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Variational Water-Wave Models and Pyramidal Freak Waves

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Variational Water-Wave Models and Pyramidal Freak Waves Variational water-wave models and pyramidal freak waves Floriane Marie Pauline Gidel Submitted in accordance with the requirements for the degree of Doctor of Philosophy The University of Leeds Department of Applied Mathematics June 2018 ii iii Declaration The candidate confirms that the work submitted is her own, except where work which has formed part of jointly authored publications has been included. The contribution of the candidate and the other authors to this work has been explicitly indicated below. The candidate confirms that appropriate credit has been given within the thesis where reference has been made to the work of others. The work in Chapter 2 of the thesis has appeared in publication as follows: [57] F. Gidel, O. Bokhove and A. Kalogirou, Variational modelling of extreme waves through oblique interaction of solitary waves: application to Mach reflection, Nonlinear Processes in Geophysics, 24 (2017), 43-60, doi: 10.5194/npg-24-43-2017. The original idea was given by O. Bokhove as an extension of his previous collaboration with A. Kalogirou [20]. Therefore, sections 2.2.2, 2.4.1 and 2.4.2 follow the method initially introduced in [20]. The other sections are directly attributable to the candidate’s work. In addition, the candidate wrote the publication fully, with proofreading from A. Kalogirou, O.Bokhove, M. Kelmanson, G. Kapsenberg and T. Bunnik. The work in Chapters 3 and 4 of the thesis has appeared in publication as follows: [58] F. Gidel, O.Bokhove and M. Kelmanson, Driven nonlinear potential flow with wave breaking at shallow-water beaches, Proc. ASME 2017 36th Int. Conf. on Ocean, Offshore and Arctic Eng., OMAE 2017, 1 (2017). The strategies presented in this paper were obtained through discussions and collaboration with O. Bokhove and M. Kelmanson. The candidate derived and implemented the mathematical and numerical models, and wrote the publication, with proofreading from O. Bokhove, M. Kelmanson, T. Bunnik and G. Kapsenberg. The experiments in Chapter 5 were planned with and supervised by T. Bunnik and G. Kapsenberg. This copy has been supplied on the understanding that it is copyright material and that no quotation from the thesis may be published without proper acknowledgement. The right of Floriane Marie iv Pauline Gidel to be identified as Author of this work has been asserted by her in accordance with the Copyright, Designs and Patents Act 1988. c 2018 The University of Leeds and Floriane Marie Pauline Gidel v A` Guy Gidel, et a` ma famille, sans qui ce projet n’aurait pas pu aboutir vi vii Acknowledgements This thesis would not have been possible without the support of many great individuals, to whom I want to dedicate the first lines of this manuscript. First, many thanks to my supervisors, Onno Bokhove and Mark Kelmanson, for their help and advice during the last four years. Onno, thank you for creating and leading this great project, and for having given me the opportunity to take part in it. I was privileged to experience such a rewarding and instructive work and that would not have been possible without your passion and infectious enthusiasm about research. This thorough project was the best way to begin my research career, so I sincerely thank you for your trust, your involvement and your guidance. Mark, thank you for your encouraging feedbacks that have been essential to keep me motivated and stimulated me to always do my best. Thanks also for your patience when correcting hundreds of times the same “Frenglish” typos; your insightful suggestions and advice have greatly improved the quality of this manuscript. More generally, thank you both for sharing your ambition and helping aim further than I could imagine. The PGR award is a great result of this ambition. Many thanks also to my MARIN supervisors, Tim Bunnik and Geert Kapsenberg, for their great help during my secondment. Thanks for sharing your knowledge and expertise about the maritime industry and its challenges; our interesting discussions helped me to define the objectives of this thesis. I am especially grateful for your assistance before and during the experiments, which considerably improved the quality and impact of this thesis. My sincere gratitude to David Hughes and Jennifer Ryan for assessing this work. Special thanks also to Rainer Hollerbach for his precious help during the first half of my PhD. I am really grateful that you took the time to go through the details with me and helped me overcome my issues. Thanks also to Bulent Duz¨ for his interest in my research and his insightful feedback on my code tutorials. My gratitude to Lawrence Mitchell for his precious help with Firedrake and the optimisation of the solvers, and with his feedback on the tutorials. This PhD thesis would not have been possible without the financial support of the European Marie Skłodowska-Curie Actions (MSCA) Fellowship. Being a MSCA fellow was a chance in viii many aspects and I am really thankful for this support. Essentially, by combining academic and industrial experience, it enabled me to address all important stages of the modelling process; that is, the specification, the modelling, and the experimental validation. I also thank the members of the School of Mathematics who, directly or indirectly, contributed to the smooth progress of this thesis. Thanks also to the “water team” for the nice meetings, during and after work, and, in particular, to Tomasz for sharing both the Leeds and MARIN adventures with me. My warmest thanks to Tom for his precious advice, his help and valuable support from first day to submission, the nice discussions, and, more importantly, all the beers! On a personal note, I want to express all my love and gratitude to my parents, Dominique and Marie-Hel´ ene,` for their unconditional support. Thank you for giving me the chance to grow in the best conditions, for your help with everything I undertake, for comforting me in the difficult moments, for your patience and so much more. In addition, thanks Mum for proofreading parts of this thesis. I am also deeply grateful to my brothers, Pierre and Gregoire,´ for their invaluable support and humour over the years. Growing next to you was the best adventure and I am the proudest sister. Many thanks to Charlotte as well for being such a great sister(-in-law), and to my second family, les Giroux, for their precious support and all the great moments. Special thanks to my dearest friend, Ana¨ıs, for her support, kindness, humour, advice, and even proofreading. Thanks to all my cousins, uncles and aunts, for their encouragements, their interest in my research, their continued support, their advice, and for the lovely family reunions. I am so proud to be part of a such a great family! Thanks to my grandmothers, Paulette and Marie-Ther´ ese,` for their love and unconditional support. I am so proud to have grandmothers who, while being over 90, Skype me when I am abroad, try to understand my research and to read my English publications, visit me in Netherlands, and always encourage me. Thank you both for being great examples. Thanks to the most courageous, Pierre, for encouraging me to embark upon this project and to finish it. Most importantly, thank you for accepting all the constraints of my three years abroad, for your advice in the most difficult moments, for believing in me when I do not, and for your priceless patience. Thanks also to your family for their warm welcome. ix I also want to address my warmest thanks to my friends for accompanying me on this journey despite living in another country. Special thanks to those who visited us in Leeds or Utrecht (or both!) for the great memories (Zoe,´ Clementine,´ Marjory, Julie, Alice, Flo(flo), Sarah, Romain, Eric,´ Pado, Flo(fleur), Marie, Paulbib, Bozzo, les Lopez...). “Bisochat” to the fan club for all the good times that boosted me. I am deeply thankful to my cousin Margaux and my friends Floriane (both!), Julie, Marie for your continued encouragements; your support has been priceless. Thanks to Amelie´ and Clemence´ for all the great moments and laughs. Thanks to all the new friends I met during this adventure and who accompanied me during the last four years; I cannot cite you all, but special thanks to Charlotte, Pauline, Kaatje and Florian for the Dutch souvenirs, and to Mathilde for sharing both UK and Dutch memories! I want to dedicate my last thoughts to my grandfathers, Andre´ Langrand and Guy Gidel, who cannot see the outcome of this project but would have been the proudest. Thank you for giving me your taste for science and for always being proud of me. x xi Abstract A little-known fact is that, every week, two ships weighing over 100 tonnes sink in oceans [32], sometimes with tragic consequences. This alarming observation suggests that maritime structures may be struck by stronger waves than those they were designed to withstand. These are the legendary rogue (or freak) waves, i.e., suddenly appearing huge waves that have traumatised mariners for centuries and currently remain an unavoidable threat to ships, and to their crews and passengers. Thus motivated, an EU-funded collaboration between the Department of Applied Mathematics (Leeds University) and the Maritime Research Institute Netherlands (MARIN) supported this project, in which the ultimate goal, of importance to the international maritime sector, is to develop reliable damage-prediction tools, leading to beneficial impact in terms of both safety and costs. To understand the behaviour of rogue waves, cost-effective water- wave models are derived in both deep and shallow water. Novel mathematical and numerical strategies are introduced to capture the dynamic air-water interface and to ensure conservation of important properties.
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