Book ReviewBOOK REVIEWS published in this section reflect the opinions of their individual authors. They are not necessarily the opinions of the Editors ofthis journal or of AIAA.

Turbulence, Coherent Structures, Dynamical Systems and SymmetryP. Holmes, J. L. Lumley, G. Berkooz, and C. W. Rowley, 2nd ed., Cambridge University Press, Cambridge, England,

U.K., 2012, 386 pp., $90

DOI: 10.2514/1.J052557

This book is a valuable (almost necessary) addition to thelibrary of students and researchers working on reduced-ordermodels (ROMs) of fluid flows. I find it onpractically allbookshelves of corresponding researchers and students.The first edition of this book was written in 1996 by P.Holmes, J. L. Lumley, and G. Berkooz, the pioneers ofmodern low-dimensional dynamical models of turbulence,and it has becomeoneof the classics in its field. The secondedition has C. W. Rowley as new author, a pioneer incontrol-oriented modeling. This edition contains a newchapter on balanced proper orthogonal decomposition(POD) targeting control-design applications. Otherwise,themain structure of the book has not been changed,whilevarious sections have been updated.Reduced-ordermodeling is emerging as a new discipline

besides experimental and theoretical fluid mechanics.ROM translates data or high-dimensional Navier–Stokesdiscretizations into low-dimensional dynamicalmodels as atestbed for the understanding of key physical processes, asa computationally inexpensive surrogate model foroptimization, and as a dynamical plant for closed-loopflow control. ROM is a difficult interdisciplinary subjectbecause it requires a solid understanding of fluidmechanics, a good mastery of computational methods,and a working knowledge of dynamical systems theory. Ihave taught the subject on numerous occasions fromsummer schools to a two-semester course (120 h) at BerlinInstitute of Technology and have learned to appreciate theteaching challenge.

The authors have elegantly addressed this challenge byan unconventional structure and exposition. The bookcomprises four parts. The first part focusses on fluidmechanics: simplifications of the Navier–Stokes equation,the phenomenology of coherent structures, (balanced)POD as low-order kinematics, and the Galerkin projectionleading to low-order dynamical models. The second partrecapitulates dynamical systems theory from a moremathematical perspective. The third part describes theirfirst pioneering model of the turbulent boundary layer indetail, building on the previous chapters. And the last partaddresses various progresses of ROM for other flows aswell as open challenges. The book contains rigoroustextbook chapters for the methodology, simple analyticalexamples that invite the reader to explore the field, andadditional sections closing up to the state of the art (i.e.,invitations to additional reading).

A single book cannot describe the vast knowledge ofrapidly evolving ROM in fluid mechanics, as the authorscorrectly emphasize in the preface. Several other bookshave beenwrittenon this subject, focusing, for instance, onmathematical system reduction, computational Galerkinmodels, control-orientedmodels, etc. Yet the current bookwill remain a standing classic in POD-based Galerkinmodels of turbulent flows.

Bernd R. NoackInstitute PPRIME/CNRS

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