Spatial development of particle-laden turbulent pipe flowF. Picano, G. Sardina, and C. M. Casciolaa�

Dipartimento di Meccanica e Aeronautica, Sapienza University of Rome, via Eudossiana 18,00184 Rome, Italy

�Received 20 July 2009; accepted 31 August 2009; published online 30 September 2009�

The inhomogeneity of turbulence in wall bounded flows induces the phenomenology calledturbophoresis whereby inertial particles of suitable mass accumulate at the solid wall. Particlesinjected near the axis of a fully turbulent pipe flow, after an initial spreading phase, undergo asegregation process which eventually leads to a pseudoequilibrium distribution sufficientlydownstream. Wall densities up to thousand times the reference value can be easily achieved. Theprocess is discussed here by analyzing the direct numerical simulation �DNS� data of a spatiallydeveloping particle laden pipe flow under the assumption of dilute suspension. Development phaseand asymptotic state are addressed in quantitative terms. A Shannon-like entropy is introduced toquantify the level of spreading/segregation achieved by the particle distributions along the pipe. Thisallows to define on a physically sound basis the length of the developing region and to summarizein a single indicator the accumulation level as a function of the particle response time. Byconditional statistics, it is unequivocally shown that particles approach the wall dragged byrelatively fast yet comparatively rare events where highly accumulating particles follow the fluidin-rush toward the wall. On the contrary, the outward particle flux takes place in the form of muchmore frequent and gentle motions away from the wall. The analysis of DNS data and a simpleargument highlight the role of the elongated clusters of particles at the wall as essential featuresresponsible for the eventual asymptotic equilibrium. © 2009 American Institute of Physics.�doi:10.1063/1.3241992�

I. INTRODUCTION

Transport of inertial particles by turbulent flows is sig-nificant in many applications. For instance filter devices, in-ternal combustion engines, and rockets involve the interac-tion between small solid particles, droplets, chemicalkinetics, and turbulence. Practical devices often includemanifolds and wall bounded flows in general. In these casesthe coupling between particles and flow dynamics in the nearwall region is crucial for the understanding and quantitativeprediction of transport.

Transport of small particle presents several nontrivialand peculiar aspects, one of which is the small-scale cluster-ing, usually addressed in homogeneous conditions �see, e.g.,Refs. 1–3 for isotropic turbulence and Refs. 4–6 for shearflows�. The clustering process is crucial in many physicalenvironments involving particle collisions, droplet evapora-tion, and also protoplanetary formations, see, e.g., Refs. 7and 8. Another relevant issue is the dispersion of small par-ticles investigated in several configurations such as isotropicturbulence, homogeneous shear flows, and jets, see, e.g.,Refs. 9–12. In addition to clustering and dispersion, the mostsignificant factor of particle dynamics in turbulent wall flowsis their preferential accumulation at the wall, a common phe-nomenon called turbophoresis. Under appropriate conditions,particles may achieve extremely large concentrations at thewall, up to thousand times the mean value. This implies thatmost particles get more or less stuck there with a substantial

overall reduction of the mobility of the dispersed phase. Thisturbulence-induced transport and the issuing preferential ac-cumulation, as have been addressed in a number of papersdealing with a variety of configurations, from boundary lay-ers to planar channels and pipes, attacked either from theexperimental �e.g., Refs. 13–15 and references therein�, thenumerical �e.g., Refs. 16–21 and references therein�, and thetheoretical side �e.g., Refs. 22 and 23 and reference therein�.

The phenomenon is attributed to inertia which preventsparticles from following the highly convoluted trajectoriesthe fluid eddies would induce on purely Lagrangian tracers.Assuming small and diluted particles, heavier than the fluid,their inertial properties are described by their relaxation—orStokes—time which depends on density ratio between trans-ported phase and carrier fluid, particle dimension, and fluidkinematic viscosity, �p=�p /� fdp

2 / �18��.24 Vanishing �p per-tains to small inertia particles that behave almost as fluidparcels �Lagrangian limit�. On the opposite limit, the hugerelaxation times of heavy particles make turbulent fluctua-tions irrelevant �ballistic behavior�. Both limiting cases leadto homogeneous distributions in wall bounded flows.

It is with �p matching certain characteristic time scalesof the turbulence that preferential accumulation occurs. Theresonance conditions are presumably set by the typical fre-quencies of quasicoherent structures populating the bufferlayer, which are able to drift tuned-in particles in turbo-phoretic motion. However the picture is not entirely clear,see Refs. 16 and 21, and a more complex selection mecha-nism may be operating to determine which kind of particleswill be able to accumulate at the wall and which will not. Ita�Electronic mail: carlomassimo.casciola@uniroma1.it.

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is presumably related to the asymmetry between sweeps andejections, the relatively smooth in-rush of fast fluid towardthe wall in contrast to the abrupt explosions corresponding tostreak breakdown in the sublayer.25 One may figure out thata critical amount of inertia allows particles to drift on smoothsweeps, at the same time preventing them from beingdragged along by abrupt ejections. At equilibrium this asym-metry is compensated by the local concentration gradient todrive the net wall-normal flux of particles to zero: A signifi-cant percentage of particles in a region of smaller concentra-tion drifts toward the wall in exchange of a small fraction ofa larger population which is dragged away. As we see, equi-librium involves dynamic properties of ejection and sweeps,their relative probability of occurrence at a given place, thelow-pass filtering attitude of inertia, and the concentrationgradient.

Substantiating the picture we have just outlined requirescomplete information on the evolution of particle popula-tions under well characterized turbulent fields. Under thisrespect, direct numerical simulation �DNS� is particularly ef-fective since it makes available various forms of conditionalstatistics. Actually, a numerical approach to turbophoresis isnot new. The typical simulation addresses time-developingparticle populations, i.e., particle statistics are homogeneousin the flow direction and evolve in time to eventually reachthe steady state. Much can be understood, however, by look-ing at spatially developing configurations in statisticallysteady conditions. In this case one can analyze how the equi-librium distributions are approached and realize how theconcentration gradient is set up. Furthermore, the developingphase of particle deposition is interesting in itself, e.g., onemay be interested in quantifying the developing length as afunction of the control parameters, see, e.g., Ref. 19 for anattempt to convert time-developing into space-developingdata.

Here we deal with the statistical analysis of a spatiallydeveloping simulation of the particle laden pipe flow at Re=UR /�=3000. In a fully developed turbulent pipe flow, dif-ferent particle populations are injected at a certain positionon the axis of the pipe and their downstream dynamics isfollowed up to 200 pipe radius R. Particle concentration andtheir velocity distributions are discussed. The developing re-gion is characterized in terms of an entropy function used toquantify the level of segregation and to define properly thedevelopment length. Conditional statistics of particle andfluid are exploited to understand the impact of the dynamicsof the near-wall region on the equilibrium concentration.

Besides the wall normal particle concentration gradient,we find that the asymptotic equilibrium also requires the seg-regation of the particles in the elongated clusters at the wall.These features, always associated with turbophoresis, will beshown to be crucial in establishing the asymptotic state. Theymake the statistical mechanics description of the system ex-tremely difficult since the reaction to turbophoresis occursvia nonlinear fluctuations in the particle distribution.

Before entering a detailed discussion of the relevantphysical aspects, a word is in order concerning the terminol-ogy here used: Complying with the accepted use for wallbounded flows, we call inner the flow region closest to the

wall and outer the one toward the bulk. Consistently we callinward the motions toward the wall and outward those awayfrom it. The former corresponds to positive radial velocity,the latter negative. To avoid misunderstandings the readershould keep in mind that the present stipulation holdsthroughout the paper.

II. METHODOLOGY

Usually, simulations of particle laden wall flows assumeperiodic boundary conditions for particles and velocityfluctuations.17,19 The issuing axial homogeneity of particledistributions deprives the transient of direct physical inter-pretation and leaves the eventual steady state as the onlymeaningful target. In fact, one may be interested in under-standing how statistically stationary systems evolve in spacetoward equilibrium conditions.

Under the basic assumption of negligible back reactionof the transported phase on the carrier fluid-dilute particles—the standard machinery of wall bounded flow DNS applies tothe fluid. Specifically, assuming the axial extent of the com-putational pipe �base domain� exceeds the correlation length,velocity fluctuations can be taken axially periodic. By repli-cating several times the base computational domain, a muchlonger pipe is assembled, see Fig. 1 for a sketch. Along thislonger pipe we should have the particle distribution develop-ing until equilibrium conditions are reached. As a specificparticle moves along from one sector to the other, enoughtime elapses to assume the velocity field the particle encoun-ters in the new sector as statistically independent from thatexperienced before. Hence, we can continuously release par-ticles near the axis of the pipe at a given station and let themproceed along until they achieve a z-independent distributionfrom a certain station onward, to eventually exit at the end.

Close to the injection, turbulent fluctuations disperse theparticles which spread through the section of the pipe. Fur-ther downstream, statistical equilibrium is eventuallyreached. The particle distribution becomes independent ofthe streamwise coordinate z, and depending on the param-eters, preferential accumulation �turbophoresis� may occur.

In a sense, our simulation resembles the famous Rey-nolds experiment, where dye is injected at a point to let it bediffused by turbulence. Here, instead of tracers, small mas-

FIG. 1. Top panel: sketch of the base computational domain for DNS andcoordinate system. Bottom panel: extended domain for particle evolutionobtained by replicating the base domain.

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sive particles are introduced so that the classical phenom-enology of turbulent dispersion is complicated by inertialeffects. The configuration reproduces quite well the experi-mental conditions with a developing near field region and afully developed far field one.

A. Carrier fluid phase

The carrier fluid is assumed to obey the incompressibleNavier–Stokes equations,

�u

�t+ � · �u � u� = −

�p

�+ ��2u , �1�

where � and � are density and kinematic viscosity of thefluid, respectively, u is the solenoidal fluid velocity�� ·u=0�, and � denotes tensor product.

We recall that the dispersed phase is highly diluted, andthe particles are much smaller than the viscous scale of theturbulence. In these conditions the feedback acting on thefluid phase can be safely neglected.

The turbulent pipe flow is reproduced by simulating acircular duct element with length Lz=2�R �R pipe radius�and periodic inlet/outlet boundary conditions. The flow ischaracterized by a nearly constant mass flux controlled by amean pressure gradient in the axial direction. The bulk Rey-nolds number is Re=U0R /�=3000, with U0 as the bulk ve-locity, corresponding to a friction Reynolds number Re�

=u�R /� of 200, where the friction velocity is defined interms of the mean shear stress at the wall, u�=��w /�. Here-after quantities expressed in wall units, i.e., made dimension-less with friction velocity and kinematic viscosity, are de-noted by +.

In our simulation we find �max+ =��max /u+

4 �0.2, where �is the turbulent kinetic energy dissipation rate. This yields aminimum Kolmogorov scale �+�1.5, see, e.g., Ref. 26 tocompare particle diameters with in order to verify our as-sumption that particles do not interfere for mere geometricalreasons with the turbulence.

The algorithm we use solves Eq. �1� in cylindrical coor-dinates with a conservative second order finite differencescheme on a staggered grid.27 Concerning diffusive terms,the axis singularity is removed as in Ref. 28. The time evo-lution is performed by an explicit third order low-storageRunge–Kutta scheme. The basic code was validated in aDNS of turbulent axisymmetric jets in Ref. 29 and for thespecific application of our present concern by comparisonwith available pipe flow data.

The base pipe element is discretized by a uniform grid of128�80�128 nodes in the axial z, radial r, and azimuthal �directions, respectively. The point closest to the wall is lo-cated at r+=1.25, while the grid spacing in the axial andazimuthal direction is �z�+= �R��+=9.8, see also Refs. 18and 30. Following standard procedures for thermalization,the simulation was run for T=300R /U0 to allow for relax-ation of the fluid velocity field to the statistical steady state.After this lapse of time, particles were introduced in the do-main. At time T=3000R /U0 after introduction, the particlepopulations apparently achieved a statistical steady state andthe data acquisition procedure to be described in Sec. II B

was initiated. Statistics was collected using 500 fields withtemporal separation of 2R /U0, enough to achieve statisticalconvergence of concentrations and of probability densityfunctions �pdfs� of fluid/particle velocities.

The mean fluid velocity profile and turbulent intensitiesare shown in Fig. 2, see the DNS of Ref. 30 at slightlysmaller Reynolds number for comparison. The viscous sub-layer is well resolved �y+= �1−r /R� /w�5,w=� /u��. Con-sistent with other low Reynolds number DNS of pipe flows,the mean profile in the region 30�y+�100 can be fittedwith a logarithmic law with a Karman constant of k=0.34,significantly smaller than the traditional value of 0.41 foundin channel flows at comparable Reynolds number, see Refs.31–33 for a thorough discussion of the concept of log lawand overlap region in wall bounded flows. Peak values ofturbulence intensities and their locations agree well with theavailable literature.

B. Particles

The particles, assumed to be spherical and smaller thanKolmogorov scale, are treated as material points with finiteinertia evolving according to Newton’s law. As anticipated,we confine the analysis to sufficiently diluted suspensions toneglect both interparticle collisions and the force feedbackon the carrier fluid. The density of the solid phase is takenmuch larger than the fluid one, �p /�=1000. In these condi-tions the only significant force acting on the particles is theviscous �Stokes� drag, e.g., Refs. 1, 17, and 19, and eachparticle evolves according to24

FIG. 2. �Color online� Statistics of the turbulent pipe flow �Re�=u�R /�

=200� normalized by friction velocity u�=��w /� and wall length w=� /u�.Top panel: mean velocity profile �red solid line�. The green dashed line isu+=y+. The blue dotted line displays the logarithmic law of the wallu+=1 /0.34 log�y+�+4.1. Bottom panel: root mean square velocity fluctua-tions by lines. In both panels DNS data of Ref. 30 at Re�=180 are denotedby symbols.

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dv

dt=

u�x� − v

�p, �2�

dx

dt= v , �3�

where v denotes the particle velocity and �p=�pdp2 / ���18� is

the particle response time �Stokes time�, with dp as the par-ticle diameter. This leaves the Stokes number, ratio of �p andthe characteristic time scale of the carrier fluid, as the mainparameter defining the particle dynamics for a given flowfield. In wall-bounded flows the viscous time � /u�

2 is thenatural choice for the fluid time scale, leading to St+

=�pu�2 /�. The outer scale Stokes number is then St=�pR /U0=St+Re�

2 /Re. The simulations we describe involveseven populations of particles, differing only for their diam-eter, hence for the Stokes number St+, see Table I.

The particles’ computational domain is obtained by as-sembling 32 pipe base elements, up to a total length of Lz

p

=32Lz�200R. Particles are introduced in the domain at z=0 with random positions within a small circular disk ofdiameter � /u� centered around the axis and velocity match-ing the local fluid one. Being transported by the fluid, par-ticles eventually leave the domain with no back reaction onthe input rate. Elastic collisions are assumed when the par-ticle surface hits the wall of the pipe, i.e., when the distancefrom the particle center to the wall equals the radius. Thenominal particle diameter has then an active role in the in-teraction with the wall, i.e., dp

+ is an additional dimensionlessparameter of the system.

Efficiency considerations led to fix the input rate at 900particles per eddy-turnover-time R /U0 for each population,corresponding to an average number of particles in the do-main ranging from 105 to 7�105 for the smallest and thelargest ones, respectively. The same three-stage third orderlow storage Runge–Kutta method used for the fluid phaseevolves particle positions and velocities, Eqs. �2� and �3�.Preliminary tests showed that significant errors arise near theaxis when evolving the velocity equation �2� in cylindricalcoordinates. Cartesian coordinates are then used in the algo-rithm for particle velocities, thereby removing the axis sin-gularity. The best choice was a combination of Cartesian andcylindrical coordinates, velocities, and positions, respec-tively, which resulted in a stable and efficient algorithm withoptimal handling of particle localization.

The interpolation of the fluid velocity u at the particleposition is carried out by a mixed linear-quadratic formulabased on Lagrange polynomials. For each velocitycomponent—uz, say, quadratic polynomials are employed inthe two normal directions—r, �—whereas linear polynomi-als are used for the remaining direction—z. As discussed in

Ref. 6 the accuracy of this interpolation is largely sufficientfor an accurate statistical analysis. On a staggered grid thismethod is faster than a standard trilinear interpolation.

III. RESULTS

A. Instantaneous visualizations

A visual impression of the whole domain is provided inthe left panel of Fig. 3, which shows a typical particleconfiguration—the three-dimensional drawing is not to scale.Particles, introduced at the axis, are dispersed by turbulentmotions and eventually reach the wall where they accumu-late due to turbophoresis. In our model, particles collide elas-tically with the wall and sufficiently downstream achieve adistribution which is axially homogeneous, i.e., statisticallyindependent of the axial coordinate z. Only three particlepopulations are shown in the figure, namely, St+=0.1, St+

=10, St+=100, in order to highlight the main typical behav-iors we have observed in our simulations.

The middle panel of Fig. 3 provides an enlarged view ofthe initial portion of the duct, focusing on the early stagedispersion. “Smoke lines” traced by particles with differentStokes time can be visualized by looking at the lining ofmarkers with same color. The different amounts of convolu-tion are due to the different inertial response time �p of thevarious populations to the same turbulent fluid velocity fluc-tuations.

TABLE I. Solid phase parameters. The density ratio �p /�=1000 is shared by all particles.

St=�pU0 /R 0.0075 0.0375 0.075 0.375 0.75 3.75 7.5

St+=�pu�2 /� 0.1 0.5 1.0 5.0 10.0 50.0 100.0

dp /R 0.000 21 0.000 47 0.000 67 0.0015 0.0021 0.0047 0.0067

dp+=dp /w 0.042 0.095 0.134 0.3 0.42 0.95 1.34

FIG. 3. �Color online� Snapshots of particle positions �St+=0.1,10,100green, blue, and red spheres�. Left panel: view of the whole computationaldomain plotted in arbitrary aspect ratio; middle panel: enlargement of thenear field region with color encoded instantaneous axial fluid velocityisocontours; right panel: thin axial slice in the far field.

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The right panel of Fig. 3 displays particle positions in adiametral thin slice of the domain where the distribution isalready axially homogeneous. Preferential accumulation atthe wall is apparent for blue particles �St+=10�, while thesmaller ones �green, St+=0.1� show a more even distribution.Our heaviest particles �red, St+=100� are still substantiallyaffected by turbophoresis, although they seem more evenlymixed in the whole section. Overall, visual inspection con-firms the axial homogeneity achieved by all species ofparticles.

Developing and far field regions are emphasized in Fig.4 where cross-flow slices in the respective zones are plotted.In the developing region �25R, top panel� aggregates of par-ticles and large void regions are present. Within the aggre-gates the alignment of particles is reminiscent of the sequen-tial injection, i.e., turbulent motions were not still able tofully mix the particles.

The impression is that spatial segregation near the wallis already starting to occur before complete mixing takesplace, at least for particles at St+=10. The heaviest particles�St+=100� are suspected of a similar behavior. Concerningthe fully developed region shown on the bottom of Fig. 4,particles fill more or less uniformly the bulk region with astrong preferential accumulation at the wall. This is particu-larly true for the heaviest particles St+=10–100, while the

accumulation is almost negligible for the lightest ones�St+=0.1�. As a further difference between red and bluepopulations, the latter is more uniformly distributed alongthe circumference. On the contrary, the red population is par-ticularly streaky at the wall, see also Refs. 16 and 17. As wewill see, such streaky features are essential aspects of theequilibrium distribution achieved by the particles.

Addressing the properties of the system in the presenceof strong fluctuations requires a detailed statistical descrip-tion, which is presented in Secs. III B–III G.

B. Particle concentration

A more quantitative information on preferential accumu-lation can be gathered by considering the mean particle num-ber normalized by volume in each of the classical four re-gions of the pipe, namely, viscous sublayer �0�y+�5�,buffer layer �5�y+�30�, log layer �30�y+�100�, and bulkregion �100�y+�200�, as a function of normalized axialdistance z /R, Fig. 5. The normalized mean particle number isdefined as C= �n� /A, where n is the axial density of particlesin each region and A is its cross-sectional area. Turbophore-sis amounts to finding a large fraction of particles in theviscous sublayer in the fully developed zone. Looking at theresults in Fig. 5, lighter particles �St+�1� exhibit an almostuniform spreading all through the cross section of the pipe,CChom. The uniform value Chom �1200 in the presentcase� is clearly proportional to the particle injection rate.Increasing the Stokes number, wall accumulation grows to amaximum, here occurring somewhere between St+=10 andSt+=50, with a normalized mean particle number in the vis-cous sublayer larger than hundred times Chom. In fact par-ticles with St+=100 apparently achieve a smaller wall con-centration, although they do not seem to have reached yet afully developed z-independent state.

In the other regions the concentration is smaller than inthe viscous sublayer and, in the buffer layer, it increasesmonotonically with the Stokes number. Conversely, in thelog and bulk regions the opposite trend is observed, withsmaller particles exceeding in number the heavier ones.

In order to appreciate dispersion and accumulationmechanisms, the mean concentrations in the bulk, log,buffer, and viscous regions are reproduced in Fig. 6 for eachtypical Stokes number in a few typical cases, namely, St+

=0.1,1.0,10.0,100.0. As already anticipated, particles withSt+=0.1 undergo a dispersion process which leads to a uni-form distribution beyond z /R50. The developing region ischaracterized by the progressive filling of the four regions insuccession, starting from the bulk down to the wall, top leftpanel. Despite that segregation is already observed at St+

=1.0 �top right panel�, with concentration in the viscous su-blayer almost three times Chom, the developing phase is simi-lar to case St+=0.1.

A completely different behavior is followed by particleswith St+=10.0, which achieve a maximum wall accumula-tion, e.g., Cviscous�Chom, top left panel. Here for 15�z /R�25 the concentration in the buffer layer is the lowest in thecross section, i.e., lower than either the log region or the

FIG. 4. �Color online� Instantaneous particle positions in a thin cross flowslice of thickness of R /10 �same color coding as in Fig. 3, symbol sizeincreasing with St+�. Top panel: snapshot in the near field �distance from thereleasing station 25R�; bottom panel: far field �distance 200R�.

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viscous sublayer. This behavior is consistent with the as-sumption that these particles tend to be expelled from thebuffer layer, keeping the local concentration to a minimum.This phenomenon is apparently related to advection by thecoherent structures characterizing the buffer region, as statedin Refs. 16 and 34. Among the populations we have consid-

ered, the effect is stronger at this Stokes number, consistentlywith a possible estimate for the resonance condition in thebuffer layer, St�y�=�pu� /y=St+y+, which yields St�ybuffer��1 for St+=10–50. Particles at St+=100.0 show a develop-ing region with an initial phase z /R�30, characterized by adefinite trend toward homogenization, with the same pro-

FIG. 5. �Color online� Mean particle concentration vs axial distance z /R. Data averaged over each of the four wall normal slices: viscous sublayer �0�y+

�5�, top left �log-lin scale�; buffer layer �5�y+�30�, top right; log layer �30�y+�100�, bottom left; bulk region �100�y+�200�, bottom right.

FIG. 6. �Color online� Same data as in Fig. 5, particle concentration in the four regions. Top left plot St+=0.1, top right plot St+=1, bottom-left plot St+

=10, and bottom right St+=100.

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gressive filling of the cross section described for St+=0.1.Further downstream, turbophoresis takes over leading to par-ticle segregation in the viscous sublayer.

From the previous picture, three different behaviorsemerge, see Ref. 23. Light particles are more or less con-vected with the fluid and show modest accumulation. Par-ticles with intermediate inertia are strongly affected by theturbulent fluctuations in the buffer layer causing substantialsegregation. Our heaviest particles still manifest a certaindegree of segregation, although they relax to the equilibriumdistribution much more slowly.

C. Shannon entropy and segregation

In order to try and summarize the above complex re-sponse of the particles into a single global indicator, able todiscriminate between dispersion and preferential accumula-tion, we resort to concepts borrowed from classical statisticalmechanics. To this purpose let us define an entropy for thesteady state particle distribution in the cross section at axialposition z. The cross section is divided into M annuli withidentical area a=��rout

2 −rinn2 �. The z-dependent entropy is

then

S�z� = − �i=1

MNi�z�N�z�

lnNi�z�N�z�

, �4�

where Ni is the mean particle number within the three-dimensional domain with axial extension z and volume Vi

=z��rout2 −rinn

2 �i and N�z�=�i=1M Ni�z�. In the actual computa-

tions we selected z=R and a�2�R2� /u�. A uniform dis-tribution in the cross section Ni�Nhom=N /M would achievethe entropy value Shom=ln�M�, which is the maximumamong all possible distributions. The minimum is zero whenall particles in the cross section are concentrated in a singledomain. The normalized entropy S�z� /Shom ranges from zeroto one, with one corresponding to the uniform spreading ofparticles.

In the top panel of Fig. 7 the normalized entropy is re-ported as a function of the axial distance. For every particlepopulation the entropy in the injection section vanishes,since all particles are concentrated within the tiny injectionregion around the axis. While carried downstream, particlesdiffuse around due to turbulent fluctuations, leading to theentropy increase downstream. An asymptotic constant en-tropy is eventually reached in the fully developed region.

For all the cases we have considered, the maximum en-tropy Smax achieved during the downstream evolution is al-ways close to Shom. In these conditions, it is easy to show thatthe location z=LSmax

, where such maximum is attained, cor-responds to the position where particles came closer to theuniform distribution, i.e., the sum of the squared fluctuationswith respect to the uniform distribution, �i�i

2 with �i= �Ni

−Nhom� /Nhom, is minimal. This follows directly by expandingthe entropy around the homogeneous distribution Ni�Nhom

up to second order in the fluctuations �i, Shom−S�1 / �2M��i�i

2.The behavior of �i�i

2 /M2 is shown in the inset of the toppanel of Fig. 7. By comparison with the behavior of theentropy, we observe that this indicator provides evidence that

entropy is well correlated with the amount of dispersion ofthe particles. Actually, where �i�i

2 is minimum, i.e., particlesare uniformly distributed, entropy is maximum. Conversely,large values corresponding to highly localized populationsare related with low entropy. In general, the two indicatorsprovide essentially equivalent information on the dispersionand segregation processes.

From Table II the maximum entropy Smax is in fact unityfor populations with St+�1.0 and for St+=100—nearly uni-form distribution. For the lightest particles, once reached,this nearly uniform state is maintained downstream. Theheaviest particles follow a different evolution. They firstspread almost uniformly in the developing region before seg-regation starts to occur asymptotically eventually leading toa significantly smaller entropy value. In the intermediaterange 5�St+�50 a lower maximum entropy value isachieved, implying a non-negligible variance of the particledistribution among the annular domains at the station wherebest mixing is achieved. For this kind of particles, entropy isnot monotonic with position z, leading to an asymptotic statewith relatively low entropy due to turbophoresis.

We observe that the distance from injection of the sec-tion where best mixing is attained is smallest at St+=10, seeTable II and Fig. 8, to increase both at smaller and largerStokes numbers. It should be remarked, however, that inter-mediate weight particles are unable to mix completely evenduring the developing phase.

Concerning the asymptotic state, the bottom panel of

FIG. 7. �Color online� Top panel: Shannon entropy of the mean particleconcentration vs z /R for each particle population. Inset: �i�i

2 /M2 vs z /R.Bottom panel: average particle number in the region from the wall to thegeneric radial distance from the wall vs 1−r /R.

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Fig. 7 provides the average particle number in the regionfrom the wall �r=R� to the generic radius r �integral of thedistribution function Ni /N� versus normalized radial distancefrom the wall, 1−r /R. In case of strong wall accumulation�St+=5–100�, a significant particle fraction is concentratedin the layer adjacent to the wall, as shown by the finite posi-tive value of the integrated distribution at the pipe wallr /R=1. In other words, the particle concentration, related tothe radial derivative of the function shown in the plot, tendsto develop a singularity at the wall. In strict terms, with ourmodel of interaction with the wall, the particles do not reachthe pipe boundary due to their finite size. This behaviorshould be contrasted with cases where turbophoresis is lessefficient, like for our lighter particles, which present a regu-lar distribution function everywhere. For comparison, theplot reports the integrated distribution for a fictitious uni-formly distributed particle ensemble, which is almost indis-tinguishable from case at St+=0.1. A remarkable feature ofthe asymptotic integrated distribution functions displayed inthe bottom panel of Fig. 7 is that similar curves correspondto nearly equal values of entropy, e.g., compare cases atSt+=5 �purple squares� and 100 �black filled squares� withcorresponding entropies S =0.57 and S =0.5, respectively�Table II�. As a final remark on the asymptotic state, weobserve that the length of the development region increasesmonotonically with the Stokes number, as qualitatively in-ferred from the top panel of Fig. 7, see, e.g., St+=5 �purplesquares� and 100 �black filled squares� where the overalllevel of accumulation at the wall is comparable. To providedata for comparison with experiments, the length of the de-veloping region as a function of the Stokes number is re-ported in Fig. 8, where a power law with exponent 0.21 is

found to fit the data with reasonable accuracy. The samefigure also shows the location where maximum mixing isachieved. As already discussed, particles with St+=10 reachtheir maximum mixing at the smallest distance from injec-tion. Nonetheless, their distribution at best mixing is the leastuniform among all the populations, as quantified by themaximum entropy Smax.

D. Particle velocity statistics

The previous analysis of particle concentrations, Fig. 6,highlighted the special role played by the buffer layer in theprocess of accumulation near the wall. For strongly segregat-ing particles, e.g., St+=10, in the developing part of the pipebefore equilibrium distributions are achieved, the bufferlayer becomes transiently the region most depleted of par-ticles. The buffer layer has been addressed by many authors�e.g., Refs. 14 and 16 and reference therein� to explain tur-bophoresis in terms of the coupling between particles andcoherent structures. Our purpose here is to provide a quanti-tative description of this effect based on the statistics of wall-normal particle velocity compared to the statistics of fluidvelocity conditioned to the particles, i.e., at particle posi-tions.

Top left and top right panels of Fig. 9 represent the pdfof the radial particle velocity for several populations at z /R=25 �developing region� and z /R=200 �far field region�, re-spectively. The statistics was acquired in the range of 5�y+�30 and we remind that Vr�0 corresponds to motiontoward the wall.

Lighter particles �St+�1� are expected to behave almostas passive tracers, i.e., the statistics of their velocity shouldreproduce that of the fluid. This is confirmed by the pdf inthe far field—top right panel—where red squares and blueand black triangles fall almost exactly on top of the solidline, which represents the unconditioned pdf of the fluid ve-locity. In the developing region—top left panel—the appar-ent bias, i.e., a positive radial velocity more frequent thanfound for the fluid, is explained in terms of the history ofparticle motion. All particles were injected at the axis of thepipe. At this location the local particle density in the wallregion is still below its equilibrium value to be achievedfurther downstream. Consistently, a net positive �i.e., towardthe wall� flux of particles is experienced. This implies thatthe statistics of particle velocities should be slightly biasedtoward positive values. Once the far field is reached, no ap-preciable difference can be detected between light particlesand carrier fluid.

Intermediate weight particles �St+=5, St+=10� are char-acterized by their inability to follow fast fluid motions away

TABLE II. Characteristic lengths of the developing region. LSmaxis the axial distance from injection where the

maximum of entropy Smax is achieved; LS is the distance beyond which the entropy reaches its asymptotic

value S .

St+ 0.1/0.5 1.0 5.0 10.0 50.0 100.0

LSmax�Smax� 50.0�1.0� 40.0�0.99� 23.18�0.93� 20.0�0.89� 27.0�0.96� 34.0�0.99�

LS �S � 50.0�1.0� 70.0�0.98� 110.0�0.57� 120.0�0.32� 170.0�0.31� �200.0� ��

FIG. 8. �Color online� Characteristic lengths of the developing region as afunction of Stokes number. The location where best mixing is achieved,LSmax

, is represented as a red solid line. The overall length of the developingregion LS

is drawn as a green dashed line. It is fitted by the power law withexponent 0.21 shown as a dotted blue line.

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from the wall. This is clearly seen in the far field—top rightpanel—where left and right triangles are not too far awayfrom the solid line in the positive tail while they are consid-erably below on the negative one. This feature is enhanced inthe developing region due to the mentioned effect of previ-ous history, top left panel. This observation supports the rea-sonable conjecture that inertia prevents these particles fromexecuting fast motions from the wall toward the outer region.On the contrary, they approach the wall with probabilitymuch closer to that of the fluid. Looking at the pdf in naturalscale—inset of both panels—one appreciates how departurefrom the wall is typically achieved through relatively slowmotions of the particles, consistent with the apparent positiveskewness of the pdf. This overall behavior is already per-ceived in the near field—top left panel—when care is takenof the bias concerning the effect of previous history.

In comparison, heaviest particles �St+=50,100� shownarrower pdfs compared to both fluid and lighter particles—top right panel. Apparently, the skewness is much smallerthan that found for lighter particles, showing a more sym-metrical attitude of the particles to follow both directions ofmotion. The large inertia acts as a filter, irrespective of thedirection of motion, so that almost no remnant of the inher-ent skewness of the fluid pdf is retained. Peculiar feature ofthe pdf for our heaviest particles along the developing region

is its clear bimodal nature with two distinct peaks, one in thepositive, the other in the negative range—see the circles�St+=100� in the top left panel. This feature is ascribed to theelastic bounce-back condition enforced at the wall combinedwith the huge particle inertia. Inertia provides a “quasiballis-tic” dynamics which conserves the velocity acquired abovethe buffer layer up to the elastic collision with the wall,where the wall-normal component is reversed.

The bottom panels of Fig. 9 show the pdfs of the fluidvelocity conditioned to the presence of a particle, i.e., theprobability distribution of fluid velocity at particle positions.This is the velocity of the fluid experienced by the trans-ported particles. This conditioned distribution conveys infor-mation as concerning the selective ability of inertial particlesto be preferentially located at positions where certain eventsoccur for the fluid. The log-linear plot evidences the tails ofthe pdf. Concerning the data for the fully developed regionshown on the right panel, strong events with negative radialfluid velocity �outward, away from the wall� are almost ac-curately sampled by particles, i.e., the frequency of a givenevent for the fluid �unconditioned statistics, solid line� coin-cides with that conditioned to particles �symbols�. We couldsay that particles sample uniformly this class of events. Onthe contrary, the right tail �large inward fluid velocity, towardthe wall� shows the substantial bias of the particles, which

FIG. 9. �Color online� pdf of particle radial velocity �top plots� and of fluid velocity experienced by particles �bottom plots� in the buffer layer. Left panels:development region z /R=25; right panels: far field z /R=200. The solid line is the z-invariant pdf of radial flow velocity.

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tend to sample strong inward fluid motion less frequently.This behavior is most pronounced at St+=10 �intermediateweight particles�, with either heavier or lighter particles ableto better reproduce the unconditioned fluid velocity statistics.The smaller probability of the events in the right tail is bal-anced by the slight increase in frequency of negative eventswith small intensity, as apparent from the inset which showsthe pdf in natural scale, hence focusing on the events aroundthe modal value. These smooth events are able to drag theparticles, as seen by the particle pdf in the inset of the topright plot. Concerning the developing region, the particlessample the fluid velocity more evenly. The rationale is that,in the asymptotic state, they tend to accumulate in regionswhere the fluid executes outward motions �toward the bulk�.This localization effect leads the particles to sample morefrequently outward fluid motions.16,17,21 Along the develop-ing phase �bottom-left panel� such preferential concentrationis not yet set up, with particles sampling the different fluidvelocity events with a reduced inherent bias, see Fig. 4 for asnapshot of the particle configurations.

Comparing the particle velocity pdf in the developingregion �top left panel� with the fluid velocity conditioned tothe particles �bottom-left panel�, it clearly appears that theinertia of the particles prevents them from acquiring the ve-locity of the fluid in its motions away from the wall. Thisfiltering effect of inertia to motions away from the wall is theexplanation for the localization of the particles in correspon-dence with outward fluid velocity events. Once theasymptotic state is achieved, for highly segregating particlewe end up with a population which frequently samples out-ward fluid velocities �bottom-right panel�, themselves acquir-ing, however, a significantly smaller speed �top right panel�.The localization of particles in the outward fluid velocityevents leads to a relatively reduced frequency of samplinginward fluid motions. On the other hand, the intermediateand low mass particles are apparently better able to followinward fluid motions �compare the positive branch of theparticle velocity pdf with that of the fluid velocity condi-tioned to particles, top and bottom-left panels�. We concludethat particles sample more often outward fluid velocity thaninward fluid velocity events, at the same time acquiring asmaller speed relative to the fluid in the former case. Theresult of this balance is a zero net particle flux in theasymptotic state which relies on the specific geometry of thenonlinear fluctuations of particle density at the wall.

E. Remarks on particle accumulation

The basic mechanism of particle accumulation describedin the previous paragraph is better grasped under the assump-tion of small relaxation time �p. In this case the particleacceleration is close to that of fluid, see, e.g., Refs. 1 and 2,

dvdt

� af p, �5�

where af p is the fluid acceleration at the particle position.Under this assumption, as we will show, the ensemble aver-age of the radial component of Eq. �2�,

vr�x,t� = ur p�x,t� − �parf p�x,t� , �6�

will prove especially insightful.As already discussed, in the developing phase, the par-

ticles sample almost uniformly the fluid velocity distribution,see the bottom-left panel of Fig. 9. Under this respect, evenfinite-inertia particles may behave like fluid tracers. Hence

arf p�x,t� = arf�x,t� =dur

2

dr+

ur2 − u�

2

r, �7�

where the first equality means that the average radial fluidacceleration conditioned to particles coincides with the un-conditioned one and the second equality is simply the aver-age radial fluid acceleration for a statistically stationary tur-bulent pipe flow. Considering that the mean radial fluidvelocity conditioned to particles vanishes �in our case ur p=ur=0, where the equality stems from the aforementionedsampling properties of particles and fluid incompressibility�,Eq. �6� becomes

vr�x,t� = − �p�dur2

dr+

ur2 − u�

2

r� . �8�

The immediate consequence is a mean particle drift towardthe wall �near the wall dur

2 /dr�0 and dur2 /dr � ur

2−u�2 /r�.

This behavior is consistent with the pdf shown in the top leftpanel of Fig. 9. Actually considering that Lagrangian tracersare recovered for vanishing �p, we have to look atintermediate-small values of St+ to observe a significant ef-fect, see, e.g., particle at St+=5 in the top panel of Fig. 9.

On the contrary, in the asymptotic equilibrium region theradial mean particle velocity vanishes by definition, leadingto

ur p�x,t� = �p�dur2 p

dr+

ur2 p − u�

2 pr

� . �9�

This equation should be carefully interpreted since the con-ditioned fluid velocity statistics is now biased by the particlelocalization in the streaky structures displayed in the bottompanel of Fig. 4. As we have already discussed, particles tendto concentrate in correspondence of outward fluid velocityevents, implying that the averaged radial fluid velocity con-ditioned to particle in the left-hand side of Eq. �9� is nega-tive, consistent with the expected value of the dominatingterm in the right-hand side of Eq. �9�, dur

2 p /dr�0.The above reasoning illustrates clearly how the cluster-

ing of particles in high concentration streaks at the wall rep-resents the counter-reaction mechanism which blocks thewall deposition and leads to asymptotic equilibrium. Westress once again the complexity of the phenomenon fromthe statistical mechanics point of view, given the strong in-tensity of the fluctuations in the particle density needed tocompensate for the turbophoretic flux and the nontrivialphase relationship with the Navier–Stokes field.

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F. Joint pdfs and conditional averages

In order to analyze in more detail the difference betweenparticle and fluid radial velocity in the buffer layer and con-firming the conclusions of the previous paragraph, joint pdfsare shown in Fig. 10 for the fully developed region. Lightparticles �St+�1, top left plot� follow the fluid velocity fluc-tuations remarkably well, as shown by the alignment of theisolevels of the joint pdf with the straight line Vr=Ur. As theinertia increases the scatter increases, top right panel. AtSt+=10, bottom left, a substantial discrepancy is apparent,with particles departing from the wall �third quadrant� takinga much smaller velocity especially for intense events. Thisobservation confirms that turbophoresis is mainly due to par-ticles hardly leaving the viscous sublayer combined withtheir easier approach to the wall, first quadrant.

For ballistic particles the joint pdf confirms that theirvelocity is much less correlated with that of the fluid. Still, abottom-left/top-right alignment is perceived. When particleinertia becomes so strong to make the collision with the wallespecially significant, a symmetrized pattern emerges due toa substantial invariance of the pdf with respect to particlevelocity reversal. The superposition of the two symmetricalorientations, bottom-left/top-right and top-left/bottom-right,gives rise to the rhomboidal shape apparent in the bottom-right panel, which should then be interpreted as a signatureof wall collisions.

A clearer view is gained by addressing the average radialparticle velocity conditioned to the velocity of the fluid, Fig.11. The top panels refer to the developing region, while thetwo bottom ones concern the fully developed state. To makethe plot easier to read, the particle populations are dividedinto two classes, light and heavy particles, with the first onesdisplayed on the left and the second ones on the right. Theimmediate impression is that the difference between devel-

oping and far field we discussed before does not emerge soneatly in the statistics conditioned to fluid events. Actually,when exposed to a given fluid velocity, the individual par-ticle dynamics is clearly the same in both regions. We rein-force our conclusion that the statistical differences, previ-ously observed between developing region and far field, aremainly ascribed to the spatial distribution of particles, whichis built up to achieve the equilibrium condition.

Lighter particles �St+�1, left panels� exhibit conditionalaverage velocities which almost exactly match the fluid ve-locity, confirming that they are carried along with the fluid atthe same speed. The only differences are found in the ex-treme events of very large fluid velocity directed away fromthe wall �negative branch�, revealing the inability of the par-ticles to accelerate promptly due to their small but finiterelaxation time. A certain data dispersion is also detectable atthe other extreme of fast fluid approaching the wall �positivebranch�. Here, however, the effect is not so neat and therather blurred result suggests contamination from an incom-pletely converged statistics in the positive tail.

Very heavy particles �St+�50, circles and diamonds inthe left panels� present an almost vanishing mean conditionalvelocity, i.e., their motion is substantially independent fromthe fluid velocity. Positive and negative events almost per-fectly balance as a consequence of the elastic rebounds at thewall. Already at St+=50 this strong signature is lost and amore significant bias is induced by the conditioning fluidvelocity.

As always, the most interesting results concern the inter-mediate range of Stokes numbers, 5�St+�10. Specifically,weak and moderate intensity positive fluid velocity eventsare followed quite accurately by the particles in the range0�Ur�0.04. Stronger positive events see the particles trail-ing the fluid with a smaller velocity. On the contrary, particleinertia considerably filters the fluid excitation away from thewall, almost independently from the intensity, as apparent inthe discontinuity in the slope of the plot at Ur=0.

G. Statistics of approaching/departing particles

It may be interesting to consider other forms of condi-tional statistics. Given the conditional probability of the ra-dial particle velocity conditioned to the radial fluid velocityp�Vr Ur�, the fraction of particles approaching/departing thewall associated with the given fluid velocity event Ur is

n��Ur� =N��Ur�N�Ur�

= �n� Ur� =� p�Vr Ur�H��Vr�dVr,

where N�Ur� is the number of particles with fluid velocity Ur

and H�Vr� is the Heaviside step function. For future use wedefine

nT� =� p�Ur�n��Ur�dUr =

�N�Ur�n��Ur�dUr

�N�Ur�dUr

=�N��Ur�dUr

�N�Ur�dUr,

where we have used p�Ur��N�Ur�. The average velocities of

FIG. 10. Joint pdf of radial velocities of fluid and particles in the bufferlayer for different Stokes numbers at z /R=200 �St+=0.1, top left; St+=1, topright; St+=10, bottom left; St+=100, bottom right�.

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particles approaching/departing the wall conditioned to agiven fluid velocity event Ur,

Vr��Ur� = �Vr Ur,Vr � 0� =� p�Vr Ur�H��Vr�VrdVr,

are plotted for the far field in the top and bottom panels ofFig. 12, respectively. The top plots confirm that light par-ticles follow accurately the fluid, reproducing the fluid veloc-ity for positive Ur and identically vanishing for negative Ur.Identical conclusion, with sign reversed, follows from thebottom plot for departing particles. Again, very heavy par-ticles tend to correlate poorly with the fluid velocity, al-though a bias is observed with vanishing Vr

+ for negative Ur,which becomes slightly positive for positive Ur �top plot�.Similar considerations can be made concerning the reversecase in the bottom plot. Intermediate Stokes number particlesmanifest the already discussed asymmetric response. Par-ticles approaching the wall follow moderate-intensitypositive-velocity fluid events rather accurately �small posi-tive Ur, bottom plot and inset�, while particles departingfrom the wall differ considerably from the fluid velocity alsofor small intensity events �small negative Ur, right top plotand inset�. Concerning n�, Fig. 13 displays only the ap-proaching particle fraction, given the obvious relation n−

=1−n+. The plot shows the non-negligible fraction of par-ticles moving opposite to the fluid increasing with particleinertia.

A typical velocity can be attached to each of the twoclasses of approaching and departing particles,

c� =�Vrp�Vr�H��Vr�dVr

�p�Vr�H��Vr�dVr=

�p�Ur��Vr Ur,Vr � 0�dUr

�p�Ur�n��Ur�dUr

=1

nT�� p�Ur��Vr Ur,Vr � 0�dUr.

Such average velocities are strictly connected to the net par-ticle fluxes M� toward/away from the wall. Actually, thefluxes m�=M� /NT, normalized by the local number of par-ticles NT, are given by

m� =NT

�

NTc� = nT

�c�.

Clearly in equilibrium conditions the net particle massflux is zero,

m = m+ + m− = 0.

It follows

FIG. 11. �Color online� Conditional particle radial velocity �Vr Ur� vs fluid velocity in the buffer layer. Top and bottom panels refer to z /R=25 and z /R=200, respectively. Left column: lighter particles �St+=0.1,0.5,1�; right column: heavier particles �St+=5,10,50,100�. The solid line gives the fluid velocity.

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nT+c+ + nT

−c− = 0.

In Fig. 14 the typical velocities c� are plotted as a functionof the radial coordinate at two axial stations, one in the de-veloping region and the other in the fully developed region.In case of fluid parcels, since �Vr Ur ,Vr�0�=H��Ur�Ur

and n��Ur�=H��Ur�, the above definitions reduce to

cF� =

�Urp�Ur�H��Ur�dUr

�p�Ur�H��Ur�dUr,

giving the average approaching/departing velocity of thefluid as experienced by perfectly Lagrangian tracers. A firstobservation is that fluid parcels cF

� manifest a systematicasymmetry: Near the wall �r /R�0.8�, inward motions occuron average with a larger velocity �c=c++c−�0, see theinset�. The opposite behavior is observed in the bulk of thepipe. Due to volume conservation which implies vanishingglobal flux, the number of observed events �nF

�� has the op-posite trend. A second observation is that the tracers aresmoothly recovered in the limit of small Stokes number. Allthese features appear more clearly in the insets of the figure,providing the difference in the modulus �algebraic sum� ofthe two averaged velocities. The particles most affected byturbophoresis show the largest difference between approachand departure velocities in the buffer layer �St+=10�. Thecomparison between developing and fully developed region�bottom versus top panel� shows that the kinetics of the pro-cess �i.e., the velocity distribution� continuously changes inthe streamwise direction until the equilibrium state isreached. The effect is explained in terms of the mentionedlocalization of particles.

FIG. 12. �Color online� Average radial velocity of particles approaching/departing the wall conditioned to the fluid velocity in the buffer layer �topand bottom panels, respectively�, z /R=196–200.

FIG. 13. �Color online� Average fraction of particles approaching the wallconditioned to the fluid velocity in the buffer layer. z /R=196–200.

FIG. 14. �Color online� Mean radial velocity of approaching/departing par-ticles in modulus c� vs radial position for three typical values of the Stokesnumber in the developing �z /R=30� and in the fully developed region�z /R=200�, top and bottom panels, respectively. Filled and open symbolsdenote departing and approaching velocities, respectively. Solid and dashedlines denote departing and approaching velocities of the fluid. In the insetthe algebraic sum of the two velocities is c=c++c−.

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IV. CONCLUDING REMARKS

Numerical simulations of turbophoresis have been re-peatedly addressed in the literature in terms of time evolutionof axially periodic configurations. Despite the much closercorrespondence to experimental conditions of the spatiallyevolving setting, to the authors’ knowledge, this kind of nu-merical simulations has never been attempted before. Actu-ally, for dilute particle populations �i.e., negligible back re-action on the fluid and interparticle collisions� one mayexploit the independence of fluid motions from particle dy-namics. This allows to use the standard tools of DNS ofturbulent channel and pipe flow to perform the simulation ofthe carrier fluid in a base turbulent domain. By replicatingseveral times the base building block of the turbulence, amuch longer pipe can be assembled, allowing to address thespatial evolution from given inlet conditions.

As expected, statistically translational-invariant condi-tions are achieved sufficiently downstream of the inlet sec-tion after a developing region, whose details strongly dependon the Stokes number. The Shannon entropy we have intro-duced to address the spatial inhomogeneity of particle distri-butions is found effective in characterizing the developingregion and in quantifying the level of wall accumulation. Itsuse naturally provides a definition of the developing lengthand a global measure of segregation. It is found that particlepopulations with similar asymptotic segregation levels �com-pare St+=5 with St+=100 or St+=10 with St+=50� maypresent substantially different developing lengths. Hence theparticle response to turbulence should be characterized by atleast two parameters, the development length and theasymptotic accumulation level. The nontrivial dependence ofthe developing length on the particle Stokes number makesimpossible a direct mapping of the rather artificial temporaldeveloping setting into the more realistic spatially evolvingone.

As expected from published temporal simulations, par-ticles with St+=10–50 are confirmed to manifest maximumcoupling with the dynamics of the buffer layer. The localStokes number of the particles in the buffer layer, based onwall normal distance and friction velocity, controls the reso-nance condition between the particles and the characteristicstructures of the buffer layer, leading to the conditionSt�ybuffer��1.

The buffer layer is confirmed as the main engine of tur-bophoresis. Actually the present simulations show beyondany doubt the systematic asymmetry of the pdf of particlevelocity. The pdf of fluid velocity sampled by the particlesclearly shows the trend toward particle accumulation in cor-respondence with fluid velocity events directed away fromthe wall. It is the origin of the streaky structures observed inthe instantaneous particle configurations. At the same time,the wall normal particle velocity pdf exhibits the filteringeffect operated on fast outward motion by particle inertia. Inthe fully developed state, the number of particles slowlydrifting outward, toward the bulk of the flow, is larger thanthe number of particles moving toward the wall at a fasterspeed, thus providing the equilibrium condition of zero netparticle flux in the wall normal direction. The effect is spe-

cially significant for resonating particles, St�ybuffer��1, St+

�10, able to follow sweeps from the bulk of the flow towardthe wall rather easily while being unable to comply withabrupt ejections. This originates the selection mechanismgiving rise to wall accumulation. To make this effect moreeasily assessed we have exploited the classical approxima-tion of the particle advection equation for small relaxationtime. The argument successfully illustrates both the particledrift toward the wall during the transient and the necessity ofthe segregation of the particles in streaky structures at thewall in the eventual equilibrium state. It follows that wallparticle streaks in correspondence with outward fluid mo-tions are the arrangement of particles needed to counteractthe inward motion induced by turbophoresis. The strong in-tensity of the local particle density fluctuations and the non-trivial phase relationship with the velocity makes the prob-lem extremely interesting from the statistical mechanicspoint of view.

For modeling purposes, since the particle distributioncontinuously changes along the developing region, robustand stable observables need being based on statistical fea-tures independent of the local concentration, like, e.g., aver-age particle velocity conditioned to given fluid velocityevents. The data, interpreted in terms of average speed ofapproaching/departing particles, indicate a substantial speeddifference between the two subensembles of approaching/departing particles in the buffer layer for particles withstrong wall accumulation. Such average speed of approach/departure does change significantly along the developing re-gion. This calls once more the attention to the developingregion, where particles with similar asymptotic accumulationat the wall present substantially different developing lengthscorresponding to different deposition rates.

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