Effective Stokes Number Research Articles

Velocity and acceleration statistics of heavy spheroidal particles in turbulence

Non-spherical particles transported by turbulent flow have a rich dynamics that combines their translational and rotational motions. Here, the focus is on small, heavy, inertial particles with a spheroidal shape fully prescribed by their aspect ratio. Such particles undergo an anisotropic, orientation-dependent viscous drag with the carrier fluid flow whose associated torque is given by the Jeffery equations. Direct numerical simulations of homogeneous, isotropic turbulence are performed to study systematically how the translational motion of such spheroidal particles depends on their shape and size. It is found that the Lagrangian statistics of both velocity and acceleration can be described in terms of an effective Stokes number obtained as an isotropic average over angles of the particle's orientation. Corrections to the translational motion of particles due to their non-sphericity and rotation can hence be recast as an effective radius obtained from such a mean.

A Mesoscale Simulation Approach to Study the Flow Field in an Axial Granular Bed Filter

In an axial granular bed filter (GBF), a new mesoscale simulation approach is obtained by combining the macroscopic calculation models, i.e., the equations of the total pressure drop and dust-removal efficiency into the porous media model and the source term of the conservation equations. After grid-dependent tests and experimental validation, the effects of different conditions, i.e., granular bed height L, superficial gas velocity ug, dust diameter dp, dust concentration cp, granular diameter dg, initial bed voidage ε0, and filtration time t, on the pressure drop and dust-removal efficiency are investigated. The results show that the pressure drop is related to the inertial and viscous resistance terms, which increase with increasing L, ug, cp and t and decreasing ε and dg. The dust-removal efficiency is related to the Reynolds number, effective Stokes number, and equivalent granular diameter ratio. It increases with increasing L, ug, dp and t (small values), and decreasing cp, ε, and dg. Moreover, the influence of different conditions coincides well with dust-removal efficiency in relevant studies, which further demonstrates the accuracy of the mesoscale simulation approach. With the application of this method, the flow field can also be obtained easily and quickly, which is expected to provide a reference for the simulation study of GBF.

Dry Deposition Methods Based on Turbulence Kinetic Energy: Part 2. Extension to Particle Deposition Using a Single-Point Model.

Magnitude of atmospheric turbulence, a key driver of several processes that contribute to aerosol (i.e., particle) deposition, is underrepresented in current models. Various formulations have been developed to model particle dry deposition; all these formulations typically rely on friction velocity and some use additional ad hoc factors to represent enhanced impacts of turbulence. However, none were formally linked with the three-dimensional (3-D) turbulence. Here, we propose a set of 3-D turbulence-dependent resistance formulations for particle dry deposition simulation and intercompare the performance of new resistance formulations with that obtained from using the existing formulations and measured dry deposition velocity. Turbulence parameters such as turbulence velocity scale, turbulence factor, intensity of turbulence, effective sedimentation velocity, and effective Stokes number are newly introduced into two different particle deposition schemes to improve turbulence strength representation. For an assumed particle size distribution, the newly proposed schemes predict stronger diurnal variation of particle dry deposition velocity and are comparable to corresponding measurements while existing formulations indicate large underpredictions. We also find that the incorporation of new turbulence parameters either introduced or added stronger diurnal variability to sedimentation velocity and collection efficiencies values, making the new schemes predict higher deposition values during daytime and nighttime when compared to existing schemes. The findings from this research may help improve the capability of dry deposition schemes in regional and global models.

Single inertial particle statistics in turbulent flows from Lagrangian velocity models

Statistical descriptions of inertial particle motion are complicated by the random and multiscale structure of the turbulent velocity field. In this paper, we develop a stochastic model for inertial particles on the basis of ``filtered'' fluid tracer particles. Introducing an effective Stokes number allows us to capture crucial features of inertial particle motion such as the sharp decrease of acceleration variance for increasing Stokes numbers (acceleration trajectories are color-coded by maximum tracer acceleration). The proposed modeling approach could yield further insights on preferential concentration and a better experimental characterization of particle-laden flows.

Clustering of inertial spheres in evolving Taylor–Green vortex flow

Clustering of inertial spheres in a statistically unsteady flow field is believed to be different from particle clustering observed in statistically steady flows. The continuously evolving three-dimensional Taylor–Green vortex (TGV) flow exhibits time-varying length and time scales, which are likely to alter the resonance of a given particle with the evolving flow structures. The tendency of homogeneously introduced spherical point-particles to cluster preferentially in the TGV flow is observed to depend on the particle inertia, parameterized in terms of the particle response time τp. The degree of the inhomogeneity of the particle distribution is measured by the variance σ2 of Voronoï volumes. The time evolution of the particle-laden TGV flow is characterized by a viscous dissipation time scale τd and the effective Stokes number Steff = τp/τd. Particles with low/little inertia do not cluster in the early stage when the TGV flow only consists of large-scale and almost inviscid structures and Steff < 1. Later, when the large structures have been broken down into smaller vortices, the least inertial particles exhibit a stronger preferential concentration than the more inertial spheres. At this stage, when the viscous energy dissipation has reached its maximum level, the effective Stokes number of these particles has reached the order of one. Particles are generally seen to cluster preferentially at strain-rate dominated locations, i.e., where the second invariant Q of the velocity gradient tensor is negative. However, a memory effect can be observed in the course of the flow evolution where high σ2 values do not always correlate with Q < 0.

Mapping spheroid rotation modes in turbulent channel flow: effects of shear, turbulence and particle inertia

The rotational behaviour of non-spherical particles in turbulent channel flow is studied by Lagrangian tracking of spheroidal point particles in a directly simulated flow. The focus is on the complex rotation modes of the spheroidal particles, in which the back reaction on the flow field is ignored. This study is a sequel to the letter by Zhao et al. (Phys. Rev. Lett., vol. 115, 2015, 244501), in which only selected results in the near-wall buffer region and the almost-isotropic channel centre were presented. Now, particle dynamics all across the channel is explored to provide a complete picture of the orientational and rotational behaviour with consideration of the effects of particle aspect ratio ranging from 0.1 to 10 and particle Stokes number from 0 (inertialess) to 30. The rotational dynamics in the innermost part of the logarithmic wall layer is particularly complex and affected not only by modest mean shear, but also by particle inertia and turbulent vorticity. While inertial disks exhibit modest preferential orientation in either the wall-normal or cross-stream direction, inertial rods show neither preferential tumbling nor spinning. Examination of the co-variances between particle orientation, particle rotation and fluid rotation vectors explains the qualitatively different ‘wall mode’ rotation and ‘centre mode’ rotation. Inertialess spheroids transition between the two modes within a narrow zone ($15<z^{+}<35$) in the buffer region. If the spheroids have inertia, the transition zone between the two modes shifts to the inner part of the logarithmic layer, i.e. $z^{+}\geqslant 40$. We ascribe the transition of inertialess spheroids from the ‘wall mode’ to the ‘centre mode’ rotation to the changeover between the time scales associated with mean shear and small-scale turbulence. Inertial spheroids, however, transition between the two rotational modes when the Kolmogorov time scale becomes comparable to the time scale for particle rotation, i.e. the effective Stokes number is of order unity. The aforementioned findings reveal, in addition to the effects of particle shape and alignment, the importance of the characteristic local time scale of fluid flow for the rotation of both tracer and inertial spheroids in turbulent channel flows.

Characterization of aerosol Stokes number in 90° bends and idealized extrathoracic airways

Prediction of aerosol deposition in the respiratory system is important for improving the efficiency of inhaled drug delivery and for assessing the toxicity of airborne pollutants. Deposition is typically reported as a function of a global Stokes number which is based on a reference flow timescale, or the ratio of the characteristic flow length and velocity scales. In reality, however, particles experience varying flow timescales as they are advected through the airways, which motivates the use of an instantaneous Stokes number based on the local properties of the flow field. We then define the effective Stokes number as the time-average of the instantaneous value. This effective Stokes number thus encapsulates the flow history and geometric variability, and provides a more detailed account of the particle trajectory in the flow. Laminar and turbulent flows in a curved pipe are examined first and provide a simplified, or canonical, configuration of the flow in the upper airways. They are followed by a study of turbulent flow in an idealized mouth–throat geometry. Our results demonstrate that the effective Stokes number can deviate significantly from the traditional value based solely on the reference flow timescale. In addition, the effective Stokes number shows a clear correlation with deposition efficiency and can therefore be used to determine optimal aerosol release locations in order to minimize extrathoracic losses.

Geometrical Optics of Dense Aerosols: Forming Dense Plasma Slabs

Assembling a freestanding, sharp-edged slab of homogeneous material that is much denser than gas, but much more rarefied than a solid, is an outstanding technological challenge. The solution may lie in focusing a dense aerosol to assume this geometry. However, whereas the geometrical optics of dilute aerosols is a well-developed field, the dense aerosol limit is mostly unexplored. Yet controlling the geometrical optics of dense aerosols is necessary in preparing such a material slab. Focusing dense aerosols is shown here to be possible, but the finite particle density reduces the effective Stokes number of the flow, a critical result for controlled focusing.

Investigating the small-scale structure of clouds using the δ-correlated closure: effect of particle inertia, condensation/evaporation and intermittency

Equations for the first and second moments of particle density are closed exactly in the large Prandtl number limit, using the δ-correlated closure whereby the turbulent velocity field is assumed to rapidly decorrelate in time. Results are summarized from two recent studies that have investigated the effect of both particle inertia and condensation/evaporation on the viscous–convective subrange. Analytic expressions for the spectrum of inertial particles are presented which show that clumping (preferential concentration) does not occur for Stokes number ( St) less than about 0.2. Also presented are analytic expressions for the scalar spectrum of cloud liquid water density derived from a simple mean-field model of condensation/evaporation. The model reproduces new experimental observations [J. Geophys. Res. 104 (1999) 6123] of cloud liquid water content (LWC) fluctuations that exhibit anomalous near-inertial scaling. For the first time, the effect of high Reynolds number ( Re λ) velocity field intermittency on preferential concentration is considered in a quantitative manner. A Re λ-dependent effective Stokes number ( St eff) is derived that is proportional to the square root of the flatness factor of the longitudinal velocity derivative. In the atmospheric boundary-layer, St eff≈2.7 St. These results support Shaw et al.'s [J. Atmos. Sci. 55 (1998) 1965] hypothesis that velocity field intermittency tends to increase preferential concentration at St<1. However, in contrast with Shaw et al., I demonstrate that, in real turbulence, vortex tubes do not statistically affect St eff and, hence, preferential concentration.

Particle Impaction on a Supersonic Cone

The theoretical collection efficiency has been determined for particles impinging on a supersonic cone with an attached shock. By defining a moving coordinate system such that the gas is stationary on the cone surface behind the shock and scaling the particle Stokes number with a shock-to-cone distance, a similarity parameter has been denned that properly accounts for the effects of freestream Mach number, cone angle, and large particle Reynolds number. The effective Stokes number, which represents the ratio of the particle stopping distance in the moving frame to the appropriate shock-to-cone length, is shown to correlate the total collection efficiency.

Inertial deposition of particles from potential flows past cylinder arrays

We exploit the computationally efficient method of images (MOI) to create steady potential flows past finite and infinite circular cylinder arrays, in order to study inertial impaction of particles on them. Although such flows are often encountered in practice (e.g. high Reynolds number “dusty gas” flows past heat exchanger tube banks and filter screens), little is yet known about the proximity effects of adjacent collectors on the capture efficiency of any particular collector in the array. Our results for linear, symmetric arrays in steady cross-flow, containing an odd number of collectors (from three to infinity), over a wide range of array spacings, support the conjecture that an appropriately defined effective Stokes number based on the stagnation region fluid deceleration rate (computed and reported here as a function of the number of collectors and their spacing) can adequately correlate, for practical purposes, the capture efficiency of a representative collector in the array. This scheme, combined with well-known correlations for an isolated collector, simplifies significantly the task of estimating inertial impaction rates on aerodynamically interacting collectors.

Particle deposition after crossing a sudden expansion

Monodispersed aerosols of 5 and 10 μm diameter DOP droplets tagged with sodium fluorescein were deposited on tube walls and the annular disc at the exit plane of expansion sections with ratios D 2/ D 1 = 1.5,1.8,2.25,3.0, and 4.5 ( D 2 = 5.72 cm). Aerosol stream flowrates varied from 6.4 to 21.3 m s −1 (7.29 × 10 3 < Re < 5.10 × 10 4 based on D 1). Tape segments attached to the wall and disc were removed after each test, washed and analysed fluorometrically to determine deposition rates and patterns, both radially and axially. Little differences in deposition were found between disc and tube walls or at various distances along the tube wall up to five diameters from the expansion. Sherwood numbers for deposition correlated well with the group Re p −0.66Stk eff −1 Velocity profiles were measured both radially and axially up to five diameters ( D 2) downstream from the expansion, and turbulent flow properties (Peclet numbers) were compared with published data. Eddy diffusivities for particle concentration and momentum were correlated with the effective Stokes number.

Particle Impaction in Axially Symmetric Supersonic Flow

The results of computations to predict the particle collection efficiency in axially symmetric supersonic flow are presented. In particular, a supersonic gas impinging on a truncated circular cylinder aligned parallel to the freestream has been analyzed. Inviscid solutions are obtained by a slight variant to MacCormack's time-dependent finite difference method. In imposing outflow boundary conditions of fixed Mach number, linear extrapolation of remaining gas properties, and a reflection technique on the cylinder walls, reasonable agreement is demonstrated for the shock detachment distance and stagnation pressure loss. It is demonstrated that the capture efficiency is influenced by particle inertia, departure from Stokes drag, and gas compressibility. By defining an effective Stokes number in terms of the particle stopping distance downstream of the shock that includes the effects of non-Stokesian drag and slip flow, and rescaling the Stokes number with the shock detachment distance, the collection efficiency is shown to be correlated with a single universal curve.

Particle Deposition in Wakes: Inertial and Coulombic Effects on Capture By Single and Multiple Spheres in Line

Ten-micrometer dioctylphthalate droplets tagged with sodium fluorescein were collected on six metal spheres, 0.6 cm in diameter and arranged in a line parallel to the axis of flow. The particles carried negative charge in the range 0.9–1.7 × 10−5 stC while the collectors were charged positively in the range 8.0–23.0 stC. The carrier stream velocities used were 600, 1200, 1500, 1800, and 2400 cm/s and spacings between targets of two through six diameters center-to-center were tested. Collection efficiencies for each target were determined by measuring the number of deposited particles, and the particle concentration in the air stream using fluorometric techniques. An effective Coulombic attraction parameter K Eeff , was defined to take into account non-Stokesian drag during particle collection at high velocities. This term was combined with an effective Stokes number to provide the empirical relation, for single spherical targets: η= 1.4[Stk eff + (2K Eeff )1/2]. A model is presented to describe the concen...

Particle sampling in supersonic streams with a thin-walled cylindrical probe

The theoretical collection efficiency of a thin-walled cylindrical probe has been determined for nondiffusing particles of low concentration suspended in a supersonic stream. It is demonstrated that the probe capture efficiency is influenced by particle inertia, departures from Stokes drag, gas compressibility, and the probe ingestion rate. By defining an effective Stokes number in terms of the particle stopping distance downstream of the shock, which includes the effects of non-Stokesian drag and slip flow, and rescaling the Stokes number with the shock detachment distance, the reduced probe collection efficiency is shown to be correlated with a single universal curve. This similarity parameter should be useful in attempts to correlate the particle deposition for any geometry in a supersonic stream.

Use of a Generalized Stokes Number to Determine the Aerodynamic Capture Efficiency of Non-Stokesian Particles from a Compressible Gas Flow

The aerodynamic capture efficiency of small but non-diffusing particles suspended in a high-speed stream flowing past a target is known to be influenced by parameters governing (a) small particle inertia, (b) departures from the Stokes drag law (associated with local particle Reynolds numbers greater than unity), and (c) carrier fluid compressibility (at nonnegligible free-stream Mach numbers). By defining an effective Stokes number in terms of the actual (prevailing) particle stopping distance, local fluid viscosity, and inviscid fluid velocity gradient at the target nose, we show that these effects are well correlated in terms of a “standard” (cylindrical collector, Stokes drag, incompressible flow, Re½ ≫ 1) capture efficiency curve. We are thus led to a correlation that (a) simplifies aerosol capture calculations in the parameter range already included in previous numerical solutions, (b) allows rational engineering predictions of deposition in situations not previously specifically calculated, (c) should facilitate the presentation of performance data for gas cleaning equipment and aerosol instruments.