Rapid distortion theory and the ‘problems’ of turbulence

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Abstract
The ‘problems’ associated with analysing different kinds of turbulent flow and different methods of solution are classified and discussed with reference to how the turbulent structure in a flow domain depends on the scale and geometry of the domain's boundary, and on the information provided in the boundary conditions. Rapid distortion theory (RDT) is a method, based on linear analysis, for calculating ‘rapidly changing turbulent’ (RCT) flows under the action of different kinds of distortion, such as large-scale velocity gradients, the effects of bounding surfaces, body forces, etc. Recent developments of the theory are reviewed, including the criteria for its validity, and new solutions allowing for the effects of inhomogeneities and boundaries.We then consider the contribution of RDT to understanding the fundamental problems of ‘slowly changing turbulent’ (SCT) flows, such as why are similar and persistent features of the local eddy structure found in different kinds of shear flow, and what are the general features of turbulent flows near boundaries. These features, which can be defined in terms of certain statistical quantities and flow patterns in individual flow realizations, are found to correspond to the form of particular solutions of RDT which change slowly over the time of the distortion. The most general, features are insensitive to the energy spectrum and to the initial anisotropy of the turbulence. A new RDT analysis of the energy spectra E(k) indicates why, in shear flows at moderate Reynolds number, the turbulence tends to have similar forms of spectra for eddies on a local scale, despite the Reynolds number not being large enough for the existence of a nonlinear cascade and there being no universal forms of spectra for unsheared turbulence; for this situation, the action of shear dU1/dx2 changes the form of the spectrum, so that, as β = (tdU1/dx2 increases, over an increasing part of the spectrum defined in terms of the integral scale L by β−1 [Gt ] kL, E(k) ∝ k−2, whatever the form of initial spectrum of E0(k) (provided E(k) = o(k−2) for kL [Gt ] 1).