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Center for Nonlinear Studies |
The concept of equilibrium similarity (ES) was introduced almost 20 years ago to describe flows whose statistics evolve in space or time. Recent experiments, computations and theoretical advances confirm that the ideas appear to be applicable to a large number of flows, including homogeneous and perhaps even boundary layer flows. It is tempting to suggest that ES might be more than simply a hypothesis, but perhaps even a governing principle for fully-developed turbulence. The basic idea is that the properly normalized averaged equations (one-point and multipoint) constrain the flow into an equilibrium in which the surviving terms in the governing equations evolve in constant ratio (either in time or space). This is quite different, for example, than the traditional theoretical approach since Kolmogorov 1941, where some kind of equilibrium and universality of the statistics themselves is sought (or assumed), generally at the dissipative scales. Instead in ES, all scales are locked together, and the statistics is determined by whatever dynamical considerations are required by the equations themselves. Recent experiments and DNS of decaying homogeneous turbulence will be used to illustrate the ideas, including the permanent effect of initial conditions. The implications, if true, are profound. For example, it can be shown that the usual turbulence single point model dissipation equation is exact (for homogenous flows), but virtually useless since the coefficients depend on the initial conditions. By contrast, LES appears in principle to retain the necessary physics, almost independent of the closure model. Questions will also be raised about the relation between theories for flows of infinite extent, and experiments and simulations which of necessity are limited by finite boundaries. The former are essential for our understanding, but the latter are the world we have to compare with.