Nonlinear water waves over varying bathymetry: Theoretical and
numerical study using variational methods

Papoutsellis, Christos

dc.contributor.author	Papoutsellis, Christos	en
dc.date.accessioned	2017-03-27T08:16:08Z
dc.date.available	2017-03-27T08:16:08Z
dc.date.issued	2017-03-27
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/44741
dc.identifier.uri	http://dx.doi.org/10.26240/heal.ntua.2614
dc.rights	Default License
dc.subject	Zakharov Hamiltonian formulation, Dirichlet to Neumann operator, coupled-mode theories, eigenfunction expansions, dispersion	el
dc.title	Nonlinear water waves over varying bathymetry: Theoretical and numerical study using variational methods	el
dc.contributor.department	Τομέας Ναυτικής και Θαλάσσιας Υδροδυναμικής	el
heal.type	doctoralThesis
heal.classification	Hydrodynamics	en
heal.language	en
heal.access	free
heal.recordProvider	ntua	el
heal.publicationDate	2016
heal.abstract	The understanding of the motion of water waves is of fundamental importance for many applications related to disciplines such Naval and Marine Hydrodynamics, Coastal and Environmental Engineering, and Oceanography. Even under the simplifying assumptions that fluid is ideal and the flow irrotational, the complete mathematical formulation of the free-boundary problem of water waves is very complicated and its theoretical 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 timeindependent Coupled Mode System (CMS), called the substrate problem, that accounts for the internal fluid kinematics. The derivation is based on the use of Luke’s variational principle in conjunction with an appropriate series representation of the velocity potential. The critical feature of this approach, initiated in (Athanassoulis & Belibassakis 1999, 2000), is the use of an enhanced vertical modal expansion that serves as an exact representation of the velocity potential in terms of horizontal modal amplitudes. Herein, we study further and justify this expansion. In particular, it is proved that under appropriate smoothness assumptions, the modal amplitudes exhibit rapid decay, ensuring that the infinite series can be termwise differentiated in the nonuniform fluid domain, including its physical boundaries. This justifies the variational procedure and proves that the resulting system, called Hamiltonian/Coupled-Mode System (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 representation of the Dirichlet to Neumann operator (DtN) operator, needed for the closure of the non-local evolution equations. No smallness assumptions are made, that is, the present approach is a non-pertubative one. In HCMS, the DtN operator is defined in terms of one of the unknown modal amplitudes, namely, the free-surface modal amplitude. Its computation avoids the numerical solution of the Laplace equation in the whole fluid domain, required in direct numerical methods, and the evaluation of higher-order horizontal derivatives, required in Boussinesq or other higher-order pertubative methods. Instead, a system of horizontal second order partial differential equations needs to be solved. In the second part of the thesis, our theoretical results are exploited for the numerical solution of various nonlinear water wave problems. The backbone of our numerical method is the computation of the DtN operator through its modal characterization which is achieved by a fourth-order finite-difference method. The accuracy and convergence of the new characterization of the DtN operator is assessed in test cases of highly non-uniform domains and our theoretical findings concerning the rate of decay of the modal amplitudes are numerically verified. This preparatory investigation demonstrates that a small number of modes suffices for the accurate computation of the DtN operator even in extremely deformed domains. Subsequently, a number of physically interesting water-wave problems, over flat as well as varying bathymetry, are considered. The first application concerns the computation of steady travelling periodic waves above a flat bottom for a wide range of nonlinearity and shallowness conditions up to the breaking limit. Next, we turn to the time integration of the new evolution equations by employing a fourth-order Runge-Kutta for the simulation of wave interactions with variable bathymetry and vertical walls. Computations are validated against predictions from laboratory experiments and other numerical methods in connection with several nonlinear phenomena. In particular we study the 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). Numerical results on the interaction of a solitary wave with an undulating bottom patch are also provided. The present method provides stable and accurate long time simulations of nonlinear waves in various depths, from deep to shallow waters, avoiding the computational burden of direct numerical methods as well as the use of filtering techniques, frequently required in pertubative approaches.	en
heal.advisorName	Athanassoulis, Gerasimos	en
heal.committeeMemberName	Belibassakis, Kostas	en
heal.committeeMemberName	Politis, Gerassimos	el
heal.committeeMemberName	Spyrou, Kostas	el
heal.committeeMemberName	Stratis, Ioannis	el
heal.committeeMemberName	Frantzeskakis, Dimitris	el
heal.committeeMemberName	Katsardi, Vanessa	el
heal.academicPublisher	Σχολή Ναυπηγών Μηχανολόγων Μηχανικών	el
heal.academicPublisherID	ntua
heal.fullTextAvailability	true