Lagrangian particle path formulation of multilayer shallow-water flows dynamically coupled to vessel motion

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 shallowwater 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.

Wetting and drying of two-layer shallow-water flow in a moving vessel via semi-Lagrangian simulation

“…We considered two examples based upon water/air (ρ 2 /ρ 1 ≈ 10 −3 ) experimental setups: a low-fill example where the upper rigid lid remains dry, and a high-fill example where the upper lid undergoes wetting/drying over each period of the forcing. The results of the low-fill example were compared to the Lagrangian Particle Path (LPP) code of [3] and the experiments of [44]. The low and moderate amplitude simulations were found to be in excellent agreement with the LPP simulations: while the larger amplitude simulations were in good agreement with the experiments.…”

“…The first example in §4.1 considers a case with no wetting/drying of the upper rigid lid, and hence provides a good test example for the semi-Lagrangian scheme, for which the results can be compared to the experiments of [44]. As there is no wetting/drying it also means that results can be directly compared to those of the two-layer Lagrangian Particle Path (LPP) code presented in [3]. The second example in §4.2 considers a case where there is wetting/drying of the upper rigid lid, and results can be compared to the experimental results of [45].…”

“…However these methods are predominantly Eulerian, and a second feature of interest in this study, after computational speed, is tracking of individual parcels of fluid, suggesting a Lagrangian Particle Path (LPP) viewpoint. The authors have had some success with LPP simulation of two-layer shallow water hydrodynamics [3]. The LPP scheme has excellent energy conservation properties, but the weaknesses…”

“…These methods are quite versatile and can easily be extended to incorporate bottom topography [27] or multiple stratified layers [28]. Another relevant study by [3] developed a numerical scheme based upon the Lagrangian Particle Path (LPP) approach, in which a mapping, determined as part of the solution, coupled the label spaces in each layer, allowing the Eulerian rigid lid constraint 1) to be satisfied at each streamwise position. Here h 1 and h 2 are the layer thicknesses in the lower and upper layers respectively, and d is the fixed height of the vessel.…”

Coupled shallow-water fluid sloshing in an upright annular vessel

“…Such schemes conserve the symplectic structure of the system and also preserves the energy partition between the fluid and the vessel, preventing a slow unphysical leaking of energy from the vessel to the fluid over time. This numerical approach has been applied successfully to related coupled sloshing problems [24][25][26].…”