LOW-DIMENSIONAL MODELS OF COHERENT STRUCTURES IN TURBULENCE

For fluid flow one has a well-accepted mathematical model: the Navier- Stokes equations. Why, then, is the problem of turbulence so intractab le? One major difficulty is that the equations appear insoluble in any reasonable sense. (A direct numerical simulation certainly yields a ' 'solution'', but it provides little understanding of the process per s e.) However, three developments are beginning to bear fruit: (1) The d iscovery, by experimental fluid mechanicians, of coherent structures i n certain fully developed turbulent flows; (2) the suggestion, by Ruel le, Takens and others, that strange attractors and other ideas from dy namical systems theory might play a role in the analysis of the govern ing equations, and (3) the introduction of the statistical technique o f Karhunen-Loeve or proper orthogonal decomposition, by Lumley in the case of turbulence. Drawing on work on modeling the dynamics of cohere nt structures in turbulent flows done over the past ten years, and con centrating on the near-wall region of the fully developed boundary lay er, we describe how these three threads can be drawn together to weave low-dimensional models which yield new qualitative understanding. We focus on low wave number phenomena of turbulence generation, appealing to simple, conventional modeling of inertial range transport and ener gy dissipation.