Computation of Lagrangian Fields from Sparse Tracers using Locally-Linear Regression in Time

Identifying Lagrangian Coherent Structures (LCS) has many advantages in the study of complex flows. Unlike Eulerian methods such as vorticity, Q-criterion, and the Okubo-Weiss parameter, LCS provide objective measures of flow dynamics over a finite time domain [1]. Unfortunately, computing typical LCS measures like the finitetime Lyapunov exponent (FTLE) and Lagrangianaveraged vorticity deviation (LAVD) from experimental data is difficult because it requires spatially dense trajectory information that is not possible to directly observe. Researchers have addressed this with various sparse techniques, but these require tuning and often lack the interpretability that traditional methods boast [2]. This work bridges the sparse and dense LCS theories by exploiting the composition property of the flow map and computing its Jacobian from a sequence of snapshots. This enables accurate LCS measurements directly from the tracer data. As these methods are refined, they will be used for real-time study and control of systems engaging with complex, unsteady flows with application to geophysical, biological, and aerodynamic problems.

Figure: On the double gyre flow using 1000 randomly placed particles (gray dots) on the time domain 𝑡𝑡 ∈ [0,15] using 150 intermediate snapshots. (a) FTLE field. (b) LAVD field. (c) Average rotation of a particle in complete revolutions. (d) Vorticity at 𝑡𝑡 = 0 computed directly from the tracers.

References

1. Haller, George. "Lagrangian coherent structures." Annual review of fluid mechanics 47 (2015): 137- 162.
2. Allshouse, Michael R., and Thomas Peacock. "Lagrangian based methods for coherent structure detection." Chaos: An Interdisciplinary Journal of Nonlinear Science 25.9 (2015): 097617.

Funding: The authors would like to thank the ARO and the ONR for their support of this work. ARO: W911NF-17-1-0306 ONR: N00014-17-1-3022