A new class of sediment transport models: Derivation and numerical solutions

Abstract

Abstract In coastal or estuarine environments the water fluctuations withgreater length of correlations are more intense and modify the meanmotion. The literature does not provide an averaged sedimenttransport model (STM) for coastal zones that accounts for both meanand fluctuating motions over an abrupt mobile bottom. Models basedon water are commonly used to describe sediment transport, but theyare not applicable when the flow becomes turbulent. These modelsonly consider the mean motion. To account the fluctuating motion ina STM, we assume $ |u -\overline{u}|^k \leq \tau^k h, \;\; k>0$ ($h= \mathcal{O}(\varepsilon)$ is the water depth, $\lambda = const$).However, distortion no exist always since during the dam break overerodible bed, the water near the bed can become dense thus weaklyturbulent. To always ensure the turbulence existence, we assume that$\tau = \mathcal{O}(\gamma^\alpha) \;\; \varepsilon^2 \ll \gamma \ll1$ and $\rho = \mathcal{O}(\gamma)$ with $\gamma = const$.Additionally we assume a weakly concentrated approximation ($d_{50}= \mathcal{O}(\varepsilon^\alpha)$, $\delta = \mathcal{O}(\epsilon),1< \alpha <3$) to derive the model. The model obtained in thisstudy builds upon previous work by authors in 2007, 2012, and 2018,while also improving upon more recent models developed in 2022 and2023. The hyperbolic STM is subject to various complexities arisingfrom turbulence and several nonlinear coupled terms. Furthermore,solving this problem still poses a computational challenge in termsof accuracy, convergence, robustness, and efficiency. To addressthis challenge, we propose a new path-conservative method called $\text{HLL}_{**}$, which we combine with a modified Averaging method.The Essentially Non-Oscillatory (AENO) nonlinear reconstructionmethod is used to approximate the model. This method is noteworthybecause it generalizes several HLL-based schemes developed in recentyears. Numerical test cases have been performed.