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 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

$\nu _{c}=0.575c_{b}|U|h$

(2)

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

$\nu _{c}=c_{1}u_{*}h$

(3)

where $c_{1}$ is approximately equal to $\kappa /6$.

Subgrid Turbulence Model

The third option for calculating $\nu _{c}$ is the subgrid turbulence model given by

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

(4)

where $c_{1}$ and $c_{2}$ are empirical coefficients related the turbulence produced by the bed and horizontal velocity gradients, and $\Delta$ is the average grid spacing. $c_{1}$ is approximately equal to 0.0667 (default) but may vary from 0.01-0.2. $c_{2}$ is equal to approximately the Smagorinsky coefficient and may vary from 0.1 to 0.3 (default is 0.2). $|{\bar {S}}|$ is equal to

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |\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 } }

(5)

and

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \bar{S}_{ij} = \frac{1}{2} \biggl( \frac{ \partial U_i} { \partial x_j} +\frac{ \partial U_j} { \partial x_i} \biggr) }

(6)

The subgrid turbulence model parameters may be changed in the advanced cards EDDY_VISCOSITY_BOTTOM, and EDDY_VISCOSITY_HORIZONTAL. Click here for further details.

Mixing Length Model

The Mixing Length Model implemented in CMS includes a component due to the vertical shear and is given by

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \nu_{c} = \sqrt{ (c_1 u_{*} h)^2 + (l_h^2 |\bar{S}|)^2}}

(7)

where the mixing length Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle l_h } is determined by Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle l_h = \kappa \min{(c_2h,y)}} , with Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle y } being the distance to the nearest wall and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \nu_w = c_3 u_w H_s + c_4 h \biggl( \frac{D_{br}}{\rho} \biggr) ^{1/3}}

(8)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c_3} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c_4} are empirical coefficients, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H_s } is the significant wave height and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c_3} is approximately equal to 0.1 and may vary from 0.05 to 0.2. The coefficient Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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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