On physical grounds there is no doubt that the Navier–Stokes equations provide an excellent model for fluid flow as long as shock waves are relatively thick (in terms of mean free paths), and in such conditions of temperature and pressure that we can regard the fluid as a continuum. The incompressible version is restricted, of course, to lower speeds and more moderate temperatures and pressures. There are some mathematical difficulties – indeed, we still lack a satisfactory existence-uniqueness theory in three dimensions – but these do not appear to compromise the equations' validity. Why then is the “problem of turbulence” so difficult? We can, of course, solve these nonlinear partial differential equations numerically for given boundary and initial conditions, to generate apparently unique turbulent solutions, but this is the only useful sense in which they are soluble, save for certain non-turbulent flows having strong symmetries and other simplifications. Unfortunately, numerical solutions do not bring much understanding.

However, three fairly recent developments offer some hope for improved understanding. (1) The discovery, by experimental fluid mechanicians, of coherent structures in certain fully developed turbulent flows, (2) the suggestion that strange attractors and other ideas from finite-dimensional dynamical systems theory might play a rôle in the analysis of the governing equations, and (3) the introduction of the statistical technique of Karhunen–Loéve or proper orthogonal decomposition. This book introduces these developments and describes how the three threads can be drawn together to weave low-dimensional models that address the rôle of coherent structures in turbulence generation.