7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Passive Scalar Characteristics along Inertial Particle Trajectory in
Turbulent Nonisothermal flows
Chao Yi, Yaming Liu, Zhu He, Zhaohui Liut and Chuguang Zheng
SState Key Laboratory of Coal Combustion, Huazhong University of Science and Technology,
Wuhan 430074, China
zliuiihust.edu.cn
Keywords: DNS, Nonisothermal flow, Gasparticle flow, Passive scalar, Isotropic turbulence
Abstract
The momentum and heat coupling between carrier fluid and particles is complex and challenge topic in turbulent reactive
gassolid flow modeling. Most of observations on such topic, either numerical or experimental, are based on Eulerian
framework, which is not enough for developing probability density function (PDF) model, for which generally a Langevin
type model is proposed (Simonin 1993).
In this paper, the instantous behavior and multiparticle statistics of passive scalar along inertial particle trajectory, in
homogenous isotropic turbulence with a mean scalar gradient, are investigated by using of direct numerical simulation (DNS).
The results show that, st~1.0 particles are easy to aggregate in high strain and low vorticity regions in fluid field, where the
scalar dissipation is usually much higher than the mean valuett and every time they move across the cliff structures, the scalar
change is much more intensive. Anyway, the selfcorrelation of scalar along particle trajectory is significantly different from
the velocities observed by particle, for which preferconcentration effect is evident. The mechanicaltothermal time scale
ratios averaged along particle, p, are approximately two times smaller than that computed in the Eulerian frame r, and stay
at nearly 1.77 with a weak dependence on particle inertia.
temperature gradient. Jaberi "I simulated the temperature
statistics in homogeneous twophase turbulent flows via
DNS, In this study, both the velocity and the temperature
fields were set to be statistically stationary via forcing of the
large scale of fluid velocity and temperature fields. Their
study indicate that the stationary value of the particle
temperature intensity is a decreasing function of the particle
time constant r,, the Prandtl number Pr, the ratio of the
specific heats, and the flow Reynolds number Re: Jaberi
and Mashayek[9]extended the work of Jaberi "' into decaying
turbulent flows, both oneway and twoway coupling
between phases considered. The influences of mass loading
ratio, the ratio of specific heats and the Pr number on the
statistics of temperature field had been investigated. These
direct numerical simulations, however, were all focus on the
Eulerian statistics of scalars (temperature) and its
dependence on fluid or particle parameters.
From viewpoint of modeling work, it would be more
enlighten to study the scalar (temperature) characteristics
along inertial particle trajectory, because such statistics are
very important to the development of models in
nonisothermal gasparticle turbulence based on the PDF
method ['014]. In these models, the kinetic equation for
probability density function (pdf) of particle velocity and
temperature [', 1 or even the fluid velocity and
temperature seen by the particles [4, was introduced.
However, the same as the fluid velocity seen by particle,
temperature (scalar) seen by particle should also be
modeled, which crucially relies on the scalar information
along particle trajectory ['5]. So following the approach of
Yeung[16], who had investigated the scalar statistics along
fluid tracer, He et al[7 had partially studied this topic. In
their study, direct numerical simulations were used to
investigate the effect of particle response time on
Introduction
Passive scalar behavior is important in turbulent mixing,
combustion and pollution, and also provides impetus for the
study of turbulence itself ']. Many research had been taken
to investigate this problem in turbulent flows, which had
been well addressed in the latest and comprehensive review
by Warhaft '].
In particle laden turbulent flows, especially in
nonisothermal or reactive gasparticle flows, the problem
become more complex. Although it has been well
understand that there are 'three effects' for particle
dispersion in isotropic turbulence, namely
crossingtrajectory 2, continuum 3, and inertia[4 effects, so
is the wellknown 'preferconcentration' [5.6] ad
'extradissipation' effects, the influence of the fluid
turbulence on heat transfer and combustion of dispersed
particles, and the reverse impact of particle on the turbulent
mixing of temperatures and species fields, are not well
understand.
The complexity of particlescalar interaction may arised
from three aspects, (1) the drift of fluid scalars
characteristics seen by particle on its trajectories due to
particle's inertia; (2) the exchange of heat/mass due to heat
up, cooling, evaporation and combustion of droplets and
particles, (3) modification of scalar mixing due to
momentum coupling, i.e. twoway coupling, between fluid
and particles. Among those, the inertia effect is the basis
and maybe prerequested to understand the other two. Up to
now, only few numerical studies can be found on such topic.
Sato et al[7 showed that in decaying isotropic turbulence
with a mean temperature gradient the particle temperature
and velocity are well correlated in the direction of the mean
Re = d'
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Lagrangian statistics of scalar (temperature) field, including
scalar intensity seen by particle, the autocorrelation scalar
seen by particle, etc.
In this paper, we will investigate temporal change of scalar
and scalar dissipation along particle trajectory firstly, to get
some qualitative view of the Lagrangian frame. Then we
pay special attention to some important statistics with the
help of DNS, such as the selfcorrelation of scalar along
particle trajectory, integral time scale of scalar and the
mechanical to thermal time scale ratio etc., both of which
are important to develop the lagrangian models. The paper
is organized as follows, the mathematical description is
briefly presented in Sec.2, including the governing
equations in Sec.2.1, the numerical method in Sec.2.2, the
results are described and discussed in Sec.3, and the final
conclusion is given in Sec.4.
Numerical method
Governing equations
The governing dimensionless equations for the fluid field
are the continuity and momentum equations:
au~= (1)
24(1 + 0.15 Rep, O 8
N r = 2 + .R 00 .6Re Po
(2)
Numerical method
The method of numerical simulation for fluid field is
pseudospectral, i.e., the spatialderivative terms in the
NavierStokes, and scalar convectiondiffusion equations
are computed in spectral (Fourier) space, while the bilinear
products in the convective terms are computed in the
physical space. The aliasing errors caused by later
operations are almost completely removed by 3/2rule.
The solution is obtained on a cubical uniform mesh with N3
grid points. In the physical space, the cube is of side length
L and the grid points are located at x(li A, l~a 13 a), where
li, lo, and 13 are integers between 0 and N1, and the grid
spacing a is equal to L/N. The grid points (or nodes) in
wave number space are at k(nizko, niko n;3kn), where >ix,
ina, in3 are integers between 1N/2 and N/2, ko (the lowest
nonzero warenumber) is equal to 2x/L. The value of N is 96
in following simulations.
The timestepping scheme is an explicit secondorder
RungeKutta method. Timestepping errors are small as
long as the Courant number, defined as
C = + v+ w(10))
dp 1 8 it,
&, Re & &,
dt &,
The scalar transport equation is given by:
+
dt dt
where p is the mean scalar gradient in y direction and its is
the corresponding velocity in this direction.
The trajectories of particles are obtained by integration of
the particle equation of motion. If the ratio of fluid density
to particle density is negligible (equal to 1:1000) and the
gravity can be neglected, the particle equation of motion is
given by:
du 01 r~U )
'"C (4)
dO, _u(Bf, B,) (5)
dt 3r Pr a
where up and B, is the particle instantaneous velocity and
temperature, u ,, and 9, is the instantaneous fluid
velocity and temperature at the particle location. r, is the
particle response time, which is defined as
d' P
7 (6)
18v p
The particle drag coefficient (CD), the Renolds number
(Re,) and Nusselt number(Nu)in the modified Stokes
relations are expressed by
is not greater than 1.0, where it, v, and w are the three
fluctuating velocities, at is the time step, and ax is the grid
spacing[19]. All the simulations in this paper used the
Courant number C = 0.5. The highest Reynolds number
flow that can be accurately simulated for certain grid size
has been determined to be that for which the highest wave
number kne > 1.0, where r is the Kolmogorov length
scale, the smallest length scale of the flow ['8]. At the same
time, the criterion k,, ,93 21.0 should also be satisfied if a
scalar field is computed, where rB is the Batchelor
length scale, its definition is expressed as r/Pr ". Since the
Pr number in our simulation equals to 1.0, so rB= q. As
can be seen in Table 1, these simulations all have kmax r>2.0
ivhich means that both the velocity field and scalar field are
well resolved.
The velocity is stochastically forced to maintain the
stationarity by adding acceleration increments to the largest
scales only, such that continuity is satisfied and on average
dissipation equals to the artificial production. Only the
wave numbers inside a sphere of radius KF are forced
(excluding the origin). This forcing scheme is the same one
developed and well tested by Eswaran and Pope ['8]. In their
work, it is shown that the resulting velocity is stationary and
isotropic, to a good approximation.
After several eddv turnover times, the fluid field reaches a
stationary state, therefore the scalar fluctuation equation
start to be computed. The scalar field is initially set to zero,
no scalar forcing is necessary since the mean scalar gradient
gives rise to a nonzero production term in the scalar
variance evolution equation. When the scalar field also
reaches a stationary state, particles are released with a
random distribution in the computational domain. All the
simulations in the present study have used 105 particles. The
initial velocities and temperature of particles are assumed to
be equal to the fluid velocity and temperature at the same
location. The secondorder RungeKutta scheme is used to
integrate the particle trajectories and compute the particle
velocity and temperature. The fluid velocity and
temperature, the scalar and their gradients at the particle
point are computed via the accurate and welltested partial
Hermite interpolation scheme developed by Balachandar
and Maxey [19]. The parameters of forced turbulence by
DNS and the primary quantities characterizing the flow are
summarized in Table 1.
Tabl. Parameters and basic statistics in the simulations
Parameters Symbol value
Grid number N 96
cubic length L 27
Reynold number Re 39
Courant number Cfl ().5
Viscosity v ).()25
radnun of forcing KF 2~12
kinematical energy K 8.94
Energy dissipation a 8.89
Kohnogorov length 7 ().()61
Scalar variance 1.9()
Scalar dissipation rate E4 2.)2
Mean scalar gradient P 1.()
Mechanics to thermal tinle r2 s(f .4
scale ratio r22E 2 E .1
S t=0.0I
0 0.5 1.0 1.5 2.0 2.5 3.0
3
3
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Results and discussion
The gas phase statistics were obtained after a time interval
required for the flow development to become independent
of its initial conditions. The value of the reference length
and time scales used in normalizing the above quantities are
Ler = 0.03 m and ter = 0.9 s, which are consistent with the
value of dimensional kinematical viscosity of the fluid (air)
vam, = 1.50 x105 m /s. The validity of our simulations had
been well tested in He et al ['7
Temporal scalar and its dissipation along particle
trajectory
Prior to quantitative analyses it may be useful to have a
qualitative view of the Lagrangian time series by looking at
some typical sample traces obtained from DNS. For this
purpose it is not necessary to consider every particle tracked
in simulation. Accordingly in figure 1 and figure 2, \ve
show the scalar value and its corresponding dissipation
following sample traces with stokes number 0.0t 1.0 and
10.0(here after, the stokes number of particle, St, is defined
as the ratio of particle response time z to the kolomogrov
time scale of fluid z,). As shown in figure 1, the lagrangian
scalar fluctuation in all of the three cases is smooth and
with a few of abrupt changes. In figure 2, \ve see that the
temporal change of scalar dissipation is more quickly than
the scalar, with intense but shortlived bursts of activity at
levels much higher than the mean. These characteristics
suggest a behavior of intennittency in the Lagrangian frame
that is consistent with the notion established in Eulerian
frame as before "10. Since the time series are recorded along
the particle trajectory, these observations also imply that the
typical residence time of a particle in a region of intense
scalar dissipation is short, and this is also consistent with
the known spottiness of scalar dissipation fluctuations and
smallscale in ilpacei r lil
[ St=0.0
AO
0.0 0.5 1.0 1.5 2.0
t
2.5 3.0
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
15
St1.0
12
0.0 0.5 1.0 1.5 2.0 2.5 3.0
,,,,,,.l.,
St1.0 O
.0 0.5 1.0 1.5 2 0 2.5 3.0
t
S St 10.0
1.5 2.0 2.5 3.0
t
0.0 0.5 1.0
5
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Fig.2. Temporal scalar dissipation along particle
trajectory with st=0.0(top), st=1.0(Middle) and
st=10.0(bottom)
Figure.3(right)).
Also accordingly in figure 1 and figure 2, we can find that
the scalar value seen by the particle st= 1.0 show more
intermittency than those of particle st=0.0and10, and the
intensity and probability of high scalar dissipation along the
sample particle with stokes number st=1.0, is much larger
than those along particle with stokes number st=0.0 and
10.0. In our opinion,this phenomenon results from the
selectivity of st~1.0 particles motion, which are easy to
aggregate in high strain and low vorticity regions in fluid
field, where the scalar change is much more intensive, and
scalar dissipation is usually much higher than the mean
value.
Fig.1. Temporal scalar value along particle trajectory
with st=0.0(top), st=1.0(Middle) and st=10.0(bottom)
Morover, through the comparison of these two figures, we
can find that there's something to tell us more. Whenever
the scalar value in Figure.1 takes a remarkable change, the
scalar dissipation in Figure 2 accordingly come out a
shortlived bursts.In the integration of these two figures
when St=1.0, see Figure.3(left),this phenomenon is quite
clear, so is it when St number equals 0 and
10.Consequently,we can image a piece of picture in which
an inertial particle moves in the scalar space with
RampCliff structure. And every time the particle moves
across the slice cliff structure, with intense scalar
dissipation, it experiences one shortlived burst, meanwhile
this leads to the step leaping of the scalar value seen by the
particle (from one ramp structure to another, see
3
A
v
II II
II ,,,
II
,I II II 1,
IIII 1 ,II ,I~I IIII
II ~, ,~,, I~
,,, II
,I :~ '' ''
rl '' r\,
.0 05 10 1. 2. 2.5 3.0olzer and Siggia, Phys.Fluids,1994,6: 18201837
Fig. 3: Scalar and scalar dissipation rate along particle trajectory (st=1.0) and schematic of turbulent mixing of scalar
~I
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
particle there are slight concentrations in its distribution,
while the number of particles around the high scalar
dissipation field isn't very considerable; However ,when it
comes to the critical particle,obviously quite a
part(100/e20%) of them concentrates intensively around
high scalar dissipation field, i.e. around cliff structure, thus
their possible times of moving cross cliff structure are more
than those of other inertial particles.
Our opinion can be verified in Figure.4, which displays the
distribution of particles and scalar dissipatin rate in a plane
at a certain time.(since the number of particles in the cubic
is approximately 10 .,the number of particle in the slice
which we select in the mid of the cubic approximates
103)We can see that the characteristic of the distribution of
these 3 different sample particles: the particle of St=0 is
relatively welldistributed in the fluid field; for the St=10
St=0
St=1.0
St=10.0
Fig. 4:The distribution of particles and scalar dissipation rate in a plane at a certain time
the selfcorrelation of scalar along particle
trajectory
In figure 5, the temporal selfcorrelation coefficients of the
scalar along particle trajectory are presented. The
coefficients reflect the memory effects of scalar and the
trajectory difference of inertial particle. These coefficients
are defined as:
<< ~fp(to) 7~, p+t 7) > >
where to is the starting time for the calculation of the
Lagrangian statistics, and calculated when particle reaches a
stationary state. As the figure shows that, with the
increasing of particle inertial, the curve of selfcorrelation
drops faster and shifts to the lower left, which means more
rapid decorrelation, or a shorter memory time for the scalar
along the trajectory. Besides, when the stokes number of
particle is large, the deviation between curves quickly
diminishes, and we found only at the time interval [1.5, 4.5],
some difference is displayed between the correlation curve
with stokes number st=5.0 and st=10.0.
In figure 7, the corresponding integral time lengths in terms
of stokes number are given out. As respected, they reduce
rapidly with the increasing of particle inertial and we found
that the 1st exponential decaying function
y=1.00265*exp(st/1.88)+0.743 fits these results very well.
Fig.7:Integral Time Scale of scalar seen by particle
1.6 ..
. fluid
0 24 6 8 0
Fig. 8:Integral Time Scale of velocity seen by particle
the mechanical to thermal scale time ratio
The time scale ratio often considered in experiments and
simulations, is the mechanical dissipation to thermal (or
scalar) dissipation ratio, usually denoted by
p=(2ic/s)/(< 2>/sq)
which relates the largescale mixing time scales of the
velocity field to scalar field. Although it was generally
taken as a universal constant for modeling purposes, there is
considerable evidence to show that it does not and is flow
dependent. Sirivat and Warhaft 12 found in their grid
turbulence experiments that r tended to decrease
downstream as the Reynolds number decreased, with
approximate values of 1.2 to 2.0. Eswaran and Pope ['8]
found in their direct numerical simulations that r kept to 2.5,
with some decreasing trend as the time further advance.
Warhaft and Lumley reviewedd a number of heated grid
experiments and found r of value in between 0.6 to 2.4.
Overholt and Pope [2]also used the direct numerical
simulation to investigate the passive scalar field and found
that r is of value 2.0, though some fluctuations existed. In
this paper, we also calculated this very important time scale
ratio and found that r is of value 2.14, which is in the range
of values reported in these former literatures. However, the
results upon were gotten in eulerian frame, it only reflected
the mean mixing state of scalar field by the turbulent flow.
On the contrary, the local mixing state and the local time
scale ratio are essential to the developments of some
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
1.8
1 5
\ y = 100265*exp(st1.88) + 0 73
0 2 4 6 8 10
Fig.5:the selfcorrelation of scalar seen by particle
1.0 = = = =
c0g.8 /= 0.1
0 1 2 3jl 4 5 B
Fig. 6: the selfcorrelation of velocity seen by particle
Figure.6 show the selfcorrelation of velocity seen by
practicle.In these figure, the selfcorrelation of velocit
reaches its maximum when St=1.0, significantly different
from the selfcorrelation of scalar seen by particle in
Figure.5. In common sense, the preferconcentration effect
of critic particle should also have affected the statistic
character of the scalar seen by the particle, but it doesn't.
We contend that this ought to have something to do with
cliff structure in the scalar field. Because of cliff structure's
slicelike character, particles can vertically move across it
very quickly, thus the correlation of scalar between two
neighboring moments disappears soon. As a result, the
selfcorrelation of scalar seen by particle is mainly
determined by crossingtrajectory and inertial effects of
particle itself, while the preferconcentration effect has no
obvious effect on the selfcorrelation of scalar seen by
particles. Through the comparison between Figure.7 and
Figure.8*t Lagrangian intergral time scale of scalar seen by
particle is apparently different from that of velocity seen by
particle, which is also stemed from the difference of their
selfcorrelation character.
lagrangian model, such as EMST model, and then need to
be taken special attentions to, so we computed value of r by
averaged along all the trajectories.
Accordingly in figure 8, it is easy to find that all these
values , along different inertial particle trajectories are
less approximately 15~20% than the one computed in
eulerian frame, r2.14, and keep at nearly 1.77. It also show
a weak dependence on particle inertial except at the stokes
number equaling to 1.0, where the maximal value is reached.
Basing on these results, we recommend that the value of
1.77 can be adopted in those lagrangian mixing models,
though 5% deviation exists at stokes number equaling to 1.0
and some decreasing trend of , also exhibits at stokes
number larger than 1.0.
0.05 '
0.90
0.85 .
C 0.80 
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
number equaling to 1.0 and some decreasing trend
when stokes number larger than 1.0.
Only inertial particle are considered in this paper, the
turbulent modulation and the scalar exchange between
particle and scalar field will be included in the future
researches.
Acknowledgements
This work was partially supported by the Natural Science
Foundation of China (Grant No. 50576027, 50721005), and
Program for New Century Excellent Talents in University,
Ministry of Education, China (Grant No. NCET040708),
and the Computing resource provided by SCTS/CGCL of
Huazhong University of Science and Technology are
gratefully acknowledged.
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conclusion
In this paper, some statistics of passive scalar along inertial
particle trajectory in homogenous isotropic turbulence with
a mean scalar gradient are investigated by using of direct
numerical simulation. As simulation results show that:
(1) The temporal change of scalar dissipation is more
quickly than the scalar, with intense but shortlived
bursts of activity at levels much higher than the mean.
St~1.0 particles are easy to aggregate in high strain and
low vorticity regions in fluid field, where the scalar
dissipation is usually much higher than the mean value.
Every time they move through the cliff structures, the
scalar change is much more intensive.
(2) The selfcorrelation coefficients of scalar along particle
trajectory decrease monotonically with the increasing
of particle inertial, therefore its corresponding integral
time lengths decrease as well, this is significantly
different from the velocities observed by particle, for
which preferconcentration effect is evident.. The
relationship between integral time scale and Stokes
number can be well fitted with the 1st exponential
decaying function y=1.00265*exp(st/1 .88)+0.743 in
our range of stoke number computed.
(3) The mechanical to thermal time scale ratios along
particle trajectory, which is very important to develop
lagrangian models in nonisothermal gasparticle flows,
are calculated and found that they are less 15~20% than
the one computed in the eulerian frame and keep at the
value of 1.77, although with a little deviation at stokes
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
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