Th. P. Gerostathis

A coupled-mode model for the refraction–diffraction of linear waves over steep three-dimensional bathymetry

Abstract A consistent coupled-mode model recently developed by Athanassoulis and Belibassakis [1] , is generalized in 2+1 dimensions and applied to the diffraction of small-amplitude water waves from localized three-dimensional scatterers lying over a parallel-contour bathymetry. The wave field is decomposed into an incident field, carrying out the effects of the background bathymetry, and a diffraction field, with forcing restricted on the surface of the localized scatterer(s). The vertical distribution of the wave potential is represented by a uniformly convergent local-mode series containing, except of the ususal propagating and evanescent modes, an additional mode, accounting for the sloping bottom boundary condition. By applying a variational principle, the problem is reduced to a coupled-mode system of differential equations in the horizontal space. To treat the unbounded domain, the Berenger perfectly matched layer model is optimized and used as an absorbing boundary condition. Computed results are compared with other simpler models and verified against experimental data. The inclusion of the sloping-bottom mode in the representation substantially accelerates its convergence, and thus, a few modes are enough to obtain accurately the wave potential and velocity up to and including the boundaries, even in steep bathymetry regions. The present method provides high-quality information concerning the pressure and the tangential velocity at the bottom, useful for the study of oscillating bottom boundary layer, sea-bed movement and sediment transport studies.

A Coupled-Mode, Phase-Resolving Model for the Transformation of Wave Spectrum Over Steep 3D Topography: Parallel-Architecture Implementation

The problem of transformation of the directional spectrum of an incident wave system over an intermediate-depth region of strongly varying 3D bottom topography is studied in the context of linear theory. The consistent coupled-mode model, developed by Athanassoulis and Belibassakis (J. Fluid Mech. 389, pp. 275-301 (1999)) and extended to three dimensions by Belibassakis et al. (Appl. Ocean Res. 23(6), pp. 319-336 (2001)) is exploited for the calculation of the linear transfer function, connecting the incident wave with the wave conditions at each point in the field. This model is fully dispersive and takes into account reflection, refraction, and diffraction phenomena, without any simplification apart the standard intermediate-depth linearization. The present approach permits the calculation of spectra of all interesting wave quantities (e.g., surface elevation, velocity, pressure) at every point in the liquid domain. The application of the present model to realistic geographical areas requires a vast amount of calculations, calling for the exploitation of advanced computational technologies. In this work, a parallel implementation of the model is developed, using the message passing programming paradigm on a commodity computer cluster. In that way, a direct numerical solution is made feasible for an area of 25 km 2 over Scripps and La Jolla submarine canyons in Southern California, where a large amount of wave measurements are available. A comparison of numerical results obtained by the present model with field measurements of free-surface frequency spectra transformation is presented, showing excellent agreement. The present approach can be extended to treat weakly nonlinear waves, and it can be further elaborated for studying wave propagation over random bottom topography.

A BEM-ISOGEOMETRIC METHOD WITH APPLICATION TO THE WAVEMAKING RESISTANCE PROBLEM OF SHIPS AT CONSTANT SPEED

In the present work IsoGeometric Analysis (IGA), initially proposed by Hughes et al (2005), is applied to the solution of the boundary integral equation associated with the Neumann-Kelvin (NK) problem and the calculation of the wave resistance of ships, following the formulation by Brard (1972) and Baar & Price (1988). As opposed to low-order panel methods, where the body is represented by a large number of quadrilateral panels and the velocity potential is assumed to be piecewise constant (or approximated by low degree polynomials) on each panel, the isogeometric concept is based on exploiting the NURBS basis, which is used for representing exactly the body geometry and adopts the very same basis functions for approximating the singularity distribution (or in general the dependent physical quantities). In order to examine the accuracy of the present method, in a previous paper Belibassakis et al (2009), numerical results obtained in the case of submerged bodies are compared against analytical and benchmark solutions and low-order panel method predictions, illustrating the superior efficiency of the isogeometric approach. In the present paper we extent previous analysis to the case of wavemaking resistance problem of surface piercing bodies. The present approach, although focusing on the linear NK problem which is more appropriate for thin ship hulls, it carries the IGA novelty of integrating CAD systems for ship-hull design with computational hydrodynamics solvers.Copyright

A COUPLED-MODE TECHNIQUE FOR THE PREDICTION OF WAVE-INDUCED SET-UP AND MEAN FLOW IN VARIABLE BATHYMETRY DOMAINS

In the present work, a complete, phase-resolving wave model is coupled with an iterative solver of the mean-flow equations in intermediate and shallow water depth, permitting an accurate calculation of wave set-up and wave-induced current in intermediate and shallow water environment with possibly steep bathymetric variations. The wave model is based on the consistent coupled-mode system of equations, developed by Athanassoulis & Belibassakis (1999) for the propagation of water waves in variable bathymetry regions. This model improves the predictions of the mild-slope equation, permitting the treatment of wave propagation in regions with steep bottom slope and/or large curvature. In addition, it supports the consistent calculation of wave velocity up to and including the bottom boundary. The above wave model has been further extended to include the effects of bottom friction and wave breaking, which are important factors for the calculation of radiation stresses on decreasing depth. The latter have been used as forcing terms to the mean flow equations in order to predict wave-induced set up and mean flow in open and closed domains. Numerical results obtained by the present model are presented and compared with predictions obtained by the mild-slope approximation (Massel & Gourlay 2000), and experimental data (Gourlay 1996).Copyright

THE POSEIDON NEARSHORE WAVE MODEL AND ITS APPLICATION TO THE PREDICTION OF THE WAVE CONDITIONS IN THE NEARSHORE/ COASTAL REGION OF THE GREEK SEAS

In the present work, the Poseidon Nearshore Wave Model (PNWM) developed in the framework of the POSEIDON project 1 , and its application to the prediction of the wave conditions in nearshore/coastal regions of Greek seas is presented. The PNWM is based on a one-way energy coupling between a third-generation, phase-averaged, nearshore wave model and a local phase-resolving model, nested in the first model. The local wave model is supported by the consistent coupled-mode theory, based on an enhanced local-mode representation of the wave velocity field, which except for the propagating and the evanescent modes includes an additional mode, permitting the exact satisfaction of the sloping-bottom boundary condition, even in areas with locally steep bottom slope and large curvature. Thus, three-dimensional diffraction effects are fully treated in the local nested area. Numerical results are presented demonstrating the application of the PNWM to nearshore and coastal sites of the Greek seas.

A WEAKLY NONLINEAR COUPLED-MODE MODEL FOR WAVE-CURRENT-SEABED INTERACTION OVER GENERAL BOTTOM TOPOGRAPHY

A weakly nonlinear, coupled-mode model is developed for the wave-current-seabed interaction problem, with application to wave scattering by steady currents over general bottom topography. Based on previous work by the authors (Athanassoulis & Belibassakis [1], Belibassakis et al [2]), the vertical distribution of the scattered wave potential is represented by a series of local vertical modes containing the propagating mode and all evanescent modes, plus an additional term accounting for the bottom boundary condition when the bottom slope is not negligible. Using the above representation, in conjunction with Luke’s [3] variational principle, the wave-current-seabed interaction problem is reduced to a coupled system of differential equations on the horizontal plane. If only the propagating mode is retained in the vertical expansion of the wave potential, and after simplifications, the present system is reduced to an one-equation model compatible with Kirby’s [4] mild-slope model with application to the problem of wave-current interaction over slowly varying topography. The present coupled-mode system is discretized on the horizontal plane by using a second-order finite difference scheme and numerically solved by iterations. Numerical results are presented for two representative test cases, demonstrating the importance of the first evanescent modes and the sloping-bottom mode. The analytical structure of the present model facilitates its extension to treat fully non-linear waves, and it can be further elaborated to study wave propagation over random bottom topography and general currents.Copyright

Transformation of wave conditions in nearshore and coastal areas by a 3D coupled-mode wave model

The fully dispersive, coupled-mode model introduced by Athanassoulis & Belibassakis (1999), and further extended to 3D by Belibassakis et al. (2001), Gerostathis et al. (2008) and modified to include effects of ambient currents by Belibassakis et al. (2011), is exploited in order to transform wave conditions from offshore to nearshore and coastal areas of interest, which are characterized by possibly non-mild spatial inhomogeneities. The present model takes fully into account reflection, refraction, and diffraction phenomena, due general bottom topography, permanent structures and ambient currents, and includes dissipation of wave energy due to bottom friction and wave breaking. Numerical results are presented and compared with other methods and experimental data for model validation, demonstrating also the usefulness and practical applicability of the present approach, as for example, for determining the wave conditions in nearshore and coastal areas for studying further wave-seabed-body interaction problems with large floating structures. by Athanassoulis and Belibassakis (1999) for harmonic waves propagating over variable bathymetry regions, and then generalised to 3D by Belibassakis et al. (2001), and applied to the transformation of wave systems over general bottom topography by Gerostathis et al (2008). The above model has been further extended to treat weakly and fully nonlinear waves over general bottom topography (Athanassoulis & Belibassakis 2002, Belibassakis & Athanassoulis 2002, 2011) scattering by shearing currents (Belibassakis 2007) and by ambient currents with general horizontal structure in 3D environment (Belibassakis et al 2011). A main feature of the above coupled-mode model is the consistent calculation of the corresponding deterministic wave field, for each frequency and direction of multichromatic wave system incident to the examined region, through which the transfer functions between any two (possibly different) physical quantities of interest, at any two points in the variable bathymetry domain are obtained. The treatment of the monochromatic wave problems is obtained by formulating a diffraction problem due to seabed and current over general bottom topography. The main features of CMS are: (a) the exact calculation of the wave field (velocity and pressure

CALCULATION OF SHIP HYDRODYNAMIC PROPULSION IN ROUGH SEAS BY NON-LINEAR BEM WITH APPLICATION TO REDUCTION OF ENERGY LOSSES IN WAVES

Environmental conditions corresponding to realistic sea states (which can be rarely considered calm) signif icantly affect ship propulsion due to added wave resistance , wind resistance and other factors, as e.g., continuous r udder motion for steering in adverse conditions. In addit ion, external factors such as ocean currents, which dete rmine the actual flow on the ship, critically affect the actu al behavior of the propulsion system. All the above cause signi ficant additional energy losses that sometimes could drive the propulsion system of a ship at its limits. On the o ther hand, the operation of ship propellers and thrusters in r eal sea conditions is quite different from their design spe cifications, usually considered in calm conditions. For example, the vertical stern motion of the ship significantly aff ects propeller efficiency and becomes dramatically worse if emergence of propeller occurs in high waves. Operat ion of the ship propulsion system in random waves causes significant variations in performance. In this work we examine in detail the effects of wave-induced motio ns of the ship on the modification of propulsive thrust and e fficiency. Our analysis is based on the non-linear Unsteady Bo undary Element Modeling Code UBEM which is applied for the analysis of an unsteadily moving propeller in a wak e field, in conjunction with seakeeping analysis in regular and irregular waves. Results from the present hydrodyna mic analysis, in conjunction with predictions of added resistance, are used to illustrate applicability in the case of an AFRAMAX tanker, investigating the benefits of small regulation of ship speed and engine RPM from the point of view of optimizing ship’s propulsive performance and reduction of energy losses. The present analysis co uld support the development of ship monitoring and deci sion support systems, integrated with engine control sys tems, aiming to maximize operating efficiency in realisti c sea conditions.

Biomimetic marine energy devices in waves and sheared currents

Oscillating hydrofoils in waves and currents are considered as a novel system for the exploitation of this kind of combined renewable energy sources. Special attention is paid to the case of waves and vertically sheared currents, as appearing in the case of nearshore tidal flows. The analysis is based on an appropriate mild-slope model, in conjunction with time-domain Boundary Element Methods. The methodology takes into account the velocity due to waves and currents on the formation of the incident flow, the free surface through the satisfaction of the corresponding boundary conditions and the effects of variable bathymetry and sheared currents. Results are presented concerning the harvested power, demonstrating that significant energy output can be obtained. Our method, after further enhancements and verification, it can be applied for the design and control of such systems, extracting energy from waves in the presence of ambient currents in nearshore regions. For the prediction of wave-seabed-current interaction quite complete and accurate coupled-mode models have been recently developed, with application to the propagation/scattering of water waves over variable bathymetry regions, in the presence of strong spatially varying currents, extending previous simplified mildslope/mild shear models that are applicable to cases of slowly varying bathymetry and current; Belibassakis et al (2011). Also, the problem of transformation of the directional spectrum of an incident wave system over a region of strongly varying three-dimensional bottom topography is further studied in Belibassakis et al (2014), where also the accuracy and efficiency of the coupled-mode method is tested, comparing numerical predictions against experimental data and calculations by phase-averaged model SWAN, Booij et al (1999). Results are shown for various representative test cases, demonstrating the importance of the first evanescent modes and the additional slopingbottom mode when the bottom slope is not negligible. The mutual existence of waves with strong following, oblique or opposing currents at various nearshore places, which otherwise is characterized by quite low wave potential, offers a motivation for comprehensive investigation of such resources and the development of hybrid technological devices, based on novel ideas such as biomimetic systems (see, e.g. BioPower Systems: www.biopowersystems. com/biowave.html). The latter are appropriate for the efficient energy extraction and exploitation of this complex type of renewable energy resources.