggl( \frac{ \partial V}{\partial y} \biggr) ^2 + | ||
\biggl( \frac{ \partial U}{\partial y} + | \biggl( \frac{ \partial U}{\partial y} + | ||
\frac{ \partial V}{\partial x} \biggr) ^2 } </math> | \frac{ \partial V}{\partial x} \biggr) ^2 } </math> |2=7}} | ||
and | and | ||
<math> \bar{S}_{ij} = \frac{1}{2} \biggl( \frac{ \partial U_i} { \partial x_j} +\frac{ \partial U_i} { \partial x_j} \biggr) </math> | {{Equation|<math> \bar{S}_{ij} = \frac{1}{2} \biggl( \frac{ \partial U_i} { \partial x_j} +\frac{ \partial U_i} { \partial x_j} \biggr) </math> |2=8}} | ||
The subgrid turbulence model parameters may be changed in the advanced cards as | The subgrid turbulence model parameters may be changed in the advanced cards as | ||
Revision as of 19:31, 25 May 2010
Subgrid Turbulence Model
In CMS-Flow eddy viscosity is calculated as the sum of a base value , the current-related eddy viscosity and the wave-related eddy viscosity
| (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
| (2) |
where is an empirical coefficient with a default value of 0.5 but may vary between 0.25 and 1.0. is the significant wave height and is bottom orbital velocity based on the significant wave height. 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
| (2) |
where is the significant wave height, is the peak wave period, is the wave number. Inside the surf zone, the turbulence due to wave breaking is considered by increasing the bottom orbital velocity as
| (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
| (4) |
where is the bottom friction coefficient, is the depth-averaged current velocity, and is the total water depth.
Parabolic Model
The second option is the parabolic model given by
| (5) |
where is approximately equal to and may be changed using the advanced card
EDDY_VISCOSITY_BOTTOM 0.0667 ![-], bottom shear coefficient
Subgrid Turbulence Model
The third option for calculating is the subgrid turbulence model given by
| (6) |
where and are empirical coefficients related the turbulence produced by the bed and horizontal velocity gradients, and is the average grid area. is approximately equal to 0.0667 (default) but may vary from 0.01-0.2. is equal to approximately the square of the Smagorinsky coefficient and may vary from 0.1 to 0.5 (default is 0.4). is equal to
| (7) |
and
| (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
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.