Modal decomposition techniques are showing a fast growth in popularity for their wide range of applications and their various properties, especially as data-driven tools. There are many modal decomposition techniques, yet Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) are the most widespread methods, especially in the field of fluid dynamics. Following their highly competent performance on various applications in several fields, numerous extensions of these techniques have been developed. In this work, we present an ambitious review comparing eight different modal decomposition techniques, including most established methods, i.e., POD, DMD, and Fast Fourier Transform; extensions of these classical methods: based either on time embedding systems, Spectral POD and Higher Order DMD, or based on scales separation, multi-scale POD (mPOD) and multi-resolution DMD (mrDMD); and also a method based on the properties of the resolvent operator, the data-driven Resolvent Analysis. The performance of all these techniques will be evaluated on four different test cases: the laminar wake around cylinder, a turbulent jet flow, the three-dimensional wake around a cylinder in transient regime, and a transient and turbulent wake around a cylinder. All these mentioned datasets are publicly available. First, we show a comparison between the performance of the eight modal decomposition techniques when the datasets are shortened. Next, all the results obtained will be explained in detail, showing both the conveniences and inconveniences of all the methods under investigation depending on the type of application and the final goal (reconstruction or identification of the flow physics). In this contribution, we aim at giving a—as fair as possible—comparison of all the techniques investigated. To the authors' knowledge, this is the first time a review paper gathering all these techniques have been produced, clarifying to the community what is the best technique to use for each application.

In this contribution, we consider the Dynamic Mode Decomposition (DMD) framework as a purely data-driven tool to investigate both standard and actuated turbulent channel databases via Direct Numerical Simulation (DNS). Both databases have comparable Reynolds number Re ≈ 3600. The actuation consists in the imposition of a streamwise-varying sinusoidal spanwise velocity at the wall, known to lead to drag reduction. Specifically, a composite-based DMD analysis is conducted, with hybrid snapshots composed by skin friction and Reynolds stresses. A small number of dynamic modes (∼3–9) are found to recover accurately the DNS Reynolds stresses near walls. Moreover, the DMD modes retrieved propagate at a range of phase speeds consistent with those reported in the literature. We conclude that composite DMD is an attractive, purely data-driven tool to study turbulent flows. On the one hand, DMD is helpful to identify features associated with the drag, and on the other hand, it reveals the changes in flow structure when actuation is imposed.

Dynamic Mode Decomposition (DMD) techniques have risen as prominent feature identification methods in the field of fluid dynamics. Any of the multiple variables of the DMD method allows to identify meaningful features from either experimental or numerical flow data on a data-driven manner. Performing a DMD analysis requires handling matrices $\mathbf{V}\in {\mathbb{R}}^{n_p\times N}$ , where $n_p$ and N are indicative of the spatial and temporal resolutions. The DMD analysis of a complex flow field requires long temporal sequences of well resolved data, and thus the memory footprint may become prohibitively large. In this contribution, the effect that principled spatial agglomeration (i.e., reduction in $n_p$ via clustering) has on the results derived from the DMD analysis is investigated. We compare twelve different clustering algorithms on three testcases, encompassing different flow regimes: a synthetic flow field, a $R{e}_D=60$ flow around a cylinder cross section, and a $R{e}_{\tau }\approx 200$ turbulent channel flow. The performance of the clustering techniques is thoroughly assessed concerning both the accuracy of the results retrieved and the computational performance. From this assessment, we identify DBSCAN/HDBSCAN as the methods to be used if only relatively high agglomeration levels are affordable. On the contrary, Mini-batch K-means arises as the method of choice whenever high agglomeration $\widetilde{n_p}/n_p\ll 1$ is possible.