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In this study, we investigate a handful of equilibrium states and their hydrodynamic stability for the incompressible flow in a square cavity driven by the tangential simultaneous motion of all of its lids at the same speed. This problem has been investigated in the recent past, albeit only partially and always disregarding the singular boundary conditions at the corners. The implications of the system symmetries have not been rigorously addressed, either. In our study, we introduce a regularized version of the boundary conditions to avoid the corner singularities, in a way that preserves the main features of the original setup. In addition, the Navier–Stokes equations are discretized by means of highly accurate Chebyshev spectral methods that provide exponential convergence of all computed flows and consistently eliminate any potential source of structural instability of the bifurcation scenario. We employ Newton–Krylov solvers, implemented within continuation algorithms, to accurately compute equilibrium solutions. Linear stability analysis of both the primary symmetric base flow and the secondary asymmetric states uncovers new branches of fully asymmetric steady states. The analysis has allowed identification of six previously undetected bifurcations, all of which are associated with the disruption of either the rotational invariance or the reflection symmetry. Some of these bifurcations have been found to be quite clustered in some regions of the parameter space, which points at the underlying action of higher codimension mechanisms. Notably, all the bifurcations reported occur within the range of low to moderate Reynolds numbers, making the regularized four-sided lid-driven cavity flow a reliable benchmark for assessing different numerical schemes in the context of Navier–Stokes equivariant bifurcation theory., Postprint (published version)

Parcial, Resolved, 2023-2024, 2n quadrimestre

Tesi en modalitat de compendi de publicacions, amb continguts parcialment retallats per drets de l’editor, (English) Turbulence is not only omnipresent in our daily lives but also in engineering applications. However, not until the middle of the XXth century researchers began to realize that simple, relatively large-scale coherent structures are embedded, and play their part, in the seemingly totally random motion. These structures are recurrently visited in turbulent dynamics and play a key role in the transition to turbulence and in its sustenance. This thesis aims at identifying exact coherent structures in shear and centrifugal flows, studying their behavior and analyzing the corresponding dynamical relevance in turbulence, as well as their role in transition and in the phenomenon of intermittency (i.e., coexistence of laminar and turbulent flows). In this research, direct numerical simulations are used to explore the rich dynamics of the Navier-Stokes equations. As a high dimensional dynamical system, turbulence can be regarded deterministically as a strange attractor embedded within the Navier-Stokes phase space, where exact coherent structures may be treated as invariant sets, and transitions can be explained in terms of bifurcation theory. The canonical flows chosen for this work are the Taylor-Couette system (flow between two coaxial cylinders counter-rotating with respect to their common axis), which embraces both centrifugal and shear instabilities, and the self-similar flows that arise between orthogonally stretching plates. In the first part, we explore the mean structure of the spiral turbulence intermittent regime arising in the Taylor-Couette system. We manage to reduce drastically the computational cost by replacing the classical orthogonal domain by parallelogram-annular domains aligned in the direction of the turbulent spiral. In the second part, a series of drifting rotating wave solutions are tracked from the subcritical region, where the self-sustained process is proved to be at work, to the supercritical regime, where the spiral turbulence appears., (Español) La turbulencia se encuentra presente no solo en nuestra vida cotidiana, sino también en múltiples aplicaciones en ingeniería. Entender los mecanismos que den lugar a la turbulencia siguiendo un reto para la Física y la Matemática Aplicada. Estas estructuras se visitan recurrentemente en la dinámica turbulenta y juegan un papel clave en la transición desde la laminaridad, y en particular en régimen turbulento subcrítico. Los objetivos de esta tesis son: identificar dichas estructuras coherentes en flujos de cizalladura y bajorotación, estudiar su comportamiento y analizar su relevancia dinámica en el régimen turbulento, así como su papel en la transición y en el fenómeno de la intermitencia (es decir, la coexistencia de flujos laminares y turbulentos). En esta investigación, se hace uso de simulaciones numéricas directas (DNS) para explorar la dinámica de las ecuaciones de Navier-Stokes. Como sistema dinámico determinista de alta dimensión, la turbulencia se puede considerar como un atractor extraño en el espacio de fases de Navier-Stokes, donde las estructuras coherentes exactas (ECS) pueden tratarse como conjuntos invariantes, y la transición laminar-turbulenta puede explicarse en términos de la teoría de bifurcaciones. Los flujos canónicos elegidos para este trabajo son el sistema de Taylor-Couette (flujo entre dos cilindros coaxiales contrarrotatorios con respecto a su eje común), que abarca tanto las inestabilidades centrífugas como de cizalladura, y los flujos autosimilares entre placas elásticas con estiramiento ortogonal. En la primera parte, exploramos la estructura promediada del régimen intermitente de la denominada espiral turbulenta (SPT) que surge en el sistema de Taylor-Couette. Se ha conseguido reducir drásticamente el coste computacional al reemplazar el dominio ortogonal clásico por dominios anulares en forma de paralelogramo, alineados en la dirección de la espiral turbulenta. En la segunda parte, se han identificado y monitorizado una se, Postprint (published version)

The large-scale laminar/turbulent spiral patterns that appear in the linearly unstable regime of counter-rotating Taylor–Couette flow are investigated from a statistical perspective by means of direct numerical simulation. Unlike the vast majority of previous numerical studies, we analyse the flow in periodic parallelogram-annular domains, following a coordinate change that aligns one of the parallelogram sides with the spiral pattern. The domain size, shape and spatial resolution have been varied and the results compared with those in a sufficiently large computational orthogonal domain with natural axial and azimuthal periodicity. We find that a minimal parallelogram of the right tilt significantly reduces the computational cost without notably compromising the statistical properties of the supercritical turbulent spiral. Its mean structure, obtained from extremely long time integrations in a co-rotating reference frame using the method of slices, bears remarkable similarity with the turbulent stripes observed in plane Couette flow, the centrifugal instability playing only a secondary route. This article is part of the theme issue ‘Taylor–Couette and related flows on the centennial of Taylor’s seminal Philosophical transactions paper (Part 2)., KD’s research was supported by Australian Research Council Discovery Early Career Researcher Award (DE170100171). BW, RA, FM and AM research was supported by the Spanish Ministerio de Economía y Competitividad (grant numbers FIS2016-77849-R and FIS2017-85794-P) and Ministerio de Ciencia e Innovación (grant number PID2020-114043GB-I00), and the Generalitat de Catalunya (grant 2017-SGR-785). BW’s research was also supported by the Chinese Scholarship Council (grant CSC No. 201806440152). FM is a Serra-Húnter Fellow., Peer Reviewed, Preprint

1r parcial amb solucions, Resolved, 2022/2023, 2n quadrimestre

2n parcial amb solucions, Resolved, 2022/2023, 2n quadrimestre

Final 1part amb solucions, Resolved, 2022/2023, 2n quadrimestre

The Spanish Pilot Centre of ERCOFTAC composes of six members. The Center is coordinated by Prof. Eugenio Oñate from the International Center for Numerical Methods in Engineering (CIMNE) and Technical University of Catalonia (UPC) and Prof. Eusebio Valero from Technical University of Madrid (UPM). Dr. Jordi Pons-Prats is acting as deputy coordinator, on behalf of Prof. Oñate, and in a close collaboration with Prof. Valero. He is also member of the Scientific Program Committee, while Prof. Valero is a member of the Knowledge Network Committee., Postprint (author's final draft)

Transition to turbulence in pipe flow is a long standing problem in the field of turbulence research. In the route to fully turbulent pipe flow, turbulence presents itself in a wide variety of forms. In recent years, a simple empirical model that reproduces many of the complex features of this phenomenon has been proposed. However, the assumptions on which this model relies are still not fully proven. In this bachelor's thesis we aim at testing said assumptions, through direct comparison of the model with results from direct numerical simulations (DNS). We will focus on the puff regime at $Re=2050$. Through a statistical analysis of the DNS data, we will present evidence that suggests that the model's claims have a connection to the physical process. However our analysis also suggests that this connection is only dominant in two short streamwise regions of the puff, the upstream front and the refractory length, i.e. the last section of the downstream front. Motivated by these findings we look for the reasons why the model fails to explain the behaviour of puffs in other streamwise sections. In particular we report the presence of recurrent structures in the flow that render the mean profile highly non-axisymmetric. Although these structures seem to dominate the dynamics of the puff in some sections, they are not captured by the model. These findings could eventually lead to improvements of the model and a better understanding of the physics of transition in pipe flow., La transición a la turbulencia en el flujo de tuberías es un problema de larga data en el campo de la investigación de la turbulencia. En la ruta hacia el flujo de tubería completamente turbulento, la turbulencia se presenta en una amplia variedad de formas. En los últimos años se ha propuesto un modelo empírico simple que reproduce muchas de las características complejas de este fenómeno. Sin embargo, los supuestos en los que se basa este modelo aún no están completamente probados. En esta tesis de licenciatura pretendemos contrastar dichos supuestos, a través de la comparación directa del modelo con resultados de simulaciones numéricas directas (DNS). Nos centraremos en el régimen de soplo en Re=2050. A través de un análisis estadístico de los datos del DNS, presentaremos evidencia que sugiere que las afirmaciones del modelo tienen una conexión con el proceso físico. Sin embargo, nuestro análisis también sugiere que esta conexión solo es dominante en dos regiones cortas del flujo del soplo, el frente de aguas arriba y la longitud refractaria, es decir, la última sección del frente de aguas abajo. Motivados por estos hallazgos, buscamos las razones por las que el modelo no logra explicar el comportamiento del puff en otras secciones de la corriente. En particular, reportamos la presencia de estructuras recurrentes en el flujo que hacen que el perfil medio sea altamente no axisimétrico. Aunque estas estructuras parecen dominar la dinámica del puff en algunas secciones, el modelo no las captura. Estos hallazgos eventualmente podrían conducir a mejoras del modelo y una mejor comprensión de la física de la transición en el flujo de tuberías., La transició a la turbulència en el flux de canonades és un problema de llarga data en el camp de la investigació de turbulències. En el camí cap al flux de tubs totalment turbulents, la turbulència es presenta en una gran varietat de formes. En els darrers anys s'ha proposat un model empíric senzill que reprodueix moltes de les característiques complexes d'aquest fenomen. Tanmateix, els supòsits en què es basa aquest model encara no estan completament demostrats. En aquesta tesi de grau pretenem provar aquests supòsits, mitjançant la comparació directa del model amb els resultats de simulacions numèriques directes (DNS). Ens centrarem en el règim de puff a $Re=2050$. Mitjançant una anàlisi estadística de les dades del DNS, presentarem proves que suggereixen que les afirmacions del model tenen una connexió amb el procés físic. No obstant això, la nostra anàlisi també suggereix que aquesta connexió només és dominant en dues regions curtes de corrent del puff, el front aigües amunt i la longitud refractària, és a dir, l'última secció del front aigües avall. Motivats per aquestes troballes, busquem les raons per les quals el model no explica el comportament de les bufades en altres seccions de corrent. En particular, informem de la presència d'estructures recurrents en el flux que fan que el perfil mitjà sigui molt axisimètric. Tot i que aquestes estructures semblen dominar la dinàmica del puff en algunes seccions, el model no les capta. Aquestes troballes podrien conduir a millores del model i a una millor comprensió de la física de la transició en el flux de canonades.

In this paper, we explore the stability and dynamical relevance of a wide variety of steady, time-periodic, quasiperiodic, and chaotic flows arising between orthogonally stretching parallel plates. We first explore the stability of all the steady flow solution families formerly identified by Ayats et al. [“Flows between orthogonally stretching parallel plates,” Phys. Fluids 33, 024103 (2021)], concluding that only the one that originates from the Stokesian approximation is actually stable. When both plates are shrinking at identical or nearly the same deceleration rates, this Stokesian flow exhibits a Hopf bifurcation that leads to stable time-periodic regimes. The resulting time-periodic orbits or flows are tracked for different Reynolds numbers and stretching rates while monitoring their Floquet exponents to identify secondary instabilities. It is found that these time-periodic flows also exhibit Neimark–Sacker bifurcations, generating stable quasiperiodic flows (tori) that may sometimes give rise to chaotic dynamics through a Ruelle–Takens–Newhouse scenario. However, chaotic dynamics is unusually observed, as the quasiperiodic flows generally become phase-locked through a resonance mechanism before a strange attractor may arise, thus restoring the time-periodicity of the flow. In this work, we have identified and tracked four different resonance regions, also known as Arnold tongues or horns. In particular, the 1¿:¿4 strong resonance region is explored in great detail, where the identified scenarios are in very good agreement with normal form theory., Postprint (author's final draft)

We investigate the local self-sustained process underlying spiral turbulence in counter-rotating Taylor–Couette flow using a periodic annular domain, shaped as a parallelogram, two of whose sides are aligned with the cylindrical helix described by the spiral pattern. The primary focus of the study is placed on the emergence of drifting–rotating waves (DRW) that capture, in a relatively small domain, the main features of coherent structures typically observed in developed turbulence. The transitional dynamics of the subcritical region, far below the first instability of the laminar circular Couette flow, is determined by the upper and lower branches of DRW solutions originated at saddle-node bifurcations. The mechanism whereby these solutions self-sustain, and the chaotic dynamics they induce, are conspicuously reminiscent of other subcritical shear flows. Remarkably, the flow properties of DRW persist even as the Reynolds number is increased beyond the linear stability threshold of the base flow. Simulations in a narrow parallelogram domain stretched in the azimuthal direction to revolve around the apparatus a full turn confirm that self-sustained vortices eventually concentrate into a localised pattern. The resulting statistical steady state satisfactorily reproduces qualitatively, and to a certain degree also quantitatively, the topology and properties of spiral turbulence as calculated in a large periodic domain of sufficient aspect ratio that is representative of the real system., Postprint (author's final draft)

Tesi en modalitat de compendi de pubicacions, Turbulence is one of the major concerns for most technological problems involving fluid motion. Specially in aeronautics, a turbulent boundary layer results in structural stresses, vibrations and higher aircraft drag, leading to a significant increase in fuel consumption. Therefore, trying to comprehend the origin of turbulence by studying its most common transition routes is a crucial first step towards its effective control. Transition to turbulence of an homogeneous flow is frequently mediated by transient visits to highly nonlinear laminar coherent structures that usually are at the threshold between laminarity and turbulence. From a dynamical systems point of view, these structures are invariant sets in the infinite-dimensional Navier-Stokes phase space that here we aim to identify in different canonical flows. With the use of direct numerical simulations together with Newton-Krylov solvers and Arnoldi iteration method for the linear stability analysis, invariant sets such as equilibria, relative equilibria or periodic orbits are accurately computed and tracked along the parameter space to understand the transition mechanisms. From a mathematical perspective, dynamical systems and bifurcation theory provide the suitable framework to understand the hydrodynamic instabilities and transition to turbulence from a deterministic point of view. In addition, the use of spectral methods for the spatial discretisation is particularly convenient due to the exponential convergence of the numerical solutions. In the first work, the onset of transition in two-dimensional Plane Poiseuille flow is analysed. A new family of Tollmien-Schlichting waves, breaking the usual half-shift and reflect symmetry of the classical ones, has been identified and tracked across parameter space. In addition, the role of another classical travelling wave family that did not participate in the localisation mechanisms has been clarified. We continue by analysing the nonlinear mode competition in p, La turbulència és una de les principals preocupacions per a la majoria de problemes tecnològics relacionats amb el moviment de fluids. Especialment en el cas de l'aeronàutica, una capa límit turbulenta produeix tensions estructurals, vibracions i una major força d'arrossegament de l'aeronau que resulten en un increment significatiu del consum de combustible. Per tant, intentar comprendre l'origen de la turbulència, tot estudiant-ne les rutes de transició més habituals, és un primer pas indispensable cap al seu control efectiu. La transició a la turbulència d'un flux homogeni sovint es caracteritza per visites transitòries a estructures coherents, laminars i altament no-lineals, que acostumen a trobar-se al llindar entre la laminaritat i la turbulència. Des del punt de vista dels sistemes dinàmics, aquestes estructures són conjunts invariants en l'espai de fase infinit-dimensional de les equacions de Navier-Stokes, que aquí es pretén identificar en diferents fluxos canònics. Mitjançant la integració temporal de les equacions, resolutors de Newton-Krylov i el mètode iteratiu d'Arnoldi per a l'anàlisi d'estabilitat lineal, els diferents conjunts invariants siguin equilibris, equilibris relatius o òrbites periòdiques són acuradament calculats i continuats al llarg de l'espai de paràmetres per tal d'entendre els mecanismes involucrats en la transició. Des d'una perspectiva matemàtica, els sistemes dinàmics i la teoria de bifurcacions proporcionen el marc adequat per a comprendre les inestabilitats hidrodinàmiques i la transició a la turbulència des d'un punt de vista determinista. A més, l'ús de mètodes espectrals per a la discretització espaial resulta particularment convenient degut a la convergència exponencial de les solucions numèriques. En el primer treball, s'analitza l'inici de la transició del flux bidimensional de Poiseuille pla. En aquest cas, una nova família d'ones de Tollmien-Schlichting, que trenca la clàssica simetria de translació i reflexió, ha estat id, Postprint (published version)

Navier–Stokes equilibrium solutions of a viscous fluid confined between two infinite parallel plates that can independently stretch or shrink in orthogonal directions are studied. It is assumed that the admissible solutions satisfy spatial self-similarity in the stretching or shrinking perpendicular coordinates. The nonlinear steady boundary-value problem is discretized using a spectral Legendre method, and equilibrium solutions are found and tracked in the two-dimensional parameter space by means of pseudo-arclength continuation Newton–Krylov schemes. Different families of solutions have been identified, some of which are two-dimensional and correspond to the classical Wang and Wu self-similar flows arising in a plane channel with one stretching–shrinking wall [Wang, C.-A. and Wu, T.-C., “Similarity solutions of steady flows in a channel with accelerating walls,” Comput. Math. Appl. 30, 1–16 (1995)]. However, a large variety of three-dimensional solutions have also been found, even for low stretching or shrinking rates. When slightly increasing those rates, some of these solutions disappear at saddle-node bifurcations. By contrast, when both plates are simultaneously stretching or shrinking at higher rates, a wide variety of new families of equilibria are created and annihilated in the neighborhood of cuspidal codimension-2 bifurcation points. This behavior has similarities with the one observed in other planar and cylindrical self-similar flows., This work was supported by the Spanish MINECO Grants No. FIS2017-85794-P and No. PRX18/00179, and the Generalitat de Catalunya Grant No. 2017-SGR-785., Peer Reviewed, Postprint (author's final draft)

The systems formed by two immiscible liquids such as emulsions are of great interest in industrial processes, the emulsion formation process occurs depending on the phenomenon of the breakup of the drops of one of the fluids within the other, and the properties depend on how this phenomenon happens. The breakup of drops in turbulent flows is investigated to determine which mechanisms are those that interact in the breakup and what is the effect of the viscosity of the drop on the dynamics of the break. For this study we used the data generated in the simulations of the work "Energetics of drop deformation with application to turbulence", to make an analysis of the average behavior of the velocity gradients, comparing one hundred simulations with different initial conditions, to identify the mechanisms as mentioned previously. In a first analysis we show the importance of the distance with respect to the interface of the drop in terms of the influence that the dynamics of the surface has on the rest of the flow. The main analysis of the work uses tensor invariants to observe how the mechanisms that regulate the dynamics of the breakup interact and change with viscosity., Los sistemas formados por dos líquidos immiscibles como son las emulsiones son de gran interés en proceso industriales, el proceso de formación de la emulsión ocurre dependiendo de como ocurre el fenómeno de la rotura de gotas de un liquido dentro del otro, y las propiedades de la emulsion dependen también de la manera en la que ocurre este fenómeno. La rotura de gotas en flujos turbulentos se investiga en este estudio para determinar qué mecanismos son los que interactúan en la rotura y cuál es el efecto de la viscosidad de la gota en la dinámica de la rotura. Para realizar este estudio se utilizan los datos generados de las simulaciones del trabajo "Energetics of drop deformation with application to turbulence? de mi supervisor, para hacer un análisis del comportamiento promedio de los gradientes de velocidad, comparando cien simulaciones con diferentes condiciones iniciales, para poder identificar los mecanismos mencionados anteriormente. En un primer análisis mostramos la importancia que tiene la distancia respecto la interfaz de la gota para observar la influencia que la dinámica de la gota tiene en el resto del fluido. El análisis principal del trabajo utiliza invariantes tensoriales para observar cómo los mecanismos que regulan la dinámica de la rotura interactúan y cambian según la viscosidad de la gota., Els sistemes formats per dos líquids immiscibles com són les emulsions són de gran interès en processos industrials, el procés de formació de l'emulsió ocorre depenent de com ocorre el fenomen del trencament de les gotes d'un dels líquids dins de l'altre, i les propietats de l'emulsió depenen també de la manera en la qual ocorre aquest fenomen. El trencament de gotes en fluxos turbulents s'investiga en Aquest treball per a determinar quins mecanismes són els que interactuen en el trencament i quin és l'efecte de la viscositat de la gota en la dinàmica del trencament. Per a realitzar aquest estudi s'utilitzen les dades generades de les simulacions del treball "Energetics of drop deformation with application to turbulence" del meu supervisor, per fer un anàlisi del comportament promig dels gradients de velocitat, comparant cent simulacions amb condicions inicials diferents, per poder identificar els mecanismes mencionats prèviament. En un primer anàlisi mostrem la importància de la distància respecte a la interfase de la gota per observar la influència de la dinàmica de la superfície de la gota sobre la resta del flux del fluid. L'anàlisi principal del treball utilitza invariants de tensors per observar com els mecanismes que regulen la dinàmica del trencament interactuen i canvien segons la viscositat de la gota.

The Kuramoto-Sivanshinsky equation is studied thoroughly by numerical integration. The rst section demonstrates the study's motivation and historical relevance. Rigorous mathematical analy- sis is made on how to approach the equation to nd possible solutions in periodic domains. Di erent numerical methods for integration have been studied along with their accuracy and computational cost. Lastly, some periodic solutions are found which follow Feigenbaum's universality principle and give a deterministic insight into the apparent chaotic behaviour, 2020/2021

In the last years the major cause of death worldwide has turned out to be because of cardiovascular diseases; for instance, owing to the presence of blood turbulence in our vessels. The problem has already been addressed from simplified points of view that ignore many blood characteristics; consequently, one of the main goals of this project is to take into account the blood shear-thinning effects in an infinite cylindrical pipe and study turbulence transition in such scenario. With this aim, the first thing to do is a non-Newtonian viscosity models research in order to find the best one to model blood behaviour. After this, an exhaustive base flow study is carried out, for steady and unsteady scenarios, so that finally the Transient Growth Analysis (TGA) provides us with the most dangerous perturbation that could trigger turbulence transition more easily in our veins., Durante los últimos años se ha detectado que la mayor causa de mortalidad en todo el mundo se debe a enfermedades cardiovasculares; por ejemplo, debido a la presencia de turbulencia en nuestras venas. El problema ya se ha encarado desde puntos de vista simplificados que ignoran muchas de las características de la sangre; consecuentemente, uno de los principales objetivos de este proyecto es tener en cuenta los efectos "shear-thinning" de la sangre en una tubería cilíndrica e infinita para estudiar la transición a turbulencia. Con este objetivo, lo primero que hay que hacer es búsqueda de modelos que describan la viscosidad de fluidos no-Newtonianos a fin de encontrar el que mejor funciona para modelar el comportamiento de la sangre. Seguidamente, se lleva a cabo un estudio exhaustivo del flujo básico, para los casos estable e inestable; finalmente el Transient Growth Analysis (TGA) nos proporciona cuál es la perturbación más peligrosa que puede desencadenar transición a turbulencia más fácilmente en nuestras venas., Durant els últims anys s'ha detectat que la major causa de mortalitat a tot el món és degut a malalties cardiovasculars; per exemple, degut a la presència de turbulència a les nostres venes. El problema ja s’ha encarat des de punts de vista simplificats que ignoren moltes de les característiques de la sang; conseqüentment, un dels principals objectius d’aquest projecte és tenir en compte els efectes ‘shear-thinning’ de la sang en una canonada cilíndrica i infinita per tal d’estudiar la transició a turbulència. Amb aquest objectiu, el primer que s’ha de fer és recerca de models que descriguin la viscositat de fluids no-Newtonians per tal de trobar el que millor funciona per modelar el comportament de la sang. Seguidament, es duu a terme un estudi exhaustiu del flux bàsic, pels casos estable i inestable, tal que finalment el Transient Growth Analysis (TGA) ens proporciona quina és la pertorbació més perillosa que pot desencadenar transició a turbulència més fàcilment a les nostres venes.

Parcial, Resolved

This project studies the flow of a fluid inside an infinite pipe with periodically forced extensional walls in the axial direction. The Navier-Stokes equations for an incompressible fluid inside the pipe are solved to find self-similar solutions of the problem. The system of PDEs is preconditioned using a spectral expansion in modulated Legendre polynomials, and is solved using an IMEX method. Anumerical exploration of time-periodic solutions in a wide parameter region has been conducted, and interesting non-linear phenomena is found for specific parameter regions. The system tends to converge to solutions with limit cycles of the same period as the oscillation. The frequency response of the system is studied. For small frequencies, the problem explores different solutions found in the ’non-forced’ problem [9], approaching stable limit cycles of high frequencies when the pipe is shrinking very slowly, and forming very fast axial jets. This behaviour is lost when the pipe oscillates at higher frequencies. Moreover, no stable solutions with azimuthal velocity are found, but a branch of unstable solutions with azimuthal velocity is born at a pitchfork bifurcation of cycles. In addition, cascade of period doubling bifurcations is found, leading the system to possible chaos.

We investigate two distinct scenarios of spatial modulation that are candidate mechanisms for streamwise localization of waves in two-dimensional plane Poiseuille flow. The first one stems from a symmetry-breaking bifurcation that disrupts the half-shift and reflect equivariance of Tollmien-Schlichting waves (TSW). A new state, an asymmetric TSW (ATSW), emerges from unstable lower-branch TSWs at subcritical Reynolds number and undergoes subharmonic Hopf bifurcations that lead to branches of asymmetric time-periodic space-modulated waves (MATSW). Streamwise modulation does not evolve into localization within the range of parameters explored. In breaking the last standing remnants of the reflectional symmetry about the channel midplane, ATSW and MATSW admit a bias toward either one of the channel walls, thus bearing a potential for explaining near-wall structures that are typical of developed turbulence. The second scenario follows the fate of a branch of time-periodic space-modulated TSWs (MTSW) initially discovered by Mellibovsky and Meseguer [J. Fluid Mech. 779, R1 (2015)]. We find that these waves can lead to localization but the mechanism is not new, as they do so through their connection, by means of a codimension-2 bifurcation point, with other known localizing MTSWs. The codimension-2 point is, however, responsible for the appearance of MTSWs that exclusively bridge upper-branch TSW-trains of different number of replicas. In this respect, these MTSWs possess all required properties that single them out as possible constituents of the strange saddle that governs domain-filling turbulent dynamics at high Reynolds numbers., Postprint (author's final draft)

We study the nonlinear mode competition of various spiral instabilities in magnetised Taylor-Couette flow. The resulting finite-amplitude mixed-mode solution branches are tracked using the annular-parallelogram periodic domain approach developed by Deguchi & Altmeyer (2013). Mode competition phenomena are studied in both the anti-cyclonic and cyclonic Rayleigh-stable regimes. In the anti-cyclonic sub-rotation regime, with the inner cylinder rotating faster than the outer, Hollerbach, Teeluck & Rüdiger (2010) found competing axisymmetric and non-axisymmetric magneto-rotational linearly unstable modes within the parameter range where experimental investigation is feasible. Here we confirm the existence of mode competition and compute the nonlinear mixed-mode solutions that result from it. In the cyclonic super-rotating regime, with the inner cylin- der rotating slower than the outer, Deguchi (2017) recently found a non-axisymmetric purely hydrodynamic linear instability that coexists with the non-axisymmetric magneto- rotational instability discovered a little earlier by Rüdiger, Schultz, Gellert & Ste- fani (2016). We show that nonlinear interactions of these instabilities give rise to rich pattern-formation phenomena leading to drastic angular momentum transport enhance- ment/reduction., Postprint (author's final draft)

Final, Resolved

En aquest treball s'explora el problema de Störmer. Aquest tracta de l'estudi de les trajectòries dels ions en influència del camp magnètic terrestre. S'ha dut a terme a partir de l'expressió del lagrangià de la partícula i, gràcies al principi de Hamilton, s'utilitza l'equació d'Euler-Lagrange per a deduïr-ne l'equació diferencial que satisfan aquestes trajectòries. S'ha fet una comparativa entre dos integradors de segon ordre, el mètode de Heun de Runge-Kutta i el mètode de Strang Splitting simplèctic de Verlet, per determinar quin dels dos preserva millor les quantitats conservades. També s'ha fet un estudi sobre la forma de les trajectòries en el pla equatorial, sobre un meridià i sobre una superfície esfèrica afegint una lligadura per a mantenir la partícula a distància constant.

final, Resolved

We study the dynamics of a viscous fluid confined within an elastic pipe that is being stretched and twisted in the axial and azimuthal directions, respectively. A comprehensive numerical exploration of self-similar flows has been conducted for a wide range of stretching and twisting acceleration rates. Up to eight different families of steady flows have been identified and tracked. For purely axial stretching, the presented computations recover the results obtained by former authors in the past. However, the twisting mechanism, seen as a homotopy parameter, has been crucial in this study to explain how different families of equilibria interact and azimuthal swirl is induced. The explorations conclude that stable time-periodic flows may also take over the dynamics when the pipe is contracting in the axial direction, with mild or absent azimuthal twist. These time-periodic flows may destabilize through different codimension-two bifurcation scenarios, sometimes leading to unstable tori that have been genuinely identified by means of edge-tracking techniques., Postprint (published version)

PARCIAL

En la última década se ha despertado un renovado interés en los cohetes híbridos debido a su bajo precio, manejo seguro y compatibilidad con carburantes verdes. Un grupo de estudiantes está desarrollando un cohete híbrido de oxígeno y parafina en el Center of Applied Space Technology and Microgravity (ZARM) en la Universidad de Bremen. Por tal de proporcionar datos que ayuden a los diseñadores del motor del cohete, hemos simulado el proceso de combustión y el flujo de materia en el motor usando simulaciones de tipo LES. Se ha generado un modelo del motor del cohete axisimétrico usando el programa de código abierto y licencia gratuita OpenFOAM. Se ha implementado el solver reactingFoam para fluidos compresibles y combustión con reacciones químicas. Se ha aplicado la condición de contorno no reflectante waveTransmissive en la salida de la tobera del cohete. Las simulaciones muestran que el flujo se separa de la pared de la tobera y que existe reflujo. Esto explica el bajo rendimiento del cohete que se ha medido en los experimentos y demuestra que el diseño de la tobera no es óptimo., Durant la darrera dècada s'ha despertat un renovat interès en els coets híbrids ja que tenen un cost reduït, són segurs de maniobrar i són compatibles amb carburants verds. Un grup d'estudiants està desenvolupant un coet híbrid d'oxigen i parafina en el Center of Applied Space Technology and Microgravity (ZARM) a la Universitat de Bremen. Per proporcionar dades que ajudin als dissenyadors del motor del coet, hem simulat el procés de combustió i el flux de matèria en el motor mitjançant simulacions de tipus LES. S'ha implementat el solver reactingFoam per fluids compressibles i combustió amb reaccions químiques. S'ha aplicat la condició de contorn no reflexant waveTransmissive a la sortida de la tovera del coet. Les simulacions mostren que el flux es separa de la paret de la tovera i que hi ha reflux. Això explica que el rendiment del coet mesurat en els experiments sigui baix i demostra que el disseny de la tovera no és òptim., In the last decade a renewed interest in hybrid rockets has arisen due to the fact that they are low-cost, safe to operate and compatible with green propellants. A group of students are developing an oxygen-paraffin hybrid rocket engine at the Center of Applied Space Technology and Microgravity (ZARM) at the University of Bremen. In order to provide helpful data to the designers of the engine, we have simulated the combustion process and the flow feld in the rocket engine using large eddy simulation (LES). An axi-symmetric model of the rocket motor has been produced using the open-source free-license CFD package OpenFOAM. The transient solver reactingFoam for compressible flow and combustion with chemical reactions has been successfully implemented. A non-reflecting boundary condition (waveTransmissive) was applied at the exit of the nozzle leading to simulation results coherent with experimental measurements. Simulations have shown the presence of flow separation and backflow at the nozzle, explaining the low performance of the rocket engine reported in test facilities and proving that the current design of the nozzle is not optimal.

Transition to turbulence is one of the oldest problems yet to be solved in physics and engineering. Certain fluid systems are known to undergo a laminar-turbulent transition through a deterministic route of laminar flows of increasing complexity. For nearly a century, the Taylor-Couette flow between two coaxial cylinders, each rotating independently, has been the paradigm of this type of route to turbulence. The aim of this work is to reconstruct a Taylor-Couette machine and to design a digital closed-loop control system in order to obtain some object-of-study flows. The initial stage of the project consisted in recovering a very old Taylor-Couette machine that had been stored for over twenty-five years. The next step was to design software and interface electronics to control the rotations of the two cylinders. This has opened a wide variety of experiments that were almost impossible to perform in the old version. The new control system is accurately programmed to act very slowly, so the process by which the cylinders arrive to the desired velocity is quasi static. This gives full control over the flow regimes studied and also robustness in terms of the noises to which the system is subjected. Observed states include Taylor vortices, wavy vortices, modulated wavy vortices, laminar spirals, interpenetrating spirals, modulated interpenetrating spirals, spiral turbulence, a flow with intermittent turbulent spots and various combinations of these flows. The documented experiments are given as functions of the inner and outer cylinder Reynolds numbers, Ri and Ro respectively. In all explored cases, the results are in very good agreement with earlier experiments and with theoretical computational predictions and simulations., La transición hacia la turbulencia es uno de los problemas más antiguos y aun sin resolver de la física y la ingeniería. Algunos sistemas de fluidos presentan una transición laminar-turbulenta como una ruta determinista de flujos laminares cada vez más complejos. Durante casi un siglo, el flujo de Taylor-Couette ha sido el paradigma de este tipo de rutas hacia la turbulencia. El objetivo de este trabajo es reconstruir una maquina de Taylor-Couette y diseñar un sistema de control digital en lazo cerrado con intención de obtener flujos que son objeto de estudio. La fase inicial del proyecto consistió en rehabilitar una maquina de Taylor-Couette muy vieja que había sido guardada durante veinticinco años. El siguiente paso fue diseñar un software y una interfaz electrónica para controlar las rotaciones de los dos cilindros de la maquina. Esto aumenta la oportunidad de una gran variedad de experimentos casi imposibles con la antigua versión de la máquina. El nuevo sistema de control está programado de manera precisa para actuar muy lentamente, de manera que el proceso a partir del cual los cilindros llegan a la velocidad deseada es cuasi estático. Así conseguimos el control total del flujo estudiado y también que el sistema sea robusto ante posible ruido. Los estados observados incluyen vórtices de Taylor, vórtices de Taylor ondulados, vórtices ondulados modulados, flujo en espiral, espirales interpenetrantes, espirales interpenetrantes modulados, espiral turbulenta, intermitencia de puntos turbulentos y otras combinaciones de estos flujos. Los experimentos documentados se dan en función de los números de Reynolds del cilindro interno y externo, Ri y Ro respectivamente. En todos los casos explorados los resultados concuerdan con autores anteriores: tanto las predicciones teóricas como las simulaciones computacionales., La transició cap a la turbulència és un dels problemes més antics i encara sense resoldre de la física i l'enginyeria. Alguns sistemes de fluids presenten una transició laminar-turbulenta com una ruta determinista de fluxos laminars cada vegada més complexos. Durant gairebé un segle, el flux de Taylor-Couette entre dos cilindres coaxials de rotació independent ha estat el paradigma d'aquest tipus de rutes cap a la turbulència. L'objectiu d'aquest treball és reconstruir una màquina de Taylor-Couette i dissenyar un sistema de control digital en llaç tancat amb intenció d'obtenir fluxos que són objecte d'estudi. La fase inicial del projecte va consistir a rehabilitar una màquina de Taylor-Couette molt vella que havia estat guardada durant vint-i-cinc anys. El següent pas va ser dissenyar un programari i una interfície electrònica per controlar les rotacions dels dos cilindres. Això ha fet augmentar l'oportunitat d'una gran varietat d'experiments que eren gairebé impossibles de dur a terme amb la versió antiga de la màquina. El nou sistema de control està programat de manera precisa per actuar molt lentament, de manera que el procés a partir del qual els cilindres arriben a la velocitat desitjada és quasi-estàtic. Així aconseguim el control total del flux estudiat i també que el sistema sigui robust davant de possibles sorolls. Els estats observats inclouen vòrtex de Taylor, vòrtex de Taylor ondulats, vòrtex ondulats modulats, flux en espiral, espirals interpenetrants, espirals interpenetrants modulats, espiral turbulenta, intermitència de punts turbulents i diverses combinacions d'aquests fluxos. Els experiments documentats es donen en funció dels nombres de Reynolds del cilindre intern i extern, Ri i Ro respectivament. En tots els casos explorats els resultats concorden amb autors anteriors: tant les prediccions teòriques com les simulacions computacionals.

Final, Resolved

Parcial, Resolved

Extensional self-similar flows in a channel are explored numerically for arbitrary stretching–shrinking rates of the confining parallel walls. The present analysis embraces time integrations, and continuations of steady and periodic solutions unfolded in the parameter space. Previous studies focused on the analysis of branches of steady solutions for particular stretching–shrinking rates, although recent studies focused also on the dynamical aspects of the problems. We have adopted a dynamical systems perspective, analysing the instabilities and bifurcations the base state undergoes when increasing the Reynolds number. It has been found that the base state becomes unstable for small Reynolds numbers, and a transitional region including complex dynamics takes place at intermediate Reynolds numbers, depending on the wall acceleration values. The base flow instabilities are constitutive parts of different codimension-two bifurcations that control the dynamics in parameter space. For large Reynolds numbers, the restriction to self-similarity results in simple flows with no realistic behaviour, but the flows obtained in the transition region can be a valuable tool for the understanding of the dynamics of realistic Navier–Stokes solutions., Postprint (author's final draft)

The interaction between fluid flows and the deformation of solid structures is important in many engineering applications, as in turbochargers, wind turbines or airplane wings. The flow of blood in the body is also an example involving very complex fluid-structure interactions. Here the flow rate pulsates cyclically, the fluid is non-Newtonian and the precise material properties are difficult to measure accurately. Hence the development of appropriate simplified models and experiments is necessary in order to determine the importance of each of these ingredients and their nature. In this project we will be dealing with a basic model for the behaviour of medium sized blood vessels consisting of a partially elastic, partially rigid channel with a Poiseuille flow as initial condition. With this model we try to simulate real life cases and try to obtain results that could bring to better understanding of the circulatory system, while also covering the basis of solid mechanics, fluid mechanics and Partial differential equations solving methods necessary to fully understand the purpose of this project., La interacción entre el flujo de fluidos y la deformación de sólidos es muy importante en diversas aplicacions de la ingeniería como los turbocompresores, turbinas eólicas o alas de avión. El flujo sanguíneo en el cuerpo humano es un claro ejemplo de una interacción fluido-estructura compleja, donde el pulso oscila de manera cíclica. Además, el fluido es a menudo no Newtoniano y las propiedades materials son difíciles de medir de forma precisa. Es por eso que el desarrollo de modelos simplificados y experimentos es necesario para determinar la importancia de cada uno de estos factores y su naturaleza. En este proyecto trataremos un modelo básico para el comportamiento de vasos sanguíneos medianos formado por una membrana parcialmente elástica y parcialmente rígida con un flujo de Poiseulle como condición inicial. Con este modelo intentaremos simular casos reales y obtener resultados que puedan llevar a un mejor entendimiento sobre el sistema circulatorio. También se cubrirán las bases de mecánica de sólidos y fluidos, así como de métodos numéricos para ecuaciones en derivadas parciales necesarias para entender el objetivo del proyecto., La interacció entre flux de fluids i la deformació d'estructures sòlides és d'importància en moltes aplicacions de l'enginyeria com podrien ser en turbocompressors, turbines eòliques o ales d'avió. El corrent sanguini en el cos és un clar exemple d'una interacció fluid-estructura complexa, on el pols oscil·la de manera cíclica, el fluid és sovint no Newtonià i les propietats materials són difícils de mesurar de forma acurada. És per això que el desenvolupament de models simplificats i experiments és necessari per a determinar la importància de cadascun d'aquests factors i la seva naturalesa. En aquest projecte tractarem un model bàsic per al comportament de vasos sanguinis de mida mitjana format per una membrana parcialment elàstica i parcialment rígida amb un flux de Poiseuille com a condició inicial. Amb aquest model intentarem simular casos reals i obtenir resultats que portin a un millor coneixement sobre el sistema circulatori, tot explicant les bases de la mecànica de sòlids, fluids i mètodes númerics per a equacions en derivades parcials necessàries per a entendre l'objectiu del projecte.

Parcial, Resolved

En este trabajo se estudia la estabilidad de las ondas Tollmien-Schlichting, que aparecen como una primera fase de la inestabilidad del flujo de Poiseuille plano como paso de laminar a turbulento. Se calcula la curva de estabilidad neutra de estas ondas tanto mediante un método directo como mediante el método de Arnoldi. Los resultados númericos son obtenidos mediante códigos Matlab proporcionados por los tutores del trabajo.

Parcial, Resolved

Final, Resolved

The aim of this Bachelor's Thesis is the study of a particular low-energy localized plane Poiseuille solution while minimizing the error introduced by the streamwise periodicity boundary condition. The fact that the instability of this shear flow is accepted to start as a 2D phenomenom, justifies the study of plane solutions that are much less computing power demanding. The main part of this work has consisted on very long time-integrations of the solution up to very long domains, and later, up to near critical Reynolds; while checking stability and performing various measurements. Finally, a domain-dependant mechanism describing the behaviour of very particular instabilities at relatively high Re was found and analyzed., El objetivo de esta tesis es el estudio de una solución particular de baja energía del flujo de Poiseuille plano a la vez que se minimiza el error introducido por la condición de contorno de periodicidad del dominio. El hecho de que la inestabilidad en este flujo de cizalla comience como un fenómeno 2D, justifica el estudio de soluciones planas que son mucho menos exigentes computacionalmente. La mayor parte de este proyecto ha consistido en integraciones temporales muy largas de la solución para dominios muy largos; y después, para Reynolds cerca del crítico; mientras se comprueba la estabilidad y se realizan otras medidas. Finalmente, un mecanismo dependiente del dominio que describe el comportamiento de inestabilidades muy particulares para Re relativamente altos ha sido descubierto y analizado., L'objectiu de la tesi de llicenciatura aquest és l'estudi d'una solució particular de Poiseuille pla localitzada de baixa energia i reduir al mínim l'error introduït per la condició de contorn de periodicitat. El fet que la inestabilitat d'aquest flux de cisallament s'accepta que comença com un fenomen 2D , justifica l'estudi de solucions planes que són molt menys exigents en computació . La part principal d'aquest treball ha consistit en molt llargs temps d'integració de la solució fins molt llarg dominis, i més tard , fins a prop de Reynolds crític; durant la comprovació de l'estabilitat i la realització de diverses mesures. Finalment , es va trobar un mecanisme dependent de domini que descriu el comportament de les inestabilitats molt particulars en relativament alts Re i es va analitzar .

The transition of strictly two-dimensional Poiseuille flow from one-frequency amplitude-modulated states to chaotic behaviour is studied through full numerical simulation of spatially periodic channels with large longitudinal aspect ratios. First, time evolution on states at different Reynolds numbers is performed. Then, linear stability techniques, namely Poincaré maps combined with Arnoldi Iteration methods, are used to attain more exact results., 2015/2016

We present the complete unfolding of streamwise localisation in a paradigm of extended shear flows, namely two-dimensional plane Poiseuille flow. Exact solutions of the Navier-Stokes equations are computed numerically and tracked in the streamwise wavenumber-Reynolds number parameter space to identify and describe the fundamental mechanism behind streamwise localisation, a ubiquitous feature of shear flow turbulence. Unlike shear flow spanwise localisation, streamwise localisation does not follow the snaking mechanism demonstrated for plane Couette flow., Peer Reviewed, Postprint (author's final draft)

In this paper, modern numerical continuation methodologies are presented as a way of understanding and computing multiplicity of solutions in undergraduate physics problems. Mechanical and thermodynamical problems are used as a storyline to introduce the mathematical formalism required to clarify the distinction between the uniqueness and multiplicity of equilibrium solutions and the critical states of a nonlinear physical problem, as well as to illustrate how these novel numerical continuation techniques are implemented in practice. The paper provides simple numerical MATLAB codes that are easily adaptable to other problems, as well as updated software and literature resources., Postprint (author’s final draft)

Nonlinear equilibrium states characterized by strongly localized vortex pairs are calculated in the linearly stable parameter region of counterrotating Taylor-Couette flow. These subcritical states are rotating waves whose region of existence is consistent with the critical threshold for relaminarization observed in experiments. For sufficiently rapid outer cylinder rotation the solutions extend beyond the static inner cylinder case to corotation, thus exceeding, for the first time, the boundary defined by the inviscid Rayleigh's stability criterion., Postprint (published version)

Rotating waves are periodic solutions in SO(2) equivariant dynamical systems. Their precession frequency changes with parameters and it may change sign, passing through zero. When this happens, the dynamical system is very sensitive to imperfections that break the SO(2) symmetry and the waves may become trapped by the imperfections, resulting in steady solutions that exist in a finite region in parameter space. This is the so-called pinning phenomenon. In this study, we analyse the breaking of the SO(2) symmetry in a dynamical system close to a Hopf bifurcation whose frequency changes sign along a curve in parameter space. The problem is very complex, as it involves the complete unfolding of high codimension. A detailed analysis of different types of imperfections indicates that a pinning region surrounded by infinite-period bifurcation curves appears in all cases. Complex bifurcational processes, strongly dependent on the specifics of the symmetry breaking, appear very close to the intersection of the Hopf bifurcation and the pinning region. Scaling laws of the pinning region width and partial breaking of SO(2) to Zm are also considered. Previous as well as new experimental and numerical studies of pinned rotating waves are reviewed in the light of the new theoretical results., Postprint (published version)

The transition from a two-dimensional to three-dimensional flow in systems with spatial O(2) symmetry and spatiotemporal Z2 symmetry happens in many fluid systems, like wakes or periodically forced flows. In most of these systems, the dynamics after the first bifurcation is very complex and involves cascades of bifurcations in a very narrow parameter range. A numerical study of a flow in an enclosed cylindrical cavity driven by axial oscillations of the sidewall, which allows a detailed study of the secondary bifurcations and the corresponding mode interactions, is presented. The study focuses on a codimension-2 point that acts as the organizing center of the dynamics for moderate values of the forcing frequency. The unraveled dynamics is very rich, including slow-fast dynamics and hysteresis, and may help understand the bifurcation cascades in more complex systems., Postprint (published version)

Even though the transition to turbulence has been studied for over a century, its complete comprehension still remains unclear even for the simplest flows and continues to be a daunting challenge for the scientific community. Among these, there is the transition from the von K\'arm\'an vortex street to turbulent wakes. The complexity of this problem poses a series of difficulties that leaves little room for manoeuvre, so other ways to tackle this question have to be sought. A reasonable option is the analysis of the instability phenomena that other flows with the same symmetry group undergo. Despite being really different, an example of such flow is the one generated in a cylindrical cavity subjected to an oscillatory shear. The purpose of the present thesis has been to provide a deeper understanding of the mechanisms that are responsible for the transition in oscillatory cylindrical cavities. Besides the potential implications of studying such systems for the transitions in wake flows, the system under consideration might be useful for any investigation involving a periodic forcing. Accurate spectral computations of the incompressible Navier-Stokes equations have been combined with equivariant bifurcation and normal form theories in an attempt to achieve our goal from different, yet complementary, perspectives. The utilisation of these techniques has produced positive results in the field under consideration. The linear stability analysis has resulted in three types of different bifurcations expected by normal form theory and previous results. The evolution in time of these bifurcating modes yield the non-linear saturated states, which can be synchronous with the forcing or acquire an additional frequency (quasiperiodic). Furthermore, the exploration of regions where two synchronous modes become unstable at the same time, has provided a wide variety of novel states that are not necessarily synchronous. The description of these phenomena via bifurcation theory and d, Encara que la transició a la turbulència s'ha estat estudiant durant més d'un segle, la comprensió completa d'aquesta, encara roman poc clara ,fins i tot en els fluxos més senzills, i contínua sent un repte d'enormes proporcions per a la comunitat científica. La complexitat d'aquest problema planteja una sèrie de dificultats que deixen poc marge de maniobra i, per tant, s'han de cercar altres mètodes per atacar aquesta qüestió. Una opció molt raonable és l'anàlisi dels fenòmens d'inestabilitats en altres sistemes amb el mateix grup de simetries. Malgrat ser molt diferents, un exemple d'aquest tipus de flux seria el que es genera en una cavitat cilíndrica sotmesa a un esforç de cisalla oscil·latori. L'objectiu de la present tesi ha estat proporcionar un coneixement més profund dels mecanismes responsables en la transició en cavitats cilíndriques. Deixant de banda les possibles implicacions d'estudiar aquests sistemes pel que fa a les transicions dels fluxos en les esteles, el sistema que s'està considerant pot resultar útil en qualsevol investigació que involucri un forçament periòdic. En un intent d'arribar a bon port des de diferents punts de vista, càlculs espectrals precisos de les equacions de Navier-Stokes s'han combinat amb la teoria de formes normals. La utilització f'aquestes tècniques ha produït resultats positius en aquest camp. L'anàlisi d'estabilitat lineal ha revelat tres tipus diferents de bifurcacions, les quals s'esperaven a causa de la teoria de formes normals i resultats anteriors. L'evolució temporal d'aquests modes que bifurquen han proporcionar estats saturats no-lineals, els quals poden ser síncrons amb el forçament o poden adquirir una freqüència addicional (quasiperiòdic). A més a més, l'exploració de les regions on dos modes síncrons esdevenen inestables a la vegada, ha proporcionat una gran varietat de nous estats que no han de ser necessàriament síncrons. La descripció d'aquest fenomen per mitjà de la teoria de bifurcacions i les tècniques, Postprint (published version)

This study aims to provide a better understanding of recently identified transition scenarios exhibited by traveling wave solutions in pipe flow. This particular family of solutions are invariant under certain reflectional symmetry transformations and they emerge from saddle-node bifurcations within a two-dimensional parameter space characterized by the length of the pipe and the Reynolds number. The present work precisely provides a detailed analysis of a codimension-two saddle-node bifurcation arising in discrete dynamical systems (maps) with Z2 symmetry. Normal form standard techniques are applied in order to obtain the reduced map up to cubic order. All possible bifurcation scenarios exhibited by this normal form are analyzed in detail. Finally, a qualitative comparison of these scenarios with the ones observed in the aforementioned hydrodynamic problem is provided., Postprint (published version)

La inestabilitat dels fluxos i la transició a la turbulència són fenòmens estesos a l’enginyeria i el medi natural. El problema de Poiseuille és un dels escenaris típics en dinàmica de fluids on la transició de flux lamina a turbulent no està plenament compresa. El problema de Poiseuille és el flux que apareix a un fluid viscós confinat entre dos plans paral•lels on el moviment és induït per un gradient de pressió. Actualment, aquest junt amb altres problemes de fluxos bàsics (com és el cas del flux de Hagen-Poiseuille) són tema de recerca per la teoria de la transició a la turbulència. La primera inestabilitat lineal apareix per Re~5800 . Tot i així, pel problema de Poiseuile 3D s’ha arribat experimentalment a la transició de flux bàsic a turbulència per Re molt menors al valor predit per la teoria lineal. Pel Poiseuille pla (2D) Jimenéz ha determinat computacionalment, per també Re menors, turbulència localitzada. Així doncs, tot i ser l’estudi lineal un dels mitjans més utilitzats en dinàmica de fluids, és evident que els resultats d’aquest anàlisi no coincideixen amb les observacions. En la present tesina s’estudia els estats de turbulència localitzada pel cas pla (2D) per Re menors i dominis menors els ja estudiat per Jimenéz. És realitza una descripció de la formulació del model físic juntament amb el desenvolupament del model numèric emprat. A partir de les equacions de Navier-Stokes, introduint la funció de corrent, es redueix el nombre de variables del problema inicial i s’obté l’equació de la funció de corrent. Mitjançant aquesta última equació, es resoldrà el problema utilitzant mètodes numèrics espectrals. Concretament per la variable espacial s’utilitza Galerkin- Numerical Integration (G-NI), Galerkin combinat amb col•locació i per la variable temporal es fa ús dels mètodes multipas (IMEX2). S’ha analitzat la variació de l’existència de fluxos preturbulents en funció del número de Reynolds i del domin de computació. De les simulacions realitzades és pot

Aunque se sigue desconociendo por qué un fl uido conducido por una tubería se vuelve turbulento, se ha dado un paso importante hacia la resolución del problema, formulado hace ya más de 125 años, Postprint (published version)

Subcritical instabilities in small gap Taylor-Couette (TCF) problem are studied numerically when both cylinders rotate in opposite directions. The computations are carried out for a radius ratio A first exploration is focused on the study of spiral flows originated from subcritical Hopf bifurcations of the basic circular Couette solution. The second exploration addresses the transition from laminar flow to the usually termed as spiral turbulence regime characterized by alternating laminar and turbulent spiral bands which coexist even in regions of the parameter space where the circular Couette flow is linearly stable., Postprint (published version)