# Notes

## Section 4: Fluid Flow

Models of turbulence

Traditional models typically view turbulence as consisting of some form of cascade of eddies. This notion was already suggested in pictures by Leonardo da Vinci from around 1510, and in Japanese pictures (notably by Katsushika Hokusai) from around 1800 showing ocean waves breaking into precisely nested tongues of water. The theoretical study of turbulence began in earnest in the early 1900s, with emphasis on issues such as energy transfer among eddies and statistical correlations between velocities. Most published work became increasingly mathematical, but particularly following the ideas of Lewis Richardson in the 1920s, the underlying physical notion was that a large eddy, formed say by fluid flowing around an object, would be unstable, and would break up into smaller eddies, which in turn would break up into still smaller eddies, until eventually the eddies would be of such a size as to be readily damped by viscosity. An important step was taken in 1941 by Andrei Kolmogorov who argued that if the eddies in such a cascade were in a statistical equilibrium, then dimensional analysis would effectively imply that the spectrum of velocity fluctuations associated with the eddies must have a k^(-5/3) distribution, with k being wavenumber. This result has turned out to be in respectable agreement with a range of experimental data, but its physical significance has remained somewhat unclear. For there appear to be no explicit entities in fluids that can be directly identified as cascades of eddies. One possibility might be that an eddy could correspond to a local patch of vorticity or rotation in the fluid. And it is a general feature of fluids that interfaces between regions of different velocity are unstable, typically first becoming wavy and then breaking into separate pieces. But physical experiments and simulations in the past few years have suggested that vorticity in turbulent fluids in practice tends to become concentrated on a complicated network of lines that stretch and twist. Perhaps some interpretation can be made involving eddies existing only in a fractal region, or interacting with each other as well as branching. And perhaps new forms of definite localized structures can be identified. But no clear understanding has yet emerged, and indeed most of the analysis that is done—which tends to be largely statistical in nature—is not likely to shed much light on the general question of why there is so much apparent randomness in turbulence.

From Stephen Wolfram: A New Kind of Science [citation]