On The Hidden Beauty of the Proper Orthogonal Decomposition

Abstract

The proper orthogonal decomposition theorem (Loève (1955)) of probability theory has been proposed and adapted by Lumley (1967) for detecting spatial coherent patterns in turbulent flows. More specifically, the decomposition extracts deterministic functions from second order statistics of a random field and converges optimally fast in the quadratic mean (I.E. in energy). Aubry et al. (1990) showed that the technique can be made completely deterministic in the sense that it can be applied to (deterministic) spatially and temporally evolving flows. The remarkable property of this deterministic decomposition is not only in its optimal convergence (as emphasized previously in the literature) but also in its space/time symmetry which permits access to the spatio-temporal dynamics. We discuss here the implications of this property for fluid mechanics studies. The flow is decomposed into both spatial and temporalorthogonal modes which are coupled: each space component is associated with a time component partner. The latter is the time evolution of the former and the former is the spatial configuration of the latter. This generalizes the notion of spatial structures to that of temporal structures both of which can be followed through the various instabilities that the flow undergoes as the Reynolds number increases (in the first flow instability stages as well as in the fully developed turbulence). It also provides a nonlinear dynamics tool for spatio-temporal dynamical systems and can be used for bifurcation detection and analysis as well as dimension and degree of complexity estimates. Simple examples are given to illustrate the theory. In particular, we describe a Hopf bifurcation occurring in time showing how the system complexifies spatially and temporally complexifies. Finally, we describe how the tool presented in this paper is ideal for analysis and compaction of experimental data obtained by simultaneous measurements at multiple locations such as laser scanning techniques, image processing of visualization and particle image velocimetry

The full text of this paper is to appear inTheoretical & Computational Fluid Dynamics