Lagrangian Coherent Structures for Flows in 2D/3D Euclidean Spaces
This tutorial explains how to use the MATLAB code for computing finite-time Lyapunov exponent (FTLE) fields for flows defined on 2D and 3D Euclidean grids. The MATLAB code is available in the GitHub repository: flow_coherent_structure. The code computes Lagrangian coherent structures by approximating the deformation gradient using finite differences. The FTLE field is then obtained from the singular values of the deformation gradient. The numerical method follows the approach described in [1].
Prerequisites
The code was developed using MATLAB R2025b on a Windows 10 system. It has also been tested on:
- macOS 15
- Ubuntu 20
The recommended installation method is through Git, since the code is hosted on GitHub. This tutorial assumes that Git is already installed and configured on your system. If Git is not installed, follow the installation instructions here: Installing Git
Installation
Navigate to the directory where you want to install the code, then clone the GitHub repository:
git clone https://github.com/SreejithSanthosh/flow_coherent_structure.git
This command creates a directory called:
flow_coherent_structure
which contains the MATLAB scripts, functions, and example files. To test the Euclidean-grid FTLE code, navigate to the folder:
ftle_from_euclidean_grid
and set it as the root directory in MATLAB.The code is under active development. To update your local copy with the latest changes, navigate to the repository directory and run:
git pull
Example: 2D Double-Gyre Flow
To test the 2D FTLE computation, run the following MATLAB script:
example_euclideangrid2d.m
This script computes the deformation field and FTLE field for the double-gyre velocity field. The output is shown below.
Here:
- $\Lambda$ is the FTLE field.
- $\xi$ is the axis of maximum deformation.
Example: 3D ABC Flow
To test the 3D FTLE computation, run:
example_euclideangrid3d.m
This script computes the deformation field and FTLE field for the ABC flow. The output is shown below.
As in the 2D case, the code computes the FTLE field from the deformation of an initially defined grid over a prescribed time interval.
Performing Deformation Analysis on Your Own Data
The example scripts are written for specific test flows:
example_euclideangrid2d.muses the 2D double-gyre flow.example_euclideangrid3d.muses the 3D ABC flow.
To apply the analysis to your own velocity field, you must first advect an initial grid from the initial time $t_0$ to the final time $t_f$.
2D and 3D Flow Data Format and Analysis
The two example scripts for 2D and 3D flows, example_euclidean2D.m and example_euclidean3D.m, are written for a specific flow field. To apply this analysis to your own flow field, you must advect an initial grid from time \(t_0\) to the final time \(t_f\). For a 2D flow, the initial grid coordinates x0, y0 are advected to the final coordinates xf, yf. The function compute_deform_euclidean2d.m takes x0, y0, xf, yf as inputs and computes the singular values and their associated directions. The resulting FTLE field can then be visualized using the code provided in the %% Visualize the result section of example_euclidean2D.m. Similarly, for a 3D flow, the initial grid coordinates x0, y0, z0 are advected to xf, yf, zf. These initial and final grids are passed to compute_deform_euclidean3d.m, which computes the deformation quantities needed to visualize the FTLE field.
Summary of Workflow
The general workflow for computing FTLE fields on Euclidean grids is:
- Define an initial grid at time $t_0$.
- Advect the grid through the velocity field from $t_0$ to $t_f$.
- Store the final grid positions at time $t_f$.
- Use the appropriate deformation-analysis function:
compute_deform_euclidean2d.mfor 2D flows.compute_deform_euclidean3d.mfor 3D flows.
- Compute the singular values of the deformation gradient.
- Visualize the FTLE field and the corresponding deformation axes.
References
[1] Haller, G. (2015). Lagrangian coherent structures. Annual Review of Fluid Mechanics, 47(1).