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Segregation of particles in incompressible random flows: singularities, intermittency and random uncorrelated motion
Lookup NU author(s)
Dr Rutgerus Ijzermans
Elena Meneguz
Emeritus Professor Mike Reeks
Author(s)
Ijzermans RHA, Meneguz E, Reeks MW
Publication type
Article
Journal
Journal of Fluid Mechanics
Year
2010
Volume
653
Issue
Pages
99-136
ISSN (print)
0022-1120
ISSN (electronic)
1469-7645
Full text is available for this publication:
Full text file 1
The results presented here are part of a long-term study in which we analyse the segregation of inertial particles in turbulent flows using the so called full Lagrangian method (FLM) to evaluate the ‘compressibility’ of the particle phase along a particle trajectory. In the present work, particles are advected by Stokes drag in a random flow field consisting of counter-rotating vortices and in a flow field composed of 200 random Fourier modes. Both flows are incompressible and, like turbulence, have structure and a distribution of scales with finite lifetime. The compressibility is obtained by first calculating the deformation tensor
J
ij associated with an infinitesimally small volume of particles following the trajectory of an individual particle. The fraction of the initial volume occupied by the particles centred around a position
x
at time
t
is denoted by |
J
|, where
J
≡ det(
J
ij) and
J
ij ≡
∂
x
i
(
x
0
, t
)
/∂x
0
,j
,
x
0 denoting the initial position of the particle. The quantity d_ln |
J
|_
/
d
t
is shown to be equal to the particle averaged compressibility of the particle velocity field _
∇ ・
v
_, which gives a measure of the rate-of-change of the total volume occupied by the particle phase as a continuum. In both flow fields the compressibility of the particle velocity field is shown to decrease continuously if the Stokes number
St
(the dimensionless particle relaxation time) is below a threshold value
St
cr
, indicating that the segregation of particles continues indefinitely. We show analytically and numerically that the long-time limit of _
∇ ・
v
_ for sufficiently small values of
St
is proportional to
St
2 in the flow field composed of random Fourier modes, and to
St
in the flow field consisting of counter-rotating vortices. If
St >St
cr
, however, the particles are ‘mixed’. The level of mixing can be quantified by the degree of random uncorrelated motion (RUM) of particles which is a measure of the decorrelation of the velocities of two nearby particles. RUM is zero for fluid particles and increases rapidly with the Stokes number if
St >St
cr
, approaching unity for
St
_ 1. The spatial averages of the higher-order moments of the particle number density are shown to diverge with time indicating that the spatial distribution of particles may be very intermittent, being associated with non-zero values of RUM and the occurrence of singularities in the particle velocity field. Our results are consistent with previous observations of the radial distribution function in Chun et al. (J. Fluid Mech., vol. 536, 2005, p. 219).
Publisher
Cambridge University Press
URL
http://dx.doi.org/10.1017/S0022112010000170
DOI
10.1017/S0022112010000170
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