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A coupled-mode model for the hydroelastic analysis of large floating bodies over variable bathymetry regions
- K. A. BELIBASSAKIS, G. A. ATHANASSOULIS
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- Journal:
- Journal of Fluid Mechanics / Volume 531 / 25 May 2005
- Published online by Cambridge University Press:
- 18 May 2005, pp. 221-249
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The consistent coupled-mode theory (Athanassoulis & Belibassakis, J. Fluid Mech. vol. 389, 1999, p. 275) is extended and applied to the hydroelastic analysis of large floating bodies of shallow draught or ice sheets of small and uniform thickness, lying over variable bathymetry regions. A parallel-contour bathymetry is assumed, characterized by a continuous depth function of the form $h( {x,y}) {=} h( x )$, attaining constant, but possibly different, values in the semi-infinite regions $x {<} a$ and $x {>} b$. We consider the scattering problem of harmonic, obliquely incident, surface waves, under the combined effects of variable bathymetry and a floating elastic plate, extending from $ x {=} a$ to $x {=} b$ and $ {-} \infty {<} y{<}\infty $. Under the assumption of small-amplitude incident waves and small plate deflections, the hydroelastic problem is formulated within the context of linearized water-wave and thin-elastic-plate theory. The problem is reformulated as a transition problem in a bounded domain, for which an equivalent, Luke-type (unconstrained), variational principle is given. In order to consistently treat the wave field beneath the elastic floating plate, down to the sloping bottom boundary, a complete, local, hydroelastic-mode series expansion of the wave field is used, enhanced by an appropriate sloping-bottom mode. The latter enables the consistent satisfaction of the Neumann bottom-boundary condition on a general topography. By introducing this expansion into the variational principle, an equivalent coupled-mode system of horizontal equations in the plate region ($a {\leq} x {\leq} b)$ is derived. Boundary conditions are also provided by the variational principle, ensuring the complete matching of the wave field at the vertical interfaces ($x{=}a$ and $x{=}b)$, and the requirements that the edges of the plate are free of moment and shear force. Numerical results concerning floating structures lying over flat, shoaling and corrugated seabeds are presented and compared, and the effects of wave direction, bottom slope and bottom corrugations on the hydroelastic response are presented and discussed. The present method can be easily extended to the fully three-dimensional hydroelastic problem, including bodies or structures characterized by variable thickness (draught), flexural rigidity and mass distributions.
Three-dimensional Green's function for harmonic water waves over a bottom topography with different depths at infinity
- K. A. BELIBASSAKIS, G. A. ATHANASSOULIS
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- Journal:
- Journal of Fluid Mechanics / Volume 510 / 10 July 2004
- Published online by Cambridge University Press:
- 23 June 2004, pp. 267-302
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The three-dimensional Green's function of water waves in variable-bathymetry regions, associated with the problem of propagation of water waves emitted from a monochromatic point source, is derived and studied. The solution is of interest in its own right but also provides useful information for the formulation and treatment of complex wave–body–seabed interaction problems in variable-bathymetry regions, especially as regards the hydrodynamic characteristics of large structures installed in the nearshore and coastal environment. Assuming a parallel-contour bathymetry, with a continuous depth function of the form $h(x,y) \,{=}\, h(x)$, attaining constant, but possibly different, values at the semi-infinite regions $x \,{<}\, a$ and $x \,{>}\, b$, the problem is reduced to a two-dimensional one, by using Fourier transform. The transformed problem is treated by applying domain decomposition and reformulating it as a transmission problem in the finite domain containing the bottom irregularity. An appropriate decomposition of the wave potential is introduced, permitting the singular part to be solved analytically, and the problem for the regular part to be reformulated as a variational problem. An enhanced local-mode series representation is used for the regular wave potential in the variable-bathymetry region, including the propagating mode, the sloping-bottom mode (see Athanassoulis & Belibassakis 1999), and a number of evanescent modes. Using this representation, in conjunction with the variational principle, a forced system of horizontal coupled-mode equations is derived for the determination of the complex modal-amplitude functions of the regular wave potential. This system is numerically solved by using a second-order central finite-difference scheme. The source-generated water-wave potential is, finally, obtained by an efficient numerical Fourier inversion based on FFT. Numerical results are presented and discussed for various bottom topographies, including smooth but steep underwater trenches and ridges, putting emphasis on the identification of the important features of the near- and far-field patterns on the horizontal plane and on the vertical plane containing the point source. Characteristic patterns of trapped (well-localized) wave propagation over ridges are predicted and discussed.
Extension of second-order Stokes theory to variable bathymetry
- K. A. BELIBASSAKIS, G. A. ATHANASSOULIS
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- Journal:
- Journal of Fluid Mechanics / Volume 464 / 10 August 2002
- Published online by Cambridge University Press:
- 21 August 2002, pp. 35-80
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In the present work second-order Stokes theory has been extended to the case of a generally shaped bottom profile connecting two half-strips of constant (but possibly different) depths, initiating a method for generalizing the Stokes hierarchy of second- and higher-order wave theory, without the assumption of spatial periodicity. In modelling the wave–bottom interaction three partial problems arise: the first order, the unsteady second order and the steady second order. The three problems are solved by using appropriate extensions of the consistent coupled-mode theory developed by the present authors for the linearized problem. Apart from the Stokes small-amplitude expansibility assumption, no additional asymptotic assumptions have been introduced. Thus, bottom slope and curvature may be arbitrary, provided that the resulting wave dynamics is Stokes-compatible. Accordingly, the present theory can be used for the study of various wave phenomena (propagation, reflection, diffraction) arising from the interaction of weakly nonlinear waves with a general bottom topography, in intermediate water depth. An interesting phenomenon, that is also very naturally resolved, is the net mass flux induced by the depth variation, which is consistently calculated by means of the steady second-order potential. The present method has been validated against experimental results and fully nonlinear numerical solutions. It has been found that it correctly predicts the second-order harmonic generation, the amplitude nonlinearity, and the amplitude variation due to non-resonant first-and-second harmonic interaction, up to the point where the energy transfer to the third and higher harmonics can no longer be neglected. Under the restriction of weak nonlinearity, the present model can be extended to treat obliquely incident waves and the resulting second-order refraction patterns, and to study bichromatic and/or bidirectional wave–wave interactions, with application to the transformation of second-order random seas in variable bathymetry regions.
A consistent coupled-mode theory for the propagation of small-amplitude water waves over variable bathymetry regions
- G. A. ATHANASSOULIS, K. A. BELIBASSAKIS
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- Journal:
- Journal of Fluid Mechanics / Volume 389 / 25 June 1999
- Published online by Cambridge University Press:
- 25 June 1999, pp. 275-301
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Extended mild-slope equations for the propagation of small-amplitude water waves over variable bathymetry regions, recently proposed by Massel (1993) and Porter & Staziker (1995), are shown to exhibit an inconsistency concerning the sloping-bottom boundary condition, which renders them non-conservative with respect to wave energy. In the present work, a consistent coupled-mode theory is derived from a variational formulation of the complete linear problem, by representing the vertical distribution of the wave potential as a uniformly convergent series of local vertical modes at each horizontal position. This series consists of the vertical eigenfunctions associated with the propagating and all evanescent modes and, when the slope of the bottom is different from zero, an additional mode, carrying information about the bottom slope. The coupled-mode system obtained in this way contains an additional equation, as well as additional interaction terms in all other equations, and reduces to the previous extended mild-slope equations when the additional mode is neglected. Extensive numerical results demonstrate that the present model leads to the exact satisfaction of the bottom boundary condition and, thus, it is energy conservative. Moreover, it is numerically shown that the rate of decay of the modal-amplitude functions is improved from O(n−2), where n is the mode number, to O(n−4), when the additional sloping-bottom mode is included in the representation. This fact substantially accelerates the convergence of the modal series and ensures the uniform convergence of the velocity field up to and including the boundaries.