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 $(v_{t})$ is equal to the sum of three parts: 1) a base value $(v_{0})$; 2) the current-related eddy viscosity $(v_{c})$; and 3) the wave-related eddy viscosity $(v_{w})$ defined as follows:

$\nu _{t}=\nu _{0}+\nu _{c}+\nu _{w}$

(1)

The base value $(v_{0})$ is approximately equal to the kinematic viscosity $(\sim 1.81x10^{-6}m^{2}/s)$ but may be changed by the user. The other two components $(v_{c}\ and\ v_{w})$ 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 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

$\nu _{c}=0.575c_{b}Uh$

(2)

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

Depth-varaged Parabolic Model

The second option is the parabolic model given by

$v_{c}=c_{v}u_{*c}h$

(3)

where $u_{*c}={\sqrt {\tau _{c}/\rho }}$ is the bed shear velocity, and $c_{v}$ is approximately equal to $\kappa /6=0.0667$ 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 $\nu _{c}$ is the subgrid turbulence model given by

$v_{c}=c_{v}u_{*c}h+(c_{h}\Delta )^{2}|{\bar {S}}|$

(4)

where:

$c_{v}$ = vertical shear coefficient [-]

$c_{h}$ = horizontal shear coefficient [-]

$\Delta$ = (average) grid size [m]

$|{\bar {S}}|={\sqrt {2e_{ij}e_{ij}}}$

$e_{ij}$= deformation (strain rate) tensor $={\frac {1}{2}}{\biggl (}{\frac {\partial V_{i}}{\partial x_{j}}}+{\frac {\partial V_{j}}{\partial x_{i}}}{\biggr )}$

The empirical coefficients $c_{v}$ and $c_{h}$ are related to the turbulence produced by the bed shear and horizontal velocity gradients. The parameter $c_{v}$ is approximately equal to $\kappa /6=0.0667$ (default) but may vary from 0.01 to 0.2. The variable $c_{h}$ 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)

$\nu _{c}={\sqrt {(c_{1}u_{*c}h)^{2}+(l_{h}^{2}|{\bar {S}}|)^{2}}}$

(7)

where the mixing length $l_{h}$ is determined by $l_{h}=\kappa \min {(c_{2}h,y)}$, with $y$ being the distance to the nearest wall and $c_{2}$ is an empirical coefficient between 0.3-1.2. Eq. (9) takes into account the effects of bed shear and horizontal velocity gradients respectively through the first and second terms on its right-hand side. 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

$\nu _{w}=c_{3}u_{w}H_{s}+c_{4}h{\biggl (}{\frac {D_{br}}{\rho }}{\biggr )}^{1/3}$

(8)

where $c_{3}$ and $c_{4}$ are empirical coefficients, $H_{s}$ is the significant wave height and $u_{w}$ is bottom orbital velocity based on the significant wave height. The first term on the R.H.S. of Eq. (10) represents the component due to bottom friction and the second term represents the component due to wave breaking. The coefficient $c_{3}$ is approximately equal to 0.1 and may vary from 0.05 to 0.2. The coefficient $c_{4}$ is approximately equal to 0.08 and may vary from 0.04 to 0.15.

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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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