We review and test twelve different approaches to the detection of finite-time coherent material structures in two-dimensional, temporally aperiodic flows. We consider both mathematical methods and diagnostic scalar fields, comparing their performance on three benchmark examples: the quasiperiodically forced Bickley jet, a two-dimensional turbulence simulation, and an observational wind velocity field from Jupiter's atmosphere. A close inspection of the results reveals that the various methods often produce very different predictions for coherent structures, once they are evaluated beyond heuristic visual assessment. As we find by passive advection of the coherent set candidates, false positives and negatives can be produced even by some of the mathematically justified methods due to the ineffectiveness of their underlying coherence principles in certain flow configurations. We summarize the inferred strengths and weaknesses of each method, and make general recommendations for minimal self-consistency requirements that any Lagrangian coherence detection technique should satisfy.

A critical comparison of Lagrangian methods for coherent structure detection

Particle image velocimetry (PIV) is used to investigate the three-dimensional wakes of rigid pitching panels with a trapezoidal geometry, chosen to model idealized fish caudal fins. Experiments are performed for Strouhal numbers from 0.17 to 0.56 for two different trailing edge pitching amplitudes. A Lagrangian coherent structure (LCS) analysis is employed to investigate the formation and evolution of the panel wake. A classic reverse von Karman vortex street pattern is observed along the mid-span of the near wake, but the vortices realign and exhibit strong interactions near the spanwise edges of the wake. At higher Strouhal numbers, the complexity of the wake increases downstream of the trailing edge as the spanwise vortices spread transversely and lose coherence as the wake splits. This wake transition is shown to correspond to a qualitative change in the LCS pattern surrounding each vortex core, and can be identified as a quantitative event that is not dependent on arbitrary threshold levels. The location of this transition is observed to depend on both the pitching amplitude and free stream velocity, but is not constant for a fixed Strouhal number. On the panel surface, the trapezoidal planform geometry is observed to create additional vortices along the swept edges that retain coherence for low Strouhal numbers or high sweep angles. These additional swept-edge structures are conjectured to add to the complex three-dimensional flow near the tips of the panel.

The unsteady three-dimensional wake produced by a trapezoidal pitching panel

This paper presents new efficient methods for computing finite-time Lyapunov exponent (FTLE) fields in unsteady flows. The methods approximate the particle flow map, eliminating redundant particle integrations in neighboring flow map calculations. Two classes of flow map approximations are investigated based on composition of intermediate flow maps; unidirectional approximation constructs a time-T map by composing a number of smaller time-h maps, while bidirectional approximation constructs a flow map by composing both positive- and negative-time maps. The unidirectional method is shown to be fast and accurate, although it is memory intensive. The bidirectional method is also fast and uses significantly less memory; however, it is prone to error which is large in regions where the opposite-time FTLE field is large, rendering it unusable. The algorithms are implemented and compared on three example fluid flows: a double gyre, a low Reynolds number pitching flat plate, and an unsteady ABC flow.

Fast computation of finite-time Lyapunov exponent fields for unsteady flows.

Review of Vortex Identification Methods

Several previous experimental and theoretical studies have shown that a leading edge vortex (LEV) on an airfoil or wing can provide lift enhancement. In this paper, unsteady 2D potential flow theory is employed to model the flow field of a flapping flat plate wing. A multi-vortices model is developed to model both the leading edge and trailing edge vortices (TEVs), which offers improved accuracy compared with using only single vortex at each separation location. The lift is obtained by integrating the unsteady Blasius equation. It is found that the motion of vortices contributes significantly to the overall aerodynamic force on the flat plate. The shedding of TEVs and the stabilization of LEVs explicitly contributes to lift enhancement. A Kutta-like condition is used to determine the vortex intensity and location at the leading edge for large angle of attack cases; however, it is proposed to relax this condition for small angle of attack cases and apply a 2D shear layer model to calculate the circulation of the new added vortex. The results of the simulation are then compared with classical numerical, modeled and experimental data for canonical unsteady flat plat problems. Good agreement with these data is observed. Moreover, these results suggested that the leading edge vortex shedding for small angles of attack should be modeled differently than that for large angles of attack. Finally, the results of vortex motion vs. lift indicate that both a motion against the streamwise direction of the LEV and a streamwise motion of the TEV contributes positive lift. This also provides the insights for future active flow control of MAVs that the formation and shedding process of LEVs and TEVs can be manipulated to provide lift enhancement.

Lift Evaluation of a 2D Flapping Flat Plate

Using invariant material manifolds and flow topology, the flow behavior and structure of flow around a two-dimensional Eppler 387 airfoil is examined with an emphasis on vortex shedding and the time-dependent reattachment profile. The examination focuses on low Reynolds number (Re = 60 000) flow at several angles of attack. Using specialized software, we identify invariant manifolds in the flow and use these structures to illuminate the process of vortex formation and the periodic behavior of the reattachment profile. Our analysis concludes with a topological view of the flow, including fixed points and a discussion of phase plots and the frequency spectrum of several key points in the flow. The behavior of invariant manifolds directly relates to the flow topology and illuminates some aspects seen in phase space during vortex shedding. Furthermore, it highlights the reattachment behavior in ways not seen before.

A Lagrangian analysis of a two-dimensional airfoil with vortex shedding

AEROSPACE LETTERS are brief communications (approximately 2000 words) that describe new and potentially important ideas or results, including critical analytical or experimental observations that justify rapid publication. They are stringently prescreened, and only a few are selected for rapid review by an Editor. They are published as soon as possible electronically and then appear in the print version of the journal.

Vortex Shedding over a Two-Dimensional Airfoil: Where the Particles Come From

Utilizing invariant material manifolds, the formation and evolution of the shed vortex on a Two-Dimensional airfoil is examined. The particles which make up this shed vortex structure are identied and their progress is tracked back to locations upstream of the airfoil leading edge. It is discovered that there exist a number of separate, well dened regions of distinct particle mixing throughout the formation of the vortices. It is also found that a large percentage of the material which makes up the vortex advects along the underside of the airfoil. The result is an intriguing view of vortex formation which may have applications for ow control.

Vortex Shedding over Two-Dimensional Airfoil: Where do the particles come from?

Utilizing ideas from dynamical system theory to find invariant material manifolds, the structure of mixing behavior in vortex formation and shedding in two dimensional low Reynolds number flow is investigated. The test case is a low Reynolds number flow over an Eppler 387 airfoil at 6 angle of attack. A software was developed, tested and validated for calculating material manifolds on unstructured grid. It is shown that the stable manifolds are an excellent method of visualizing the reattachment profile, and that the manifold associated with reattachment behaves in a periodic shedding type manner. Structural elements of the flow, not found in traditional flow maps, are discovered and found to have important interactions in the vortex roll up process. Structures are formed ahead of the airfoil and transported above and below the airfoil before they interact at the trailing edge of the airfoil to form the wake vortices. It is expected that these structures could play a role in flow control of the shedding process. Finally, it is shown that the combination of stable and unstable manifolds allow for a time dependent definition of the separation bubble.

A Lagrangian View of Vortex Shedding and Reattachment Behavior in the Wake of a 2D Airfoil

Lagrangian Coherent Structures in the Wake of an Airfoil

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