Particles In Homogeneous Isotropic Turbulence Research Articles

An efficient parallel simulation of interacting inertial particles in homogeneous isotropic turbulence

This study has conducted parallel simulations of interacting inertial particles in statistically-steady isotropic turbulence using a newly-developed efficient parallel simulation code. Flow is computed with a fourth-order finite-difference method and particles are tracked with the Lagrangian method. A binary-based superposition method has been developed and implemented in the code in order to investigate the hydrodynamic interaction among many particles. The code adopts an MPI library for a distributed-memory parallelization and is designed to minimize the MPI communication, which leads to a high parallel performance. The code has been run to obtain collision statistics of a monodisperse system with St=0.4 particles, where St is the Stokes number representing the particle relaxation time relative to the Kolmogorov time. The attained Taylor-microscale based Reynolds number Rλ ranges from 54.9 to 527. The largest simulation computed the flow on 20003 grids and 10003 (one billion) particles. Numerical results have shown that the collision kernel increases for Rλ<100 then decreases as Rλ increases. This Reynolds dependency is attributed to that of the radial distribution function at contact, which measures the contribution of particle clustering to the collision kernel. The results have also shown that the hydrodynamic interaction for St=0.4 particles decreases both the radial relative velocity and radial distribution function at contact, leading the collision efficiency less than unity. The collision efficiency increases from 0.65 to 0.75 as Rλ increases for Rλ<200 and then saturates.

Particle and particle pair dispersion in turbulence modeled with spatially and temporally correlated stochastic processes

In this paper we present a model for modeling the diffusion and relative dispersion of particles in homogeneous isotropic turbulence. We use a Heisenberg-like Hamiltonian to incorporate spatial correlations between fluid particles, which are modeled by stochastic processes correlated in time. We are able to reproduce the ballistic regime in the mean square displacement of single particles and the transition to a normal diffusion regime for long times. For the dispersion of particle pairs we find a t2 dependence of the mean square separation at short times and a t dependence for long ones. For intermediate times indications for a Richardson t3 law are observed in certain situations. Finally, the influence of inertia of real particles on the dispersion is investigated.

Shape effects on turbulent modulation by large nearly neutrally buoyant particles

AbstractWe investigate dilute suspensions of Taylor-microscale-sized particles in homogeneous isotropic turbulence. In particular, we focus on the effect of particle shape on particle–fluid interaction. We conduct laboratory experiments using a novel experimental technique to simultaneously measure the kinematics of fluid and particle phases. This uses transparent particles having the same refractive index as water, whose motion we track via embedded optical tracers. We compare the turbulent statistics of a single-phase flow to the turbulent statistics of the fluid phase in a particle–laden suspension. Two suspensions are compared, one in which the particles are spheres and the other in which they are prolate ellipsoids with aspect ratio 2. We find that spherical particles at volume fraction ${\phi }_{v} = 0. 14\hspace{0.167em} \% $ reduce the turbulent kinetic energy (TKE) by 15 % relative to the single-phase flow. At the same volume fraction (and slightly smaller total surface area), ellipsoidal particles have a much smaller effect: they reduce the TKE by 3 % relative to the single-phase flow. Spectral analysis shows the details of TKE reduction and redistribution across spatial scales: spherical particles remove energy from large scales and reinsert it at small scales, while ellipsoids remove relatively less TKE from large scales and reinsert relatively more at small scales. Shape effects are far less evident in the statistics of particle rotation, which are very similar for ellipsoids and spheres. Comparing these with fluid enstrophy statistics, we find that particle rotation is dominated by velocity gradients on scales much larger than the particle characteristic length scales.

Effect of the concentration of particles on their clustering in an isotropic turbulent field

A statistical model of clustering (accumulation) of monodispersed particles in homogeneous isotropic turbulence taking into account absolutely elastic collisions has been presented. The effect of collisions on clustering of particles has been analyzed in a wide range of variation of their concentration, inertia, and size.

Three-dimensional Lagrangian Voronoï analysis for clustering of particles and bubbles in turbulence

AbstractThree-dimensional Voronoï analysis is used to quantify the clustering of inertial particles in homogeneous isotropic turbulence using data sets from numerics in the point particle limit and one experimental data set. We study the clustering behaviour at different density ratios, particle response times (i.e. Stokes numbers $\mathit{St}$) and two Taylor–Reynolds numbers (${\mathit{Re}}_{\lambda } = 75$ and 180). The probability density functions (p.d.f.s) of the Voronoï cell volumes of light and heavy particles show different behaviour from that of randomly distributed particles, i.e. fluid tracers, implying that clustering is present. The standard deviation of the p.d.f. normalized by that of randomly distributed particles is used to quantify the clustering. The clustering for both light and heavy particles is stronger for higher ${\mathit{Re}}_{\lambda } $. Light particles show maximum clustering for $\mathit{St}$ around 1–2 for both Taylor–Reynolds numbers. The experimental data set shows reasonable agreement with the numerical results. The results are consistent with previous investigations employing other approaches to quantify the clustering. We also present the joint p.d.f.s of enstrophy and Voronoï volumes and their Lagrangian autocorrelations. The small Voronoï volumes of light particles correspond to regions of higher enstrophy than those of heavy particles, indicating that light particles cluster in higher vorticity regions. The Lagrangian temporal autocorrelation function of Voronoï volumes shows that the clustering of light particles lasts much longer than that of heavy or neutrally buoyant particles. Due to inertial effects arising from the density contrast with the surrounding liquid, light and heavy particles remain clustered for much longer times than the flow structures which cause the clustering.

Direct numerical simulations of aggregation of monosized spherical particles in homogeneous isotropic turbulence

AbstractDirect numerical simulations of turbulent solid–liquid suspensions have been performed. The liquid is Newtonian, and the particles are identical spheres. The spheres have a tendency to aggregate since they are attracted to one another as a result of a square‐well potential. The size of the particles is typically larger than the Kolmogorov scale, albeit of the same order of magnitude. In such situations, the particle dynamics (including the aggregation process), and turbulence strongly interact which explains the need for direct simulations. The lattice‐Boltzmann method combined with an immersed boundary method for representing the no‐slip conditions at the spherical solid–liquid interfaces was used. The results show that the aggregate size distributions depend on both the strength of particle–particle interactions and the intensity of the turbulence. © 2011 American Institute of Chemical Engineers AIChE J, 58: 2589–2600, 2012

The dynamics of small inertial particles in weakly stratified turbulence

We present an overview of a numerical study on the small-scale dynamics and the large-scale dispersion of small inertial particles in stably stratified turbulence. Three types of particles are examined: fluid particles, light inertial particles (with particle-to-fluid density ratio 1 ≲ ρ p / ρ f ≲ 25 ) and heavy inertial particles ( ρ p / ρ f ≫ 1 ) . Stratification suppresses the vertical dispersion of all three types of particles compared to isotropic turbulence. The horizontal dispersion, on the other hand, is enhanced. The importance of the hydrodynamic forces, which act on (light) inertial particles and directly affect the small-scale dynamics of the particles, is examined for particles with ρ p/ ρ f in the range from 1 to 25. These results primarily concern the forces on light particles in homogeneous isotropic turbulence, as at small scales in weakly stratified turbulence isotropy is almost restored. The effect of the presence of a density stratification on the particle dynamics is thus assumed to be almost negligible, which is supported by the similarity of the results from computations of the forces on small inertial particles in homogeneous isotropic turbulence and in weakly stratified turbulence. The inertia of the particles results in a non-uniform particle distribution over the domain. This preferential concentration effect is briefly reviewed here for heavy particles, and it is found that the effect is stronger in isotropic turbulence than in stratified turbulence. It is shown that preferential concentration also occurs for light inertial particles in homogeneous isotropic turbulence.

On the quantification of preferential accumulation

It is well-known that heavy particles in homogeneous isotropic turbulence tend to collect in certain regions of the flow, leading to preferential accumulation. In this study, we investigate different measures of preferential accumulation, some of which show a dependence on Reynolds number. All the measures used in this study confirm that this phenomenon is most pronounced for a Stokes number, St ∼ 1, based on the Kolmogorov time-scale. The variation with Stokes number has been studied over a wider range for all the measures than in previous studies. Though the effect of changing St, for a particular Re, has been well understood, the scaling with Re is not so obvious. The measures used in this study seem to suggest a structure to the particle concentration field which scales with Re. To investigate this structure, we introduce a concentration distribution length scale which shows a dependence on Re, especially at higher St. The results in the present work agree with several previous computational and experimental works and provide a more unified picture of the dependence of preferential accumulation on St and Re.

The effect of Brownian motion on collisions between aerosol particles in turbulent flow

A statistical model is given for determining the kernel of collisions of small aerosol particles in homogeneous isotropic turbulence in view of Brownian motion. The suggested model is based on the kinetic equation for the probability density function of distribution of relative velocity of two particles. Analysis is performed of the effect of Brownian motion on the relative velocity of two particles (transport effect) and on the radial distribution function (accumulation effect).

Clustering of low-inertia particles in isotropic turbulence

A statistical model is given for describing the clustering of small heavy particles in homogeneous isotropic turbulence. The suggested model is based on the kinetic equation for the probability density function of distribution of relative velocity of two particles. Analysis is performed of the phenomenon of clustering of low-inertia particles at high values of the Reynolds number.

A statistical theory of turbulent relative dispersion

The laws governing the spread of a cluster of particles in homogeneous isotropic turbulence are derived using a theoretical approach based on inertial subrange scaling and statistical diffusion theory. The equations for the mean square dispersion of a puff admit an analytical solution in the inertial subrange and at large scales. The solution is consistent with Taylor's theory of absolute dispersion. An analytical derivation of the Richardson–Obukhov constant of relative dispersion is presented. A time scale for relative dispersion is identified, as well as relations between Lagrangian and Eulerian structure functions. The results are extended to turbulence at finite Reynolds number. A closure assumption for the relative kinetic energy, based on Taylor's theory, is presented. Comparisons with direct numerical simulations and laboratory experiments are reported.

A hierarchy of models for turbulent dispersed two-phase flows derived from a kinetic equation for the joint particle-gas pdf

This paper deals with the statistical modeling of turbulent two-phase flows consisting of particles or droplets immersed in a gas. The problem of gaseous turbulence alone being very complex, we concentrate here on the simpler case of an a priori given forced isotropic homogeneous turbulence acting on the particles, whose mean square velocity and integral Lagrangian time-scale are given constants. Our main objective is to derive a hierarchy of reduced models from the joint particle-gas pdf (probability density function). The latter equation may therefore be regarded as a master equation for our problem. The reduced models describe the dispersion of a cloud of particles observed at di®erent time scales compared to the dynamic response time of the particles and the characteristic time scale of the turbulence along their trajectories. These derivations rely on very classical Chapman-Enskog expansions. We recover in particular the result of Tchen [C. M. Tchen, /i>Mean value and correlation problems connected with the motion of small particles suspended in a turbulent fluid, PhD thesis, Delft, The Hague, Martinus Nijhoff, 1947] stating that the diffusion rate is the same for small or large particles in homogeneous turbulence, under the assumption that the lagrangian statistical properties along their paths are the same. Moreover, our approach allows us to prove that the long-time limit of the joint particle-gas distribution function is a bi-maxwellian distribution, whatever the size of the particles. This is consistent with some usual assumptions made in the literature for the derivation of particle collision models [J. Lavieville, E. Deutsch and O. Simonin, Large eddy simulation of interactions between colliding particles and a homogeneous isotropic turbulence field, Gas-Solid Flows, ASME, 228, 347-358, 1995], [Leonid I. Zaichik, Olivier Simonin and Vladimir M. Alipchenkov, Two statistical models for predicting collision rates of inertia particles in homogeneous isotropic turbulence, Phys. Fluids, 15(10), 2995-3005, 2003].

Particle behavior in homogeneous isotropic turbulence

Direct numerical simulations were conducted to investigate the behavior of heavy particles in homogeneous isotropic turbulence. The present study focused on the effect of particle inertia and drift on the autocorrelations of the particle velocity and the fluid seen by particles and the dispersion characteristics of particles. The Lagrangian integral time scale of particles monotonically increased as the magnitude of the particle response time increased, while that of the fluid seen by particles remained relatively constant; it reached a maximum when the particle response time was close to the Kolmolgorov time scale of the flow. Particle dispersion increased as the particle inertia increased for small particles, while for larger particles, it decreased as particle inertia increased; particle eddy diffusion coefficient was maximal, and greater than that of the fluid by about 30%, at the preferential concentration. The concentration field of the particles with τp/τk≈1.0 showed that particles tend to collect in regions of low vorticity (high strain) due to preferential concentration. As the drift velocity of a particle is increased it crosses the paths of fluid elements more rapidly and will tend to lose correlation with its previous velocity faster than a fluid element will. And the correlation of particle velocities along the drift direction is more persistent than that perpendicular to the direction of drift. Simulations also showed that the continuity effect and the crossing-trajectory effect are weakened for particles with infinite inertia.

Two statistical models for predicting collision rates of inertial particles in homogeneous isotropic turbulence

The objective of the paper is to present and compare two models for the collision rate of inertial particles immersed in homogeneous isotropic turbulence. The merits and demerits of several known collision models are discussed. One of the models proposed in the paper is based on the assumption that the velocities of the fluid and a particle obey a correlated Gaussian distribution. The other model stems from a kinetic equation for the probability density function of the relative velocity distribution of two particles. The predictions obtained by means of these two models are compared with numerical simulations published in the literature.

Particle Clustering in Isotropic Turbulent Flow

A kinetic statistical model for describing the dispersion and clustering of heavy particles in homogeneous isotropic turbulence is presented. The model developed is used for calculating the relative velocities and the radial distribution function of a pair of particles in a steady suspension. The results obtained are compared with the known data obtained by direct numerical simulation.

Effect of nonlinear drag on the settling velocity of particles in homogeneous isotropic turbulence

We study the average settling velocity of small spherical particles under gravity through a Gaussian random velocity field generated by Fourier modes with a von Kárman energy spectrum. The particles are subjected to the effects of a linear (Stokes) or nonlinear drag force, inertia and gravity. It is shown that the effect of drag nonlinearity is a function of the particle to fluid density ratio ρp/ρf and a function of the ratio τp/τk of the particle time constant to the Kolmogorov timescale of the fluid. Simulations show that as ρp/ρf decreases from 877 to 2.65 or as τp/τk increases from 1 to 2.74, the drag nonlinearity increases as a result of the increase in particle Reynolds numbers. Hence the settling velocity changes from larger to smaller as compared with the still fluid settling velocity, showing that one of the major mechanisms governing the fall velocity reduction in a turbulent flow is the drag nonlinearity. The maximum increase in settling rate occurs at VT/υk ≈ 2 for ρp/ρf ≥ 87 (where VT is the terminal velocity of the particle and υk is the Kolmogorov velocity); this is consistent with the results of Wang and Maxey [1993]. The maximum decrease in settling rate occurs at VT/VK ≈ 1 for ρp/ρf ≈ 2.65, consistent with the results of Fung [1993]. In addition, the role of the spatial and temporal variations of the flow field on the settling rate is investigated. Finally, the Gaussian velocity field is also simulated with an exponential energy spectrum, and similar results are observed.

The role of the turbulent scales in the settling velocity of heavy particles in homogeneous isotropic turbulence

The average settling velocity of heavy particles under a body force field is studied numerically in stationary homogeneous isotropic turbulent flows generated by the direct numerical simulation and the large eddy simulation of the continuity and Navier–Stokes equations. The flow fields corresponding to different selected ranges of turbulent scales are obtained by filtering the results of the direct numerical simulation, and employed for calculating the particle motion. Wang &amp; Maxey (1993) showed that as a consequence of the particle accumulation in the low vorticity region and the preferential sweeping phenomenon, the average settling rate in turbulence is greater than that in still fluid. In the present study, the phenomenon of particle accumulation in the low vorticity region is found to be controlled mainly by the small scales with wavenumber kω corresponding to the maximum of the dissipation (vorticity) spectrum. However, the increase of the average settling rate, 〈ΔvS〉, also depends strongly on the large energetic eddies. The small eddies with wavenumber greater than 2.5kω have essentially no effect on the particle accumulation and the average settling velocity. The large eddy simulation is thus adequate for the present study provided the smallest resolved scale is greater than 1/(2.5kω). Detailed calculations show that 〈ΔvS〉 is maximized and is of order u′/10 when τp/τk≈1 and vd/u′≈0.5 for Rλ=22.6–153, where τp is the particle's relaxation time, τk is the Kolmogorov timescale, vd is the settling rate in still fluid, u′ is the root mean square of the velocity fluctuation, and Rλ is the Reynolds number based on the Taylor microscale.

Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence

The average settling velocity in homogeneous turbulence of a small rigid spherical particle, subject to a Stokes drag force, has been shown to differ from that in still fluid owing to a bias from the particle inertia (Maxey 1987). Previous numerical results for particles in a random flow field, where the flow dynamics were not considered, showed an increase in the average settling velocity. Direct numerical simulations of the motion of heavy particles in isotropic homogeneous turbulence have been performed where the flow dynamics are included. These show that a significant increase in the average settling velocity can occur for particles with inertial response time and still-fluid terminal velocity comparable to the Kolmogorov scales of the turbulence. This increase may be as much as 50% of the terminal velocity, which is much larger than was previously found. The concentration field of the heavy particles, obtained from direct numerical simulations, shows the importance of the inertial bias with particles tending to collect in elongated sheets on the peripheries of local vortical structures. This is coupled then to a preferential sweeping of the particles in downward moving fluid. Again the importance of Kolmogorov scaling to these processes is demonstrated. Finally, some consideration is given to larger particles that are subject to a nonlinear drag force where it is found that the nonlinearity reduces the net increase in settling velocity.

On the Diffusion of Fluid Particles by the Combined Model of Random Fourier Modes and Random Flight.

To investigate the motion of fluid particles round a cylinder in turbulent flows, we have developed a new model simulating the trajectory of particles: the combined model of Kinematic Simulation (KS) and Random Flight (RF). The large-scale turbulence is simulated by a sum of random Fourier modes varying in space and time, and the small-scale turbulent fluctuation is simply modelled by ITO type of stochastic differential equation with a memory time comparable to a Lagrangian integral time scale TLS of small-scale motion. The Lagrangian properties of fluid particles in homogeneous isotropic turbulence with uniform mean flow has been examined. Some new results have been obtained using this new model: (a) the cross-correlation of the displacements by the large and small scale motions is negligibly small; (b) the random small-scale motion causes the decorrelation of the large-scale velocity; (c) the value of TLS used in RF is estimated from pure KS.