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) over tilde near fixed line vortices with radius (R) over tilde (v) and circulation <(<Gamma>)over tilde>, 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 V-T = (V) over tilde (T)(R) over tilde (v)/<(<Gamma>)over tilde> 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 of order unity or small. The average time Delta(T) over tilde for particles to settle between two levels a distance (Y) over tilde (0) above and below the vortex (where (Y) over tilde (0) much greater than <(<Gamma>)over tilde>/(V) over tilde (T)) and the average vertical velocity of particles <(V) over tilde > (L) along their trajectories depends on the dimensionless parameters V-T and F-p. The bulk 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 taken 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 within which the trajectories of the particles are distorted, alpha = (<(<Gamma>)over tilde>/(V) over tilde (T))(2)/(Y) over tilde (2)(0). It is shown here how <(V) over tilde > (B) is related to Delta<(<eta>)over tilde>((X) over tilde (0)), the difference between the vertical settling distances with and without the vortex for particles starting on ((X) over tilde (0), (Y) over 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)(-infinity)(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) over tilde (y)> (B) for any constant value of V-T occurs when F-p = F-pM similar 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) over tilde (T)/(u) over tilde ' > 4 the differences are negligible.