2D Shallow Water Equations

The depth-averaged Shallow Water Equations (SWE) model solves volume and momentum conservation equations and includes temporal and spatial accelerations as well as horizontal mixing while the DWE model ignores these processes but is therefore simpler and more computationally efficient. The 2D volume conservation of the water-solid mixture is given by:

$\begin{array}{l}\displaystyle \frac{\partial \eta}{\partial t} + \nabla \cdot (h\boldsymbol{V}) = q\end{array}$

where η is the flow surface elevation, t is time, h is the water depth, V is the velocity vector, and q is a source or sink term, to account for external and internal fluxes. The depth-averaged momentum conservation equations may be written as (Hergarten and Robl, 2015):

$\begin{array}{l}\displaystyle \frac{\partial \textbf{V}}{\partial t} + (\boldsymbol{V} \cdot \nabla ) \boldsymbol{V} = -g \cos ^2 \varphi \nabla \eta + \frac{1}{h} \nabla \cdot (\boldsymbol{\nu}_t h \nabla \boldsymbol{V}) - \frac{\boldsymbol{\tau} }{\rho_m R} \frac{cos \psi}{\cos \varphi} \frac{\boldsymbol{V}}{| \boldsymbol{V} |}\end{array}$

in which $\begin{array}{l}g\end{array}$ is the gravitational acceleration, $\begin{array}{l}\nu_t\end{array}$ is a turbulent eddy viscosity, $\begin{array}{l}\tau\end{array}$ is the total basal stress, $\begin{array}{l}\rho_m\end{array}$ is the water-solid mixture density, $\begin{array}{l}R\end{array}$ is the hydraulic radius, $\begin{array}{l}V\end{array}$ is the magnitude of the velocity vector, $\begin{array}{l}\varphi\end{array}$ is the water surface slope, and $\begin{array}{l}\psi\end{array}$ is the inclination angle of the current velocity direction. In the above equations, the second term on the right-hand-side represents the horizontal mixing due to turbulence and also in the case of a debris flow, horizontal mixing due to particle collisions. Utilizing the conservative form of the mixing terms is essential for accurate momentum conservation. The bottom friction coefficient is computed utilizing the Manning's roughness coefficient as

$\begin{array}{l}\displaystyle \tau = \tau_b + \tau_{MD}\end{array}$

where $\begin{array}{l}\tau_b\end{array}$ is the bottom turbulent shear stress and $\begin{array}{l}\tau_{MD}\end{array}$ is the mud and debris stress which includes all non-Newtonian stresses. The turbulence bottom shear stress is computed as a function of the Manning's roughness coefficient

$\begin{array}{l}\displaystyle \tau_b = \rho_m C_d | \boldsymbol{V} |^2\end{array}$

$\begin{array}{l}\displaystyle C_d = \frac{g n^2}{R^{1/3}}\end{array}$

where ρm is the density water-particle mixture and n is the Manning's roughness coefficient. The mud and debris stress is described in detail in the section "Rheological Models".

When the non-Newtonian stress is equal to zero and the cosine functions (slope corrections) are removed, the above 2D SWE equations reduce to the clear-water equations utilized in HEC-RAS.

When simulating hyperconcentratedted flows, the longitudinal and transverse components of the turbulent eddy viscosity are computed with the shear velocity from total shear stress (i.e. u*=τ/ρm). There is no existing research on the appropriate values for the turbulence coefficients for hyperconcentratedted flows. However, testing has shown that using similar values to those for clear-water produce reasonable results. This is a subject which requires further research. The current guidance is to start with "clear-water" values for the turbulence coefficients and calibrate them as best as possible with measurements.