Statistical representations of coherent structures in turbulent flow fields

The detection and measurement of coherent structures in turbulent flows rely heavily on flow visualization and conditional averaging. Conditional averaging requires specifying measurement conditions based on assumed properties of the coherent structures. It is difficult to determine the bias in these measurements caused by the choice of conditions. The shot-noise decomposition and the Karhunen-Loeve expansion are mathematical representations of turbulent fields have been proposed as methods of measuring coherent structures without using any conditions. These were used to analyze numerical data generated from a simple turbulent model, Burgers' equation with random forcing. Statistically homogeneous solutions were generated with periodic boundary conditions on a fixed interval. Inhomogeneous solutions were obtained with zero boundary conditions on a fixed interval. The inhomogeneous solutions were found to be nearly homogeneous in the interior of the interval.;The shot-noise decomposition for homogeneous random fields u(x) is written in one dimension as a convolution u = f*g, where f(x) is an eddy function which is distributed throughout the domain by the random scattering function g(x). The eddy functions obtained for the homogeneous solutions to Burgers' equation depended strongly on assumptions necessary to specify the scattering function. This makes the shot-noise decomposition equivalent to conditional averaging in requiring prior conditions to specify the coherent structures. A second representation, the conditional shot-noise decomposition, is developed which uses the scattering function g(x) as an indicator of the positions of the coherent structures. This defines a unique shot-noise representation for which the eddy function closely resembles the coherent structures in Burgers' equation.;The Karhunen-Loeve expansion for an inhomogeneous random field is a generalized Fourier expansion with basis functions obtained by solving a Fredholm integral equation whose kernel is the correlation. It was computed using the correlation of the inhomogeneous solutions to Burgers' equation. The results show that the expansion is similar to an ordinary Fourier expansion and requires approximately 20 terms to represent 95% of the energy of u(x). The basis functions in the expansion are sinusoidal in the interior of the interval where the solutions are nearly homogeneous.