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This paper develops a spatial input-output approach to investigate the dynamics of a turbulent boundary layer subject to a localized single frequency excitation. This method uses one-way spatial integration to reformulate the problem in terms of spatial evolution equations. The technique is used to examine the effect of localized periodic actuation at a given temporal frequency, based on an experimental set-up in which an active large-scale is introduced into the outer layer of a turbulent boundary layer. First, the large-scale structures associated with the phase-locked modal velocity field obtained from spatial input-output analysis are shown to closely match those computed based on hot-wire measurements. The approach is then used to further investigate the response of the boundary layer to the synthetically generated large-scale. A quadrant trajectory analysis indicates that the spatial input-output response produces shear stress distributions consistent with those in canonical wall-bounded turbulent flows in terms of both the order and types of events observed. The expected correspondence between the dominance of different quadrant behavior and actuation frequency is also observed. These results highlight the promise of a spatial input-output framework for analyzing the formation and streamwise evolution of structures in actuated wall-bounded turbulent flows., Comment: 34 pages, 12 figures, preliminary work presented in AIAA Aviation 2021 with paper no. 2021-2873

Simple aerodynamic configurations under even modest conditions can exhibit complex flows with a wide range of temporal and spatial features. It has become common practice in the analysis of these flows to look for and extract physically important features, or modes, as a first step in the analysis. This step typically starts with a modal decomposition of an experimental or numerical dataset of the flow field, or of an operator relevant to the system. We describe herein some of the dominant techniques for accomplishing these modal decompositions and analyses that have seen a surge of activity in recent decades. For a non-expert, keeping track of recent developments can be daunting, and the intent of this document is to provide an introduction to modal analysis that is accessible to the larger fluid dynamics community. In particular, we present a brief overview of several of the well-established techniques and clearly lay the framework of these methods using familiar linear algebra. The modal analysis techniques covered in this paper include the proper orthogonal decomposition (POD), balanced proper orthogonal decomposition (Balanced POD), dynamic mode decomposition (DMD), Koopman analysis, global linear stability analysis, and resolvent analysis., Comment: 43 pages, 17 figures