Dynamic coupling between shallow-water sloshing and horizontal vehicle motion

Abstract

The coupled motion between shallow-water sloshing in a moving vehicle and the vehicle dynamics is considered, with the vehicle dynamics restricted to horizontal motion. The paper is motivated by Cooker's experiments and theory for water waves in a suspended container. A new derivation of the coupled problem in the Eulerian fluid representation is given. However, it is found that transformation to a Lagrangian representation leads to a formulation which has nice properties for numerical simulation. In the Lagrangian representation, a simple and fast numerical algorithm with excellent energy conservation over long times, based on the Störmer-Verlet method, is implemented. Numerical simulations of the coupled dynamics in both the linear and non-linear case are presented.

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

“…ρ [1,3] , where H N is the discretized form of the Hamiltonian (8.14). The slight increase in the error over the duration of the simulation is believed to be due to the Kelvin-Helmholtz instability, but this error growth is not large enough to affect the result and hence is tolerable.…”

“…writing q = q +δq with δq(τ 1 ) = δq(τ 2 ) = 0) leads to (2.17) and (2.18) respectively. Note that in the case ρ 2 = 0 (with ν 2 = 0), (2.19) reduces to the one-layer coupled Lagrangian given in [1], i.e. in this case the fluid does not feel the effect of the rigid lid.…”

“…This energy conservation property is particularly important when the fluid motion is inside a vessel, and it is coupled to the vessel motion, as then it is of interest to accurately capture the energy partition between vessel and fluid. This strategy was used for simulating the dynamic coupling with a single-layer fluid in [1,2]. 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.…”

“…Therefore this approach is not able to model the interior workings of the OWEL WEC, where the air/water density ratio is ρ 2 /ρ 1 ≈ 10 −3 . The current paper formulates a simple numerical approach based upon a discretization of the LPP scheme, generalizing the numerics of [1] to two layers with nonlinear vessel motion. The LPP approach allows two-layer simulations with ρ 2 /ρ 1 = 10 −3 to be produced.…”

Instability of sloshing motion in a vessel undergoing pivoted oscillations

“…Cooker (1994) compared the results of his experiments to a theoretical model in which the fluid was assumed to be shallow, and the vessel motion was modelled using a linear pendulum model. Various extensions of this problem have been considered such as introducing a finite depth fluid (Yu, 2010), having a nonlinear shallow water fluid (Alemi Ardakani and Bridges, 2010) and including the fully nonlinear vessel motion (Alemi Ardakani et al, 2012a). Alemi Ardakani et al (2012a,b) also highlighted the existence of an internal 1 : 1 resonance in the Cooker experiment, where the symmetric sloshing modes are dynamically coupled to the antisymmetric sloshing modes, and hence the vessel motion.…”