Abstract
The coupled motion—between multiple inviscid, incompressible, immiscible fluid layers in a rectangular vessel with a rigid lid and the vessel dynamics—is considered. The fluid layers are assumed to be thin and the shallow-water assumption is applied. The governing form of the Lagrangian functional in the Lagrangian particle path (LPP) framework is derived for an arbitrary number of layers, while the corresponding Hamiltonian is explicitly derived in the case of two- and three-layer fluids. The Hamiltonian formulation has nice properties for numerical simulations, and a fast, effective and symplectic numerical scheme is presented in the two- and three-layer cases, based upon the implicit-midpoint rule. Results of the simulations are compared with linear solutions and with the existing results of Alemi Ardakani et al. (J Fluid Struct 59:432–460, 2015) which were obtained using a finite volume approach in the Eulerian representation. The latter results are extended to non-Boussinesq regimes. The advantages and limitations of the LPP formulation and variational discretization are highlighted.
Highlights
The Lagrangian and Eulerian descriptions of fluid motion are two viewpoints for representing fluid motion, with the Eulerian description being the most widely used in theoretical fluid dynamics
This paper documents the Lagrangian variational formulation of the Lagrangian particle path (LPP) representation of multilayer shallowwater sloshing, coupled to horizontal vessel motion governed by a nonlinear spring
The Lagrangian variational formulation was transformed to a Hamiltonian formulation which has nice properties for numerical simulation
Summary
The Lagrangian and Eulerian descriptions of fluid motion are two viewpoints for representing fluid motion, with the Eulerian description being the most widely used in theoretical fluid dynamics. The aim of this paper is to derive the LPP formulation to shallow water flows, with multiple layers of differing density, in a vessel with dynamic coupling, and use it as a basis for a variational formulation and numerical scheme This generalisation is straightforward in principle, the variational formulation has complex subtleties due to the integration over different label spaces. The studies most relevant to the one in this paper examine the two-layer sloshing problem using a numerical scheme based upon a class of high resolution wave-propagating finite volume methods known as f-wave methods for both the forced [12] and the coupled problem [13] This f-wave approach is very effective and can be readily be extended to multilayer systems [14] and systems with bottom topography [15], but [13] find the scheme is limited to layer density ratios of ρ2/ρ1 0.7, where ρ1 and ρ2 are the fluid densities in the lower and upper layers, respectively, due to a linear growth in the system constraint error.
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