Nonautonomous control of stable and unstable manifolds in two-dimensional flows

We outline a method for controlling the location of stable and unstable manifolds in the following sense. From a known location of the stable and unstable manifolds in a steady two-dimensional flow, the primary segments of the manifolds are to be moved to a user-specified time-varying location which is near the steady location. We determine the nonautonomous perturbation to the vector field required to achieve this control, and give a theoretical bound for the error in the manifolds resulting from applying this control. The efficacy of the control strategy is illustrated via a numerical example

Impulsive perturbations to differential equations: stable/unstable pseudo-manifolds, heteroclinic connections, and flux

State-dependent time-impulsive perturbations to a two-dimensional autonomous flow with stable and unstable manifolds are analysed by posing in terms of an integral equation which is valid in both forwards- and backwards-time. The impulses destroy the smooth invariant manifolds, necessitating new definitions for stable and unstable pseudo-manifolds. Their time-evolution is characterised by solving a Volterra integral equation of the second kind with discontinuous inhomogeniety. A criteria for heteroclinic trajectory persistence in this impulsive context is developed, as is a quantification of an instantaneous flux across broken heteroclinic manifolds. Several examples, including a kicked Duffing oscillator and an underwater explosion in the vicinity of an eddy, are used to illustrate the theory

Local stable and unstable manifolds and their control in nonautonomous finite-time flows

It is well-known that stable and unstable manifolds strongly influence fluid motion in unsteady flows. These emanate from hyperbolic trajectories, with the structures moving nonautonomously in time. The local directions of emanation at each instance in time is the focus of this article. Within a nearly autonomous setting, it is shown that these time-varying directions can be characterised through the accumulated effect of velocity shear. Connections to Oseledets spaces and projection operators in exponential dichotomies are established. Availability of data for both infinite and finite time-intervals is considered. With microfluidic flow control in mind, a methodology for manipulating these directions in any prescribed time-varying fashion by applying a local velocity shear is developed. The results are verified for both smoothly and discontinuously time-varying directions using finite-time Lyapunov exponent fields, and excellent agreement is obtained.Comment: Under consideration for publication in the Journal of Nonlinear Science

Controlling the unsteady analogue of saddle stagnation points

It is well known that saddle stagnation points are crucial flow organizers in steady (autonomous) flows due to their accompanying stable and unstable manifolds. These have been extensively investigated experimentally, numerically, and theoretically in situations related to macroand micromixers in order to either restrict or enhance mixing. Saddle points are also important players in the dynamics of mechanical oscillators, in which such points and their associated invariant manifolds form boundaries of basins of attraction corresponding to qualitatively different types of behavior. The entity analogous to a saddle point in an unsteady (nonautonomous) flow is a timevarying hyperbolic trajectory with accompanying stable and unstable manifolds which move in time. Within the context of nearly steady flows, the unsteady velocity perturbation required to ensure that such a hyperbolic (saddle) trajectory follows a specified trajectory in space is derived and shown to be equivalent to that which can be obtained via a heuristic approach. An expression for the error in the hyperbolic trajectory's motion is also derived. This provides a new tool for the control of both fluid transport and mechanical oscillators. The method is applied to two examples-a four-roll mill and a Duffing oscillator-and the performance of the control strategy is shown to be excellent in both instances. © 2013 Society for Industrial and Applied Mathematics.Sanjeeva Balasuriya and Kathrin Padberg-Gehl

Vortex detection and tracking in massively separated and turbulent flows

The vortex produced at the leading edge of the wing, known as the leading edge vortex (LEV), plays an important role in enhancing or destroying aerodynamic force, especially lift, upon its formation or shedding during the flapping flight of birds and insects. In this thesis, we integrate multiple new and traditional vortex identification approaches to visualize and track the LEV dynamics during its shedding process. The study is carried out using a 2D simulation of a flat plate undergoing a 45 degree pitch-up maneuver. The Eulerian 1 function and criterion are used along with the Lagrangian coherent structures (LCS) analyses including the finite-time Lyapunov exponent (FTLE), the geodesic LCS, and the Lagrangian-Averaged Vorticity Deviation (LAVD). Each of \h{these} Lagrangian methods \h{is} applied at the centers and boundaries of the vortices to detect the vortex dynamics. The techniques enable the tracking of identifiable features in the flow organization using the FTLE-saddles and -saddles. The FTLE-saddle traces have shown potential to identify the timing and location of vortex shedding, more precisely than by only studying the vortex cores as identified by Eulerian techniques. The traces and the shedding times of the FTLE-saddles on the LEV boundary matches well with the plate lift fluctuation, and indicates a consistent timing of LEV formation, growth, shedding. The formation number and vortex shedding mechanisms are compared in the thesis with the shedding time and location by the FTLE-saddle, which validates the result of the FTLE-saddles and provide explanations of vortex shedding in different aspects (vortex strength and flow dynamics). The techniques are applied to more cases involving vortex dominated flows to explore and expand their application in providing insight of flow physics. For a set of experimental two-component PIV data in the wake of a purely pitching trapezoidal panel, the Lagrangian analysis of FTLE-saddle tracking identifies and tracks the vortex breakdown location with relatively less user interaction and provide a more direct and consistent analysis. For a simulation of wall-bounded turbulence in a channel flow, tracking FTLE-saddles shows that the average structure convection speed exhibits a similar trend as a previously published result based on velocity and pressure correlations, giving validity to the method. When these Lagrangian techniques are applied in a study of the evolution of an isolated hairpin vortex, it shows the connection between primary and secondary hairpin heads of their circulation and position, and the contribution to the generation of the secondary hairpin by the flow characteristics at the channel wall. The current method of tracking vortices yields insight into the behavior of the vortices in all of the diverse flows presented, highlighting the breadth of its potential application

The deep structure of Gaussian scale space images

In order to be able to deal with the discrete nature of images in a continuous way, one can use results of the mathematical field of 'distribution theory'. Under almost trivial assumptions, like 'we know nothing', one ends up with convolving the image with a Gaussian filter. In this manner scale is introduced by means of the filter's width. The ensemble of the image and its convolved versions at al scales is called a 'Gaussian scale space image'. The filter's main property is that the scale derivative equals the Laplacean of the spatial variables: it is the Greens function of the so-called Heat, or Diffusion, Equation. The investigation of the image all scales simultaneously is called 'deep structure'. In this thesis I focus on the behaviour of the elementary topological items 'spatial critical points' and 'iso-intensity manifolds'. The spatial critical points are traced over scale. Generically they are annihilated and sometimes created pair wise, involving extrema and saddles. The locations of these so-called 'catastrophe events' are calculated with sub-pixel precision. Regarded in the scale space image, these spatial critical points form one-dimensional manifolds, the so-called critical curves. A second type of critical points is formed by the scale space saddles. They are the only possible critical points in the scale space image. At these points the iso-intensity manifolds exhibit special behaviour: they consist of two touching parts, of which one intersects an extremum that is part of the critical curve containing the scale space saddle. This part of the manifold uniquely assigns an area in scale space to this extremum. The remaining part uniquely assigns it to 'other structure'. Since this can be repeated, automatically an algorithm is obtained that reveals the 'hidden' structure present in the scale space image. This topological structure can be hierarchically presented as a binary tree, enabling one to (de-)select parts of it, sweeping out parts, simplify, etc. This structure can easily be projected to the initial image resulting in an uncommitted 'pre-segmentation': a segmentation of the image based on the topological properties without any user-defined parameters or whatsoever. Investigation of non-generic catastrophes shows that symmetries can easily be dealt with. Furthermore, the appearance of creations is shown to be nothing but (instable) protuberances at critical curves. There is also biological inspiration for using a Gaussian scale space, since the visual system seems to use Gaussian-like filters: we are able of seeing and interpreting multi-scale

Lagrangian Visualization and Real-Time Identification of the Vortex Shedding Time in the Wake of a Circular Cylinder

The flow around a circular cylinder, a canonical bluff body, has been extensively studied in the literature to determine the mechanisms that cause the formation of vortices in the cylinder wake. Understanding of these mechanisms has led to myriad attempts to control the vortices either to mitigate the oscillating forces they cause, or to augment them in order to enhance mixing in the near-wake. While these flow control techniques have been effective at low Reynolds numbers, they generally lose effectiveness or require excessive power at Reynolds numbers commonly experienced in practical applications. For this reason, new methods for identifying the locations of vortices and their shedding time could increase the effectiveness of the control techniques. In the current work, two-dimensional, two-component velocity data was collected in the wake of a circular cylinder using a planar digital particle image velocimetry (DPIV) measurement system at Reynolds numbers of 9,000 and 19,000. This experimental data, as well as two-dimensional simulation data at a Reynolds number of 150, and three-dimensional simulation data at a Reynolds number of 400, is used to calculate the finite-time Lyapunov exponent (FTLE) field. The locations of Lagrangian saddles, identified as non-parallel intersections of positive and negative time FTLE ridges, are shown to indicate the timing of von Kármán vortex shedding in the wake of a circular cylinder. The Lagrangian saddle found upstream of a forming and subsequently shedding vortex is shown to clearly accelerate away from the cylinder surface as the vortex begins to shed. This provides a novel, objective method to determine the timing of vortex shedding. The saddles are impossible to track in real-time, however, since future flow field data is needed for the computation of the FTLE fields. In order to detect the Lagrangian saddle acceleration without direct access to the FTLE, the saddle dynamics are connected to measurable surface quantities on a circular cylinder in crossflow. The acceleration of the Lagrangian saddle occurs simultaneously with a maximum in lift in both numerical cases, and with a minimum in the static pressure at a location slightly upstream of the mean separation location in the numerical cases, as well as the experimental data at a Reynolds number of 19,000. This allows the von Kármán vortex shedding time, determined objectively by the acceleration of the Lagrangian saddle away from the circular cylinder, to be detected by a minimum in the static pressure at one location on the cylinder, a quantity that can be measured in real-time using available pressure sensors. These results can be used to place sensors in optimal locations on bluff bodies to inform closed-loop flow control algorithms of the timing of von Kármán vortex shedding

Meridional and zonal wavenumber dependence in tracer flux in Rossby waves

Eddy-driven jets are of importance in the ocean and atmosphere, and to a first approximation are governed by Rossby wave dynamics. This study addresses the time-dependent flux of fluid and a passive tracer between such a jet and an adjacent eddy, with specific regard to determining zonal and meridional wavenumber dependence. The flux amplitude in wavenumber space is obtained, which is easily computable for a given jet geometry, speed and latitude, and which provides instant information on the wavenumbers of the Rossby waves which maximize the flux. This new tool enables the quick determination of which modes are most influential in imparting fluid exchange, which in the long term will homogenize the tracer concentration between the eddy and the jet. The results are validated by computing backward- and forward-time finite-time Lyapunov exponent fields, and also stable and unstable manifolds; the intermingling of these entities defines the region of chaotic transport between the eddy and the jet. The relationship of all of these to the time-varying transport flux between the eddy and the jet is carefully elucidated. The flux quantification presented here works for general time-dependence, whether or not lobes (intersection regions between stable and unstable manifolds) are present in the mixing region, and is therefore also easily computable for wave packets consisting of infinitely many wavenumbers.Sanjeeva Balasuriy