This monograph applies dynamical system ideas to open turbulent flows dominated by coherent structures. It centers around the work of the authors and their students and colleagues in deriving low-dimensional models for turbulent flows resolving only the coherent structures. Altogether, the book consists of four parts. The first two parts are fairly general and introduce key ideas from fluid mechanics of turbulence, dynamical system theory and mathematical techniques needed for treatment of particular flows in parts three and four.
More detailed, the first part gives some background on turbulence and coherent structures. Order-of-magnitude and scaling estimates are reviewed for turbulent mixing and boundary layers. The method of Karhunen-Loève (or proper orthogonal decomposition) is presented to provide empirical basis functions for low-dimensional projections via the Galerkin method. The second part is a mini-treatise on dynamical system theory discussing the main ideas and tools. Particular attention is paid to the influence of symmetries on bifurcations and the local and global dynamical behavior of low-dimensional systems of ODEs. The one-dimensional Kuramoto-Sivashinsky equation serves as a simple model problem to illustrate the described methods. In part three the relevant Galerkin projection and modelling issues are discussed for the main case study of the wall region of a turbulent boundary layer. A complete analysis of the bifurcations and dynamical behavior of the low-dimensional models is presented and compared with variously truncated numerical simulations. Finally, in part four related works using the same general approach to jets, wakes, mixing layers as well as more complex geometries, such as the grooved channel, are reviewed. Somehow surprising is the fact that even for wakes there is no mentioning of an absolute instability.
The authors come to the conclusion that the empirical eigenfunctions obtained via proper orthogonal decomposition are very useful for identifying coherent structures in low-dimensional models and can lead to an improved understanding of issues like the bursting mechanism, but they lose their advantage as one goes to higher dimensions.