The Eddy-Diffusivity in Turbulent Two-Particle Dispersion

R. H. Kraichnan (1966) and T. S. Lundgren (1981) derived a formula for the eddy-diffusivity in Richardson's theory of turbulent 2-particle dispersion: $$\eta _{ij}({\bf r}, t)=\int ^{t}_{0} ds \langle(u_{i} ({\bf x} + {\bf r}, t) - u_{i} ({\bf x}, t)) (u_{i}({\bf x} + {\bf r}, t\vert s) - u_{j}({\bf x}, t \vert s)\rangle. $$ This formula involves the Lagrangian velocity field ${\bf u}({\bf x}, t \vert s)$ experienced at time $s < t$ by the fluid particle which is at point ${\bf x}$ at time t. Evaluating this formula requires tracking fluid particles backward in time, a difficult task with standard DNS. We compute the formula using the JHU Turbulence Database,\footnote{http://turbulence.pha.jhu.edu/} which stores the entire spacetime history of a 1024$^{3}$ DNS of homogeneous, isotropic turbulence at $Re_{\lambda }=433$. We average over particle pairs started at many different initial positions ${\bf x}$ with initial separations ${\bf r}$. We obtain a time-dependent eddy-diffusivity $\eta_{ij}({\bf r}, t)$ which has Batchelor scaling $(\varepsilon r)^{2/3}t$ for short time and Richardson scaling $\varepsilon^{1/3 }r^{4/3}$ for long time. Our resulting diffusion model describes both the Batchelor and Richardson regimes and also predicts new phenomena not yet seen in experiment or simulations.