###### Senior Lecturer in Mathematics

Department of Mathematics

University of Exeter

Penryn Campus

Cornwall TR10 9FE

ENGLAND

Background and History

The OWEL Wave Energy Converter (WEC) comprises a floating horizontal duct, with angled roof and bottom plates, supported by buoyancy tanks. The duct in each unit is open at one end, and the mooring system of the platform takes account of wind and tides to ensure that this open end is presented to the incoming waves. The waves repeatedly compress air trapped within the ducts which is directed to drive an air turbine that will generate electricity. At the end of each duct, behind the air-take-off, is a baffle system which disperses any remnant energy in spent waves so that they do not reflect back along the duct to interfere with following waves. This concept was originally proposed by Professor John Kemp. It is the basis of the international patents established and appropriate regional patents have now been granted and are held by OWEL. The figure on the right shows the completed design of the Marine Demonstrator that is being sent out to various shipyards and fabricators to tender. In the OWEL-Surrey project, EPSRC funded project (EP/K008188/1), mathematical models are being developed for simulating the dynamics, interior sloshing, power take off, and control of the OWEL WEC.

Leybourne M, Batten WMJ, Bahaj AS, Minns N, O'Nians J (2014) Preliminary design of the OWEL wave energy converter pre-commercial demonstrator, Renewable Energy, Volume 61, Pages 51-56.

Feasibility of a gravity current model for the OWEL WEC

The OWEL wave energy converter is a floating rectangular device open at one end to capture the incoming wave field. The trapped waves in the duct hit the upper rigid lid and create a seal resulting in a moving trapped pocket of air ahead of the wave front which drives the power take off. Understanding the dynamics of the two phase flow created by the wave input is key to the energy optimisation of the power take off. A photo of a model OWEL vessel is shown on the right at the point where the wave has been trapped and begins to drive the pocket of air along the upper lid of the vessel with air takeoff at the upper right edge. The interior two-phase flow field in OWEL is very similar to a gravity-current configuration. The purpose of this document is to review gravity current theory and propose a theoretical and experimental strategy for adapting and modifying existing gravity current theory to the OWEL setting.

H. Alemi Ardakani. A gravity-current model for the OWEL WEC: literature review and feasibility study, Internal Report, OWEL Project. (2015). [PDF]

Variational principles for interactions between water waves and a floating rigid-body containing fluid

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.

Alemi Ardakani H. (2017) A coupled variational principle for 2D interactions between water waves and a rigid body containing fluid, Journal of Fluid Mechanics, volume 827, DOI:10.1017/jfm.2017.517. [PDF]

Symplectic integration and fluid-structure interaction

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 Stormer-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.

H. Alemi Ardakani. A symplectic integrator for dynamic coupling between nonlinear vessel motion with variable cross-section and bottom topography and interior shallow-water sloshing, J. Fluids & Struct. 65, 30-43 (2016) [PDF]

F-wave finite volume methods for the reduced two-layer shallow-water equations

A numerical method is proposed based on the high-resolution f-wave finite volume methods due to Bale et al. (2002) for the reduced two-layer inviscid, incompressible and immiscible shallow interfacial sloshing equations with a rigid-lid in the non-Boussinesq limit due to Boonkasame and Milewski (2011) The horizontal surge forcing function is added to the fluid equations. Numerical simulations are presented for the sloshing waves in a stationary vessel and also for the sloshing equations in a vessel under a prescribed surge forcing function. The proposed non-Boussinesq two-fluid finite volume solver is coupled to a Runge–Kutta solver for the vessel motion which is free to undergo horizontal motion governed by a nonlinear spring. The coupled numerical solutions with simulations near the internal 1:1resonance are presented. Of particular interest is the partition of energy between the vessel and fluid motion. The new solvers are fast, stable, to machine accuracy, and can cover a wide range of parameters. The motivation is modeling of ocean wave energy harvesters.

Alemi Ardakani H. (2016) Adaptation of f-wave finite volume methods to the Boonkasame–Milewski non-Boussinesq two-layer shallow interfacial sloshing equations coupled to the vessel motion, European Journal of Mechanics, B/Fluids, volume 60, pages 33-47, DOI:10.1016/j.euromechflu.2016.04.012.

Simulation of the nonlinear pendulum-slosh problem

The exact equations for the nonlinear pendulum-slosh system are derived with the fluid motion governed by the Euler equations relative to the moving frame of the vessel, and the vessel motion governed by a modified forced pendulum equation. The equations of motion for the fluid are solved numerically via a time-dependent conformal mapping, and the coupled system is integrated in time with a fourth-order Runge-Kutta method. The starting point for the simulations is the linear neutral stability contour discovered by Turner, Alemi Ardakani & Bridges (2014, J. Fluid Struct. 52, 166-180). Near the contour the nonlinear results confirm the instability boundary, and far from the neutral curve (parameterised by longer pole lengths) nonlinearity is found to significantly alter the vessel response.

Turner MR, Bridges TJ, Alemi Ardakani H. (2015) The pendulum-slosh problem: Simulation using a time-dependent conformal mapping, Journalof Fluids and Structures, volume 59, pages 202-223, DOI:10.1016/j.jfluidstructs.2015.09.007.

Shallow water sloshing with wetting and drying

A class of augmented approximate Riemann solvers due to George (J. Comp. Phys. 227 2008: 3089-3113) 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 are derived which respect 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 is coupled to a Runge-Kutta integrator for the vessel motion.

Alemi Ardakani H, Bridges TJ, Turner MR. (2016) 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, Journal of Computational Physics, volume 314, pages 590-617, DOI:10.1016/j.jcp.2016.03.037.

D.L. George. Augmented Riemann solvers for the shallow water equations over variable topography with steady states and inundation, J. Comp. Phys. 227, 3089-3113 (2008) [PDF]

Dynamic coupling of vessel motion with a two-layer fluid

In this paper, the shallow water equations from the sloshing project are extended to two-layers while retaining coupling with the simplest horizontal vessel motion. Numerical and analytical results are presented for fluid sloshing, of the two-layer inviscid, incompressible and immiscible fluid with thin layers and a rigid lid, coupled to the vessel motion. Exact analytical results are obtained for the linear problem, giving the natural frequencies and the resonance structure, particularly between fluid and vessel. A numerical method for the linear and nonlinear equations is developed based on the high-resolution f-wave-propagation finite volume methods due to Bale, LeVeque, Mitran and Rossmanith (2002), adapted to include the pressure gradient at the rigid-lid, and coupled to a Runge-Kutta solver for the vessel motion. The numerical simulations in the linear limit are compared with the exact analytical solutions. The coupled nonlinear numerical solutions with simulations near the internal $1:1$ resonance are presented. Of particular interest is the partition of energy between vessel and fluid motion. A preprint on this work is available below.

Alemi Ardakani H, Bridges TJ, Turner MR. (2015) Dynamic coupling between horizontal vessel motion and two-layer shallow-water sloshing, Journal of Fluids and Structures, volume 59, pages 432-460, DOI:10.1016/j.jfluidstructs.2015.10.002.

Simulating fluid sloshing coupled to a vessel with baffles

The coupled motion between shallow water sloshing in a moving vessel with baffles and the vessel dynamics is considered. Here the vessel dynamics is restricted to horizontal motion such as in tuned liquid dampers. It was shown previously that partitioning a moving vessel into n separate compartments leads to an interesting dynamical behaviour of the system. Also, under particular input parameter values an internal (n+1)-fold 1:...:1 resonance can be generated, where the frequency of the sloshing fluid in each compartment is equal, and equal to the frequency of the vessel itself. Here the form of the sloshing eigenmodes at this resonance are derived in the shallow-water limit. Using the Lagrangian formulation of the problem, an efficient numerical algorithm is implemented to solve the fully nonlinear system of equations based on the implicit midpoint rule. This algorithm is simple, fast and maintains the energy partition between the vessel and the fluid over long times. In this work numerical results are presented for dynamical vessel/sloshing motion attached to a nonlinear spring.

Alemi Ardakani H, Turner MR. (2016) Numerical simulations of dynamic coupling between shallow-water sloshing and horizontal vessel motion with baffles, Fluid Dynamics Research, volume 48, no. 3, pages 035504-035504, DOI:10.1088/0169-5983/48/3/035504.

F-wave finite-volume methods for OWEL simulation

In the sloshing project, forced sloshing was simulated using an implicit finite-difference method and the coupled problem was simulated using a variational geometric integrator. For the OWEL project a key feature is wetting and drying which requires a re-think of the appropriate numerical method. In this paper a numerical method is proposed to solve the two-layer inviscid, incompressible and immiscible shallow-water equations in a moving vessel with a rigid-lid in one dimension with different boundary conditions based on the high-resolution f-wave finite volume methods due to Bale, LeVeque, Mitran and Rossmanith (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. A preprint is available for downloading below.

Alemi Ardakani H, Bridges TJ, Turner MR. (2016) Adaptation of f-wave finite volume methods to the two-layer shallow-water equations in a moving vessel with a rigid-lid, Journal of Computational and Applied Mathematics, volume 296, pages 462-479, DOI:10.1016/j.cam.2015.09.026.

Instability of sloshing motion in a vessel undergoing pivoted oscillations

Suspending a rectangular vessel partially filled with an inviscid fluid from a single rigid pivoting rod produces an interesting physical model for investigating the dynamic coupling between the fluid and vessel motion. The fluid motion is governed by the Euler equations relative to the moving frame of the vessel, and the vessel motion is given by a modified forced pendulum equation. The fully nonlinear, two-dimensional, equations of motion are derived and linearised for small-amplitude vessel and free-surface motions, and the natural frequencies of the system analysed. It is found that the linear problem exhibits an unstable solution if the rod length is shorter than a critical length which depends on the length of the vessel, the fluid height and the ratio of the fluid and vessel masses. In addition, we identify the existence of 1:1 resonances in the system where the symmetric sloshing modes oscillate with the same frequency as the coupled fluid/vessel motion. The implications of instability and resonance on the nonlinear problem are also briefly discussed.

Turner MR, Alemi Ardakani H, Bridges TJ. (2015) Instability of sloshing motion in a vessel undergoing pivoted oscillations, Journal of Fluids and Structures, volume 52, pages 166-180, DOI:10.1016/j.jfluidstructs.2014.10.012.

UK success story in industrial mathematics

The OWEL project featured as one of the case studies in the recent book UK Success Stories in Industrial Mathematics, edited by Philip Aston, Anthony Mulholland, & Katherine Tant, published by Springer. The title of the chapter is "Modelling and analysis of floating ocean wave energy extraction devices", and it reviews the industrial collaboration of the Surrey Team- when Hamid Alemi Ardakani was the Principal Researcher of the project. The modelling requirements of the ocean wave energy device at Offshore Wave Energy Ltd dovetail with research on interior fluid sloshing, external water wave dynamics, coupling between vessel and fluid motion, and modelling of the PTO as a gravity current interaction. The outcome of the interaction is direct impact on the wave energy industry and indirect impact on the environment and the economy.

Bridges TJ, Turner MR, Alemi Ardakani H. (2016) Modelling and analysis of floating ocean wave energy extraction devices, UK Success Stories in Industrial Mathematics, P. Aston, T. Mulholland, K. Tant (Eds.), Springer International Publishing Switzerland 77-82.