Gahl Berkooz

Turbulence, Coherent Structures, Dynamical Systems and Symmetry: Proper orthogonal decomposition

The proper orthogonal decomposition (POD) provides a basis for the modal decomposition of an ensemble of functions, such as data obtained in the course of experiments. Its properties suggest that it is the preferred basis to use in various applications. The most striking of these is optimality : it provides the most efficient way of capturing the dominant components of an infinite-dimensional process with only finitely many, and often surprisingly few, “modes.” The POD was introduced in the context of turbulence by Lumley in. In other disciplines the same procedure goes by the names: Karhunen–Loeve decomposition, principal components analysis, singular systems analysis, and singular value decomposition. The basis functions it yields are variously called: empirical eigenfunctions, empirical basis functions, and empirical orthogonal functions. According to Yaglom (see), the POD was introduced independently by numerous people at different times, including Kosambi, Loeve, Karhunen, Pougachev, and Obukhov. Lorenz, whose name we have already met in another context, suggested its use in weather prediction. The procedure has been used in various disciplines other than fluid mechanics, including random variables, image processing, signal analysis, data compression, process identification and control in chemical engineering, and oceanography. Computational packages based on the POD are now becoming available. In the bulk of these applications, the POD is used to analyse experimental data with a view to extracting dominant features and trends: coherent structures.

Viewpoint: Steven Ray on Information Standards in Manufacturing

Steven R. Ray, Chief Manufacturing Systems Integration Division at the US National Institute of Standards and Technology to give his perspectives on the changing role of Information Standards in Manufacturing. As the person directing the national US response in the area of information standards for manufacturing, Steve enjoys a unique vantage point on the field. Steve notes that our times are characterised by an increase in the importance of mastering information standards as a competitive enabler. However, the scope of activities covered by information standards is growing, making it harder for companies to stay abreast of all the developments and opportunities. Steves organisation at NIST plays a key role in assuring that US manufacturers can efficiently incorporate information standards for manufacturing into their development plans.

Randomly perturbed systems

As we have described in Part I, attempts to build low-dimensional models of truly turbulent processes are likely to involve averaging or, more generally, modelling to account for neglected modes that are dynamically active in the sense that their states cannot be expressed as an algebraic function of the modes included in the model. Such models are in turn likely to involve probabilistic elements. Here, “neglected modes” may refer to (high wavenumber) modes in the inertial and dissipative ranges or to mid-range, active modes whose wavenumbers might be linearly unstable. They also may refer to spatial locations that are omitted, in selecting a subdomain of a large or infinite physical spatial extent. The boundary layer model of Chapter 9, for example, contains a forcing term representing a pressure field, unknown a priori , imposed on the outer edge of the wall region. While estimates of this term can be obtained from direct numerical simulations (e.g.), a natural simplification is to replace it with an external random perturbation of suitably small magnitude and appropriate power spectral content. More generally, many processes modelled by non-linear differential equations involve random effects, in either multiplicative form (coefficient variations) or additive form, and it is therefore worth making a brief foray into the field of stochastic dynamical systems to sample some of the tools available. In this chapter we give a very selective and cursory description of how one can analyse the effect of additive white noise on a system linearised near an equilibrium point.