A robust model for rapidly varying flows over movable bottom with suspended and bedload transport: Modelling and numerical approach

Abstract

We propose a coupled model for suspended and bedload sediment transport in the shallow water framework. The model is deduced under hydrostatic pressure assumptions and will not assume any Bossinesq hypothesis. The numerical resolution is carried out in a segregated way. First the underlying system of conservation laws is solved by using a first order path-conservative Riemann solver. Then, the source terms corresponding with the erosion and depositions rates are approximated in a semi-implicit way. The final scheme preserves the positivity of the density. Several numerical experiments were carried out in order to validate the model and the numerical scheme. The results obtained are in good agreement with the experimental data.

Shallow Water Moment Models for Bedload Transport Problems

“…Only bedload transport is assumed here for the sake of simplicity, since the goal is to improve the vertical description of the velocity, leading to an improvement of the bedload sediment transport. However, erosion and deposition effects may be relevant in some situations as it was shown in [24]. Many empirical formulae may be found for erosion and deposition fluxes, which depend in different parameters to be calibrated.…”

“…We consider here a method in the framework of Polynomial Viscosity Methods (PVM) introduced in [9], where the numerical diffusion D i+1/2 is defined in terms of a polynomial evaluation on the Roe matrix of the full non-conservative system. In particular, the IFCP method [20], that is used in [24] to simulate sediment transport problems with erosion-deposition effects, will be used here as well. In [24], the ideas introduced in [12] were used to approximate the eigenvalues of the system.…”

“…In particular, the IFCP method [20], that is used in [24] to simulate sediment transport problems with erosion-deposition effects, will be used here as well. In [24], the ideas introduced in [12] were used to approximate the eigenvalues of the system. This is crucial, namely in the case of a strong fluid-sediment interaction.…”

“…Now, the eigenvalues are the values λ * verifying f (λ * ) = d(λ * ). To find these values, we use the approach in [24]. One iteration of the Newton's method is performed, taking as initial seeds the roots of f (λ), i.e.,…”

“…Moreover, the explicit expressions of the system eigenvalues are known (assuming that the eigenvalues for Exner's system are calculated, for instance by Cardano-Vieta's formula). Another contribution of this paper is a modification and extension of the IFCP method proposed in [24]. The original one fails in the case F r ≈ 1, so that the modification proposed here ensures the stability of the numerical scheme in such situations.…”