Velocity correlation considered as a 'scale average' in a hierarchical particle model of turbulence: Part I. Unbounded hierarchy

An analysis of the average of sets of 'correlation' functions, which are identical in shape but whose arguments are linearly scaled is pursued in the context of a model of inertial range turbulence based on the kinematics of the compactly-supported impulse density variable. In the limit of scale continuum, this averaging procedure implies a k-2 ln k behaviour for large wave number k in the energy spectrum. We draw attention, in both one- and three-dimensional contexts, to the numerical resemblance of this spectrum to a k-5/3 power law. Furthermore, we examine the spectrum which arises for a shape function intended to model that typically encountered in measured data. We discuss the possible implications of this in understanding the inertial range turbulence.