On the motion of suspended particles in stationary homogeneous turbulence

Abstract

The closed equations for the velocity correlation tensor and for the mean-squared displacement of a particle suspended in a stationary homogeneous turbulent flow, with an arbitrary linear law of fluid-particle interaction, are obtained using two assumptions suggested previously for the problem of turbulent self-diffusion: the ‘independence approximation’ and the Gaussian property of the functional distribution of particle velocities. The numerical solution of the derived equations is given for an isotropic system with a model turbulence spectrum. The following characteristics of the particle motion are obtained: (a) the mean kinetic energy, (b) diffusivity, (c) rate of energy dissipation, (d) velocity correlation function, and (e) the correlation function of the relative fluid-particle velocity. The impact of various spectral modes on the characteristics of the particle motion is discussed.

Transport properties of heavy particles in high Reynolds number turbulence

“…(15), neglecting the conditioning on u F would mean that any correlation among velocity increments along fluid trajectories (longitudinal) and from trajectory separation (transverse), is disregarded. (It is to be mentioned that, contrary to the prediction of [4,6], the renormalization of the turbulent correlation time, necessary to the fluid velocity amplitude constant, leads to a decrease rather than to an increment of τ L ). Notice that approximating spatial correlations with exponentials [10,13] is not sufficient to cure this problem; the only gain is eliminating the spurious noise dependence on the discretization ∆.…”

“…An Eulerian time-statistics would suggest a faster fluid velocity variation in a Lagrangian frame [25], which is the physical motivation of the results obtained in [4,6]. Following the same argument, a time-statistics given along trajectories would lead to the fixed particles feeling the fastest velocity variation.…”

“…It is to be noticed that the renormalization of C 0 is negative in all the inertia dominated situations, with the exception of the τ S ≥ 1 cases, in which the linear spectrum is essentially absent. A negative renormalization of C 0 corresponds to an inhibition of the time variation for u(t), which is akin to the increase of the correlation time for u(t), predicted in [4,6] in the case of a purely large scale flow. It is not clear, however, how much these phenomena depend upon modelling assumptions.…”

“…This problem, associated with the small turbulent scales being seen as frozen by the drifting particle, was not present in [4,5], who dealt essentially with a turbulent flow with only large scales present. The same turbulent regime was considered in [6], based on an expansion around the case of an infinite inertia particle.…”

Lagrangian statistics for fluid particles and bubbles in turbulence

“…Later, analytical investigations have been carried out for heavy particles that, owing to inertia, do not follow all the turbulence fluctuations [4,5], and, most recently, also for air bubbles [6]- [8].…”