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# Understanding the equations key to taming turbulence

"The weather is very large-scale turbulence. That's why it's so hard to predict," says Professor Fabian Waleffe.

Almost every flow we know is turbulent, he says, citing not only the weather, but other common examples such as blood flow, boiling water, air rushing around a moving vehicle, and oil traveling through a pipeline.

But while turbulence is ubiquitous, it is not well understood, despite studies that began as early as the 1500s, with artist Leonardo da Vinci's observations of and drawings of water flows.

Then in the 1880s, scientists Claude-Louis Navier and George Stokes derived the equations that govern fluid flow and describe it well, but which are tremendously complicated to solve, even with the help of a supercomputer. "We're not able to derive from the equations when turbulence occurs, or what is turbulence, or how to describe it," says Waleffe.

As a result, engineers must resort to ad hoc empirical formulas of limited validity and scientists must study turbulence at its most basic, or slowest-moving, level. "Even if you just walk around, we cannot fully calculate the air flow around you," he says.

He is using the Navier-Stokes equations to calculate solutions that describe the turbulent structures scientists have long observed in channel flows and boundary layers.

Although the very idea of turbulence suggests randomness and disorder, Waleffe says scientists' experiments half a century ago pointed to an underlying order. "Most natural turbulent flow shows eddies — like you see when there's a lot of wind and you see leaves spiraling around each other," he says. "The flow's clearly not random because it has these organized motions — such as the eddies — which we call coherent structures."

And Waleffe's studies reveal that the equations have underlying coherent solutions that capture the average features of turbulence. In other words, there's an order to the disorder. That's why, he says, scientists can observe structures embedded within the turbulence — like wavy streaks and horseshoe vortices — in one area of a flow and then watch them disappear and return elsewhere.

Given some 40 years' worth of supporting experimental evidence, pinpointing these coherent structures in the equations is a big step forward, he says, because experimentalists argued about what they were seeing. "It turned out to be all different pieces of the same structure," says Waleffe. "Once you put it into mathematical terms and once you can actually compute the solution, everything sort of falls into place."

Now that a mathematical description exists, many scientists are rethinking their notions of this order and how it affects flow. "It's really these underlying coherent structures that increase the transport — of momentum and heat, for instance — and the disorder that comes on top of it actually reduces the transport," he says.

The next step in Waleffe's research, which is funded mainly by the National Science Foundation Division of Mathematical Sciences, is to investigate the relationship between a coherent structure and the disorder, and to try to learn why a flow becomes turbulent.

Like the most recent discoveries, the answer won't come easily or quickly. "Turbulence is a very hard problem to crack," he says. "I think of it as a succession of hard shells, and we certainly have broken a shell. Now we can dig deeper into the problem, but we may not get to the core. There may be another shell to crack."

Studies of turbulence began as early as the 1500s, with artist Leonardo da Vinci's observations of and drawings of water flows, including this one of a series of 10 "deluge" drawings done circa 1515. (large image)

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8/11/2003