Asymptotic analysis of homogeneous isotropic decaying turbulence with unknown initial conditions

Abstract

In decaying grid turbulence there is a transition from the initial state immediately behind the grid to the state of fully developed turbulence downstream, which is believed to be self-similar. This state is characterized by a power law decay of the turbulent kinetic energy with a time-independent decay exponent n. The value of this exponent, however, depends on the initial distribution of the velocity, about which we have only general information at the very best. For homogeneous isotropic decaying turbulence, the evolution of the two-point velocity correlation is described by the von Kármán–Howarth equation. In the non-dimensionalized form of this equation a decay exponent dependent term occurs, whose coefficient will be called δ. We exploit the fact that δ vanishes for n→2, which is shown to correspond to the limit d→∞, where d denotes the dimensionality of space to formulate a singular perturbation problem. It is shown that a distinguished limit exists for d→∞ and δ→0. We obtain in the limit of infinitely large Reynolds numbers an outer layer of limited, but a priori unknown extension, as well as an inner layer of the thickness of the order , where the Kolmogorov scaling is valid. To leading order, we obtain an algebraic balance in the outer layer between the two-point correlation and the third-order structure function. In the inner layer the analysis yields the emergence of higher order terms to the classical K41 scaling. All leading order solutions are shown to be subject to a band of uncertainty of the order , which is argued to be due to intrinsically unknown initial conditions.