## Abstract

This paper is a study of the random motions of small, spherical inertial particles in spatially and temporarily random turbulent–like velocity fields. The results show how the diffusivity *D*^{p} of a small, dense particle depends on the structure of the flow field. The method is to use general, analytical or scaling arguments, model problems and numerical simulations. It is assumed that the particles are so small that the only significant forces are inertia, drag and body force, characterized by the relaxation time (*τ*_{p}) and the terminal velocity of the particle (V_{T}). It is shown that *only* if the flow is spatially non–uniform is the diffusivity of a solid particle *D*^{p} different from that of a fluid particle D^{f}. This small difference, when *τ*_{p} is small, is calculated in terms of how the *smallest scales* contribute to D^{f}and *D*^{f}. We construct an idealized one–dimensional model consisting of random sets of small ‘eddies’ of length *l* separated by a distance of order *s* with step–like velocity profiles with amplitude *u*_{0}(*l*), superimposed on large–scale eddy motions with length and velocity scales *L* and *U*_{0}, to show the effects of spatio–temporal structure of turbulence on the particle statistics. In order to examine the significance of the time dependence of the velocity field, firstly, the ‘null’ effect is studied; the velocity field is ‘frozen’ in time. It is found that the difference between the diffusivities (*D* ^{p} − *D* ^{f} ) for inertial and fluid particles, defined for this narrow range of eddy scales, is of the order of (*Ll*/*s*)*u*^{2}_{0}*τ*_{p}(for *τ*_{p} ≪ *l*/*U*_{0}).

Then two unsteady effects are considered.

(i) The small eddies are advected by the large–scale motion and move relative to it at a velocity of order *γU*_{0} (where γ≪ 1 *u*_{0}/*U*_{0}); this strength also varies over a time–scale *τ*_{e} (γ *l*/*U*_{0}). The result is that for larger eddies and smaller particle relaxation times *τ*_{p}⩽ *l*/*U*_{0}, *D*^{p} is greater than *D*^{f} by *β*(*L*/*s*)*u*^{2}_{0}*τ*_{p}, where 1 < *β* < *U*_{0}/*u*_{0}. This increased difference, compared with the frozen turbulence case, is caused by the slower passage of particles through the eddies.

(ii) The second unsteady effect is that of eddy decay, so that particles with ‘sufficient’ inertia do not have time to accelerate within the lifetime of the eddy. Then for *τ*_{p} ≳ *l*/*U*_{0}, *D*^{p} *less* than *D*^{f} by (*Ll*/*s*)*u*_{0}. By summing over the whole spectrum of eddies in high–Reynolds–number turbulence, the total diffusivities *D*^{p}, *D*^{f} are calculated. It is concluded that for small particle relaxation times (*τ*_{p} = *τ*_{p}*U*_{0} /*L* ≪ 1)_{u}, λ_{F} are coefficients for the unsteady and structured components. In frozen turbulence and in low–Reynolds–number turbulence without an inertial subrange, λ_{u} ≃ 0. These order–of–magnitude concepts are used to derive the inertial–range frequency spectra of the velocity of inertial particles *ϕ*^{pp} (*ω*) and of the fluid velocity of the locations of the particles *ϕ*^{fp} (*ω*) in high–Reynolds–number turbulence when *τ _{p}*≪ 1.

The predictions are tested and largely confirmed using the technique for kinematic simulation of inertial turbulence. In particular, the prediction is confirmed that *D*^{p}– D^{f} is negative for the light particle (*τ*≪ 1) as the magnitude of the unsteady component of the velocity field increases and is positive for particles with greater inertia (*τ*⩾ 1); in addition, it is shown that the coefficients λ_{u} ≃λ_{F} ≃ 1/3. The effects of particle settling under gravity are also calculated in the idealized model and in the numerical simulation. The results of all the predictions are consistent with the results of direct numerical simulations of particles in three–dimensional turbulence and with some experiments.

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