1991 Small Particles in Air

The effect of small particles on fluid turbulence in a flat-plate, turbulent boundary layer in air

C. B. Rogers@ and J. K. Eaton Department of Mechanical Engineering, Stanford University, Stanford, California, 943053030

(Received 11 April 1990; accepted 7 January 1991)

This paper describes the result of both an experimental and an analytical investigation of the response of a two-dimensional, turbulent boundary layer in air to the presence of particles. Copper shot, 70 ,um in diameter, were uniformly introduced into a vertical boundary layer, at a momentum thickness Reynolds number of about 1000. The particle mass flux was set at 20% of the fluid mass flux, and all measurements were made using a single-component, forward- scatter laser Doppler anemometer. The measurements clearly demonstrated that the particles damped fluid turbulence, apparently affecting all scales equally. The measurements further showed a strong correlation between the degree of damping and the particle concentration in the log region of the boundary layer.

I. INTRODUCTION Though common throughout industry, particle-laden

flows have received relatively little attention in the laborato- ry until the past decade. The two-phase environment has been traditionally hostile to the presence of even,simple velocity probes and has therefore hindered accurate velocity measurements. The advent of the laser Doppler anemometer allowed two-phase data to be taken with comparative ease and has greatly enhanced the volume of experimental data. Even with these new data, a coherent understanding of turbulence attenuation by particles allowing accurate flow pre- diction has not emerged.

Turbulence attenuation (i.e., the reduction of the gas- phase turbulent stress levels) has been very difficult to inves- tigate experimentally. One problem is the requirement to measure the gas-phase velocity statistics in the presence of at least moderate particle concentration. Another problem is that nonuniform particle loading will modify the fluid mean flow. This can in turn modify the turbulence level, an effect that cannot be attributed to particle/turbulence interaction. Because of these difficulties, there are only two flows, fully developed pipe flow and axisymmetric jet flow, where turbulence modification has been extensively studied. It is the objective of this work to extend the understanding to another simple shear flow, the flat-plate boundary layer.

One of the problems encountered in investigating particle-laden flows is the large parameter space. An incompressible, single-phase flow can be defined by a flow Reynolds number and geometric parameters. However, the addition of a particulate phase adds several new parameters. In particular, the particle Reynolds number, particle time constant, particle gravitational drift velocity, particle mass loading, particle loading distribution, particle diameter, and particle- to-fluid density ratio may all play an important role in determining both particle and fluid behavior. An experiment can only fill a small niche in the parameter space, but possibly will illuminate the flow behavior over a wider range of parameters. In the following paragraphs, we define where the

a) Present address: Department of Mechanical Engineering, Tufts Universi- ty, Medford, Massachusetts 02155.

present experiments fit within the parameter space. We also examine the presumed effects of varying each parameter in hopes of extending the usefulness of the data.

The particle Reynolds number is defined as

Rep = Urel d,/vp (1)

where Cr,, is the particle relative velocity ( V - U), dP is the particle diameter, and vf is the fluid kinematic viscosity. For the remainder of this paper, we will use the customary notation of U for the fluid velocity and V for the particle velocity, and the subscripts f and p will refer to fluid and particle properties, respectively. This Reynolds number characterizes the flow around the particle; small Reynolds numbers correspond to attached laminar flow and large Reynolds numbers corresponding to fully turbulent particle wakes. Particles with small Reynolds numbers are expected to act only as point sources of momentum exchange between the phases while large particles with high Reynolds numbers are expected to increase turbulence levels by producing turbulent wakes. The particles in the present study had a particle Reynolds number ranging from 4 to 6. Therefore, the results are expected to be representative of the broad class of problems where the particles act only as point sources of momentum exchange. Stokes showed that for Reynolds numbers less than 0.1, the particle drag is linearly related to the relative velocity as

CD = 24/Re,, (2) where C, is the particle drag coefficient. While the present particles had larger Reynolds numbers, Rogers and Eaton showed that the assumption of linear drag was adequate to model the response of the particles to a turbulent flow.

The particle time constant, or particle relaxation time, characterizes particle inertia. For particles with Stokes drag, the time constant rp is defined as

7;, = ppd ;/18/q, (3) where pp is the particle density and ,u~ is the fluid viscosity. Rogers and Eaton showed that using Eq. (3) for particles with Reynolds numbers greater than 0.1, however, leads to overestimation of the particle drag. Therefore, the time con-

928 Phys. Fluids A 3 (5). May 1991 0899-8213/91/050928-10$02.00 @I 1991 American Institute of Physics 928

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stant for these non-Stokesian particles can be estimated from the measured particle drag. Using Stokes formulation of the particle transport equation,

dV,- 1 dt -c v,, - g>

and evaluating it at the particle terminal relative velocity U,, one can estimate a corrected time constant to be

9;, = u,/g. (5) Conventionally, the time constant is presented nondi-

mensionally as the Stokes number which is defined as the ratio of the particle to fluid time scale. Particles with small Stokes numbers ( < 0.01) will follow the flow exactly and will not affect the turbulence except to the degree that they modify the fluid properties. Particles with large Stokes numbers ( > 100) will not respond significantly to turbulent velocity fluctuations. The particles in the present experiment had Stokes numbers between 1 and 10, implying that the particles would follow some, but not all, of the fluid turbulence. Therefore, this experiment is representative of a broad class of situations where the particles respond to turbulent fluctuations but there is a significant relative velocity between the particles and the flow. Unfortunately, it cannot be determined a priori how variation of the Stokes number may change turbulence attenuation.

Gravitational force, too, keeps particles from exactly following the path of a fluid point. Combined with the iner- tial effects described above, these forces will pull the particle through a series of different fluid neighborhoods, making it difficult to predict the behavior of the fluid surroundings in the particle reference frame. This effect is called the crossing-trajectories effect and was first reported by Yudines3 The crossing trajectories effect acts to reduce the fluctuation levels of the particles.4 The related continuity effect5 re- duces the particle fluctuations even more in directions normal to the direction of the gravitational drift. Possible dimensional parameters describing this effect are the ratio of the particle terminal velocity to the mean free-stream velocity or a characteristic turbulent velocity. In the present case, the ratio of the particle terminal velocity to the free-stream velocity is about 10%. More significant, perhaps, is the ratio of the particle terminal velocity to the fluid streamwise tur-

bulence, u,/m), which varies across the boundary layer but is close to 1 at the position of peak turbulenceintensi- ty. The presumed effects of varying this ratio are described in the analysis section.

The particle mass loading, that is the ratio of the total mass flux of particles to the mass flux of the fluid, characterizes the influence of the particles on their surroundings. Low mass loadings imply that the total particle drag is small compared to other forces involved and therefore the fluid behavior remains unaffected by the particle presence. The results presented here were at a mass loading of about 20%. Based on previous experiments, such loadings should have a minor effect on the mean flow behavior. However, as will be shown below they do have a significant effect on the turbulence. For moderate particle loadings, we may guess that the particle loading affects the turbulence linearly; that is if a 10% parti-

cle loading causes a 5.% turbulence attenuation, a 20% loading would cause a 10% attenuation. Such a linear relation would probably not hold if the attenuation became large or if the particle concentration became so large that particle-to- particle interactions were significant.

Nonuniform particle loading will cause mean flow variations leading to changes in the turbulence. A dramatic example would be if all the particles collected in the center of an upward pipe flow. The mean fluid velocity would thus be retarded in the pipe center making a dip in the velocity profile. The velocity gradient would serve as a source of turbulence production. A key objective of the present experiment was to have a uniform particle loading. Nonuniformities are treated as sources of uncertainty. It is assumed that calcula- tion of mean-velocity variations and turbulence modification produced only by nonuniform loading can be estimated analytically.

Variations in particle diameter, with respect to the fluid length scale, will also affect the particle and flow behavior. If the particle diameter is substantially larger than the Kolmo- gorov length scale of the surrounding fluid, then the simple presence of the particle will affect the energy distribution of the 0ow. Turbulent eddies will be locally strained in the vicinity of each particle. In addition, the particle will be ex- posed to a turbulent rather than laminar approach flow. The particles used in this experiment were smaller than the Kol- mogorov length scales of the flow. Therefore, the present results should be representative of all cases where individual particles do not interact directly with turbulent eddies.

Last, the ratio of the particle-to-fluid density is important in determining which forces on the particle are important and which are not. The particle transport equation above [ Eq. (4) ] neglects a large number of forces. The full transport equation would be the one first put forth by Tchen, with the addition of acceleration forces, the Magnus force, spin forces, the Saffman lift force, and electrical forces. For particles with low Reynolds numbers and densities much greater than the fluid, one can show that the affect of these forces is small and therefore Eq. (4) closely approximates the particle behavior.

Once having defined our parameter space, the objective of this research was to examine the interaction between the fluid turbulence and the particle motion in fluid shear. This interaction was isolated from changes in the fluid turbulence due to variations in the fluid mean behavior. For the remainder of this paper, we will refer to fluid turbulence simply as turbulence, since particle turbulence is an unrealistic concept for these low-volume loadings. Using these results, we examine the concept of modeling this interaction as an increased viscosity.

II. RELATED EXPERIMENTAL WORK The problem of turbulence modification by dispersed

particles is receiving increasing attention in recent years.6 The majority of the experimental work has concentrated on two flows, the axisymmetric jet and the axisymmetric pipe flow. Gore and Crowe recently reviewed the available experimental data for these flows. They identified the ratio of the particle diameter to a characteristic length scale of the

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turbulence (d,/l, ) as a key parameter and found that turbulence was attenuated for small values of d/Z, and amplified for large values. The critical value of d,/l, separating regions of attenuation and amplification was of the order of 0.1. This critical value varied with radius inn the pipe flows but not in jet flows. It is surprising that this simple correlation works at all since the particle diameter-to-length scale ratio is just one of the many relevant parameters of the problem. The Gore and Crowe correlation cannot be used to predict the level of turbulence modification. However, it is clear that the present experiment is well within the size range expected to cause turbulence attenuation.

In the remainder of this paper, we will concentrate only on the pipe flow studies since they are directly relevant to the present work on wall-bounded turbulent flows. The pipe flow studies have used a variety of fluid and particle types the most relevant ones being those examining solid particles dispersed in gas flows. In many cases, the particles were introduced nonuniformly, causing mean-velocity variations that overwhelmed the direct effects of particles on turbulence. As a result, there is little uniformity in the data and no generally accepted correlation describing turbulence modulation by particles.

Key papers describing early work in vertical pipe flows are those by Soo et a1.,8 Doig and Roper, and Reddy and Pei. lo Reddy and Pei also provided an extensive review of all of the work prior to 1969. Most of the experiments examined vertical pipe flows with spherical particles at mass loadings ranging up to about 5. Typical instruments used were special pitot probes to measure the mean gas velocity and photo- graphic techniques to measure particle velocities. Most of the results showed that the mean gas velocity profiles were flattened by the presence of particles for loadings above about 1. The particle concentration profile was uniform in some experiments while other experiments showed a higher concentration near the wall. Soo (quoted in Reddy and Pei) reasoned that nonuniform distribution was caused by electrostatic effects. Direct measurements of the gas-phase turbulence were impossible, but Soo et al. inferred the gas- phase turbulence behavior based on helium diffusion measurements. Their results, valid only near the centerline of the pipe, showed that the intensity of the turbulence was not a&cted by the particles, but the Lagrangian integral scale was decreased.

Boothroyd and Walton12 examined the displacement of CO, -He tracers in the presence of 0-40pm zinc powder in a vertical pipe flow of air. Using a Kathometer to measure species concentration they found that for a mass loading of 300%, the particles suppressed turbulent diffusion, especial- ly near the wall. More recent investigations of particle-laden pipe flows have used modified laser anemometers to make measurements of both the particle and gas-phase velocities. In one of the earliest such studies, Carlson and Pesl&3 measured particle velocities in a 7.62 cm square duct. The flow was laden with either 44- or 2 14pm glass beads at mass loadings less than 1. They found that the bulk of particle velocity fluctuations could be attributed to particle size variations, suggesting that such variation may be responsible for increases in gas-phase turbulence when the mass loading is

substantial. Lee and Durst14 made axial velocity measurements in a

2 cm diam pipe laden with 100, 200, 400, or 800 pm glass beads. Mass loadings increased from about 1 for the smallest particles to about 2.5 for the largest. They found that the largest particles had a nearly uniform velocity profile across the pipe and caused a flattening of the gas velocity profile and increases in the turbulence intensity. They also found that smaller particles damped the turbulence. Finally, they documented particle mean velocities exceeding those of the surrounding flow near the wall, implying that perhaps the particles were still accelerating. They proposed a simple model in which the particles either totally ignore the fluid

turbulence, or follow it exactly. Dividing their flow into these two regimes, they used their model to predict their results with a fair degree-of accuracy. :

Arnason and Stock injected 57 ym glass beads on the centerline of a vertically downward pipe flow. They noticed an increase in the flow turbulence as they increased the particle loading. However, the nonuniform particle distribution would create a distorted mean-velocity profile which may have been responsible for the increase in the turbulence.

The most complete experiment for particle-laden pipe flows was performed by Tsuji et al. and examined the flow in both horizontal6 and vertical pipes. They examined the fluid turbulence in the presence of particles ranging from 200 to 30qOpm and for mass loadings up to 610%. They found that the small particles tended to suppress the iluid turbulence whereas the large particles enhanced the fluid turbulence, The. intermediate particles enhanced the fluid turbulence near the pipe centerline and reduced the fluid turbulence near the wall. They also showed that the large particles added broadband energy, increasing the energy content at all frequencies equally, whereas the small particles increased the high-frequency energy content of the ffow and decreased the low-frequency- content.

Analytical and numerical studies of turbulence attenuation have been restricted by a lack of fundamental understanding of particle-turbulence interactions. Elghobashi and co-workersi9 modified the transport equations for the kinetic energy and the dissipation rate in the k-epsilon model to account for damping of turbulence by particles, The model worked well for the jet flow but may be quite difficult to generalize to different flows. Leems2 and Lee and Borne? developed a model based on theory and empirical input from the pipe flow experiments of Lee and Durst and Tsuji and Morikawa. The model divided the flow into regions where the dominant flow frequency was above or below the particles cutoff frequency. In the low-frequency region, particles were assumed to diffuse like 0uid points while in high- frequency regions, they respond only to the mean-velocity field. Another element of the model was an empirical correlation for an effective turbulent viscosity seen by the particles. The model represented the pipe flow data well but cannot be generalized to other flow geometries easily. In particular, Rogers and Eaton showed that the model did not correctly predict the present data.

.-

In a sister study to the present work, Squires and Ea- ton23*24mhave used direct numerical simulation of the incom-

930 Phys. Fluids A, Vol. 3, No. 5, May 1QQl C. B. Rogers and J. K. Eaton 930


pressible Navier-Stokes equations coupled with Lagrangian tracking of up to one million particles to study turbulence damping in homogeneous, isotropic turbulence. The particle Stokes number ranged from 0.14 to 1.5 for the computations. Their work was restricted to relatively low-turbulence Reyn- olds numbers, but showed significant turbulence damping for a particle mass loading of 10%. At 100% particle mass loading the turbulence kinetic energy was reduced by nearly one-half.

III. EXPERIMENTAL SETUP Floi Direction

In order to examine a particle-laden boundary layer, we built a low-turbulence air boundary layer wind tunnel with a unique particle feed mechanism for uniform particle loading. Figure 1 is a schematic of the tunnel itself, showing the vertical orientation of the test section, minimizing the effect of gravity on the local particle concentrations. Air was drawn through a filter into a seeding chamber where talcum powder Aow tracer was dispersed using an air brush. The talcum powder was suspended inalcohol which evaporated before the seed reached the test section. The flow was drawn into a centrifugal blower then passed through a diffuser, a flow conditioning section, and a turn to the vertical orientation.

FIG. 2. Particle feeder schematic.

loading uniformity was ensured through visual checks using a laser sheet.

The particle phase was added downstream of the flow conditioning using a mechanism designed for optimal uni- forrnity.and steadiness of the mass loading. The particles were dispersed into the flow by 1.3 cm tall, 1.0 cm diam buckets (see Fig. 2)) having a 1 .O cm opening on the top and a 0.15 cm hole in the bottom. This geometry allowed the buckets to fill rapidly on either side of the wind tunnel before slowly draining their contents as they crossed the flow. These buckets were traversed across the wind tunnel by two sets of belts. Since the draining rate of the particles out of the bucket remained constant as the bucket was drawn across the tunnel, the particle flux proved to be extremely uniform under each belt. Grids downstream of the feeder enhanced the lateral mixing of the particles. Two hundred buckets were used to get the required mass loading rate of 20%. The

The flow then passed through a 3:1 area ratio, two-dimensional contraction. A section of honeycomb was in- stalled at the exit of the contraction to eliminate the small normal velocity of the particles. In the absence of the honeycomb, the contraction curvature accelerated the particles to- ward the tunnel center. With the honeycomb in place, the concentration was uniform at the test section entrance. The free-stream turbulence level (of 1%) was higher than de- sired but apparently had little effect on the boundary-layer development.

The boundary layer developed in a 7.6 by 46 cm rectangular duct that was 114 cm long. A 0.8 mm high by 13 mm long rectangular boundary-layer trip was mounted at the test section entrance to ensure a uniform, two-dimensional transition in the relatively low Reynolds number boundary layer.

The particles used in this experiment were nominally 70 pm diam spherical copper beads with a material density of 8800 kg[m3. They were commercially sized classified to a t 5 ,um window to limit,the effects due to particle diameter

variations. Microscopic examination showed that most of the particles were nearly spherical with some particles more elongated. The actual size distribution and the corresponding aerodynamic time constants are shown in Fig. 3. These data were measured using a Coulter counter. An actual photograph of the particles is shown in Fig. 4.

Velocity statistics were obtained using a single-compo- m EXHAUST DUCT

CYCLONE

PARTICLE HOPPERS

PARTICLE FEED

Bucket Cross-section:

1.0 CIll

1.3 cm

B 5 5

a 0 40 60 a0

Diameter [pm]

FIG. 3. Diameter distribution of particles. FIG. 1. Wind tunnel schematic.

931 Phys. Fluids A, Vol. 3, No. 5, May 1991 C. B. Rogers and J. K. Eaton 931


FIG. 4. Photograph of the particles.

nent, forward-scatter laser Doppler anemometer which was used to measure the gas flow without particles (i.e., unladen flow), the gas flow in the presence of particles (i.e., laden flow), and the particles themselves. Measurements of the gas phase required flow tracer particles that were talcum powder with a nominal diameter of 1 pm. Measured velocity statistics for the unladen flow agreed to those taken by a hot wire to within l%, implying that the tracers followed the flow exactly for the velocities used in this experiment.

The local density approximation LDA system used a low-power, helium-neon laser, TSI transmitting optics, and on-axis forward-scattering, collection optics, with a TSI model 1980B counter-type signal processor. Table I shows the specific LDA parameters used for thedata-taking pro- cess. The key parameters are a measuring volume diameter of approximately 0.5 mm and no Bragg shifting .used for axial velocity measurements.

The LDA transmitting and receiving optics were rigidly coupled on a computer-controlled traverse system. The measuring position was read using an Accurite miniscale linear encoder with better than 13 ,om resolution. The traverse was controlled by an IBM PC which also collected the data digi- tally from the LDA signal processor and controlled the wind tunnel speed.

Discrimination between gas-phase tracers and the copper beads was done on the basis of signal amplitude. This method was effective due to the distinct size gap between the

largest tracer particles and the smallest copper particles. To measure the particle phase, the preamplifier gain on the counter processor was reduced until flow tracers could not be detected. Therefore, there was no contamination of particle-phase measurements. To measure the gas-phase in the presence of copper particles the preamplifier gain was increased and an amplitude discriminator circuit used. High- amplitude signals were not processed. Cross talk can occur in such a system when a large particle grazes the measuring volume producing a small signal amplitude. This effect is minimized by requiring a relatively large number of fringe crossings for a vali-d measurement. In adjusting the system, we relied on the fact that the mean-velocity difference between the particles and gas in the free stream was larger than the standard deviation of the velocity. Therefore, cross talk was easily detected. Extensive qualification experiments are reported in Rogers and Eaton. These experiments showed that the effect of cross talk was minimal for particle mass loadings up to 20%.

Fluid power spectra were measured using the method of Gaster and Roberts. Due to the random arrival of a velocity measurement (i.e., a tracer particle), one cannot use the conventional method of performing a fast Fourier transform on incoming data if the data rate is close to the frequencies present in the flow. Therefore, we performed a direct Fourier transform on autocorrelations formed from 60 000 instantaneous velocity measurements, separated into 1000 time bins, each 0.0005 set wide. Figure 5 favorably compares the results of this method with conventionally processed, hot-wire measurements. . The experimental uncertainty m the mean-velocity statistics presented in this paper is close to 6%. Measurement uncertainty arose from inaccurate phase discrimination, slight loading nonuniformity, small signal-to-noise ratio, and from limitations inherent in finite data records. The mass loading of 20% used in this experiment was the maxi- mum loading obtainable before the increase in background noise due to the copper particles became too large for accurate data validation.

Finally, the problem of particle loading uniformity be- comes somewhat more complex in a shear layer. Although both the particle flux and particle concentration were uniform across the test section entrance, as the boundary layer grew, the particle flux initially remained constant, therefore

TABLE I. Operating conditions for LDA measurements. I

Laser settings Laser beam Wavelength 632.8 nm (red) Diameter 480.5 mm Power 4mW

Measuring Beam half-angle 2.98 volume Diameter 0.48 mm

Fringe spacing 6.1 pm Number of fringes 80

Counter settings Streamwise Number of cycles 16 velocities Shift frequency OHz

Comparison setting 1% Exponent setting Automatic Filter settings 0.3-3 MHz Data transfer Digital IO

FIG. 5. Spectral check: power spectrum at X = 55 cm and y+ = 300.

932 Phys. Fluids A, Vol. 3, No. 5, May 1991 C. B. Rogers and J. K. Eaton 932


increasing the local particle concentration in the low-velocity regions. The particle concentration eventually became uniform across the boundary layer farther downstream due to turbulent mixing. Thus initially in the boundary layer the particle flux is constant, and farther downstream the particle concentration approaches uniformity. Figure 6 compares constant particle flux to constant particle concentration drag profiles of a cloud of particles as a function of position in the boundary layer. These curves result from simple argu- ments assuming that the particles were at their terminal velocity and that the total drag force of the particles is simply the product of the particle concentration and the weight of an individual particle. Therefore, for the case of uniform particle concentration, the total drag of the particles is uniform across the boundary layer and for uniform particle flux, the drag varies like the inverse of the local particle velocity. At the measuring locations presented in this paper, the particle concentrations were close to uniform, as evi- denced by the similarity between the laser anemometer data rate profile and the mean-velocity profile.

IV. RESULTS Boundary-layer profiles were measured at two stream-

wise locations: 55 and 85 cm downstream from the trip. Each profile consisted of 25 measurement points and each point was the result of 4000 individual velocity samples. Integral boundary-layer parameters are compiled in Table II. In order to establish that the boundary layer measured in this experiment is similar to the others appearing in the litera- ture, we compared our velocity profiles with those of Purtell and Klebanoe6 and our velocity spectra with those taken by Johnson and Johnston.7 Figure 7 compares the mean fluid velocity profiles, showing excellent agreement between our data and that of Purtell and Klebanoff. The streamwise turbulence intensity (Fig. 8) measured in our boundary layer agrees well with the data of Purtell and Klebanoff in the boundary layer but the effect of the honeycomb at the contraction exit can be seen in the higher free-stream turbulence intensity values measured in our tunnel. The comparison of velocity spectral data between our boundary layer and that of Johnson and Johnston (Fig. 9) shows that the turbulence in the boundary layer is characteristic of a-low Reynolds number turbulent boundary layer.

Figures 10 and 11 show the laden and unladen fluid

-Uniform Parficle Flux -.-_-Uniform Partlcle Concentration

0 2 4 6 8 10 12

TABLE II. Fluid parameters (dissipation from Murlis et ol., Ref. 28: Re, = 1089).

Flow parameters X = 55 cm Boundary-layer thickness (S,, ) 20 mm Free-stream velocity 8.0 m/set Displacement thickness (a*) 3.0 mm Momentum thickness (6) 2.1 mm Shape factor (H) 1.4 Rep 1550 R% 1090 UT 0.38 m/set Skin friction coefficient ( Cf/2) 0.0022

X = 85 cm Boundary-layer thickness (S,, ) 24 mm Free-stream velocity 8.2 trdsec Displacement thickness (S*) 3.8 mm Momentum thickness (0) 2.6 mm Shape factor (H) 1.4 Re, 2020 Ree 1410 cr, 0.37 m/set Skin friction coefficient (C/2) 0.0020

Flow scales X= 55 cm, yf =300 Dissipation 5.0 m/set

Kolmogorov leligth scale 0.16 mm Integral scale (streamwise) 7.3 mm Integral scale (normal) 3.8 mm

mean-velocity profile at both X locations downstream of the boundary-layer trip. The tunnel was readjusted in the case of the laden flow to account for the bulk drag of all the particles in the tunnel. That is, the free-stream fluid velocity at the tunnel inlet was set to be the same in both the laden and unladen flows. Since the boundary layers at the inlet are thin, this is equivalent to holding the mass flow of air constant. Due to the high experimental uncertainty, it is difficult to ascertain whether the particles actually caused a change in the shape of the fluid mean-velocity profile. Close examina-

8 2

0.9

3

0.8 __ Purtall Data 0 Present Experiment

0.7 /

y/S* YIP

FIG. 6. Drag variation across the boundary layer. FIG. 7. Qualification: mean-velocity profile comparison.

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FIG. 8. Qualification: fluctuating velocity profile comparison. FIG. 10. Mean-velocity profiles: X = 55 cm.

tion of the local probability density function where the curves appear to deviate suggests that the difference results from experimental uncertainty. If the presence of the particles changes the turbulent shear stress, then the mean-velocity profile must also change. Such changes, however, develop slowly and would probably not be apparent in the relatively short test section.

Since changes in the mean flow velocity statistics appear to be minor, any decrease in fluid turbulence is most likely due to a turbulence-particle interaction. The comparison of the laden and unladen flow streamwise turbulence intensities across the boundary layer at X = 55 cm [Fig. 12) shows no evidence of turbulence-particle interaction, whereas the downstream velocity profile at the X = 85 cm location (Fig. 13 ) shows up to a 35% decrease in the measured fluid turbulence intensities. This substantial damping of the fluid turbulence cannot result solely from the high measurement uncertainty because the possible sources of uncertainty would tend to increase rather than decrease the measured turbulence fluctuations. The increase in the measured free-stream turbulence, however, does fall within the range of this measurement uncertainty.

The turbulence/particle interaction seen in the present experiment had many similarities to that documented by Tsuji et al. for a vertical fully developed pipe flow. Their data show the same trend of larger turbulence suppression by the particles in the end of the log region and the beginning of the wake region of the boundary layer. This uneven suppression

k B 5 s 10-l 7 z I -Johnson Data E - - - -Current Experiment \

lo?

is most likely a result of the uneven drag loading discussed previously and possibly variations in the particle initial conditions, fluid length scales, and particle/wall interactions. The fluid power spectra of both the present study and the work of Tsuji et al., too, showed similar behavior. Figures 14 and 15 show the fluid power spectra at two positions in the boundary layer at theX = 85 cm location. Both experiments clearly showed particle-induced turbulence attenuation of the low-frequency content and an increase in the high-frequency content of the streamwise component of the fluid energy.

The increase in the high-frequency content of the flow is most likely not a real phenomenon but rather a measurement phenomenon. One would expect to measure energy at frequencies corresponding to the reciprocal of the time between particles crossing the measuring volume; about IO3 Hz. Most of the fluid velocity readings measured in the vicinity of the particle would not be validated since both the particle and

the tracer would be scattering light simultaneously and the resulting beat frequency would inhibit signal validation. Cases would exist, however, where the particle would be outside the measuring volume and the tracer would be inside. It is these cases that most likely increase the high-frequency content of the measured fluid energy. Furthermore, any contamination of the fluid signal by particles would show up in the high frequencies. Tsuji et al. noticed that as the particle concentration was increased, this high-frequency content correspondingly increased, possibly a result of an

0.08

8 0.06

$

4.J 0.04

0.02

0 0 2 4 6 a 10 12

y/S

FIG. 9. Qualification: power spectra comparison. FIG. 11. Fluctuating velocity profiles: X= 55 cm.

934 Phys. Fluids A, Vol. 3, No. 5, May 1991 C. EL Rogers and J. K. Eaton 934


1.0

3. 0.0

3

0.8 --

0.7 ~ I- i.

Ll-LJ-L-- d 1 1 1 1 1 1 I 1 1 I 0 2 4 6 8 10 12

y/S'

1o.J-- 1 1 10 1.111 102 -----JJ 103 _= .~ .".. f [Hz] .- FIG. 12. Mean-velocity profiles: A= 85 cm. FIG. 14. Flow power spectra: X = 85 cm, y + .= 100.

increased number of particles skirting the measuring volume with a tracer inside. Full numerical simulations. (e.g., Squires and Eaton) that resolve all scales of fluid turbulence but do not resolve the local perturbations of the flow around the particles, do not measure any increase in the high-frequency content of the fluid power spectrum. In fact, they measured broadband attenuation across all frequencies due to the particles.

._ .- V. ANALYTICAL EXAMINATION

The momentum equation for an incompressible fluid bearing fine solid particles is

?f!L + upi,, = - -LpJ + vfui,ii - at

LFi, Pf ff ._

where indicial notation is used and commas imply deriva- tives. Fi in this equation is the instantaneous drag force per unit volume applied by the particles to the fluid. Assuming that the particles occupy a negligible volume fraction and do not react with the fluid (i.e., the present case), the continuity equation for the fluid remains unchanged by the addition of particles. Further limiting the scope of this analysis to particles obeying a linear drag law, the drag force can be:expressed as

F, = tWrp)(ut -oil, (7) where Q, is the local particle concentration (mass of particles per unit volume). Note that this assumption does not require the use of the Stokes drag coefficient.

O.OT I a i I I h I 1 A 0.08 0 Unladen Flow ($=O)

0 Laden f%w (cpJ.2)

8 0.06

$ 3 0.04

I. 0.02

0 0 i 4 6 8 10 12

YIP

The Reynolds decomposit ion, in which all quantities are expressed as the sum of mean and fluctuating parts, is applied to develop the transport equation for the turbulent kinetic energy. We assume that the flow is homogeneous so we can neglect transport and diffusion terms that will compli- cate the equation and obscure the. turbulence damping terms. Since the particles will only indirectly affect these terms; by varying the fluid velocity, the bulk of.the particle/ turbulence interaction will appear in an additional term in the transport equation for kinetic energy. Because the flow is incompressible and the particle volume is negligible, the fluid density is constant. The transport equation for the turbulent kinetic energy is then

B q -irtT=P-E- $-[.@ (&m,. rP

+ (Q --Fi) q .L. ___ ~

+ ( wu;u; - wu;q, 1, (8) where the turbulence production term, P and the viscous dissipation term E are identical to the production and dissipation terms in a single-phase flow. This equation can be further simplified for the conditions of the present experiment. Using direct numerical simulation, Squires and Eaton showed that W is very small for Stokes numbers greater than one. Therefore the triple correlations terms are negligible compared to the other terms. Also, from the data in the present experiments, fi, - 7, is about the same as ( z) , n2 - p2 is very small, and g3 - F3, is identically zero (for

-----.--L- B 2 ?: g

lo.3 ? ---Unladen F6.Q (t$=O.O) I -- - Laden Flow ($=O.i?)

a

_. lo.4 1

10 102 103

f WI

FIG. 13. Fluctuating velocity profiles: X = 85 cm. FIG. 15. Flow power spectra: X= 85 cm, .J = 300.

935 Phys. Fluids A, Vol. 3, No. 5, May 1991 C. El. Rogers and J. K. Eaton 935 Downloaded 11 Apr 2005 to 132.77.4.129. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp

the purposes of discussion, we took X, as the streamwise coordinate, X, as the normal coordinate, and X, as the span- wise coordinate). Therefore the term ( ni - F{ ) @u; is of the same order as the triple correlation terms and can be ignored. The appropriate transport equation for the turbulent kinetic energy in the present conditions is then

DC :y=P-e-- , I - UiVi). '"r(&y -7-T (9) ut 2 PfTP

This equation is identical. to the single-phase equation with the addition of the last term due to particle drag. It should be noted that the form of the viscous dissipation term, E is identical to the equivalent term in the single-phase equation. The transport equation for dissipation, however, will be affected by the particles as it is a function of the fluid turbulence. Any change in the fluid turbulence due to the presence of the particles will affect the amount of viscous dissipation. Under the present conditions, the greatest change in the fluid turbulence is of the order of 10% and therefore the major cause of turbulence modification will be the drag term in Eq. (9) because it appears directly in the transport equation of the turbulent kinetic energy.

The modification term can be expressed alternately as

(@/pf~p)[ G - RBi( -$)""( $,"'I, (10)

where R,,, is the correlation coefficient between fluid and particle velocity fluctuations. In stationary, homogeneous turbulence, ( e) 12 must be less than or equal to ( z) 12. Therefore, particles that obey a linear drag law and do not produce turbulent wakes must cause turbulence attenuation in such flows. In inhomogeneous flows, however, the particle velocity fluctuation levels could possibly exceed the fluid turbulence intensity (e.g., resulting from wall interactions).

In the present case, the streamwise particle fluctuation levels are nearly the same as the streamwise turbulence intensity. This does not imply that the fluid and particle velocities are perfectly correlated but we might expect that u; u; is a large fraction of u; u; . Previous measurements of a more lightly laden flow in the same facility (Rogers and Eaton) showed that the normal particle velocity fluctuations are much less than the corresponding fluid velocity fluctuations - - so the terms u; u; and u; v; must be small relative to -u; u; and u; u;, respectively. Thus U;LJ; is less than ufu; and therefore the particle-associated term in Eq. (9) should lead to turbulence damping.

In order to be more quantitative, the fluid/particle velocity correlation ulvf must be estimated. Following the discussion of the previous paragraph, we can guess that uiv; is between 20% and 50% of U;ul. Evaluating the production and turbulence kinetic energy using the data of Murlis et ai. for a similar single-phase boundary layer at y/S = 0.4 yields a production level of 11 m/sec and a particle damping term between 1.2 and 0.8 mz/sec3. In fact, if the particles are unwrrelated with the fluid, the damping term would be 1.5 m2/sec3, or 14% of the production. Thus the particle damping term is on the order of 10% of the production term, corresponding roughly to the observed level of turbulence damping.

Using the analysis above and the present data we can estimate the effects of varying the particle parameters on turbulence attenuation. First we consider varying the particle terminal velocity independent.of other parameters. This could be done, for example, by varying the particle material density. Rogers and Eaton showed that particle response to turbulence fluctuations is controlled by the ratio of the particle relative velocity to the gas velocity. As the terminal velocity increases, the spectrum of turbulence fluctuations observed by the particles shifts to a higher frequency. This decreases the particle/fluid velocity correlation and increases the turbulence attenuation.

The analysis is not so simple when the particle time constant is varied, holding the terminal velocity constant. In- creasing the time constant reduced the particle/fluid velocity correlation, which should increase the turbulence attenuation. However, the particle time constant also appears in the denominator of the turbulence attenuation term. Thus, the variation of the turbulence attenuation with time, constant cannot be predicted without a model of the fluid turbulence spectrum, as observed by the particles. Squires and Eaton showed that the turbulence attenuation is insensi- tive to the particle time constant for homogeneous flows with Stokes numbers between 0.14 and 1.5. Further experimental and numerical studies are needed to determine how the attenuation will vary over a larger range of Stokes number and in inhomogeneous flows.

VI. CONCLUSION The results from this work showed four important

points. First, the measurements presented here are, as far as we knoti, the first measurements that isolate particle/turbulence interaction in a flat-plate boundary layer. Second, for Stokes numbers of order 1 and Reynolds numbers around 5, mass loadings of 20% will cause a significant suppression of the fluid turbulence. Third, the suppression of the turbulence appears to be a strong function of the local particle concentration in the near-wall regions of high turbulent kinetic energy of the fluid. And fourth, the particles appear to take energy from all fluid scales equally.

ACKNOWLEDGMENTS The authors are grateful for sponsorship by a Presiden-

tial Young Investigator Award from the National Science Foundation (Grant No. MEA-83-51417). The grant was generously matched by the General Motors Corporation.

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1991 Small Particles in Air

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