7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Relations between preferential sampling and turbophoresis of inertial particles
G. SardinaT F. PicanoT P. GualtieriT and C.M. Casciola*
Dipartimento di Meccanica e Aeronautica Sapienza Universith di Roma via Eudossiana 18, 00184 Roma, Italy
gaetano.sardina ~uniromal1.it
Keywords: Particleladen gas, turbophoresis, turbulent pipe flow, DNS
Abstract
In wall flows turbophoresis amounts to a large accumulation of inertial particle close to the wall. This complex
phenomenon, although deeply studied, has not been sufficiently understood. In this paper we want to stress the role
of particle preferential sampling of certain fluid events in the turbophoretic mechanism. In order to investigate both
the transient and the asymptotic equilibrium states a spatial evolving DNS of particle laden turbulent pipe flow has
been run. Particles with relaxation time comparable to buffer layer characteristic flow turbulent time scales have been
found as the most accumulating ones. From the data, it emerges that different inertia exhibit comparable level of wall
accumulation although at very different developing lengths. In the equilibrium asymptotic region, a balance between
the turbophoretic drift and the particle preferential sampling of fluid events occurs since the mean particle flux in
the wall normal direction vanishes. The transient phase is characterized by particle dispersion, which is influenced
by both turbophoretic drift and the preferential sampling. Increasing the particle inertia the relevance of the latter
phenomenon decreases.
Introduction
The carrier phase obeys to the incompressible Navier
Stokes equations,
Particle laden wall flows have been investigated in sev
eral configurations both through experimental and nu
merical approaches Balachandar & Eaton (2010); Sol
dati & Marchioli (2009); Young & Leeming (1997).
Main feature of these multiphase flows is the socalled
turbophoresis, a curious phenomenology consisting of
a substantial particle concentration increase at the wall.
The classical numerical approach used to address this
process, is based on time evolving simulations where
the flow is evolved in time up to the eventual steady
state for particle distributions. This approach can not
provide information on the transient dynamics and the
interrelated features which block the turbophoretic drift
leading to the asymptotic equilibrium conditions. In this
context, the preferential localization of particles induc
ing the preferential sampling of certain fluid events is
shown to be crucial.
A new spatial evolving configuration was considered
to follow the particle dynamics through the developing
region up to the asymptotic farfield, more details in Pi
cano, Sardina & Casciola (2009). Aim of present work
is to expand on the role of preferential sampling of spe
cific fluid events as key feature of the whole accumula
tion process.
+u
8= xy
1l 8p 8 Ui
+vi~djz
where p and v are density and kinematic viscosity of the
fluid, respectively, with as the fluid velocity. As usual
the control parameter of the flow is the Reynolds num
ber, Re UbRIV, where R is the pipe radius and Ub
the bulk velocity across the section. It is useful to in
troduce the friction Reynolds number Re, U,R/V,
where U, Jr is the friction velocity, based on
the shear stress at the wall 7,. The simulation we dis
cuss concerns a fully turbulent flow with Re 3000
corresponding to Re, = 200.
Assuming small and diluted particles with density
much larger than the fluid, the particle feedback on the
fluid and interparticle collisions can be neglected (one
way coupling). The viscous Stokes drag remains as
the only force acting on a particle Armenio & Fiorotto
(2001). The dynamics of each particle is described by a
Methodology
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
.. +
S..
..
*:
Lagrangian formulation, Maxey & Riley (1983),
(2)
di V, (3
where v; denotes the particle velocity and 7,
P, d /(Pvl8) is the particle response time (Stokes time),
with di, the particle diameter. So the socalled Stokes
number, ratio of ri, and the characteristic time scale of
the carrier fluid, represents the only parameter influenc
ing the particle dynamics in a given flow field. For
wallbounded flows the characteristic fluid time is con
veniently taken as the viscous time scale (v/U, ), leading
to define the viscous Stokes number St+ = U,/v.
Concerning wallparticle interaction, we assume here a
pure elastic rebound when the particle surface hits the
wall.
The fluid solver is based on the discretization of the
equations (1) in cylindrical coordinates with a conser
vative second order finite difference scheme on a stag
gered grid. The simulation evolves in time by an ex
plicit third order lowstorage RungeKutta scheme. The
primary pipe element, see below for a complete descrip
tion of the geometry, has the length L, = 2i7R and is
discretized by a uniform grid of 128 x 80 x 128 nodes in
the axial z, radial r and azimuthal 9 directions, respec
tively.
The particle positions and velocities evolve with the
same time scheme used for the fluid ((2)(3)) and the
fluid velocity u is interpolated at the particle position by
linearquadratic Lagrange polynomials. The simulation
involves seven different populations of particles (St
0.1, 0.5, 1, 5, 10, 50, 100) which are injected at fixed rate
near the axis at the inlet section. The particle dynamics
is followed through in a very long domain (~ 200R)
consisting of 32 replications of the primary periodic pipe
flow element (see Picano, Sardina & Casciola 2009, for
details).
In order to provide a qualitative view of the simula
tion, figure 1 shows an instantaneous visualization of
the particle near field just beyond the injection point
(only 1/8 of the whole axial extension of the domain
is drawn). The particles injected at the axis, cross the
first pipe element (colors correspond to isolevels of ax
ial fluid velocity) and disperse through the domain. The
wall accumulation process is already operating at the
furthest station displayed in the figure, z/R ~ 25.
Results
For a quantitative analysis of the dispersion and accu
mulation, the mean concentrations in the classical four
regions characteristic of wall bounded flows, namely
Figure 1: Instantaneous particle configuration (St+ =
10 population) in the early stages after the injection.
Contours represent the instantaneous axial fluid veloc
ity field in the primary pipe element
wake, log, buffer and viscous regions, respectively, are
plotted in figure 2. The data are normalized by the con
centration achieved by the lightest particles (almost La
grangian tracers) in the wake region of the farfield.
Lightest particles St+ 0.5, 1. undergo dispersion
until reaching a quasiuniform distribution across the
section beyond z/R ~ 50. Despite the small Stokes
number a slight amount of wall accumulation still ex
ists as shown by the concentration in the viscous region.
Along the transient a progressive filling of the four re
gions is observed, from the wake down to the wall.
A completely different mechanism is ap.nemhll'll act
ing on particles with Stt 5., 10., which are found
to achieve huge wall accumulations, middle panels. For
these particles, the buffer layer concentration (blue sym
wake 
3000 log 
buffer
e06t
0 50 100 150 2E
zlR
wake 
3000~ log
buffer
VISOU C 
100~
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
wake 
log
buffer  *; 
teUD
0 50 100 150 200
lR
wake 
10000~ log
buffer
le04
le06
0 50 100 150 200
O 50 100
z/R
150 200
wake 
log
buffer
V15cous
le04
le04 ~
le06
0 50 100
z/R
150 200
wake 
log
buffer
v15COUS
100DD
le04i
le06
0 50 100 150 200
Figure 2: Mean particle concentration vs axial distance z/R in every region characteristic of wall bounded flows. Top
left plot Stt 0.5, top right Stt 1 ., middle left Stt 5., middle right Stt 10., bottom left Stt 50. and
bottom right Stt 100.
bols) in the developing region, 15 < z/R < 25, is the
lowest across the section, i.e. below the values found ei
ther in the log and viscous layers. This behavior particles
tend to be expelled from the buffer layer, keeping the lo
cal concentration to a minimum. The effect is strongest
at St+ 10, consistently with a rough estimate of the
resonance condition in the buffer layer, St( . r fer)
1, based on the the typical buffer time scale ,.". ,fer/v,
yielding resonance for St+ 5 + 50. Heavier parti
cles, Stt = 50., 100., initially show a definite trend to
wards homogenization with the same progressive filling
of the cross section described for the lightest particles
St+ .5. Further downstream, z/R > 30, turbophore
sis takes over leading to particle segregation in the vis
cous sublayer.
Overall three different behaviors can be distinguished.
Light particles are more or less convected with the fluid
and show modest accumulation. Particles with interme
.~1 0 U
50
0.005
20 80 140 200
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
0.1
0.01 0.01 Vq
0.00 0.05 .0
0.005 0.005
20 80 140 200 I 20 80 140 200
0.05
0.1
50 100 150 200 0 50 100 150 200
zlR zlR
50 100 150 200 0 50 100
zlR zlR
0 50 100
2/R
0
0.0
2DD
0.0
0.
0.1
150 200
50 100
zlR
150 200
Figure 3: Particle mean radial velocity V,, fluid mean radial velocity
velocity drift V, 7, ap vs z/R in the buffer layer region 5 < y
Stt 1 ., middle left Stt 5., middle right Stt 10., bottom left St
diate inertia are strongly coupled to the turbulent fluc
tuations in the buffer layer, to end up with a substantial v
wall segregation and wall concentrations of the order of
hundred times the wake concentration. The heaviest par where vr
ticles still manifest a certain level of segregation, which is the mear
is reached, however, further downstream. and a,, is
dial directic
The mechanism of particle accumulation can be spot ties toward
ted by addressing the wall normal component of the en V, 7, a
semble average of equation (2) towards the
sampled by particles Uf and turbophoretic
< 20. Top left plot Stt 0.5, top right
50. and bottom right Stt 100.
r(r, z) = 1. (r, z) 7parp(r, z), (4)
Sis the mean particle radial velocity, .I.
n fluid radial velocity sampled by particles
the mean acceleration of particles in the ra
on (hereafter positive values denote veloci
s the wall). The turbophoretic drift velocity
r (r, z) promotes the motion of the particles
:wall. In the quasiLagrangian limit, parti
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
cles with small 7,, the turbophoretic velocity can be es
timated in terms of fluid acceleration. Since the particle
and fluid accelerations are almost identical in this case,
it follows
~(, )~f()dr t 5
Hence the maximum fluid acceleration occurring close
to the wall in the bufferlayer region, produces a strong
turbophoretic drift V, directed towards the wall. The ar
guments presented here hold for small relaxation time
and provide a linear dependence of the turbophoretic
drift on 7,. In the general case, the particle acceleration
differs from that of the fluid leading to a more complex
dependence (see Young & Leeming 1997, for a collec
tion of experimental results). The three different behav
iors discussed previously concerning the axial variation
of the particle concentrations, figure 2, can be explained
by looking at figure 3. There the three terms of (4),
namely the average particle radial velocity 17, v,,
the average fluid radial velocity sampled by particles
LT;  ... and the turbophoretic drift Vt q,<,1,, are
reported as a function of the axial coordinate z/R in the
buffer layer. More precisely in the figure 1 is plotted
in red, LT, in blue and VT in green (note the change
of sign introduced for better readability). The accumu
lation process progressively changes from the injection
section to the fully developed region as shown by the
different terms locally dominating the balance (4).
For the smallest particles, Stt 0.5 + 1 top pan
els, in the near field at z/R ~ 15, the mean particle
velocity is essentially given by the mean fluid velocity
sampled by the particles, T, Uf Particles, injected at
z/R 0 on the axis, initially tend to sample more fre
quently fluid motions from the bulk of the pipe towards
the near wall region. In this region, z/R < 25, the tur
bophoretic velocity V, does not play any relevant role
and the mean particle velocity is dominated by turbu
lent dispersion. The dynamics changes in the developed
region where the axiallyhomogeneous equilibrium dis
tribution is achieved and the mean particle velocity nec
essarily vanishes. Here, downstream of z/R ~ 50 + 60,
the small turbophoretic drift, directed towards the wall,
is balanced by the mean fluid velocity sampled by the
particles which is directed in the opposite direction. We
see how crucial is the preferential sampling of outward
fluid velocity events in establishing the equilibrium con
ditions. The low level of wall accumulation found in this
case, is somehow associated to the small intensity of 15
induced by the modest 7,.
The dynamics is different for intermediate particles,
Stt 5 + 10, middle panels of figure 3. The devel
oping stage is here characterized by a combined effect
of the preferential sampling of fluid velocity and the tur
bophoretic velocity Vt, which are both directed towards
the wall. For Stt 10 particles, right middle panel,
the turbophoretic drift is the leading term in the whole
axial span of the domain. This comparatively strong tur
bophoretic effect is presumably the reason for the min
imum concentration achieved by these particles in the
buffer layer of the developing region, see figure 2. As
always at equilibrium, here z/R > 80, preferential sam
pling of outward fluid motions exactly balances the tur
bophoretic drift Vt.
Concerning heavy particles, Stt 50 + 100 bottom
panels, the dynamics in the developing region, 10 <
z/R < 20, is dominated by the turbophoretic drift Vt
with negligible influence of preferential sampling which
eventually becomes effective in the far field, z/R > 50,
where 177 is bound to balance Vt Fully developed con
ditions are achieved further downstream than in the other
cases (z/R ~ 180 for St+ 50 and at z/R > 200 for
the other ones).
Preferential sampling is realized as an anomalous spa
tial concentration of particles in correspondence of co
herent structures where the fluid departs slowly from the
wall. This effect is graphically shown in figure 4 where
the sign of the instantaneous particlesampled fluid ve
locity is represented as a colored circle (blue departing,
red approaching) in the buffer layer of both the develop
ing region (25 < z/R < 35, top panel) and the farfield
(190 < z/R < 200, bottom panel). In the farfield,
the axially elongated particle aggregates which prefer
entially sample departing fluid motion are clearly con
nected with the classical lowspeed streaks which are
known to be associated with wall departing fluid velocity
fluctuations. Such localization effect is absent in the de
veloping region where particles are not segregated into
well defined patterns and show no net preferential sam
pling effect.
Conclusions
A detailed analysis of particle preferential sampling has
been discussed in relation with wall accumulation us
ing data from a direct numerical simulation of a particle
laden turbulent pipe flow at Re v = 200. The simulation
concerns the spatial evolution of particle populations
along the axial coordinate. After the initial injection,
near the axis, particles are subjected to both dispersion
and turbophoretic drift which are both directed towards
the wall in the near field region. These two different
mechanism contribute in different proportions to wall
migration depending on particle inertia. The lightest
particles are dominated by the dispersion while migra
tion of heaviest particles is controlled by turbophoretic
drift. After the developing phase, inertial particles be
gin to accumulate in the near wall region with the in
.
I, I: t *,
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
state. Particles tend to be preferentially localized in
streamwise elongated aggregates which are well corre
lated with the classical lowspeed streaks populating the
near wall region. We stress that such coherent state is
established only in the asymptotic state, whereas more
random spatial distribution characterizes the first devel
oping stages of the accumulation process. we feel that
the above considerations may have a certain relevance
for the closure theories for panticle laden flow.
References
Balachandar S. and Eaton J.K., Turbulence Dispersed
Multiphase Flow, Annual Review of Fluid Mechanics
,2010
Young J. and Leeming A., A theory of particle deposi
tion in turbulent pipe flow, Journal of Fluid Mechanics,
Vol. 340, pp. 129159, 1997
Soldati A., Marchioli C., Physics and modeling of tur
bulent particle deposition and entrainment: Review of a
systematic study, Int. J. of Multiphase Flow, Vol. 35, pp.
827, 2009
Picano F., Sardina G. and Casciola C.M., Spatial devel
opment of panticleladen turbulent pipe flow, Physics of
Fluids, Vol. 21, 2009
Armenio V. and Fiorotto V., The importance of the
forces acting on particles in turbulent flows, Physics of
Fluids, Vol. 13, 2001
Maxey M.R. and Riley J.J., Equation of motion for a
small rigid sphere in a nonuniform flow, Physics of Flu
ids, Vol. 26, 1983
34
N 30
26
4 2 0
2 4
Figure 4: Instantaneous snapshot of particles with
Stt 10 in the buffer layer. Developing region, top
panel; developed asymptotic region, bottom panel. Red
and blue circles denote particles sampling approaching
and departing fluid velocity, respectively.
termediate particles St+ = 10, 50 reaching maximum
levels of wall concentration even hundred times larger
than the average value. Particles keep on accumulat
ing until an axially independent statistical equilibrium
is reached sufficiently far from the injection point. The
equilibrium in the asymptotic region, is characterized
by the balance between turbophoretic drift and average
particlesampled fluid velocity. The occurrence of an
outwards average panticlesampled fluid velocity corre
sponds to the preferential sampling of slow fluid depart
ing motions. The competition between turbophoretic
drift and preferential sampling eventually drives the wall
normal panticle flux to zero establishing the equilibrium
