Of cream and clouds

Fluid Dynamics for Physicists - Turbulence
November 8, 1996

These are very different books. Tom Faber hopes that his "will slightly increase the chance that future generations of physicists will be taught the subject systematically in a way that I and my contemporaries were not". Uriel Frisch's is based on lectures to first-year graduate students who "had some knowledge of fluid mechanics, but little or no training in probability theory". A hypothetical reader who mastered Faber's book before starting Frisch's would also need a fair mathematical background to be unperturbed by the undefined use of self-adjoint projections, maps, etc.

The books do have two features in common: both are works of enthusiasm, and both recommend the reader to my colleague Milton Van Dyke's collection of photographs, An Album of Fluid Motion. Faber refers to the "curious and beautiful natural phenomena, visible every day in the world about us, which a physicist with no knowledge of fluid mechanics is unable to appreciate to the full'', while Frisch speaks of the "rich world of turbulence phenomena". Frisch does not claim that turbulence is beautiful as well as rich, but perhaps this is granted as rarely to phenomena as to people.

Faber is not a professional fluid dynamicist and, although his account is lively and interesting, it may give the reader an excessive trust in the applicability of Bernoulli's equation (valid only when viscous stresses have negligible effect) and there are a few misleading remarks. A surface does not have to be rough to impose the no-slip condition on fluid motion; aircraft wing boundary layers are turbulent, not laminar, unless heroic and expensive measures are taken; and the mushroom cloud from an atomic bomb explosion is not coextensive with the initial shock wave. The illustrations from real life are otherwise excellent.

Faber rightly laments the compendiousness of engineering textbooks, which sometimes give so many applications that one loses sight of the basic principles. But an engineer would at least be familiar with the fact that the partition between normal and shear stresses depends on the orientation of the axes, whereas Faber for some time conceals the existence of viscous normal stresses altogether. If this book goes to a second edition, as it should, perhaps a professional fluid dynamicist - even an engineer - could bring it a little more into the mainstream without losing the freshness of the approach. Certainly it covers all the main topics in fluid dynamics, including compressible flow, surface waves and non-Newtonian fluids, and even the present edition would be very valuable in the hands of a careful instructor.

Frisch has written his book around a seminal paper written by Andrei Nikolaevich Kolmogorov in 1941. The book is more general than the subtitle might imply, and covers most of the modern developments in the mathematical "theory" of turbulence. "Theory" belongs in quotes because turbulence, the nearly random fluid motion that mixes the cream in your coffee and the clouds in the sky, is governed by the Navier-Stokes equations, which express Newton's law of momentum conservation in the form "acceleration equals force per unit mass" (Newtonian viscous stresses also appear in the Navier-Stokes equations but have remarkably little effect in turbulence). Therefore a true theory would have to be a simpler alternative to Newton's laws, which seems rather unlikely to be achieved.

However, there is plenty of room for mathematics in the study of turbulence, and the recent rise of chaos studies has impacted on work on the Navier-Stokes equations. Frisch gives a nice but, as he admits, pathological, model of chaos, which he calls the "poor man's Navier-Stokes equation" and uses to illustrate the concept of "unpredictability". The Navier-Stokes equations are deterministic, not random, and they can be (expensively) solved numerically. But in turbulence, which is their most general solution, very small changes in the initial conditions - the state of the flow at the start of the calculation - inevitably produce large changes later on. The statistical-average properties of the turbulence (probability distributions and the like) are not sensitive to initial conditions; but the details of the solutions are. A small, transient change in the atmosphere over Stanford, California will leave the statistical-average rainfall at Epsom, Surrey, unaltered, but might make the difference between blue sky and a downpour next Derby Day. Interestingly, it is Faber who forthrightly - and I believe correctly - states that the gulf between (simple examples of) chaotic motion and (the far more complicated) genuinely turbulent motion, is wide and may ultimately prove unbridgeable.

Frisch reviews the Kolmogorov theory and later developments and then discusses "intermittency. Kolmogorov assumed that the smaller eddies in turbulence received their energy supply from the larger eddies in what might be described as plain envelopes: all that mattered was the rate of energy supply, and the details of the source were irrelevant. Landau pointed out the rate of energy supply to the small-scale motion near a given point in space will fluctuate in time as more or less intense large eddies pass by - mail delivery is an intermittent process and envelopes come in different sizes. This intermittency of energy transfer complicates statistical averaging: we need to accumulate averages over many periods of the small eddies, but the required averaging time may be so long that the energy transfer rate has changed.

Both books are slightly idiosyncratic: the authors have written the texts they wanted to, rather than mass-market books. It is a pity their readerships will not overlap much: Faber's is for the near-beginner in general fluid dynamics, Frisch's is for the turbulence expert. Together they show the wide range of topics covered by the endlessly fascinating subject of fluid dynamics.

Peter Bradshaw is emeritus professor of engineering, Stanford University.

Fluid Dynamics for Physicists

Author - T. E. Faber
ISBN - 0 521 41943 3 and 42969 2
Publisher - Cambridge University Press
Price - £50.00 and £22.95
Pages - 440

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