3 results
Statistical properties of particle segregation in homogeneous isotropic turbulence
- Elena Meneguz, Michael W. Reeks
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- Journal:
- Journal of Fluid Mechanics / Volume 686 / 10 November 2011
- Published online by Cambridge University Press:
- 03 October 2011, pp. 338-351
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A full Lagrangian method (FLM) is used in direct numerical simulations (DNS) of incompressible homogeneous isotropic and statistically stationary turbulent flow to measure the statistical properties of the segregation of small inertial particles advected with Stokes drag by the flow. Qualitative good agreement is observed with previous kinematic simulations (KS) (IJzermans, Meneguz & Reeks, J. Fluid Mech., vol. 653, 2010, pp. 99–136): in particular, the existence of singularities in the particle concentration field and a threshold value for the particle Stokes number above which the net compressibility of the particle concentration changes sign (from compression to dilation). A further KS analysis is carried out by examining the distribution in time of the compression of an elemental volume of particles, which shows that it is close to Gaussian as far as the third and fourth moments but non-Gaussian (within the uncertainties of the measurements) for higher-order moments when the contribution of singularities in the tails of the distribution increasingly dominates the statistics. Measurements of the rate of occurrence of singularities show that it reaches a maximum at , with the distribution of times between singularities following a Poisson process. Following the approach used by Fevrier, Simonin & Squires (J. Fluid Mech., vol. 553, 2005, pp. 1–46), we also measured the random uncorrelated motion (RUM) and mesoscopic components of the compression for and show that the non-Gaussian highly intermittent part of the distribution of the compression is associated with the RUM component and ultimately with the occurrence of singularities. This result is consistent with the formation of caustics (Wilkinson et al. Phys. Fluids, vol. 19, 2007, p. 113303), where the activation of singularities precedes the crossing of trajectories (RUM).
Segregation of particles in incompressible random flows: singularities, intermittency and random uncorrelated motion
- RUTGER H. A. IJZERMANS, ELENA MENEGUZ, MICHAEL W. REEKS
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- Journal:
- Journal of Fluid Mechanics / Volume 653 / 25 June 2010
- Published online by Cambridge University Press:
- 13 April 2010, pp. 99-136
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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 Jij 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(Jij) and Jij ≡ ∂xi(x0, t)/∂x0,j, x0 denoting the initial position of the particle. The quantity d〈ln|J|〉/dt 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 Stcr, 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 St2 in the flow field composed of random Fourier modes, and to St in the flow field consisting of counter-rotating vortices. If St > Stcr, 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 > Stcr, 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).
On probability density function equations for particle dispersion in a uniform shear flow
- MICHAEL W. REEKS
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- Journal:
- Journal of Fluid Mechanics / Volume 522 / 10 January 2005
- Published online by Cambridge University Press:
- 13 January 2005, pp. 263-302
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The paper examines a fundamental discrepancy between two probability density function (PDF) models, the kinetic model (KM) and generalized Langevin model (GLM), currently used to model the dispersion of particles in turbulent flows. This discrepancy is manifest in particle dispersion in an unbounded simple shear flow where model predictions for the values of the streamwise fluid-particle diffusion coefficients are not only different but are of opposite sign. It is shown that this discrepancy arises through a neglect of the inertial convection term in the GLM equation for the mean carrier flow velocity local to a particle which eventually leads to algebraic forms for the particle-fluid diffusion coefficients. Evaluating this term for a Gaussian process leads to identical results for both PDF formulations. This also resolves a fundamental long-standing discrepancy in previous forms reported for the passive scalar diffusion coefficients in a simple shear flow where similar assumptions were made. Avoiding this assumption, the exact solutions are given for the dispersion of particles in this simple shear flow case derived from the solution of the GLM PDF equation which show explicitly the dependence on the particle response time and the strain rate, both normalized on the integral timescale of the turbulence. The analysis shows that the particle diffusion coefficient in the streamwise direction is negative when the strain rate $\,{\geq}\,$ a certain value. The origin of negative diffusion coefficients is explained and their influence is shown in the way in which the mean concentration and mean velocity flow fields of the particle and carrier flow (seen by the particle) evolve with time for particles released from the centre of the shear.