Stochastic Estimation of the Structure of Turbulent Fields
The stochastic estimation method educes structure by approximating an average field in terms of event data that are given. The estimated fields satisfy the continuity equation, and they possess the correct scales of length and/or time. The fundamental con
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R.J. Adrian University of IIlinois, Urbana, IL, USA
ABSTRACT The stochastic estimation method educes structure by approximating an average field in terms of event data that are given. The estimated fields satisfy the continuity equation, and they possess the correct sc ales of length and/or time. The fundamental concepts of general stochastic estimation and the specific application of this technique to the estimation of conditional averages are discussed. Linear stochastic estimation of random fields and of their conditional averages is developed as the principal tool, and its accuracy is demonstrated. The
linear stochastic estimate is expressible in terms of second order correlation functions between the given event data and the quantity being estimated. This establishes a simple link between conditional averages, the coherent structure that they represent and correlation functions. The related problems of selecting events and interpreting the estimates that result from a given set of events are explored by considering events of increasing complexity: single-point vectors, two-point vectors, local deformation tensors, multi-point vectors, space-time vectors, and space-wave-number events. General kinematic and statistical properties are derived, and stochastically estimated structures from various types of turbulent flows are described and related to the underlying coherent structures.
1. INTRODUCTION 1.1 Estimation of random processes In general, stochastic estimation deals with the estimation of one random variable, say y, in terms of a set of other variables EJ, ... ,EN about which some information is known. We shall denote this set as the data vector E= (EJ, ... ,EN) . The data vector E may
J. P. Bonnet (ed.), Eddy Structure Identification © Springer-Verlag Wien 1996
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RJ. Adrian
or may not contain all of the variables that affect y, and it may even contain some that are irrelevant, but in any case it represents a list of the available information that is thought to be important. The problem is to find a function F(E) that approximates y in some sense. Estimating a quantity in terms of other parameters is, of course, ubiquitous in engineering and science, and for this reason stochastic estimation is a weIl known process, especially in communications and information theory-related disciplines [Papoulis 1984], [Deutsch 1965]. 1t also arises routinely in our daily lives, as for example when one makes the decision to walk across astreet based on an estimate of the arrival time of the approaching automobiles in terms of visual image data and engine sounds, or when one estimates the future weather in terms of current conditions. A common element in stochastic estimation methods is that they employ information on the relationships between the variables that is based largely on empirical experience as embodied in a statistical relationship between y and E. This should be contrasted to mathematical models in which empirical information is ultimately condensed into the form of the governing differential equatio
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