Turbulence Modeling

Eddy Viscosity

Turbulence is a complex phenomenon of chaotic (turbulent) fluid motion and eddies spanning a wide range of length scales. Many of the length scales are too small to be feasibly resolved by a discrete numerical model, so turbulent flow mixing is modeled as a gradient diffusion process. In this approach, the diffusion rate is cast as the eddy viscosity $//<![CDATA[ \begin{array}{l}\nu_t\end{array} //]]>$ . The eddy viscosity is computed as follows,

where the tensor $//<![CDATA[ \begin{array}{l}\boldsymbol{D}\end{array} //]]>$ is the mixing coefficient tensor, $//<![CDATA[ \begin{array}{l}u_*\end{array} //]]>$ is the shear velocity, $//<![CDATA[ \begin{array}{l}h\end{array} //]]>$ is the water depth, $//<![CDATA[ \begin{array}{l}C_s\end{array} //]]>$ is the Smagorinsky coefficient (approximately between 0.05 and 0.2), $//<![CDATA[ \begin{array}{l}\Delta\end{array} //]]>$ is the filter width equal to local grid resolution, and $//<![CDATA[ \begin{array}{l}\left| \overline{S} \right|\end{array} //]]>$ is the strain rate. The first term on the right-hand-side represents the turbulence produced by vertical shear, and more specifically bottom shear in longitudinal direction and secondary flows in the transverse direction. The mixing coefficients also represent the mixing due to momentum dispersion and not just turbulence. The second term on the right-hand-side of equation2-136 represents the turbulence produced by horizontal shear in the flow. The second term in equation 2-136 is the Smagorinsky-Lilly eddy viscosity model (Smagorinsky 1963; Deardorff 1970). The Smagorinsky-Lilly model assumes that the turbulent energy production and dissipation at small scales are in equilibrium. The Smagorinsky-Lilly model is somewhat expensive to compute because it requires computing the velocity gradients. However, it is more physically accurate, especially in regions of high shear such as close to solid/dry boundaries. It is noted that the velocity gradients are computed at cells using the Green-Gauss divergence theorem and then interpolated at the faces with the weighting coefficients $//<![CDATA[ \begin{array}{l}\alpha _L\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}\alpha _R\end{array} //]]>$ . The strain rate is given as:

The diffusion coefficient tensor is given by:

and

$//<![CDATA[ \begin{array}{l}D_{xx} = D_L \cos^2\theta + D_T \sin^2\theta\end{array} //]]>$
$//<![CDATA[ \begin{array}{l}D_{yy} = D_L \sin^2\theta + D_T \cos^2\theta\end{array} //]]>$

The parameters $//<![CDATA[ \begin{array}{l}D_L\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}D_T\end{array} //]]>$ are user-specified mixing coefficients in the longitudinal and transverse directions, respectively. $//<![CDATA[ \begin{array}{l}\theta\end{array} //]]>$ is the velocity direction. If $//<![CDATA[ \begin{array}{l}D_L\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}D_T\end{array} //]]>$ are equal, then the mixing is isotropic. Some values for $//<![CDATA[ \begin{array}{l}D_L\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}D_T\end{array} //]]>$ are provided in the tables below:

Table 1. Longitudinal Coefficients.

Table 2. Transverse Mixing Coefficients.