Animated Attractor

Turbulent times for fluids

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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New Scientist 11Nov 1989 File Info: Created 8/2/2002 Updated 9/9/2017 Page Address: