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)

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

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]

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

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

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

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

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

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

When the implicit weighting factor $\begin{array}{l}\theta\end{array}$ 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.