OWEL-SURREY Wave Energy Modelling Project Department of Mathematics University of Surrey and Offshore Wave Energy Ltd

Background and history       The OWEL Wave Energy Convertor (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, mathematical models are being developed for simulating the dynamics, interior sloshing, power take off, and control of the OWEL WEC.

Surrey shallow water sloshing project       In the Surrey sloshing project, shallow water equations for sloshing in 2D and 3D vessels undergoing translation and rotation, are derived. Both the cases where the vessel motion is prescribed and the case where the vessel motion is dynamically coupled, and therefore calculated along with the fluid motion, are considered. This project dovetails with an analysis of the OWEL WEC because the main chamber traps incoming waves which then "slosh" before the power take off and discharge. A key additional feature in the OWEL modelling is the fact that it is a two layer fluid, with the upper layer being trapped air, and the upper layer plays an important role in the dynamics and the power take off. Surrey Sloshing Website         M. R. Turner, H. Alemi Ardakani & T.J. Bridges. Instability of sloshing motion in a vessel undergoing pivoted oscillations, J. Fluids Struct. 52, 166-180 (2015).     M. R. Turner & T.J. Bridges. Nonlinear energy transfer between fluid sloshing and vessel motion, J. Fluid Mech. 719, 606-636 (2013).     H. Alemi Ardakani & T.J. Bridges. Shallow-water sloshing in vessels undergoing prescribed rigid-body motion in three dimensions, J. Fluid Mech. 667, 474-519 (2011).

   

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).

   

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 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.       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).   JFS Website

   

Time-dependent conformal mapping of doubly-connected regions       The interest in this project is the evolution of the conformal modulus Q(t), associated with conformal mapping of doubly-connected regions, and the boundary transformation generalizing the Hilbert transform. The theory is applied to an unsteady free surface flow. Focusing on inviscid, incompressible, irrotational fluid sloshing in a rectangular vessel, it is shown that the explicit calculation of the conformal modulus is essential to correctly predict features of the flow. Results are also presented for fully dynamic simulations which use a time-dependent conformal mapping and the Garrick generalization of the Hilbert transform to map the physical domain to a time-dependent rectangle in the computational domain. The results of this new approach are compared to the complementary numerical scheme of Frandsen and it is shown that correct calculation of the conformal modulus is essential in order to obtain agreement between the two methods. The theory is extended to the case where the bottom topography is variable in Turner (2016).       M.R. Turner & T.J. Bridges. Time-dependent conformal mapping of doubly-connected regions, Adv. Comp. Math., (2016, in press).   Link for open access   M.R. Turner. Liquid sloshing in a horizontally forced vessel with bottom topography, J. Fluids & Struct. 64, 1-26 (2016).   JFS Link

   

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.       M.R. Turner, T.J. Bridges, & H. Alemi Ardakani. The pendulum-slosh problem: simulation using time-dependent conformal mapping, J. Fluids & Struct. 59, 202-223 (2015).

   

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.       H. Alemi Ardakani, T.J. Bridges & M.R. Turner. 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, J. Comp. Phys. 314, 590-617 (2016).   JCP Website   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).   JCP Website

   

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 retainling 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 presure 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.       H. Alemi Ardakani, T.J. Bridges & M.R. Turner. Dynamic coupling between horizontal vessel motion and two-layer shallow-water sloshing. J. Fluids & Struct. 59, 432-460 (2015).     H. Alemi Ardakani. Adaptation of f-wave finite volume methods to the Boonkasame-Milewski non-Boussinesq two-layer shallow interfacial sloshing equations coupled to the vessel motion. Euro. J. Mech. B/Fluids (in press, 2016).   EJMBF Website

   

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.       H. Alemi Ardakani & M.R. Turner. Numerical simulations of dynamic coupling between shallow-water sloshing and horizontal vessel motion with baffles, Fluid Dyn. Res. 48, 035504 (28pp) (2016).

   

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.       H. Alemi Ardakani, T.J. Bridges & M.R. Turner. Adaption of f-wave finite volume methods to the two-layer shallow-water equations in a moving vessel with a rigid lid. J. Comp. Appl. Math. 296, 462-479 (2016).

   

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.       M.R. Turner, H. Alemi Ardakani & T.J. Bridges. Instability of sloshing motion in a vessel undergoing pivoted oscillations, J. Fluids & Struct. 52, 166-180 (2015).     JFS Website

   

Nonlinear energy transfer between vessel and fluid       This project examined the dynamic coupling between a sloshing fluid and the motion of the vessel containing the fluid. A mechanism is identified that leads to an energy exchange between the vessel dynamics and fluid motion. It is based on a 1:1 resonance in the linearized equations, but nonlinearity is essential for the energy transfer. For definiteness, the theory is developed for Cooker's pendulous sloshing experiment. The vessel has a rectangular cross-section, is partially filled with a fluid and is suspended by two cables. A nonlinear normal form is derived close to an internal 1:1 resonance, with the energy transfer manifested by a heteroclinic connection, which connects the purely symmetric sloshing modes to the purely antisymmetric sloshing modes. Parameter values where this pure energy transfer occurs are identified. In practice, this energy transfer can lead to sloshing-induced destabilization of fluid-carrying vessels       M.R. Turner & T.J. Bridges. Nonlinear energy transfer between fluid sloshing and vessel motion, J. Fluid Mech. 719, 606-636 (2013).     JFM Website

   

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. The modelling requirements of the ocean wave energy device at Offshore Wave Energy Ltd dovetail with research at the University of Surrey 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.       T.J. Bridges, M.R. Turner & H. Alemi Ardakani. Modelling and analysis of floating ocean wave energy extraction devices, UK Success Stories in Industrial Mathematics, Springer (2016).     Springer website for chapter download

   

EPSRC: Dynamics of floating water-wave energy extraction devices       The aim of this EPSRC-funded project is to tackle three main mathematical modelling issues: (1) the dynamics of the waves once they are trapped inside the vessel, (2) the effect of resonance between components, and (3) the dynamic feedback due to multi-fold coupling between interior waves, the ambient ocean wave field, vessel dynamics and mooring dynamics.       EP/K008188/1 Dynamics of floating water-wave energy extraction devices     EPSRC link

   

Additional references             A.F. de O. Falcao. Wave energy utilization: a review of the technologies, Renew. Sustain. Energy Rev. 14, 899-918 (2010).     Journal website   J. Falnes. Ocean Waves and Oscillating Systems, Cambridge University Press, (2002).     CUP website   S. Herring. Design and evaluation of turbines for use in OWC power plants, PhD Thesis, Cranfield University, (2007).     Cranfield website   J.J. Keller & Y.P. Chyou. On the hydraulic lock-exchange problem, ZAMP, 42 874-909 (1991).     Springer website   M. Leybourne, W. Batten, A.S. Bahaj, N. Minns, & J. O'Nians. Experimental and computational modelling of the OWEL wave energy converter, Third Int. Conf. Ocean Energy, ICOE2010 (2010).     ICOE link   M. Leybourne, W. Batten, A.S. Bahaj, N. Minns, & J. O'Nians. Preliminary design of the OWEL wave energy converter pre-commercial demonstration, Renewable Energy, 61 (2014).     ICOE link   K.T. Mandli. A numerical method for the two-layer shallow water equations with dry states, Ocean Modelling, 72 80-91 (2013).     OM link   J.E.S. Minns. Comparative performance of a novel oscillating water column wave energy converter, PhD Thesis, Loughborough University, (2012).     Lboro link   T.C. Smith & W.L. Thomas III. A survey of ship motion reduction devices, David Taylor Research Center, DTRC/SHD-1338-01 (1990).     DTRC website   Wave Hub Ltd. Advancing Offshore Renewable Energy.     Wavehub website   OWEL CFD Modelling with ANSYS. Water waves and 3 DoF rigid body motion.     Youtube video   HC 1624 Energy and Climate Change Committee. Memorandum submitted to House of Commons by Offshore Wave Energy Ltd, (2011).     Parliament website

Surrey Sloshing Project     Offshore Wave Energy Limited