Large Stokes Number Research Articles

Numerical computation of turbulent gas‐solid particle flow in a 90° bend

AbstractA numerical computation of the LDV results of Kliafas and Holt is reported for a turbulent gas‐solid particle flow in a square‐sectioned 90° bend. A Eulerian model with generalized Eulerian solid surface boundary conditions for the particulate‐phase momentum exchanges with solid walls are included. The turbulent closure is effected by using the gas‐phase RNG‐based k‐ϵ turbulence model, and the particulate turbulence diffusivity is related to the turbulent viscosity of the gas phase. Comparisons are made with experimental data for the mean streamwise velocities of both phases, the streamwise turbulence intensity of the gas phase, and the particulate concentration distribution in the bend. The localized high particulate concentration near the outer curve of the bend that occurs at large Stokes number is accurately predicted. Empirical computational evidence is presented for a relaxation of the minimum particle number density required to allow the use of a continuum model.

Structure of a methanol/air coaxial reacting spray near the stabilization region

Planar laser Mie scattering, planar laser-induced fluorescence and two component phase-doppler interferometry have been used to study the reaction zone structure near the stabilization region of a coaxial methanol/air spray flame. The configuration of the experiment was chosen to approximate the atomizer geometry, surface tension and Weber number of a single coaxial rocket injector. The measurements are made in a water-cooled, optically accessible confinement chamber at a pressure of 1 atm. Data are reported for two atomizing air velocity conditions. One yields a flame length of approximately 1 m, the other, half that value. Both the Weber number (characterizing the atomization process) and the Reynolds number (characterizing the gas-phase mixing process) vary between the cases, but the data suggest that it is the Weber number which has the dominant effect. In both cases OH imaging shows that the reaction zone is confined to a narrow region, with the OH field being similar in appearance to that of a single-phase turbulent mixing-controlled (diffusion) flame. Size-classified mean velocity vectors derived from the phase-doppler data show striking differences in the flow pattern for low and high Stokes number droplets. Droplets 5 μm in diameter and below (Stokes number less than 3) appear to follow the recirculating eddies that provide flame stabilization while droplets of large Stokes number travel ballistically through the flow. Increasing the Weber number by a factor of 2.5 decreased the Sauter mean diameter of the spray by as much as one-third, and the arithmetic mean diameter by as much as one-half. We believe that it is this decrease in the spray droplet diameter that is primarily responsible for the very different flame lengths in the two cases.

Flow due to an oscillating sphere and an expression for unsteady drag on the sphere at finite Reynolds number

Unsteady flow due to an oscillating sphere with a velocity U0cosωt’, in which U0 and ω are the amplitude and frequency of the oscillation and t’ is time, is investigated at finite Reynolds number. The methods used are: (i) Fourier mode expansion in the frequency domain; (ii) a time-dependent finite difference technique in the time domain; and (iii) a matched asymptotic expansion for high-frequency oscillation. The flow fields of the steady streaming component, the second and third harmonic components are obtained with the fundamental component. The dependence of the unsteady drag on ω is examined at small and finite Reynolds numbers. For large Stokes number, ε = (ωa2/2v)½ [Gt ] 1, in which a is the radius of the sphere and v is the kinematic viscosity, the numerical result for the unsteady drag agrees well with the high-frequency asymptotic solution; and the Stokes (1851) solution is valid for finite Re at ε [Gt ] 1. For small Strouhal number, St = ωa/U0 [Lt ] 1, the imaginary component of the unsteady drag (Scaled by 6πU0pfva, in which Pf is the fluid density) behaves as Dml ∼ (h0Stlog St–h1St), m = 1,3,5… This is in direct contrast to an earlier result obtained for an unsteady flow over a stationary sphere with a small-amplitude oscillation in the free-stream velocity (hereinafter referred to as the SA case) in which D1∼ –h1St (Mei, Lawrence & Adrian 1991). Computations for flow over a sphere with a free-stream velocity U0(1–α1+α1cosωt’) at Re = U02a/v = 0.2 and St [Lt ] 1 show that h0 for the first mode varies from 0 (at α1 = 0) to around 0.5 (at α1 = 1) and that the SA case is a degenerated case in which the logarithmic dependence of the drag in St is suppressed by the strong mean uniform flow.The numerical results for the unsteady drag are used to examine an approximate particle dynamic equation proposed for spherical particles with finite Reynolds number. The equation includes a quasi-steady drag, an added-mass force, and a modified history force. The approximate expression for the history force in the time domain compares very well with the numerical results of the SA case for all frequencies; it compares favourably for the PO case for moderate and high frequencies; it underestimates slightly the history force for the PO case at low frequency. For a solid sphere settling in a stagnant liquid with zero initial velocity, the velocity history is computed using the proposed particle dynamic equation. The results compare very well with experimental data of Moorman (1955) over a large range of Reynolds numbers. The present particle dynamic equation at finite Re performs consistently better than that proposed by Odar & Hamilton (1964) both qualitatively and quantitatively for three different types of spatially uniform unsteady flows.

The effect of a mean fluid velocity gradient on the streamwise velocity variance of a particle suspended in a turbulent flow

The effect of a mean fluid velocity gradient on the motion of a small solid particle suspended in a turbulent gas is analyzed using Fourier transform techniques. The presence of a mean fluid velocity gradient is shown to elevate the streamwise particle velocity variance above the level predicted without such gradients; the particle velocity variance in the direction normal to the flow is shown to be only indirectly affected by the existence of fluid velocity gradients. When the particle Stokes number is small, the streamwise particle velocity variance is elevated above the level predicted in flows without mean velocity gradients at a rate which is linearly proportional to both the particle Stokes number, α, and the ratio of the streamwise velocity gradient to the characteristic frequency of the energy-containing eddies in the turbulent field, G/ω e , For particles with large Stokes numbers, the dominant mechanism by which particle streamwise velocity fluctuations are generated is through random interaction between the particle path and the mean fluid velocity gradient.

Kinetic theory for a monodisperse gas–solid suspension

The fluid-dynamic and solid-body interactions among a suspension of perfectly elastic particles settling in a viscous gas are studied. The Reynolds number of the particles, Re≡ρfUa/μ, is small but their Stokes number St≡mŪ/(6πμa2) is large, indicating that particle inertia and viscous forces in the fluid are important. Here, ρf is the density of the fluid, m is the mass of a particle, Ū is the average velocity of the particles, a is their radius, and μ is the fluid viscosity. Equations for the particle velocity distribution and averages of the fluid and particle velocities are derived. For very large Stokes numbers, St≫φ−3/2, where φ is the particle volume fraction, solid-body collisions lead to a nearly Maxwellian velocity distribution. On the other hand, at smaller Stokes numbers, St≪φ−3/2, fluid-dynamic interactions play a more important role in determining the particle velocity distribution and the distribution is not Maxwellian. The amount of energy contained in the particle velocity fluctuations is determined by a balance involving fluid-dynamic interactions, even in the case where the solid-body collisions lead to a nearly Maxwellian velocity distribution. The viscous interactions are found to be similar to those in a fixed bed at high Stokes numbers, St≫φ−3/4, where the particles do not respond quickly to changes in the local fluid velocity. A stability analysis of the averaged equations indicates that the suspension is unstable to particle density waves for St≫φ−3/2. The inertia of the particles is destabilizing, while the energy introduced into the fluctuating motions of the particles by viscous flow interactions is stabilizing.

Effects of vortex pairing on particle dispersion in turbulent shear flows

Particle dispersion in large-scale dominated turbulent shear flow is investigated numerically with special emphasis on the effects of the vortex-pairing phenomenon. The particle dispersion is visualized numerically by following the particle trajectories in a flow consisting of large vortices which are undergoing pairing interaction. The flow field is generated by a discrete vortex method. Important global and local fiow quantities from the numerical simulation compare reasonably well with experimental measurements. For both cases of point sources with continuous particle release and an initially distributed line source, the particle dispersion results demonstrate that the extent of particle dispersion depends strongly on the Stokes number, the ratio of the particle aerodynamic response time to the characteristic time of the vortex-pairing flow field. Particles with relatively small Stokes numbers disperse laterally at approximately the saine rate as that of the fluid particles and particles with large Stokes numbers disperse much less than the fluid particles. Particles with intermediate Stokes numbers (0.5-5) may be dispersed laterally farther than the fiuid particles and may actually be flung out of the vortex structures. Due to the strong particie entrainment power, the flow during the vortex-pairing process seems to produce higher particle lateral dispersion than the pre-pairing and post-pairing flows.

The effect of stokes and reynolds numbers on the collection efficiency of grid filters

The collection efficiencies and pressure drops across model grid filters were measured as a function of particle size, gas composition, and flow rate. The monodisperse, latex spheres ranged from 0.36 to 1.0 μm. Measurements indicated that the pressure drop could be correlated with the Fuch-Stechkina equation with a constant C HK ⋍ 0.52 . These results for the argon-helium mixtures had been previously reported for air and nitrogen. A modified channeling model was developed to relate the overall efficiency measurements to a single fiber efficiency. Electron micrographs and measurements with filters of different thicknesses were consistent with the model. The single fiber efficiency was found to be a function of Stokes number alone for Stk > 0.2. At lower Stokes number, the efficiency was a function of both the Reynolds and Stokes numbers. Qualitative agreement was found with the shape of the curve between experiment and trajectory calculations based on the Kuwabara model. However, these calculations considerably underestimated the collection efficiency which was found to agree quantitatively with a trajectory calculation based on potential flow at large Stokes numbers.

Inertial impaction in stagnation flow

Particle motion in the stagnation flow at the front of an obstacle is investigated analytically. The particle motion is described by the Basset-Boussinesq-Oseen equation. An exact solution for the particle motion is obtained. Unlike the normal aerosol theory, this analysis yields no critical Stokes number for impaction. For sufficiently large Stokes number however, buoyancy resulting from fluid deceleration and directed away from the obstacle can exceed the Stokes drag on the particle.

Added Mass of a Sphere in a Bounded Viscous Fluid

The validity of potential flow and Stokes viscous flow theory was investigated experimentally to determine the effect of a real fluid on an oscillating rigid sphere bounded by a concentric spherical shell. Added mass and damping coefficients were determined over a range of Stokes numbers of 400 to 3.5 × 105. Results of the experiments indicate that Stokes viscous flow solution is valid over this entire range of Stokes numbers for small amplitudes of oscillation and that the potential solution is approached asymptotically for large Stokes numbers. A simple apparatus is described that was used to accurately determine both added mass and damping coefficients.