B.8 Horizontal Scalar Mixing Model
These models set the horizontal turbulent eddy diffusivity \(D_{t,h}\).
B.8.1 None
The horizontal turbulent eddy diffusivity \(D_{t,h}\) is set to 0. This results in no turbulent scalar fluxes being calculated by the model. Use of this model is discouraged.
B.8.2 Constant
A user specified value is assigned to \(D_{t,h}\). This is applied constant throughout the model, irrespective of velocity gradients and variations.
B.8.3 Elder
The Elder formulation for calculation of \(D_{t,h}\) is described by Falconer et al. (2005). This model calculates a non-isotropic diffusivity tensor that accounts for velocity dispersion processes not resolved in 2D depth-averaged models.
\[\begin{equation} \begin{aligned} D_{xx} &= \frac{\left(D_lu^2+D_tv^2\right)h}{u_\ast} \\ D_{yy} &= \frac{\left(D_lv^2+D_tu^2\right)h}{u_\ast} \\ D_{xy} &= \frac{\left(D_l-D_t\right)uvh}{u_\ast} = D_{yx} \\ u_\ast &= \sqrt{\frac{ \left| \tau_b \right| }{\rho}} \end{aligned} \tag{B.105} \end{equation}\]
where:
- \(D_l\) is the Elder coefficient in the lateral direction to the local current
- \(D_t\) is the Elder coefficient in the transverse direction to the local current
- \(\tau_b\) is the bed shear stress
The ranges of values for \(D_l\) and \(D_t\) derived from measurements are discussed in Fischer et al. (1979). This Elder model is less applicable in 3D simulations than that of Smagorinsky (Appendix B.8.4).
B.8.4 Smagorinsky
The Smagorinsky model is used to estimate horizontal eddy diffusivity as a function of the local strain rate in the flow field. The horizontal eddy diffusivity \(D_{t,h}\) is calculated via Equation (B.106).
This formulation ensures that turbulent diffusivity increases in regions with high velocity gradients, enhancing numerical stability and realism in resolved eddy behaviour.
\[\begin{equation} D_{t,h} = C_s^2 \, l^2 \, \sqrt{ \left( \frac{\partial u}{\partial x} \right)^2 + \left( \frac{\partial v}{\partial y} \right)^2 + \frac{1}{2} \left( \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \right)^2 } \tag{B.106} \end{equation}\]
| Symbol | Description |
|---|---|
| \(D_{t,h}\) | Turbulent eddy diffusivity |
| \(C_s\) | Smagorinsky coefficient |
| \(l\) | Mixing length scale, distance between adjacent cell centroids |
| \(u\), \(v\) | Horizontal velocity components in the \(x\) and \(y\) directions |
| \(\frac{\partial u}{\partial x}\), \(\frac{\partial v}{\partial y}\) | Velocity gradients in the direction of flow (normal strain rates) |
| \(\frac{\partial u}{\partial y}\), \(\frac{\partial v}{\partial x}\) | Cross-direction velocity gradients (shear strain rates) |
It is recommended that user specified minimum and maximum eddy diffusivity limits (m\(^2\)/s) are included to provide bounds to the formulation. This formulation can be used in 2D or 3D HD Simulation Classes.