Welcome back. I hope my readers are having a great third week of the semester. In this installment of “Introduction to Flight,” I am very excited to discuss finite-time Lyapunov exponents (FTLEs), which are powerful tools in the fluids research community. When we think of turbulence, we might imagine an infinitely jumbled mess of fluid moving erratically throughout space. This image correctly suggests that turbulence is one of the most difficult and messy topics in physics. However, there can also be fascinating patterns that develop and sustain themselves in messy flows. Many of these patterns are called Lagrangian Coherent Structures (LCSs), and they are important to researchers because they provide some order to turbulence and insight into how matter is transported in flows. One fundamental tool to identify coherent structures is an FTLE field, which is a collection of many finite-time Lyapunov exponents, each of which corresponds to one point in the given flow field. In short, the FTLE is a pointwise number that quantifies the stretching and squishing of particle trajectories that pass near a given point. However, the goal of this blog is to go deeper, and derive the precise mathematical formula for the FTLE to provide insight into what it specifically represents.
To begin, let’s suppose we have a 3D incompressible fluid flow. We can safely assume that the continuum hypothesis holds, and thus, we can write down the Navier-Stokes equations that govern our fluid’s motion:
∇ • u = 0
ρ ∂u/∂t + ρ (u • ∇) u = – ∇p + µ ∆u + ρ g.
This system of coupled nonlinear second-order partial differential equations admits a velocity field u(x,t), where x represents a point in 3D space. This velocity field also represents a 3D nonlinear dynamical system governed by the equation
dx/dt = u(x,t).
This concept will play a key role later. For now, suppose we want to analyze the flow by looking at the trajectories of many small particles that move along the velocity field from an array of initial positions. This analytical perspective is called the Lagrangian point of view, and we can begin our analysis by looking at what happens to just one particle. So let us consider a particle that begins at position x0 at time t = 0. To find the position of this particle at time t > 0, we can integrate along the velocity field using the equation
x(t) = x0 + ∫[0, t] u(x(t), t) dt.
We give this integration process a name, and call it the flow map. Specifically, the flow map Φt takes a particle at position x0 at time t = 0 to its position x(t) at time t > 0. So,
x(t) = Φt(x0).
Now suppose that we perturbed our particle away from x0 by a small displacement ξ0. Since the particle’s initial position is x0 + ξ0, we can denote the perturbed particle’s final position by x(t) + ξ(t). Thus, ξ(t) is a vector that measures the difference between the unperturbed particle’s position at time t and the perturbed particle’s position at time t. Let’s now perform a Taylor series expansion of the flow map around x0 to obtain
x(t) + ξ(t) = Φt(x0 + ξ0) = Φt(x0) + DΦt(x0) ξ0 + O(|ξ0|2),
where DΦt(x0) is the Jacobian matrix of Φt evaluated at x0 that acts on the initial perturbation vector ξ0. We can assume from this moment forward that ξ0 is small enough to ignore the higher-order terms, represented by the symbol O(|ξ0|2). Since x(t) = Φt(x0), we can simplify the expression above to
ξ(t) = DΦt(x0) ξ0.
Since ξ0 is a small disturbance, we also know that the evolution of ξ is primarily governed by the linearized dynamical system
dξ /dt = Du(x(t), t) ξ(t),
where the linear operator Du(x(t), t) is the Jacobian of the velocity field that acts on the perturbation ξ(t) to give dξ /dt. Because the perturbation is governed by the linearized dynamical equations, it is reasonable to assume that the magnitude of ξ(t) obeys an exponential relationship
|ξ(t)| = |ξ0| exp(b t),
b = limt → ∞ 1/t log( |ξ(t)| / |ξ0| ).
If b > 0, then particle trajectories will spread away from each other at x0, but if b < 0, particle trajectories will scrunch together. The FTLE will be our best approximation to “b,” and hence will directly capture the exponential divergence or convergence of fluid particle trajectories near the point x0. To compute our estimate of λ, let’s first notice that
|ξ(t)|2 = (ξ(t), ξ(t)) = (DΦt(x0) ξ0, DΦt(x0) ξ0)
= (ξ0, [DΦt(x0)]T DΦt(x0) ξ0),
where ( –,–) is the Euclidean inner product and [DΦt(x0)]T is the transpose of DΦt(x0). To ease notation, let’s define the Cauchy-Green strain tensor
Ct(x0) = [DΦt(x0)]T DΦt(x0).
|ξ(t)|2 = (ξ0, Ct(x0) ξ0).
For most flows, the Cauchy-Green strain tensor often has one eigenvalue λmax that is much larger than the other eigenvalues. In this setting, we can approximate
|ξ(t)|2 ≈ λmax |ξ0|2,
log(|ξ(t)| / |ξ0|) ≈ ½ log(λmax).
The true exponent b is computed by dividing log(|ξ(t)| / |ξ0|) by t and letting t approach infinity. Since there are no true infinities in experiments, we can do the next best thing by dividing our approximation to log(|ξ(t)| / |ξ0|) by the integration time T, which is the total time duration our data lasts for. Thus, we can define our finite-time Lyapunov exponent at x0 to be
σt(x0) = 1/(2T) log(λmax).
So there we have it! The FTLE value for one point in our flow, x0. To identify coherent patterns, we can then compute σt(x0) for a whole bunch of different x0’s, then plot them all on a grid, and observe any ridges or valleys of very high FTLE values or very low FTLE values. This process is best explained using pictures and videos, so I will refer my readers to the excellent resources on YouTube for more information.
For now, I hope this blog was educational and provided a flavor of the rich mathematical analysis that we can perform on fluid flows. Next week, we will continue our conversation with another exciting topic in fluid dynamics or partial differential equations. Until then, take care.