Efficient mixing can be achieved in flows where turbulence is absent, if the trajectories of passively advected particles in the flow are chaotic. The chaotic nature of particle trajectories results in exponential stretching of material lines in the flow. Thus the interface along which diffusion occurs is stretched exponentially leading to efficient mixing. It has been demonstrated recently that regions in flow fields that exhibit poor mixing and non-chaotic particle trajectories can have an important bearing on the overall dynamics and transport of the entire domain. The space-time trajectories of physical stirrers or elliptic points in two dimensional flows can be classified according to braid groups. One can predict a lower bound on the topological entropy (i.e. exponential rate of stretching of material lines) of flows (h_f) by applying the Thurston-Nielsen classification theorems to these braids. This gives a reduced order model for the dynamics of transport of the entire flow field using just a few points. Recent work has shown that this methodology can be used to estimate a lower bound on h_f using the braids formed by Almost Cyclic Sets (ACS) in certain periodic Stokes' flows. These ACS are closely related to Almost Invariant Sets (AIS) which are identified using a probabilistic set oriented method that makes use of the descritised Perron-Frobenius operator of the flow map. This work extends this approach to flows at non-zero Reynolds numbers, which take into account the effects of inertia. The role of Finite Time Coherent Structures (FTCS) in the dynamics of flow fields is also investigated. Unlike ACS, the FTCS approach is more general as it can be applied to aperiodic flow fields. Further, the relationship between mixing efficiency and the topological entropy of flow fields at non-zero Reynolds numbers is also studied. / Ph. D.

We are interested in examining the long-time decay rate of a passive scalar in two-dimensional flows. The focus is on the effect of boundary conditions for kinematically prescribed velocity fields with random or periodic time dependence. Scalar evolution is followed numerically in a periodic geometry for families of flows that have either a slip or a no-slip boundary condition on a square or plane layer subdomain D. The boundary conditions on the passive scalar are imposed on the boundary C of the domain D by restricting to a subclass invariant under certain symmetry transformations. The scalar field obeys constant (Dirichlet) or no-flux (Neumann) conditions exactly for a flow with the slip boundary condition and approximately in the no-slip case. At late times the decay of a passive scalar, for example temperature, is exponential in time with a decay rate gamma(kappa), where kappa is the molecular diffusivity. Scaling laws of the form gamma(kappa) ~ C*kappa^alpha for small kappa are obtained numerically for a variety of boundary conditions on flow and scalar, and supporting theoretical arguments are presented. In particular when the scalar field satisfies a Neumann condition on all boundaries, alpha ~ 0 for a slip flow condition; for a no-slip condition we confirm results in the literature that alpha ~ 1/2 for a plane layer, but find alpha ~ 2/3 in a square subdomain D where the decay is controlled by stagnant flow in the corners. For cases where there is a Dirichlet boundary condition on one or more sides of the subdomain D, the exponent measuring the decay of the scalar field is alpha ~ 1/2 for a slip flow condition and alpha ~ 3/4 for a no-slip condition. The scaling law exponents alpha for chaotic time-periodic flows are compared with those for similarly constructed random flows. Motivated by the theory of passive scalar field, in Part II of this work we extend the investigation of the evolution of passive scalar for the flows addressed specifically in Part I. Based on an ensemble averaging over random velocity fields, the theoretical results obtained confirm the scaling laws computed numerically for a single, long realisation of random flows. In analogy with Lebedev and Turitsyn (2004) and Salman and Haynes (2007) our results show very good agreement between such an ensemble theory and applications. In part III of our study, we expand upon the work set out in the previous parts of this thesis in terms of the polar-co-ordinate system. We analyse the structures of flows driven near to a corner with a link to Moffatt corner eddies. A long-time exponential decay rate gamma(kappa)=C*kappa^alpha has been obtained confirming our numerical and theoretical results predicted in Part I and Part II in this work. The exponent alpha is determined in a structure of Moffatt corner eddies.

In many engineering applications involving mixing of highly viscous fluids or mixing at micro-scales, efficient mixing must be accomplished in the absence of turbulence. Similarly in geophysical flows large-scale, deterministic flow structures can account for a considerable portion of global transport and mixing. For these types of problems, concepts from non-linear dynamical systems and the theory of chaotic advection provide the tools for understanding, quantifying, and optimizing transport and mixing processes. In this thesis chaotic advection is studied numerically in three, steady, experimentally realizable, three-dimensional flows: 1) steady vortex breakdown flow in a cylindrical container with bottom rotating lid, 2) flow in a cylindrical container with exactly counter rotating lids, and 3) flow in a new model stirred-tank with counter-rotating disks. For all cases the three-dimensional Navier-Stokes equations are solved numerically and the Lagrangian properties of the computed velocity fields are analyzed using a variety of computational and theoretical tools. For the flow in the interior of vortex breakdown bubbles it is shown that even though from the Eulerian viewpoint the simulated flow fields are steady and nearly axisymmetric the Lagrangian dynamics could be chaotic. Silnikovs mechanism is shown to play a critical role in breaking up the invariance of the bubble and giving rise to chaotic dynamics. The computations for the steady flow in a cylindrical container with two exactly counter-rotating lids confirm for the first time the findings of recent linear stability studies. Above a threshold Reynolds number the equatorial shear layer becomes unstable to azimuthal modes and an intricate web of radial (cats eyes) and axial, azimuthally-inclined vortices emerge in the flow paving the way for extremely complex chaotic dynamics. Using these fundamental insights, a new stirring tank device with exactly counter-rotating disks is proposed. Results show for the first time that counter rotation of the middle disk in a three-disk stirred tank can create a flow with large chaotic regions. The results of this thesis serve to demonstrate that fundamental studies of chaotic mixing are both important from a theoretical standpoint and can potentially lead to valuable technological breakthroughs.

The effect of finite Reynolds numbers on chaotic advection is investigated for two dimensional lid-driven cavity flows that exhibit topological chaos in the creeping flow regime. The emphasis in this endeavor is to study how the inertial effects present due to small, but non-zero, Reynolds number influence the efficacy of mixing. A spectral method code based on the Fourier-Chebyshev method for two-dimensional flows is developed to solve the Navier-Stokes and species transport equations. The high sensitivity to initial conditions and the exponentional growth of errors in chaotic flows necessitate an accurate solution of the flow variables, which is provided by the exponentially convergent spectral methods. Using the spectral coefficients of the basis functions as solved through the conservation equations, exponentially accurate values of velocity everywhere in the flow domain are obtained as required for the Lagrangian particle tracking. Techniques such as Poincare maps, the stirring index based on the box counting method, and the tracking of passive scalars in the flow are used to analyze the topological chaos and quantify the mixing efficiency.

Theoretical and numerical studies of chaotic mixing are performed to circumvent the difficulties of efficient mixing, which come from the lack of turbulence in microfluidic devices. In order to carry out efficient and accurate parametric studies and to identify a fully chaotic state, a spectral element algorithm for solution of the incompressible Navier-Stokes and species transport equations is developed. Using Taylor series expansions in time marching, the new algorithm employs an algebraic factorization scheme on multi-dimensional staggered spectral element grids, and extends classical conforming Galerkin formulations to nonconforming spectral elements. Lagrangian particle tracking methods are utilized to study particle dispersion in the mixing device using spectral element and fourth order Runge-Kutta discretizations in space and time, respectively. Comparative studies of five different techniques commonly employed to identify the chaotic strength and mixing efficiency in microfluidic systems are presented to demonstrate the competitive advantages and shortcomings of each method. These are the stirring index based on the box counting method, Poincare sections, finite time Lyapunov exponents, the probability density function of the stretching field, and mixing index inverse, based on the standard deviation of scalar species distribution. Series of numerical simulations are performed by varying the Peclet number (Pe) at fixed kinematic conditions. The mixing length (lm) is characterized as function of the Pe number, and lm ∝ ln(Pe) scaling is demonstrated for fully chaotic cases. Employing the aforementioned techniques, optimum kinematic conditions and the actuation frequency of the stirrer that result in the highest mixing/stirring efficiency are identified in a zeta potential patterned straight micro channel, where a continuous flow is generated by superposition of a steady pressure driven flow and time periodic electroosmotic flow induced by a stream-wise AC electric field. Finally, it is shown that the invariant manifold of hyperbolic periodic point determines the geometry of fast mixing zones in oscillatory flows in two-dimensional cavity.

Creating chaotic advection is the most efficient strategy to achieve mixing in a microscale or in a very viscous fluid, and it has many important applications in microfluidic devices, material processing and so on. In this paper, we present a quantitative long-term theory of resonant mixing in 3-D near-integrable flows. We use the flow in the annulus between two coaxial elliptic counter-rotating cylinders as a demonstrative model. We illustrate that such resonance phenomena as resonance and separatrix crossings accelerate mixing by causing the jumps of adiabatic invariants. We calculate the width of the mixing domain and estimate a characteristic time of mixing. We show that the resulting mixing can be described in terms of a single diffusion-type equation with a diffusion coefficient depending on the averaged effect of multiple passages through resonances. We discuss what must be done to accommodate the effects of the boundaries of the chaotic domain. / Mechanical Engineering

Hydraulic structures such as dykes and groynes are commonly used to help control river flows and reduce flood risk. The present research aims to develop an idealized model of the hydrodynamics in the vicinity of a large river bend, and the advection and mixing processes where groynes are located. In this study a curvilinear model of shallow water equations is applied to investigate chaotic advection of particles in a river bend similar in dimensions to a typical bend in the River Danube, Hungary. First, a curvilinear grid generator is developed based on Poisson-type elliptic partial differential equations. The grid generator is verified for benchmark tests concerning a circular domain and for distorted grids in a rectangular domain. It is found that multi-grid (MG) and conjugate gradient (CG) methods performed better computationally than successive over-relaxation (SOR) in generating the curvilinear grids. The open channel hydrodynamics are modelled using the shallow water equations (SWEs) derived by depth-averaging the continuity and Navier-Stokes momentum equations. Both Cartesian and curvilinear forms of the shallow water equations are presented. Both sets of equations are discretized spatially using finite differences and the solution marched forward in time using fourth-order Runge-Kutta scheme. The shallow water solvers are verified and validated for uniform flow in the rectangular channel, wind-induced set up in rectangular and circular basins, flow past a sidewall expansion, and Shallow flow in a rectangular channel with single groyne. A Lagrangian particle tracking model is used to predict the trajectories of tracer particles, and bilinear interpolation is used to provide a representation of the continuous flow field from discrete results. The particle tracking model is verified for trajectories in the flow field of a single free vortex and in the alternating flow field of a pair of blinking vortices. Excellent agreement is obtained with analytical solutions, previously published results in the literature. The combined shallow flow and Lagrangian particle tracking model is then used to simulate particle advection in the flow past a side-wall cavity containing a groyne and reasonable agreement is obtained with published experimental and alternative numerical data. Finally, the combined model is applied to simulate the shallow flow hydrodynamics, advection and mixing processes in the vicinity of groynes in river bend, the dimensions representative of a typical bend in the Danube River, Hungary.

The research presented in this thesis was motivated by the desire to understand the flow field within a new digital microfluidic device currently under development. This required an investigation of the dynamics of a droplet migrating along the surface of another fluid due to interfacial surface tension gradients. The quantitative analysis of the flow field presented in this thesis provides the first known solution for the velocity field in a migrating droplet confined to an interface. The first step towards gaining insight into the flow field was accomplished by using the method of reflections to obtain an analytical model for a submerged droplet migrating near a free surface. The submerged droplet model enabled the analysis of the velocity field and droplet migration speed and their dependence on the fluid properties. In general, the migration velocity of a submerged droplet was found to differ dramatically from the classic problem of thermocapillary migration in an unbounded substrate. A boundary-collocation scheme was developed to determine the flow field and migration velocity of a droplet floating trapped at the air-substrate interface. The numerical method was found to produce accurate solutions for the velocity and temperature fields for nearly all parameters. This numerical scheme was used to judge the accuracy of the flow field obtained by the submerged droplet model. In particular, the model was tested using parameter values taken from a digital microfluidic device. It was determined that the submerged droplet model captured most of the flow structure within the microfluidic droplet. However, for a slightly different choice of parameters, agreement between the two methods was lost. In this case, the numerical scheme was used to uncover novel flow structures.

Cette thèse est une étude numérique d'un écoulement d'air dans une cavité différentiellement chauffée bidimensionnelle en présence de gravité. Pour un rapport hauteur/largeur de deux et des parois horizontales supposées adiabatiques, l'écoulement de base correspond à une recirculation autour de la cavité avec un coeur stratifié et des couches limites verticales. Les équations de Navier-Stokes sont résolues par un code de simulation numérique directe spectrale instationnaire basé sur l’hypothèse de Boussinesq couplé à un algorithme de suivi de particules avec interpolation. Le nombre de Rayleigh basé sur la différence de température est choisi comme paramètre de contrôle de l’écoulement. La transition vers le chaos au sein de cet écoulement est explorée à la fois du point de vue eulérien (développement de l’instationnarité) et lagrangien (mélange chaotique).L'approche lagrangienne considère le mélange de traceurs passifs infinitésimaux non diffusifs. L'étude se base sur l'identification d'objets invariants de la dynamiques : points fixes, orbites périodiques et leurs variétés stable/instable, connections homoclines et hétéroclines, trajectoires toroïdales. Le mélange des traceurs est partiel lorsque l'écoulement subit une première bifurcation de Hopf. La dispersion globale des traceurs résulte d'un compromis entre la présence de tores Kolmogorov-Arnold-Moser qui jouent le rôle de barrières au mélange, et d'enchevêtrements homoclines/hétéroclines responsables du chaos lagrangien. L'étude statistique des temps de retour et du taux d'homogénéisation révèle la présence de zones où la dynamique est non hyperbolique. En augmentant le nombre de Rayleigh, le mélange devient progressivement complet avant que l'écoulement ne devienne quasi-périodique en temps. L'approche eulérienne considère les divers scénarios de transition vers le chaos par l'identification numérique d'attracteurs et des bifurcations associées lorsque le nombre de Rayleigh varie. Deux routes principales se distinguent en fonction des symétries associées aux deux premières bifurcations de Hopf du système, contenant chacune plusieurs branches hystérétiques. Trente trois régimes différents sont identifiés et analysés depuis l'écoulement stationnaire jusqu'à un écoulement chaotique voire hyperchaotique. Parmi ceux-ci, des branches de tores à deux et trois fréquences incommensurables, ainsi que des régimes intermittents sont examinés. Des diagrammes de bifurcations qualitatifs et quantitatifs sont proposés pour résumer l'ensemble des dynamiques observées. / This thesis is about the numerical study of an air flow inside a two dimensionally heated cavity. The aspect ratio height/width is set to two. Boundary conditions on horizontal walls are taken as adiabatic. In this case, the base flow consists of a recirculation around the stratified core of the cavity and of boundary layers along the vertical walls. The Navier-Stokes equations are solved using a spectral direct numerical simulation code under the Boussinesq assumption coupled with a particle tracking scheme based on interpolation. The Rayleigh number, based on the temperature difference is chosen as the control parameter of the system. The transition to chaos in this flow is considered both from the Eulerian and Lagrangian point of view.The Lagrangian point of view considers the mixing of point-wise non-diffusive passive tracers. The study is based on the identification of invariant objects: fixed points, periodic orbits and their stable/unstable manifolds,homoclinic and heteroclinic connections, toroidal trajectories.The mixing of tracers is partial when the flow undergoes the first Hopf bifurcation. The complete mixing of tracers results from a compromise between Kolmogorov-Arnold-Moser's tori, which act as barriers to mixing, and homoclinic/heteroclinic tangles which are responsible for the mixing.The statistical study of return times and the homogenisation rate shows regionswhere the dynamics is non-hyperbolic. When the Rayleigh number is increased, mixing is increasingly complete before the flow becomes quasi-periodic in time.The Eulerian description considers the transition to chaos via the numerical identification of attractors and their associated bifurcations when the Rayleigh number is varied. Two main routes are found depending on the symmetries associated with the first two Hopf bifurcations of the system. A total of thirty three different regimes are identified from steady to hyperchaotic, among which two- and three-frequency tori as well as intermittent dynamics. Both quantitative and qualitative bifurcation diagrams are suggested for the system.

Le mélange de solutés par les écoulements en milieux poreux contrôle les réactions chimiques dans un grand nombre d'applications souterraines, dont le transport et la remédiation des contaminants, le stockage et l'extraction souterrains d'énergie, et la séquestration du CO2. Nous étudions les mécanismes du mélange à l'échelle du pore et plus précisément comment la topologie de l'écoulement est reliée à la dynamique du mélange d'espèces conservatives; en particulier, l'émergence d'un mélange chaotique est-elle possible dans un milieu poreux tridimensionnel (3D) ? Nous calculons donc numériquement ou mesurons expérimentalement les vitesses d'écoulement et l'évolution temporelle des champs de concentration afin de caractériser la déformation et le mélange à l'échelle du pore. Une première étude, expérimentale, permet de caractériser le mélange dans un fluide s’écoulant à travers un milieu poreux bidimensionnel (2D). Nous mesurons les vitesses par suivi de microparticules solides (''PTV''). L’évolution temporelle de la distance séparant deux particules permet de caractériser la dynamique de la déformation lagrangienne. Des mesures de transport conservatif dans le même milieu fournissent l'évolution temporelle du gradient de concentration moyen (une mesure du mélange). À partir de ces résultats expérimentaux nous proposons la première validation expérimentale à l'échelle du pore de la théorie lamellaire du mélange, reliant les propriétés de la déformation du fluide à la dynamique du mélange. Dans une deuxième étude nous examinons les conditions d'apparition du mélange chaotique dans l’écoulement dans des milieux poreux 3D granulaires ordonnés. Nous effectuons des calculs numériques hautement résolus de d'écoulement de Stokes entre des sphères empilées selon une structure cristalline (cubique simple ou cubique centrée), périodique. La déformation lagrangienne, obtenue à partir des champs de vitesse à l'aide d'outils numériques développés spécifiquement, met en lumière une large variété de dynamiques de la déformation dans ces milieux 3D, selon l'orientation de l'écoulement. Quand la direction de l'écoulement n'est pas normale à l'un des plans de symétrie de réflection du cristal, l'évolution temporelle de la déformation est exponentielle, traduisant une advection chaotique. L’émergence (ou non) du chaos est contrôlée par un mécanisme similaire à la ''transformation du boulanger'': les particules fluides se déplaçant autour d'un grain solide se retrouvent séparées par une surface virtuelle (appelée “variété”) qui émerge de la surface du grain. De multiples variétés existent dans l’écoulement, et la façon dont elles s'intersectent contrôle la nature - chaotique ou non - du mélange, et l'intensité du chaos. En particulier, l'exposant de Lyapunov (une mesure du chaos), est contrôlé par la fréquence spatiale des intersections appropriées à la génération du chaos, nommées ''connections hétéroclinines'' entre variétés. L'image conventionnelle, 2D, des mécanismes du mélange, impose des contraintes topologiques qui ne permettent pas le développement de ces mécanismes 3D. Elle pourrait donc être inadaptée aux milieux poreux naturels. La troisième étude a deux objectifs: (i) fournir une preuve expérimentale de la nature chaotique de l'advection, par la visualisation des variétés et par l'obtention d'une mesure de l'exposant de Lyapunov; et (ii), évaluer si nos résultats numériques obtenus pour des milieux granulaires ordonnés peuvent être généralisés à des milieux désordonnés, plus proches des milieux naturels. L’expérience est fondée sur un empilement désordonné de sphères rendu transparent par l'ajustement optique du liquide avec les sphères. La fluorescence induite par laser (''LIF'') permet de détecter les variétés au sein de l'écoulement, et des techniques PTV de mesurer les vitesses d'écoulement et quantifier l'exposant de Lyapunov. Les premiers résultats expérimentaux sont prometteurs. / Solute mixing in porous media flows plays a central role in driving chemical reactions in a number of subsurface applications, including contaminant transport and remediation, subsurface energy storage and extraction, and CO2 sequestration. We study the mechanisms of solute mixing, in particular how the pore scale flow topology is related to the mixing dynamics of conservative solutes, with a particular emphasis on the possible emergence of chaotic mixing processes in three-dimensional (3D) porous media. To do so, we perform numerical computations or experimental measurements of the flow velocities and temporal evolution of the concentration fields, and characterize fluid deformation and mixing at the pore scale. This PhD work consists of three main studies. In the first study, we experimentally characterize mixing in a fluid flowing through a two-dimensional (2D) porous medium built by lithography. We measure the velocity distributions from Particle Tracking Velocimetry (PTV). The time evolution of the separation distance between two particles is analyzed to characterize the Lagrangian deformation dynamics. In parallel we perform conservative transport experiments with the same porous media, and quantify the temporal evolution of the mean concentration gradient, which is a measure of the mixing rate. From these experimental results we obtain the first experimental pore scale validation of the lamella mixing theory, which relates the fluid deformation properties to the mixing dynamics. In the second study, we investigate the conditions of emergence of chaotic mixing in the flow through 3D ordered granular porous media. In these periodic cubic crystalline packings (Simple Cubic - SC - and Body-Centered Cubic - BCC) of spheres, we are able to perform highly resolved computations of the 3D Stokes flow. Using custom-developed numerical tools to measure the Lagrangian deformation from the computed velocity fields, we uncover the existence of a rich array of Lagrangian deformation dynamics in these 3D media, depending on the flow orientation. When the flow direction is not normal to one of the reflection symmetry planes of the crystalline lattice, we find that the Lagrangian deformation dynamics follow an exponential law, which indicates chaotic advection. This chaotic behavior is controlled by a mechanism akin to the baker's transformation: fluid particles traveling around a solid grain along different paths end up either separated by, or on the same side of, a virtual surface projecting from the grain surface and called a manifold. Multiple such manifolds exist within the flow, and the way they intersect controls the nature of mixing (that is, either non-chaotic or chaotic), and the strength of chaos. We show in particular that the magnitude of the Lyapunov exponent (a measure of the vigor of chaos) is controlled by the spatial frequency of transverse connections between the manifolds (called heteroclinic intersections). We thus demonstrate that the conventional 2D picture of the mechanisms of mixing may not be adapted for natural porous media because that picture imposes topological constraints which cannot account for these important 3D mechanisms. The third study has two objectives: (i) provide experimental evidence of the chaotic nature of pore scale advection/mixing, both by visualizing the manifolds and by obtaining a quantitative estimate of the Lyapunov exponent; and (ii) assess if the results obtained numerically in ordered packings of spheres extend to random packings, which are closer to natural porous media. The experiment features a random packing of glass beads rendered transparent by optical index-matching between the fluid and solid grains. We use Laser Induced Fluorescence (LIF) to detect the manifolds, and PTV techniques to measure flow velocities and subsequently quantify Lyapunov exponent. The first experimental results are promising.