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:

$l_{h}$ = the mixing length $(=\kappa \ \min(c_{h}h,y^{'}))\ [m]$

$y^{'}$ =distance to the nearest wall [m]

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

The empirical coefficient $c_{h}$ 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

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

(8)

where

$c_{wf}$ = wave bottom friction coefficient for eddy viscosity [-]

$u_{ws}$ = peak bottom orbital velocity [m/s] based on the significant wave height $H_{s}$ [m] and peak wave period $T_{p}$ [s]

$c_{br}$ = wave breaking coefficient for eddy viscosity [-]

$D_{br}$ = 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 $c_{wf}$ is approximately equal to 0.5 and may vary from 0.5 to 2.0. The coefficient $c_{br}$ is approximately equal to 0.1 and may vary from 0.04 to 0.15.

Wave Radiation Stresses

The wave radiation stresses $(S_{ij})$ are calculated using linear wave theory as (Longuet-Higgins and Stewart 1961; Dean and Dalrymple 1984)

$S_{ij}=\int \int E_{w}(f,\theta )\left[n_{g}\omega _{i}\omega _{j}+\delta _{ij}\left(n_{g}-{\frac {1}{2}}\right)\right]dfd\theta$

(9)

where:

f = the wave frequency [1/s]

$\theta$ = the wave direction [rad]

$E_{e}$ = wave energy = $1/16\ \rho gH_{s}^{2}$ [N/m]

$H_{s}$ = significant wave height [m]

$w_{i}$ = wave unit vector = $(cos\ \theta ,sin\ \theta )$[-]

$\delta _{ij}=\left\{{\begin{aligned}&1\ for\ i=j\\&0\ for\ i\neq j\end{aligned}}\right.$

$n_{g}={\frac {c_{g}}{c}}={\frac {1}{2}}\left(1+{\frac {2kh}{sinh\ 2kn}}\right)[-]$

$c_{g}$ = wave group velocity [m/s]

$c$ = wave ce;erotu [m/s]

$k$ = wave number [rad/s]

The wave radiation stresses and their gradients are computed within the wave model and interpolated in space and time in the flow model.

Roller Stresses

As a wave transitions from non-breaking to fully-breaking, some of the energy is converted into momentum that goes into the aerated region of the water column. This phenomenon is known as the surface roller. The surface roller contribution to the wave stresses $(R_{ij})$ is given by

$R_{ij}=2E_{sr}w_{i}w_{j}$

(10)

where:

$E_{sr}$ = surface roller energy density [N/m]

$w_{j}$ = wave unit vector = $(cos\ \theta _{m},sin\ \theta _{m})[-]$

The surface roller is calculated within CMS-Wave. An effect of the surface roller is to shift the peak alongshore current velocity closer to shore. Another side effect of the surface roller is to improve model stability (Sánchez et al. 2011a). The influence of the surface roller on the mean water surface elevation is relatively minor (Sánchez et al. 2011a).

Wave Flux Velocity

In the presence of waves, the oscillatory wave motion produces a net time-averaged mass (volume) transport referred to as Stokes drift. In the surf zone, the surface roller also provides a contribution to the mean wave mass flux. The mean wave mass flux velocity, or simply the mass flux velocity, is defined as the mean wave volume flux divided by the local water depth and is approximated here as (Phillips 1977; Svendsen 2006)

$U_{wi}={\frac {(E_{w}+2E_{sr})w_{i}}{\rho hc}}$

(11)

where:

$E_{w}$ = wave energy = $1/16\ \rho gH_{s}^{2}[N/m]$

$H_{s}$ = significant wave height [m]

$E_{sr}$= surface roller energy density [N/m]

$w_{i}=(cos\ \theta ,sin\ \theta )$ = wave unit vector [-]

$c$= wave celerity [m/s].

The first component is due to the Stokes velocity while the second component is due to the surface roller (only present in the surf zone).

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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