7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Inertial particle dynamics in turbulent premixed flames
F. Battista*, F. Picano*, G. Troianit and C.M. Casciola*
Dip. Meccanica e Aeronautica, Sapienza University, via Eudossiana 18, 00184, Rome Italy
t ENEA C.R., Casaccia, Rome, Italy
battista.f@tiscali.it
Keywords: particleladen Bunsen flame, inertial particle, premixed turbulent flames
Abstract
The heat release, occurring in reacting flows, induces a sudden fluid acceleration which particles follow with a certain
lag, due to their finite inertia. Clearly, the particle response time is crucial for the accuracy achieved in Particle Image
Velocimetry, where the fluid velocity is measured by the displacement experienced by small dispersed particles.
There are however other subtle effects, such as small scale clustering, by which inertia affects the particle dynamics.
Small scale clustering is important to determine the collision and coalescence rates of a dispersed particulate phase
which may e.g. contribute to form soot or other kinds of solid pollutants. Actually, inertia strongly biases the spatial
distribution of the particles, by inducing the formation of localized clouds with different dimensions downstream the
thin flame front. A possible indicator of this preferential localization is the socalled Clustering Index, quantifying
the departure of the actual particle distribution from the Poissonian, which would correspond to a purely random
spatial arrangement. Most of the clustering is found in the flame brush region, which is spanned by the fluctuating
instantaneous flame front. The effect is iniiilik.ai also for very light particles. In this case a simple model based on
the BrayMossLibby formalism is able to account for most of the deviation from the Poissonian. When the particle
inertia increases, the effect is found to increases and persist well within the region of burned gases. The effect is
maximum when the particle relaxation time is of the order of the flame front time scale. It expected that these findings
could be important to accurately model the formation of large aggregates by coalescence processes in the burned
region.
Introduction
Reacting flows, in which particles of different inertia and
dimensions are transported, are characteristics of many
fields of engineering and physics. Dispersed particles
are found in solidpropellant rockets where solid parti
cles are introduced to enhance the combustion process
or in reciprocating engines where carbon particles form
due to combustion. Besides, Particle Image Velocime
try (PIV) estimates the fluid velocity field measuring the
displacements of small seeding particles between two
images separated by a small time interval.
The study of particles behavior in several flows, i.e.
homogeneous and isotropic flow, shear, pipe and channel
flow, is addressed in many works to understand the phe
nomena that induce the turbophoresis in wall bounded
flows (Picano et al. 2009) or the preferential accumula
tion due to the small scales of turbulence, i.e. clustering,
(Gualtieri et al. 2009; Goto and Vassilicos 2006).
The non uniform spatial distribution of particles (clus
tering) is crucial to address the mechanisms influenc
ing growth of droplets in clouds via collisions and co
alescence (Kostinski and Shaw (2001)), particle settling
(Wang and Maxey (1993)) and inter particle collisions
(Wang et al. (2000)). Moreover, the spatial distribution
of droplets or particles in reactive jets is a topic of inter
est in combustion processes. The effects of combustion
on inertial particle dynamics is still poor understood, de
spite its relevance for its effects on particles collisions
and coalescence, phenomena which have a large influ
ence in soot formation (Hu et al. 2003; Mueller et al.
2009).
Apart from the classical effects which are observed
also in cold flows, in turbulent premixed combustion in
ertial particles are affected also by new phenomenolo
gies such as the interaction with the thermal expansion
induced by the heat release in the reaction region or
thermophoresis caused by intense temperature gradients,
see Sung et al. (1994). As a matter of fact, the flame
front induces abrupt accelerations of the fluid in a very
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
simplifying assumptions are made about the particle sus
pensions we are dealing with. Small spherical diluted
particles are considered. Interparticle collisions and
force feedback on the fluid are neglected. Under these
conditions and with particle density much larger than
the fluid pp/pf > 1, the equation of motion for dis
persed particles reads Maxey and Riley (1983); Stella
et al. (2001):
mp d = Fs + FG + FT ,
Figure 1: Instantaneous field of axial velocity fluid,
four azimuthal planes, with contours and isolevel of re
actants at Y,/Y = 0.5. Black dots corresponds to the
lighter particles position Stif 0.022. For graphical
needs the particles represented are less of about 40 times
than the evolved ones.
thin region which particles follow with different lags de
pending on their finite inertia. This phenomenon has
a large impact on the particle spatial arrangement that
present high level of clustering.
In this framework, aim of this work is to assess the ef
fects of finite inertia on particle preferential localization.
To this purpose a DNS of a Bunsen jet coupled with par
ticle Lagrangian tracking has been carried out. In order
to isolate the role of inertia from concurrent effects, ther
mophoresis was neglected. As we shall see, the proper
parameter can be defined as the flamelet Stokes num
ber Stfl based on the characteristic time scale of the
flame front and the particle relaxation time. To evalu
ate the amount of particle localization we consider the
clustering index to measure the inertia induced segrega
tion of the particles. An analytical model is proposed to
show that a ,igIilik.iIn contribution to clustering is as
sociated with the fluctuation of the instantaneous flame
front position. The model works accurately for tracers
and light particles, predicting the clustering peak in the
flame brush. Increasing the inertia, the model becomes
inappropriate and clustering effects are found to persist
in the product region, away from the flame brush.
Theoretical Background
In order to gain insight into particle motions and to dis
cuss several issues arising from finite inertia, certain
where mp and V are the particle mass and velocity, Fs
is the Stokes viscous force, FG is the gravitational force
and FT is the thermophoretic one. To focus only on
the effects of the particle inertia in turbulent premixed
flames we neglect FG and FT. Hence equation (1) sim
plifies to
UV
V1 (2)
Tp
where U is the fluid velocity at particle position, while
18 dppP
is the particle relaxation time representing the response
time to fluid velocity fluctuations (here dp is the particle
diameter, pp and pf the particle and fluid density, v the
fluid kinematic viscosity).
To characterize the particle dynamics a Stokes num
ber should be defined as the ratio of the typical time scale
of the phenomenon and particle relaxation time. In tur
bulent premixed combustion the strong fluid accelera
tions due to the abrupt density variation across the flame
front determines the relevant time scale, which, assum
ing the flamelet regime, can be expressed in terms of the
laminar characteristics of the flame, i.e.
Tfl A= ,
where 6f1 is the thermal thickness of the laminar flame
and A.., the velocity jump across the front. The ther
mal thickness is estimated as 61f = (Tb T,)/VTs.p,
where Tb and T, are the temperature of burned and un
burned mixtures respectively, while VTs,, is the largest
temperature gradient. The thermal expansion drives a
velocity jump across the front A.., Sf (Tb/T, 1),
with Sfl the laminar flame speed. Hence we define the
relevant Stokes number as
Tp Sf (Tb/T, 1)
(5)
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Figure 2: Thin slice of width R/20 in the axialradial plane of the instantaneous particle configuration for different
Stokes time. Topleft: Stif 0.022; Topright: Stif 0.54; Bottomleft St1 2.16; Bottomright Sti; 8.65.
Colors denote the norm of the difference between particle and fluid velocity: IV UI/Uo.
Numerical method
The particle dynamics in a premixed turbulent reacting
flow is studied by means of an Eulerian DNS of a turbu
lent Bunsen flame coupled with a Lagrangian solver for
particle evolution.
The Eulerian algorithm discretizes the LowMach num
ber formulation of the exact NavierStokes equations
(no subgrid model) in a cylindrical domain to address
a reacting low Mach number flow with arbitrary density
variations neglecting acoustics.
Spatial discretization is based on central second order
finite differences in conservative form on a staggered
grid while the convective term of scalar equations is dis
cretized by a bounded central difference scheme to avoid
spurious oscillations. Temporal evolution is performed
by a lowstorage third order RungeKutta scheme.
Dirichlet (prescribed velocity) conditions are en
forced at the inflow, by using a crosssectional plane of
a periodic turbulent pipe flow, obtained by a companion
timeevolving DNS. A convective condition is adopted
at the outflow, while a tractionfree condition is used for
the side boundary. More details on the code and tests for
incompressible jets can be found in Picano and Casciola
(2007).
The chemical kinetics is given by a simple onestep ir
reversible reaction transforming premixed fresh mixture
R into exhaust gas, or combustion products P, with an
Arrhenius model for the reaction rate:
R w P,
S= oo (pYR)e6T .
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
The direct numerical simulation reproduces a premixed
Bunsen flame with a a Reynolds number based on the
diameter, equal to ReD UoD/Iv 6000, with U0
the bulk velocity.
The parameters reproduce a lean premixed methaneair
Bunsen flame (equivalence ratio p 0.7, Tb/Tu 5.3)
with constant heat capacity ratio y cp/c, 1.33,
and Sutherland's law for dynamic viscosity, p oc T1/2.
The ratio of laminar flame speed to the bulk velocity
is SL/Uo 0.05, with laminar flame thickness 61f 
0.019D.
The computational domain, [Omax x Rmax x Zmax] =
[27r x 6.2D x 7D] is discretized by No x N, x N,
128 x 201 x 560 nodes with a stretched mesh in radial
direction to assure a resolved shear layer. The grid size
in the jet region is about two/three times the Kolmogorov
scale, and is able to accurately resolves the instantaneous
flame front, for details Battista et al. (2009).
Particles evolve by a Lagrangian tracking method
which integrates eq. (2) by means of the same Runge
Kutta scheme adopted for the fluid phase. The fluid ve
locities at particles positions are interpolated with sec
ond order Lagrangian polynomials. Particles do not
counteract on the flow (oneway coupling) and inter
particle collisions are neglected. Details and tests on
an incompressible pipe flow can be found in Picano
et al. (2009). Four particle populations are considered to
mimic laboratory Bunsen premixed flame seeded with
alumina particles (pp 4000 kg/m3) with diameters:
dp = 1/m, dp = 5/m, dp = 10pm, dp = 20pm.
In these conditions the flamelet Stokes numbers are re
spectively: St1f 0.022, St1f 0.54, St1f 2.16,
Stif 8.65.
Particles are introduced in the field at fixed rate with
homogeneous distribution at the inlet section of the jet.
We assign the local fluid velocity as initial condition for
the particle. Overall, about six millions particles are
evolved in the simulation.
After reaching the statistical steady state, one hundred
complete fields, separated by 0.125 D/Uo, are collected
for statistical analysis.
A snapshot of axial fluid velocity and particle posi
tions (Stfl 0.022) is shown in figure 1.
Results
In figure 2 instantaneous configurations of the four parti
cle populations is presented to illustrate the effect on the
particles of the fluid density variation induced by heat re
lease. The differences in particle concentration between
fresh and exhaust gas regions are apparent. While for
particles with Stfi 0.022 the normalized velocity dif
ference IV UI/Uo remains always below 1%, particles
with Stf = 0.54 already show a bias of the order 5'.
N
4
2
0
2.000
0.603
0.182
0.055
0.017
0.005
I
5
r/R
Figure 3: Cluster Index field, contour, computed
through the equation (7); solid lines indicates two
isolevels of the reactant concentration YR/Y = 0.05
and YR/Y 0.95
in the proximity of the instantaneous flame front, see top
panels of figure 2.
The velocity difference increases increasing the
flamelet Stokes number and eventually persists also in
the product region for sufficiently massive particles. In
any case the largest deviation from the local fluid ve
locity is localized in the flame front, where the abrupt
acceleration of fluid occurs on the fastest time scale set
by chemical reactions which is shorter than the parti
cle relaxation time, bottom panels of figure 2. We re
inforce that, unlike lighter particles, the heavier parti
cles do not recover the fluid velocity in the hot gases
region, as promptly. We remark that a sigiilik.llI ve
locity difference is found also in the outer part of the
jet, in the region spanned by the interface separating hot
products and ambient fluid. This region is characterized
by a shear layer with large vortical structures, forced
also by buoyancy, that produce intense fluctuations and
strong intermittency. These phenomena promote the dif
10 10
8 8
6 6
N N
4 4
2 2
O 0
1 1
r/R r/R
0.01 0.02 0.03 0.04 0.07 0.11 0.18 0.30
Figure 4: Comparison of the clustering index field com
puted by the analytical form (12), left panel, through
the equation (7), right panel; solid lines indicates two
isolevels of the reactant concentration YR/Y = 0.05
and YR/Y 0.95.
ferences between fluid and particle velocity in the exter
nal part of the Bunsen jet.
The particle spatial distribution is also affected by
inertia. Figure 2 shows clustering phenomena taking
place at smaller and smaller scales as the Stokes num
ber is reduced. Dealing with the intermediate particles,
Stf = 0.54 and 2.16, clusters are particularly apparent
in the burned region, where voids have dimensions large
enough to be clearly detected by the eye. The lightest
particles, Stfi 0.022, due to their negligible inertia,
are almost uniformly distributed in each of the two re
gions separated by the flame front (burned and unbumed
regions), and locally behave more or less as a pure tracer.
In the following we will focus on the analysis of the
particle spatial distribution and its dependence on par
ticle inertia and flow features. To this aim we exploit
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
the Clustering Index CI (Kostinski and Shaw 2001),
to quantify the deviation of the actual particle distribu
tion from the corresponding perfectly random arrange
ment. For a perfectly random distribution the posi
tions of different particles are independently distributed
within the fluid volume. This state is described by
the Poissonian distribution that expresses the probabil
ity p(n, AV) to find n particles in a fixed volume AV.
The average number of particle in the volume n(AV)
f np(n, AV) dn k AV, where k No/Vo is the par
ticle density with No and Vo the total number of parti
cles and the whole volume, respectively, determines the
probability density function,
p (n, n)
nn n
126
The distinguishing feature of a Poissonian process is
that variance equals mean value, (6n)2 = (n n)2 n
and it describes particle populations homogeneous at all
scales, with no clustering nor preferential localization.
The amount of clustering can thus conveniently evalu
ated by a measure of the departure from a Poissonian
distribution by defining the clustering index,
r=(6n)
which vanishes identically for Poissonian distributions.
In general, CI depends on the control volume AV via
n and (6n)2. For deterministic systems where the num
ber of particles found in AV systematically equals n the
clustering index attains its minimum value 1. On the
contrary, where particles aggregate in clusters, the vari
ance of the number of particles found in AV is larger
than the mean value, resulting in a positive clustering
index, CI > 0.
For nonhomogeneous systems the clustering index
becomes a scalar field CI(). It is reported in figure 3
for particles with flamelet Stokes number 0.022 and con
trol volume corresponding to the computational cell. CI
is positive within the flame brush and in the outer region
of the jet. In the outer region the huge cluster index is in
duced by the large intermittency of the flow. This region
is spanned by the interface separating the burned gases
seeded with particles and the surrounding gases with
out particles. The statistical arrangement originated by
such process differs substantially from a purely random
homogeneous distribution and gives reason for the very
large value of CI. Analogous considerations concern
the effect of the fluctuating instantaneous flame front
which separates fresh and hot gases. In this case the
different concentration on the two sides of the front is
due to the expansion of the fluid across the flame. The
region spanned by the instantaneous front is called the
flame brush. Again the intermittency associated with the
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
different instantaneous configurations of the flame front
generates, igiikiik.iiI clustering detected by positive val
ues of CI. Away from the two regions just discussed,
where the intermittency tends to conform to the classi
cal picture of homogeneous turbulence, the cluster in
dex reduces to almost vanish for very light particles. For
these quasiLagrangian tracers, i.e. Stfi < 1, a possi
ble model may be based on the assumption of Poisso
nian distributions for the particles in the two nearly ho
mogeneous regions separated by the instantaneous thin
front (e.g. burned and unburned gas), .i niisl'l\ with
the flamelet assumptions. In this framework the Bray
MossLibby (BML) formalism provides the proper for
mulation. Defining the instantaneous progress variable
c 1 YR/Yj with c 0 in the reactants and c 1
in the products the probability distribution of the state of
the mixture isp(7; c) = a()6(c) + [1 a(i)]6(1 c),
where 6 denotes the Dirac function, and a = 1 c the
probability to find the unburned state at Y. The average
particle number in the control volume in each of the two
states, n, = k, AV and ~b kb AV, for the unburned
c 0 and burned c = 1 state, respectively can be ex
pressed in terms of expansion ratio
T = Tb/Tu = ku/kb = n/'. (8)
where we assumed ku/kb pu/pb on the basis of the
nearly Lagrangian behavior of the particles. The prob
ability to find n particles in the control volume at 7 as
a function of the instantaneous progress variable c, then
reads
pc(; c, n)
a(Y)6(c)p[n; l] + (1 a())6(1 c)p[n; Fb] (9)
Starting from the equation (9) we can express the first
and second order moments of the particle number pdf,
n(X) = a(x' + [1 a()]}b (10)
n2 (x) = a( + [1 Y( ] (11)
Introducing equations (8), (10) and (11), into the defini
tion of clustering index (7), it follows
Sk a(Y)(l a(()) (1 1/T)2 *
CI() = k, AV (12)
a(a) + (1 a(x)/T
As easily verified, according to this model the clustering
index deviates from zero only in the flame brush where
0 < c = 1 a < 1, and comes back to the Poissonian
null value elsewhere.
Despite its crudeness the model shows the basic
mechanism by which the fluctuating thin interface, sep
arating two Poissonian particle distributions with differ
ent mean particle number induces a positive clustering
index.
A part from the trivial behavior for T 1, the limit
of T 00 is SigSilik.lli since it describes the parti
cle distribution originating from a thin interface sep
arating the empty space and a region with a Poisso
nian particle arrangement with a finite mean value, n,.
This limit is useful to model the behavior found in the
outer region of the Bunsen jet where the prediction is
CI ku AV (1 a), with ku the mean particle con
centration in the product region and a suitably ranging
from zero in the products to 1 in the external environ
ment. Before comparing the model prediction with the
actual numerical data we remark that the model is ex
pected to be accurate for tracers and very light particles,
with inertia adding new phenomenologies.
The left panel of figure 4 reports the clustering index
for the smallest particle population, Stif 0.022, in
comparison with the reference field obtained by eq. (12).
The clear correspondence between the two fields show
that the deviation from the Poissonian distribution ex
hibited by these very small particles in the flame brush
is entirely explained in terms of the concentrations jump
across the flamelet.
As anticipated increasing inertia larger and larger de
partures from the proposed model should be expected.
These effects are clearly seen in figure 5 where the clus
ter index is reported for all particles. Notably, increasing
the inertia, the departure from the Poissonian behavior
is no more limited to the flame brush and occurs also in
the product region. The maximum deviation occurs for
Stfl 2.16 particles, still in the flame brush. Heaviest
particles exhibit a more uniform field of the clustering
index and are less influenced by the fluid expansion in
the flame brush region.
In conclusion we found that maximum clustering is
achieved for order one flamelet Stokes number, i.e.
when the particle relaxation time is of the order of the
typical time scale of the flame front. In such conditions
maximum clustering occurs within the flame brush with
a ,ig nilk .ll direct effect of inertia which can not be ex
plained only in terms of the concentration jump across
the front.
Final Remarks
The paper discusses several issues concerning the behav
ior of particles in a reacting turbulent Bunsen jet with
special emphasis on the clustering which has been quan
tified for several particle populations in the different re
gions of the flow.
The cluster index CI proved useful to quantify the
clustering by measuring the departure of the particle dis
tribution from a perfect random arrangement.
It is shown that the regions which are mostly influ
enced by preferential accumulation are the flame brush
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
10
8
c6
N
',
0.01 0.(
ir
N
1
r/R r/
)2 0.03 0.04 0.07 0.11 0.
1
R
1
r/R
.18 0.30
Figure 5: Contours of the Cluster Index (7) for all particles, from left Stfl 0.022, Stfl 0.54, Stfl 2.16 and
Stfi 8.65, respectively. The control volumes corresponds to the computational mesh cells. The lines indicate two
isolevels of the reactant concentration YR/Yj = 0.05 and YR/YR = 0.95.
and the outer part of the jet. Light particles deviate
from the random homogeneous spatial arrangement only
in the flame brush, exhibiting a Poissonian distribution
both in the reactant and product regions. This behav
ior is perfectly captured by a model based on flamelet
assumptions and tracerlike dynamics for the particles.
The model reproduce the field of clustering index with
very good accuracy.
Increasing inertia influences the clustering process
leading to a maximum clustering level in the flame brush
for particles with Stfi = 0(1). In this case clustering
persists well beyond the flame brush and is clearly ap
parent in the product region. The heavier particles are
too massive and are less influenced by the flame brush
region, exhibiting a more uniform clustering distribution
on the whole jet region.
We expect that the clustering evidenced in the flame
brush and in the product region could play a crucial role
in coalescence processes, at the origin of particulate and
soot formation (Hu et al. 2003; Mueller et al. 2009). Pre
sumably these findings could prove useful to improve
models for solid pollutant prediction in reacting envi
N
N
1
r/R
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
ronments. L.P. Wang, A.S. Wexler, and Y. Zhou. Statistical me
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