A symplectic integrator for dynamic coupling between nonlinear vessel motion with variable cross-section and bottom topography and interior shallow-water sloshing

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.The coupled motion between shallow-water sloshing in a moving vessel with variable cross-section and bottom topography, and the vessel dynamics is considered, with the vessel dynamics restricted to horizontal motion governed by a nonlinear spring. The coupled fluid and vessel equations in Eulerian coordinates are transformed to the Lagrangian particle path setting which leads to a formulation with nice properties for numerical simulation. In the Lagrangian representation, a simple and fast numerical algorithm based on the Störmer-Verlet method, is implemented. The numerical scheme conserves the total energy in the system, as well as giving the partition of energy between the fluid and vessel. Numerical simulations of the coupled nonlinear dynamics are presented.The author is grateful to Thomas J. Bridges and Matthew R. Turner for useful discussions. The research reported in this paper is supported by the EPSRC under Grant number EP/K008188/1. Due to confidentiality agreements with research collaborators, supporting data can only be made available to bona fide researchers subject to a non-disclosure agreement. Details of the data and how to request access are available from the University of Surrey publications repository: [email protected]

A coupled variational principle for 2D interactions between water waves and a rigid body containing fluid

This is the author accepted manuscript. The final version is available from CUP via the DOI in this record.New variational principles are given for the two-dimensional interactions between gravity-driven water waves and a rotating and translating rectangular vessel dynamically coupled to its interior potential flow with uniform vorticity. The complete set of equations of motion for the exterior water waves, the exact nonlinear hydrodynamic equations of motion for the vessel in the roll/pitch, sway/surge and heave directions, and also the full set of equations of motion for the interior fluid of the vessel, relative to the body coordinate system attached to the rotating–translating vessel, are derived from two Lagrangian functionals.This work is partially supported by the EPSRC under grant no. EP/K008188/1. The author is grateful to the University of Surrey for the award of a two-year Visiting Researcher Fellowship between 2010 and 2013

Variational generalization of the Green–Naghdi and Whitham equations for fluid sloshing in three-dimensional rotating and translating coordinates

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this recordThis paper derives an averaged Lagrangian functional for dynamic coupling between rigid-body motion and its interior shallow-water sloshing in three-dimensional rotating and translating coordinates; with a time-dependent rotation vector. A new set of variational shallow-water equations (SWEs) and generalized Green–Naghdi equations for the interior fluid sloshing with 3–D rotation vector and translations, and also the equations of motion for the linear momentum and angular momentum of the rigid-body containing shallow water, are derived from the averaged Lagrangian functional, which describes a columnar motion, by using Hamilton’s principle and the Euler–Poincare variational ´ framework. The generalized Green–Naghdi equations have a form of potential vorticity (PV) conservation, which can be obtained from the particle-relabeling symmetry, and is a combination of the PV derived by Miles and Salmon (1985) and the PV derived by Dellar & Salmon (2005) for geophysical fluid dynamics problems, where the rotation vector varies spatially. By applying the assumption of zero-potential-vorticity flow to the averaged Lagrangian functional, a new set of Boussinesq-like evolution equations are derived, which are a generalization of the Whitham equations for fluid sloshing in three-dimensional rotating and translating coordinates. Moreover, the new variational principles are appended to Luke’s variational principle to present a unified variational framework for the hydrodynamic problem of interactions between gravity-driven potential-flow water waves and a freely floating rigid-body, dynamically coupled to its interior weakly dispersive nonlinear shallow-water sloshing in three dimensions

Shallow-water sloshing in a moving vessel with variable cross-section and wetting-drying using an extension of George's well-balanced finite volume solver

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.A class of augmented approximate Riemann solvers due to George (2008) is extended to solve the shallow-water equations in a moving vessel with variable bottom topography and variable cross-section with wetting and drying. A class of Roe-type upwind solvers for the system of balance laws is derived which respects the steady-state solutions. The numerical solutions of the new adapted augmented f-wave solvers are validated against the Roe-type solvers. The theory is extended to solve the shallow-water flows in moving vessels with arbitrary cross-section with influx-efflux boundary conditions motivated by the shallow-water sloshing in the ocean wave energy converter (WEC) proposed by Offshore Wave Energy Ltd. (OWEL). A fractional step approach is used to handle the time-dependent forcing functions. The numerical solutions are compared to an extended new Roe-type solver for the system of balance laws with a time-dependent source function. The shallow-water sloshing finite volume solver can be coupled to a Runge-Kutta integrator for the vessel motion.The research reported in this paper is supported by the EPSRC under Grant number EP/K008188/1. Due to confidentiality agreements with research collaborators, supporting data can only be made available to bona fide researchers subject to a non-disclosure agreement. Details of the data and how to request access are available from the University of Surrey publications repository: [email protected]. The authors are grateful to both referees for their valuable comments

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

This is the author accepted manuscript. The final version is available from Springer Verlag via the DOI in this record.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.This work is supported by the EPSRC under Grant number EP/K008188/1

Adaptation of f-wave finite volume methods to the two-layer shallow-water equations in a moving vessel with a rigid-lid

This is the author accepted manuscript. The final version is available from the publisher via the DOI in this recordA numerical method is proposed to solve the two-layer inviscid, incompressible and immiscible 1D shallow-water equations in a moving vessel with a rigid-lid with different boundary conditions based on the high-resolution f-wave finite volume methods due to Bale et al. (2002). The method splits the jump in the fluxes and source terms including the pressure gradient at the rigid-lid into waves propagating away from each grid cell interface. For the influx-efflux boundary conditions the time dependent source terms are handled via a fractional step approach. In the linear case the numerical solutions are validated by comparison with the exact analytical solutions. Numerical solutions presented for the nonlinear case include shallow-water sloshing waves due to prescribed surge motion of the vessel.The research reported in this paper is supported by the Engineering and Physical Sciences Research Council Grant EP/K008188/1. Due to confidentiality agreements with research collaborators, supporting data can only be made available to bona fide researchers subject to a non-disclosure agreement. Details of the data and how to request access are available from the University of Surrey publications repository: [email protected]

A variational principle for fluid sloshing with vorticity, dynamically coupled to vessel motion

A variational principle is derived for two-dimensional incompressible rotational fluid flow with a free surface in a moving vessel when both the vessel and fluid motion are to be determined. The fluid is represented by a stream function and the vessel motion is represented by a path in the planar Euclidean group. Novelties in the formulation include how the pressure boundary condition is treated, the introduction of a stream function into the Euler-Poincar\'e variations, the derivation of free surface variations, and how the equations for the vessel path in the Euclidean group, coupled to the fluid motion, are generated automatically.Comment: 19 pages, 3 figure