The Decay of Energy and Scalar Variance in Axisymmetric Turbulence

We review the recent progress in our understanding of the large scales in homogeneous (but anisotropic) turbulence. We focus on turbulence which emerges from Saffman-like initial conditions, in which the vortices possess a finite linear impulse. Such turbulence supports long-range velocity correlations of the form < u(i)u(j)(')>= O(r(-3)), where u and u' are separated by a distance r, and these long-range interactions dominate the dynamics of large eddies. We show that, for axisymmetric turbulence, the energy and integral scales evolve as u(perpendicular to)(2)similar to u(//)(2)similar to t(-6/5) and l(perpendicular to)similar to l(//)similar to t(2/5), where perpendicular to and // indicate directions that are perpendicular and parallel to the symmetry axis, respectively. These predictions are consistent with the evidence of direct numerical simulations. Similar results are obtained for the passive scalar variance, where we find that similar to t(-6/5). The primary point of novelty in our discussion of passive scalar decay is that it is based in real (rather than spectral) space, making use of an integral invariant which is a generalization of the isotropic Corrsin integral.

Automated summary

In this article, we review the recent progress made in understanding large scales in homogeneous but anisotropic turbulence. We specifically focus on turbulence that arises from Saffman-like initial conditions, where vortices possess finite linear impulse. These turbulent flows support long-range velocity correlations and we discuss the evolution of energy, integral scales, and passive scalar variance using real space methods.