Unified theory of relative turbulent diffusion

Anomalous relative diffusion (pair separation) in the inertial, as well as in the viscous, subranges of fully developed turbulent flow is considered. Using the Lagrangian velocity-correlation function in lowest-order continued-fraction approximation based on the Navier-Stokes equations and a phenomenological closure assumption, we express the variance in terms of the static structure function. This closed form for the turbulent diffusivity unifies scaling concepts with fluid mechanics and is free of fitting parameters. We obtain the following conclusions: Various regimes of variance versus time $t$ are identified, $\propto t^2$ initially, $t^1$ and exponential (viscous subrange), $t^{3+\gamma }$ (inertial subrange). Intermittency (well known to alter the scaling exponent) implies a fairly strong effect upon the magnitude of the diffusion depending on the intial separation and on the Reynolds number; diffusion is delayed with increasing intermittency. Incompressibility is expected to lead to differences in transverse versus parallel separation. Molecular diffusivity may show up even in the universal regime as a permanent enhancement of diffusion due to different incubation times.