2D Diffusion Wave Equation

HEC-RAS also includes a simplified, unsteady, hydrodynamic model, which replaces momentum with the Diffusive-Wave Equation (DWE):

$\begin{array}{l}\displaystyle \frac{\partial \eta}{\partial t} = \nabla \cdot \left( \beta \nabla \eta \right) + q\end{array}$

where $\begin{array}{l}\beta\end{array}$ is a non-linear "diffusion" coefficient which is a function of the bottom friction and non-Newtonian stress

$\begin{array}{l}\displaystyle \beta = \cos^{1/2} \psi \cos \varphi \frac{K}{A} \frac{h}{|\nabla \eta|^{1/2}}\end{array}$

in which

$\begin{array}{l}\displaystyle \frac{K}{A} = \left[ \frac{n^2}{(R \cos \varphi )^{4/3}} + \frac{\tau_{MD}}{\gamma_m R \cos \varphi |V|^2}\right]^{-1/2}\end{array}$

In the above equations, K is the conveyance, and A is the vertical area. The diffusion equation has been modified for steep slopes following an approach similar to that of Hergarten and Robl (2015). Again, it is noted that when the non-Newtonian stress is equal to zero and the cosine functions (slope corrections) are removed, the above DWE equation reduce to the clear-water equations utilized in HEC-RAS.

For many of the types of non-Newtonian flows the DWE may not be applicable and in fact most 2D non-Newtonian models are not based on the DWE. However, there are some types of applications where the DWE model is useful and there are some examples in literature such as Lin et al. (2011).