Settling of small particles near vortices and in turbulence

The trajectories of small heavy particles in a gravitational field, having fall-Speed in still fluid (V) over tilde (T) and moving with velocity (V) o ver tilde near fixed line vortices with radius (R) over tilde (v) and circu lation <(<Gamma>)over tilde>, are determined by a balance between the settl ing process and the centrifugal effects of the particles' inertia. We show that the main characteristics are determined by two parameters: the dimensi onless ratio V-T = (V) over tilde (T)(R) over tilde (v)/<(<Gamma>)over tild e> and a new parameter (F-p) given by the ratio between the relaxation time of the particle (t) over tilde (p)) and the time (<(<Gamma>)over tilde>/(V ) over tilde (2)(T)) for the particle to move around a vortex when V-T is o f order unity or small. The average time Delta(T) over tilde for particles to settle between two le vels a distance (Y) over tilde (0) above and below the vortex (where (Y) ov er tilde (0) much greater than <(<Gamma>)over tilde>/(V) over tilde (T)) an d the average vertical velocity of particles <(V) over tilde > (L) along th eir trajectories depends on the dimensionless parameters V-T and F-p. The b ulk settling velocity <(V) over tilde > (B) = 2 (Y) over tilde (0)/< Delta( T) over tildeT >, where the average value of < Delta(T) over tilde > is tak en over all initial particle positions of the upper level, is only equal to <(v) over tilde > (L) for small values of the effective volume fraction wi thin which the trajectories of the particles are distorted, alpha = (<(<Gam ma>)over tilde>/(V) over tilde (T))(2)/(Y) over tilde (2)(0). It is shown h ere how <(V) over tilde > (B) is related to Delta<(<eta>)over tilde>((X) ov er tilde (0)), the difference between the vertical settling distances with and without the vortex for particles starting on ((X) over tilde (0), (Y) o ver tilde (0)) and falling for a fixed period Delta(T) over tilde (T) much greater than <(<Gamma>)over tilde>/(V) over tilde (T)(2); <(V) over tilde > (B) = (V) over tilde (T) [1 - alphaD], where D = integral (infinity)(-infi nity)(Delta<(<eta>)over tilde>d (X) over tilde (0)/(<(<Gamma>)over tilde>/( V) over tilde/ (T))(2)) is the drift integral. The maximum value of <(V) ov er tilde (y)> (B) for any constant value of V-T occurs when F-p = F-pM simi lar to 1 and the minimum when F-p = F-p > F-pM, where typically 3 < F-pm < 5. Individual trajectories and the bulk quantities D and (V) over tildey > (B) have been calculated analytically in two limits, first F-p --> 0, finite V -T, and secondly V-T much greater than 1. They have also been computed for the range 0 < F-p < 10(2), 0 < V-T < 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 ( F-p similar to 1) whose terminal velocity is less than the root mean square of the fluid velocity, (u) over tilde ', and typically up to 20% less for particles with a terminal velocity larger than (u) over tilde '. If (V) ove r tilde (T)/(u) over tilde ' > 4 the differences are negligible.