Mathematics Helps Identify Building Blocks of Turbulence

April 4, 2007

A paper about to be published in an upcoming "Physical Review Letters" makes a bold claim: to have clarified the convoluted tangle underlying turbulence. The mathematical underpinnings of this work, by MIT researchers and colleagues, may lead to more efficient engines, planes, cars, trains, submarines, and so on.

Using involved mathematical tools, researchers uncovered the convoluted tangle embedded in a flow. They isolated what they described as the "very source of turbulent mixing," not just its effect on dye or smoke as earlier studies have.

Researchers have long suspected that there's a hidden, coherent structure underlying turbulene's complexity, but there has been no objective way of identifying it until now, said MIT research group leader George Haller, professor of mechanical engineering, who also heads Morgan Stanley's Mathematical Modeling Center, in Hungary.

"The fluid mechanics community has not reached a consensus even on an objective definition of a vortex, or whirlpool effect, let alone the definition of structures forming turbulence. The mathematical techniques we have developed give a systematic way to identify the material building blocks of a turbulent flow," Haller said.

To picture the skeleton of turbulence, the MIT researchers analyzed experimental data obtained by researchers from the University of Texas, at Austin. The Texas group used water jets to force water from below into a rotating tank of fluid. They seeded the resulting, complicated flow with luminescent, buoyant particles. When illuminated by a laser, the miniscule polystyrene spheres were visible as they raced around the vortices and jets.

"Most important to our analysis were the particles' velocities, which our collaborators obtained by recording the particles' motion with a high-resolution camera, then using a software tool to figure out which particle moved where in a split second," Haller said. "This gave us a high-quality map of the whole velocity field of the turbulent flow at each time instance."

The science team discovered that complicated, constantly evolving flow patterns were driven by two competing armies of particles constantly being pulled together and pushed apart.

The MIT researchers called their discovery the "Lagrangian skeleton" of turbulence because their particle-based approach is motivated by the work of 19th-century mathematician Joseph-Louis Lagrange.

Lagrangian mathematical tools are still used today for calculating mechanical and fluid motion.

Source: MIT News (http://web.mit.edu/neewsoffice/2007/turbulence.html