B.7 Horizontal Momentum Mixing Model
These models set the horizontal turbulent eddy viscosity \(\nu_{t,h}\).
B.7.1 None
The horizontal turbulent eddy viscosity \(\nu_{t,h}\) is set to 0. This results in no turbulent fluxes being calculated by the model. Use of this model is discouraged.
B.7.2 Constant
A user specified value is assigned to \(\nu_{t,h}\). This is applied constant throughout the model, irrespective of velocity gradients and variations.
B.7.3 Smagorinsky
The Smagorinsky model is used to estimate horizontal eddy viscosity as a function of the local strain rate in the flow field. The horizontal eddy viscosity \(\nu_{t,h}\) is calculated via Equation (B.102).
This formulation ensures that turbulent viscosity increases in regions with high velocity gradients, enhancing numerical stability and realism in resolved eddy behaviour.
\[\begin{equation} \nu_{t,h} = C_s^2 \, l^2 \, \sqrt{ \left( \frac{\partial u}{\partial x} \right)^2 + \left( \frac{\partial v}{\partial y} \right)^2 + \frac{1}{2} \left( \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \right)^2 } \tag{B.102} \end{equation}\]
| Symbol | Description |
|---|---|
| \(\nu_{t,h}\) | Turbulent eddy viscosity |
| \(C_s\) | 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 viscosity limits (m\(^2\)/s) are included to provide bounds to the formulation. This formulation can be used in 2D or 3D HD Simulation Classes.
B.7.4 Wu
The Wu horizontal momentum model is based on W. Wu et al. (2005). The Wu model is a zero equation model whereby the eddy viscosity coefficient is diagnostically computed from the mean depth and velocity fields.
\[\begin{equation} \nu_{t,h} = C_{3D} \cdot U^* \cdot L_m \tag{B.103} \end{equation}\]
\[\begin{equation} U^* = |U| \cdot n \cdot \frac{\sqrt{g}}{h^{1/6}} \tag{B.104} \end{equation}\]
The Wu momentum mixing model should not be used for 3D HD simulation class models, or simulations that rely on 3D hydrodynamic fields. If used, it is recommended that user specified minimum and maximum eddy viscosity limits (m\(^2\)/s) are included to provide bounds to the formulation. The Wu model is sometimes preferred in areas where the 2D cell size is smaller or close to the water depth.
| Symbol | Description |
|---|---|
| \(\nu_{t,h}\) | Turbulent eddy viscosity |
| \(C_{3D}\) | Wu coefficient |
| \(U_*\) | Friction velocity |
| \(L_m\) | Turbulent length scale (set to water depth, \(h\)) |
| \(n\) | Manning’s roughness coefficient |
| \(g\) | Acceleration due to gravity |
| \(h\) | Water depth |
If the model uses the ks Bottom Drag model than the Nikaradse roughness length is internally converted to a an equivalent Manning’s ‘n’ using Equation (B.45).