Toward a turbulent constitutive relation

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Abstract
In an attempt to explain the failure of the various pure homogeneous strain experiments to reach equilibrium (and consequently to support the contention of Townsend of an equilibrium structure of the Reynolds stress dependent only on geometry), the nature of the general Reynolds stress-mean velocity relation is examined. It is shown that if homogeneous flows become asymptotically independent of initial conditions, and if the Reynolds stress bearing structure can be characterized by a single time scale (i.e.–at sufficiently high Reynolds number) then these flows behave like classical non-linear viscoelastic media, with the Reynolds stress structure dependent on the (strain-rate) (time scale) product. Thus, the existence of an equilibrium structure implies the existence of an equilibrium time scale and a universal value of the product. The ideas permitting Reynolds stress and mean velocity to be related are applied to the dissipative structure in homogeneous flows, and it is found that in such flow the time scale never ceases to grow, so that these flows can never reach an equilibrium structure. With the aid of an ad-hoc assumption these flows are examined in some detail, and the results of experiments are predicted with considerable accuracy. It is suggested that (inhomogeneous) flows having an equilibrium time scale may, in the homogeneous limit, be expected to display a universal structure. The small departure from universality induced by the large eddies associated with inhomogeneity may be adequately predicted by this same ad-hoc model.

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