Stochastic Tools in Turbulence by John L. Lumey (PDF)

Description

Stochastic Tools in Turbulence discusses the available mathematical tools to describe stochastic vector fields to solve problems related to these fields. The book deals with the needs of turbulence in relation to stochastic vector fields, particularly, on three-dimensional aspects, linear problems, and stochastic model building. The text describes probability distributions and densities, including Lebesgue integration, conditional probabilities, conditional expectations, statistical independence, lack of correlation. The book also explains the significance of the moments, the properties of the characteristic function, and the Gaussian distribution from a more physical point of view. In considering fields, one must account for single-valued functions of one or more parameters, or collections of single-valued functions of one or more parameters such as time or space coordinates. The text also discusses multidimensional vector fields of finite energy, the characteristic eddies for a homogenous vector field, as well as, the distribution of solutions of an algebraic equation. Engineers, algebra students, and professors of statistics and advanced mathematics will find the book highly useful.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book is a good math reference or math primer if you plan to take a course in turbulence or stochastic processes. Other books might have the same information sprinkled over 1200 pages, but this book is just the math which makes it easier to look up the relevant topics/properties. The appendices on tensors and Fourier transforms are nice. The text was a reasonable size and the printing is as clear as could be expected.The ordering of topics is good in general, but occasionally somewhat odd within chapters. The book would benefit from more concrete numerical examples to better drive home the behavior/ meaning of various statistical properties.

⭐Books by Lumley have generally a high stature and impact, namely if written together with Tennekes. The present book is also valuable. In the Preface and the Acknowledgements the author indicates that it is part of a broad spectrum of mathematical literature around the topic of turbulence, dominated by schools around Kolmogorov, Doob, Monin, Yaglom, Batchelor and others; most of them are mathematicians. The irony here is that the most successful contributions (i.e. those which observable/measurable) to turbulence are mathematically trivial: Prandtl’s concept of eddy diffusivity (so-called Kolmogorov-Prandtl relation between turbulent kinetic energy, a certain scale, and eddy diffusivity; not too far from Einstein’s ideas about Brownian motion), and Kolmogorov’s arguments about the form of the energy spectrum, based solely on dimensional arguments. There is a new theory which is free of empirical parameters and essentially of a certain geometrical nature. It follows those classic approaches and remains mathematically simple but physically it is a complex series of arguments (see doi:10.1088/0031-8949/2013/T155/014001) using early stochastic concepts (Langevin, Fokker, Planck) and theory elements of complex systems (Debye, Landau, Feynman). It is best summarized unter the headline “synergetics” (H. Haken). Today we may state that the new theory of turbulence is comparable with Copernican revolution. While the mathematical efforts for the Ptolemaian picture were substantial, the Copernican picture was much simpler and eventually the winner. Therefore at the end of the day the value of all the tools in Lumley’s book might be limited. However, there are questions left open also by the new theory, and these migth need filigrane, sophisticated tools like those well described in the present book. Nevertheless it will remain true what Falkovich & Sreenivasan wrote in 2006: “The diversity of problems in turbulence should not obscur the fact that the heart of the subject belongs to physics.” And the adventure goes on! For scholars interested in mathematics beyond turbulence this book is clearly a good buy!

⭐John L Lumley has written a number of well regarded books on the general subject of turbulence, but “Stochastic Tools in Turbulence” is severely misnamed. A better title would have been “Mathematics of Three-Dimensional Stochastic Vector Fields.” If you are looking for careful definitions of probabalistic ideas in terms of measures and Lebesgue integrals, and a derivation of the Characteristic functional for the multipoint probabilities, this book might be a good choice. If, on the other hand, you (like me) were just trying to make sense out of estimating the Reynolds Stress Tensor from 20 Hz anemometer data – not so much.There is no discussion of applications, much less application to turbulence or fluid flow whatsoever. There are a number of long appendices, but since the only one I read was the (very nice) one on generalized functions, I won’t comment further on them. Monin and Yaglom’s two volume “Statistical Fluid Mechanics” which covers much of the same ground plus much, much more is far more fluid oriented, but still a bit theoretical (and long) for my purposes, so I’m still looking for a good discussion of my questions.

⭐The book is good and easy to read. A few errors but you can figure them out easily. In general, covers fundamental concepts in a way that you can follow the main idea

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