Diffuselet method in a rotating tilted cylinder

Known as the `Diffuselet’ method, a numerical scheme for calculating the evolution of a passive scalar field within a prescribed flow-field has recently been proposed (P. Meunier & E. Villermaux, J. Fluid Mech. 951, 2022). The method builds-up on Ranz transformation (Ranz, AiChE J., 1979) to couple advection with molecular diffusion, and accounts explicitly (and nearly exactly) for the stretching history of each individual fluid element. Thus, a complex scalar field is represented as a sum of independent Lagrangian tracers uniformly spread-out in the stirring volume.

We have applied this method to the flow created by a rotating cylinder whose axis is tilted with respect to gravity, a configuration which happens to be an efficient mixer (Meunier , J. Fluid Mech. 903, 2020) and whose non-trivial flow-flied is nevertheless described analytically.

There is a broad variability of the mixing characteristics (mixing time, concentration distribution) from one tracer to the other within the volume, depending on the initial location of the tracers: recirculating and highly divergent regions behave quite differently. We are therefore able to compute the overall statistics of the mixture in terms of the contributions of the different sub-regions of the flow, thereby explaining the origin of the broad tails of the concentration distribution in its route to uniformity. Eventually, diffuselets overlap diffusively, explaining again the nature of the ultimate mixture homogenization process.