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

A space-time hybridizable discontinuous Galerkin method for linear free-surface waves

We present and analyze a novel space-time hybridizable discontinuous Galerkin (HDG) method for the linear free-surface problem on prismatic space-time meshes. We consider a mixed formulation which immediately allows us to compute the velocity of the fluid. In order to show well-posedness, our space-time HDG formulation makes use of weighted inner products. We perform an a priori error analysis in which the dependence on the time step and spatial mesh size is explicit. We provide two numerical examples: one that verifies our analysis and a wave maker simulation

On variational and symplectic time integrators for Hamiltonian systems

Various systems in nature have a Hamiltonian structure and therefore accurate time integrators for those systems are of great practical use. In this paper, a fi nite element method will be explored to derive symplectic time stepping schemes for (non-)autonomous systems in a systematic way. The technique used is a variational discontinuous Galerkin nite element method in time. This approach provides a uni ed framework to derive known and new symplectic time integrators. An extended analysis for the new time integrators will be provided. The analysis shows that a novel third order time integrator presented in this paper has excellent dispersion properties. These new time stepping schemes are necessary to get accurate and stable simulations of (forced) water waves and other non-autonomous variational systems, which we illustrate in our numerical results

Hamiltonian Finite Element Discretization for Nonlinear Free Surface Water Waves

A novel finite element discretization for nonlinear potential flow water waves is presented. Starting from Luke’s Lagrangian formulation we prove that an appropriate finite element discretization preserves the Hamiltonian structure of the potential flow water wave equations, even on general time-dependent, deforming and unstructured meshes. For the time-integration we use a modified Störmer–Verlet method, since the Hamiltonian system is non-autonomous due to boundary surfaces with a prescribed motion, such as a wave maker. This results in a stable and accurate numerical discretization, even for large amplitude nonlinear water waves. The numerical algorithm is tested on various wave problems, including a comparison with experiments containing wave interactions resulting in a large amplitude splash

Variational space-time (dis)continuous Galerkin method for nonlinear free surface water waves

A new variational finite element method is developed for nonlinear free surface gravity water waves using the potential flow approximation. This method also handles waves generated by a wave maker. Its formulation stems from Miles' variational principle for water waves together with a finite element discretization that is continuous in space and discontinuous in time. One novel feature of this variational finite element approach is that the free surface evolution is variationally dependent on the mesh deformation vis-à-vis the mesh deformation being geometrically dependent on free surface evolution. Another key feature is the use of a variational (dis)continuous Galerkin finite element discretization in time. Moreover, in the absence of a wave maker, it is shown to be equivalent to the second order symplectic Störmer-Verlet time stepping scheme for the free-surface degrees of freedom. These key features add to the stability of the numerical method. Finally, the resulting numerical scheme is verified against nonlinear analytical solutions with long time simulations and validated against experimental measurements of driven wave solutions in a wave basin of the Maritime Research Institute Netherlands. © 2014 Elsevier Inc

On variational modelling of wave slamming by water waves

This thesis is concerned with the development of both mathematical (variational formulation) models and simulation (finite-element Galerkin) tools for describing a physical system consisting of water waves interacting with an offshore wind-turbine mast. In the first approach, the starting point is an action functional describing a dual system comprising a potential-flow fluid, a solid structure modelled with nonlinear elasticity, and the coupling between them. Novel numerical results for the linear case indicate that our variational approach yields a stable numerical discretization of a fully coupled model of water waves and an elastic beam. The drawback of the incompressible potential flow model is that it inevitably does not allow for wave-breaking. Therefore another approach loosely based on a van-der-Waals gas is proposed. The starting point is again an action functional, but with an extra term representing internal energy. The flow can be assumed to have no rotation, so although it is again described with a potential, compressibility is now introduced. The free surface is embedded within the compressible fluid for an appropriate van-der-Waals-inspired equation of state, which enables a pseudo-phase transition between the water and air phases separated by a sharp or steep transition variation in density. Due to the compressibility, in addition to gravity waves the model enables acoustic ones, which is confirmed by a dispersion relation. Higher-frequency acoustic waves can be dampened by the appropriate choice of time integrators. Hydrostatic and linearized models have been examined as verification steps. The model also matches incompressible linear potential flow. However, at the nonlinear level, the acoustic noise remains significant

Space-time hybridizable discontinuous Galerkin methods for free-surface wave problems

Free-surface problems arise in many real-world applications such as in the design of ships and offshore structures, modeling of tsunamis, and dam breaking. Mathematically, free-surface wave problems are described by a set of partial differential equations that govern the movement of the fluid together with certain boundary conditions that describe the free-surface. The numerical solution of such problems is challenging because the boundary of the computational domain depends on the solution of the problem. This implies that there is a strong coupling between the fluid and the free-surface, and the domain must be continuously updated to track the changes in the free-surface. In this thesis we explore and develop space-time hybridizable discontinuous Galerkin (HDG) methods for free-surface problems. First, we focus on a linear free-surface problem in which the amplitude of the waves is assumed to be small enough so that the domain can remain fixed. We initially consider a traditional approach for the numerical discretization of time-dependent partial differential equations: we discretize in space using, in this case, an HDG method to obtain an ordinary differential equation. Then, we use a second order backward differentiation formula to discretize in time. We see that in comparison to an interior penalty discontinuous Galerkin discretization, this HDG discretization results in smaller linear systems (in general), and produces better approximations to the velocity of the fluid. Next, we consider the solution of the same linear free-surface problem with a space-time hybridizable discontinuous Galerkin method. Unlike previous finite element discretizations of this problem, we consider a mixed formulation in which the velocity of the flow can be approximated with an optimal order of convergence. We develop a set of space-time analysis tools that allow us to obtain a priori error estimates in which the dependency on the spatial mesh size and the time step is explicit. This is in contrast to previous space-time error analyses in which the error bounds depend on the size of the space-time elements. Finally, we move on to incompressible nonlinear free-surface flow. We consider the two-fluid (gas and liquid) Navier-Stokes equations and use a level set method in which the flow and the level set equations are solved subsequently until a certain stopping criterion has been met. The flow equations are solved with a space-time HDG method which is exactly mass conserving. Furthermore, a space-time embedded discontinuous Galerkin method is employed for the solution of the level set equation. This discretization possesses the same conservation and stability properties as discontinuous Galerkin methods, but produces a continuous approximation to the free-surface elevation. When a discontinuous approximation to the free-surface elevation is obtained, smoothing techniques have to be applied in order to move the mesh and track the interface. It has been shown in the past that such techniques can lead to instabilities and stabilization terms have to be added to the discretization. Therefore, obtaining a continuous approximation to the free-surface elevation in our discretization is crucial: not only can the mesh be deformed in a straightforward manner, but it can also be done without introducing any potential sources of instabilities. We present two numerical results that demonstrate the capabilities of the method. In the first test case we compare against an analytical solution and we demonstrate how the mesh conforms to the interface between the two fluids. Finally, we present a simulation of waves generated by a submerged obstacle