The energy decay in self-preserving isotropic turbulence revisited

Abstract

The assumption of self-preservation permits an analytical determination of the energy decay in isotropic turbulence. Batchelor (1948), who was the first to carry out a detailed study of this problem, based his analysis on the assumption that the Loitsianskii integral is a dynamic invariant – a widely accepted hypothesis that was later discovered to be invalid. Nonetheless, it appears that the self-preserving isotropic decay problem has never been reinvestigated in depth subsequent to this earlier work. In the present paper such an analysis is carried out, yielding a much more complete picture of self-preserving isotropic turbulence. It is proven rigorously that complete self-preserving isotropic turbulence admits two general types of asymptotic solutions: one where the turbulent kinetic energy K ∼ t−1 and one where K ∼ t−α with an exponent α > 1 that is determined explicitly by the initial conditions. By a fixed-point analysis and numerical integration of the exact one-point equations, it is demonstrated that the K ∼ t−1 power law decay is the asymptotically consistent high-Reynolds-number solution; the K ∼ t−α decay law is only achieved in the limit as t → ∞ and the turbulence Reynolds number Rt vanishes. Arguments are provided which indicate that a t−1 power law decay is the asymptotic state toward which a complete self-preserving isotropic turbulence is driven at high Reynolds numbers in order to resolve an O(R1½) imbalance between vortex stretching and viscous diffusion. Unlike in previous studies, the asymptotic approach to a complete self-preserving state is investigated which uncovers some surprising results.