Lance R. Collins

Collision statistics in an isotropic particle-laden turbulent suspension. Part 1. Direct numerical simulations

Direct numerical simulations of heavy particles suspended in a turbulent fluid are performed to study the rate of inter-particle collisions as a function of the turbulence parameters and particle properties. The particle volume fractions are kept small (∼10 −4 ) so that the system is well within the dilute limit. The fluid velocities are updated using a pseudo-spectral algorithm while the particle forces are approximated by Stokes drag. One unique aspect of the present simulations is that the particles have finite volumes (as opposed to point masses) and therefore particle collisions must be accounted for. The collision frequency is monitored over several eddy turnover times. It is found that particles with small Stokes numbers behave similarly to the prediction of Saffman & Turner (1956). On the other hand, particles with very large Stokes numbers have collision frequencies similar to kinetic theory (Abrahamson 1975). For intermediate Stokes numbers, the behaviour is complicated by two effects: (i) particles tend to collect in regions of low vorticity (high strain) due to a centrifugal effect (preferential concentration); (ii) particle pairs are less strongly correlated with each other, resulting in an increase in their relative velocity. Both effects tend to increase collision rates, however the scalings of the two effects are different, leading to the observed complex behaviour. An explanation for the entire range of Stokes numbers can be found by considering the relationship between the collision frequency and two statistical properties of the particle phase: the radial distribution function and the relative velocity probability density function. Statistical analysis of the data, in the context of this relationship, confirms the relationship and provides a quantitative description of how preferential concentration and particle decorrelation ultimately affect the collision frequency.

Versatility in lipid compositions showing prolonged circulation with sterically stabilized liposomes

Efforts to overcome rapid uptake of liposomes by cells of the mononuclear phagocytic system (MPS) have identified that lipids derivatized with the hydrophilic polymer poly(ethylene glycol) (PEG) have many advantages. The structure-function relationship of PEG-derivatized phosphatidylethanolamine (PEG-PE) has been examined by studies of blood lifetime and tissue distribution in both mice and rats. Liposomes composed of phosphatidylcholine (PC), cholesterol, and 7.5 mol% of PEG-PE show prolonged circulation and reduced MPS uptake when the PEG has a molecular weight in the range of 1000 to 5000. Up to 35% of the injected dose remains in the blood and less than 10% is taken up by the MPS (liver plus spleen) after 24 h in the best cases as compared to 1% and 40%, respectively, for liposomes without PEG-PE. Prolonged circulation with PEG-PE is independent of cholesterol, degree of saturation in either the PC or the PE lipid anchor, lipid dose, or addition of other negatively charged lipids, phosphatidylglycerol or cholesterol sulfate. This versatility in lipid composition and dose without alteration of blood lifetime or tissue distribution is essential for controlling drug dosage and release properties in a liposome-based therapeutic agent.

Effect of preferential concentration on turbulent collision rates

The effect of particle inertia on the interparticle collision rates of a turbulent aerosol was investigated recently by Sundaram and Collins (1997) using direct numerical simulation (DNS). They observed that for values of the particle Stokes number (here defined as the ratio of the particle response time to Kolmogorov time scale) near unity, the collision frequency was enhanced by between one and two orders of magnitude. This enhancement was attributed in part to the local enrichment of the particle concentration in low-vorticity regions of the flow due to the centrifuge effect commonly referred to as preferential concentration (Eaton and Fessler 1994). Sundaram and Collins (1997) showed that the correction factor for the collision kernel in a preferentially concentrated system is g(σ), where g(r) is the particle radial distribution function and σ is the collision diameter. This paper uses DNS, in combination with statistical analysis, to study the dependence of the radial distribution function on the tur...

A numerical study of the modulation of isotropic turbulence by suspended particles

Direct numerical simulations of a turbulent fluid laden with finite-sized particles are performed. The computations, on a 128 3 grid along with a maximum of 262 144 particles, incorporated both direct particle interactions via hard-sphere collisions and particle feedback. The ‘reverse’ coupling is accomplished in a manner ensuring correct discrete energy conservation (Sundaram & Collins 1996). A novel two-field formalism (Sundaram & Collins 1994 a ) is employed to calculate two-point correlations and equivalent spectral densities. An important consideration in these simulations is the initial state of fluid and particles. That is, the initial conditions must be chosen so as to allow a meaningful comparison of the different runs. Using such a carefully initialized set of runs, particle inertia was observed to increase both the viscous and drag dissipations; however, simultaneously, it also caused particle velocities to correlate for longer distances. The combination of effects suggests a mechanism for turbulence enhancement or suppression that depends on the parameter values. Like previous investigators, ‘pivoting’ or crossover of the fluid energy spectra was observed. A possible new scaling for this phenomenon is suggested. Furthermore, investigations of the influence of particle mass and number densities on turbulence modulation are also carried out.

Clustering of aerosol particles in isotropic turbulence

It has been recognized that particle inertia throws dense particles out of regions of high vorticity and leads to an accumulation of particles in the straining-flow regions of a turbulent flow field. However, recent direct numerical simulations (DNS) indicate that the tendency to cluster is evident even at particle separations smaller than the size of the smallest eddy. Indeed, the particle radial distribution function (RDF), an important measure of clustering, increases as an inverse power of the interparticle separation for separations much smaller than the Kolmogorov length scale. Motivated by this observation, we have developed an analytical theory to predict the RDF in a turbulent flow for particles with a small, but non-zero Stokes number. Here, the Stokes number (St) is the ratio of the particles viscous relaxation time to the Kolmogorov time. The theory approximates the turbulent flow in a reference frame following an aerosol particle as a local linear flow field with a velocity gradient tensor and acceleration that vary stochastically in time

Sterically stabilized liposomes. Reduction in electrophoretic mobility but not electrostatic surface potential

The electrophoretic mobility of liposomes containing a negatively charged derivative of phosphatidylethanolamine with a large headgroup composed of the hydrophilic polymer polyethylene glycol (PEG-PE) was determined by Doppler electrophoretic light scattering. The results show that this method is improved by the use of measurements at multiple angles to eliminate artifacts and that very small mobilities can be measured. The electrophoretic mobility of liposomes with 5 to 10 mol% PEG-PE is approximately -0.5 mu ms-1/Vcm-1 regardless of PEG-PE content compared with approximately -2 mu ms-1/Vcm-1 for similar liposomes but containing 7.5% phosphatidylglycerol (PG) instead of PEG-PE. Measurements of surface potential by distribution of an anionic fluorescent probe show that the PEG-PE imparts a negative charge identical to that by PG, consistent with the expectation of similar locations of the ionized phosphate responsible for the charge. The reduced mobility imparted by the surface bound PEG is attributed to a mechanism similar to that described for colloidal steric stabilization: hydrodynamic drag moves the hydrodynamic plane of shear, or the hydrodynamic radius, away from the charge-bearing plane, that of the phosphate moities. An extended length of approximately 50 A for the 2,000 molecular weight PEG is estimated from the reduction in electrophoretic mobility.

Preferential concentration of cloud droplets by turbulence : Effects on the early evolution of cumulus cloud droplet spectra

Abstract A mechanism is presented, based on the inherent turbulent nature of cumulus clouds, for the broadening of cloud droplet spectra during condensational growth. This mechanism operates independent of entrainment and, therefore, can operate in adiabatic cloud cores. Cloud droplets of sufficient size are not randomly dispersed in a cloud but are preferentially concentrated in regions of low vorticity in the turbulent flow field. Regions of high vorticity (low droplet concentration) develop higher supersaturation than regions of low vorticity (high droplet concentration). Therefore, on small spatial scales cloud droplets are growing in a strongly fluctuating supersaturation field. These fluctuations in supersaturation exist independent of large-scale vertical velocity fluctuations. Droplets growing in regions of high vorticity will experience enhanced growth rates, allowing some droplets to grow larger than predicted by the classic theory of condensational growth. This mechanism helps to account for tw...

Reynolds number scaling of particle clustering in turbulent aerosols

Particles with finite inertia in turbulent flow cluster in low-vorticity regions of the fluid due to the inertial imbalance between the denser particles and the lighter surrounding fluid. This effect, sometimes referred to as preferential concentration, has been investigated in several recent numerical studies. Sundaram and Collins (1997 J. Fluid Mech. 335 75) considered the effect of particle clustering on the interparticle collision rate and showed that the radial distribution function, evaluated at contact, precisely corrects the collision kernel for this effect. An open question is how preferential concentration scales with Reynolds number. We investigate this question using direct numerical simulations (DNS). Over the limited range accessible by DNS, it appears that the radial distribution function approaches a plateau with increasing Reynolds number. This contradicts earlier studies that predicted linear growth with Reynolds number. The implications of these findings for very high Reynolds number applications such as cloud formation in the upper atmosphere is briefly discussed.

A numerical study of the particle size distribution of an aerosol undergoing turbulent coagulation

Coagulation and growth of aerosol particles subject to isotropic turbulence has been explored using direct numerical simulations. The computations follow the trajectories of 262 144 initial particles as they are convected by the turbulent flow field. Collision between two parent particles leads to the formation of a new daughter particle with the mass and momentum (but not necessarily the energy) of the parent particles. The initially monodisperse population of particles will develop a size distribution over time that is controlled by the collision dynamics. In an earlier study, Sundaram & Collins (1997) showed that collision rates in isotropic turbulence are controlled by two statistics: (i) the radial distribution of the particles and (ii) the relative velocity probability density function. Their study considered particles that rebound elastically; however, we find that the formula that they derived is equally valid in a coagulating system. However, coagulation alters the numerical values of these statistics from the values they attain for the elastic rebound case. This difference is substantial and must be taken into consideration to properly predict the evolution of the size distribution of a population of particles. The DNS results also show surprising trends in the relative breadth of the particle size distribution. First, in all cases, the standard deviation of the particle size distribution of particles with finite Stokes numbers is much larger than the standard deviation for either the zero-Stokes-number or infinite-Stokes-number limits. Secondly, for particles with small initial Stokes numbers, the standard deviation of the final particle size distribution decreases with increasing initial particle size; however, the opposite trend is observed for particles with slightly larger initial Stokes numbers. An explanation for these phenomena can be found by carefully examining the functional dependence of the radial distribution function on the particle size and Stokes number.

Numerical approach to simulating turbulent flow of a viscoelastic polymer solution

In this paper, we present two new numerical algorithms for updating the equations of motion for a viscoelastic fluid that can be described by the finite extensible nonlinear elastic polymer model with the closure proposed by Peterlin (so called FENE-P model) in a transient calculation. In particular, our algorithms address two difficulties found in earlier formulations. First, the polymer extension, represented by the trace of the conformation tensor, can numerically exceed the finite extensible length causing the restoring spring force to change sign and the calculation to rapidly diverge. In our formulations, we have redefined the conformation tensor so that this possibility no longer exists. Secondly, the conformation tensor must remain symmetric and positive definite at all times for the calculation to remain stable. The accumulation of numerical errors can cause loss of this property, leading to the growth of Hadamard instabilities [J. Non-Newtonian Fluid Mech. 60 (1995) 53]. We present two matrix decompositions that enable us to construct the conformational tensor in a manner that ensures positive definiteness. Numerical tests of the new algorithms show significant departures from other approaches that rely on filtering to remove the instabilities.