Taming turbulence: Understanding the equations
he weather is very largescale turbulence. That’s why it is 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.

One of a series of
10 Leonardo da Vinci "deluge" drawings done circa 1515.
(36K
JPG) 
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 ClaudeLouis
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 slowestmoving, level. “Even if you just
walk around, we cannot fully calculate the air flow around you,”
he says. He is using the NavierStokes 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 40 years 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.”