Monday, December 1, 2008

Transitory Turbulence

Turbulence is often called the last unsolved problem of classical mechanics. It is a ubiquitous phenomenon that anyone who has ever flown on a airplane knows about, but there have been exactly two major breakthroughs in turbulence theory in the past 70 years. Both of these breakthroughs came in the early 1940's and both were made by a Russian mathematician and physicist named Andrey Kolmogorov. However, there has been a lot of work on the problem, especially since the development of the theory of dynamical chaos in the 1990's. One of the big questions, in the language of dynamical chaos, is whether turbulent states are attractors or repellers. Put another way, if you get a system into a turbulent state, will it remain turbulent indefinitely or will it eventually decay into a laminar state? There are, of course, three possible answers to this question: yes, no, or maybe.

We can throw out the yes answer by common counter examples. If you pour water into a glass the flow is normally turbulent, however it quickly becomes laminar if you stop pouring. The real debate then, is whether turbulent systems always decay into laminar systems or if there are some special cases where turbulence is a chaotic attractor.

Before we go further, I need to introduce you to a parameter known as the Reynolds number. Fluid dynamics is full of dimensionless parameters and in turbulence the Reynolds number (abbreviated Re) is the most important one. Essentially, the bigger the Reynolds number the more turbulent the system is. More specifically, the Reynolds number is the ratio of inertial forces to viscous forces. In laminar flows, small chaotic motions are damped by viscous forces (low Re), while in turbulent flows viscosity is insufficient to damp out the small chaotic motions (high Re).

The maybe answer comes from the argument that above some Reynolds number, turbulence is self-sustaining. However, a new paper in Physical Review Letters by Bjorn Hof, Alberto de Lozar, Dirk Jan Kuik, and Jerry Westerweel (and nicely summarized by the APS at claims that the correct answer is that there is no such thing as a turbulent attractor. They conduct a number of experiments that show that the decay time from turbulence to laminar flow goes as a super-exponential function of the Reynolds number ( tau = k*exp[exp[Re]] ). This trend fits very nicely over 6 orders of magnitude in decay times from ~5 seconds to ~150 days as seen in the figure to the right. This gives strong evidence that turbulence always decays into laminar flow, it just takes a while in some cases. In fact, for a system with a Reynolds number of 2200, the decay time is longer than the age of the universe.

By comparison, in the solar convection zone the Reynolds number is something like 10^10. That's a decay time of 10^[10^(10^9)] seconds. As they say in Brazil, don't wait standing.

While the debate may not be concluded quite yet, it appears that turbulence is really a temporary phenomena - if by temporary one means it commonly lasts orders of magnitude longer than the age of the universe. More meaningfully, it means that from a theoretical standpoint, turbulence is a chaotic repeller rather than a chaotic attractor. Perhaps most importantly, this means that we are still making progress on the last major puzzle of classical mechanics.

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