## Context

This research project is to further develop the analysis tools used to study flow simulations and non-invasive fluid flow measurements based on PIV (*Particle Image Velocimetry*). For example, the lack of a universally accepted mathematical definition of a vortex structure has led to a considerable number of Eulerian criteria to identify coherent structures. In fluid mechanics, an alternative is the Lyapunov exponent computed over a finite-time horizon (the so-called FTLE). However, the current computational FTLE methodology requires the numerical evaluation of a great number of trajectories over a Cartesian grid laid upon the measured velocity fields. The number of nodes used to define an accurate FTLE field in 3D transient time-resolved flow field measurement easily reaches more than a million, thus requiring significant computing resources for proper analysis.

The development of the FTLE (and real life applications) is well detailed in Shawn C. Shadden, tutorial website. The following velocity field comes from HF-Radar measurements of the Monterey Bay in California during the month of August 2003. (click the image to see the animation!)

The structures of the flow are hard to extract and visualize from a Eulerian perspective. By calculating the FTLE, shown on red in the following animations, we can easily distinguish coherent structures of the flow field. Theses structures can be compared to material boundaries and it is shown by the next animation. By adding black and white particles in the Bay, we can visualize that particles on one side of a FTLE will never cross the structure, since it acts has a material boundary with no mass flux trough it. This technique could be use to estimate the dispersion of pollution or debris lost at sea.

## Thesis objectives

The main objective of this research is the development of a standalone tools for the analysis of flow simulations and experimental data, making the identification of vortical coherent structures in the flow possible. With these tools, the analysis of complex phenomena in otherwise simple geometries, like axisymmetric expansions or impinging jet, could help reveal the inner working physical mechanisms. The study of coherent structures in these cases will lead to a better understanding of the physics in transitional and turbulent flows.

## Methodology

The development of numerical algorithms needs a lot of testing, validation and finishing touches. My research combine different numerical techniques used in domains such has mathematics, engineering, signal analysis and experimental physics into one standalone software. These combination made the originality of the research and provide a new interesting way of calculating the FTLE field. Using data generated from analytical solution allow the validation of the algorithm before introducing noise normally present in the experimental data. Furthermore, once the method was validated using two dimensional data, I extended the principles to support three dimension flow data. With new experimental measurements techniques able to estimate the velocity field in a fixed volume, the three dimensional extension was a necessity for future work at the Laboratory of Fluid Dynamics (LADYF) at Polytechnique Montreal.

## Results and publications

The result of my thesis provide a tool that can be use by students and teacher working at LADYF. It generates visualization by extracting structures in chaotic turbulent flows that can help to understand the flow physic and create better design and more efficient geometries. This type of analysis are crucial in the air, land and marine transport, in biomedical, in the energy production and many other domains.

My research has led to two scientific articles in the Journal of Computational Physics and Experiments in Fluid. The first publication introduce the originality of the method I have implemented and present the advantages over other more traditional method. The second publication present visualization of a flow through a mechanical heart valve. The detail and information that it is made possible to extract from turbulent flows with the developed method is not comparable with any other tools. I also presented at the 43rd AIAA Fluid Dynamics Conference this new visualization tool.

## References

Philippe Miron, Jérôme Vétel, André Garon, Michel Delfour and Mouhammad El Hassan

Anisotropic mesh adaptation on Lagrangian Coherent Structures, Journal of Computational Physics, Volume 231, Issue 19, August 2012, Pages 6419-6437, ISSN 0021-9991)

Keywords: Finite time Lyapunov exponent; Lagrangian Coherent Structure; Continuous metric; Anisotropic mesh adaptation; Error estimation; Particle image velocimetry

Philippe Miron, Jérôme Vétel, and André Garon

Efficient computation of the finite-time Lyapunov exponent

21st AIAA Computational Fluid Dynamics Conference. June 2013, San Diego, USA

Philippe Miron, Jérôme Vétel, and André Garon

On the use of the finite-time Lyapunov exponent to reveal complex flow physics in the wake of a mechanical valve

Experiments in Fluids, EID 1814, Volume 55, Number 9, September 2014, Pages 6419-6437, ISSN 0723-4864)

André Garon, André Fortin, Thomas Briffard, Jérôme Vétel and Philippe Miron

Anisotropic mesh adaptation for the computation of Lagrangian coherent structures

Conference, Coupled Problems 2015, Venice, Italia.