Further insights into self-similarity and self-preservation in freely decaying isotropic turbulence

Abstract

An investigation of some open issues in homogeneous isotropic turbulence (HIT) decay is proposed in the present paper. More specifically, several apparent paradoxical or controversial issues, among which the existence of solutions satisfying complete self-similarity and complete self-preservation, are investigated via theoretical analysis and numerical results. The investigated range of Reynolds numbers encompasses values from Reλ = 10−4 to Reλ = 105, which are far beyond capabilities of existing wind tunnels and DNS. Thus, this analysis offers an almost unique opportunity to cover at the same time all possible decay regimes, from the high-Reynolds initial decay period to the low-Reynolds final decay period. The numerical analysis of such an extended range of Reλ is achieved by the use of the Eddy-Damped Quasi-Normal Markovian (EDQNM) model. Moreover, the high versatility of this model enabled a complete screening of HIT decay in terms of the initial conditions, and in particular the influence of the slope of the energy spectrum at large scales so that σ ∈ [1,4], E(k → 0) ∼ kσ has been extensively investigated. The analysis of the Lin equation shows that a complete self-similar decay regime occurs for σ = 1 only, and that all existing theories, including George’s theory, collapse in this case. Moreover, numerical results indicate that HIT does not lose memory of its initial conditions, i.e. no universal asymptotic behaviour derived by fixed-point analysis is recovered after very long time decay. As a confirmation, it is shown that a decay regime such that occurs for σ = 1 only, and from the initial time. Consequently, results of the fixed-point analysis of equations must not be considered as the evidence of a possible universal asymptotic behaviour, as they are valid in the sole case σ = 1.