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Image: James Marvin Phelps Laminar flow turns turbulent.Image: James Marvin Phelps

Turbulence has famously been called "the last great unsolved problem in classical physics" by Richard Feynman. Indeed, the chaotic nature of turbulence makes it nearly impossible to predict, and we are forced to talk about averages and probabilities. A recent paper in the journal Physical Review Letters takes a stab at the problem by making more precise measurements of the persistence of turbulence than any previous study. In particular, they focused on the probability that a puff of turbulence will decay back into laminar, non-turbulent, flow.

Fluid flow will tend to be laminar when viscous forces dominate over the inertial forces. A highly viscous fluid, such as honey, will tend towards laminar flow, while a less viscous fluid, such as water, will be more likely to flow turbulently. Of course, the intertial forces are also important. The faster a fluid flows, the more likely it is to flow turbulently. To quantify this relationship, we use the so-called Reynolds number. For flow in a pipe, the Reynolds number can be calculated as the fluid velocity times the pipe diameter divided by the kinematic viscosity.

To study the persistence of turbulence, Hof et al. set up a narrow pipe with water flowing through it, with a location at which they could inject a burst of water that would induce turbulence. They could then observe whether or not that turbulence persisted as it traveled down the pipe. They made their set up such that they could vary the Reynolds number around a critical value determined by a previous study.

That previous study, by Willis and Kerswell, postulated that there was a critical Reynolds number (about 1875) at which turbulence would persist indefinitely. The present study suggests that turbulence would not persist, but the probability of it decaying would become very small.

To measure these small probabilities, one must be able to repeat a measurement a large number of times. Past studies haven't been able to do this because they were not able to maintain a constant Reynolds number. The problem stems not from maintaining a constant velocity or pipe diameter; instead, maintaining a constant viscosity is a challenge.

Viscosity is related to the temperature of the fluid, and minor temperature fluctuations over time produce changes in the Reynolds number. The current study managed an amazing level of precision. The researchers controlled the temperature of the water to within 0.05 Kelvin over several days of measurements, all of which involved water flowing through a 14m long pipe that was a mere 4mm in diameter. This resulted in an increase in the range of measurable lifetimes by five orders of magnitude!

Their measurements showed that, even at higher Reynolds numbers, the turbulence could still collapse back to laminar flow with a finite probability. They suggest that this could be true at even higher Reynolds numbers than they were able to measure, thus making turbulence a chaotic repellor instead of an attractor.

Physical Review Letters, 2008. DOI: 10.1103/PhysRevLett.101.214501