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Turbulence

One of the most important and baffling phenomena in fluid dynamics is turbulence: irregular, twisting flow-patterns far removed from the smooth ‘laminar’ flows beloved of the classical analysts. Until recently, turbulence has been studied by a variety of ad hoc analytical methods and probabilistic models, but relatively little attention has been paid to the geometry of turbulence. Yet the geometry contains hints of a deeper structure that the analytic approach misses. Turbulence involves motion on a wide range of scales, large and small. As Lewis Fry Richardson put it in 1922:

Big whorls have little whorls,

Which feed on their velocity;

And little whorls have lesser whorls,

And so on to viscosity.

Could fractals be involved in the geometry of turbulence?

This suggestion was made by Mandelbrot in about 1960. It re-emerged in a very different guise from the topological dynamics of the mid-1970s, and it now appears to be firmly established by careful experiments using a variety of ‘small-scale’ laboratory systems. Of course the theory met with fierce resistance along the way, as occurs with anything genuinely new in science, especially when it is advocated by interlopers from another field. To be fair, these experiments establish the occurrence of fractal geometry in weak turbulence; fully developed turbulence is quite another matter — but if anything, that looks even more fractal.

Turbulence may be confined to certain regions of an otherwise smooth flow, or it may appear suddenly everywhere. It can appear and disappear intermittently. In the Taylor vortex experiment, where fluid is observed in the region between two concentric rotating cylinders, spiral turbulence occurs in patches on a predominantly helical flow like a spinning barber’s pole. The boundaries of turbulent regions typically have a complex local structure: billows upon billows, whorls upon whorls. The region around Jupiter’s Great Red Spot is typical of such behaviour.

The topological approach to turbulence was initiated by Ruelle and Takens (1971), who suggested a scenario for the transition to turbulence in terms of the creation of a so-called ‘fractal attractor’ in the dynamics. Harry Swinney, Jerry Gollub, and others carried out experiments using lasers to measure the speed of the fluid, and confirmed the general conceptual framework, though not the precise scenario originally proposed.

In larger systems, the transition to turbulence is a much more complex affair. So we still have much to learn about turbulence. Fractal geometry can help us make advances, but it cannot answer everything.

What can?