Browsing by Author "Aref, Hassan"
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- Chaos in Pulsed Laminar FlowKumar, Pankaj (Virginia Tech, 2010-08-09)Fluid mixing is a challenging problem in laminar flow systems. Chaotic advection can play an important role in enhancing mixing in such flow. In this thesis, different approaches are used to enhance fluid mixing in two laminar flow systems. In the first system, chaos is generated in a flow between two closely spaced parallel circular plates by pulsed operation of fluid extraction and reinjection through singularities in the domain. A singularity through which fluid is injected (or extracted) is called a source (or a sink). In a bounded domain, one source and one sink with equal strength operate together as a source-sink pair to conserve the fluid volume. Fluid flow between two closely spaced parallel plates is modeled as Hele-Shaw flow with the depth averaged velocity proportional to the gradient of the pressure. So, with the depth-averaged velocity, the flow between the parallel plates can effectively be modeled as two-dimensional potential flow. This thesis discusses pulsed source-sink systems with two source-sink pairs operating alternately to generate zig-zag trajectories of fluid particles in the domain. For reinjection purpose, fluid extracted through a sink-type singularity can either be relocated to a source-type one, or the same sink-type singularity can be activated as a source to reinject it without relocation. Relocation of fluid can be accomplished using either "first out first in" or "last out first in" scheme. Both relocation methods add delay to the pulse time of the system. This thesis analyzes mixing in pulsed source-sink systems both with and without fluid relocation. It is shown that a pulsed source-sink system with "first out first in" scheme generates comparatively complex fluid flow than pulsed source-sink systems with "last out first in" scheme. It is also shown that a pulsed source-sink system without fluid relocation can generate complex fluid flow. In the second system, mixing and transport is analyzed in a two-dimensional Stokes flow system. Appropriate periodic motions of three rods or periodic points in a two-dimensional flow are determined using the Thurston-Nielsen Classification Theorem (TNCT), which also predicts a lower bound on the complexity generated in the fluid flow. This thesis extends the TNCT -based framework by demonstrating that, in a perturbed system with no lower order fixed points, almost invariant sets are natural objects on which to apply the TNCT. In addition, a method is presented to compute line stretching by tracking appropriate motion of finite size rods. This method accounts for the effect of the rod size in computing the complexity generated in the fluid flow. The last section verifies the existence of almost invariant sets in a two-dimensional flow at finite Reynolds number. The almost invariant set structures move with appropriate periodic motion validating the application of the TNCT to predict a lower bound on the complexity generated in the fluid flow.
- Chaotic scattering of two identical point vortex pairs revisitedTophoj, L.; Aref, Hassan (American Institute of Physics, 2008-09-01)A new numerical exploration suggests that the motion of two vortex pairs, with constituent vortices all of the same absolute circulation, displays chaotic scattering regimes. The mechanisms leading to chaotic scattering are different from the "slingshot effect" identified by Price [Phys. Fluids A 5, 2479 (1993)] and occur in a different region of the four-vortex phase space. They may, in many cases, be understood by appealing to the solutions of the three-vortex problem obtained by merging two like-signed vortices into one of twice the strength and by assuming that the four-vortex problem has unstable periodic solutions similar to those seen in the thereby associated three-vortex problems. The integrals of motion, linear impulse and Hamiltonian are recast in a form appropriate for vortex pair scattering interactions that provides constraints on the parameters characterizing the outgoing vortex pairs in terms of the initial conditions.
- Close pairs of relative equilibria for identical point vorticesDirksen, T.; Aref, Hassan (American Institute of Physics, 2011-05-01)Numerical solution of the classical problem of relative equilibria for identical point vortices on the unbounded plane reveals configurations that are very close to the analytically known, centered, symmetrically arranged, nested equilateral triangles. New numerical solutions of this kind are found for 3n+1 vortices, where n = 2, 3, ..., 30. A sufficient, although apparently not necessary, condition for this phenomenon of close solutions is that the "core" of the configuration is marginally stable, as occurs for a central vortex surrounded by an equilateral triangle. The open, regular heptagon also has this property, and new relative equilibria close to the nested, symmetrically arranged, regular heptagons have been found. The centered regular nonagon is also marginally stable. Again, a new family of close relative equilibria has been found. The closest relative equilibrium pairs occur, however, for symmetrically nested equilateral triangles. (C) 2011 American Institute of Physics.
- Instability of vortex pair leapfroggingTophoj, L.; Aref, Hassan (American Institute of Physics, 2013-01-01)Leapfrogging is a periodic solution of the four-vortex problem with two positive and two negative point vortices all of the same absolute circulation arranged as co-axial vortex pairs. The set of co-axial motions can be parameterized by the ratio 0 < alpha < 1 of vortex pair sizes at the time when one pair passes through the other. Leapfrogging occurs for alpha > sigma(2), where sigma = root 2 - 1 is the silver ratio. The motion is known in full analytical detail since the 1877 thesis of Grobli and a well known 1894 paper by Love. Acheson ["Instability of vortex leapfrogging," Eur. J. Phys. 21, 269-273 (2000)] determined by numerical experiments that leapfrogging is linearly unstable for sigma(2) < alpha < 0.382, but apparently stable for larger alpha. Here we derive a linear system of equations governing small perturbations of the leapfrogging motion. We show that symmetry-breaking perturbations are essentially governed by a 2D linear system with time-periodic coefficients and perform a Floquet analysis. We find transition from linearly unstable to stable leapfrogging at alpha = phi(2) approximate to 0.381966, where phi = 1/2 (root 5 - 1) is the golden ratio. Acheson also suggested that there was a sharp transition between a "disintegration" instability mode, where two pairs fly off to infinity, and a "walk-about" mode, where the vortices depart from leapfrogging but still remain within a finite distance of one another. We show numerically that this transition is more gradual, a result that we relate to earlier investigations of chaotic scattering of vortex pairs [L. Tophoj and H. Aref, "Chaotic scattering of two identical point vortex pairs revisited," Phys. Fluids 20, 093605 (2008)]. Both leapfrogging and "walkabout" motions can appear as intermediate states in chaotic scattering at the same values of linear impulse and energy.
- A note on the energy of relative equilibria of point vorticesAref, Hassan (American Institute of Physics, 2007-10-01)Analytical formulas are derived for the energy of simple relative equilibria of identical point vortices such as the regular polygons, both open and centered, and the various known configurations consisting of nested regular polygons with or without a vortex at the center.
- Numerical Modeling of Microscale Mixing Using Lattice Boltzmann MethodDe, Anindya Kanti (Virginia Tech, 2008-04-28)Recent advancements in microfabrication technology have led to the development of micro-total analytical systems (μ-TAS), more popularly known as lab-on-a-chip (LOC) devices. These devices have a relatively small size and are capable of performing sample and reagent handling steps together with analytical measurements. Rapid mixing is essential in such microfluidic systems for various applications e.g., biochemical analysis, sequencing or synthesis of nucleic acids, and for reproducible biological processes that involve cell activation, enzyme reactions, and protein folding. In this work a numerical model is developed using a lattice Boltzmann method (LBM) to study microscale mixing. The study involves two mixing methods, namely, electroosmotic mixing and magnetic assisted mixing. A single component LBM model is developed to study electroosmotic flow in a square cavity. Mixing is studied by introducing two types of tracer particles in the steady electroosmotic flow and characterized by various mixing parameters. The results show that rapid mixing can be achieved by using a steady electric field and a homogeneous zeta potential. A multicomponent LBM method is also developed to study magnetic assisted mixing in a channel configuration. The ferrofluid flow is influenced by two magnets placed across a microchannel. The interacting field induced by these magnets promotes cross-stream motion of the ferrofluid, which induces its mixing with the other nonmagnetic fluid. Two fluids, one magnetic and another non-magnetic fluid, are introduced in a channel, when two magnets are placed across it at a distance apart. In the presence of the magnetic field, the magnetic fluid tries to follow a zig-zag motion generating two rolls of vortices thereby enhancing mixing. A parametric study characterizes the effects of diffusivity, magnetic field strength, and relative magnet positions on a mixing parameter. Mixing is enhanced when the magnetic field strength and diffusivity are increased. However, contrary to the observed trend, placing the magnets very close to each other axially results in local ferrofluid agglomeration rather than promoting mixing.
- On the atmosphere of a moving bodyRoenby, J.; Aref, Hassan (American Institute of Physics, 2010-05-01)We explore whether a rigid body moving freely with no circulation around it in a two-dimensional ideal fluid can carry a fluid "atmosphere" with it in its motion. Somewhat surprisingly, the answer appears to be "yes." When the body is elongated and the motion is dominated by rotation, we demonstrate numerically that, indeed, regions of fluid follow the body in its motion. We see this as an example of the stability of Kolmogorov-Arnold-Moser tori. These observations of an atmosphere around a moving body with no circulation around it appear to be new.
- Point vortex dynamics: A classical mathematics playgroundAref, Hassan (AIP Publishing, 2007-06-01)The idealization of a two-dimensional, ideal flow as a collection of point vortices embedded in otherwise irrotational flow yields a surprisingly large number of mathematical insights and connects to a large number of areas of classical mathematics. Several examples are given including the integrability of the three-vortex problem, the interplay of relative equilibria of identical vortices and the roots of certain polynomials, addition formulas for the cotangent and the Weierstrass zeta function, projective geometry, and other topics. The hope and intent of the article is to garner further participation in the exploration of this intriguing dynamical system from the mathematical physics community. (c) 2007 American Institute of Physics.
- Self-similar motion of three point vorticesAref, Hassan (American Institute of Physics, 2010-05-01)One of the counter-intuitive results in the three-vortex problem is that the vortices can converge on and meet at a point in a finite time for certain sets of vortex circulations and for certain initial conditions. This result was already included in Groumlbli's thesis of 1877 and has since been elaborated by several authors. It arises from an investigation of motions where the vortex triangle retains its shape for all time, but not its size. We revisit these self-similar motions, develop a new derivation of the initial conditions that lead to them, and derive a number of formulae pertaining to the rate of expansion or collapse and the angular frequency of rotation, some of which appear to be new. We also pursue the problem of linear stability of these motions in detail and, again, provide a number of formulae, some of which are new. In particular, we determine all eigenmodes analytically.
- Stability of relative equilibria of three vorticesAref, Hassan (American Institute of Physics, 2009-09-01)Three point vortices on the unbounded plane have relative equilibria wherein the vortices either form an equilateral triangle or are collinear. While the stability analysis of the equilateral triangle configurations is straightforward, that of the collinear relative equilibria is considerably more involved. The only comprehensive analysis available in the literature, by Tavantzis and Ting [Phys. Fluids 31, 1392 (1988)], is not easy to follow nor is it very physically intuitive. The symmetry between the three vortices is lost in this analysis. A different analysis is given based on explicit formulas for the three eigenvalues determining the stability, including a new formula for the angular velocity of rotation of a collinear relative equilibrium. A graphical representation of the space of vortex circulations is introduced, and the resultants between various polynomials that enter the problem are used. This approach adds considerable transparency to the solution of the stability problem and provides more physical understanding. The main results are summarized in a diagram that gives both the stability and instability of the various collinear relative equilibria and their sense of rotation.
- Topological chaos and chaotic mixing of viscous flowsGheisarieha, Mohsen (Virginia Tech, 2011-04-27)Since it is difficult or impossible to generate turbulent flow in a highly viscous fluid or a microfluidic system, efficient mixing becomes a challenge. However, it is possible in a laminar flow to generate chaotic particle trajectories (well-known as chaotic advection), that can lead to effective mixing. This dissertation studies mixing in flows with the limiting case of zero Reynolds numbers that are called Stokes flows and illustrates the practical use of different theories, namely the topological chaos theory, the set-oriented analysis and lobe dynamics in the analysis, design and optimization of different laminar-flow mixing systems. In a recent development, the topological chaos theory has been used to explain the chaos built in the flow only based on the topology of boundary motions. Without considering any details of the fluid dynamics, this novel method uses the Thurston-Nielsen (TN) classification theorem to predict and describe the stretching of material lines both qualitatively and quantitatively. The practical application of this theory toward design and optimization of a viscous-flow mixer and the important role of periodic orbits as "ghost rods" are studied. The relationship between stretching of material lines (chaos) and the homogenization of a scalar (mixing) in chaotic Stokes flows is examined in this work. This study helps determining the extent to which the stretching can represent real mixing. Using a set-oriented approach to describe the stirring in the flow, invariance or leakiness of the Almost Invariant Sets (AIS) playing the role of ghost rods is found to be in a direct relationship with the rate of homogenization of a scalar. The mixing caused by these AIS and the variations of their structure are explained from the point of view of geometric mechanics using transport through lobes. These lobes are made of segments of invariant manifolds of the periodic points that are generators of the ghost rods. A variety of the concentration-based measures, the important parameters of their calculation, and the implicit effect of diffusion are described. The studies, measures and methods of this dissertation help in the evaluation and understanding of chaotic mixing systems in nature and in industrial applications. They provide theoretical and numerical grounds for selection of the appropriate mixing protocol and design and optimization of mixing systems, examples of which can be seen throughout the dissertation.
- Topological Chaos and Mixing in Lid-Driven Cavities and Rectangular ChannelsChen, Jie (Virginia Tech, 2008-11-17)Fluid mixing is a challenging problem in laminar flow systems. Even in microfluidic systems, diffusion is often negligible compared to advection in the flow. The idea of chaotic advection can be applied in these systems to enhance mixing efficiency. Topological chaos can also lead to efficient and rapid mixing. In this dissertation, an approach to enhance fluid mixing in laminar flows without internal rods is demonstrated by using the idea of topological chaos. Periodic motion of three stirrers in a two-dimensional flow can lead to chaotic transport of the surrounding fluid. For certain stirrer motions, the generation of chaos is guaranteed solely by the topology of that motion and continuity of the fluid. This approach is in contrast to standard techniques. Appropriate stirrer motions are determined using the Thurston-Nielsen classification theorem, which also predicts a lower bound on the complexity of the dynamics in the flow. Work in this area has focused largely on using physical rods as stirrers, but the theorem also applies when the ``stirrers'' are passive fluid particles. In this thesis, topological chaos is theoretically and numerically investigated in lid-driven cavities and rectangular channels without internal rods. When a two-dimensional incompressible Newtonian flow is in the Stokes flow regime, the stream function satisfies the two-dimensional biharmonic equation. When the flow occurs in a lid-driven cavity with solid side walls, this equation can be solved using a method that is similar to the traditional Fourier expansion but uses an asymptotic approximation for the sum of high order terms. When the flow occurs between two infinite plates with spatially periodic boundary conditions, an exact solution in a rectangle with finite width, which represents the flow in this infinitely-wide cavity, can be obtained by using the principle of superposition. A fully developed, three-dimensional flow in a rectangular channel can be decomposed into an unperturbed Poiseuille flow in the axial direction and a lid-driven cavity secondary flow in the cross section. This model can be applied to numerically simulate either pressure-driven flow in a rectangular channel with surface grooves or electro-osmotic flow in a rectangular channel with variations in surface potential. In this dissertation, the occurrence of topological chaos in unsteady two-dimensional flows as well as steady three-dimensional flows without internal rods is demonstrated. For appropriate choices of boundary velocity on the top and/or bottom walls, there exist three periodic points in the flows that produce a chaos-generating motion. In steady flow through a three-dimensional rectangular channel, the axial direction plays the role of time and the periodic points lie on streamtubes that "braid" the surrounding fluid as it moves through the duct. When appropriate motion is applied on the boundary of the wide cavity or channel, topological chaos can also be generated in the flow. The stretching rate of non-trivial material lines in all these flows agrees with the prediction of the lower bound of topological entropy provided by the Thurston-Nielsen theorem. Ghost rod structures are found and analyzed in the lid-driven cavity and rectangular channel flows with solid side walls. The results suggest that the no-slip boundary condition on the stationary internal surfaces is one of the reasons for poor mixing in steady laminar three-dimensional flows considered previously with solid braided internal rods.
- Vortex Synchronization Regions in Shedding From an Oscillating CylinderPonta, F. L.; Aref, Hassan (AIP Publishing, 2005-01-01)A classification of the vortex patterns in the wake of a normally oscillating cylinder was given by Williamson and Roshko who also studied when the various vortex patterns would occur. Using a symbolic code of letters and numbers that describes the combination of pairs and single vortices shed during each cycle of the forced oscillation of the cylinder, they gave a "map" of vortex-synchronization regions with the wavelength and amplitude of the oscillation as coordinates. In this study, we provide a theoretical basis for the experimental Williamson-Roshko map of vortex synchronization. We show that the region boundaries consist of two families of curves given by specific relations between the values of two nondimensional parameters defining the map. We construct a synthetic map that compares favorably with the experimental data. (C) 2005 American Institute of Physics.
- Vortex triple ringsAref, Hassan; van Buren, M. (American Institute of Physics, 2005-05-01)We determine equilibrium configurations of interacting point vortices in which identical vortices are arranged on concentric circles and/or form a set of nested regular polygons with or without a vortex at the center. A new analytical method is developed that uses moments of the vortex positions and yields particularly simple results for equilibria of this kind. A complete determination of all triple-ring equilibria is given and numerous previously unknown configurations are identified. Several equilibria previously reported in the literature, found by numerical solution, are characterized analytically.