Rapid Distortion Theory and the structure of turbulence

Abstract

This is a review of how, when linear distortions are applied to turbulent velocity fields, certain changes to some or all components of the turbulence can be calculated using linear theory. Important examples of such distortions are mean and random straining motions, body forces, interactions with other flows (eg. waves). This theory is usually known as Rapid Distortion Theory (RDT) because it is valid for all kinds of rapidly changing turbulent flows (RCT), when the distortion is applied for a time (defined in a Lagrangian frame) T _D that is short compared to the ‘turn-over’ or decorrelation time scales T _L or τ _D (k) of the energy containing eddies or smaller eddies of scale k ^-l, respectively. However, for certain kinds of distortion the theory is also applicable to slowly changing turbulence (SCT) where T _D is of the order or greater than

New insight about the structure of slowly changing turbulence has been derived from RDT by considering different strain rates, initial conditions and time scales. RDT calculations show that in shear flows, whatever the initial form of the energy spectrum E(k) (provided it decreases with wavenumberk k faster than k ^-2 ) or of the anisotropy, for a long enough period of strain, E(k) always tends to the limiting form where it is proportional to k ^-2 for the small scales. Other statistical properties, such as ratios of Reynolds stresses, are also insensitive to the initial conditions. By contrast RDT shows how turbulent flows without mean strain or with irrotational strain are significantly more sensitive to initial and external conditions. These conclusions are consistent with those drawn from Direct Numerical Simulations (DNS) and experiments at moderate Reynolds numbers.

The eddy structure of small scale turbulence can be studied by calculating the distortion of random Fourier components in a velocity field caused by different kinds of large scale straining motions. This method is used here in conjunction with an analysis of how the different types of straining motions in different regions of the small-scale turbulence are affected by the distortion. Interestingly, linear analysis shows that the vorticity vector tends to become aligned with the middle eigenvector of the rate of strain tensor, which is consistent with DNS for turbulent flows having a continuous spectrum.

At sufficiently high values of the Reynolds number, even in homogeneous and in-homogeneous distorted turbulence (mean straining flows, body forces and boundaries), the nonlinear processes act over a wide enough spectrum to ensure that at small scales the energy spectrum E(k) has an approximately universal form (as proposed by Kolmogorov). However, at the same time different components of the spectrum have an anisotropic structure. This can be estimated by applying RDT to each wavenumber component of the spectrum and taking the time of distortion to be approximately equal to the turnover time t(k) appropriate to the value of k.