Scaling properties of particle density fields formed in simulated turbulent flows

Direct numerical simulations of particle concentrations in fully developed three-dimensional turbulence were carried out in order to study the nonuniform structure of the particle density field. Three steady-state turbulent fluid fields with Taylor microscale Reynolds numbers $\left({\mathrm{Re}}_{\lambda }\right)$ of 40, 80, and 140 were generated by solving the Navier-Stokes equations with pseudospectral methods. Large-scale forcing was used to drive the turbulence and maintain temporal stationarity. The response of the particles to the fluid was parametrized by the particle Stokes number St, defined as the ratio of the particle’s stopping time to the mean period of eddies on the Kolmogorov scale $\left(\eta \right).\mathrm{}$ In this paper, we consider only passive particles optimally coupled to these eddies $\left(\mathrm{St}\approx 1\right)$ because of their tendency to concentrate more than particles with lesser or greater St values. The trajectories of up to $70\times {10}^6$ particles were tracked in the equilibrated turbulent flows until the particle concentration field reached a statistically stationary state. The nonuniform structure of the concentration fields was characterized by the multifractal singularity spectrum $f\left(\alpha \right),$ derived from measures obtained after binning particles into cells ranging from $2\eta$ to $15\eta$ in size. We observed strong systematic variations of $f\left(\alpha \right)$ across this scale range in all three simulations and conclude that the particle concentration field is not statistically self-similar across the scale range explored. However, spectra obtained at the $2\eta ,$ $4\eta ,$ and $8\eta$ scales of each flow case were found to be qualitatively similar. This result suggests that the local structure of the particle concentration field may be flow independent. The singularity spectra found for $2\eta$-sized cells were used to predict concentration distributions in good agreement with those obtained directly from the particle data. This singularity spectrum has a shape similar to the analogous spectrum derived for the inertial-range energy dissipation fields of experimental turbulent flows at ${\mathrm{Re}}_{\lambda }=110\mathrm{}$ and 1100. Based on this agreement, and the expectation that both dissipation and particle concentration are controlled by the same cascade process, we hypothesize that singularity spectra similar to the ones found in this work provide a good characterization of the spatially averaged statistical properties of preferentially concentrated particles in higher ${\mathrm{Re}}_{\lambda }$ turbulent flows.