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,

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 v_t = v_0 + v_c + v_w} (2-21)

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

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 v_c = 0.575c_{b}Uh } (2-22)

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

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 v_c = c_{v}u_{*}h } (2-23)

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

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 v_c = c_{v} u_{*} h + (c_s \Delta)^2 \mid\bar{S}\mid } (2-24)

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)

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 v_c = \sqrt {(c_v u_{*c} h)^2 + (l_{h}^2)|\bar S|)^2} } (2-25)

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]

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

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_h = } horizontal shear coefficient [-]

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

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 v_w = c_{wf}u_w H_s + c_{br}h \left(\frac{D_{br}}{\rho} \right)^{1/3} } (2-26)

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

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]

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_{br}} wave breaking coefficient for eddy viscosity [-]

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 D_{br}} = wave breaking dissipation [N/m/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]

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 \theta } = the wave direction [rad]

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

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 n_g = \frac{c_g}{c}=\frac{1}{2}\left( 1+\frac{2kh}{sinh 2kh} \right)} (2-28)

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)

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_{wi}= \frac{ (E_w + 2 E_{s\gamma}) w_i} {\rho hc}} (2-29)

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]

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_{s\gamma}} = surface roller energy density [N/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]

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 \theta = } mean wave direction, [rad]

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

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 \tau_{si} = \rho_a c_D WW_i } (2-30)

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 \rho_a } = air density at sea level [~1.2 kg/m³]

$c_{D}$ = wind drag 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 W_i } = 10-m wind speed [m/s]

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

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 W_i = W_i^E - \gamma_w U_i } (2-31)

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 W_i^E = } 10-m atmospheric wind speed relative to the solid earth (Eulerian wind speed) [m/s]

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

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_i} = current velocity [m/s]

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.

fig_2.2.png Figure 2.2 Modieifed Hsu (1988) wind drag coefficient