An unstructured finite-volume semi-coupled projection model for bed load sediment transport in shallow-water flows

Abstract

The aim of this work is to develop an original semi-coupled approach for solving the Saint-Venant and Exner equation system. The shallow-water equations are solved first to calculate the full evolution of flow fields, and then the Exner equation is solved to model the morphodynamic response. The numerical method is based on a projection method, which consists in combining the momentum and continuity equations in order to establish a Poisson-type equation for water surface levels. The present formulation is identified as a semi-coupled approach because the projection method incorporates the change in the bed evolution in its approximation. A second-order numerical scheme has been proposed using an unstructured finite-volume technique and an implicit time integration method. Several benchmarks are used to demonstrate the capabilities, accuracy and performance of the model.

Keywords:

• Exner equation
• projection method
• Saint-Venant equations
• second-order accuracy
• semi-coupled approach
• unstructured finite-volume method

Notation

$A_g$	=	constant coefficient (s${}^2$ m${}^{-1}$)
$C_h$	=	Chézy coefficient (–)
$d_s$	=	sediment grain size (m)
$\mathit{f}$	=	vector of source terms (–)
f	=	Coriolis parameter (–)
F	=	bed friction force (N)
g	=	gravity acceleration (m s${}^{-2}$)
h	=	total water depth (m)
H	=	depth of reference (m)
$\mathfrak{n}$	=	Manning's coefficient (–)
$\mathit{q}$	=	vector of unit discharge (m${}^2$ s${}^{-1}$)
$q_x,q_y$	=	components of $\mathit{b}$ in x and y directions (–)
${\mathit{q}}_{\mathit{b}}$	=	vector of sediment transport discharge (m${}^2$ s${}^{-1}$)
$q_{bx},q_{by}$	=	components of $\mathit{b}$ in x and y directions (–)
t	=	time (s)
$\mathit{u}$	=	vector of depth-averaged velocity (–)
u,v	=	components of $\mathit{u}$ in x and y directions (–)
$u^{\ast }$	=	friction velocity (m $s^{-1}$)
$z_b$	=	bed thickness of non-erodible bottom (m)
$z_s$	=	water surface (m)
δ	=	time difference of water surface elevation (m)
ν	=	horizontal dispersion coefficient (m${}^2$ s${}^{-1}$)
ρ	=	water density (kg m${}^{-3}$)
${\rho }_s$	=	sediment density (kg m${}^{-3}$)
σ	=	porosity of the bed material (–)
${\tau }_b$	=	bed friction force (N)
${\tau }_b^{\ast }$	=	Shields parameter (–)
${\tau }_b^c$	=	critical shear stress (–)
ξ	=	model porosity (–)

Additional information

Funding

This work was partially supported by Mexican Council of Science and Technology, CONACYT [grant CB number 256252].