Resumen de Empirical eigenfunctions: applications in unsteady aerodynamics
The main aim of modal decompositions is to obtain a set of functions which can describe in a compact way the variability contained in a set of observables/data. While this can be easily obtained by means of the eigenfunctions of the operator from which the observables depends, theempirical eigenfunctions allow to obtain a similar result from a set of data, without the knowledge of the problem operator.
In Fluid Mechanics and related sciences one of the most relevant techniques to obtain \emph{empirical eigenfunctions} is referred to as Proper Orthogonal Decomposition (POD).
This thesis contains several novel applications of the empirical eigenfunctions related to the field of (Experimental) Aerodynamics. The mathematical framework of the POD is introduced following the bi-orthogonal approach by Aubry (1991). The mathematical derivation of the POD is given, wherever possible, in the most general framework, without bounding it to the decomposition of a specific quantity. This choice of the author depends on the variety of POD applications which are included in this manuscript, ranging from signal processing problems to applications more strictly related with the flow physics.
The mathematical framework includes also one of the POD extensions, the Extended POD (EPOD), which allows to extract modes linearly correlated to the \emph{empirical eigenfunctions} of a second quantity.
The first two applications of the empirical eigenfunctions are strictly connected with the signal treatment in experimental techniques for Fluid Mechanics. In Chapter 3, the empirical eigenfunctions are identified as an optimal basis in which perform a "low-pass" spectral filter of the fluid mechanics field measurements, such as velocity fields measured with Particle Image Velocimetry (PIV). This filtering is extremely beneficial to reduce the random errors contained in the PIV fields and obtain a more accurate estimate of derivative quantities (such as, for instance, vorticity), which are more affected by random errors. In Chapter 4 the POD is exploited for the pre-treatment of a sequence of PIV images. The aim is to remove background and reflections which are sources of uncertainty in PIV measurements. In this case a "high-pass" spectral filtering is applied to the PIV image ensemble in order to remove the highly-coherent part of the signal corresponding to the background.
In the third and fourth applications, the POD is applied to recover the underlying dynamics of a flow.
More specifically, in Chapter 5 the POD is applied to the complex wake of a pair of cylinders in tandem arrangement with the additional perturbation of the wall proximity. Through this technique it is possible to track the changes in the oscillatory behaviour of the wake instabilities ascribed to different geometrical configurations of the cylinders.
In Chapter 6 the POD and the EPOD are applied respectively to the flow fields around an airfoil in plunging and pitching motion and to the unsteady aerodynamic forces acting on the airfoil. The decomposition allows to extract a reduced set of modes of the flow field which are related to the force generation mechanism. These modes correspond to well-recognizable phenomena of the flow which can be identified for a wide variety of airfoil kinematics. The force associated with these flow field modes can be related to force models already present in literature, thus enabling their reinterpretation.
The final application is devoted to overcome the low temporal resolution of typical flow field measurements, such as PIV, by proposing a robust estimation of turbulent flows dynamics. The method employs a modified version of the EPOD to identify the correlation between a non-time-resolved field measurement and a time-resolved point measurement. The estimation of the time-resolved flow fields is obtained exploiting the correlation of the flow fields with the temporal information contained in the point measurements.
