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

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

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

A tangential displacement theory for locating perturbed saddles and their manifolds

The stable and unstable manifolds associated with a saddle point in two-dimensional non–area-preserving flows under general time-aperiodic perturbations are examined. An improvement to existing geometric Melnikov theory on the normal displacement of these manifolds is presented. A new theory on the previously neglected tangential displacement is developed. Together, these enable locating the perturbed invariant manifolds to leading order. An easily usable Laplace transform expression for the location of the perturbed time-dependent saddle is also obtained. The theory is illustrated with an application to the Duffing equation.Sanjeeva Balasuriy

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

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