Rapid distortion theory and the ‘problems’ of turbulence

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(L) indicates why, in shear flows a t 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 dU,/dx, changes the form of the spectrum, so that, as /? = (tdU,/dx,) increases, over an increasing part of the spectrum defined in terms of the integral scale L by f ' >> k L , E (k) cc k-2, whatever the form of initial spectrum of E,(k) (provided E (k) = o(k-2) for kL B 1).

Intermittency in fractal Fourier hydrodynamics: Lessons from the Burgers equation

“…It is not unphysical to suppose that due to the power-law dependence on the wavenumber, a different weight between local and non-local interactions in Fourier space is introduced, making the latter more important then in the usual D = 1 case. Given all this, it is conceivable that an extra decorrelation time of the order of τ dec (k) ∝ 1/k appears, leading to a slow down of the energy transfer mechanisms, as is the case for Alfvén waves in MHD [23] or in the presence of a rapid distortion mechanism [24]. Typically, this leads to an estimate for the energy flux = kE(k)/τ tr (k), where the transfer time is given in terms of a golden mean between the eddy-turn-over time τ eddy (k) and the decorrelation time τ dec (k), τ tr (k) = τ 2 eddy /τ dec (k).…”

Refined Eddy Viscosity Schemes and Large Eddy Simulations for Ascending Mixed Convection Flows

“…(13) mimics some aspects of Rapid Distortion Theory (RDT), see for example Hunt and Carruthers [1990]. Eddy viscosity in the CI model is obtained as:…”

Rapid distortion theory for compressible homogeneous turbulence under isotropic mean strain

“…9 RDT is a linearized theory in which the time scale of the mean flow is assumed much smaller than the time scale of the turbulence at a given length scale. Based on this assumption, the nonlinear terms in the equations of motion can be neglected relative to the terms containing the mean deformation rate.…”