| Inertial particle pair diffusion has received much less attention than fluid particle pair diffusion, even though it is arguably more relevant to real world applications, such as sand storms, and pollen dispersion. Only the DNS work of Bec et al [1] has been reported. A non-local theory of fluid particle pair diffusion has recently been proposed [2,3]; but the question is, can non-locality be extended to inertial particle pair diffusion? Here, we investigate it using Kinematic Simulations [4,5], in the limit of Stokes' drag where the transport is given by,
\begin{equation}
\frac{d{\bf x}}{dt}={\bf v}(t), &\qquad& \frac{d{\bf v}}{dt} = -\frac{1}{\tau}({\bf v}(t)-{\bf u})
\end{equation}
${\bf x}(t)$ is the particle position at time $t$, ${\bf v}(t)$ is the particle velocity, ${\bf u}({\bf x},t)$ is the Eulerian velocity field generated by the KS model, $\tau$ is the particle response time. The Stokes number is, $St=\tau/t_\eta$, where $t_\eta$ is the Kolmogorov time scale, $\sigma_l(t)=\langle l(t)^2\rangle^{1/2}$, where $l(t)=|{\bf x}_1(t)-{\bf x}_2(t)|$ is the distance between particles in a pair, in an ensemble of particle pairs released at time $t=0$ such that $l(t=0) =l_0 2/3$.
KS was used in a frame of reference moving with the (virtual) large scale sweeping velocities with spectrum, $E(k)\sim k^{-5/3}$, for $1\le k $\le$10^4$, and $E(k)=0$, for $k |