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TURBULENCE is probably the most important and yet least understood problem
in classical physics.

The majority of fluid flows that are interesting from a practical point
of view – from the movement of air in the atmosphere to the flow of water
in central heating systems – behave in a disordered way. Turbulence has
always worried physicists because it is so difficult to model. In 1932,
the British physicist, Horace Lamb, told a meeting of the British Association
for the Advancement of Science: ‘I am an old man now, and when I die and
go to Heaven thereare two matters on which I hope forenlightenment. One
is quantum electrodynamics, and the other is the turbulent motion of fluids.
And about the former I am really rather optimistic.’

Nearly 60 years later, the fundamental nature of turbulence remains
a mystery although new and exciting developments continue to emerge. One
of these developments is the application of some of the ideas of chaos.
Fluid dynamics has proved to be a useful test bed for mathematical theories
modelling the transition from order to chaos.

For more than a century, fluid dynamicists and mathematicians have been
trying to understand turbulence in fluids, by analysing the mechanisms that
generate disordered motion. As a starting point for their investigations,
they have used a set of equations that describe how both liquids and gases
move. These equations, which were developed independently by Claude Navier
and George Stokes in the first half of the last century, are based on Newton’s
laws of motion so they are deterministic. You might, therefore, expect thatby
putting a set of measurements – velocity, time and soon, into these equations
you would be able to producesolutions that describe the motion of the fluid
for all time.

Unfortunately, this is not so. If you read the recent article in this
series by Franco Vivaldi on the logistic map (‘An experiment with mathematics’,
New ÐÓ°ÉÔ­´´, 28 October) you might have been alarmed to discover that
even extremely simple setsof deterministic equations, when allowed to evolve
over time, quickly give chaotic answers. The reason is that the equations
are ‘nonlinear’ (the variables in the equations are not directly proportional
to one another but vary as the square or some higher power). The Navier-Stokes
equations are also nonlinear and much more complicated, so you should not
be surprised to find that chaos, or turbulence, is the rule rather than
the exception.

Furthermore, even with today’s most powerful computers, we cannot usually
solve these equations. Take a simple example, such as the flow from a tap.
The parameter that describes the flow is called the Reynolds number, R,
after the British physicist Osborne Reynolds. It is the ratio of the inertia
of the fluid, as defined byits mass and velocity, to the thickness, or viscosity,
of the fluid. This means that R is small when the viscosity is highand the
velocity low. A high viscositytends to damp out any disturbances in the
motion of the fluid, which are dissipated as heat. You do not usually see
turbulence in treacle.

So, for small values of R, when the tap is only slightly open, the water
falls smoothly as shown in photograph a below. We describe this as laminar
flow because the water flows in parallel sheets. Even calculating this simple
state of affairs would test the limits of a supercomputer if we started
from the full Navier-Stokes equations. What is more, laminar flow is not
usually found in nature so it does not have much practical value.

If we increase the Reynolds number by opening the tap further, then
we obtain the resulting flow in photograph b. This disordered motion is
called turbulent and is the most type of fluid flow encountered. In this
case, it is impossible to calculate the flow from the Navier-Stokes equations.
Engineers can, however, make some progress with ad hoc models that capture
some of the essential features of the flow, but only rarely do these models
apply to more than one situation. One way to understand how and why turbulence
happens is to study the fundamental processes causing it. The transition
to turbulence in water from a tap is as sudden as it is in the case for
flow in a pipe – which was the problem on which Reynolds carried out his
pioneering work. Both these situations are extremely difficult tocontrol
in the laboratory so, although they are important practically, exactinglaboratory
set-ups with taps make unattractive experiments.

Instead, some physicists prefer to study a classical system that is
much easier to work with. The Taylor-Couette system, first studied at Cambridge
University by Geoffrey Ingram Taylor in the 1920s, shows the onset of disordered
motion through a sequence of stages. In this experiment, we look at the
movement of fluid in a gap between two concentric cylinders where the inner
cylinder rotates and the outer one is kept still. At small values of the
Reynolds number R, the motion is mainly in concentric circles around the
axis of the cylinders, except for some inevitable motion in other directions
near the ends of the apparatus. If we increase R by speeding up the rotation
of the cylinder so that the fluid moves faster, a rather strange thing happens.
At a certain critical point, a secondary motion appears superimposed on
the concentric motion. This secondary motion is in the form of cells that
look like stacked Swiss rolls (see photograph c). There, we have made the
flow visible by adding metal particles that reflect the incident light.
The photograph taken with a time exposure of four seconds shows the paths
of the spiralling particles caused by the secondary motion superposed on
the main flow round the cylinder.

If we now view a cross section through the flow, then we can see the
circular motion shown in photograph d. This is done by illuminating a section
through the cylindrical gap with a slit of light from the side, and viewing
the flow in a direction at right angles to the light. Compare this experimental
result with the picture above which shows the stream lines calculated from
the Navier-Stokes equations by Andrew Cliffe using the Cray supercomputer
at Harwell Laboratory.

When we increase R a little more, then waves appear inthe cells as shown
in photograph e. The waves travelaround the cylindrical gap at some fraction
of the speed ofthe inner cylinder (typically in the range between 0.1 and0.5).
If we were to measure the velocity at a point in sucha flow using a laser
as a probe, then the change in velocityover time would appear as a simple
wave (a sine wave)when shown on an oscilloscope.

If we now increase the Reynolds number above a second critical value,
then another frequency will appear, which is not related to the first. In
1944, the famous Soviet physicist Lev Landau had suggested that further
increases in R would produce more and more frequencies in the flow with
each new wave appearing at diminishing intervals of R. Thus, R would rapidly
reach a value where all possible frequencies arose in the flow. According
to Landau, this is turbulence. Photograph f of ‘turbulent’ flow in a Taylor-Couette
system shows that although the fluid flow is disordered in time, there is
still order in space. So it is very different from, say, a fast-flowing
river where the flow of water is completely disordered in both time and
space.

Landau’s description of turbulence turned out not to be quite right.
Two mathematicians, David Ruelle at Bures-sur-Yvette, near Paris, and Floris
Takens at the University of Amsterdam put forward an alternative view. They
were studying sets of equations in the abstract, and suggested that chaos
would ensue after only a few frequencies had appeared in the motion. To
understand their point we need to consider a way of representing the data
in terms of the ‘phase portraits’ that Ian Stewart described recently in
his article (‘Portraits of chaos’, New ÐÓ°ÉÔ­´´, 4 November).

The traditional way of analysing how a variable quantity, such as velocity,
varies periodically with time – a time series – is to break down the overall
complex signal into a series of sine waves of differing frequencies. This
is called Fourier analysis. Takens suggested an alternative way of portraying
the time series as a two- or three-dimensional map in what is called phase
space. In its simplest form, you plot the time series against a time-delayed
version of itself. A sine wave would give a circle in two-dimensional phase
space. The time taken for the trajectory to rotate around the loop would
then be the period of the oscillation. When there are two frequencies that
are not related to each other, we need a second delayed time coordinate
and another dimension to show it. We then obtain a torus, or doughnut. David
Broomhead, Robin Jones and Greg King at the Royal Signals and Radar Establishment
at Malvern have developed a powerful practical method which we have used
to construct the results presented here.

If we disturb the trajectories in our phase space (this would correspond
to perturbing the rotation of the inner cylinder, say) then a short transient
phenomenon appears before the cyclic motion on the circle or torus is regained.
We call these objects attractors in phase space.

Ruelle and Takens suggested that, instead of the appearance of a third
wave as suggested by the Landau picture, chaos would arise through the emergence
of a ‘strange attractor’ in phase space. We can consider the strange attractor
as a region of phase space that attracts nearby trajectories but inside
the region neighbouring trajectories diverge and are chaotic in form. Harry
Swinney and Jerry Gollub at the University of Texas at Austin have carried
out experiments on the Taylor-Couette problem, which support this picture.

Is there any more evidence to show the connection between observations
in this fluid dynamical system and the chaos found in the solutions of simple
sets of equations? An essential feature of any nonlinear system is that
many different states can exist under the same operating conditions. In
Taylor-Couette flow, these states would have different numbers of cells
or waves or both. The state that you obtain depends on the history of its
creation. Thus, if the system is large, there may be literally thousands
of different states available – all of which can interact with each other.
Controlling the experiment and interpreting the outcome becomes impossible,
in any practical sense. We have to be vigilant when extracting qualitative
features because the picture could be confused by the presence of many competing
attractors.

One way to cope with this situation is to restrict the number of available
states by limiting the physical size of the system. Gerd Pfister at the
University of Kiel in West Germany experimented on the onset of chaos in
a miniature version of the Taylor-Couette system containing a single cell.
He uncovered a period doubling route of the type already described in Vivaldi’s
article. This suggests that chaos plays an important role in the understanding
of the results of this restricted flow.

If we now consider the sequence of events leading to chaos when the
system is made a little larger, then new types of behaviour happen, each
of which appears to be linked directly with modern thinking on chaos. The
results from one of these studies is presented in the phase portraits in
the two photographs g and h. Photograph g shows motion on a torus in our
reconstructed ‘phase space’ and corresponds to the presence of two waves
in the flow with very different time-scales. It has qualitatively the same
kind of structure as the output from some simple ordinary differential equations
that you can solve on a microcomputer. If we change the control parameters
of the experiment just a little, then chaos appears as shown in photograph
h. The ‘shell’ of the structure begins to fill in and the trajectories join
the core at irregular intervals. However, the overall structure of the portrait
is maintained and so we appear to have behaviour of the strange-attractor
kind similar to that outlined above.

There does, therefore, appear to be a connection between observations
of the onset of chaos in this fluid dynamical system and the onset of chaos
in much simpler sets of equations. What is more, the chaos appears to arise
through mechanisms that researchers have found in other situations such
as chemical oscillators and lasers. You should remember, however, that the
results are concerned only with the evolution of disorder in time in a very
special flow: do not think that the challenging and exciting problem of
turbulence of the type shown in photograph i is solved. Nevertheless, mathematical
ideas of chaos may have found a chink in the armour of this intriguing problem.

Tom Mullin is leader of a research group studying nonlinear systems
in the Clarendon Laboratory at Oxford.

Further Reading: Physical Fluid Dynamics, B. J. Tritton, Oxford Science
Publications, 1988.