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 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 model.
Highlights
Hydrostatic models such as the nonlinear shallow water equations [26] or the hydrostatic Navier-Stokes equations [3] are based on the assumption q ≡ 0
The second goal of this paper is to extend the aforementioned numerical method designed for L D N H2(1) to its multilayer counterpart L D N H2(L) as derived in [35]
The output clearly shows better performance for Model L D N H2, which is expected according to the linear dispersion relation of the continuum models
Summary
Hydrostatic models such as the nonlinear shallow water equations [26] or the hydrostatic Navier-Stokes equations [3] are based on the assumption q ≡ 0. In [21] a semi-implicit finite difference model for non-hydrostatic free-surface flows is presented based on a vertical discretisation of the 3D free-surface Navier-Stokes equations. A more recent work on the subject is [22], where semi-implicit methods are extended to complex free-surface flows that are governed by the full incompressible Navier-Stokes equations and are delimited by solid boundaries and arbitrarily shaped free-surfaces. These approaches are comparable to the multilayer or layer-averaged framework, see for example [5,50,60]. In [6] a numerical study of the linear dispersive celerity shows convergence of the multilayer model
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
