## Final Report Summary - SAFEMILLS (Increasing safety of offshore wind turbines operation: Study of the violent wave loads)

The project’s proposal had succinctly two major goals: i) the training of the Researcher in the scientific area of hydrodynamic slamming and ii) the development of new methodologies, solution methods and procedures for relevant problems. The training of the researcher aimed at gaining and transferring the new knowledge in his home institution in an attempt to make this new area at home attractive to engineering students and to the home academic community at large. This goal will be further developed as the Researcher has already returned to his home institution assuming again responsibilities and he is under the process of creating his own research group. The results of the endavour are already developing: the Researcher has awarded his first PhD that was completed under his supervision in the concerned area. The PhD was awarded to Dr.-Eng. S. Katifeoglou who completed a thesis under the title “Hydroelastic Behaviour of Structural Elements under Extreme Loading Conditions, Focusing on Marine Applications”, January 2016. In addition, as stated in the proposal of the project the Researcher was aiming to be trained and increase his scientific capacity in order to ask for a promotion to the rank of full-professorship. The procedure is now under way and the final decision will be taken very soon by the responsible decision committee. The scientific discipline which the Researcher will cover as a Professor will be “Hydroelasticity”, an area which is wider than the “hydrodynamic slamming problems” in which he was actually trained during his stay in UEA. However “Hydroelasticity” explicitly dictates knowledge of hydrodynamic slamming. A second PhD thesis will commence very soon under the Researcher’s supervision. The title of the thesis will be in a relevant subject.

In terms of the core of the subject, the Researcher under the guidance and supervision of the Scientist in Change and the Steering Committee dealt with several issues which could be summarized as follows:

1. Methods for 2D steep wave impact on solid, porous and perforated walls

2. Methods for 2D impact problems with mixed conditions

3. Methods for 3D water entry problems

4. Methods for 3D steep wave impact problems on plates and convex surfaces

5. Solutions of mixed-boundary value problems involving triple trigonometrical series

6. Solutions of mixed boundary value problems involving triple integral equations

Items 5 and 6 were performed in an explicit mathematical fashion, so as to be general and be used for other problems as well, but they were required to tackle hydrodynamic slamming problems involved in items 2-4. In terms of one and two dimensional mixed-boundary value problems the Researcher re-discovered ingenious approaches for dual trigonometrical and integral equations series and extended them to triple series, so as to tackle hydrodynamic slamming cases that resemble impact of breaking waves forming an air pocket between the upper and lower liquid regions. The same approximations were used for solid walls having an open area allowing the discharge of the liquid.

In general, the work was carried out analytically, focusing on theoretical and novel mathematical developments. The studies on porous and perforated structures were performed using linear and non-linear forms of unconventional body boundary conditions, respectively. These non-classical relations, required numerical elaboration of the involved boundary value problems and the eigenfunction expansions of the solutions.

Most of the time during the project was invested for finding solutions for the 3D hydrodynamic slamming problems on convex surfaces such as cylinders. That was due to the fact that by general confession, 3D solutions of boundary value problems of mixed type are scarce in the literature and they exist only for very simple geometries, such as paraboloids. Even the numerical implementation poses significant challenges on the treatment of the difficulties that arise on the treatment of singularities. In most of the cases, they result in hyper-singular integral equations which are not easy to deal with.

The project emphasized on the advanced mathematical treatment of relevant problems using both von Karman and Wagner conditions for the instantaneous wetted (impacted) part of the surface. The von Karman approach was proven relatively easy to implement, so focus was given to the Wagner approach. To this end, unavoidably, some simplifications had to be assumed such as the assumption of nearly vertical instantaneous contact line in order to secure that the Field equation (Laplace domain) is not violated. Relevant approaches resulted in robust and closed-form solutions expressed by purely mathematical – analytical relations.

The study on 3D problems was extended to tackle slamming problems due to steep waves on 3D plates of finite width installed on finite water depths for which, so far, only 2D analytical solutions existed. The goal was to provide a solution that was flexible enough to accommodate at a later stage the von Karman and the Wagner approaches for wave impact in order to derive the time varying components of loading, including force and pressure-impulse. It is noted that wave impact problems is different (and more complicated) than the classical water entry problems due to the simple fact that they involve two free surfaces in which the linearized free-surface boundary conditions should be satisfied. The solution to the plate wave impact problems was achieved by assuming an unconventional approach, assimilating the plates as elliptical cylinders with marginal semi-minor axes, change the coordinate system of the Laplace equation and seeking solutions in terms of elliptical harmonics. The work was further extended by considering plates with intermediate open areas that allow the discharge of the liquid. In this context, a perturbation technique was applied while two overlapping boundary value problems with mixed conditions in two dimensions had to be considered. The obtained results were indeed significant in value showing why 3D effects are concentrated at the edges (near the contact line). In fact it was verified that this is due to the higher order terms of the velocity potential

(see Fig. 1). Fig. 1 Higher order term of the velocity potential during the steep wave impact on a plate with an intermediate open area.

Finally, the project was directed to water entry problems of complicated shapes such as spheroids. Effort was made, although with not much success, to extend the renowned Galin’s theorem from elliptical to arbitrary contact lines. Emphasizing explicitly on elliptical contact lines, effort was made to achieve a solution in terms of the unfamiliar Lame functions and ellipsoidal harmonics. The work is still under way despite the completion of the project while the Researcher aspires to supervise several PhD titles under this challenge.

Impact of the project

The impact of the project is mostly academic and research focused. Despite the work done and the experience and additional knowledge that the Researcher gained, there are many open issues which need to be addressed in terms of hydrodynamic slamming problems and mixed-boundary value problems at large. The situation will remain challengeable until proper approximations and efficient methodologies will be developed. The main beneficiary out of the project is, aside from the Researcher, his own institution. The home institution now has an additional discipline to treat and it is hoped that it will attract ambitious students who will boost the research towards hydrodynamic slamming problems, violent wave impact, hydroelasticity of ships and offshore structures etc.

In terms of the core of the subject, the Researcher under the guidance and supervision of the Scientist in Change and the Steering Committee dealt with several issues which could be summarized as follows:

1. Methods for 2D steep wave impact on solid, porous and perforated walls

2. Methods for 2D impact problems with mixed conditions

3. Methods for 3D water entry problems

4. Methods for 3D steep wave impact problems on plates and convex surfaces

5. Solutions of mixed-boundary value problems involving triple trigonometrical series

6. Solutions of mixed boundary value problems involving triple integral equations

Items 5 and 6 were performed in an explicit mathematical fashion, so as to be general and be used for other problems as well, but they were required to tackle hydrodynamic slamming problems involved in items 2-4. In terms of one and two dimensional mixed-boundary value problems the Researcher re-discovered ingenious approaches for dual trigonometrical and integral equations series and extended them to triple series, so as to tackle hydrodynamic slamming cases that resemble impact of breaking waves forming an air pocket between the upper and lower liquid regions. The same approximations were used for solid walls having an open area allowing the discharge of the liquid.

In general, the work was carried out analytically, focusing on theoretical and novel mathematical developments. The studies on porous and perforated structures were performed using linear and non-linear forms of unconventional body boundary conditions, respectively. These non-classical relations, required numerical elaboration of the involved boundary value problems and the eigenfunction expansions of the solutions.

Most of the time during the project was invested for finding solutions for the 3D hydrodynamic slamming problems on convex surfaces such as cylinders. That was due to the fact that by general confession, 3D solutions of boundary value problems of mixed type are scarce in the literature and they exist only for very simple geometries, such as paraboloids. Even the numerical implementation poses significant challenges on the treatment of the difficulties that arise on the treatment of singularities. In most of the cases, they result in hyper-singular integral equations which are not easy to deal with.

The project emphasized on the advanced mathematical treatment of relevant problems using both von Karman and Wagner conditions for the instantaneous wetted (impacted) part of the surface. The von Karman approach was proven relatively easy to implement, so focus was given to the Wagner approach. To this end, unavoidably, some simplifications had to be assumed such as the assumption of nearly vertical instantaneous contact line in order to secure that the Field equation (Laplace domain) is not violated. Relevant approaches resulted in robust and closed-form solutions expressed by purely mathematical – analytical relations.

The study on 3D problems was extended to tackle slamming problems due to steep waves on 3D plates of finite width installed on finite water depths for which, so far, only 2D analytical solutions existed. The goal was to provide a solution that was flexible enough to accommodate at a later stage the von Karman and the Wagner approaches for wave impact in order to derive the time varying components of loading, including force and pressure-impulse. It is noted that wave impact problems is different (and more complicated) than the classical water entry problems due to the simple fact that they involve two free surfaces in which the linearized free-surface boundary conditions should be satisfied. The solution to the plate wave impact problems was achieved by assuming an unconventional approach, assimilating the plates as elliptical cylinders with marginal semi-minor axes, change the coordinate system of the Laplace equation and seeking solutions in terms of elliptical harmonics. The work was further extended by considering plates with intermediate open areas that allow the discharge of the liquid. In this context, a perturbation technique was applied while two overlapping boundary value problems with mixed conditions in two dimensions had to be considered. The obtained results were indeed significant in value showing why 3D effects are concentrated at the edges (near the contact line). In fact it was verified that this is due to the higher order terms of the velocity potential

(see Fig. 1). Fig. 1 Higher order term of the velocity potential during the steep wave impact on a plate with an intermediate open area.

Finally, the project was directed to water entry problems of complicated shapes such as spheroids. Effort was made, although with not much success, to extend the renowned Galin’s theorem from elliptical to arbitrary contact lines. Emphasizing explicitly on elliptical contact lines, effort was made to achieve a solution in terms of the unfamiliar Lame functions and ellipsoidal harmonics. The work is still under way despite the completion of the project while the Researcher aspires to supervise several PhD titles under this challenge.

Impact of the project

The impact of the project is mostly academic and research focused. Despite the work done and the experience and additional knowledge that the Researcher gained, there are many open issues which need to be addressed in terms of hydrodynamic slamming problems and mixed-boundary value problems at large. The situation will remain challengeable until proper approximations and efficient methodologies will be developed. The main beneficiary out of the project is, aside from the Researcher, his own institution. The home institution now has an additional discipline to treat and it is hoped that it will attract ambitious students who will boost the research towards hydrodynamic slamming problems, violent wave impact, hydroelasticity of ships and offshore structures etc.