Objective quantification of Particle Pair Diffusion in Homogeneous Isotropic Turbulence

Turbulence consists of interacting flow structures covering a wide range of length and time scales. A long-standing question looms over pair diffusion of particles in close proximity i.e. particle pair diffusion at small separations: what range of turbulence length scales governs pair diffusion? Here, we attempt to answer this question by addressing pair diffusion by both fine scales and larger scale coherent structures in which the fine scales are embedded - we unavoidably encounter a combination of both local and non-local interactions associated with the small and large length scales. The local structures possess length scales of the same order of magnitude as the pair separation $l$, and they induce strong relative motion between the particle pair. However, the non-local structures, possessing length scales much larger than $l$, also induce (via Biot-Savart) significant relative motion, an effect ignored in prior studies (based on Richardson-Obukhov R-O theory). This fundamentally changes the interpretation of the pair diffusion process, giving the pair diffusivity $K$ scaling as $K\sim l^{1.556}$ -- agreeing within $1\%$ of data. The `R-O constant' $g_l$ is shown to be not a constant, although widely assumed to be a constant. However, new constants (representing pair diffusivity $G_K$ and pair separation $G_l$) are identified, which we show to asymptote to $G_K\approx 0.73$ and $G_l\sim 0.01$ at high Reynolds numbers. As an application, we show that the radius of a cloud of droplets in a spray is smaller by an order of magnitude as compared to R-O theory.

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