# CMS-Flow:Eddy Viscosity

The term eddy viscosity arises from the fact that small-scale vortices or eddies on the order of the grid cell size are not resolved, and only the large-scale flow is simulated. The eddy viscosity is intended to simulate the dissipation of energy at smaller scales than the model can simulate. In the nearshore environment, large mixing or turbulence occurs due to waves, wind, bottom shear, and strong horizontal gradients. Therefore, the eddy viscosity is an important parameter which can have a large influence on the calculated flow field and resulting sediment transport. In CMS-Flow, the total eddy viscosity $\displaystyle (v_t )$ is equal to the sum of three parts: 1) a base value $\displaystyle (v_0 )$ ; 2) the current-related eddy viscosity $\displaystyle (v_c )$ ; and 3) the wave-related eddy viscosity $\displaystyle (v_w)$ defined as follows:

 $\displaystyle \nu_t = \nu_0 + \nu_c + \nu_w$ (1)

The base value $\displaystyle (v_0 )$ is approximately equal to the kinematic viscosity $\displaystyle (\sim 1.81 \ x \ 10^{-6} \ m^2 /s)$ but may be changed by the user. The other two components $\displaystyle (v_c \ and\ v_w )$ are described in the sections below.

## Current-Related Eddy Viscosity Component

There are four algebraic models for the current-related eddy viscosity: 1) Falconer Equation; 2) depth-averaged parabolic; 3) subgrid; and 4) mixing-length. The default turbulence model is the subgrid model but may be changed by the user.

### Falconer Equation

The Falconer (1980) equation was default method used in earlier versions of CMS (Militello et al. 2004)for the current-related eddy viscosity. The equation is given by

 $\displaystyle \nu_c = 0.575c_b Uh$ (2)

where $\displaystyle c_b$ is the bottom friction coefficient, $\displaystyle U$ is the depth-averaged current velocity magnitude, and h is the total water depth.

### Depth-averaged Parabolic Model

The second option is the parabolic model given by

 $\displaystyle v_c = c_v u_{*c} h$ (3)

where $\displaystyle u_{*c} = \sqrt{\tau_c / \rho}$ is the bed shear velocity, and $\displaystyle c_v$ is approximately equal to $\displaystyle \kappa/6=0.0667$ but is set as a calibrated parameter whose value can be up to 1.0 in irregular waterways with weak meanders or even larger for strongly curved waterways.

### Subgrid Model

The third option for calculating $\displaystyle \nu_c$ is the subgrid turbulence model given by

 $\displaystyle v_c = c_v u_{*c} h + (c_h \Delta)^2 |\bar{S}|$ (4)

where:

$\displaystyle c_v$ = vertical shear coefficient [-]
$\displaystyle c_h$ = horizontal shear coefficient [-]
$\displaystyle \Delta$ = (average) grid size [m]
$\displaystyle |\bar{S}| = \sqrt{2e_{ij}e_{ij}}$
$\displaystyle e_{ij}$ = deformation (strain rate) tensor $\displaystyle = \frac{1}{2} \biggl( \frac{ \partial V_i} { \partial x_j} +\frac{ \partial V_j} { \partial x_i} \biggr)$

The empirical coefficients $\displaystyle c_v$ and $\displaystyle c_h$ are related to the turbulence produced by the bed shear and horizontal velocity gradients. The parameter $\displaystyle c_v$ is approximately equal to $\displaystyle \kappa/6=0.0667$ (default) but may vary from 0.01 to 0.2. The variable $\displaystyle c_h$ is equal to approximately the Smagorinsky coefficient (Smagorinsky 1963) and may vary between 0.1 and 0.3 (default is 0.2).

### Mixing Length Model

The Mixing Length Model implemented in CMS for the current-related eddy viscosity includes a component due to the vertical shear and is given by (Wu 2007)

 $\displaystyle \nu_{c} = \sqrt{ (c_v u_{*c} h)^2 + (l_h^2 |\bar{S}|)^2}$ (7)

where:

$\displaystyle l_h$ = the mixing length $\displaystyle ( = \kappa\ \min(c_h h,y^'))\ [m]$
$\displaystyle y^'$ =distance to the nearest wall [m]
$\displaystyle c_h$ = horizontal shear coefficient [-]

The empirical coefficient $\displaystyle c_h$ is usually between 0.3 and1.2. The effects of bed shear and horizontal velocity gradients, respectively, are taken into account through the first and second terms on the right-hand side of Equation (7). It has been found that the modified mixing length model is better than the depth-averaged parabolic eddy viscosity model that accounts for only the bed shear effect.

## Wave-Related Eddy Viscosity

The wave component of the eddy viscosity is separated into two components

 $\displaystyle \nu_w = c_{wf} u_{ws} H_s + c_{br} h \biggl( \frac{D_{br}}{\rho} \biggr) ^{1/3}$ (8)

where

$\displaystyle c_{wf}$ = wave bottom friction coefficient for eddy viscosity [-]
$\displaystyle u_{ws}$ = peak bottom orbital velocity [m/s] based on the significant wave height $\displaystyle H_s$ [m] and peak wave period $\displaystyle T_p$ [s]
$\displaystyle c_{br}$ = wave breaking coefficient for eddy viscosity [-]
$\displaystyle D_{br}$ = wave breaking dissipation [N/m/s].

The first term on the righ-hand side of Equation (8) represents the component due to bottom friction and the second term represents the component due to wave breaking. The coefficient $\displaystyle c_{wf}$ is approximately equal to 0.5 and may vary from 0.5 to 2.0. The coefficient $\displaystyle c_{br}$ is approximately equal to 0.1 and may vary from 0.04 to 0.15.

## 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.
• Militello, A., C. W. Reed, A. K. Zundel, and N. C. Kraus. 2004. Two-dimensional depth-averaged circulation model M2D: Version 2.0, Report 1, Technical documentation and user's guide. ERDC/CHL TR-04-02. Vicksburg, MS: US Army Engineer Research and Development Center.
• Smagorinsky, J. 1963. General circulation experiments with the primitive equations. Monthly Weather Review 93(3):99–164.