Fluid dynamics

Flows like smoke and honey

Observe wafting cigarette smoke, and you are watching some complex physics: close to the lit end, the smoke rises smoothly; farther away, it unfurls into swirls. This second type of behaviour is an example of chaotic, turbulent flow, the course of which — because of the huge number of interacting particles involved — cannot be precisely predicted.

Denis Bernard et al. instead perform numerical simulations of turbulent flow (Nature Phys. 2, 124–128; 2006). Surprisingly, they find a connection between two-dimensional turbulence and a simpler, hitherto unrelated phenomenon — that of critical percolation.

Percolation describes the flow of a fluid through a porous material, such as honey seeping through beeswax. In a honeycomb, percolation reaches a critical threshold at which half the cells are filled with honey and the probability that a row of honey-filled cells spans the whole layer is not zero (although it might be infinitesimally small). At criticality, the filled and empty clusters of cells assume a ‘fractal’ pattern that follows a principle known as conformal invariance: as long as the angles don't change, magnifying different sections of the pattern by different amounts results in a pattern that is indistinguishable from the original.

The principle is demonstrated in this image of Bernard and colleagues' turbulent flow. Here, connected clusters of vortices rotating in one direction are coloured and those rotating in the other are black; the resulting pattern is indistinguishable on small and large scales. The authors calculate a fractal ‘dimension’, a measure of the degree to which the pattern seems to fill space as one looks at ever finer scales. The value they find, 4/3, corresponds exactly to that found in the distribution of a fluid such as honey at the critical percolation threshold.

The discovery of this link could open the field of turbulence to the full theoretical artillery of conformal mapping, a technique that is central to a whole range of physical theories, such as string theory, besides critical percolation. But more work is required to smoke out the exact extent to which these ideas can be applied.

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Chiao, M. Flows like smoke and honey. Nature 439, 547 (2006). https://doi.org/10.1038/439547a

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