Transport Equation

A brief description of the Finite-Volume discretization of the total-load transport equation is provided here without any derivation or details. For addditional information including advection schemes, gradient operators, etc., the reader is referred to the HEC-RAS 2D Transport Module Technical Reference document (Sánchez et al. 2020). The 2D Transport Module solves generic Advection-Diffusion equations using explicit and implicit Finite-Volume methods. The final form of the discretized total-load advection-diffusion equation is given by

1)

$\begin{array}{l}\displaystyle \frac{\Omega _{P}^{n+1}C_{tk,P}^{n+1}}{\Delta t\beta _{tk,P}^{n+1}}=\frac{\Omega _{P}^{n}C_{tk,P}^{n}}{\Delta t\beta _{tk,P}^{n}}+\sum _{f}\left[\frac{A_{f}\varepsilon _{tk,f}}{\delta _{PN}}\left(C_{tk,N}^{n+\theta }-C_{tk,P}^{n+\theta }\right)-F_{f}C_{tk,f}^{n+\theta }\right]+\left(E_{tk}^{HF}-D_{tk}^{HF}+S_{tk}\right)_{P}A_{P}^{W}\end{array}$

where

$\begin{array}{l}\Omega\end{array}$ = cell volume

$\begin{array}{l}P\end{array}$ = subscript indicating cell

$\begin{array}{l}f\end{array}$ = subscript indicating face between cells P and N

$\begin{array}{l}N\end{array}$ = subscript indicating neighboring cell to P and sharing face f

$\begin{array}{l}n\end{array}$ = superscript indicating time step

$\begin{array}{l}C_{tk}\end{array}$ = total-load sediment concentration of the k^th grain class [M/L³]

$\begin{array}{l}β_{tk}\end{array}$ = total-load correction factor for the k^th grain class

$\begin{array}{l}ε_{tk}\end{array}$ = total-load diffusion coefficient corresponding to the k^th grain class

$\begin{array}{l}A_f\end{array}$ = face vertical area [L²/T]

$\begin{array}{l}F_{f}\end{array}$ = face-normal water flow [L³/T]

$\begin{array}{l}{E_{tk}}^{HF}\end{array}$ = total-load erosion rate in hydraulic flow [M/T/L²]

$\begin{array}{l}{D_{tk}}^{HF}\end{array}$ = total-load deposition rate in hydraulic flow [M/T/L²]

$\begin{array}{l}S_{tk}\end{array}$ = total-load source/sink term [M/T/L²]

$\begin{array}{l}A_P^W\end{array}$ = cell wetted horizontal area [L²]

$\begin{array}{l}δ_{PN}\end{array}$ = distance between cell points N and P [L]

$\begin{array}{l}θ\end{array}$ = implicit weighting factor [-]

$\begin{array}{l}n + θ\end{array}$ = superscript representing the temporal weighting $\begin{array}{l}X^{n + θ}=θX^{n + 1} + (1-θ)X^n\end{array}$ (Generalized Euler scheme)

When the implicit weighting factor is equal to 1, the scheme reduces to the first-order fully implicit Backward Euler scheme. When the implicit weighting factor is equal to 0.5, the scheme is the second-order Crank-Nicholson scheme. An implicit discretization is utilized for robustness. However, future versions will have the option to use an explicit scheme.