NASA Technical Reports Server (NTRS) 20000062307: A Priori Subgrid Scale Modeling for a Droplet Laden Temporal Mixing Layer

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by  NASA Technical Reports Server (NTRS)

A Priori Subgrid Scale Modeling for a Droplet Laden Temporal Mixing Layer
Nora Okong'o and Josette Bellan
JetPropulsionLaboratory
CaliforniaInstituteofTechnology
Pasadena CA 91109-8099
Abstract filtered variables are substituted for the unfiltered vari-
ables; however, this assumption may be substantially in-
Subgrid analysisof a transitionaltemporal mixing layer accurate for droplets with small Stokes numbers. With an
with evaporatingdropletshas been performed usinga di- increasing body of DNS computations [5][6][7][8] [13][14], it
rectnumerical simulation(DNS) database. The DNS is is now possible to assess subgrid scale quantities at mod-
fora Reynolds number (based on initialvorticitythick- erate Reynolds numbers, with good prospects for devising
ness)of 600,with dropletmass loadingof0.2. The gas subgrid scale models.
phase iscomputed usinga Eulerianformulation,with L_- Recently, Miller and Bellan [9][10] have performed
grangian droplettracking.Since Large Eddy Simulation DNS of droplet laden mixing layers. They use 'DNS' to
(LES) of thisflowrequiresthe computation of unfiltered refer to computations in which all length scales of the
gas-phasevariablesat dropletlocationsfrom filteredga_- gas-phase are resolved but the effect of the gas on each
phase variablesatthe gridpoints,itisproposed tomodel droplet is modeled using Stokes drag, and the effect of
theseby assuming the gas-phasevariablestobe givenby the droplets on the gas are modeled as source terms in
thefilterevdariablesplusa correctionbasedon thefiltered the gas-phase equations. The present paper addresses the
standard deviation,which can be computed from thesub- use of the DNS database of Miller and Bellan [9] to eval-
gridscale(SGS) standard deviation.This model predicts uate subgridscaleclosures.Specificallyw,e examine the
the unfilteredvariablesat dropletlocationsbetter than largestReynolds number (based on initialvorticitythick-
simply interpolatingthefilteredvariables.Three methods ness,6_,0,and initialvelocitydifference,AU0) of 600,
areinvestigatedformodeling the SGS standard deviation: with mass loadingof0.2(3x10_drops)on a300x 332x180
Smagorinsky, gradient and scale-similarityW.hen prop- (0.25mx0.22mx0.12m) grid,which they denote as case
erlycalibrated,the gradientand scale-similaritmyethods TP600.
giveresultsinexcellentagreement with theDNS.
Governing Equations
Introduction
The governing equations for the gas phase density (p), ve-
Droplet-laden turbulent flows occur in many problems of locity (ui), total energy (E) and vapor mass fraction (Yv)
practical interest, such as spray combustion and atomiza- are given by:
tion. The interaction of particles and turbulence is an inte-
gral feature of such flows, and hence the topic of much re- Op+ _0 [_I=s_ (1)
search [2][3][5][12]. Large Eddy Simulation (LES), in which
the flow field is spatially filtered, is emerging as a powerful
Opus+ 0
tool in modelling unsteady turbulent flows. It is expected -&- _ L_ + P_,J- _,J]= s,., (2)
to be more generally applicable than Reynolds-Averaged
Navier Stokes (RANS), since the large scale structures
apE O[ _OT_ ]
axe computed, and the more universal small scale struc- + _x: (pE + P) u: - cgxj u_i.I = SHz (3)
tures are modelled. LES is also less computationally in-
tensive than direct numerical simulation (DNS) in which OpYv + _O [,Pw": - pr_a-Yjv ] =s, (4)
Ml length scales are resolved, and has the additional ad-
vantage of being able to accommodate considerably larger
Reynolds numbers (Re). Whereas much research has been aU = I_ + dxi 3
devoted to LES modeling for single phase incompressible
P = pRT (6)
flows, only moderate attention has been given to compress-
ible shear flows [ll[t5], with focus now turning two-phase E = ½u,u, + C,_T + h°Yv (7)
flows [4][tt][16]. In addition to modeling subgrid scale
R = YvRv + (1 - Yv) Rc (8)
(SGS) terms h)r the gas phase, an LES of a droplet-laden
flow would require modeling the unfiltered gas phase vari- c. = YvC_.v + (t - Yv)co,c (9)
ed)los at the droplet locations. In simplistic models, the
wlwr.._uhscrtpt V denotes the vapor, subscript C denotes o, = T,,-"-]- Ta, (22)
t.llt;carrier g.'u_,;utd the nla._s fraction of the carrier gas is
% =Yv%-Yv_j (23)
Y_: = t - Yr.
The Lagrangian particle equations for the position
(24)
(X,), velocity (v,) temperature (Td) and mass (rod) are: aU = _ \dx.s + dx, 3_ o,,)
dX, _,=f'vc,,+,(v_-f'v)c,,,c (25)
-- = v_ (10)
dt
(26)
dv_ Fi
(11)
dt ma + - eo
(27)
a,,t_r.
dT_= Q+ _,-. (12) and it has been assumed that
dt maCL
(28)
Computation of the drag force F_, the heat flux Q and the
evaporation rate _tt require knowledge of the gas phase _(_ -_,_) = -_,,_ (29)
variables (ui,T, Yv, P) at the droplet locations, and in-
(30)
volve validated relations [10]. l/,iO'i_ = uio'o
The source terms are
pYvT = pYv T (31)
St = -Z [--_t] (13) (32)
SH.i = - Z [Fi + d-_tvi ] (14) P = _RT (33)
= ½_ +,9,,_+h_, +½,,, (34)
SHz = - _ [Fir, + Q + d_t (_v,v, + C,.vTa + h°v)]
(15) The terms that need to be modeled are the subgrid stress
where _ indicates appropriately weighted summations vO, the subgrid heat flux Oj and the subgrid species flux
over droplets within a discretization volume associated
with each grid point. The droplet equations remain as before, except that
now the gas-phase variables (u_, T, Yv, P) at the droplet lo-
Filtered Governing Equations cations are no longer immediately available, and will need
to be derived from the filtered variables (_,T, Yv, P).
The filtering operation is defined as: Thus the (unfiltered) gas-phaee variables also need to be
modeled.
"$(_) =/v ¢(T')G(-_ - _)dT' (16)
Model for Instantaneous Variables
where G is the filter function, with the property that i = 1,
The droplet model requires modeling gas phase variables
and V is the filtering volume. We use a cubic top-hat
at droplet locations. To guide the modeling, we will first
filter, in which V is a cube of sides A, and G is simply
consider the known DNS generic variable ¢ and its filtered
a volume-average. For compmmsible flow, we use Favre
form _, where the bar denotes Favre filtering for u_ and
filtering, defined as _ --- _/_, to simplify the notation.
Yv and regular filtering for T and P. The definition of the
After filtering, the gu phase equations become:
standard deviation is
0_+0 = (35)
0"-m_+ 0 Thus the relation between $ and _ is
-k- _ [_,_j +_6,, - _,, +_,,] --_.,, (18)
¢ = _ + f_r (36)
%-+_ _E+ _- O_ where from the definition of a, f = ±1. The goal of the
modeling is to compute, from the filtered flowtield, the
e, +O, : (19)
form of fer that provides a better approximation to ¢ than
does fa = O. In this formulation, fa can be viewed as a
correctionto_with signf and magnitude or.
T + _ _Pv_,- or_7__+N, : _ (20) Itistempting to assume that f randomly takeson
valuesof-I and i. However, ifthe filterinogperation is
where viewed as a volume average,a relationbetween _band
r O = u,u'--_- _u_ (21) can be derivedasfollows.Consider thethird-orderTaylor
,xl)ansion of ,/_ill the filtering volume V of size A with i.e. f = -sign(V2_), ct = _'. To assess this model, we will
(:entr¢)id at .to -- (x I,,. xz,, x3.) integrated over the volume: use the "exact" _Yobtained by filtering the DNS flowfield,
and then turn our attention to modeling it based on the
filtered flowfield. The proposed model will be computed
_(.r.=)P _(_)dV (37)
at the grid points, and then interpolated to the droplet
locations.
_(_')= ¢(_0)+ _-_(z0)(z, Fig_ure 1shows the probability density function(PDF)
for (_b- _b)/'_ for case TP600 at the end of the simulation.
These results were obtained using a cubic top-hat (box)
+__ ,2, -x_)(z_-_o)+O(a 3) (38) filter with filter width A = 4Ax. Notably, for all quantities
there are peaks near 1, and for the velocity components
From the definition of the centroid T_fv(x_- Xio) dV = o. there are also peaks near -1, similar to the PDF of (_b-
If v is symmetric, then -¢)la, which takes on values of =t:1.
V (x, - X_o)(X-j zjo)_ = o; i# j (39)
0.1
and
O£9
(4o)
O_
_07
is the (positive) moment of inertia, so
O£6
2 IoA2
(41)
0.04
where terms of O(A 3)vanish due to the symmetry of the
0.a3 Ii
filtering volume.
Thus f will generally be -s/gn (V2¢). From the
available filtered quantities, we can compute V2_ rather
than V2¢; so to model f we assume that V2_ and V2_b
have the same sign. To model a, we note that for the gas
phase we will be modeling terms of the form ¢¢ - _b¢),
which are the subgrid scale fluctuations. Defining crscs as
the SGS standard deviation,
0.12
/.__._ 0.11
ascs = _/_ - ¢4) • (42)
0.1
The relationship_between a and ascs can be illuminated 0.0_
by considering or2: 0.08
(107
_= (¢__2 =_- 2¢_+_ (43)
i!
m (I06 iI
We note that ¢-¢ = cr_(:s +_ and that ¢¢ can be written
(104 ,_ !l]
in terms of the local correlation between 0band
0.03
(I02
(44) :2¢
(I01
R(_,_) = v_v/_
.,,, .... ,...... .__-,_f..., .... "_',_._r-_._
4 -,3 -2 -1(_. _.0_ _1 2 3 4 o
If we assume that R(_, _) -_ l, then
Figure 1: PDF of (¢ - ¢)/_
(45)
Defining _ = _'_2 and using _ as a model for a, we Figures 2 and 3 show the comparison between inter-
arrive at a model for ¢pof the form polating the DNS flowfield to the droplet locations (the
"exact" quantities) and interpolating the models to the
(46) droplet locations. Results are presented in terms of av-
,'r;u4,'(s)v,r
dr_q_k'tswithin a given y-interval; this aver-
aging is ,htsa)tt_'d by <<>>. First, using ¢r as given by :mr
Equation 35, we comp,tre the models for f. It is seen
I
dlat f = 0 leads to significant discrepancy between the
exact _md mode[ interpolated variable. Any model with
f with mean 0 will not perform any better as the devia-
tions toward the exact value will be just as likely as the
deviations away from the exact values. This is seen in the
case where f is randomly taken to be -1 or +1, which per- I I /, /'q kL, ,, t,
forms slightly worse than ]"= 0 despite having the exact QQI ..
or. Using f = -sign(V2¢) gives significant improvement
in view of the two assumptions made i.e. f = -sign(Va4_) i aolsF I;I _l_',' '11
and sign(V2¢) -- sign(V2_). For the data shown in Fig-
ures 2 and 3, this is true (86%, 86%, 86%, 85%, 88% and
93%) of the time for ul, u2, us, T, Yv, and P respectively.
When cr is replaced by Y (denoted [a] in the figures), with
f = -sign(V2¢), there is considerable improvement over
Y
f -- 0, for all quantities except T, where all the models give
similar results. For T, Yv and P, S = -xign(V_) can be
f,@
replaced by f = i ifthe signed _ = V_ - _/_b_bisused a-r ........f_.md.ml[.-1,1}
instead of the positive _ = V/_- of Equation __:..__.: f.O,iualol i_
45. An alternative expression for f uses ascale-similarity
.f = s,gn(, - _) = _(;_ - _). Th_ gives _¢_ .......
idea, i.e.
similar results to using f = -si-gsing(n__72¢_)72¢) ffoorr aallll s,iixx vwairi-i- a_
ables. An analysis of the correl.alatitoioni RR((__bb,,¢)¢) ooff E_quatio_nti, "_ _
44 showed that R = 1 for T and P, 0.97 < R < I for ul,u2
and u3 and 0.7 < R < 1 for Yrr'.v. FI)oruri,uu2l,ua aanndd uu_3, tth_ee
iP,(]97<R<it;zi,,",2 l=r,=Ii_ _
greatest deviation from 1 was imn tthhee mmiiddddllee ooff tthh,e mmixxi_ngg _'I _._"_ _I[
layer. For Yv, the greatest devviaiattiiocn z wwaass aatt tthhee ditr(oppleht-_- .-_ _ "' '
laden/droplet-flee interface. However, even in this region,
owev_r,even inthi.,r_liei, i am
R = 1 led to the best prediction of Yv at droplet locations £
compared to smaller values. Most importantly, R = 1
is the only value that provides the correct Y _- 0 in the O_ _, I .... i .... i .... I ,
-_1_'_" -_(31_ O 0.05 0.I
laminar freestream.Finally,we notethattheerrorfor the Y
approximation ofEquation 46 isdetermined by thatofthe
velocitycomponents, which havethelargesterrorofabout
z.5%.
Models for Subgrid Cross-terms
(ie3 .......
For LES in the gas-phase, models are required for the
subgrid stresses _'ij = _ -u_u-#,A_heatfluxes 9j =
_uj - T_j and species fluxes r/# = Yvuj - _vvu_. For the ,r, ^ft
°° i2 ,t,
droplet part, models are required for the subgrid variances
Yv Yv - Yv Yv and P P- P P. Since these two sets of terms
(mi5 /
axe of the same form, it seems reasonable and consistent
to use the same type of model for both. We consider three
models, Smagorinsky, Gradient, and Scale-Similarity [15].
Smagorinsky SGS Model
Y
Figure 2: Error in unfiltered variable model _b= _ + fa
_)j= CnA2 /77_ (48)
-Pr-----_-v ">_'_'Ox---_ interpolated to droplet locations
rb = -CRA "V °_.*°__x J (49)
with model constant Ca and filter width A, and where the
rate-of-strain tensor is defined as
( o_, o%) (50)
Although this is forms the basis for most SGS modeling,
it cannot be easily extended to compute the subgrid vari-
ances for T, P and Yv.
r_j = Ca_'2 _z'k azk (51)
y ej = CaA 2aT o_j (52)
Ozk c3zk
.... ,--_.z_ _ =cca -E_=_k_=, (_)
°_[ I:::0:-_._.,:w: T_,mode;l_--_ye=e,dteodcomputtheseub_d_,_-
J " I ,Orv<
O.(:EgE- I--_-- f,4d'_[ol
.bA ¢1_:_[ "?l---_'---'_l_{l_,141Ydltt4_l ances for any quantity _bas
'N" "\_ ''! Theoretically, CGA2 is the moment of the filtering volume,
(_Cl_[r ,''1__" _,.
0,(:04.'_- ///'_- _z_--c_", ']_';; /cA 2 of Equation 40, as can be seen by integrating the
V
a0_-_ ----_r/,_ %y._._ sthqeuarfielteroinfgthevoTluamyloe,r ebxuptanussioinng ffioltrere¢d, Eqquuaantitointies 38,inovtheer
V
l_ Q(_'t '__":-_:_-. ii derivatives. Thus for a cubic top-hat filter CG = _, This
QC01l"_ _'r--'_'"/-." _...'." model has the advantage that the derivatives are already
available from the computation of the resolved part.
- a_ - " -o,o_ o ass a_
Y Scale-Similarity SGS Model
This model involvesrefilterintgheflow-fieladta testfilter
-- f,0
,--,,h_.,., £ _>
_" _(x)r_ -_ -
(-_ _-_)
a_m:. , _ = Cs _,Yv - _Yv (57)
_1 I
o.ao:_S "/" ' Theoretically, Cs should be 1. This model is easily ex-
"-_J_x , tended to compute the subgrid variances for any quantity
Cas
, ,,..
aoco5 . ,. -x Model Coefficients
0 _'----'_,'1-( , , , i , , . . * .....
_0S 0 (_0S 0_I The validation of the models involves comparing the values
Y predicted by the models to those obtained from the DNS
database. Standard deviations from Equation 42 will be
referred to as "exact". First, the correlation between the
Figure 3: Error in unfiltered variable model 4) = _ + fa "exact" and model standard deviations will be computed.
interpolated to droplet locations
Vari;dde Slope=VCc; Correlation a)
u, 0.4037 0.9855
8OO
,t2 0.4087 0.9847
u:_ 0.4112 0.9835
T 0.3886 0.8727
Yv 0.4155 0.9806 eC0 tz
P 0.3980 0.9902
_0 ,,/'rl' \!,._:,,/\
Combined 0.40
1,_ i_\
Table 1: Gradient SGS Model A = 4Az
ii x
The correlationsaxecomputed by averagingoverhomoge-
,o f
neous (xL-x3) planes
<XY>
R(X,Y;x2) =
_/_ X 2 >_ (59)
Y
or over the whole domain
[XY] (60)
R(X,Y) =
By definitionR isbetween -I and I. Values near
1 indicatestrong positivecorrelation,valuesnear -I in-
dicatestrong negativecorrelation,whereas valuesnear 0
indicatepoor correlation.This allowspointwiseassess-
ment ofthe correlations.Next, the relationshipbetween
the two variables(i.e.the model coefficientn)eeds to be
determined. The simplestisa constant-coe_cientmodel
where CR, Ca, Cs arethe same forallflowvariablesover
the whole (spatial,temporal) domain. In that case,the
coefficientcan be determined usinga least-squaresfitto
Y = bX which leadstob= [XY]/[XX]. IfX isthemodel
standard deviationand Y isthe "exact"standard devi_
tion,then b isthe square root ofthe model coefficient. .200 -_1i i , '-0_(]5' , i , b i t i i0J35' , , , 0I.1
More sophisticatedmodels would have the model coeff- Y
cientsas functionsofspaceand time.
Results
All three models were evaluated for Reynolds number of
600 and mass loading 0.2 (case TP600 of Miller and BeUan
[9]) at the end of the simulation, using cubic top-hat filters.
Figure 4 shows the subgrid scale stresses predicted by
the Smagorinsky model. It can be seen that there is little
|ocal correlation between the exact and predicted stresses.
Thus, this model was not assessed further.
Figures 5 and 6 show the plane-averaged SGS stan-
414
daxd deviations for ut,u2, u._ and T, Yv,P respectively.
Deviations from both models show good correlations with -(18
the exact deviations, although the models tend to under-
_8
predict the temperature deviations. For these figures, val-
ues of Ca = 0-42 and Cs = 0.792 were used. These values -1 I_1.... -O_{._... O .... n_35'' ' 'dl
were obtained from linear fits of the SGS standard devia- Y
tions, which produced the coefficients presented in Tables
l and 2.
Figure 4: Subgrid stresses a)exact b)Smagorinsky c) cor-
relation
007 0011
! i "
i i
.I i O.I_ ..
I / Y
/ i
•¢ i -_-_."",,,._.• '0.C04
o.r.o O.OOQ
0.01_
o!_, .am o am Q1 'dos'' ' ' G.... a_'
Y Y
ooe
o.f_
0,007 i"i
oo. _i
o.o4
o.03 oo[ / y,
O,Ol2
o.01
o -01 _C6 0 (_05 Q1 " -0,1 -GIO5_ J _ _ 0I a , , ,0.0!5, , ,0,I1,
Y Y
O.07 (101
o._! .,\ _.,J
o.08 .... Qmdlmt
ao_I ..,I_..',.-......
a05 oo_i V 'V_'.,
°4 / Y
0O4
0.03 _,i
0.02 o.cm_- ; \,.
0,01
• . | ,
o 1 -0.05 0 0,05 ' O,1
Figure 5: Plane Averages of SGS Standard Deviations, Figure 6: Plane Averages of SGS Standard Deviations,
Velocity Components Temperature, Vapor Mass Fraction, and Pressure
V;tri,tl_l,. SIope=_ Correlation S_npo._zum on Turb,dcnt Shear Flow._, p_tges 10-1:1-
ul (}.7465 0.9475 6, September 1991.
u_ 0.7772 0.9498
[5] S. Elghobashi and G.C. Truesdeil. On the Two-Way
u:l 0.8008 0.9498
Interaction Between Homogeneous Turbulence and
T 0.9143 0.8958
Dispersed Solid Particles. I: Turbulence Modification.
Yv 0.7856 0.9181
Physics of Fluids A, 5(7):1790-1801, 1993.
P 0.7350 0.9476
Combined 0.79 [6] F.F Grinstein and K. Kailasanath. Three-
Dimensional Numerical Simulations of Unsteady Re-
Table 2: Scale-Similarity Model A = 4Ax; _ = 8Ax active Square Jets. Combustion and Flame, 100:2-10,
1995.
Conclusions [7] D. Harmell, I.M. Kennedy, and W. KoUmann. A Sim-
ulation of Particle Dispersion in a Turbulent Jet. In-
An a priorisubgridanalysishas been conducted fora tem- ternational Journal of Multiphase Flow, 18(4):559--
porallydevelopingmixing layerwith evaporatingdroplets. 576, 1992.
This analysiswas performed on a DNS database for
Reynolds number (basedon vorticitythickness)of600 and [8] F. Mashaayek. Droplet-Turbulence Interactions in
Low-Mach-Number Homogeneous Shear Two-Phase
mass loading of 0.2. Two models forthe subgrid scale
Flows. Journal of Fluid Mechanics, 367:163-203,
(SGS) standard deviations,the gradientand scalesimi-
1998.
laxitymodels, where found togiveexcellentresultswhen
the model constant was properlycalibrated.A model to
[9] R.S. Miller and J. Bellan. Direct Numerical Simula-
recoverthe unfilteredvariablesfrom thefilteredvariables
tion and Subgrid Analysis of a Transitional Droplet
was alsoexamined. In thismodel, theunfilteredvariables
Laden Mixing Layer. Submitted to Physics of Fluids,
are taken to be filteredvariablesplusa correctionterm
1999.
which canbe computed from theSGS standarddeviations.
Predictionsforthe unfilteredvariablesat thedropletlo- [10] R.S. Miller and J. Bellan. Direct Numerical Simu-
cationswere found to be improved compared to simply lation of a Confined Three-Dimensional Gas Mixing
interpolatingthe filteredvariables.Future work involves Layer with One Evaporating Hydrocarbon-Droplet
both testingthesemodels on a DNS databasefora higher Laden Stream. Journal of Fluid Mechanics, 384:293-
mass loadingand a posterioritestingofthesemodels inan 338, 1999.
LES.
[11] O. Simonin, E. Deutsch, and M. Boivin. Large Eddy
Simulation and Second-Moment Closure of Particle
Acknowledgments
Fluctuating Motion in Two-Phase Turbulent Shear
Flows. Turbulent Shear Flows, 9:85-115, 1993.
This research was conducted at the Jet Propulsion Lab-
oratory (JPL) of the California Institute of Technology [12] K.D. Squires and J.K. Eaton. Particle Response
(Caltech). Computations were performed usingthe Cal- and Turbulence Modification in Isotropic Turbulence.
techCenter forAdvanced Computing Research (CACR) Physics of Fluids A, 2(7):1191-1203, 1990.
parallelHP Exemplar.
[13] G.C. Truesdell and S. Elghobashi. On the Two-
Way Interaction Between Homogeneous Turbulence
References
and Dispersed Solid Particles. II. Particle Dispersion.
Physics of Fluids A, 6(3):1405--1407, 1994.
[].]A. Ansari and W.Z. Strang. Large-Eddy Simulation
of Turbulent Mixing Layers. TechnicalReport 96-
[14] B. Vreman, B. Geurts, and H. Kuerten. A Priori Tests
0684,AIAA, 1996. of Large Eddy Simulation of the Compressible Plane
Mixing Layer. Journal of Engineering Mathematics,
[2]M. Boivin, O. Simonin, and K.D. Squires. Direct
29:299-327, 1995.
Numerical Simulation ofTurbulence Modulation by
Particlesin IsotropicTurbulence. Journal ofFluid [15] B. Vreman, B. Geurts, and H. Kuerten. Large-Eddy
Mechanics, 375:235-263, 1998. Simulation of the Turbulent Mixing Layer. Journal
of Fluid Mechanics, 339:357-390, 1997.
[3] C.T. Crowe, T.R. Troutt, and J.N. Chung. Numeri-
cal Models for Two-Phase Turbulent Flows. Annual [16] Q. Wang, K.D. Squires, and O. Simonin. Large Eddy
Rewew of Fluid Mechanics, 28:11-43, 1996. Simulation of Turbulent Gas-Solid Flows in a Verti-
cal Channel and Evaluation of Second-Order Models.
[4] E. Deutsch and O. Simonin. Large Eddy Simulation International Journal of Heat and Fluid Flow, 19:505---
Applied to the Modelling of Particulate "Pransport Co-
5tl, 1998.
efficients in Turbulent Two-Phase Flows. In Eighth