a Study of Low-Dimensional Models for the Wall Region of a Turbulent Boundary Layer

Fluid flow turbulence bas been a subject of intensive research over the past 100 years or more. Up until the last 15 years approaches to the problem of fluid turbulence have relied heavily on a statistical description of the phenomenon. Recently the advent of dynamical systems theory and a rapid increase in computing power has brought new approaches to the problem of turbulence. If a connection between the statistical description of turbulence and deterministic dynamics could be made, then fluid turbulence might be viewed not as an infinitely complicated process, but as one governed by a finite dimensional deterministic system with sensitive dependence on initial conditions. This, of course, is a far from straight forward proposition and there has been no work conclusively linking fully developed turbulent flows and low order dynamical systems. A projection method was formulated by Lumley (1967) in a version of the Karhunen-Loeve decomposition of statistics, which he called the proper orthogonal decomposition. The proper orthogonal decomposition capitalizes on the existence of large scale organized structures in the flow to create empirical eigenfunctions. Low order O.D.E.s are then derived that describe the evolution, interaction and dynamics of the structures. It is in the analysis of these strongly nonlinear O.D.E.s that dynamical systems theory joins with statistical methods to help solve the problem of fully developed turbulent fluid flow. This thesis addresses the application of the proper orthogonal decomposition to the wall region of a turbulent boundary layer, which was studied in Aubry et al. (1988). The effect of the addition of both streamwise and spanwise modes on the dynamics of the model is documented and analyzed, while, in particular, the presence of symmetry induced structurally stable heteroclinic cycles is qualified. The connection of these deterministic cycles to turbulent bursting in the wall layer is further elucidated by the derivation of a scaling law for burst frequency on inner and outer variables from the properties of the model, as well as an expression for the probability distribution of these time intervals.