Quasi two-dimensional turbulent flows, like mesoscale oceanic and large-scale atmospheric flows, create finite-time coherent structures, compact fluid masses that resist mixing for finite-time despite the turbulent nature of the ambient flow. These coherent structures significantly affect the mixing and transport of fluid elements. In return, the transport of passive scalars like heat, humidity, salinity, chemical concentration, nutrients and even algae has a substantial impact on countless geophysical phenomena. Thus, in order to understand these effects reliable methods for coherent structure detection and the identification of their boundaries are necessary. Here, in this thesis, we present two contributions in this regard. First, we present improvements of the transfer operator approach, an established stochastic approach for the detection of almost-invariant and coherent sets. The approach approximates the transport properties of a complicated flow by a linear transfer operator and aims to partition a given domain in multiple sets such that the inter-set mass transport is minimized. The improvements include the introduction of mixing boundary conditions in stationary and time-dependent flows. By modifying the transfer operator we couple the filaments that surround the coherent and jointly rotating fluid volume such that effectively only two non-communicating sets remain: the coherent eddy core and the ambient flow. This significantly stabilizes the inference of coherent eddy cores and makes the use of popular but error-prone clustering techniques unnecessary. In addition, we discuss the identification of temporally consistent areas of increased coherence. Instead of coherent structures that are defined by advected non-filamenting masses, the concept describes consistent and moving patches of reduced mixing whose mass can change over time. This permits the decoupling of the coherence time scale from the time window under consideration. Both modifications are used to study the transport properties of eight selected Baltic Sea eddies. Secondly, we introduce the MSCS-search, a new algorithm for the inference of finite-time coherent volumes that is solely based upon the concept of convexity. Persistent convexity is a sufficient condition for coherence in two-dimensional flows if coherent structures are understood as non-filamenting volumes. However, convexity has never been considered as condition of coherence, even though some methods use it, for practical reasons, as an explicit constraint. The approach identifies the largest structure inside a given volume that remains star-convex with respect to a given reference trajectory within a given time window. We test the approach thoroughly and our results show that the approach yields good and reliable estimations of coherent structures in all test cases. Moreover, since the results depend explicitly on the considered time window, the results are intuitive and enable the identification and study of filaments. The novel approach is then used to re-evaluate transport processes in the data set of Baltic eddies.:1 Introduction 2 Theoretical Background 2.1 Methods 2.1.1 Okubo-Weiss criterion 2.1.2 Finite-Time Lyapunov Exponents 2.1.3 Lagrangian Descriptors and Lagrangian averaged vorticity deviation 2.2 Models 2.2.1 Euler equation 2.2.2 Stationary Gaussian Blob Model 2.2.3 Periodically Perturbed Gaussian Blob Model 2.2.4 Bickley-Jet 3 Transfer Operator Approach I: State of the Art 3.1 Frobenius-Perron Operator 3.2 Markov-Chains 3.3 Laplacian matrices 3.4 Finding Almost-Invariant Sets 3.5 Finding Coherent Sets 3.5.1 Coherent pairs as tuples 3.5.2 Coherent pairs as triples 3.6 Spectral Clustering 3.7 Critique and Discussion 4 Transfer Operator Approach II: Mixing Boundary Conditions in Stationary Flows 4.1 Overview 4.2 Estimation of Transfer Probability Matrices P 4.3 Mixing Boundary Conditions 4.4 Thresholding 4.5 Results 4.5.1 Robustness with respect to parameters 4.5.2 Comparison and alternatives 4.5.3 Handling of false positives 4.5.4 Periodically perturbed flow 4.6 Closing Remark 5 Transfer operator approach III: Analysis of Oceanic Transport 5.1 Mixing Boundary Conditions: Time-Dependent Flows 5.1.1 Method 5.1.2 Results 5.2 Temporal consistency 5.2.1 Method 5.2.2 Results 5.3 Treatment of Coastal Effects 5.4 Application to Baltic velocity fields 5.4.1 Method 5.4.2 Results 5.5 Closing Remark 6 Prototypes of Coherent Sets: Star-Convex Structures 6.1 Mathematical Considerations 6.2 MSCS-Algorithm 6.3 Extracting star-convex sub-volumes 6.4 Results 6.4.1 Stationary Gaussian blob model 6.4.2 Bickley-Jet 6.4.3 Two dimensional inviscid flow 6.4.4 Real data sets 6.5 Closing Remark 7 Conclusion and Outlook / Es ist bekannt, dass quasi-zweidimensionale turbulente Strömungen, wie etwa Strömungen an der Meeresoberfläche oder großskalige Atmosphärenbewegungen, kohärente Strukturen ausbilden, kompakte Volumina, welche einer Mischung mit dem umgebenden Material für endliche Zeit widerstehen, obwohl die Strömung als solche turbulent ist. Diese Strukturen haben einen signifikanten Einfluss auf den Transport von Fluidelementen. Der Transport von passiven skalaren Größen, wie etwa Wärme, Feuchtigkeit, Salzgehalt, Nährstoffgehalt, jegliche Konzentration che- mischer Stoffe und sogar Algendichte hat wiederum einen Effekt auf unzählige geophysikalische Phänomene. Um diese Phänomene im Detail zu verstehen, sind zuverlässige Methoden für die Detektion von kohärenten Strukturen und die Identifikation ihrer Grenzen notwendig. In der vorliegenden Arbeit präsentieren wir zwei Beiträge zur Lösung dieses Problems. Als erstes präsentieren wir Verbesserungen der Transferoperatormethode. Diese etablierte Methode zur Identifikation von fast-invarianten und kohärenten Mengen approximiert die Transporteigenschaften eines Flusses mithilfe eines linearen Transferoperators, mit dem Ziel, das betrachtete Gebiet in Bereiche zu unterteilen, welche den Materialaustausch zwischen den Bereichen minimieren. Die Verbesserungen beinhalten die Einführung von mischenden Randbedingungen in stationären und nicht-stationären Flüssen. Dabei wird der Transferoperator so modifiziert, dass Filamente, welche die kohärente Struktur umgeben, gekoppelt werden, was zur Folge hat, dass effektiv nur zwei nicht kommunizierende Strukturen übrig bleiben: die kohärente Struktur und das sie umgebende Volumen. Dies führt zu einer signifikant erhöhten Stabilität der Methode ohne auf die sonst üblichen aber fehleranfälligen clustering-Techniken zurückgreifen zu müssen. Des Weiteren diskutieren wir die Identifikation von zeitlich konsistenten Regionen erhöhter Kohärenz. Anstatt von kohärenten Strukturen als bewegliche nicht filamentierende Massen zu sprechen, beschreibt dieses Konzept kohärente Strukturen als konsistente sich bewegende Regionen reduzierten Mischverhaltens deren beteiligte Masse zeitlich veränderlich ist. Dies erlaubt die Entkopplung der Kohärenzzeitskala vom betrachteten Zeitfenster. Im Anschluss verwenden wir beide Modifikationen, um die Transporteigenschaften von acht ausgewählten Ostseewirbeln zu untersuchen. Als zweiten Beitrag stellen wir den MSCS-Algorithmus vor, eine neue Methode zur Identifikation von kohärenten Strukturen, welche einzig und allein auf dem Prinzip von Konvexität basiert. Versteht man kohärente Strukturen als nicht filamentierende Massen, ist persistente Konvexität eine hinreichende Bedingung für Kohärenz. Konvexität wurde bisher noch nie als Grundlage für Kohärenz untersucht, obwohl sie aus praktischen Gründen in einigen Methoden bereits als einschränkende Bedingung verwendet wird. Die Methode identifiziert die größte Struktur innerhalb eines gegebenen Volumens, welches während eines gegebenen Zeitraums stern-konvex bezüglich einer gegebenen Referenztrajektor bleibt. Alle studierten Testszenarien zeigen, dass der Ansatz gute und zuverlässige Ergebnisse liefert. Diese Ergebnisse hängen darüber hinaus direkt von den Eingangsparametern ab, was die Interpretation der Ergebnisse stark erleichtert und zusätzlich die Untersuchung von Filamentbildung erlaubt. Die neue Methode wird verwendet um den kohärenten Transport in den bereits untersuchten Ostseewirbeln zu reevaluieren.:1 Introduction 2 Theoretical Background 2.1 Methods 2.1.1 Okubo-Weiss criterion 2.1.2 Finite-Time Lyapunov Exponents 2.1.3 Lagrangian Descriptors and Lagrangian averaged vorticity deviation 2.2 Models 2.2.1 Euler equation 2.2.2 Stationary Gaussian Blob Model 2.2.3 Periodically Perturbed Gaussian Blob Model 2.2.4 Bickley-Jet 3 Transfer Operator Approach I: State of the Art 3.1 Frobenius-Perron Operator 3.2 Markov-Chains 3.3 Laplacian matrices 3.4 Finding Almost-Invariant Sets 3.5 Finding Coherent Sets 3.5.1 Coherent pairs as tuples 3.5.2 Coherent pairs as triples 3.6 Spectral Clustering 3.7 Critique and Discussion 4 Transfer Operator Approach II: Mixing Boundary Conditions in Stationary Flows 4.1 Overview 4.2 Estimation of Transfer Probability Matrices P 4.3 Mixing Boundary Conditions 4.4 Thresholding 4.5 Results 4.5.1 Robustness with respect to parameters 4.5.2 Comparison and alternatives 4.5.3 Handling of false positives 4.5.4 Periodically perturbed flow 4.6 Closing Remark 5 Transfer operator approach III: Analysis of Oceanic Transport 5.1 Mixing Boundary Conditions: Time-Dependent Flows 5.1.1 Method 5.1.2 Results 5.2 Temporal consistency 5.2.1 Method 5.2.2 Results 5.3 Treatment of Coastal Effects 5.4 Application to Baltic velocity fields 5.4.1 Method 5.4.2 Results 5.5 Closing Remark 6 Prototypes of Coherent Sets: Star-Convex Structures 6.1 Mathematical Considerations 6.2 MSCS-Algorithm 6.3 Extracting star-convex sub-volumes 6.4 Results 6.4.1 Stationary Gaussian blob model 6.4.2 Bickley-Jet 6.4.3 Two dimensional inviscid flow 6.4.4 Real data sets 6.5 Closing Remark 7 Conclusion and Outlook

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In many problems in dynamical systems one is interested in the identification of sets which have qualitatively different fates. The finite-time Lyapunov exponent (FTLE) method is a general and equation-free method that identifies codimension-one sets which have a locally high rate of stretching around which maximal exponential expansion of line elements occurs. These codimension-one sets thus act as transport barriers. This geometric framework of transport barriers is used to study various problems in phase space transport, specifically problems of separation in flows that can vary in scale from the micro to the geophysical. The first problem which we study is of the nontrivial motion of inertial particles in a two-dimensional fluid flow. We use the method of FTLE to identify transport barriers that produce segregation of inertial particles by size. The second problem we study is the long range advective transport of plant pathogen spores in the atmosphere. We compute the FTLE field for isobaric atmospheric flow and identify atmospheric transport barriers (ATBs). We find that rapid temporal changes in the spore concentrations at a sampling point occur due to the passage of these ATBs across the sampling point. We also investigate the theory behind the computation of the FTLE and devise a new method to compute the FTLE which does not rely on the tangent linearization. We do this using the 925 matrix of a probability density function. This method of computing the geometric quantities of stretching and FTLE also heuristically bridge the gap between the geometric and probabilistic methods of studying phase space transport. We show this with two examples. / Ph. D.

This thesis focused on investigating flow, coherent structures, and mechanisms leading to scour around a series of three submerged groynes in a high curvature (radius of curvature (R)/channel width (B)=1.5) channel bend using a Large Eddy Simulation Numerical (LES) model. Flow was investigated during both an initial and a later stage of scour. The results showed that the groynes appeared effective in keeping the main core of high streamwise velocity away from the outer bank wall in the region where they were installed, although high potential still existed for local scour around the groynes. During the initial stage of scour, horseshoe vortices (HVs) showed the greatest propensity to induce scour immediately upstream of the groyne tips. During the later stage of scour, the HV in front of the first upstream groyne (G1) induced very high mean pressure fluctuations on the outer bank wall. Scour was also of very great concern around the tip of G1 due to severe mean bed pressure fluctuations. Downstream of the groyne field, the presence of a counter-rotating outer bank cell was capable of endangering the stability of the outer bank. The second focus of this thesis was to investigate flow in a 135° channel bend using both Reynolds Averaged Navier Stokes (RANS) and LES numerical models. The RANS study examined the effects of curvature ratio (R/B), and aspect ratio (B/H, where H is the inlet flow depth), on secondary circulation strength, and bed shear stresses. The study revealed that a decrease in R/B was associated with an increase in secondary circulation strength and peak bed shear stress. A change in B/H also substantially affected cross stream circulation strength. The LES study was conducted in a 135° (R/B = 1.5) bend flume with a fixed bed corresponding to near equilibrium scour conditions, and the results were compared to a similar high curvature 193° bend numerical study. Inner bank vortices and shear layers were present in both cases although their characteristics were substantially different. Distributions of boundary friction velocities, and turbulence were also quite different for each case.

Esta tese trata o escoamento ao redor de um cilindro logo após a sua primeira instabilidade, dentro do seu regime bidimensional periódico. A abordagem é principalmente teórica, passa por experimentos e culmina em uma importante parte numérica que complementa a teoria com evidências e ilustrações. As principais contribuições são a análise sobre a composição modal da solução dentro do regime periódico e o método desenvolvido para identificar autovetores de uma linearização da equação de Navier-Stokes presentes em uma dada solução. As bases compostas pelos autovetores identificados servem para a projeção da equação de Navier-Stokes e dão a essência dos modelos reduzidos deste estudo. A aplicação numérica apresentada para Re = 60 traz duas iterações do processo, com duas bases de autovetores de dimensões 12 e 24. Os modelos reduzidos são numericamente estáveis e a sua integração apresenta custo várias ordens mais baixo que o da simulação numérica completa. As séries temporais das coordenadas e as bases de autovetores possibilitam a recomposição do escoamento e a sua comparação com a simulação numérica de referência. A análise de aderência foi baseada nas médias temporais, nos valores de Strouhal e na estrutura dos harmônicos. Ambos modelos reduzidos têm correspondência próxima com o comportamento assintótico do escoamento e a tendência convergente das iterações é clara. As simetrias espaciais e temporais dos harmônicos são facilmente identificadas na estrutura dos modelos, de forma que as bases construídas podem ser entendidas como conjuntos de estruturas coerentes do fenômeno. / This thesis concerns the flow past a cylinder just after its first bifurcation, within its two-dimensional periodic regime. The approach is mainly theoretical, goes through experiments and is concluded by an important numerical part which complements the theory with evidences and illustrations. The main contributions are the analysis concerning the modal composition of the solution within the periodic regime and a method to identify eigenvectors of some linearizaton of the Navier-Stokes equation participating on a given solution. The bases spanned by the identified eigenvectors are employed in the projection of the Navier-Stokes equation and are central to the reduced models of this study. The numerical results for Re = 60 present two iterations of the process, with two bases of dimensions 12 e 24. The reduced models are numerically stable and their integration is many orders less costly than that of the full simulation. The time series of the modal coordinates and the eigenvectors bases allow the recomposition of the flow and its comparison with the full simulation results. The convergence analysis was based on the time averages, the Strouhal number values and the harmonic structure. Both reduced models have close correspondence with the asymptotic behavior of the flow and the convergent trend of the iterations is clear. The space and time symmetries of the harmonics have a simple representation within the structure of the models, therefore the identified bases can be understood as sets of coherent structures of the phenomenon.

<p>The magnetospheric cusps are two funnel-like regions of Earth's magnetosphere where solar wind plasma can have direct access to the ionosphere. The cusps are very dynamic regions where wave-particle interactions continuously take place and redistribute energy among different particle populations. In this thesis, both low and high frequency plasma waves in the cusp have been studied in detail using data from the Cluster spacecraft mission. The waves were studied with respect to frequency, Poynting flux and polarization. Wavelengths have also been estimated using multi-spacecraft techniques. At low frequencies, kinetic Alfvén waves and nonpotential ion cyclotron waves are identified and at high frequencies, electron cyclotron waves, whistler waves, upper-hybrid and RX-waves are observed. A common generation mechanism called the shell-instability is proposed for several of the wave modes present in the cusp, both at low and high frequencies. </p><p>The plasma in the cusp is shown to be strongly inhomogeneous. In an inhomogeneous low-frequency magnetoplasma, kinetic Alfvén waves couple to drift-waves. Such drift-kinetic Alfvén waves have long been believed to nonlinearly self-interact and form coherent structures in the form of drift-kinetic Alfvén vortices. In this thesis the first unambiguous direct measurements confirming the existence of such vortices in a turbulent space plasma are presented. Some of the crucial parameters such as the vortex radius are determined. </p><p>Plasma theory is electrodynamics applied to a large collection of charged particles. In this thesis a new way of looking at the fundamental Maxwell tensor is presented. A covariant spectral density tensor containing information on electromagnetic waves is formed. This tensor is then decomposed into irreducible components by using the spinor formalism for an arbitrary metric. The obtained fundamental tensors are shown to correspond both to well known tensors in Maxwell's theory, as well as several physically interesting new tensors.</p>

QC 20110926

The magnetospheric cusps are two funnel-like regions of Earth's magnetosphere where solar wind plasma can have direct access to the ionosphere. The cusps are very dynamic regions where wave-particle interactions continuously take place and redistribute energy among different particle populations. In this thesis, both low and high frequency plasma waves in the cusp have been studied in detail using data from the Cluster spacecraft mission. The waves were studied with respect to frequency, Poynting flux and polarization. Wavelengths have also been estimated using multi-spacecraft techniques. At low frequencies, kinetic Alfvén waves and nonpotential ion cyclotron waves are identified and at high frequencies, electron cyclotron waves, whistler waves, upper-hybrid and RX-waves are observed. A common generation mechanism called the shell-instability is proposed for several of the wave modes present in the cusp, both at low and high frequencies. The plasma in the cusp is shown to be strongly inhomogeneous. In an inhomogeneous low-frequency magnetoplasma, kinetic Alfvén waves couple to drift-waves. Such drift-kinetic Alfvén waves have long been believed to nonlinearly self-interact and form coherent structures in the form of drift-kinetic Alfvén vortices. In this thesis the first unambiguous direct measurements confirming the existence of such vortices in a turbulent space plasma are presented. Some of the crucial parameters such as the vortex radius are determined. Plasma theory is electrodynamics applied to a large collection of charged particles. In this thesis a new way of looking at the fundamental Maxwell tensor is presented. A covariant spectral density tensor containing information on electromagnetic waves is formed. This tensor is then decomposed into irreducible components by using the spinor formalism for an arbitrary metric. The obtained fundamental tensors are shown to correspond both to well known tensors in Maxwell's theory, as well as several physically interesting new tensors.

In this work, flow through synthetic arrangements of contacting spheres is studied as a model problem for porous media and packed bed type flows. Direct numerical simulations are performed for moderate pore Reynolds numbers in the range, 10 ≤ Re ≤ 600, where non-linear porescale flow features are known to contribute significantly to macroscale properties of engineering interest. To first choose and validate appropriate computational models for this problem, the relative performance of two numerical approaches involving body conforming and non-conforming grids for simulating porescale flows is examined. In the first approach, an unstructured solver is used with tetrahedral meshes, which conform to the boundaries of the porespace. In the second approach, a fictitious domain formulation (Apte et al., 2009. J Comput. Phys. 228 (8), 2712-2738) is used, which employs non-body conforming Cartesian grids and enforces the no-slip conditions on the pore boundaries implicitly through a rigidity constraint force. Detailed grid convergence studies of both steady and unsteady flow through prototypical arrangements of spheres indicate that for a fixed level of uncertainty, significantly lower grid densities may be used with the fictitious domain approach, which also does not require complex grid generation techniques. Next, flows through both random and structured arrangements of spheres are simulated at pore Reynolds numbers in the steady inertial ( 10 ≲ Re ≲ 200) and unsteady inertial (Re ≈ 600) regimes, and used to analyze the characteristics of porescale vortical structures. Even at similar Reynolds numbers, the vortical structures observed in structured and random packings are remarkably different. The interior of the structured packings are dominated by multi-lobed vortex rings structures that align with the principal axes of the packing, but perpendicular to the mean flow. The random packing is dominated by helical vortices, elongated parallel to the mean flow direction. The unsteady dynamics observed in random and structured arrangements are also distinct, and are linked to the behavior of the porescale vortices. Finally, to investigate the existence and behavior of transport barriers in packed beds, a numerical tool is developed to compute high resolution finite-time Lyapunov exponent (FTLE) fields on-the-fly during DNS of unsteady flows. Ridges in this field are known to correspond to Lagrangian Coherent Structures (LCS), which are invariant barriers to transport and form the skeleton of time dependent Lagrangian fluid motion. The algorithm and its implementation into a parallel DNS solver are described in detail and used to explore several flows, including unsteady inertial flow in a random sphere packing. The resulting FTLE fields unambiguously define the boundaries of dynamically distinct porescale features such as counter rotating helical vortices and jets, and capture time dependent phenomena including vortex shedding at the pore level. / Graduation date: 2013

This thesis focused on investigating flow, coherent structures, and mechanisms leading to scour around a series of three submerged groynes in a high curvature (radius of curvature (R)/channel width (B)=1.5) channel bend using a Large Eddy Simulation Numerical (LES) model. Flow was investigated during both an initial and a later stage of scour. The results showed that the groynes appeared effective in keeping the main core of high streamwise velocity away from the outer bank wall in the region where they were installed, although high potential still existed for local scour around the groynes. During the initial stage of scour, horseshoe vortices (HVs) showed the greatest propensity to induce scour immediately upstream of the groyne tips. During the later stage of scour, the HV in front of the first upstream groyne (G1) induced very high mean pressure fluctuations on the outer bank wall. Scour was also of very great concern around the tip of G1 due to severe mean bed pressure fluctuations. Downstream of the groyne field, the presence of a counter-rotating outer bank cell was capable of endangering the stability of the outer bank. The second focus of this thesis was to investigate flow in a 135° channel bend using both Reynolds Averaged Navier Stokes (RANS) and LES numerical models. The RANS study examined the effects of curvature ratio (R/B), and aspect ratio (B/H, where H is the inlet flow depth), on secondary circulation strength, and bed shear stresses. The study revealed that a decrease in R/B was associated with an increase in secondary circulation strength and peak bed shear stress. A change in B/H also substantially affected cross stream circulation strength. The LES study was conducted in a 135° (R/B = 1.5) bend flume with a fixed bed corresponding to near equilibrium scour conditions, and the results were compared to a similar high curvature 193° bend numerical study. Inner bank vortices and shear layers were present in both cases although their characteristics were substantially different. Distributions of boundary friction velocities, and turbulence were also quite different for each case.