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Breakwaters are used to provide sheltered areas for loading and unloading of ships, and coastal protection. Often the breakwaters are bottom mounted such as rubble mound breakwaters. However, there can be several advantages to use a Floating Breakwater (FB). Therefore, the objective of this paper is to study the effect of two different damping mechanisms of a floating breakwater. Three basic cross-sections of FBs were tested and analysed in 2D; a regular pontoon (RG), a regular pontoon with wing plates attached (WP), and a regular pontoon with wing plates and porous media attached to the sides (WP P100). The damping of the FB motions was due to wave radiation and viscous damping. The viscous damping originated mainly from vortex generation around the edges of the structure and due to energy loss inside the porous material attached to the vertical sides of the floating breakwater. Attaching wing plates to the floating breakwater significantly reduced the motion, which was also anticipated. When the porous sides were attached the motion of the FB increased compared to the (WP) cross-section, but the wave transmission was reduced. The possibility for incorporating the effect of the damping in the radiation/diffraction code WAMIT was assessed. The study showed that the cross section with wing plates reduced the motions of the breakwater to the largest extend, while the cross section with wing plates and porous media attached to the sides reduced the reflection and transmission most effectively.

Numerical and physical modeling are the two main tools available for predicting the influence of water waves on coastal or offshore structures. Both models have their strengths and weaknesses. An integrated use of numerical and physical modeling which exploits their advantages can provide an optimal description of full-scale, realistic engineering problems. In this series of two papers, we report on a generalized three-dimensional (3D) deterministic coupling theory for multidirectional nonlinear wave propagation from a numerical model to a physical model up to second order. In this work, the second-order coupling theory developed by [Z. Yang, S. Liu, H.B. Bingham and J. Li, 2014a. Second-order coupling of numerical and physical wave tanks for 2D irregular waves. Part I Formulation, implementation and numerical properties. Coast. Eng. 92, 48–60] and the ad hoc unified wave generation theory developed by [H. Zhang, H.A., Schaffer, K.P., Jakobsen, 2007. Deterministic combination of numerical and physical coastal wave models. Coast. Eng. 54, 171–186.] have been extended to include the coupling between multidirectional nonlinear waves. In part I of this article series, the full second-order 3D coupling theory for multidirectional nonlinear waves is been described in detail. A novel generalized fully-nonlinear motion boundary equation has been derived, which allows the interface between the numerical and physical wave domains to be a 3D arbitrarily shaped wavemaker system. The corresponding 3D wave coupling equation is given for I-, L-, and O-shaped wavemaker layouts. The new formulation is presented in a unified form for an arbitrarily shaped layout of planar wavemakers, and covers both hinged and piston wavemaker types. For the second-order dispersive correction of the paddle stroke, the super-harmonic and sub-harmonic wave-wave interactions have been taken into account. For practical implementation, a typical discretization method for the 3D coupling equations is derived by considering the case of the I-shaped piston wavemaker. The new 3D coupling equations are solved by a combined five-point Lagrange interpolation and the fourth-order Runge–Kutta scheme. Numerical evaluations of the implementation have been conducted by considering a theoretical second-order unidirectional wave field over a range of spatial and time resolutions. The results thus obtained indicate that the proposed discretization scheme is accurate and effective. In addition, the precision of the discrete scheme is observed to be closely related to the spatial and temporal resolution. A separate experimental validation of the theory is presented in Part II.

In this series of two papers, we report on the irregular wave extension of the second-order coupling theory of numerical and physical wave model described in [Z. Yang, S. Liu, H.B. Bingham and J. Li. Second-order theory for coupling numerical and physical wave tanks: Derivation, evaluation and experimental validation. Coast. Eng. 71, 37–51]. We also correct several errors which unfortunately appeared in that manuscript. In the present part I, the full second-order coupling theory for irregular wave is described in detail. The new second-order coupling signal is presented including both superharmonics and subharmonics and covering wavemaker configurations of the piston- and flap-types. The second-order dispersive correction allows for an improved nonlinear transfer of wave information between the two models. For practical implementation, the coupling equations are solved by a combined five-point Lagrange interpolation and the fourth-order Runge–Kutta scheme, with a numerical velocity time series which is decomposed by the Newton–Raphson iterative method. Analytical evaluations on the suppression of spurious free waves and the relative errors of the resultant bound waves have been conducted by considering a 2nd-order, bi-chromatic wave over a range of dimensionless water depth and oscillation frequency combinations, indicating that the resultant wave quality is significantly improved using the second-order coupling theory. A separate verification combining numerical and experimental model of the theory will be presented in Part II by the same authors.

A fully nonlinear domain decomposed solver is proposed for efficient computations of wave loads on surface piercing structures in the time domain. A fully nonlinear potential flow solver was combined with a fully nonlinear Navier–Stokes/VOF solver via generalized coupling zones of arbitrary shape. Sensitivity tests of the extent of the inner Navier–Stokes/VOF domain were carried out. Numerical computations of wave loads on surface piercing circular cylinders at intermediate water depths are presented. Four different test cases of increasing complexity were considered; 1) weakly nonlinear regular waves on a sloping bed, 2) phase-focused irregular waves on a flat bed, 3) irregular waves on a sloping bed and 4) multidirectional irregular waves on a sloping bed. For all cases, the free surface elevation and the inline force were successfully compared against experimental measurements.

The highly accurate Boussinesq-type equations of Madsen et al. (Madsen, P.A., Bingham, H.B., Schaffer, H.A., 2003. Boussinesq-type formulations for fully nonlinear and extremely dispersive water waves: Derivation and analysis. Proc. R. Soc. Lond. A 459, 1075–1104; Madsen, P.A., Fuhrman, D.R., Wang, B., 2006. A Boussinesq-type method for fully nonlinear waves interacting with a rapidly varying bathymetry. Coast. Eng. 53, 487–504); Jamois et al. (Jamois, E., Fuhrman, D.R., Bingham, H.B., Molin, B., 2006. Wave-structure interactions and nonlinear wave processes on the weather side of reflective structures. Coast. Eng. 53, 929–945) are re-derived in a more general framework which establishes the correct relationship between the model in a velocity formulation and a velocity potential formulation. Although most work with this model has used the velocity formulation, the potential formulation is of interest because it reduces the computational effort by approximately a factor of two and facilitates a coupling to other potential flow solvers. A new shoaling enhancement operator is introduced to derive new models (in both formulations) with a velocity profile which is always consistent with the kinematic bottom boundary condition. The true behaviour of the velocity potential formulation with respect to linear shoaling is given for the first time, correcting errors made by Jamois et al. (Jamois, E., Fuhrman, D.R., Bingham, H.B., Molin, B., 2006. Wave-structure interactions and nonlinear wave processes on the weather side of reflective structures. Coast. Eng. 53, 929–945). An exact infinite series solution for the potential is obtained via a Taylor expansion about an arbitrary vertical position z = zˆ. For practical implementation however, the solution is expanded based on a slow variation of zˆ and terms are retained to first-order. With shoaling enhancement, the new models obtain a comparable accuracy in linear shoaling to the original velocity formulation. General consistency relations are also derived which are convenient for verifying that the differential operators satisfy a potential flow and/or conserve mass up to the order of truncation of the model. The performance of the new formulation is validated using computations of linear and nonlinear shoaling problems. The behaviour on a rapidly varying bathymetry is also checked using linear wave reflection from a shelf and Bragg scattering from an undulating bottom. Although the new models perform equally well for Bragg scattering they fail earlier than the existing model for reflection/transmission problems in very deep water.

International audience; A finite difference model based on a recently derived highly-accurate Boussinesq-type formulation is presented. Up to the third-order space derivatives in terms of the velocity variables are retained, and the horizontal velocity variables are re-formulated in terms of a velocity potential. This decreases the total number of unknowns in two horizontal dimensions from seven to five, simplifying the implementation, and leading to increased computational efficiency. Analysis of the embedded properties demonstrates that the resulting model has applications with errors of 2 to 3% for (wavenumber times depth) kh ≤ 10 in terms of dispersion and kh ≤ 4 in terms of internal kinematics. The stability and accuracy of the discrete linearised systems are also analysed for both potential and velocity formulations and the advantages and disadvantages of each are discussed. The velocity potential model is then used to study physically demanding problems involving highly nonlinear wave run-up on a bottom-mounted (surface-piercing) plate. New cases involving oblique incidence are considered. In all cases, comparisons with recent physical experiments demonstrate good quantitative accuracy, even in the most demanding cases, where the local wave steepness can exceed (waveheight divided by wavelength) H / L = 0.20. The velocity potential model is additionally shown to have numerical advantages when dealing with wave–structure interactions, requiring less smoothing around exterior structural corners.

This paper describes the extension of a finite difference model based on a recently derived highly accurate Boussinesq formulation to include domains having arbitrary piecewise-rectangular bottom-mounted (surface-piercing) structures. The resulting linearized system is analyzed for stability on a structurally divided domain, and it is shown that exterior corner points pose potential stability problems, as well as other numerical difficulties. These are mainly due to the discretization of high-order mixed-derivative terms near these points, where the flow is theoretically singular. Fortunately, the system is receptive to dissipation, and these problems can be overcome in practice using high-order filtering techniques. The resulting model is verified through numerical simulations involving classical linear wave diffraction around a semi-infinite breakwater, linear and nonlinear gap diffraction, and highly nonlinear deep water wave run-up on a vertical plate. These cases demonstrate the applicability of the model over a wide range of water depth and nonlinearity.

This paper describes a new computational technique for predicting the wave-induced motion of a restrained floating body in restricted water. A combination of established methods is used in an attempt to account for the most important physical processes involved in this complicated problem, while keeping the computational burden modest. Potential theory is invoked to describe both the wave transformation over the bathymetry of the harbour, and the hydrodynamic interaction between the waves and the floating structure. Modified Boussinesq theory is used to predict the transformation of the waves as they propagate from deep water into to the harbour or bay where the body is moored. This model includes the effects of shoaling, refraction and non-linear wave–wave interaction, and most importantly sub-harmonic generation. This flow is then linearised locally, at the structure, to provide the incident wave forcing term in the equation of motion which is solved in the time-domain. Linear wave radiation and diffraction forces are computed using a constant-strength panel method, while the instantaneous, non-linear, point mooring forces are included exactly. The model is validated for the linear problem, and non-linear calculations are compared with experimental measurements for a ship moored in an L-shaped harbour. Qualitatively, the non-linear features of the dynamical system are successfully captured by the model. Some tuning, in the form of empirically obtained damping coefficients, is, however, necessary in order to get a reasonable prediction of the response amplitudes near resonance, when the linear hydrodynamic damping is very small.

DOI of original article: http://dx.doi.org/10.1016/j.coastaleng.2012.07.003. ⁎ Correspondence to: Z. Yang, State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116023, China. Fax: +86 411 84708526. ⁎⁎ Corresponding author. Fax: +86 411 84708526. ⁎⁎⁎ Corresponding author. Fax: +45 4588 4325. E-mail addresses: ouyangzhiwen@gmail.com (Z. Yang), liusx@dlut.edu.cn (S. Liu), hbb@dtu.mek.dk (H.B. Bingham), lijx@dlut.edu.cn (J. Li).