# CMS-Flow:Subgrid Turbulence Model

In CMS-Flow eddy viscosity is calculated as the sum of a base value $\nu_{0}$, the current-related eddy viscosity $\nu_c$ and the wave-related eddy viscosity $\nu_w$ $\nu_t = \nu_0 + \nu_c + \nu_w$ (1)

The base value for the eddy viscosity is approximately equal to the kinematic eddy viscosity can be changed using the advanced cards (Click [here] for further details).

## Current-Related Eddy Viscosity Component

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

### 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 $\kappa/6$.

### Subgrid Turbulence Model

The third option for calculating $\nu_c$ is the subgrid turbulence model given by $\nu_{c} = \sqrt{ (c_0 u_{*} h)^2 + (c_1 \Delta |\bar{S}|)^2}$ (6)

where $c_0$ and $c_1$ are empirical coefficients related the turbulence produced by the bed and horizontal velocity gradients, and $\Delta$ is the average grid area. $c_0$ is approximately equal to 0.0667 (default) but may vary from 0.01-0.2. $c_{1}$ is equal to approximately the square of the Smagorinsky coefficient and may vary from 0.1 to 0.5 (default is 0.4). $|\bar{S}|$ is equal to $|\bar{S}| = \sqrt{2\bar{S}_{ij}\bar{S}_{ij}} = \sqrt{ 2\biggl( \frac{ \partial U}{\partial x} \biggr) ^2 + 2\biggl( \frac{ \partial V}{\partial y} \biggr) ^2 + \biggl( \frac{ \partial U}{\partial y} + \frac{ \partial V}{\partial x} \biggr) ^2 }$ (7)

and $\bar{S}_{ij} = \frac{1}{2} \biggl( \frac{ \partial U_i} { \partial x_j} +\frac{ \partial U_j} { \partial x_i} \biggr)$ (8)

The subgrid turbulence model parameters may be changed in the advanced cards EDDY_VISCOSITY_BOTTOM, and EDDY_VISCOSITY_HORIZONTAL.

## Wave-Related Eddy Viscosity

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.

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)