Abstract for the 29th Intl Workshop on Water Waves and Floating Bodies, Osaka (Japan), March 30 - April 02, 2014 Development of a highly nonlinear model for wave propagation over a variable bathymetry M. Gouin* 1,2 , G. Ducrozet 1 , P. Ferrant 1 1. Ecole Centrale Nantes, LHEEA Lab., UMR CNRS 6598, Nantes, France 2. Institut de Recherche Technologique Jules Verne, Bouguenais, France E-mail: maite.gouin@ec-nantes.fr Highlights • Development of two strategies to model nonlinear waves propagating over variable bathymetry with HOS method. • Presentation of a validation case to assess accuracy and eﬃciency of both approaches which are compared. Introduction Modeling surface gravity waves is a major concern in ocean engineering, and especially in the ﬁeld of marine renewable energy. These marine structures (wave and tidal energy converters, oﬀshore wind turbines, . . . ) are intended to be installed in limited water depth, where the inﬂuence of variable bathymetry is very signiﬁcant on local wave conditions. In this paper two diﬀerent schemes for modeling bathymetry in the High-Order Spectral (HOS) method are presented. This highly non-linear potential model has been initially developed by West et al. [9] and Dommermuth & Yue [3] for a ﬂat bottom and extensively validated for diﬀerent conﬁgurations. A few HOS studies [5] consider a variable bathymetry, but more has been done in Dirichlet to Neumann Operator method (DNO) [2, 8, 4]. Sch¨ aﬀer [7] proved that HOS and DNO methods are identical thus we propose to characterize these diﬀerent approaches. The ﬁrst presented scheme resolves the bottom condition without any approximation whereas the second scheme [5] uses the boundary condition by expanding the surface potential in Taylor series with respect to the mean water depth. These two schemes are presented here in details and then compared on a validation case considering propagation of nonlinear regular waves. A comparison of their accuracy and eﬃciency is made with respect to parameters of the problem (steepness, relative water depth, bottom variation . . . ). 1 Methods and Algorithms 1.1 Hypothesis and formulation of the problem A 2D rectangular ﬂuid domain with Cartesian coordinate system is considered. The z axis is vertical and oriented upwards, with the level z = 0 corresponding to the mean water level. z = η (x, t) represents the free surface elevation, h the total depth, h 0 the mean depth and β the bottom variation, such as -h (x)= -h 0 + β (x)(cf Fig.1). We assume periodic boundary conditions in the horizontal plane so that the domain is considered inﬁnite. A potential ﬂow formalism is used (incompressible and inviscid ﬂuid, irrotational ﬂow). Given these assumptions, the velocity ﬁeld ~ V derives from a potential ~ V (x,z,t)= ~ ∇φ and the continuity equation becomes the Laplace equation in the ﬂuid domain: Δφ =0 (1) Following Zakharov [10], both fully nonlinear free-surface boundary conditions (kinematic and dynamic) are written in terms of surface quantities η and e φ: ∂η ∂t =  1+ |∇η| 2  ∂φ ∂z -∇ e φ.∇η on z = η (x, t) (2) ∂ e φ ∂t = -gη - 1 2   ∇ e φ    2 + 1 2  1+ |∇η| 2   ∂φ ∂z  2 on z = η (x, t) (3) with e φ (x, t)= φ (x, z = η,t) standing for the free surface velocity potential and ∇ the horizontal gradient. To account for the time evolution of the quantities of interest η and e φ one only needs to evaluate the vertical velocity at the free surface W (x, t)= ∂φ ∂z (x, z = η (x, t) ,t). Note that HOS method was initially developed for a ﬂat bottom, while here the bottom boundary condition reads: ∂φ ∂x ∂β ∂x - ∂φ ∂z =0 on z = -h 0 + β (x) (4)