Experimental investigation of turbulent diffusion of slightly buoyantdroplets in locally isotropic turbulence

Balaji Gopalan, Edwin Malkiel, and Joseph KatzDepartment of Mechanical Engineering, The Johns Hopkins University, Baltimore,Maryland 21218-7736, USA

�Received 7 January 2008; accepted 26 June 2008; published online 2 September 2008�

High-speed inline digital holographic cinematography is used for studying turbulent diffusion ofslightly buoyant 0.5–1.2 mm diameter diesel droplets and 50 �m diameter neutral density particles.Experiments are performed in a 50�50�70 mm3 sample volume in a controlled, nearly isotropicturbulence facility, which is characterized by two dimensional particle image velocimetry. Anautomated tracking program has been used for measuring velocity time history of more than 17 000droplets and 15 000 particles. For most of the present conditions, rms values of horizontal dropletvelocity exceed those of the fluid. The rms values of droplet vertical velocity are higher than thoseof the fluid only for the highest turbulence level. The turbulent diffusion coefficient is calculated byintegration of the ensemble-averaged Lagrangian velocity autocovariance. Trends of the asymptoticdroplet diffusion coefficient are examined by noting that it can be viewed as a product of a meansquare velocity and a diffusion time scale. To compare the effects of turbulence and buoyancy, theturbulence intensity �ui�� is scaled by the droplet quiescent rise velocity �Uq�. The droplet diffusioncoefficients in horizontal and vertical directions are lower than those of the fluid at low normalizedturbulence intensity, but exceed it with increasing normalized turbulence intensity. For most of thepresent conditions the droplet horizontal diffusion coefficient is higher than the vertical diffusioncoefficient, consistent with trends of the droplet velocity fluctuations and in contrast to the trends ofthe diffusion timescales. The droplet diffusion coefficients scaled by the product of turbulenceintensity and an integral length scale are a monotonically increasing function of ui� /Uq. © 2008American Institute of Physics. �DOI: 10.1063/1.2969470�

I. INTRODUCTION

A. Motive

Currently most crude oil is transported in ships throughoceans, seas, and other waterways. Oil spills are not onlyexpensive to clean up but also cause significant damage tothe environment and marine life. Furthermore, after the ini-tial cleanup effort, which clears large patches, the remainingoil is broken up by waves and turbulence and by the use ofdispersants into small droplets, which increases their surfacearea, thus presumably increasing the rate of decomposition.Quantitative data on the dispersion rate of these droplets byoceanic turbulence are needed for predicting and modelingthe environmental impact of oil spills. Most of the presentdata on oceanic dispersion rates of fuel come from the dyebased experiments �e.g., Talbot and Talbot1 and Moraleset al.2�, which essentially measures the dispersion rate ofpassive scalars rather than that of the buoyant oil droplets.Thus, we have no field data on droplet dispersion. In lookingfor data in previous laboratory or computational studies, onenotes that most of the attention has been paid to transport andturbulent dispersion of bubbles or heavy particles, e.g.,Csanady,3 Reeks,4 Wells and Stock,5 Yeung and Pope,6 Wangand Stock,7 Longmire and Eaton,8 Wang and Maxey,9

Fung,10 Stout et al.,11 Yang and Lei,12 Spelt andBiesheuvel,13 Yang et al.,14 Poorte and Biesheuvel,15

Mazzitelli and Lohse,16 and Snyder et al.17 However, thedensity of most liquid oils deviates only slightly from that of

water, more commonly being slightly buoyant. Experimentsperformed by Friedman and Katz18 show that trends of themean rise rate of diesel oil droplets in locally isotropic tur-bulence differ significantly from that of bubbles or heavyparticles. Hence, one should also expect differences in thedispersion rates of these slightly buoyant droplets. Data andunderstanding of scaling trends of oil droplet dispersion areessential for predicting and modeling of oil spill.

B. Background

The idea of looking at mixing and dispersion by turbu-lence from a Lagrangian viewpoint, which we utilize in thispaper, has been first proposed by Taylor.19 Subsequently, dis-persion of particles has been studied primarily in the contextof atmospheric diffusion �Csanady3 and Yudine20�, where thedensity of the dispersed phase is much greater than that ofthe continuous phase. Theoretical studies, e.g., Reeks4 andPismen and Nir,21 have been performed assuming that inertiadominates, hence simplifying the force balance equation. Inthe present slightly buoyant droplet case, both inertia andbuoyancy are equally important, complicating the analysis.More recent computational studies, involving one-way cou-pling, have examined dispersion of both dense particles andlight bubbles, e.g., Squires and Eaton22 and Wang andStock,7 Elghobashi and Truesdell,23 Mei et al.,24 Spelt andBiesheuvel,13 Mazzitelli and Lohse,16 and Snyder et al.17 Inthese studies, a force balance equation is solved for the La-

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grangian motion of particles or bubbles. These simulationsinherently require force coefficients, e.g., for lift and drag,which are either measured experimentally or obtained fromdetailed numerical simulations of flows around isolated par-ticles, e.g., Mei and Adrian,25 Takagi et al.,26 Magnaudet andEames,27 and Bagchi and Balachander.28 There are a fewcomputations of turbulent diffusion that involve two-waycoupling, e.g., Druzhinin and Elghobashi29 and Mazzitelliand Lohse,16 that show significant differences from one-waycoupling results. Even in this case, particles and bubbles aremodeled as point forces.

The main difficulty in experimental studies of particledispersion is the need to follow their three dimensional �3D�trajectories. Further complications arise from using windtunnels that have a large mean velocity to create nearly iso-tropic turbulence conditions. Snyder and Lumley30 overcomethis problem by using ten cameras placed successively toincrease their field of view. They have observed that as theparticle density increases, its diffusion timescale and conse-quently its diffusion coefficient decrease. Sato andYamamotto31 mounted their camera on a 3D translation sys-tem in order to follow tracer fluid particles in real time,which inherently limits them to low Reynolds numbers. Theyhave calculated the Lagrangian autocorrelation function ofthe fluid particle and use it to obtain the ratio of the Lagrang-ian to Eulerian diffusion timescale as a function of Reynoldsnumber. Wells and Stock5 have measured the concentrationof particles originating from a fixed source. They usecharged particles, whose sizes are smaller than that of theKolmogorov scale, in an electric field as a means of varyingthe body force. They are thus able to isolate the effect ofinertia �size� and settling velocity, which gives rise to cross-ing of trajectories. They also observe that as the settling ve-locity increases the diffusion coefficient decreases. Poorteand Biesheuvel15 have performed extensive measurements ofdispersion of single bubbles in isotropic turbulence in a wa-ter tunnel using 3D position sensitive detectors and observedsome skewness in the probability density function �PDF� ofbubble velocity and bubble displacement.

Recently, with the advent of high-speed measurementtechniques, several experimental studies �Voth et al.,32,33

La Porta et al.,34 and Luthi et al.35� have examined turbu-lence intermittency using 3D acceleration measurements in aLagrangian framework over a small sample volume. As wediscuss below, measurement of diffusion requires followingthe particles for times exceeding the integral time scale,which in turn requires a large sample volume.

C. Theory of diffusion

In this paper we focus on turbulent transport of oil drop-lets at very low void fraction ��0.02%�, where droplet-droplet interactions are negligible. Transport of particles inturbulence has been traditionally modeled as a diffusionproblem, i.e., by assuming that the flux is proportional to theconcentration gradient. The resulting Eulerian passive scalartransport equation is

�c

�t+ u · �c = � · �D · �c� , �1�

where c denotes scalar concentration, u refers to fluid veloc-ity, D refers to the turbulent diffusion tensor, and the overbardenotes some type of averaging, e.g., volume averaging orensemble averaging. Modeling of turbulent particle diffusionas a Fickian diffusion process with a constant diffusion co-efficient is problematic since the scale of the diffusingmechanism is of the same order as the property being dif-fused �Tennekes and Lumley36�. A Lagrangian method fordetermining the diffusion rate of a scalar in stationary, ho-mogeneous, and isotropic turbulence has been introduced byTaylor19 and extended to particles by Csanady.3 This theoryshows that the diffusion coefficient can be calculated fromthe Lagrangian velocity autocorrelation function Rii,

Rii��� = �t=0

�

Vi�t�Vi�t + ��dt/�t=0

�

Vi2�t�dt , �2�

where � is the time, Vi�t� is the fluctuating component offluid or particle velocity, and the subscript i refers to thedirection. Since experimental systems are limited in the pe-riod over which the autocorrelation can be calculated, fol-lowing Snyder and Lumley,30 the time averaging is replacedby ensemble averaging. Assuming a Fickian diffusion pro-cess, the diffusion coefficient of the oil droplets Ddii��� is

Ddii��� = �t=0

�

�Ui�t��Ui�t� + t��dt , �3�

where Ui is the fluctuating component of the droplet velocityin direction i and � � denotes an ensemble average over alldroplets of the same size and flow conditions. The meansquare displacement or dispersion is calculated using

Xi2��� = 2�

0

�

Dii�t�dt . �4�

This approach gives us a time-dependent diffusion coeffi-cient, which is sometimes referred to as “quasi-Fickian”�Deng and Cushman37� and can be used in a scalar like dif-fusion equation. At large �, as the integral becomes constant,the droplet diffusion coefficient converges into the classicalFickian diffusion coefficient �Ddii�. Typical scaling assumesthat Ddii=Ui�

2Tdi, where Ui�= �Ui2�0.5 is droplet velocity rms

and Tdi is the droplet diffusion timescale.Following Spelt and Biesheuvel13 and Friedman and

Katz,18 scaling of the Fickian diffusion coefficient of dropletsin turbulent flows can have the following form:

Ddii/Lui� or Ddii/LUq = f�ui�/Uq,Re�,St� , �5�

where L is the turbulence integral length scale and ui� is thefluid velocity rms. The turbulence level is characterized bythe Taylor microscale Reynolds number, Re�=u�� /�, where

u�=�13 �ux�

2+uy�2+uz�

2�, � is the Taylor microscale �definedlater� and � is the kinematic viscosity. The effect of dropletproperties is accounted for in part by the Stokes number �St�,which is defined as St=�d /��, where �d is the droplet re-sponse time and �� is the Kolmogorov time scale. The re-

095102-2 Gopalan, Malkiel, and Katz Phys. Fluids 20, 095102 �2008�

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sponse time of the droplet is calculated using a Stokes flowassumption, giving �d=dd2 /18 �c �Sec. 2.3 of Croweet al.38�, where d is the droplet density, d is the dropletdiameter, and �c is the dynamic viscosity of the fluid. Drop-let properties are also affected by ui� /Uq, where Uq is thequiescent rise velocity of the droplets, which is obtained byfollowing Friedman and Katz.18 They have estimated Uq, bytreating the droplets as rigid particles and equating the netbuoyant force with the drag force, and then confirmed it bymeasurements, especially for larger droplets. As we showlater, ui� /Uq is the dominant scaling parameter for the currentdroplet sizes and turbulent intensities.

In the case of droplets, both finite inertia and the influ-ence of gravity impact the diffusion process. The effect ofinertia is to increase the diffusion time scale and reduce thedroplet velocity fluctuation �Squires and Eaton22�. A key con-tributor, which arises due to the droplet rise velocity, is theso-called crossing trajectory effect, namely, shifting of drop-lets from one eddy to another while a fluid particle stays inthe same eddy until it decays. Csanady3 and Yudine20

showed that crossing trajectory lowers the diffusion timescale of a solid dense particle in the atmosphere compared tothat of the continuous phase. The reduction is higher in thehorizontal direction due to the so-called continuity effect�Csanady3�. Spelt and Biesheuvel13 reported similar effectsfor bubbles in water. Inertia and crossing trajectories are byno means the only phenomenon affecting the droplet or par-ticle dynamics. Nonlinear drag, lift, added mass, and pre-ferred locations, i.e., trajectory biasing, also play significantroles �Fung et al.10 Stout et al.,11 Friedman and Katz,18

Elghobashi and Truesdell,23 Clift et al.,39 Maxey and Riley,40

Mei and Adrian,25 Tunstall and Houghton,41 Sridhar andKatz,42 and Brucato et al.43�.

To measure the Fickian diffusion coefficient using theLagrangian autocovariance, one has to follow the same drop-lets over periods that are comparable to their integral time-scale. Consequently, we perform the measurements in a setupproviding nearly isotropic turbulence with low mean fluidvelocity �umean�, i.e., u� /umean8, achieved by generatingthe turbulence using four symmetrically located spinninggrids �Friedman and Katz18�. The trajectories are measuredusing high-speed digital holographic cinematography. To ac-count for the finite �20%� anisotropy in the rms values of thefluid velocity in the horizontal and vertical directions, wecompare the diffusion of slightly buoyant droplets to those ofneutrally buoyant very small particles that could be treated asfluid particles, following the same procedure. This approachenables us to distinguish between effects of buoyancy andanisotropy.

We obtain data for a wide range of Stokes number �oru� /Uq� and show that the droplet diffusion coefficient scaledby the product of the turbulence intensity and the turbulenceintegral length scale monotonically increases as a function ofu� /Uq. Difference between vertical and horizontal diffusionrates of the droplets is analyzed based on correspondingdroplet velocity rms and diffusion time scales.

II. EXPERIMENTS

The facility shown in Fig. 1, and described in detail inFriedman and Katz,18 generates nearly isotropic turbulencewith weak mean flow by using four symmetrically locatedrotating grids. Each grid has a 40% blockage factor and theyare attached to separate ac synchronous motors whose speed,and consequently the turbulence level, can be adjusted bystatic inverters. The experiments are performed at 225,337.5, and 506.3 rpm of the rotating grids. The researchgrade diesel fuel �LSRD-4� has been provided by SpecifiedFuels and Chemicals of Channel-view Texas. This diesel fuelhas a specific gravity of 0.85, viscosity of 5.5 mPa s, andsurface tension with water of 13 mN/m. The diesel dropletsare injected from 0.15 mm diameter needles. The continuousmedium is de-ionized tap water and experiments are per-formed at a mean temperature of 20 °C �19–21 °C�.

A. Turbulence characterization

The turbulence in the central part of the tank, whichextends beyond the sample volume, has been characterizedusing two dimensional �2D� particle image velocimetry�PIV� in several planes and in two perpendicular directions.The light source is a neodymium doped yttrium aluminumgarnet laser and images are acquired using Kodak ES 4.0,2k�2k digital cameras with a 105 and 210 mm lenses forthe shorter �x-y� and longer �y-z� sides, respectively. Thedelay between exposures varies between 2.5 and 4 ms, de-pending on turbulence intensity. Hollow glass beads with7–10 �m diameter are used as tracers. The digital imagesare enhanced using modified histogram equalization, and thevelocity vectors are determined using cross correlationanalysis �Roth and Katz44�. With 50% overlap betweenneighboring windows, the number of vectors in a PIV planevaries from 62�62 to 40�40 and, based on magnification,the vector spacing varies from 0.9 to 1.7 mm. At least 1000vector maps are averaged to obtain the turbulent statistics foreach location and condition, and consequently the associateduncertainty is at least an order of magnitude smaller than themeasured parameters. Measured turbulence parameters forthe three mixer speeds are provided in Table I.

Figure 2 shows a sample of mean vertical velocity dis-tribution in the central x-y plane. The corresponding distri-butions of rms of horizontal and vertical velocities are pre-sented in Figs. 3�a� and 3�b�, respectively. Similar data areavailable for nine x-y and seven y-z planes for the mixerspeed of 225 rpm. For 337.5 and 506.3 rpm, we have ana-lyzed five planes and three planes, respectively, in each di-rection due to the small variability that we observed for the225 rpm case. The fluid mean and rms velocity provided inTable I are calculated by averaging these quantities over allthe planes. As is evident, the rms values are at least eighttimes higher than the mean velocity, i.e., the facility providesthe desired flow features.

Stationarity of turbulence has been confirmed by repeat-ing the experiment at the same location after stopping andrestarting the mixers. Results show that the spatially aver-aged rms varies by less than 3%. The extent of turbulencehomogeneity of the flow can be observed from the two rms

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plots. The spatial rms’s of ux� and uy� are 0.21 and 0.15, i.e.,5.2% and 3% of the mean rms values, respectively. The en-ergy spectra have a −5 /3 slope and very similar to those ofFriedman and Katz,18 and as a result not shown. Since thespatial resolution is not fine enough to estimate the dissipa-tion rate from velocity gradients, it is estimated by fitting−5 /3 slope lines to the inertial part of the kinetic energyspectra. The turbulence scales are calculated using

� = ��3/�0.25, �� = ��/�0.5, L = �u�3/� ,

and

� = u��15�/�0.5. �6�

Statistics for the present three turbulence levels calcu-lated using the above equations are also summarized in

Table I. The 20%–25% anisotropy is characterized here bythe ratio of the spatially averaged rms velocity components.At small and inertial scales, the spectra �not shown in thispaper� overlap, suggesting that the anisotropy is mostlyassociated with large scales. Given these results, in the restof the paper we assume that the turbulence in the facilityis homogeneous, stationary, and nearly isotropic. As notedbefore, the effects of anisotropy are accounted for in theanalysis.

TABLE I. Turbulence parameters for the three mixer rpm’s

Mixer rpm 225 337.5 506.3

Vertical mean velocity uy mean �cm/s� 0.56 0.78 1.02

rms velocity u� �cm/s� u�=�13 �ux�

2+uy�2+uz�

2� 4.63 7.1 9.4

Anisotropy ratio uy� /ux� 1.24 1.26 1.2

Dissipation �m2 /s3� 0.0019 0.0099 0.0256

Integral length scale L �mm� 52 35 32

Integral time scale Tf =L /u� �s� 1.12 0.49 0.34

Kolmogorov length scale � �mm� 0.151 0.1 0.079

Kolmogorov time scale �� �s� 0.0229 0.01 0.0063

Taylor microscale � �mm� 4.11 2.75 2.28

Taylor scale Reynolds number Re� 190 195 214

FIG. 1. �Color online� Turbulence generating facility with the optical setup of one view digital holography.

FIG. 2. Mean vertical velocity distribution in the central x-y plane �z=0� atRe�=190.

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B. Droplets and fluid particle track measurements

Our objectives require measurements of the time historyof a target moving in a “random” 3D trajectory over scalesthat exceed the integral scale of turbulence. At the sametime, it will be beneficial to obtain statistics such as dropletacceleration and velocity structure function requiring us toachieve a time resolution up to the Kolmogorov scales. Wewould also like to measure the droplet size accurately. Holo-graphic PIV is particularly suited for measuring the instanta-neous 3D location, shape, size, and velocity of many par-ticles located in a sample volume with an extended depth,e.g., Barnhart et al.,45 Meng and Hussain,46,47 Zhang et al.,48

Pu and Meng,49 Tao et al.,50 Sheng et al.,51–53 and Malkielet al.54–56 Holography involves convolving an object beam,lights scattered from the objects in the region of interest,with a reference beam and recording the resulting interfer-ence pattern. In the present study we use inline digital ho-lography using the setup illustrated in Fig. 1. Here, the partof the beam which is not scattered by the particles acts as areference beam, simplifying the optical setup. Forward scat-tering from particles greatly reduces the power requirement

of the laser, by more than two orders of magnitude, allowinghigh-speed imaging with a relatively low cost laser.

Figure 1 shows the optical setup for recording a singleview inline hologram. A 0.1 mJ/pulse, 523 nm �green�,neodymium doped yttrium lithium fluoride laser beam firstpasses through a spatial filter, consisting of a microscopicobjective �40X�, which focuses the laser beam, allowing it topass through a 25 �m pinhole. Scattered parts of the beamare filtered out, providing a wave front with substantiallyimproved uniformity. This beam then passes through a colli-mating lens before illuminating the 50�50�70 mm3

sample volume in the center of the tank. The beam is thendemagnified by 2.9:1 and recorded by the high-speed digitalcomplementary metal oxide semiconductor camera with aresolution of 1k�1k pixels �pixel size of 17 �m� andspeeds ranging between 250 to 1000 frames/s, depending onthe mixer rpm. At the present magnification, the lateral res-olution is around 50 �m /pixel. To maintain a uniform back-ground, the recorded hologram is divided by a backgroundimage obtained by averaging thousands of images.

Numerical reconstruction of the recorded digital inlineholograms is performed using the Fresnel approximation,which involves convolution of the intensity distribution witha far-field source function. Details of the reconstruction pro-cedure are available in Refs. 52 and 57. The hologram isreconstructed plane by plane along the beam axis �Shenget al.52�. In each reconstructed image all the particles locatedwithin the “depth of focus” come into focus, i.e., the edge ofthe particle becomes sharp. Hence, using a maximum inten-sity gradient criterion, we can obtain the particle location inthe beam axis direction. Due to the finite depth of focus, theaccuracy with which we can obtain the object’s axial locationis lower than the lateral 17 �m pixel resolution. The prob-lem arises primarily not only from the limited numerical ap-erture of the system but also due to the finite resolution ofthe camera �Sheng et al.52�. Reconstruction is repeated every2 mm and consequently each instantaneous 3D image con-sists of 35 planes. The spacing between the reconstructionplanes is governed by the depth of focus, about 4 mm atthe present magnification and resolution. The computationalcost for reconstructing 35 planes using a Pentium IV com-puter with two 3.4 GHz processors is 23 s. We initially re-corded two perpendicular hologram which provided x-y andy-z planes with an intersecting volume of 5�5�5 cm3, fol-lowing Tao et al.50 and Sheng et al.51 in order to obtainaccurate 3D tracks. Later we shifted to a single view sincewe need to resolve the dispersion in vertical and only onehorizontal direction. Consequently the measurements pro-vide a time series of the 3D distribution of two velocitycomponents, Ux and Uy.

1. Tracking of oil droplets

Hybrid LABVIEW and C�� based processing softwarehas been developed for an automated analysis of the holo-grams to obtain the droplet tracks. The software is dividedinto two parts, the first procedure measures the location ofthe droplets as well as their size, and the second matches thetraces of the same droplet in successive frames, providing us

FIG. 3. Distributions of �a� ux� �cm/s� and �b� uy� �cm/s� in the central x-yplane �z=0� at Re�=190.

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the droplet velocity. The droplets in the reconstructed holo-grams have pixel values in the range �total range of 0–255�of 25–75, irrespective of their spatial �x-y� or beam �z� axislocation. Hence, a double threshold between 25 and 75 isapplied on the reconstructed hologram which reduces it to abinary image for further segmentation. Since we do not ob-serve significant deformation of the droplet from the spheri-cal shape, we use the circularity filter of the LABVIEW soft-ware, which performs segmentation of the binary image andgives us the location and size of each droplet. An additionaladvantage is that the circularity filter can recognize overlap-ping droplets to a certain degree, making the droplet trackingmore efficient. This circularity filter assumes that all objectsin the field of view are circles. When an odd shaped blobforms as a result of overlapping droplets, it does not satisfythe circularity conditions, and consequently the filter tries tofit two or more circles into this blob. As long as the droplettraces only partially overlap, in the majority of cases thisfilter correctly predicts the spatial location of each of theoverlapping droplets. With the droplet parameters obtained,we match the traces of droplets in successive frames by usingthe following criteria: �1� closest point in the next time step;�2� radius within a certain tolerance range; �3� depth within acertain tolerance range; �4� limitation on maximum accelera-tion; �5� avoiding sharp jitter �sudden change in accelera-tion�; �6� extrapolation of tracks based on previous slopes tofind subsequent images in cases where the droplets at differ-ent depths cross each other.

In order to obtain 3D tracks �for the limited two viewdata� we perform the steps mentioned before on the two per-pendicular views and then match the 2D projection of tracksby using least squared difference in the common verticaldirection. Figures 4�a� and 4�b� show a sample 2D projectionof 3D droplet tracks and 3D droplet tracks at Re�=190, re-spectively, with the shading indicating the magnitude of thevelocity. Since the shape and size of the droplet do notchange appreciably, cross correlation of the in-focus dropletimages is used to obtain the droplet velocity. We first apply a3�3 Sobel high pass filter on the droplet image so that onlythe edges are cross correlated. According to Manduchi andMian,58 using a high pass filter increases the robustness indetecting the correlation maxima by sharpening the correla-tion peaks as long as the noise to signal �image in our case�ratio is minimal. The peak of the cross correlation curve isdetermined using a Gaussian subpixel curve fitting. Sridharand Katz42 have shown that use of cross correlation, withouthigh pass filtering but, with subpixel curve fitting increasesthe accuracy in determining the displacement to about 0.4pixels.

As we know the droplet velocity approximately prior tothe correlation analysis, from the subtraction of the dropletdisplacement, we vary the time gaps between images used incross correlation to ensure sufficient displacement betweendroplet locations. Our criterion maintains the uncertainty ofthe droplet velocity to be at most 25% of the Kolmogorovvelocity, not taking into account the extra accuracy gained byhigh pass filtering of the images. The local mean fluid veloc-ity, which is measured using PIV, is subtracted from eachinstantaneous droplet velocity to obtain the droplet velocity

relative to the mean fluid flow. However, the difference in thediffusion coefficient obtained by subtracting the local meanfluid velocity, in comparison to no fluid velocity subtraction,is less than 2% and does not bias the result. Thus the effectof local mean fluid flow on the droplet dispersion is negli-gible. In order to estimate the effect of local mean fluid ve-locity on the droplet mean rise velocity, we have recalculatedthe rise velocity of droplets located only in a symmetric sub-sample around the injector, where the local mean verticalfluid velocity is less than 10% of the local rms velocity. Acomparison to the entire data set shows that the maximumdeviations of the recalculated values of droplet mean risevelocity from those obtained over the entire sample area isless than 9%. Hence the variability of the mean fluid flowshould be accounted for, but the difference has little impacton the trends discussed in this paper.

Using the above mentioned procedure we have obtainedthe velocity time series of more than 17 000 droplets in oneview and 4000 droplets in two views, ranging in size be-tween 600 and 1200 �m. Analysis of dispersion is based onthe one view measurements. During analysis, the droplets arebinned based on size in steps of 100 �m, consistent with our

FIG. 4. A sample �a� 2D projection and �b� 3D matched droplet tracks withgrayscale showing velocity magnitude.

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spatial resolution. While analyzing the variation of dropletdispersion with size, we use a three-bin running average inorder to increase the number of droplets per bin, giving thecentral bin twice the weightage. Hence, the data of dropletdispersion coefficient are presented for five different sizes inthe 700–1100 �m range for Re�=190 and 195, as well asfor three different sizes in the 700–900 �m range for Re�

=214 �due to insufficient droplets in the last two bins�. TheReynolds number, based on the droplet diameter and dropletquiescent rise velocity, varies from 14 to 37.

Sample ensemble-averaged droplet autocorrelation func-tions Rdii���= �Ui�t�Ui�t+��� /Ui�

2 extending to time scalesfor which the diffusion becomes almost Fickian, i.e., the au-tocorrelation decreases to zero, are presented in Fig. 5. Fig-ure 6 shows the variation with time of the horizontal dropletdiffusion coefficients Ddxx�t� calculated using Eq. �5� for dif-ferent sizes, and the corresponding horizontal fluid particlediffusion coefficient Dfxx�t� at Re�=190 and 214. The mea-surement of fluid particle data will be elaborated in the nextsection. In some cases Ddxx�t� does not reach a plateau, sincethe tracks are not sufficiently long to reach convergence to afixed Fickian diffusion coefficient. For these cases �only�, toestimate the Fickian diffusion coefficient, we use a lowestorder �four to six� polynomial fit to extrapolate the availabledata to a plateau. In evaluating the accuracy of this extrapo-lation method, using cases for which we have a convergedvalue, we find that the maximum uncertainty in the extrapo-lated Fickian diffusion coefficient is about 6%. In somecases, the autocorrelation function decreases to small nega-tive values �not shown�, which would cause a slight decreasein Ddii�t� beyond its maximum value. In these cases, we usethe maximum value for the diffusion coefficient to preventthe finite size of the field of view, which might induce nega-tive correlations, from biasing the data toward lower values.

We have also performed a “bootstrap uncertainty analy-sis” for the droplet diffusion coefficient by taking a randomsample of 67% of the droplets in a bin and recalculating thediffusion coefficient. The uncertainties of the subsamples areestimated by repeating this process for 250 realizations foreach of the bins. Based on the ratio of twice the rms of the

variations of the diffusion coefficient to the mean value, theuncertainty is in the range of 3%–6%. It is included as errorbars in Fig. 16�a�. In this paper we are mainly concernedwith trends of the asymptotic value, i.e., the Fickian diffu-sion coefficient and henceforth we refer to it simply as dif-fusion coefficient. To avoid confusion, the time varying dif-fusion coefficient will be referred to as quasi-Fickiandiffusion coefficient.

2. Tracking of fluid particles

We have also obtained Lagrangian data of neutral den-sity �specific gravity of 1.03�, 50 �m diameter polyamideparticles, which will be treated as Lagrangian fluid data, inorder to completely account for the effect of anisotropy. TheLagrangian fluid data are also obtained using one view inlineholography with the experimental setup identical to that usedfor obtaining the droplet tracks �Fig. 1�. During the experi-ments an extra water filtering loop is added with a 25 �mfilter to further filter the de-ionized water between subse-quent data acquisitions to minimize the impurities in the wa-ter and reduce the noise in the recorded holograms. We havehad to modify the software to obtain the spatial location, asthe small size of the particles makes the circularity filter unfit

FIG. 5. Sample ensemble-averaged droplet Lagrangian velocity autocorre-lations at Re�=214.

FIG. 6. Fluid and droplet �different sizes� horizontal diffusion coefficients asa function of time for �a� Re�=190 �low ui� /Uq� and �b� Re�=214 �highui� /Uq�.

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for identification. The analysis consists of increasing the par-ticle images, as the segmentation routines do not pick upvery small particles, by thresholding to identify particle cen-ters, application of a low pass filter and a second threshold toproduce a binary image of the enlarged particle.

A particle detection routine in LABVIEW is used to obtainthe spatial coordinates of the particles from the binary im-ages. Matching of particle traces between subsequent framesis performed following the procedures adopted for droplets.Overall we have tracked more than 15 000 “fluid particles”for the three different turbulence levels. We have performedextensive manual checking by observing reconstructed mov-ies of the tracked data, to confirm its success in picking upparticles and tracking them. The accuracy of the tracks isalso confirmed independently by the fact that the maximumdeviation between measured fluid velocity rms �Lagrangian�and the mean rms value obtained from PIV measurements�Eulerian� is 6%. Since the subpixel curve fitting is notuseful for the very small, 1–4 pixels, particles, the accuracyin measuring fluid particle displacement is 0.5 pixel. Wemaintain the uncertainty of the fluid particle velocity below33% of the Kolmogorov velocity by varying the time gapbetween frames used for calculating this velocity. We use theLagrangian fluid velocity data to calculate the time varying�quasi-Fickian� diffusion coefficient of the fluid particles,Dfii���, using Eq. �5�, where the droplet velocity is replacedby the fluid particle velocity and use it to obtain theasymptotic fluid diffusion coefficient �Dfii�. In experimentsperformed by Sato and Yamamoto31 in homogeneous isotro-pic turbulence, the fluid diffusion coefficient scaled by theproduct of the rms velocity and the turbulence integral lengthscale Dfii /ui�L decreases from 0.6 to 0.3, as Re� increasesfrom 20 to 70. In the present experiment Dfii /ui�L variesbetween 0.23 and 0.33, for the present limited range of Re�,without clear trends. Thus, the scaled fluid diffusion coeffi-cient in the present experiment is similar to that of Sato andYamamoto31 for their highest Re�.

Since we have been performing measurements within anearly homogeneous turbulent flow with an inertial range,the Lagrangian energy spectral density �Eii��t�� should vary

as Eii��t� �t−2 �Inoue59 and Tennekes60�, where �t is the

angular frequency. As shown in Fig. 7, the fluid particle La-grangian spectra follow the −2 slope very closely, except forthe high frequency range, where the noise dominates. Thedroplet Lagrangian spectra have similar trends but are shiftedaccording to differences in rms values, as discussed later.

III. MEAN RISE RATE OF DROPLETS

From the Lagrangian droplet velocity time series data wefirst obtain the mean droplet velocity for each size bin aftersubtracting the mean fluid velocity, as mentioned before. Thehorizontal mean droplet velocity components are an order ofmagnitude smaller than the vertical components. Deviationsfrom zero are attributed to finite data sample and, conse-quently, are not discussed further. Figure 8 shows a sampleaveraged droplet mean vertical displacement with time overthe entire sample volume. The constant slope confirms that atleast the mean displacement is not affected by slight inho-mogeneities in the flow, consistent with the comparison be-tween rise velocities in different sections of the sample vol-ume discussed in Sec. II. The slope of this line is the meanrise velocity. The difference between droplet mean rise ve-

FIG. 7. Sample Lagrangian kinetic energy spectra of fluid particles forRe�=190, showing the presence of inertial scale with a −2 slope.

FIG. 8. Mean vertical displacement of a 0.7 mm droplet at Re�=190.

FIG. 9. Droplet mean rise velocity normalized by the quiescent rise velocityas a function of Stokes number. The corresponding Re� are 190, 195, and214.

095102-8 Gopalan, Malkiel, and Katz Phys. Fluids 20, 095102 �2008�

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locity �Uslip� and the droplet quiescent rise velocity, normal-ized by the droplet quiescent velocity, is shown in Fig. 9.Figure 10 compares the current data with that of Friedmanand Katz18 using their scaling. Mostly in agreement withtheir results, which have been obtained in part in the samefacility �except for the highest ui� /Uq�, we observe that forui� /Uq�3.5, the mean rise velocity is enhanced by turbu-lence regardless of Stokes number. Also, at a given Stokesnumber, the droplet’s rise velocity increases with turbulencelevel �Fig. 9�, from values that can be lower than Uq to levelsthat are much higher than the quiescent rise rate. As ex-plained and analyzed in detail by Friedman and Katz,18 thisenhancement is most probably a result of trajectory biasing,i.e., rising droplets are preferably swept toward upward flow-ing region of an eddy. The present rise rates agree with thefindings of Friedman and Katz18 for similar values of St andui� /Uq. Since the data in this section have previously beenobserved and discussed by them, we would not address thesefindings further in this paper.

IV. PDF OF VELOCITY FLUCTUATIONS

To obtain the time history of the droplet and neutrallybuoyant particle velocity fluctuations, we subtract the meanvelocity from the instantaneous values. For the purpose ofobtaining the PDF of the droplet velocity fluctuations only,with a large database, the data at each turbulence level areseparated into two bins, with mean sizes of 0.75 and 1.05mm. Sample results, shown in Figs. 11�a� and 11�b� forRe�=195, confirm that all the distributions are close toGaussian. The nearly Gaussian distribution is essential forthe present analysis because it implies that the PDF of thedroplet displacement in any time step is also Gaussian�Batchelor61�. Indeed, PDFs of droplet displacement over pe-riods that are an order of magnitude higher �not shown�,compared to the period used to obtain the droplet velocity,also have a Gaussian distribution. A Gaussian process, i.e., atime varying random process, where the distribution at anyinstance is Gaussian, can be completely determined by amean, variance, and an autocorrelation function �Pope62�.Hence applying Taylor’s19 analysis provides us statisticallycomplete information on the dispersion of droplets. The

slight deviation of the PDFs of the fluid velocity from Gauss-ian distribution at low velocities can be mostly traced to theuncertainty in velocity. The asymmetry in the PDF of verticaldroplet velocity, which is particularly evident for the 0.75mm droplets, consists of a slight skewness toward negativevalues. This effect seems to diminish with increasing Re�

and droplet size. A similar behavior is observed in the simu-lations of Spelt and Biesheuvel13 and the experiments ofPoorte and Biesheuvel15 for the PDFs of bubble vertical ve-locity, which they attribute to preferred location of bubbles.

V. DROPLET VELOCITY RMS

Figure 12 shows the horizontal and vertical droplet ve-locity rms values scaled by those of the corresponding fluidvelocity fluctuations in order to account for anisotropy. Theuncertainty in Stokes number, around 15%, is shown as errorbars in Fig. 12�a�. This uncertainty is estimated by assumingthat the uncertainty in droplet size is 25 �m, i.e., 50% of thepixel resolution, and that the dissipation has an uncertaintyof 20%, following Ott and Mann.63 For a fixed Reynolds

FIG. 10. Droplet mean rise velocity compared to the results of Friedmanand Katz �Ref. 18�.

FIG. 11. Sample PDF of droplet horizontal and vertical velocities for �a�0.75 mm droplets and �b� 1.05 mm droplets compared to PDF of fluidparticles and Gaussian distribution at Re�=195.

095102-9 Experimental investigation of turbulent diffusion Phys. Fluids 20, 095102 �2008�

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number, both Ux� and Uy� decrease with increasing size orquiescent velocity. This trend presumably occurs since thedroplet response time increases with size, reducing its abilityto respond to the local turbulence. When the same data areplotted in Fig. 13 as a function of ui� /Uq, they appear tocollapse onto a single curve for the horizontal direction, fordifferent droplet sizes and turbulence levels, showing that

Ux� /ux� increases with increasing ux� /Uq. In the vertical direc-tion, the data do not collapse as well as the horizontal direc-tion, but the trend of increasing Uy� /uy� with increasing uy� /Uq

still persists. Clearly, the rms values of droplet horizontalvelocity fluctuations exceed those of the fluid, for all of thepresent cases, while the droplet vertical velocity fluctuationsare higher than those of the fluid only for the highest turbu-lence level, but lower in the other two cases. Thus, the scaledhorizontal droplet velocity fluctuations are higher than thevertical values, but the difference decreases with increasingui� /Uq. Possible specific reasons for these trends are dis-cussed in Sec. VIII. Rms of velocity fluctuations of a dis-persed phase exceeding those of the continuous phase hasbeen observed before only for bubbles, experimentally byPoorte64 and Colin and Legendre65 and numerically by Speltand Biesheuvel.13,66 Note that as the droplet diameter �d�becomes very small, well below the present range, oneshould expect that the scaled rms values of the droplet ve-locity approach unity. Thus, the present trends cannot persistfor very large ui� /Uq or very small St.

VI. SCALING OF DROPLET DIFFUSION COEFFICIENT

As we are interested in the application of the current datafor oceanic oil spill modeling, where the length scales aremuch larger than the laboratory length scales, we have tried

FIG. 12. Droplet �a� horizontal and �b� vertical rms velocities, normalizedby corresponding rms values of fluid velocity.

FIG. 13. Variations, with ui� /Uq, of the droplet velocity fluctuations normal-ized by corresponding fluid velocity fluctuations.

FIG. 14. Variation of normalized �a� horizontal and �b� vertical diffusioncoefficients with normalized turbulence intensity.

095102-10 Gopalan, Malkiel, and Katz Phys. Fluids 20, 095102 �2008�

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to scale the droplet diffusion coefficients with several mea-sured Eulerian and Lagrangian turbulence parameters. Figure14 shows that for all Reynolds numbers, directions, and sizes�St�, the asymptotic droplet diffusion coefficient scaled bythe product of the turbulent intensity and turbulence integrallength scale monotonically increases with increasing u� /Uq,

Ddii/u�L = f�u�/Uq� . �7�

This formulation uses u� and L, which are based on tur-bulent kinetic energy, for both directions, i.e., we assume thatthe turbulence is isotropic. If we use turbulence parametersaligned in the same direction, i.e., ui� /Uq and Li=ui�

3 /, thedata become more scattered in the horizontal direction, sug-gesting that Eulerian directional scaling is not appropriate.This trend is explained when we examine the fluid diffusiontime scale, i.e., the fluid diffusion coefficient divided by themean squared fluid velocity, Tfi=Dfii /ui�

2, in Fig. 15. ClearlyTfx and Tfy are almost equal to each other and are consis-tently very close but slightly lower than the period of ourmixers. The method that we use to artificially force the tur-bulence to remain stationary, by the rotation of the mixers,influences the turbulent diffusion timescale �Mordantet al.67�. Using this timescale, we can define new lengthscales Lfi=ui�Tfi or Lqi=UqTfi, which accounts for the La-grangian nature of dispersion, and hence the artificial forc-ing, that would be more appropriate than the Eulerian eddysize �L�. Before proceeding, the Eulerian directional scalingand the Eulerian isotropic scaling of the droplet diffusioncoefficient do not differ by much in the vertical direction dueto the fact that the variation in anisotropy �uy� /ux�� for thepresent three Reynolds numbers is due to the horizontal rms,i.e., uy� /u� and hence Ly /L remain constant in our measure-ments. Also, the Eulerian isotropic scaling of the droplet dif-fusion coefficient works better than the directional scalingfor the horizontal direction because the deviations of L /Lfx

are smaller than those of Lx /Lfx.

When we scale the droplet diffusion coefficient using ui�and the two new length scales, as shown for Ddii /ui�Lqi inFig. 16, in linear and log scales, we observe a collapse of theentire data onto a single curve, suggesting that

Ddii/ui�Lfi = f�ui�/Uq�

or

Ddii/ui�Lqi = f�ui�/Uq� . �8�

The former relation is equivalent to scaling the dropletdiffusion coefficient by the fluid diffusion coefficient andtreated in detail in the next section. Also, as mentioned inSec. III, error bars for the droplet diffusion coefficient esti-mated using bootstrap uncertainty analysis are shown in Fig.16�a�. The data for which the error bars are not shown, theuncertainty is less than the symbol size. To facilitate the ap-plication of these data for oceanic modeling purposes, wehave tried to fit a curve to our data. A straight line fitted tothe log plot, i.e., a power law with exponents of 1.44 for thehorizontal direction and 1.52 for the vertical direction asshown in Fig. 16�b� seem appropriate. For the present rangesof St and ui� /Uq, the empirical equations for the droplet hori-zontal and vertical diffusion coefficients, when scaled usingLfi, are

FIG. 15. Comparison of Lagrangian fluid diffusion timescale in horizontaland vertical directions with the rotation time of the mixer.

FIG. 16. Variations, with ui� /Uq, of the droplet horizontal and vertical dif-fusion coefficients normalized by ui�Lqi in �a� linear and �b� log scales.

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Ddxx/ux�2Tfx = 0.66�ux�/Uq�0.44

and

Ddyy/uy�2Tfy = 0.51�uy�/Uq�0.52. �9�

The power law behavior implies that the functional relation-ships using Lfi and Lqi are similar, as can be seen from Eq.�9� and Fig. 16�b�, but with different exponents. The standarddeviations of the data from this relation are 0.034 and 0.069for the horizontal and the vertical directions, respectively.Note that this equation may not be necessarily valid in thelimit of ui� /Uq�1, where one would expect that Ddii

=ui�2Tfi, i.e., equal to fluid diffusion. Also, for ui� /Uq�1,

following Csanady3 and neglecting the difference betweenEulerian and Lagrangian length scales, one would expect toobtain Ddii=ui�

2L /Uq, i.e., Ddii /ui�2Tf =ui� /Uq.

VII. DROPLET DIFFUSION VERSUS FLUID DIFFUSION

Figures 17 and 18 show the diffusion coefficient and thediffusion timescale of the droplet, normalized by the corre-sponding diffusion coefficient and diffusion timescale of thefluid, as a function of ui� /Uq. It is evident that at low turbu-lence levels and/or large droplet size, i.e., ui� /Uq�3, the

droplet diffusion coefficients, in both directions, are lowerthan those of the fluid. Recognizing that Ddii=Ui�

2Tdi, trendsof both the rms values and the diffusion timescales affectDdii. In the horizontal direction Ux� /ux��1, but Tdx /Tfx issubstantially lower than 1, resulting in Ddxx�Dfxx. In thevertical direction both the timescale and the velocity fluctua-tions are lower than those of the fluid at low ui� /Uq, eachcontributing to Ddyy �Dfyy. When ui� /Uq�3, Ddxx first ex-ceeds the corresponding fluid diffusion coefficient due tocombined effects of increasing rms velocity and increasingdiffusion timescale. For of ui� /Uq�4 the vertical droplet dif-fusion coefficient also exceeds that of the correspondingfluid diffusion coefficient, again due to combined effects ofincreasing Uy� /uy� and the droplet diffusion timescale. Overallwith increasing ui� /Uq, the normalized horizontal and verticaldirection droplet diffusion coefficients increase and also be-come closer to each other. The latter trend can also be ob-served from the two straight line fits in Fig. 16�b�. The otherclear observation from Fig. 17 is that for most of the presentcases, the normalized droplet horizontal diffusion coefficientis higher than the vertical one. This trend is inconsistent withpreviously published data, both for heavy particles, e.g., nu-merical simulations by Squires and Eaton,22 and for bubbles,e.g., the numerical simulations of Spelt and Biesheuvel13 andMazzitelli and Lohse,16 as well as experimental data reportedby Poorte.64 The discussion in the next section examines thisissue. Note that the changing diffusion timescales are alsoevident from trend of quasi-Fickian diffusion coefficients. AsFig. 6 shows for low ui� /Uq, the droplet quasi-Fickian diffu-sion coefficient reaches a plateau at shorter times comparedto that of the fluid. This trend is clearly reversed for largeui� /Uq.

As shown in the Introduction, the droplet dispersion canbe obtained by integrating the quasi-Fickian diffusion coef-ficient. Figure 19 shows sample horizontal dispersions �Xdx

2 �of 0.7–0.9 mm droplets for Re�=214. As expected fromTaylor’s19 predictions, at short times, Xdx

2 �t� t2, but at longertimes the relation becomes linear.

FIG. 17. Droplet horizontal and vertical diffusion coefficients normalized bythe corresponding fluid diffusion coefficients.

FIG. 18. Droplet horizontal and vertical diffusion time scales normalized bythe corresponding fluid diffusion time scales.

FIG. 19. Sample droplet horizontal dispersion �mean square displacement�at Re�=214.

095102-12 Gopalan, Malkiel, and Katz Phys. Fluids 20, 095102 �2008�

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VIII. DISCUSSION

In agreement with previous observations, for both heavyparticles and bubbles, as Uq increases the droplet diffusioncoefficient decreases, e.g., Csanady,3 Snyder and Lumley,30

Wells and Stock,5 and Squires and Eaton.22 As alluded to inthe Introduction, this behavior is a result of crossing of tra-jectories by droplets due to the effect of buoyancy, as firstproposed by Csanady.3

To the best of our knowledge, we provide the first ex-perimental data showing a diffusion coefficient of a buoyantdroplet/particle exceeding that of a fluid particle. Further-more, we are not aware of other studies showing that thescaled horizontal diffusion coefficient exceeds the verticalone. This latter trend mainly stems from that of the dropletvelocity fluctuations. A possible explanation for the differ-ence between Ux� and Uy� is provided in a theoretical modelfor bubble velocity fluctuation introduced by Spelt andBiesheuvel.13,66 They show that the horizontal force balanceequation contains a “lift” term, Uq��, where � is the in-stantaneous vorticity, which does not exist in the verticalforce balance equation. For the cases of ui� /Uq�1 andui� /Uq1 they show that this term causes Ux��ux� They alsoshow that �a� for ui� /Uq1 the bubble horizontal velocityfluctuations exceeds that of the fluid in all cases, but thevertical velocity fluctuations exceed those of the fluid onlywhen ui� /Uq�0.2 and �b� with increasing turbulence level,the difference between Ux� /ux� and Uy� /uy� diminishes. Bothtrends agree qualitatively, but not quantitatively with thepresent data, which may be caused by differences in densityor assumptions associated with the model.

We observe that, for ui� /Uq�3, the diffusion timescaleof the fluid exceeds that of the droplet and that the dropletvertical diffusion timescales exceed the horizontal one.These behaviors have been explained by Csanady3 as an out-come of “crossing trajectories” and the associated “continu-ity effect.” Csanady3 argued that the droplets migrate fromone eddy to another due to the combined effects of inertiaand buoyancy, resulting in the droplets “losing theirmemory” faster than a fluid particle, which remains withinthe same eddy during its entire lifetime. Hence the diffusiontimescale of the droplets, which physically means how wellit remembers its past history, is lower than that of the fluid.

An interesting observation is the increasing-decreasing-increasing trend of the droplet diffusion timescale with vary-ing ui� /Uq in both directions, which occur at different condi-tions �Fig. 18�. It has been established that inertia increasesthe diffusion timescale and crossing trajectories decrease it�Wells and Stock5 and Squires and Eaton22�. With increasingui� /Uq, the effects of both inertia and crossing trajectoriesdecrease, but not necessarily in the same rate. Thus, it ispossible that the variations in the trend are a result of con-fluence of opposing effects of inertia and rise velocity, thelatter dominating the crossing trajectories.

For ui� /Uq�3, trends of the droplet diffusion timescalebecome monotonic, and the difference between the verticaland horizontal timescales seem to diminish. The latter indi-cates a diminishing effect of crossing trajectories, whichwould increase the difference in diffusion timescale between

the two directions. We currently cannot provide explanationfor the observation, Tdy /Tfy �1 at high ui� /Uq. However,similar trends, namely, that the vertical diffusion timescale ofbubbles exceed that of the fluid, have been observed in thedirect numerical simulation and Lagrangian tracking ofbubbles by Mazzitelli and Lohse.16 It has also been observedfor heavy particles, in both directions, in the theoreticalmodel of Wang and Stock,7 at high Stokes number. Hence,beyond the current ranges of St and ui� /Uq the droplet diffu-sion parameters might have a complex dependence on boththe variables. For example, the normalized vertical dropletdiffusion coefficient at ui� /Uq2.4, which deviates signifi-cantly from the generally observed trend, is the lowestStokes number data of 1. Also, as stated before, inertia andcrossing trajectories are not the only factors that might affectdispersion, e.g., we have seen that lift force probably plays arole. Hence, future analysis, numerical and experimental,will have to identify the effects of lift force, nonlinear drag,added mass, history force, and also preferential distributions,e.g., due to trajectory biasing �Friedman and Katz18�, in orderto completely understand the complex phenomena of disper-sion due to turbulence.

IX. CONCLUSIONS

High-speed inline digital holographic cinematography isused to investigate the diesel droplet diffusion in nearly ho-mogeneous and isotropic turbulence. Droplet tracks havebeen recorded in 50�50�70 mm3 sample volume in afully characterized turbulence facility. An automated codehas been developed to track the droplets, which provides usthe time history of the droplet velocity. Data of over 17 000droplets of varying sizes and 15 000, 50 �m diameter, neu-tral density particles have been used to obtain Lagrangianstatistics for both.

Similar to Friedman and Katz,18 our current results con-firm that at low turbulence levels, the mean rise velocitytends toward the quiescent value, but becomes higher thanUq with increasing ui� /Uq. The PDFs of the droplet velocityare close to Gaussian, which justify use of Taylor’s19 theoryto calculate dispersion parameters. The rms values of dropletvelocities decrease with increasing droplet size and increasewith increasing turbulence level. When the droplet velocityfluctuations are scaled by those of the fluid, they monotoni-cally increase with increasing ui� /Uq. For most of the presentdroplet sizes and turbulence level, the horizontal droplet ve-locity fluctuations exceed those of the fluid. The verticaldroplet velocity fluctuations are higher than those of the fluidonly for the highest ui� /Uq. These results are consistent withthe previous observations of bubble velocity rms, e.g., Speltand Biesheuvel,13,66 Poorte,64 and Colin and Legendre,65 andthe theoretical model proposed by Spelt and Biesheuvel13,66

might explain these observations.Diffusion rate is calculated by integration of the

ensemble-averaged Lagrangian velocity autocovariance.Both vertical and horizontal droplet diffusion coefficients arelower than those of the fluid for ui� /Uq�3, mainly since thediffusion timescale of the droplet is lower than that of thefluid. Under the effects of increasing Ui� /ui� and diffusion

095102-13 Experimental investigation of turbulent diffusion Phys. Fluids 20, 095102 �2008�

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timescale, for ui� /Uq�3, the horizontal diffusion coefficientexceeds that of the fluid, while in the vertical directionDdyy /Dfyy �1 only for ui� /Uq�4. The normalized horizontaldiffusion coefficient is larger than the vertical one for mostof the present data. This result does not agree with trendsobserved for bubbles �Spelt and Biesheuvel13 and Mazzitelliand Lohse16� or heavy particles �Squires and Eaton22 andWang and Stock7�. Trends of the diffusion timescale are ingeneral agreement with previous literatures, e.g., Csanady,3

Wang and Stock,7 and Spelt and Biesheuvel,13 and can beexplained by the effect of droplet inertia and crossing trajec-tories. Finally and most important, the droplet diffusion co-efficient scaled by the product of the turbulence intensity anda suitable length scale, either UqTfi or ui�Tfi, monotonicallyincreases as a function of ui� /Uq. We obtain empirical corre-lations of the form Ddxx /ux�

2Tfx=0.66�ux� /Uq�0.44 andDdyy /uy�

2Tfy =0.51�uy� /Uq�0.52 for the horizontal and verticaldirections, respectively, for the current ranges of St andui� /Uq.

ACKNOWLEDGMENTS

Funding for this project has been provided in part by theU.S. National Oceanic and Atmospheric Administration/University of New Hampshire Coastal Response ResearchCenter �CRRC�, under Contract No. 07–059. Early phaseswere performed under a Department of Energy Grant No.DE-FG02-03ER46047.

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