Numerical Simulations of a Dispersive Model Approximating Free-Surface Euler Equations

Abstract

In some configurations, dispersion effects must be taken into account to improve the simulation of complex fluid flows. A family of free-surface dispersive models has been derived in Fernández-Nieto et al. (Commun Math Sci 16(05):1169–1202, 2018). The hierarchy of models is based on a Galerkin approach and parameterised by the number of discrete layers along the vertical axis. In this paper we propose some numerical schemes designed for these models in a 1D open channel. The cornerstone of this family of models is the Serre – Green-Naghdi model which has been extensively studied in the literature from both theoretical and numerical points of view. More precisely, the goal is to propose a numerical method for the $$LDNH_2$$ L D N H 2 model that is based on a projection method extended from the one-layer case to any number of layers. To do so, the one-layer case is addressed by means of a projection-correction method applied to a non-standard differential operator. A special attention is paid to boundary conditions. This case is extended to several layers thanks to an original relabelling of the unknowns. In the numerical tests we show the convergence of the method and its accuracy compared to the $$LDNH_0$$ L D N H 0 model.

Layer-averaged approximation of Navier–Stokes system with complex rheologies

“…The models presented in this work have to be analyzed before discretizing them, for instance following the results introduced in [14] for a multilayer model with a simpler definition of the viscous terms. In the future, it would be also interesting to develop efficient numerical schemes, following the strategies introduced in [17], as well as the extension of this framework to approximate 3D Navier-Stokes systems and more general stress tensor definitions, such as the case of turbulent models.…”

“…Remark 2. Concerning the numerical approximation of non-hydrostatic layer-averaged models, in [17] authors introduced a specific method for LDNH models (see [23]), which consider a layerwise constant horizontal velocity to approximate Euler system. It consists of a projection method based on a duality relation, which is similar to (38).…”

“…The design of an efficient numerical scheme for the full non-hydrostatic model is a key but hard problem, which in particular is necessary to make these models usable from the practical point of view. Then, due to the difficulty of such schemes, they deserve special attention (see [17]) and it is out of the scope of this work.…”