Distribution of inertial particles in flows

This is the Batchelor spectrum, that is valid in the viscous-convective range that extends from the Kolmogorov scale down to the diffusive scale, where the scalar variance is finally dissipated by molecular diffusion. The diffusive scale in this case is the length scale at which the diffusion time l2/D is comparable to the timescale of advection corresponding to the Kolmogorov scale eddies, that gives 

The structure functions in this regime are logarithmic of the form 

Note that the inertial range and high Re are not necessary for the existence of this type of Batchelor scaling and it can be also produced by chaotic advection in spatially smooth unsteady large scale flows with relatively small or moderate Reynolds number, or in two-dimensional flows (Jullien et ah, 2000 Pierrehumbert, 2000). 

Particles of finite size or with a density different from that of the surrounding fluid (e.g. liquid droplets or dust particles suspended in a fluid), due to their inertia and non-vanishing size, have an instantaneous velocity that is somewhat different from the local velocity of the fluid. Therefore such inertial effects can have a significant influence on the distribution of suspended particles. If the Reynolds number based on the size of the particle and its velocity relative to the fluid is small, the flow around the particle can be approximated 

Another interesting effect of the particle inertia is that it can transform non-attracting chaotic sets into chaotic attractors. This has been shown by Benczik et al. (2002) who studied the motion of inertial particles in the time-periodic Karman vortex flow that produces transient chaotic advection of non-inertial particles, while inertial particles are trapped indefinitely in the wake indicating the presence of an attractor. 

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