Abstract
The understanding of the motion of water waves is of fundamental importance for manyapplications related to disciplines such Naval and Marine Hydrodynamics, Coastaland Environmental Engineering, and Oceanography. Even under the simplifyingassumptions that fluid is ideal and the flow irrotational, the complete mathematicalformulation of the free-boundary problem of water waves is very complicated and itstheoretical and numerical study comprises a contemporary direction of research.In the first part of this thesis, a new system of two Hamiltonian equations is derived,governing the evolution of free-surface waves. This system is coupled with a time independent Coupled Mode System (CMS), called the substrate problem, that accountsfor the internal fluid kinematics. The derivation is based on the use of Luke’svariational principle in conjunction with an appropriate series representation of thevelocity potential. The critical feature of this approach, initiated in (Athanassoulis &Belibassakis ...
The understanding of the motion of water waves is of fundamental importance for manyapplications related to disciplines such Naval and Marine Hydrodynamics, Coastaland Environmental Engineering, and Oceanography. Even under the simplifyingassumptions that fluid is ideal and the flow irrotational, the complete mathematicalformulation of the free-boundary problem of water waves is very complicated and itstheoretical and numerical study comprises a contemporary direction of research.In the first part of this thesis, a new system of two Hamiltonian equations is derived,governing the evolution of free-surface waves. This system is coupled with a time independent Coupled Mode System (CMS), called the substrate problem, that accountsfor the internal fluid kinematics. The derivation is based on the use of Luke’svariational principle in conjunction with an appropriate series representation of thevelocity potential. The critical feature of this approach, initiated in (Athanassoulis &Belibassakis 1999, 2000), is the use of an enhanced vertical modal expansion thatserves as an exact representation of the velocity potential in terms of horizontal modalamplitudes. Herein, we study further and justify this expansion. In particular, it isproved that under appropriate smoothness assumptions, the modal amplitudes exhibitrapid decay, ensuring that the infinite series can be termwise differentiated in the nonuniformfluid domain, including its physical boundaries. This justifies the variationalprocedure and proves that the resulting system, called Hamiltonian/Coupled-ModeSystem (HCMS), is an exact reformulation of the complete hydrodynamic problem,and therefore it is valid for fully nonlinear waves and significantly varying seabeds.In fact, it is a modal version of Zakharov/Craig-Sulem Hamiltonian formulation(Zakharov 1968, Craig & Sulem 1993) with a new, versatile and efficient representationof the Dirichlet to Neumann operator (DtN) operator, needed for the closure of thenon-local evolution equations. No smallness assumptions are made, that is, thepresent approach is a non-pertubative one. In HCMS, the DtN operator is definedin terms of one of the unknown modal amplitudes, namely, the free-surface modalamplitude. Its computation avoids the numerical solution of the Laplace equation inthe whole fluid domain, required in direct numerical methods, and the evaluationof higher-order horizontal derivatives, required in Boussinesq or other higher-orderpertubative methods. Instead, a system of horizontal second order partial differentialequations needs to be solved.In the second part of the thesis, our theoretical results are exploited for the numericalsolution of various nonlinear water wave problems. The backbone of our numericalmethod is the computation of the DtN operator through its modal characterizationwhich is achieved by a fourth-order finite-difference method. The accuracy andconvergence of the new characterization of the DtN operator is assessed in test casesof highly non-uniform domains and our theoretical findings concerning the rate ofdecay of the modal amplitudes are numerically verified. This preparatory investigationdemonstrates that a small number of modes suffices for the accurate computation ofthe DtN operator even in extremely deformed domains. Subsequently, a number ofphysically interesting water-wave problems, over flat as well as varying bathymetry,are considered. The first application concerns the computation of steady travellingperiodic waves above a flat bottom for a wide range of nonlinearity and shallownessconditions up to the breaking limit. Next, we turn to the time integration of thenew evolution equations by employing a fourth-order Runge-Kutta for the simulationof wave interactions with variable bathymetry and vertical walls. Computationsare validated against predictions from laboratory experiments and other numericalmethods in connection with several nonlinear phenomena. In particular we studythe interaction of solitary waves with a vertical wall (reflection) and a plane beach(shoaling) and the transformation of regular incident waves past submerged obstacles(harmonic generation) or undulating bathymetry (Bragg scattering). Numericalresults on the interaction of a solitary wave with an undulating bottom patch are alsoprovided. The present method provides stable and accurate long time simulationsof nonlinear waves in various depths, from deep to shallow waters, avoiding thecomputational burden of direct numerical methods as well as the use of filteringtechniques, frequently required in pertubative approaches.
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