Settling of small particles near vortices and in turbulence

Abstract

The trajectories of small heavy particles in a gravitational field, having fall speed in still fluid V˜T and moving with velocity V˜ near fixed line vortices with radius R˜v and circulation Γ˜, are determined by a balance between the settling process and the centrifugal effects of the particles' inertia. We show that the main characteristics are determined by two parameters: the dimensionless ratio VT = V˜TR˜v/Γ˜ and a new parameter (Fp) given by the ratio between the relaxation time of the particle (t˜p) and the time (Γ˜/V˜2T) for the particle to move around a vortex when VT is of order unity or small.The average time ΔT˜ for particles to settle between two levels a distance Y˜0 above and below the vortex (where Y˜0 [Gt ] Γ˜/VT) and the average vertical velocity of particles 〈Vy〉L along their trajectories depends on the dimensionless parameters VT and Fp. The bulk settling velocity 〈Vy〉B = 2Y˜0/〈ΔT˜〉, where the average value of 〈ΔT˜〉 is taken over all initial particle positions of the upper level, is only equal to 〈V˜y〉L for small values of the effective volume fraction within which the trajectories of the particles are distorted, α = (Γ˜/V˜T)2/ Y˜20. It is shown here how 〈Vy〉B is related to Δ&η;(X˜0), the difference between the vertical settling distances with and without the vortex for particles starting on (X˜0, Y˜0) and falling for a fixed period Δt˜T [Gt ] &Γtilde;/V˜2T; 〈V˜y〉B = V˜T[1 − αD], where D = ∫∞−∞(Δ&η;dX˜0/ (&Γtilde;/V˜T)2) is the drift integral. The maximum value of 〈V˜〉B for any constant value of VT occurs when Fp = FpM ∼ 1 and the minimum when Fp = Fpm > FpM, where typically 3 < Fpm < 5.Individual trajectories and the bulk quantities D and 〈Vy〉B have been calculated analytically in two limits, first Fp → 0, finite VT, and secondly VT [Gt ] 1. They have also been computed for the range 0 < Fp < 102, 0 < VT < 5 in the case of a Rankine vortex. The results of this study are consistent with experimental observations of the pattern of particle motion and on how the fall speed of inertial particles in turbulent flows (where the vorticity is concentrated in small regions) is typically up to 80% greater than in still fluid for inertial particles (Fp ∼ 1) whose terminal velocity is less than the root mean square of the fluid velocity, ũ′, and typically up to 20% less for particles with a terminal velocity larger than ũ′. If V˜T/ũ′ > 4 the differences are negligible.