ALBERT

Notes: Advection of small particles with inertia in two-dimensional ideal flows is studied both numerically and analytically. It is assumed that the flow disturbance around the particle corresponds to a potential dipole, so that the motion is driven by pressure gradient, inertial, and added-mass forces. It is found that in general the motion is nonintegrable, but particular exact solutions can be obtained. The problem is then studied for the cases of axisymmetric flow, when the motion proves to be completely integrable, and of a cellular flow, for which both regular and stochastic (bounded and unbounded) trajectories are calculated. In the latter case, the unbounded stochastic motion is of Brownian-like character, and the results derived show that the particle dispersion process is generally anomalous.

Notes: The processes of the accumulation of solid heavy particles settling under gravity and modification of the carrier flow are studied analytically and numerically. The dilute limit is considered when the effects of particle–particle interactions are neglected. The flow Reynolds number is assumed to be large enough to disregard effects related to the viscous dissipation. Particles with a small Stokes (St) number are considered, when approximate solution for the particulate velocity in the form of a series in St can be derived.Analytical solutions are obtained for initially uniform distribution of particles in a cellular flow (formed by a lattice of two-dimensional vortices with different circulations) describing dynamics of the particulate concentration and corresponding modification of the fluid flow. Two different regimes of large and moderate flow Froude number (Fr) are considered. Solutions obtained show that in the first case (Fr(very-much-greater-than)1), when the gravitational settling of the particles is insignificant, the concentration of particles drops at the centers and grows at the periphery of the vortices, so that sheets of the increased concentration are formed in the vicinity of the flow separatrices. Due to the coupling between the particulate and fluid dynamics, the flow vorticity is reduced at the cells centers. The flow gradients grow due to the drag forces between the two phases and sheets of increased vorticity are generated in the vicinity of the concentration sheets. In the case Fr∼1 the concentration sheets correspond to the particle settling paths. Vorticity is effectively generated in the vicinity of these paths, where the drag force based on the local slip velocity is enhanced by the settling. As a result, vortex centers are shifted toward the settling paths, so that the whole symmetry of the flow is changed by the coupling. Results of direct numerical simulations performed for St∼1 show a good agreement with analytical solutions.It is observed also that the average value of particle settling velocity may exceed that in the still fluid (similar to the earlier result obtained by Maxey [Philos. Trans. R. Soc. London Ser. A 333, 289 (1990)] under assumption that the influence of the particulate on the carrier flow remains negligible). © 1995 American Institute of Physics.

Notes: The objective of this paper is to examine the modulation of isotropic decaying turbulence by particles whose diameter is smaller than the Kolmogorov length scale, and their response time, τp, is smaller than the Kolmogorov time scale, τk (hence microparticles), and the influence of increasing the particle inertia on the two-way coupling. The particle volume fraction is considered small enough, so that particle–particle interactions are neglected. On the other hand, the particle material density ρp(very-much-greater-than)ρf, the fluid density, and the mass loading of the particles is large enough to modify the carrier flow. The particle Reynolds number is smaller than unity, and the gravitational settling of the particles is neglected. We obtain an asymptotic analytical solution describing the spectrum of the instantaneous two-way coupling source term, Ψp(k,t), in the equation for the fluid turbulence kinetic energy (TKE) spectrum, E(k,t), as a series in powers of the ratio (τp/τk). Recent results of Druzhinin and Elghobashi [Phys. Fluids 11, 602 (1999)] for particles whose τp(very-much-less-than)τk show that to the zeroth order in (τp/τk), Ψp(k,t) is proportional to the fluid spectral dissipation function, ε(k,t). In the present paper, the asymptotic solution is extended up to the first order in (τp/τk) and is applicable for particles with small but finite inertia. We also perform direct numerical simulation (DNS) of particle-laden isotropic turbulence using the Eulerian–Lagrangian approach. The results obtained for particles whose τp≤0.4τk show that both the TKE and its dissipation rate, ε(t), as well as the spectral transfer of the fluid kinetic energy, are increased by the two-way coupling as compared to the particle-free case, and the increase is more pronounced for smaller τp. The asymptotic solution for the two-way coupling source term spectrum, Ψp(k,t), is found in good qualitative and quantitative agreement with the numerical results. Both the asymptotic solution and the DNS results for the instantaneous source term spectrum, Ψp(k,t), show that as the particle response time is increased, the magnitude of the maximum of Ψp(k,t) is reduced and its location is shifted toward higher wave numbers, as compared to the limiting case τp(very-much-less-than)τk. The DNS results also show that for particles with sufficiently high inertia (whose τp≥0.5τk), a negative peak of Ψp(k,t) is created at low wave numbers, whereas the fluid spectral energy transfer is reduced, as compared to the one-way coupling case. The development of the negative peak of Ψp(k,t) is accompanied by a well-pronounced preferential accumulation of particles. The net two-way coupling effect is the reduction of the TKE by particles with sufficiently high inertia (whose τp=0.8τk in our DNS), as compared to the particle-free flow. In this case, our results are in qualitative agreement with the DNS results of Boivin et al. [J. Fluid Mech. 375, 235 (1998)], who considered particles whose τp≥1.26τk. Therefore, our results show that there occurs a qualitative transition in the two-way coupling effect of particles on isotropic turbulence as the particle response time is increased from τp(very-much-less-than)τk, in the limit of microparticles, to τp(similar, equals)τk, for particles with finite inertia. In the case of microparticles (whose τp(very-much-less-than)τk), the instantaneous spectrum of the two-way coupling source term, Ψp(k,t), is positive at all wave numbers so that the particles add the energy to the fluid motion and increase the turbulence kinetic energy, as compared to the one-way coupling case. On the other hand, in the case of particles with higher inertia (whose τp(similar, equals)τk), the positive contribution of the source term, Ψp(k,t), is reduced at high wave numbers whereas a negative peak of Ψp(k,t) is created at low wave numbers. In this case, the net two-way coupling effect is the reduction of the TKE by the particles, as compared to the one-way coupling case. © 2001 American Institute of Physics.

Notes: Evolution of a two-dimensional axisymmetrical vortex laden with solid heavy particles is studied analytically and numerically. The particulate phase is assumed to be dilute enough to neglect the effects of particle–particle collisions. Only sufficiently small particle Stokes (St) and Reynolds numbers are considered, for which an approximate solution for the particle velocity can be derived. An analytical solution to a Cauchy problem is obtained for initially uniform concentration of particles in a circular flow describing the accumulation of particles in the form of a kinematic wave and the corresponding modification of the carrier flow. According to this solution, a steep peak of the concentration develops forming the wave crest which propagates out of the vortex. Due to the interaction between the two phases, a fluid velocity component directed towards the vortex center is generated, so that in the vicinity of the crest the vortex acquires a spiral-like shape. At later stages, the growth of the crest is inhibited and its propagation velocity decreases. Analysis of the problem for particles with larger Stokes numbers shows that the accumulation process is most intense when St is close to a critical value St* which generally depends on the vortex structure and, for the flow considered, is of the order unity.

Notes: This paper is concerned with the two-way coupling effects on the decay rate of isotropic turbulence laden with solid spherical microparticles whose response time, τp, is much smaller than the Kolmogorov time scale (τp(very-much-less-than)τk). The particles volumetric concentration, C0, is small enough (C0∼10−4) to neglect particle–particle interactions, and the material density of the particle is much larger than the fluid density (δ=ρp/ρf(very-much-greater-than)1). We obtain asymptotic analytical solutions for the instantaneous particle velocity and kinetic energy spectrum, in the limit τp(very-much-less-than)τk, which indicate that the two-way coupling increases the fluid inertia term in the fluid momentum equation by the factor (1+C0δ). Consequently, the high-wave number components of the spectra of turbulence energy and dissipation develop in time as ∼exp[−2νk2t/(1+C0δ)]. The net result is a reduction of the decay rate of turbulence energy compared to that of particle-free turbulence (i.e., the one-way coupling case where C0δ≡0). We also perform direct numerical simulation (DNS) of isotropic turbulence laden with microparticles, using the two-fluid (TF or Eulerian–Eulerian) approach developed in an earlier study [Druzhinin and Elghobashi, Phys. Fluids 3 (1998)]. Excellent agreement is achieved between the DNS results and the analytical solution for the particle kinetic energy spectrum. The DNS results show that the two-way coupling reduces the decay rate of turbulence energy compared to that of one-way coupling. In addition, we compare the temporal developments of the turbulence kinetic energy and its dissipation rate, obtained from DNS using TF, with those from DNS using the trajectory (Eulerian–Lagrangian) approach. Satisfactory agreement is achieved between the two approaches, with TF requiring considerably less computational time. © 1999 American Institute of Physics.

Notes: This note examines the stability of a stationary velocity solution of the Tchen's equation, describing the motion of a spherical particle with a Reynolds number much less than unity in a uniform unsteady fluid flow. The results show that the stationary solution is unstable if the ratio of the particle and fluid densities ρp/ρf〈7/4. Therefore, applying the stationary solution is not justified in this case. © 2000 American Institute of Physics.

Notes: Direct numerical simulations (DNS) of bubble-laden isotropic decaying turbulence are performed using the two-fluid approach (TF) instead of the Eulerian–Lagrangian approach (EL). The motivation for the study is that EL requires considerable computational resources, especially for the case of two-way coupling, where the instantaneous trajectories of a large number of individual bubbles need to be computed. The TF formulation is developed by spatially averaging the instantaneous equations of the carrier flow and bubble phase over a scale of the order of the Kolmogorov length scale, which, in our case, is much larger than the bubble diameter. On that scale, the bubbles are treated as a continuum (without molecular diffusivity) characterized by the bubble phase velocity field and concentration (volume fraction). The bubble concentration, C, is assumed small enough (C≤10−3) to neglect the bubble–bubble interactions. As a test case, direct simulation of a bubble-laden Taylor–Green vortex with one-way coupling is performed with a bubble response time of the order of the flow time scale (inverse of the mean vorticity). This simple flow allows a direct examination of the effects of the preferential accumulation of bubbles in the high-enstrophy regions of the flow on the accuracy of the two-fluid formulation. The temporal development of the maximum bubble concentration obtained from DNS agrees well with the analytical solution. DNS of the bubble-laden decaying turbulence are also performed for both cases of one-way and two-way coupling. Here, the bubble diameter and response time are much smaller than the Kolmogorov length and time scales, respectively. In this case, as expected, the effects of the preferential accumulation of the bubbles are not pronounced. The results also show that the bubble-laden flow is analogous to a stratified flow with an effective density =(1−C)ρf. Thus, due to the two-way interaction between the bubbles and carrier flow, the turbulence decay is enhanced with stable stratification, and reduced with unstable stratification.© 1998 American Institute of Physics.

Notes: Gravitational settling of solid heavy particles in a dilute suspension is studied analytically and numerically. The particle Reynolds number is assumed to be less than unity, for which the viscous drag force on the particle is well approximated by the linear Stokes law. The particulate volume fraction (or concentration) c is assumed to be small enough for the effects of particle–particle interactions to be negligible. The ratio δ=ρp/ρf of the particle and fluid densities is considered large enough however, so that the momentum exchange between the two phases caused by the viscous drag forces (which is of the order of the particulate mass loading factor cδ) is significant. The particulate base concentration, c0(y), is assumed to be a smooth function of the vertical coordinate y (hence, a stratified suspension) and a perturbation of the initially stationary settling regime is considered in the form of a horizontally propagating monochromatic wave with wavenumber k and frequency ω(k). Analytical solutions for the perturbations in the limit of small particle inertia (such that ωτp(very-much-less-than)1, where τp is the particle response time) are found to be similar to those for internal waves propagating in a stratified fluid with effective density ρeff=ρf(1+c0(y)δ). On the other hand, it is found that in the opposite limit of large particle inertia (ωτp(very-much-greater-than)1) the perturbations are damped. As an example, we consider a suspension consisting of two layers with uniform concentrations of particles c1 (for y〉+h/2) and c2 (for y〈−h/2) separated by the interface layer of thickness h, where the concentration gradient is substantial. The solutions obtained in the long-wave limit kh(very-much-less-than)1 show that if the concentration in the lower layer exceeds that in the upper layer (c2〉c1), the disturbance of the interface brings about wavy motions analogous to internal waves in a two-layer fluid. In the case of inverse stratification (c2〈c1) the disturbance grows exponentially and generates plume-like "bubbles,'' similar to those produced due to the Rayleigh–Taylor instability in a two-layer fluid. The results of the numerical simulations show that, as expected, the waves are damped and the instability growth rate is reduced for particles having larger inertia. © 1997 American Institute of Physics.