Asymptotic analysis of homogeneous isotropic decaying turbulence with unknown initial conditions

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 become self-similar and is characterized by a power law decay of the turbulent kinetic energy with a decay exponent $n$. The value of this exponent however depends on the initial distribution of the velocity moments. In the non-dimensionalized form of the von K\'arm\'an-Howarth equation a decay exponent dependant term occurs whose coefficient will be called $\delta$. We exploit the fact that $\delta$ vanishes for $n=2$ to formulate a singular perturbation problem, where another small number in the equation, namely 1/4, is assumed to be of the same order as of magnitude as $\delta$. In the limit of infinitely large Reynolds numbers, we obtain an outer layer as well as an inner layer of the thickness of the order ${\cal O}(\delta^{\frac{3}{2}})$, where the Kolmogorov scaling is valid. To leading order, we obtain in the outer layer an algebraic balance between the two-point correlation and the third order structure function.