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FEDSM98-5036
Proceedings of FEDSM 98
1998 ASME Fluids Engineering Division Summer Meeting
June 21-25, 1998, Washington, DC
FEDSM98-5036
1 Copyright
©
1998 by ASME
NUMERICAL SIMULATION OF TURBULENT PARTICLE DISPERSION USING A
MONTE CARLO APPROACH
Vibhor Mehrotra, Geoffrey D. Silcox, Philip J. Smith
Department of Chemical and Fuels Engineering, University of Utah
3290, Merill Engineering Building, Salt Lake City, Utah-84112, USA
Telephone: 801-585-1237: Fax: 801-585-1456: email: vibhor@karma.che.utah.edu
ABSTRACT
In the present paper, a Lagrangian three-eddy interaction
model is developed which accounts for anisotropy due to the
differences in length and time scales in the different coordinate
direction. The new model as well as the single-eddy interaction
model is numerically simulated and the results are compared
with the experimental data of Snyder & Lumley (1971). It is
shown that the two models give similar predictions for trans-
verse particle dispersion in decaying turbulence. It is also
shown that the particle disperse faster in the gravity direction
compared to the other two directions using the three-eddy
interaction model. This is called the continuity effect which
arises due to the differences in length scales in the lateral and
the longitudinal directions. The three-eddy interaction model is
also used to numerically simulate dispersion of uid and non-
uid particles in a uniform homogeneous shear ow. The pre-
dictions of the new model are compared with the experimental
data of Huang & Stock (1997). Fair agreement is observed for
both uid and non-uid particles.
INTRODUCTION
Lagrangian particle tracking approaches have been used exten-
sively to model gas-solid systems where the concentration of
particles is small. In such systems, particle-particle interactions
can be neglected and the uid and solid phases can be treated
separately. Single-phase uid ow solutions are often obtained
in terms of averaged quantities such as mean velocity and mean
pressure. The uid phase turbulence is often modeled using
one of the standard turbulence models. A particle equation of
motion that includes drag and body forces is solved to obtain
the instantaneous velocity and position of the particles. This in
turn requires the specication of the instantaneous uid velocity
along a particle path. Lagrangian models differ in the manner in
which they compute the instantaneous velocity of the uid phase
surrounding the particle. A brief discussion of different
approaches is given by Graham (1996a,1996b) and Graham &
James (1996). Fluid and non-uid particle motion in turbulent
ows are random in nature and a large number of individual tra-
jectory computations are required to obtain a statistically stable
average over all particle trajectories.
A simple approach to obtain the instantaneous velocity of the
uid phase, was rst proposed by Hutchinson et. al. (1971) and
subsequently developed by Gosman & Ionnides (1981) called the
eddy interaction model. A complete description of the eddy inter-
action model in homogeneous isotropic stationary turbulence is
given by Graham(1996b). In this paper the eddy interaction
model of Graham will be referred to as a single-eddy interaction
model(SEIM). In SEIM the instantaneous velocity of the uid
phase is obtained by adding to the mean uid velocity, a random
uctuating velocity which is sampled from an assumed probabil-
ity density function(pdf). The various moments of the assumed
pdf are computed from a standard turbulence model. The particle
moves through a succession of instantaneous uid velocities. The
interaction time of the particle with each instantaneous velocity
can be prescribed from another assumed pdf of interaction times.
In general the instantaneous particle velocity is not the same as
the instantaneous uid velocity.
Experimental investigations (Csanady, 1963), have demonstrated
that for nite inertia particles the following three effects are
important and have to be accounted for to predict particle disper-
sion accurately. 2 Copyright
©
1998 by ASME
(i) The inertia effect (Wells & Stock, 1983; Reeks, 1977),
predicts that the dispersion of solid particles might exceed
the dispersion of the uid particles in the absence of body
forces.
(ii)
The crossing trajectories effect (Yudine, 1959), pre-
dicts that in the presence of a drift velocity a nite inertia
particle will disperse less than a uid particle.
(iii) The continuity effect (Csanady, 1963), where the dis-
persion in the direction of the drift velocity exceeds the dis-
persion in the other two directions.
Graham(1996a) has modied the single-eddy interaction
model, to account for the three effects described above. The
improved single-eddy interaction model of Graham can
account for the inertia and the crossing trajectories, but it is not
clear how the continuity effect can be accounted for by assum-
ing a spherical eddy. From here onwards the improved eddy
interaction model will be referred to as a single-eddy interac-
tion model.
The aim of the present paper is to develop a three-eddy interac-
tion model(TEIM) which can account for all of the above
effects and is able to predict particle dispersion in a homoge-
neous turbulent shear ow. In order to aid the development of
the model, Stokes law will be assumed to be applicable in the
particle equation of motion. All forces on the particle accept
drag and gravity will be neglected (Maxey & Riley, 1983).
Throughout the development of the model a homogeneous iso-
tropic turbulence eld will be assumed. In Section , the single-
eddy interaction model will be described. In the next section
the three-eddy interaction model will be outlined in detail. The
subsequent section will compare the predictions from both the
single-eddy and the three-eddy interaction model with the
experimental data of Snyder & Lumley (1971) and Huang &
Stock(1997). Finally in some conclusions will be drawn and a
summary will be presented.
THE SINGLE EDDY INTERACTION MODEL
In this section a brief description of the single-eddy interaction
model which accounts for the three main effects pertaining to
particles will be discussed. For a complete description of this
model the reader is referred to Graham (1996a,1996b) and
Graham & James (1996). At the start of the particle/uid inter-
action(
), the particle will be assumed to be sitting at the
center of the eddy. Within an eddy the instantaneous uid
velocity is assumed to be a constant and is given by
(1)
t
0
=
u
e
U
u'
+
=
The instantaneous eddy velocity comprises of a mean velocity
and a uctuating velocity
. In a homogeneous turbulence eld
we can assume the mean velocity to be zero. The uctuating
velocity is sampled from an assumed Gaussian pdf with zero
mean and the variance equal to the mean square velocity.
Throughout the duration of an eddy the instantaneous velocity
remains a constant in space and time as long as the non-uid par-
ticle remains in that eddy. At some later time (
), both the
eddy and the non-uid particle must have moved in space. The
eddy gets convected with its instantaneous uid velocity while
the non-uid particle movement is governed by the particle equa-
tion of motion. The non-uid particle remains under the inuence
of that eddy until the interaction time exceeds the eddy lifetime
(
), or the separation of the non-uid particle and the center of
the eddy exceeds the eddy length (
). The choice of the pdf for
sampling the length scales and time scales is described in Gra-
ham & James(1996) and Wang & Stock(1992).
Particle Motion Effects
Assuming that the uid/particle density ratio is low and the only
forces acting on the particle are drag and gravity, the equation of
motion of an individual particle can be written as
(2)
where
and
denote uid and particle velocity vectors
respectively.
is the gravity vector and
denotes the particle
relaxation time. For a constant uid velocity the equation of
motion can be integrated analytically to give the particle position
at time
, given that the solid particle is coincident with the
uid particle at the center of the eddy at time . Integrating Eq.
(2) we get
(3)
Integrating Eq. (3) we obtain
(4)
U
u'
t
0 T
e
L
e
du
p
dt
---------
1 r
---- u
f
u
p (
)
g
+
=
u
f
u
p
g r
t
t +
t
u
p
t
t