The underpinning mathematics for a novel wave energy converter: the FlexSlosh WEC

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The purpose of this small grant is to complete two steps in an important study that brings the power of mathematics to the problem of clean energy derived from waves. Ocean waves are a perpetual source of clean energy. Harvesting of this energy via Wave Energy Convertors (WECs) is one of the great challenges of the sustainable energy agenda. The proof of concept has been achieved, and the current overarching aim is to achieve power take-off (PTO) with commercial efficiencies, and this involves new GEOMETRIC modelling. Our proposed contribution to this agenda is to develop the underpinning mathematics for a class of next-generation floating WECs, in particular ducted wave energy converterswith flexible bottom topography, named FlexSlosh WEC.The FlexSlosh WEC is a freely floating rigid body in hydrodynamic interaction with exterior ocean surface waves, which extracts energy from its interior fluid sloshing over a flexible bottom topography. The flexible bottom may exploit novel use of deformable materials enabling the use of new distributed embedded energy converter technologies (DEEC-Tec) utilising distributed bellows action. The underpinning mathematics of the FlexSlosh WEC is based on a generalised Lie-Poisson bracket formulation of nonlinear partial differential equations for `dynamic coupling' between rigid-body motion and its interior dissipative shallow-water sloshing in two horizontal space dimensions, and computationally based on geometric and structure-preserving numerical analysis.Geometric and structure-preserving methods, which respect the underlying mathematical structure, i.e. specific geometric or topological, and conservation laws of the partial differential equations (PDEs) they solve, are a new generation of advanced numerical simulation techniques for evolutionary PDEs. Their advantages are being robust, stable, fast and precise for `long-time' computational modelling of highly-coupled nonlinear systems, which is so important for the development of a geometric optimisation tool for wave energy extraction with commercial efficiency.The fully coupled nonlinear system involves four subsystems: the interior fluid motion, the rigid body motion of the WEC, the elastic body modelling the flexible bottom topography, and the exterior wave motion. This project will concentrate on the first three. Poisson brackets, Lagrangians, Hamiltonians, and structure-preserving numerical schemes have been derived for these components as independent systems. Dynamic coupling brings in new challenges, in particular it is important to maintain the correct energy and momentum partition between components over long-time integration.The aim of the proposed research is (1) to develop new generalised Poisson bracket and Casimir invariants for the FlexSlosh WEC dynamics, i.e. dynamic coupling between rigid-body motion and its interior shallow-water sloshing and boundary coupling with the flexible bottom topography; and (2) to develop new finite difference energy- and potential-enstrophy-conserving symplectic scheme for long-time integration of the coupled nonlinear system.The proposed mathematical advances will develop the needed aspects of wave energy modelling in rigorous theoretical and numerical frameworks. By developing new continuum and discrete differential geometric pathways to transformation of ocean wave energy, the proposed underpinning mathematics project will contribute to the 2050 net zero target.