Welcome back. In this blog, I was hoping to take a short break from hydrodynamic stability, and share an old essay I have written about computing finite-time Lyapunov exponent (FTLE) fields to analyze Lagrangian coherent structures (LCSs):
My project was to develop numerical integration code that could compute particle trajectories in unsteady flows. While previous studies made use of built-in MATLAB functions, I was looking for a more efficient program that could simultaneously integrate x- and y-coordinates of thousands of particles at a time. To address this challenge, I adapted the 4th-order Runge-Kutta scheme from standard MATLAB code into a custom script that performed column-wise and row-wise operations across matrices, instead of just element-wise operations. The result of this effort was a program that was more accurate and faster than default MATLAB solutions.
By computing these trajectories, we could construct a flow map, 𝛷t0t, that mapped a fluid particle’s initial position x0 at time t0 to its position x(t) at an arbitrary time t. To understand how this flow map worked, it was helpful to view a particle’s trajectory x(t) as a solution to the differential equation
dx/dt = u(x(t), t)
where u was the velocity vector at position x and time t. The initial condition for x(t) was then the initial position x0. Hence, the flow map could be naturally defined as
𝛷t0t(x0) = x0 + ∫t0t u(x(t), t) dt
For any integration time t, the outputs of my MATLAB code were the outputs of the flow map evaluated at the initial coordinates, x0, of the particles I integrated.
I could then compute the Jacobian of the flow map to find the deformation rates of particle trajectories around an airfoil. The maximum deformation at a point x0 was proportional to the largest singular value of the Jacobian of the flow map, 𝛻𝛷t0t(x0). This leading singular value gave the maximum amount of stretching a circular curve of fluid particles would undergo under the action of 𝛻𝛷t0t(x0).
With these singular values, I could plot a finite-time Lyapunov exponent (FTLE) field, which highlighted regions of particle attraction and repulsion throughout our flow. Ridges of high FTLE values marked repelling Lagrangian coherent structures (LCSs) that formed the backbones of material transport in the flow. Since we had flow separating from the upper surface of a wing, there was a distinct LCS that separated the fluid particles that were ejected up and away from the wing and the fluid particles that were trapped in the recirculation zone behind the region of flow separation. By changing the angles and locations of these Lagrangian coherent structures through flow control methods, there was a possibility to improve wing performance.
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