BOOK REVIEW

# Smooth Turbulence

*Turbulence, Coherent Structures, Dynamical Systems and Symmetry*. Philip Holmes, John L. Lumley and Gal Berkooz. 420 pp. Cambridge University Press, 1996. $69.95.

Philip Holmes, John L. Lumley and Gal Berkooz have written a charming and stimulating extended essay on the possible—or, if one is optimistic, probable—intersection of turbulence and nonlinear dynamical systems theory. The authors, well-known and highly regarded investigators in these areas, have provided a valuable roadmap of what is known, surmised and reasonably speculated. The book is organized into four parts, in the first of which the authors review much of what is known from years of experimental, theoretical and—more recently—numerical research on turbulence. Specifically they focus on coherent, relatively large-scale structures to suggest the possibility that a few such canonical structures—modes determined from proper orthogonal decomposition—may be used to describe turbulent flows to some useful level of approximation. In the second part, they provide a compact, clear discussion of the analysis of the interaction of such modes as determined from modern (and not-so-modern) nonlinear dynamical systems theory. In the third part, they address directly the turbulent boundary layer, perhaps the most famous and certainly one of the most challenging of turbulent flows. Finally in part four, they address other turbulent flows.

The authors express some concern that "experts in turbulence and dynamical systems may find our treatments of their respective fields superficial." But surely their modesty and worry, although becoming, are misplaced. Many readers from a wide variety of backgrounds including engineering, fluid mechanics, mathematics and beyond will find this volume a treasure-trove of information and insight. The authors' straightforward acknowledgments of the limitations of our current knowledge of turbulence are certainly appropriate, but it is also probable that their book will serve as a significant advance in our penetration to a deeper understanding of this notoriously difficult subject.

This volume is valuable, if for no other reason, in that it reminds us that to understand turbulence at a fundamental level, we must understand it as a nonlinear, dynamical process. Whether a modal decomposition will prove quantitatively useful—or simply suggest qualitative analogies to simpler dynamical systems—remains to be seen.

In a book of this scope and erudition, it is almost absurd to speak of omissions. It is surprising, however, that more emphasis was not given to the realization that the complexity of dynamical response may be enormous even for low-dimensional (a small number of modes) systems, an observation that has become clear in recent years through the work of many scholars. Hence it should not be surprising that at least some forms of turbulence may be described by models of relatively low-dimensional nonlinear dynamical systems. Perhaps, because this view is now commonly held among members of the nonlinear dynamical systems research community, the authors chose not to emphasize it here, although they do mention the work of H. C. Swinney, J. Gollub and their colleagues in the introduction. Reference is also made to the classical work of Heisenberg, Lin and others on the linear instability of laminar flows, which featured the study of eigenmodes of the fluid, and served as a precursor to the study of turbulence. Thus modes have been involved in the early history of turbulence. J. T. Stuart, among others, examined simple, nonlinear modal systems in this light, as the authors also briefly mention.

Finally, although these results were too new to be included in this book, we now know that rather complex fluid flows of initially very high dimension (but still with the turbulence described by empirical models) may be accurately described by modal decompositions. Hence the evidence to support the use of modal decompositions to describe turbulence is perhaps even more compelling now than the authors suggest in this book.

When turbulence is finally understood or at least much better understood than it is at the present time, it is probable that this book will continue to be regarded as a significant landmark. The reviewer heartily recommends this lucid and wholly admirable account of a new approach to an ancient riddle.—*Earl H. Dowell, Engineering, Duke University*

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