203 research outputs found
Mathematical modelling in Swirling flows; a Hamiltonian perspective
No full textDe aanwezigheid van wervels is een essentieel onderdeel in veel industri¨ele processen zoalsmenging, scheiding, stabilisatie, etc. Maar het onbedoeld optreden van wervelingen, al dan niet door toedoen van de mens, kan ook grote schade aanrichten; tornado's zijn waarschijnlijk het meest bekend, maar ook de enorme vleugeltip-wervels die zich achter grote vliegtuigen vormen kunnen op z'n minst tot vertraging maar ook tot ongelukken leiden. Het proefschrift begint met het schetsen van het fenomeen `vortex-breakdown' in roterende pijpstromingen. Hierna worden de wiskundige vergelijkingen ten tonele gevoerd, en in het bijzonder haar Hamiltonse structuur. Deze structuur ligt ten grondslag aan een niet-lineaire spectraal methode welke gebruikt wordt om de invloed van verstoringen op speciale stromingen te beschrijven. Meer specifiek, om een spectraal methode te starten, is een (rijke) familie met geparametriseerde basisfuncties nodig, eventueel gerelateerd aan het ongestoorde probleem: in ons geval de stroming van een niet-visceus flu¨idum door een rechte pijp. Deze familie wordt in hoofdstuk 2 geconstrueerd waarbij technieken uit de variatierekening centraal staan. Deze technieken geven de mogelijkheid om stromingen te beschouwen die tot nu toe niet veel aandacht hebben gekregen: de niet-axisymmetrische, tijdsafhankelijke roterende stromingen. In hoofdstuk 3 wordt de lineaire en niet-lineaire stabiliteit van de corresponderende stromingen onderzocht. Hierbij wordt gebruik gemaakt van ide¨een die ontwikkeld zijn door Arnol'd en mensen die door hem ge¨inspireerd zijn. In de hoofdstukken 4 en 5 gebruiken we de geparametriseerde basisfuncties om twee problemen te analyseren: een niet-visceus flu¨idum in een (langzaam) expanderende pijp en de effecten van viscositeit in zowel een rechte als een expanderende pijp. De idee om de dynamica van de parameters te vinden is het best te beschrijven als we het (abstracte) definitie van een Hamiltons of Poisson systeem voor ogen hebben: het is een dynamisch systeem voor functionalen, met bepaalde eigenschappen. Van de benadering wordt geeist dat een aantal goed te kiezen functionalen een dynamica hebben die consistent is. Deze functionalen zijn niet zomaar willekeurig maar corresponderen met duidelijk fysische grootheden en zijn bovendien constanten van beweging. Dit heeft als grote voordeel dat er een eenvoudige terugkoppeling naar experimenten kan zijn. Bovendien is in het algemeen deze eis equivalent met oplosbaarheids voorwaarden (niet-resonantie condities) voor de vergelijking voor de fout. Als resultaat moet een stelsel gekoppelde, niet-autonome gewone differentiaal vergelijkingen worden opgelost, waarbij de dynamische variabele de straal van de pijp (hoofdstuk 4) of een (geschaalde) co¨ordinaat langs de wand van de pijp (hoofdstuk 5) is. Op deze manier wordt met een laag-dimensionaal model een goede beschrijving gegeven van het snelheidsveld tot aan `breakdown' en de loslating van de grenslaag in een expansie. Bovendien worden er suggesties gedaan om modellen op te stellen voor de beschrijving van het snelheidsveld voorbij `breakdown' en loslating
The splitting of solitary waves over shallower water
No full textThe Korteweg-de Vries type of equation (called KdV-top) for uni-directional waves over a slowly varying bottom that has been derived by Van Groesen and Pudjaprasetya [E. van Groesen, S.R. Pudjaprasetya, Uni-directional waves over slowly varying bottom. Part I. Derivation of a KdV-type of equation, Wave Motion 18 (1993) 345¿370.] is used to describe the splitting of solitary waves, running over shallower water, into two (or more) waves. Results of numerical computations with KdV-top are presented; qualitative and quantitative comparisons between the analytical and numerical results show a good agreement
Derivation of the NLS breather solutions using displaced phase-amplitude variables
No full textBreather solutions of the nonlinear Schrödinger equation are derived in this paper: the Soliton on Finite Background, the Ma breather and the rational breather. A special Ansatz of a displaced phase-amplitude equation with respect to a background is used as has been proposed by van Groesen et. al. (2006). Requiring the displaced phase to be temporally independent, has as consequence that the dynamics at each position is described by the motion of a nonlinear autonomous oscillator in a potential energy that depends on the phase and on the spatial phase change. The relation among the breather solutions is confirmed by explicit expressions, and illustrated with the amplitude amplification factor. Additionally, the corresponding physical wave field is also studied and wavefront dislocation together with phase singularity at vanishing amplitude are observed in all three cases
Characterization and Simulation of localized stated in Optical Structures
No full textThis thesis deals with the characterization of localized states of resonant optical structures. We restrict ourself to one and two dimensional photonic crystal structures
Optimization of variational Boussinesq models
No full textThe research presented in this thesis is concentrated mostly around the properties of the Variational Boussinesq Model (VBM). The VBM is a model for waves above a layer of ideal fluid, which conserves mass, momentum, energy, and has decreased dimensionality compared to the full problem. It is derived from the Hamiltonian formulation via an approximation of the kinetic energy, and can provide approximate dispersion characteristics. In particular, dispersive properties of a model are important for a large number of practical applications, especially if modelled waves have a broad spectrum. The most important assessment for a model is a comparison between results obtained from the model and experimental data. We show how well the VBM performs in a few complex cases, such as focussing wave groups with broad spectra. The real-life experiments with this type of waves were performed at the facilities of the MARIN hydrodynamic laboratory (Maritime Research Institute Netherlands). In order to put the work into the context of existing research, we compare the obtained results to those of other models, in particular the AB-equation which is briefly discussed in the thesis as well. Having free parameters in the VBM gives opportunities to optimize the parameters depending on the specifcs of the application. We explore possibilities of such optimization, interchanging norms in different optimization criteria. Then the question rises, which of these norms is the best? We come up with the novel kinetic energy optimization criterium that is natural for the VBM and gives seemingly the best result in the considered test cases. Another important property from the practical viewpoint is how well a model simulates reflection. We study the reflective properties of the VBM and compare them to previous results by other authors. We also derive and investigate an analytical reflection model of the WKB (Wentzel–Kramers–Brillouin) type, whose performance is surprisingly good. For the numerical implementation of the VBM we employ a Finite Element Method (FEM), and a pseudo-spectral method in case of the AB-equation. In the thesis we concentrate on errors caused by the modelling process, and provide details of the numerical implementation in the first chapters and in the Appendix. This includes an embedded influx condition, which we use in signalling problems for both the AB-equation and VBM
Hawassi-AB modelling and simulation of fully dispersive nonlinear waves above bathymetry
No full textA challenge in the study of water waves is that the motion can exhibit qualitative differences at different scales such as deep water versus shallow water, long wavelength versus short wavelength, etc.. That leads to a restriction in the applicability of the existing wave models. This dissertation concerns the development of an accurate and efficient model that can simulate wave propagation in any range of wave lengths, in any water depth and moreover can deal with various inhomogeneous problems such as bathymetry and walls, leading to wave structure interactions. Based on a variational principle of water waves, the dynamic equations are of Hamiltonian form for wave elevation and surface potential with non-local operators applied to the canonical surface variables. Since the kinetic energy cannot be expressed explicitly in the basic variables an approximation is required. The approximate Hamilton equations are expressed in pseudo-differential operators (PDO) applied to the surface variables. The PDO has a physical interpretation related to the phase velocity. The phase velocity as function of wave length is specified by a dispersion relation. A spatial-spectral implementation with the global PDO or a generalization with global Fourier integral operators (FIO) can retain the exact dispersion property of the model. To deal with practical applications, the model with localization methods in the FIO can deal with localized effects such as breaking waves, partially or fully reflective walls, submerged bars, run-up on shores, etc. The inclusion of a fixed-structure in the spatial-spectral setting is a challenging task. The method as presented here perhaps serves as a first contribution in this topic. Performance of the model is shown by comparing the simulation result with measurement data of various long crested cases of breaking and non-breaking waves. The model has been extensively tested against at least 50 measurement data. Moreover, 30 measurement data of wave breaking experiments were designed by the accurate wave model. The models and methods presented in this dissertation have been packaged as software under the name HAWASSI-AB; here HAWASSI stands for Hamiltonian Wave-Ship-Structure Interaction, while AB stands for Analytic Boussinesq. Further information of the software can be found on http://hawassi.labmath-indonesia.org
Mathematical modelling of generation and forward propagation of dispersive waves
No full textThis dissertation concerns the mathematical theory of forward propagation and generation of dispersive waves. We derive the AB2-equation which describes forward traveling waves in two horizontal dimension. It is the generalization of the Kadomtsev-Petviashvilli (KP) equation. The derivation is based on the variational principle of water waves. Similar to its predecessor, the AB-equation, the AB2-equation is dispersive, accurate in second order and can be adjusted for any water depth. Using pseudo-spectral method, the numerical implementation of the AB2-equation can be done easily since exact dispersion is described by a nonpolynomial pseudo-differential operator that can easily be dealt with in spectral space. For wave generation, we derive various models that describe excitation if the wave elevation (or fluid potential) at a certain position is given. The wave generation discussed in this dissertation is done by an embedded source term added to the equation(s) of water wave motion. In this way, we transform the problem of homogeneous boundary value problem into an inhomogeneous problem. We derive the source functions for any kind of waves to be generated and for any dispersive equation including the general case of (linear) dispersive Boussinesq equations. For a dispersive wave equation, the source is not unique; many choices can be taken as long as they satisfy a certain source - influx signal relation. This is different from the actual condition in a hydrodynamic laboratory where there is a one to one correspondence between influx signal and the generated waves. We designed a set of experiments for oblique wave interaction. The aim of the experiment is to test the applicability and the performance of the AB2-equation and the influxing technique. These experiments were executed in a water tank of MARIN hydrodynamic laboratory. The experiments are performed by generating two oblique waves from two sides of the basin and let the waves collide. We compare the measurements from the experiments and the AB2 simulation results. The AB2 simulations and the MARIN measurements are in satisfactory agreement, showing the bichromatic beat wave pattern, even for large nonlinear effects
FREAK WAVE: prediction and its generation from phase coherence
No full textThe processes that lead to the appearance of an extreme wave are not unique: one extreme wave may occur due to different mechanisms than another extreme wave. This gives challenges in the study of extreme waves, which are also called ’freak’ waves, or ’rogue’ waves when they satisfy certain conditions on the wave height compared to (the average of) neighbouring waves. After a freak wave with 18.5 m crest height hit the Draupner oil platform in the North sea on 1 January 1995, the investigation in the topic of freak wave has become more intense. It has been widely recognized that freak waves in the ocean are an important cause of accidents, and that they occur more frequent than expected. It is therefore important to understand the freak wave appearance. This dissertation is intended to understand the development of irregular waves into freak waves, restricting ourselves to waves travelling in one horizontal direction, corresponding to long-crested waves in the ocean. It contains new concepts that can explain the mechanism that can lead to a freak wave. The mechanism for freak wave appearance that is investigated is that of phase coherence: the more coherent the phases of waves contributing to the freak wave, the higher the crest of the freak wave will be. A freak wave for which all waves contribute to the highest possible crest is the so-called maximal wave. Although this concept is used in hydrodynamic laboratories to generate high waves in a tank, much higher than can be achieved with a single stroke of the wave flap, it is extremely unlikely to occur in the real open seas so that the occurrence of freak waves in a random wave field has to be investigated. It turns out that the more flexible notion of pseudo-maximal wave as a description of an extreme wave with less coherent phase is more applicable for extreme wave occurrence in the ocean. Even less restrictive, a weak pseudo-maximal wave that only takes into account the most energy carrying waves can be used to describe an extreme wave as well. These proposed concepts are based on linear wave theory, while nonlinear contributions are added by the Stokes correction. By understanding that an extreme wave may occur as a consequence of linear coherence, a linear prediction method based on minimizing the total wave phase can estimate the time and position of an extreme wave. A further contribution of this dissertation investigates the local energy propagation that leads to a freak wave. A freak wave is mostly developed from a localised wave group that contains a considerable amount of energy that evolves into successive states with even higher coherence. The wavelet transformation is used effectively for identifying the spectral energy distribution of the group events and its evolution. The local energy of waves in a wave group interact and build a larger amplitude. This interaction is based on local dispersive effects within the wave group. A high correlation between the local coherence and the wave amplitude showed that the local coherence can be a good indicator of the appearance of freak waves
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