It is difficult to mix solutions in microchannels. Under typical operating conditions, flows in these channels are laminar—the spontaneous fluctuations of velocity that tend to homogenize fluids in turbulent flows are absent, and molecular diffusion across the channels is slow. We present a passive method for mixing streams of steady pressure-driven flows in microchannels at low Reynolds number. Using this method, the length of the channel required for mixing grows only logarithmically with the Péclet number, and hydrodynamic dispersion along the channel is reduced relative to that in a simple, smooth channel. This method uses bas-relief structures on the floor of the channel that are easily fabricated with commonly used methods of planar lithography.

This article reviews theory and applications of Koopman modes in fluid mechanics. Koopman mode decomposition is based on the surprising fact, discovered in Mezić (2005) , that normal modes of linear oscillations have their natural analogs—Koopman modes—in the context of nonlinear dynamics. To pursue this analogy, one must change the representation of the system from the state-space representation to the dynamics governed by the linear Koopman operator on an infinite-dimensional space of observables. Whereas Koopman in his original paper dealt only with measure-preserving transformations, the discussion here is predominantly on dissipative systems arising from Navier-Stokes evolution. The analysis is based on spectral properties of the Koopman operator. Aspects of point and continuous parts of the spectrum are discussed. The point spectrum corresponds to isolated frequencies of oscillation present in the fluid flow, and also to growth rates of stable and unstable modes. The continuous part of the spectrum corresponds to chaotic motion on the attractor. A method of computation of the spectrum and the associated Koopman modes is discussed in terms of generalized Laplace analysis. When applied to a generic observable, this method uncovers the full point spectrum. A computational alternative is given by Arnoldi-type methods, leading to so-called dynamic mode decomposition, and I discuss the connection and differences between these two methods. A number of applications are reviewed in which decompositions of this type have been pursued. Koopman mode theory unifies and provides a rigorous background for a number of different concepts that have been advanced in fluid mechanics, including global mode analysis, triple decomposition, and dynamic mode decomposition.

A majority of methods from dynamical system analysis, especially those in applied settings, rely on Poincaré's geometric picture that focuses on “dynamics of states.” While this picture has fueled our field for a century, it has shown difficulties in handling high-dimensional, ill-described, and uncertain systems, which are more and more common in engineered systems design and analysis of “big data” measurements. This overview article presents an alternative framework for dynamical systems, based on the “dynamics of observables” picture. The central object is the Koopman operator: an infinite-dimensional, linear operator that is nonetheless capable of capturing the full nonlinear dynamics. The first goal of this paper is to make it clear how methods that appeared in different papers and contexts all relate to each other through spectral properties of the Koopman operator. The second goal is to present these methods in a concise manner in an effort to make the framework accessible to researchers who would like to apply them, but also, expand and improve them. Finally, we aim to provide a road map through the literature where each of the topics was described in detail. We describe three main concepts: Koopman mode analysis, Koopman eigenquotients, and continuous indicators of ergodicity. For each concept, we provide a summary of theoretical concepts required to define and study them, numerical methods that have been developed for their analysis, and, when possible, applications that made use of them. The Koopman framework is showing potential for crossing over from academic and theoretical use to industrial practice. Therefore, the paper highlights its strengths in applied and numerical contexts. Additionally, we point out areas where an additional research push is needed before the approach is adopted as an off-the-shelf framework for analysis and design.

We establish the convergence of a class of numerical algorithms, known as dynamic mode decomposition (DMD), for computation of the eigenvalues and eigenfunctions of the infinite-dimensional Koopman operator. The algorithms act on data coming from observables on a state space, arranged in Hankel-type matrices. The proofs utilize the assumption that the underlying dynamical system is ergodic. This includes the classical measure-preserving systems, as well as systems whose attractors support a physical measure. Our approach relies on the observation that vector projections in DMD can be used to approximate the function projections by the virtue of Birkhoff's ergodic theorem. Using this fact, we show that applying DMD to Hankel data matrices in the limit of infinite-time observations yields the true Koopman eigenfunctions and eigenvalues. We also show that the singular value decomposition, which is the central part of most DMD algorithms, converges to the proper orthogonal decomposition of observables. We use...