Mathl. Comput. ModelZing Vol. 21, No. 9, pp. 31-37, 1995
Copyright@1995
Elsevier Science Ltd
Printed in Great Britain. All rights reserved
0895-7177/95 $9.50 + 0.00
Pergamon
08957177(95)00049-6
A Lagrangian Stochastic Model
for Nonpassive Particle Diffusion
in Turbulent Flows
Y.-P.
SHAO
Centre for Environmental Mechanics
CSIRO, Canberra, Australia
Abstract-In
this paper, we present a Lagrangian stochastic model for heavy particle dispersion in
turbulence. The model includes the equation of motion for a heavy particle and a stochastic approach
to predicting the velocity of fluid elements along the heavy particle trajectory. The trajectory crossing
effect of heavy particles is described by using an Ito type stochastic differential equation combined with
a fractional Langevin equation. The comparison of the predicted dispersion of four heavy particles
with the observations shows that the model is potentially useful but requires further development.
Keywords-Lagrangian
stochastic modelling, Turbulent dispersion, Heavy particles, Fractional
Brownian motion, Wind erosion.
1.
INTRODUCTION
There
is a variety of nonpassive particles with a density differing substantially
from the surrounding fluid, such as dust, spray droplets, smoke plumes in the atmosphere,
and plankton in water
bodies. While the diffusion of passive scalars is determined
by the flow properties
alone, that
of nonpassive
particles (hereafter simply particles) depends both on the fluid motion and the
fluid-particle
interaction.
Particle diffusion in this case is thus a much more complicated
process.
One approach to the problem is to solve the particle diffusion equation under given initial and
boundary
conditions,
which in a two-dimensional
(2, z) system can be simplified to
where c is the mean particle concentration,
i-i is the mean flow speed and wt the particle fall
velocity. The major difficulty of this approach is that the particle eddy diffusivity, Ic,, is not well
understood
and is difficult to determine.
A second possible approach is to consider the particle
trajectories
and to determine
the ensemble statistics of the particle motion using Lagrangian
stochastic
models [1,2]. The fundamental
difficulty encountered
by the Lagrangian
approach
is the requirement
of the simulation
of turbulent
fluctuations
along the trajectories
of heavy
particles. In the existing models, there are aspects which are inconsistent
with either the theory
of stochastic
processes or the theory of turbulence,
as discussed in [3]. This paper presents
The key new ingredient
of the model is a
an attempt
to remedy some of the inconsistencies.
geometric representation
of Eulerian turbulence
using fractional Brownian motions.
We give a
brief description
of the model in Section 2 and discuss the numerical simulations
in Section 3.
In decaying homogeneous
turbulence,
the simulation
of heavy particle dispersion
is shown to
be comparable
with wind tunnel observations.
An example of the model’s application
to wind
erosion studies is also given in Section 3.
31
32
Y.-l=‘.
2. DESCRIPTION
SHAO
OF THE MODEL
It has been long recognised [4] that heavy particles and fluid elements follow different trajectories as a consequence
of their different inertial response times (the inertial effect) and their
different responses to gravity, which forces a heavy particle to continuously
change its fluid element neighborhood
(the trajectory crossing effect). For a two-dimensional
for the velocity and position of a heavy particle can be explicitly written
problem,
as
the equations
dt - S,,sg dt
dv, = -:
(2)
dyi = vi dt,
where vi is the particle
is the fluid velocity
velocity,
v,+ = vi - 2~: is the particle
at the location
relative
velocity
to fluid (u: = ui(y)
of the particle y), 7 is the particle response time, t is time,
Although equation (2) appears simple, it is an extremely
and g is gravitational
acceleration.
complicated
system because of the nonlinearity
embedded
in r which depends
on the particle
Reynolds number, the particle relative velocity, the particle diameter and the particle to fluid
density ratio. Equation (2) implies that it is necessary to predict u; so that the particle trajectory
can be determined.
In the following, we combine the Lagrangian
stochastic model for the fluid
element [5] with the theory of fractional Brownian motion to simulate the velocity fluctuations
along the particle trajectories.
The velocity of a passive scalar
equation
(or fluid element)
can be described
by a stochastic
differential
[5]
dui = ai dt -I-JZ;;;;dSZiy
(3)
where ai is the drift coefficient determined
by the physical constraints
of turbulent
diffusion,
Cc is the Kolmogorov constant,
E is the dissipation
rate of turbulent
kinetic energy, and dS2, is
the increment
of a Wiener process. The determination
of ai involves solving the Fokker-Planck
equation
dP
dt=---where P = P(u,x,
t) is a specified
d&P
da,P
1 d2C#P
dxi
dui
+ Zdu,dzlj’
probability
density
function
F,W)
d-t--
1
I
I
(4)
of Eulerian
--
velocity
u at (x, t).
*
Fluid
trajectory
vdt
Particle
trajectory
Figure 1. An illustration
turbulent PLOW.
of the trajectory-crossing
effect for a heavy particle in a
A direct application
of the above model predicts the velocity of a marked fluid element, rather
than ut. However, we expect that the uf can be obtained from a modification
of the scheme.
Figure 1 illustrates
the trajectory-crossing
effect in a small time interval.
A heavy particle and
a fluid element start at the same location 0 at time t = to. After a small time dt, the particle
arrives at P while the fluid element arrives at F. It follows that the velocity increments
du; can
be expressed in two terms
du; = dui + 6ui,
(5)
A Lagrangian Stochastic
where du, is the Lagrangian
ment
(in space).
velocity
The velocity
increment
increments
Model
(in time)
33
and 6ui is the Eulerian
dui and 6~ have fundamentally
Following Kolmogorov’s theory of local isotropy, the Lagrangian velocity
is proportional
to the time increment dt, if dt is sufficiently small,
structure
DiJ = (dU,(t) dUj(t)) = C’ocdt.
The Eulerian
structure
velocity
different
incre-
behaviour.
function,
Dij,
(6)
E,j, is defined by
function,
Ezj = (hi (to + dt) SUj (to + dt)) = ([u:(t) - u%(t)] [UT(t) - Uj(t)]) .
For sufficiently heavy particles, 21,.is approximately
the particle fall velocity wt and the particle
and fluid is separated by a vertical distance wt dt. Therefore, the Eulerian structure function Eij
obeys
Eij N Si,Ci (curt dt)2’3.
The situation
is more complicated
variable and thus E,j cannot be
is v, and the distance separating
ensemble of r can be divided into
for the kth sub-ensemble
is
&k(r)
The structure
respectively,
functions
(7)
for lighter particles, since TJ, must be considered as a random
prescribed.
In general, the particle relative velocity to fluid
the particle and the fluid elements after dt is r = v, dt. If the
K sub-ensembles,
then the Eulerian structure
function valid
1
= --Cl
3
(crle) 21s 99
for the horizontal
11k =
E ,
+ sij f Ci (cr#/3.
TE
and vertical
velocity
Cl (w-,kdt)2’3
4
3
(
_
components
&,le
-)V,“,k
(8)
El1.k and Ess,]~ are,
(9)
and
33 k _
E,
-
cl (%,k
W213
3
for r being in the kth sub-ensemble
If the probability
K
~3%= -&'k
cl (E%-,k
3 W 2’3
(&!$_)
(10)
is pk, then
=~~2,3(~~,3~-~))dt2,3.
(11)
k=l
Thus, in general, Ei3 remains being proportional
dt213.
The small-time Eulerian behaviour cannot be adequately described using an Ito type stochastic
differential equation such as (3), because the Gaussian white noise inevitably results in a structure
function being linearly proportional
to the time increment.
This difficulty leads to two different
approaches in Lagrangian particle diffusion models. The first approach (e.g., [2]) assumes that the
Lagrangian
and Eulerian auto-correlation
functions of velocity have the same exponential
shape
and the Lagrangian
and Euranian integral time scales (TL and TE, respectively)
are proportional.
This approach implies that the Eulerian structure function is proportional
to the time increment,
and is thus inconsistent
with the theory of local isotropy. The second approach uses a stochastic
differential
equation system similar to (3), but allows the drift and diffusion coefficients to be
dependent
on dt. This approach is inconsistent
with the theory of stochastic processes (see (31
for details).
Y.-P.
34
SHAO
In a recent attempt to remedy these inconsistencies, Shao (61 proposed a fractional Langevin
equation for a stochastic representation of Eulerian turbulence,
6u, = -kbu, dt + fidi-&,
(12)
where df&, is the increment of a fractional Brownian motion [7], which obeys
(di&(t)dRb(t + 7)) = fr dt2H [(~+~)‘~+~-$~‘“-2(-3”“],
(13)
where H is the Hurst parameter between 0 and 1, and kb and Db are the model parameters to be
constrained by physical considerations. Equation (12) determines a stationary Gaussian process
to be called the fractional Ornstein-Uhlenbeck process. Equations (12) and (13) imply that
(&&u,)
= Db dt213.
(14)
The parameters H and Db are chosen such that (12) reproduces the small time behaviour specified
by (11). As can be seen from equations (11) and (14), an adequate choice of H is l/3.
Since
Eij cannot be specified a priori, Db cannot be treated as a pre-specified function or constant
as in the Lagrangian stochastic diffusion models for passive scalars (e.g., [5]). Instead, Db is
determined for each time step by
(15)
with u, being predicted by the model for each time step. Equation (15) is a sufficient constraint on Dg, so that (11) is always satisfied. The coefficient kb is chosen so that the fractional
O-U process has the desired statistical moments. It is shown in [6] for homogeneous turbulence
that kb is given by kb = 0.8516(Db/2a2)3/2, with c2 being the variance of velocity.
3. HEAVY PARTICLE MOTION IN DECAYING
TURBULENCE
AND IN THE ATMOSPHERIC
SURFACE LAYER
The model has been tested against heavy particle dispersion data in decaying grid turbulence [7]. The positions and velocities of four kinds of heavy particles (Table 1) were measured
in a vertical wind tunnel. The model is used to simulate the dispersion of these four kinds of
particles, using the measured turbulence characteristics reported in [7]. In the numerical experiments, a number of combinations of CO and Ci are tested. The predicted mean-square dispersions
with Ce = 10, Ci = 3 and dt = 0.0005s are compared with the observations in Figure 2. The
predicted mean-square dispersions for all four particles agree well with the observations. Similar
comparisons were made in some previous studies (e.g., [2]), but it was found that the model predictions are in disagreement with the measurements for either one or the other particle. In this
work, it is possible to adjust both Cc and Ci so that a good agreement between the simulation
and observations can be obtained for all particles. However, the values of Cs and Ci used in this
study appear to be larger than expected (Cc = 6 and Ci = 2 are widely used in the literature).
This possibly unrealistic choice of Ce and Ci is probably associated with a small over prediction
of velocity variance along the heavy particle trajectories. Further investigations are necessary to
validate the model.
Table 1. Diameter and density of the particles used in the experiment
numerical simulation.
of [8] and
A Lagrangjan Stochastic Model
35
6
-
CO, simulation
0
0.1
0.0
0.3
0.2
0.5
0.4
Time (s)
F’lgure 2. Comparison of simulation with observations of [7] for mean square particle
dispersion in decaying homogeneous turbulence. Points are observed data and curves
are model results.
I
,
-
0
0.0
/
,
0.3
0.4
CO, dt=O.OOO§s
0.1
0.2
0.5
Time (s)
Figure 3. Comparison of simulations of particle dispersion with dt = 0.0005s and
dt = 0.001s. As in Figure 2, the simulations are for the decaying homogeneous
turbulence as reported m [7]. In these tests, CO = 11 and Cl = 2.8 are used.
t
is also important
to test
whether
the choice of dt would influence
the numerical
results.
The
merical simulations
with dt = 0.0005s and 0.001s for the dispersions of the four particies are
npared in Figure 3. This comparison shows that the model results do not significantly
depend
dt.
We now apply the model to study the hopping
motion of sand grains (saltation)
in atmospheric
rface layers. The trajectory
of a saltating particle can be adequately described by (2). For a
‘en mean wind profile, the behaviour of saltating particles without turbulence is deterministic.
this case, particles with the same initial condition follow the same trajectory
191, while under
: influence of turbulence the trajectories of saltating particles are stochastic [3]. The variability
particle trajectories
is studied in a neutral surface layer, where the the mean wind profile is
carithmic and the turbulent velocity is Gaussian. The streamwise and vertical velocity standard
viations 01 and ~3, and the dissipation rate 6, have the following simple relationships
with the
36
Y.-P.
%-IA0
surface friction velocity u*:
(TV= 2.4u,,
cl3
=
1.2u*,
22
e=_T_
fC.z’
where K. is the von Karman constant and z is height.
To illustrate the stochastic variability induced by turbulence, Figure 4 shows a set of particle
trajectories with identical initial conditions. The initial vertical velocity of all particles is set
to 0.63~~ and the lift-off angle is set to 50 degrees (based on wind tunnel observations (e.g., [lo]).
The corresponding trajectory without turbulence is indicated by the thick line. In the presence
of turbulence, particles with identical initial conditions no longer follow a single trajectory
band of trajectories which must be described statistically.
but a
2o
:
100
50
0
Figure 4. Trajectories
of sand grains in turbulence.
cated by the thick line.
The nonturbulent
case 1s indi-
As pointed out in [ll], saltation of sand grains is a key process in wind erosion. The purpose
of Figure 4 is merely to provide a qualitative example of the model’s application, but it has
been shown quantitatively in [3] that the stochastic nature of particle trajectories in turbulent
conditions has a significant influence on bulk properties of the saltation cloud, in particular the
profiles with height of the mean particle concentration and streamwise flux density, and the
vertically-integrated
streamwise particle flux. These properties may differ substantially from
those obtained using nonturbulent saltation models in which particle trajectories depend only on
the mean wind profile and are therefore deterministic for given initial conditions.
4. CONCLUDING
REMARKS
In this study, we presented a Lagrangian stochastic model for the dispersion of heavy particles.
The model includes the equation of motion for a heavy particle and a stochastic equation system
for predicting the velocity of fluid elements along the heavy particle trajectory. The trajectory
crossing effect of heavy particles is described by using an Ito type stochastic differential equation
combined with a fractional Langevin equation. The introduction of the fractional Gaussian noise
is to overcome the inconsistencies in previous Lagrangian stochastic models. The comparison
of the predicted dispersion of four heavy particles with the observations showed the potential
of the model. The numerical experiments also showed that the predictions are not significantly
influenced by the time increment of the numerical integration. However, the model over predicted
the variance of velocity along the heavy particle trajectory and thus further improvements are
necessary. As a qualitative example of the model’s application, we calculated the trajectories of
saltating sand grains in turbulent wind. The stochastic nature of particle trajectories in turbulent
conditions may have a significant influence on wind erosion processes.
A Lagrangian Stochastic
Model
37
REFERENCES
1. J.C.R. Hunt and P. Nalpanis, Saltating and suspended particles over flat and sloping surfaces, In Proceedzngs
of the International Workshop on the Physzcs of Blown Sand, Memoirs, (Edited by O.E. Barndorff-Nielsen),
Vol. 8, pp. 36-66, Dept. Theoretical Statistics, Aarhus University, Denmark, (1985).
2. B.L. Sawford and F.M. Guest, Lagrangian statistical simulation of the turbulent motion of heavy particles,
Boundary-Layer Meteorol. 54, 147-166 (1991)
3. Y. Shso and M.R. Raupach, A Lagrangian stochastic model for the dispersion of heavy particles with
application to saltation in a turbulent flow, Boundary-Layer h1eteorol. (submitted) (1995).
4. M.I. Yudine, Physical considerations on heavy-particle diffusion, Adv. Geophys. 6, 185-191 (1959).
5. D.J. Thomson, Criteria for the selection of stochastic models of particle trajectories in turbulent flows,
.I. Fluid Mech. 180, 529-556 (1987).
6 Y. Shao, The fractional Ornstein-Uhlenbeck process as a representation of homogeneous Eulerian turbulent
velocity fields, Physzca D (submitted) (1994).
7. W.H. Snyder and J.L. Lumley, Some measurements of particle velocity auto-correlation
functions in a
turbulent flow, J. Fluid Mech. 48, 41-71 (1971)
8. B.B. Mandelbrot and J.W. Van Nees, Fractional Brownian motions, fractional noises and applications, SIAM
Rev. 10, 422-437 (1968).
9. P.R. Owen, Saltation of uniform grains in air, J. Fluzd Mech. 20, 225-242 (1964).
10. B.R. White and J.C. Schulz, Magnus effect in saltation, J. F&d Mech. 81, 497-512 (1977).
11. Y Shao, M.R. Raupach and P.A. Findlater, Effect of saltation bombardment on the entrainment of dust by
wind, J. Geophy. Res. 98, 12719912726 (1993).
MCA 21:9-D