Conservation laws of rotational free-surface flows by means of variational principles

Sachinidou, Anastasia; Σαχινίδου, Αναστασία

dc.contributor.author	Sachinidou, Anastasia	en
dc.contributor.author	Σαχινίδου, Αναστασία	el
dc.date.accessioned	2024-02-07T08:38:38Z
dc.date.available	2024-02-07T08:38:38Z
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/58794
dc.identifier.uri	http://dx.doi.org/10.26240/heal.ntua.26490
dc.rights	Default License
dc.subject	Rotational Gravity Waves	en
dc.subject	Variational Principles	en
dc.subject	Noether's Theorem	en
dc.subject	Symmetries	en
dc.subject	Conservation Laws	en
dc.title	Conservation laws of rotational free-surface flows by means of variational principles	en
heal.type	bachelorThesis
heal.classification	Fluid Dynamics	en
heal.language	en
heal.access	free
heal.recordProvider	ntua	el
heal.publicationDate	2023-10-06
heal.abstract	Variational principles have provided, since Lagrange’s monumental work (Mécanique Analytique, 1815), an effective framework for the systematic study of fluid dynamics. Hamilton’s principle and Lagrangian action symmetries are two fundamental tools encountered in variational methods that give rise to the variational formulation of the complete fluid flow problem and to the variational derivation of conservation laws. Since the understanding of the motion of rotational free-surface waves is of fundamental importance for a plethora of applications in naval, coastal and marine hydrodynamics, the question of applying the variational methods to such flows seems to be, not only interesting, but justified too. In this work, we address this question in the Lagrangian framework (approach), where the fluid continuum is modeled as continuously distributed fluid parcels. Then, the variational formulation for the free-surface flow problem is a straightforward extension of Hamilton’s principle for discrete systems. Conservation of mass is embedded in the principle via two distinct ways, which results in either a primitive or a constrained action functional. In either case, the differential equations governing the bulk fluid motion are derived by using the standard methods of Calculus of Variations, while the kinematic and dynamic boundary conditions are obtained by considering and treating appropriately the boundary terms that result after the variation of the action functional. Detailed investigation is next made in the class of variational problems which involve functionals defined on a variable region. In this case, the variational formula contains two types of variations, the local ones due to variations of the fields at a fixed point of the domain, and the boundary ones, due to the variation of the boundary. Continuous infinitesimal transformations leaving invariant the functional (symmetries) result in the well-known Noether’s First Theorem. Under the assumption that the fields satisfy the equations of motion, Noether’s Theorem states that, for each symmetry there exists a conservation law, and provides the formula to obtain it. One of the propositions of this work is that, in fact, along the extremals, the Weak Formulation of the Euler-Lagrange equations of motion, if taken in the specific forms driven by the symmetries the functional, provides an alternative (apparently equivalent) systematic approach to obtain invariant quantities. To address the question regarding the efficiency of the two methods, we explore a very important symmetry in the context of vortical free-surface flows, arising from the relabeling of fluid parcels. The primitive action functional is invariant (relabeling symmetry), and this gives rise to Cauchy’s invariant equations. This is proved both by using the Weak Formulation approach and Noether’s Theorem. The former proof is straightforward, simpler than the second one, and provides us with Cauchy’s invariants without any additional assumptions (except the standard smoothness assumptions). The analogous procedure with Noether’s Theorem seems to need additional assumptions, concerning the variations on the boundary and the initial state of the fluid, restricting unnecessarily the class of flows considered. Therefore, our approach seems to be simpler and more general in the treatment of this case. The analysis is completed with a thorough investigation of Cauchy’s invariant equations and their consequences. It is shown that a number of well-known vorticity theorems and conservation laws (e.g., Ertel’s Potential Vorticity, Circulation Theorem, Weber’s transformation) can be derived as consequences of Cauchy’s invariants (written either in Lagrangian framework or in Eulerian terms), showcasing the significance of the relabeling symmetry.	en
heal.advisorName	Belibassakis, Kostas	en
heal.advisorName	Athanassoulis, Gerassimos	en
heal.committeeMemberName	Belibassakis, Kostas	en
heal.committeeMemberName	Grigoropoulos, Grigoris	en
heal.committeeMemberName	Angelou, Manolis	en
heal.academicPublisher	Εθνικό Μετσόβιο Πολυτεχνείο. Σχολή Ναυπηγών Μηχανολόγων Μηχανικών. Τομέας Ναυτικής και Θαλάσσιας Υδροδυναμικής	el
heal.academicPublisherID	ntua
heal.numberOfPages	227 σ.	el
heal.fullTextAvailability	false