Investigation of geometric nonlinear potential flow effects on free surface flows Chapchap, A. C.*, Chen, Y.*, Temarel, P*. and Hirdaris S**. *Ship Science, University of Southampton, UK ** Lloyds Register Strategic Research Group, London, UK Corresponding/Presenting Author: acc1e09@soton.ac.uk 1 Summary The aim of this work is to report the first steps of ongoing research towards the investigation of geometric non linear potential flow effects on free surface flows and, eventually, on the seakeeping problem in time domain using the mixed Eulerian Lagrangian (MEL) description of the fluid flow. In particular, once the method is working properly, special interest is going to be devoted to the effects of non linear effects on the radiation potential. The foundation of the MEL method was established originally to simulate steep waves in two dimensions by Longuet-Higgins and Cokelet [1]. The main idea behind the MEL scheme is to approximate the nonlinear solution by solving a linear problem at each time step, the so called mixed boundary value problem. MEL schemes, because of their flexibility, have been applied to a broad range of hydrodynamic problems, eg. Subramani et al [2] and Liu et al [3]. Unfortunately, although relatively simple in theory, MEL implementations bring their own problems, especially in the presence of the floating body, due to the mixed nature of the boundary value problem, to instabilities associated with the free surface and to wave breaking phenomena. Some of these problems are discussed by Bai and Eatock Taylor [4] . In order to try to overcome some of the problems associated with the MEL description of the fluid flow and enhance its applicability, the present methodology describes the geometric domain by means of signed distance functions. In this context, grid (re)generation is performed using the algorithm develloped by Persson [5]. This method allows for flexible and simple grid (re)generation schemes to be implemented, which are not only capable of accounting for the instantaneous water line intersection between the free surface and the floating body but also provide ways of handling different kinds of domain geometry once intersections and unions of different domains are performed by basic operations between the corresponding distance functions. On the other hand, the additional problem of evolving signed distance functions in time, on a tridimensional background grid, is introduced once the meshing scheme relies on this domain representation. In this work two alternatives are currently being investigated to tackle this problem: in the first approach, for a given velocity field, the single level set method is used to advect the distance function in time; in the second approach, the interface is explicitly moved in a Lagrangian fashion and represented by means of a family of radial basis function (RBF) from which a new signed distance function is then derived for the free surface. In what follows of the present work a brief description of how the present methodology can be incorporated in the context of the mixed Eulerian Lagrangian description of the fluid flow is presented. In addition, numerical tests using both approaches are also presented for simple waves highlighting the issues currently being addressed. 2 Methodology In the current context the geometric domain is represented by a fixed cartesian grid equiped with the three- dimensional Euclidean norm ||.|| in which the fluid domain Ω is embeded. Thus, for a given point x g on the background grid, the signed distance function d(x g ,x, t) is defined as: d(x g ,x, t) = ||x g − x|| forx/ ∈ Ω d(x g ,x, t) = −||x g − x|| forx ∈ Ω (1) d(x g ,x, t) = 0 forx ∈ ∂ Ω. From the above defintion it also follows that |∇d(x g ,x, t)| =1. In the fluid domain Ω , the formulation of the problem is made in a similar fashion to the usual MEL description of the fluid flow ( see for instance Liu et al [3]). In this context, Laplace’s equation governs the fluid flow, and the velocity potential φ(x,y,z,t) satisfies: