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Variational Approaches to Water Wave Simulations Variational Approaches To Water Wave Simulations Wave Water To Approaches Variational Variational Approaches To Water Wave Simulations Elena Gagarina Elena Elena Gagarina ISBN: 978-90-36-3754-4 Variational Approaches To Water Wave Simulations Elena Gagarina University of Twente 2014 Composition of the Graduation Committee: Chairman and Secretary: Prof. Dr. P. M. G. Apers University of Twente Promotors: Prof. Dr. Ir. J. J. W. van der Vegt University of Twente Prof. Dr. Ir. O. Bokhove University of Leeds Members: Prof. Dr. Ir. E. W. C. van Groesen University of Twente Prof. Dr. H. J. Zwart University of Twente Prof. Dr. Ir. R. H. M. Huijsmans Delft University of Technology Prof. Dr. A. E. P. Veldman University of Groningen Referee: Dr. T. Bunnink MARIN University of Twente, MaCS Group P.O. Box 217, 7500 AE, Enschede, The Netherlands. ISBN: 978-90-36-3754-4 DOI: 10.3990/1.9789036537544 http://dx.doi.org/10.3990/1.9789036537544 Printed by CPI-Wohrmann¨ Print Service, Zutphen. Copyright c 2014 Elena Gagarina, Enschede, The Netherlands. VARIATIONAL APPROACHES TO WATER WAVE SIMULATIONS DISSERTATION to obtain the degree of doctor the University of Twente, on the authority of the Rector Magnificus, Prof. Dr. H. Brinksma, on account of the decision of the Graduation Committee, to be publicly defended on Friday 3 October 2014 at 12:45 by Elena Vitalyevna Gagarina born on 25th of November 1987 in Kiev, USSR The dissertation is approved by: Prof. Dr. Ir. J. J. W. van der Vegt (promotor) Prof. Dr. Ir. O. Bokhove (promotor) To the memory of Sergei Ivanovich Gagarin vi Preface The research presented in this dissertation was carried out at the Mathematics of Com- putational Science (MaCS) group, Department of Applied Mathematics, Faculty of Elec- trical Engineering, Mathematics and Computer Science pf the University of Twente, The Netherlands. The work has been a part of the STW project ”Complex wave-current interactions in a numerical wave tank” and NWO project ”Compatible Mathematical Models for Coastal Hydrodynamics”. The support from the Technology Foundation STW, the Netherlands Organization for Scientific Research (NWO), the Maritime Research Institute Netherlands MARIN is gratefully acknowledged. The project has been associated with the J. M. Burgers Research School for Fluid Dy- namics and the Nanotechnology Research institute of the Univeristy of Twente MESA+. vii viii Contents 1 Introduction 1 1.1 Motivation . .1 1.2 Objectives . .4 1.3 Understanding water waves . .5 1.3.1 Experiments . .5 1.4 Modeling water waves . .6 1.4.1 Mathematical wave models . .6 1.4.2 Numerical methods . .7 1.4.3 Numerical wave tank . .8 1.4.4 Variational dynamics . .9 1.5 Outline . 10 2 Horizontal circulation and jumps in Hamiltonian wave models 13 2.1 Introduction . 13 2.2 New water wave model . 15 2.2.1 Variational principle . 15 2.2.2 Reduction of Hamiltonian dynamics . 18 2.2.3 Hamiltonian dynamics of new water wave model . 19 2.3 Shallow water and potential flow limits . 22 2.4 Hamiltonian Boussinesq reductions of new model . 23 2.4.1 Variational Boussinesq model . 24 2.4.2 Green-Naghdi limit . 26 2.5 Jump conditions for bores . 28 2.5.1 1D Jump conditions for shallow water equations . 29 2.5.2 2D Jump conditions for new water wave model . 33 2.5.3 2D Jump conditions for shallow water equations . 38 2.5.4 1D Jump conditions for new water wave model . 39 ix CONTENTS 2.5.5 1D Jump conditions for variational Boussinesq and Green-Naghdi equations . 40 2.6 Conclusions . 40 2.7 Appendix . 42 2.7.1 Integrals . 42 2.7.2 Variational Reynolds’ transport theorem . 43 3 Symplectic Time Discontinuous Galerkin Discretizations for Hamiltonian Sys- tems 49 3.1 Introduction . 49 3.2 Dynamics of a Hamiltonian system with one degree of freedom . 51 3.3 Variational discontinuous Galerkin time discretization of a Hamiltonian system . 52 3.3.1 Discrete functional . 52 3.3.2 First order variational time discretization . 54 3.3.3 Second order variational time discretization: Stormer-Verlet¨ . 55 3.3.4 Second order time discretization: Modified symplectic midpoint scheme . 58 3.4 New variational time integrators . 60 3.4.1 Explicit second order accurate time integration scheme . 61 3.4.2 Third order scheme . 62 3.5 Applications to non-autonomous systems . 69 3.5.1 Damped oscillator . 71 3.5.2 Forced oscillator . 74 3.6 Conclusions . 78 3.7 Appendix . 78 3.7.1 Continuum harmonic oscillator . 78 3.7.2 Stormer-Verlet¨ scheme . 80 3.7.3 New explicit second order variational time integration scheme . 82 3.7.4 Extended continuum system . 89 4 Variational space-time (dis)continuous Galerkin method for nonlinear free surface water waves 93 4.1 Introduction . 94 4.2 Variational Description of Nonlinear Inviscid Water Waves . 96 4.3 Variational Finite Element Formulation . 99 4.3.1 Tessellation . 99 4.3.2 Function spaces and approximations . 100 4.3.3 Space-plus-time formulation . 102 4.3.4 Dynamics . 109 4.4 Numerical Results . 113 4.4.1 Verification . 113 4.4.2 Validation against laboratory data . 115 x CONTENTS 4.5 Application of new third order symplectic time integrator . 123 4.5.1 Nonlinear potential flow water waves . 127 4.6 Conclusions . 132 4.7 Appendix . 133 4.7.1 Evaluation of finite element matrices . 133 4.7.2 Variations of the kinetic energy . 134 5 Conclusions 137 5.1 Recommendations . 137 Summary 139 Samenvatting 141 Bibliography 143 Acknowledgments 157 xi CONTENTS xii Chapter 1 Introduction 1.1 Motivation A wise man once said that one can observe fire and water forever. The elements are fascinating indeed, what is their secret? Perhaps, the reason is the subtle interplay between life and death. We don’t exist without water, but it can also easily kill. Water has always been a crucial part in the progress of mankind. The largest, richest, and most important cities are those that have easy access to a convenient water transportation route. This allows the exchange of goods and knowledge between civilizations. For many centuries explorers and merchants were sacrificing their lives in a battle with nature to explore new shipping routes and distant countries. It is no surprise, that understanding the behavior of water waves became of great importance to humanity. This was also motivated by the many natural disasters that had a great impact on society. One of the most dangerous events for coastal towns is a tsunami wave. The Santorini eruption and subsequent tsunami are believed to be the cause of the Minoan civilization downfall. The disastrous 1755 Lisbon earthquake and tsunami, Fig. 1.1(a), influenced the mind of many Europeans. For instance, Voltaire, Rousseau, and Kant were trying to comprehend in their works a power so supernatural and unfriendly. An extensive overview of floods, tsunamis and other natural disasters can be found in [117]. These events have also extensively motivated the visual arts, see [84]. The recent tragic events of the Boxing Day Tsunami in 2004 and the 2011 Tohoku Tsunami reveal how unprotected modern society still can be when disaster strikes. Possi- ble ways of protection against disastrous waves, such as dikes, wave breakers, and man- grove forests, are therefore of great importance. A novel way to protect people from a devastating tsunami is an alarm system that is based on accurate and fast methods to simulate wave propagation and run-up in the near coastal zone. On the open seas tsunamis are hardly noticeable, but severe storms, resulting in large 1 CHAPTER 1. INTRODUCTION Figure 1.1: (a) A copper engraving showing Lisbon, devastated by the earthquake and subsequent tsunami in 1755. Original in: Museu da Cidade, Lisbon. Source: Wikipedia.org. (b) ”The Ninth Wave” – a famous painting of Russian Armenian marin- ist Ivan Aivazovsky. Collection of the State Russian Museum, St. Petersburg. Source: Wikipedia.org. Figure 1.2: (a) Tsunami wave enters Miyako city in Japan in 2011. Authorship: Mainichi Shimbun /Reuters. Source: reuters.com. (b) The Draupner wave or New Year’s wave is one of the first measured rogue waves. It occurred at the Draupner platform in the North Sea on 1 January 1995. Source: Wikipedia.org. and sometimes extreme waves, present a serious danger to ships and offshore structures. Recently, special attention has been given to rare extreme wave events, called freak or rogue waves. In 1978 the German carrier Munchen¨ sank. The investigation into this acci- dent revealed that the ship was most likely damaged by a wave of up to 30 m height. In 1942 the passenger ship ”Queen Mary” was transporting troops from the USA to Europe. Sixteen thousand people were on board when a wave of 28 meters height stroke the ship’s side and tilted it by 52 degrees [95]. Those people were just lucky to survive. Never- theless, the actual causes of ship damages and losses in deep water often remain unclear. Freak waves, which used to be a scary fisherman tale, now are proven to exist [76]. One of the first recorded freak waves is the Draupner wave, see Fig. 1.2(b), it had a maxi- 2 1.1. MOTIVATION mum height of 25.6m, which more than twice exceeded the significant wave height. The southern tip of Africa is also a place where extreme waves happen more frequently, see Fig. 1.3(b). One of the possible mechanisms to explain this is the special kind of wave- current interaction occurring in this region due to the Agulhas current, which is opposite to the direction of predominant winds. In order to understand extreme wave phenomena it is important to define which sea states can result in the occurrence of extreme waves.
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