Babbling brooks and bracing breezes may please poets but they bother physicists. These natural examples of turbulence are difficult to analyse mathematically. Now, theories of chaos combined with some simple laboratory experiments may provide some answers.
Tom Mullin
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 there are two matters on which I hope for enlightenment.
One is quantum electrodynamics, and the other is the turbulent motion of
fluids. And about the former I am really rather optimistic."
(A)When the tap is only slightly open, waterfalls in a streamlined way. Even so, calulating what is happening is extremely difficult. (B)Turning the tap further causes the water to flow turbulently. It is virtually impossible to describe this state of affairs as a mathematical equation. |
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 that by putting a set of measurements-velocity, time and so on, into
these equations you would be able to produce solutions 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 Scientist,
28 October) you might have been alarmed to discover that even extremely simple
sets of 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 by its mass and velocity, to the thickness, or
viscosity, of the fluid. This means that R is small when the viscosity is
high and the velocity low. A high viscosity tends 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 above. 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.
(C) A front view (at 90 to photo D) of circular cells that appear in a Taylor-Couette system as fluid flows between the two cylinders. | (D) A cross section of the circular cells in a Taylor-Couette system compared with the same streamlines calculated on a computer. |
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 to control
in the laboratory so, although they are important practically, exacting
laboratory 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 spiraling particles caused by the secondary motion superposed
on the main flow round the cylinder.
(E) As a fluid flows faster in the Taylor- Couette system, waves start to appear as shown in this front view. | (F) A further increase in the flow rate causes turbulence. Although the fluid flow is disordered in time, it is ordered in space. |
If we now view a cross section through the flow, then we can
see the arcular motion shown in photograph d. This is done by illuminating
a section through the cylindrical gap with a slit of lightfrom 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 in the cells
as shown in photograph e. The waves travel around the cylindrical gap at
some fraction of the speed of the inner cylinder (typically in the range
between 0.1 and 0.5). If we were to measure the velocity at a point in such
a flow using a laser as a probe, then the change in velocity over 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 frequenaes 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.
This strange computer graphic shows a reconstructed "phase portrait" of a regular flow in the Taylor-Couette system. | This is a phase portrait of chaotic flow. The "shell" of the structure begins to fill in to give a "strange attractor". |
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 Scientist,
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.
(I) "Real" turbulence in a system where the fluid flows between a rotating inner cylinder and a square outer shell. |
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. |
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