The Turbulence Problem

Build an accelerator and physicists will tell you all about the beautiful mathematics of quantum theory. Build a telescope and physicists will tell you all about the beautiful mathematics of general relativity. Turn on a faucet and physicists may tell you that whatever the mathematics of turbulence turn out to be they probably won’t be beautiful. Perhaps mathematical beauty is a measure of the distance from everyday experience.

 

The persistent riddle in the title of Eckert’s short monograph is the transition from laminar to turbulent flow. I, not unlike another reviewer of a book on turbulence (see SIAM Reviews, 55(3), p. 579), recommend that you start with the last chapter, “Turbulence as a Challenge for the Historian,” because Eckert is short on details on the nature of the turbulence problem while being long on details of the many attempts to solve it. Indeed, a notable strength of the monograph is the book’s bibliography of over 300 entries covering these efforts. The mathematics included in the book is largely unexplained and is there, one suspects, to refresh the memory of readers already steeped in the problem.

 

The historical perspective of the book is that of the twentieth century as both experimental and theoretical physicists tried to either resolve or escape the nineteenth-century work of Navier, Stokes, and Reynolds. The exposition is linear in time, from Klein’s theories and Prandtl’s experiments at the turn of the century through Kármán’s and Kolmogorov’s statistical approaches, Osager’s point-vortex dynamics, Dryden’s and Lin’s work on shearing flows, the numerical simulations of von Neumann and Deardorff, Ruelle’s strange attractors, and Mandelbrot’s fractals.

 

For readers not sufficiently steeped in fluid dynamics to fully appreciate Eckert’s Knickerbocker tour of twentieth-century turbulence study, the book can be fruitfully read on another level; namely, the sociology of the interaction between theoreticians and experimentalists in building mathematical descriptions of physical phenomena. Laboratory Life: The Construction of Scientific Facts by Latour et al. is a seminal work of this genre but there are others. In fact, David Turnbull wrote explicitly about the sociology of the study of turbulence in 1995 (“Rendering Turbulence Orderly”, Social Studies of Science, 25(1), 9-33) and doubtless would have referenced Eckert’s monograph had it existed at the time.

 

While Latour isn’t cited, The Turbulence Problem is as much about the sociology of the search for a solution to the riddle as it is about the mathematics. Eckert writes in his Preface “I do not aim at a comprehensive account but rather at an exemplary exposition of the environments in which problems become items of research agendas. From this perspective, the turbulence problem also provides more general lessons for the history and epistemology of science and technology in the twentieth century.” Eckert and Turnbull agree that the scientific study of turbulence does not fit comfortably in the Kuhnian framework but neither offers a strong argument as to why this might be the case. It might fit fine once a paradigm is established or there might be non-Kuhnian scientific evolutions.

 

That turbulence is not only a hard problem but still very much an open problem makes it a compelling laboratory subject for this purpose. As Latour and others have observed, when consensus on a scientific framework congeals alternatives lose their voice and thereby become difficult to interview. Before a winner has been declared, all voices are eager to be heard.

 

For a complementary historical analysis of the turbulence problem, I’d suggest Terry Reynolds’ Stronger than a Hundred Men: A History of the Vertical Water Wheel. Some mathematicians mentioned by Eckert — Euler, Prony, Daniel Bernoulli, d’Alembert, Borda, and Bossut to name a few — are also heard from in Reynolds’ book. For readers interested in learning more about fluid dynamics it might not be out of place to call attention to a series of films produced in 1961 by the National Committee for Fluid Mechanics under the leadership of its founder, Ascher Shaprio. 

 

Putting down The Turbulence Problem one is left wondering if the mathematics needed to describe turbulence hasn’t been invented yet or the insistence on beauty might be over constraining the use of existing mathematics. My own view is that we need something between Newton’s calculus and Knuth’s algorithms. This is a third level on which one can enjoy Eckert’s short monograph and the on which I recommend it to the reader’s attention.

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Scott Guthery is the founder and editor of Docent Press and co-founder of Life-Notes.