Mathematics – Dynamical Systems
Scientific paper
Feb 1991
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1991phdt........78c&link_type=abstract
Thesis (PH.D.)--THE UNIVERSITY OF UTAH, 1991.Source: Dissertation Abstracts International, Volume: 52-04, Section: B, page: 223
Mathematics
Dynamical Systems
Self Organization, Chaos, Turbulence
Scientific paper
This dissertation presents a study of the correlation between self-organization and spectral entropy in transition flows. The spectral entropy algorithm is applied to three theoretical dynamical systems, to experimental data taken from an internal cavity shear layer and to a novel model describing transition in a wall boundary layer. The results show a decrease in spectral entropy when self-organization occurs. The spectral entropy is plotted against a control parameter for the three dynamical systems. In two of the systems, the logistic and Lorenz equations, the spectral entropy decreases across the initial bifurcation. This bifurcation marks the transition to a far from equilibrium regime where self-organization occurs. In the third dynamical system, a modified set of Townsend equations applied to a free shear layer, the spectral entropy decreases slightly downstream from the forward restrictor. Experimental observations confirm the presence of vortex structures in this region. By analyzing the experimental time-series data with spectral entropy techniques, one finds that coherent structures within the flow field contain a lower spectral entropy than the surrounding fluid. These structures are then considered more ordered than the encompassing field. The portion of the flow upstream of the forward restrictor lies within the linear nonequilibrium regime, and the region immediately downstream of the restrictor is the nonlinear nonequilibrium regime. The results show a decrease in the spectral entropy from the linear to the nonlinear regions, where spiral vortices are formed. As the vortices collapse, the spectral entropy increases. Finally, the spectral entropy is applied to a novel model of a wall boundary layer. The model is designed to understand the transition process for flow over a flat plate. A set of three complex nonlinear ordinary differential equations for the velocity fluctuations is obtained by reducing the turbulent Navier-Stokes equations through a Fourier decomposition and employing assumptions characteristic of a wall boundary layer. Phase space portraits help determine the behavior of the fluctuations. The results indicate an initiation of ordered structures at a Reynolds number below the accepted value of the critical Reynolds number.
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