Lagrangian Coherent Structures in 2D Turbulence

In two-dimensional turbulence, coherent structures tend to emerge. While the existence of these structures is
clear from visual observations, their mathematical description is far more difficult. Several instantaneous physical quantities have been proposed and used to describe coherent structure, all yielding different answers. For example, coherent structures are sometimes defined as regions of high vorticity, such as those highlighted on the left for a two-dimensional turbulence simulation.

In this project, we have sought a frame-independent description of coherent structures, so that we can unambiguously identify them in numerical simulations and laboratory experiments. The applications of such an identification scheme range from large-scale geophysical data analysis to the design of mixers.
 

Any Lagrangian (or material) description of coherent structures is inherently frame-independent, which prompted us to develop a Lagrangian coherent structure (LCS) theory. We have obtained analytic criteria that reveal complex material structures that would normally remain hidden in instantaneous velocity or pressure plots. Shown in the figure below, these LCS are responsible for chaotic mixing in turbulent flows.

Specifically, fluid is attracted to, then stretched by, a convoluted web of LCS. In the language of nonlinear dynamical systems, these LCS are finite-time unstable manifolds: While they attract fluid, their internal dynamics are unstable, leading to exponential stretching over the finite life-span of the LCS web.

 

 

 

 

The techniques we have developed also work remarkably well on experimental data sets. In joint work with with G. Voth and J. Gollub (Haverford College), we have been able to identify the LCS causing chaotic mixing in a forced rectangular tank. These structures have long been predicted theoretically, but no direct experimental verification has been available. The image below shows details of attracting and repelling material lines we found experimentally.

 

Finally, this image on the left shows repelling LCS extracted from a flow experiment past a backward facing step. The step is located at x=0, ranging from y=0 to y=0.9. The green LCS is a complex nonlinear structure extracted from the post-processed experimental data. Invisible to the naked eye, this LCS contains fluid particles that converge to the x=1 point of the horizontal wall behind the step. This LCS is the Lagrangian signature of reattachment behind the step. The experiment was conducted by J. Cohen (United Technologies Research Center).

 

 

Our current research on Lagrangian coherent structures includes:

(1) Extraction of LCS from measured geophysical flow data

(2) Effect of LCS on the alignment of passive scalar gradients in turbulence

(3) Frame-independent description of LCS in three-dimensional flows

 

References:

1. G. Haller & R. Iacono, Stretching, alignment, and shear in slowly varying velocity fields, Phys. Rev. E 68 (2003) 056304. [PDF]
2. G. A. Voth, G. Haller, & J.P. Gollub, Experimental measurements of stretching fields in fluid mixing, Phys. Rev. Lett., 88 (2002) 254501. [PDF]
3. G. Haller, Lagrangian coherent structures from approximate velocity data, Phys. Fluids A14 (2002) 1851-1861 [PDF]
4. G. Haller, Lagrangian coherent structures and the rate of strain in two-dimensional turbulence, Phys. Fluids A 13 (2001) 3365-3385. [PDF]
5. G. Haller & G. Yuan, Lagrangian coherent structures and mixing in two-dimensional turbulence, Physica D 147 (2000) 352-370. [PDF]
6. G. Haller, Finding finite-time invariant manifolds in two-dimensional velocity fields, Chaos 10 (2000) 99-108. [PDF]
7. A. Poje, G. Haller, & I. Mezic, The geometry and statistics of mixing in aperiodic flows, Phys. Fluids A 11 (1999) 2963-2968. [PDF]
8. A. Poje & G. Haller, Geometry of cross-stream mixing in a double-gyre ocean model, J. Phys. Oceanogr. 29 (1999) 1649-1665. [PDF]
9. G. Haller & A. Poje, Finite time mixing in aperiodic flows, Physica D 119 (1998) 352-380. [PDF]