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; and therefore the eddy viscosity is an important aspect which can have a large influence on the calculated flow field and resulting sediment transport. In CMS-Flow, the total eddy viscosity, ν_t , is equal to the sum of three parts: 1) a base value ν₀, 2) the current-related eddy viscosity ν_c, and 3) the wave-related eddy viscosity ν_w and is defined as,

The base value (ν₀) is approximately equal to the kinematic eddy viscosity (~1×10^-6 m²/s), but may be changed by user. The other two components (ν_c and ν_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) mix-ing-length. The default turbulence model is the subgrid model, but may be changed in the user.

Falconer Equation

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

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

Depth-averaged Parabolic Model

The second option for the current-related eddy viscosity is the depth-averaged parabolic model given by

where c_v is approximately 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 \kappa/6=0.0667.}

Subgrid Model The third option for calculating the current-related eddy viscosity, ν_c , is the subgrid turbulence model given by

in which

c_v = vertical shear coefficient [-]

c_h = horizontal shear 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 \mid\bar{S}\mid} = magnitude of the deformation (strain rate) tensor 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 e_{ij} = \left(\sqrt {2e_{ij}e_{ij}} \right) }

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 e_y} = deformation (strain rate) tensor = 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 = \frac{1}{2} \left(\frac{\partial V_i}{\partial x_j} + \frac{\partial V_j}{\partial x_i} \right) }

The empirical coefficients c_v and c_s are related to the turbulence pro-duced by the bed shear and horizontal velocity gradients, and Δ is the (average) grid size. The parameter c_v is approximately equal to κ/6=0.0667 (default) but may vary from 0.01-0.2. The variable c_s is equal to approximately the Smagorinsky coefficient (Smagorisnky 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 2008)

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 l_h } = 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 = \kappa min(c_h h, y^')) } [m]

y^{'}=

distance to the nearest wall [m]

The empirical coefficient c_h is usually between 0.3 and 1.2. 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 (2-26). 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

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 u_w } = peak bottom orbital velocity [m/s] based on the significant wave height H_i [m] and peak wave period T_p [s]

The first term on the right-hand side of Equation (2-27) represents the component due to wave bottom friction and the second term represents the component due to wave breaking. The coefficient c_wf is approximately equal to 0.1 and may vary from 0.05 to 0.2. The coefficient c_br is approxi-mately equal to 0.08 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)

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 S_{ij} = \iint E_w (f,\Theta) \left[n_g w_i w_j + \delta_{ij} \left(n_g - \frac{1}{2} \right) \right]df d\theta} (2-27)

where

f = the wave frequency [1/s]

w_i = wave unit vector = 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 (cos \theta, sin \theta)} [-]

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 \delta_{ij} = \left\{ \begin{align} &1 \quad \text{for i} = j \\ &0 \quad \text{for i} \neq j \end{align} \right. }

and

in which c_g is the wave group velocity, c is the wave celerity, and k is the wave number.

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 surfzone, the surface roller also provides a contribution to the mean wave mass flux. The mean wave mass flux velocity, or simply the mass flux ve-locity for short, is defined as the mean wave volume flux divided by the local water depth and is approximated here as (Phillips 1977; Ruessink et al. 2001; Svendsen 2006)

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 E_w} = wave energy 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 1/16 \rho gH_s^2 } [N/m}

H_s = significant wave height [m]

w_i = wave unit vector 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 = (cos \theta, sin \theta)} [-]

c = wave speed [m/s]

The first component is due to the wave energy ( is the Stokes velocity), while the second component is due to the surface roller ( is only pre-sent in the surfzone).

Wind Surface Stress

The wind surface stress is calculated as

where

W = 10-m wind velocity magnitude 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 = \sqrt{W_i W_i}} [m/s]

The wind speed is calculated using either an Eulerian or Lagrangian reference frame as

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 \gamma_w } = equal to 0 for the Eulerian reference frame or 1 for the Lagrangian reference frame

Using the Lagrangian reference frame or relative wind speed is more accurate and realistic for field applications (Bye 1985; Pacanowski 1987; Dawe and Thompson 2006), but the option to use the Eulerian wind speed is provided for idealized cases. The drag coefficient is calculated using the formula of Hsu (1988) and modified for high wind speeds based on field data by Powell et al. (2003)

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_D = \left\{ \begin{align} &\left(\frac{\kappa}{14.56 - 2 ln W}\right)^2 for W \leq 30 m/s \\ &10^{-3} max(3.86 - 0.04 W, 1.5) for W > 30 m/s \end{align} \right. } (2-32) Powell et al. (2003) speculate that the reason for the decrease in drag coefficient with higher wind speeds is due to increasing foam coverage leading to the formation of a “slip” surface at the air-sea interface.

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