Turbulence, coherent structures, dynamical systems and symmetry. 2nd ed.

*(English)*Zbl 1251.76001
Cambridge Monographs on Mechanics. Cambridge: Cambridge University Press (ISBN 978-1-107-00825-0/hbk; 978-1-139-22768-1/ebook). xvi, 386 p. (2012).

This advanced textbook on turbulence, coherent structures, dynamical systems and symmetry is the second edition of the book first published in 1996 [see for the review Zbl 0890.76001 and Zbl 0923.76002]. The goal is to provide tools to reduce the complexity of the governing Navier-Stokes equations in the turbulent regime. The existence of coherent structures in most turbulent flows allows for a projection onto new building blocks which govern the dynamics using only a strongly reduced number of modes, the so-called low-dimensional models.

The book is organized into four parts. The first part devoted to turbulence introduces general ideas from fluid mechanics and turbulence theory, including basic features of generic flows, like mixing and boundary layers. Coherent structures, the proper orthogonal decomposition (POD) of flows fields and the Galerkin projection onto the resulting basis functions yield the context of low-dimensional models proposed in this work. With respect to the first edition, the authors introduce the balanced POD, which is of some interest for the control theory community.

Dynamical systems are reviewed from a qualitative point of view in Part two. Geometry and symmetries play hereby a particular role for the considered finite-dimensional ordinary differential equations and iterated maps. Different tools, like linearization, invariant manifolds, structural stability, bifurcations and attractors are presented. For illustration, the developed techniques are applied to the one-dimensional Kuramoto-Sivashinsky equation with periodic boundary conditions. Randomly perturbed dynamical systems are also considered to provide additional modeling tools.

Part three is focused on low-dimensional models of the boundary layer. Equations for coherent structures are derived, and the interaction with unresolved modes is discussed. Various symmetries that the low-dimensional models inherit from the original system are also studied. The authors analyze the behavior of the models by describing the qualitative structure in phase space and their physical implications.

Finally, the fourth part presents other applications and a survey on related work dealing with different flows, like jets, wakes and transition in boundary layers. Conclusions are drawn, and speculations on the use of low-dimensional models, e.g., for active turbulence control, terminate this part. An impressive list of about 400 updated references follows.

The book is organized into four parts. The first part devoted to turbulence introduces general ideas from fluid mechanics and turbulence theory, including basic features of generic flows, like mixing and boundary layers. Coherent structures, the proper orthogonal decomposition (POD) of flows fields and the Galerkin projection onto the resulting basis functions yield the context of low-dimensional models proposed in this work. With respect to the first edition, the authors introduce the balanced POD, which is of some interest for the control theory community.

Dynamical systems are reviewed from a qualitative point of view in Part two. Geometry and symmetries play hereby a particular role for the considered finite-dimensional ordinary differential equations and iterated maps. Different tools, like linearization, invariant manifolds, structural stability, bifurcations and attractors are presented. For illustration, the developed techniques are applied to the one-dimensional Kuramoto-Sivashinsky equation with periodic boundary conditions. Randomly perturbed dynamical systems are also considered to provide additional modeling tools.

Part three is focused on low-dimensional models of the boundary layer. Equations for coherent structures are derived, and the interaction with unresolved modes is discussed. Various symmetries that the low-dimensional models inherit from the original system are also studied. The authors analyze the behavior of the models by describing the qualitative structure in phase space and their physical implications.

Finally, the fourth part presents other applications and a survey on related work dealing with different flows, like jets, wakes and transition in boundary layers. Conclusions are drawn, and speculations on the use of low-dimensional models, e.g., for active turbulence control, terminate this part. An impressive list of about 400 updated references follows.

Reviewer: Kai Schneider (Marseille)

##### MSC:

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

76F02 | Fundamentals of turbulence |

76F40 | Turbulent boundary layers |

76F20 | Dynamical systems approach to turbulence |