A two‐dimensional finite volume morphodynamic model on unstructured triangular grids

Abstract

SUMMARYWe discuss the application of a finite volume method to morphodynamic models on unstructured triangular meshes. The model is based on coupling the shallow water equations for the hydrodynamics with a sediment transport equation for the morphodynamics. The finite volume method is formulated for the quasi-steady approach and the coupled approach. In the first approach, the steady hydrodynamic state is calculated first and the corresponding water velocity is used in the sediment transport equation to be solved subsequently. The second approach solves the coupled hydrodynamics and sediment transport system within the same time step. The gradient fluxes are discretized using a modified Roe's scheme incorporating the sign of the Jacobian matrix in the morphodynamic system. A well-balanced discretization is used for the treatment of source terms. We also describe an adaptive procedure in the finite volume method by monitoring the bed-load in the computational domain during its transport process. The method uses unstructured meshes, incorporates upwinded numerical fluxes and slope limiters to provide sharp resolution of steep bed gradients that may form in the approximate solution. Numerical results are shown for a test problem in the evolution of an initially hump-shaped bed in a squared channel. For the considered morphodynamical regimes, the obtained results point out that the coupled approach performs better than the quasi-steady approach only when the bed-load rapidly interacts with the hydrodynamics.

An unstructured finite‐volume method for coupled models of suspended sediment and bed load transport in shallow‐water flows

“…Such practical coupled hydrodynamical and morphodynamical problems are not trivial to simulate since the geometry can be complex and the topography irregular. It should be pointed out that other mathematical models for morphodynamical systems have also been studied in [31,23,5] among others. In these systems, the shallow water equations have been coupled to an Exner-type equation for the bed-load without accounting for suspended sediments as it is shown in the current study.…”

A non-homogeneous Riemann solver for shallow water equations in porous media

“…When slopes are based on centroid values, the three points used to estimate the slopes are the centroid and the two neighboring centroids inside the computational domain. For further details on the implementation of boundary conditions for finite volume methods we refer to [2,5] among others.…”

“…The sign matrix of the Jacobian matrix is used in the reconstruction of the numerical fluxes. Most of these techniques have been recently investigated in [2][3][4][5]18] for solving the canonical shallow water models without accounting for porosity variation. The current study presents an extension of this method to transient flows involving porosity variation in the water flows.…”

Coupled flood and sediment transport modelling with adaptive mesh refinement

“…In this regard, it is noted that the extra data storage or computation may be highly costly as the array is huge. Benkhaldoun et al [18,24] employed the normalized pollution concentration or bed load transport rate as an adaptation indicator. In this method, if the normalized value exceeds the threshold, the cell is marked for refinement or coarsening.…”

“…To date, however, most studies using AMR have been limited to cases either without involving sediment transport at all [4, [9][10][11]13,17] or featuring the assumption of sediment transport capacity, which is not normally justified. Benkhaldoun et al [18] have developed a model based on an unstructured AMR for modelling of flood flows over mobile beds. Compared with the structured AMR, the generation of the unstructured AMR is more complex and the discretization of the governing equations becomes much more cumbersome.…”