Linear Response Theory of Single Particle Diffusion in Turbulence

A theory is proposed for the statistics of single particle diffusion in stationary homogenous isotropic turbulence of incompressible fluid. The theory is based on a generalization of the idea of linear response theory that is known in the statistical mechanics for systems at or near the thermal equilibrium state. The theory gives $< |\Delta {\bf v}(t)|^2 >/(\epsilon t) = C_0+ B_1+ \cdots$, for the inertial time interval $t$ such that $ T \gg t \gg \tau$, where $\Delta {\bf v}(t)$ is the velocity increment of a particle during the time interval $t$, $\epsilon$ is the kinetic-energy dissipation rate per unit mass, $T$ is the integral time scale, $\tau$ is the Kolmogorov micro-time-scale, $B_1=C_{1T}(t/T)+C_{1\tau}(\tau/t)$, and $C_0,C_{1T}, C_{1\tau}$ are non-dimensional universal constants. $B_1$ represents the effect of small but finite $t/T$ and $\tau/t$. An examination of the theory by comparison with the data of direct numerical simulation (DNS) [B. L. Sawford and P. K. Yeung, Phys. Fluids 23, 091704 (2011)] suggests that the theory is in good agreement with the DNS.