An Overview of the NCC Spray/Monte-Carlo-PDF Computations

36 Pages·2000·2.1 MB·English

by  Raju, M. S

AIAA-2000-0337
An Overviewof the NCCSpray/Monte-Carlo-PDF
Computations
M.S.Raju
DynacsEngineeringCompany,Inc.
Brook Park,OH
38th Aerospace Sciences
Meeting & Exhibit
January 10-13, 2000 / Reno,NV
For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics
1801 Alexander Bell Drive, Suite 500, Reston, VA 20191-4344
1
!
E
W
An Overview of the NCC Spray/Monte-Carlo-PDF Computations
M.S. Raju*
Dynacs Engineering Co., Inc., NASA Glenn Research Center
2001 Aerospace Parkway
Brook Park, Ohio-44142
1 ABSTRACT by summarizing the results of a 3D test case with
periodic boundary conditions. For the 3D case, the
This paper advances the state-of-the-art in parallel performance of all the three solvers (CFD,
spray computations with some of our recent contri- PDF, and spray) has been found to be good when
butions involving scalar Monte Carlo PDF (Proba- the computations were performed on a 24-processor
bility Density Function), unstructured grids and par- SGI Origin work-station.
allel computing. It provides a complete overview of
2 NOMENCLATURE
the scalar Monte Carlo PDF and Lagrangian spray
computer codes developed for application with un- A pre-exponent of an Arrhenius
structured grids and parallel computing. Detailed reaction rate term
comparisons for the case of a reacting non-swirling a non-unity exponent of an Arrh-
spray clearly highlight the important role that chem- enius reaction rate term
istry/turbulence interactions play in the modeling of an outward area normal vector
reacting sprays. The results from the PDF and non- of the nth surface, ms
PDF methods were found to be markedly different Bk Spalding transfer number
and the PDF solution is closer to the reported exper- b non-unity exponent of an
imental data. The PDF computations predict that Arrhenius reaction rate term
some of the combustion occurs in a predominantly CD drag coefficient
premixed-ftame environment and the rest in a pre- Cv specific heat, J/(kg K)
dominantly diffusion-flame environment. However, C¢ a constant in Eq. (39)
the non-PDF solution predicts wrongly for the com- cn convection/diffusion coefficient
bustion to occur in a vaporization-controlled regime. of the nth face, kg/s
Near the premixed flame, the Monte Carlo particle D turbulent diffusion coefficient, m2/s
temperature distribution shows two distinct peaks: d droplet diameter, m
one centered around the flame temperature and the Ea activation energy of an
other around the surrounding-gas temperature. Near Arrhenius reaction rate term
the diffusion flame, the Monte Carlo particle tem- h specific enthalpy, J/kg
perature distribution shows a single peak. In both Jia diffusive mass flux vector, kg/ms
cases, the computed PDF's shape and strength are k turbulence kinetic energy, m2/s 2
found to vary substantially depending upon the prox- I_ latent heat of evaporation, J/kg
imity to the flame surface. The results bring to the I_,_]1 effective latent heat of evaporation,
fore some of the deficiencies associated with the use J/kg (defined in Eq. (11))
of assumed-shape PDF methods in spray computa- Mi molecular weight of ith
tions. Finally, we end the paper by demonstrating species, kg/kg-mole
the computational viability of the present solution mk droplet vaporization rate, kg/s
procedure for its use in 3D combustor calculations m_o initial mass flow rate associated
with kth droplet group, kg/s
*Engineering Specialist, Associate fellow AIAA. Copyright
Na_ number of time steps employed in
(c) 1999 by the author. Published by the AIAA with
permission. the PDF time-averaging scheme
¢ represents a set of scalars of the joint PDF
Nf number of surfaces contained in
_¢ independent composition space
a given computational cell p density, kg/m 3
Nffq total number of Monte Carlo o" dimensionality of C-space
particles per grid cell r stress tensor term, kg/ms 2
total number of computational cells 0 void fraction
nk number of droplets in kth group
P pressure, N/m 2 Subscripts
Prandtl number
P joint scalar PDF f represents conditions associated with fuel
gas constant, J/(kg K) g global or gas-phase
Re Reynolds number
i index for the coordinate or species components
rk droplet radius, m j index for the species component
rko initial droplet radial location, m k droplet group or liquid-phase
8k droplet radius squaredl r_, m2 l liquid-phase
8talc liquid source contribution of the m conditions associated with Nrn
gas-phase continuity equation
n nth-face of the computational cell
Srnle liquid source contribution of the o initial conditions or oxidizer
gas-phase energy equation p conditions associated with the
liquid source contribution of the
Smlm properties of a grid cell
gas-phase momentum equations
s represents conditions at the droplet
liquid source contribution of the
Srnl$ surface or adjacent computational cell
gas-phase species equations t conditions associated with time
liquid source contribution of the a variable
Sc_ index for the scalar component of
T temperature, K the joint PDF equation
t time, s
, partial differentiation with respect
ith velocity component, m/s
to the variable followed by it
ith velocity component of
kth droplet group, m/s Superscripts
½
volume of the computational cell, m3
Wo chemical reaction rate, 1/s
" Favre averaging
gas-phase chemical reaction rate, 1/s
- time averaging or average based on the Monte
Cartesian coordinate in the ith direction m
Carlo particles present in a given cell
mass fraction of jth species II fluctuations
spatial vector
X. mole fraction
At local time step used in the 3 INTRODUCTION
PDF computationsl _-..........
Spray combustion=is =:of interest in a wide'
At/ local time step in the flow solver, s
variety of applications: g_turbine combustors, in-:
At91 global time step in the spray solver, s
At. fuel injection time step, s ternal combustion engines (diesel and spark-ignition),
Atml allowable time step in the spray solver, s liquid-rocket motors, and industrial furnaces. A large
AV computational cell volume, m3 fraction of the world's energy needs axe met by the
6 Dirac-delta function combustion of the liquid fuels. Because of its abun-
dant use, the need for developing better predictive
rate of turbulence dissipation, m2/s 3
tools for aiding in the design of ever more efficient,
ej species mass fraction at the droplet surface
pollution-free, and stable combustion devices has re-
species mass fraction at the droplet surface
ceived considerable attention. 1-i°
re turbulent diffusion coefficient, kg/ms
A thermal conductivity, J/(ms K) The physical modeling of turbulent spray flames
requires consideration of various complex and rate-
dynamic viscosity, kg/ms
controlling processes associated with turbulent trans-
turbulence frequency, 1/s
port, mixing, and chemical kinetics, fluid-dynamic
characteristicosffuelinjectionandsprayformation,siderations but also depends on the computational
transporat ndvaporizatiocnharacteristicosfindivid- efficiency considerations as determined by the com-
ualdropletsa,ndtheinteractionofturbulencewith
puter memory and turnaround times afforded by the
chemicaklinetics,amongothers.Theinteractionof
present-day computers. With the aim of developing
turbulencewith chemicaklineticsmayoccurover an efficient solution procedure for use in multidimen-
a wide-rangeofdisparatetime andlengthscales.sional combustor modeling, we extended the scalar
Turbulencpelaysanimportantrolein determiningMonte Carlo PDF approach to the modeling of sprays
theratesofmassandheattransferc,hemicarleac-
with parallel computing in order to facilitate large-
tions,andliquid-phaseevaporatioinnmanypracti- scale combustor applications/In this approach, the
calcombustiodnevicesT.heinfluenceofturbulence
mean gas-phase velocity and turbulence fields are de-
in agaseoudsiffusionflamemanifestsitselfin sev- termined from the solution of a conventional CFD
eralforms,rangingfromtheso-calledwrinkledor method, the scalar fields of species and enthalpy from
stretchefdlameletrsegimetothedistributedcombus-a modeled PDF transport equation using a Monte
tionregime,dependinugponhowturbulenceinter- Carlo method, and a Lagrangian-based dilute spray
actswithvariousflamescale1s.1-12Althoughmost model is used for the liquid-phase representation.
oftheturbulentspraycombustiomnodels
are based The application of this method showed reasonable
on either diffusion or premixed gaseous flame theo- agreement when detailed comparisons were made for
ries, combustion in a spray flame is more complex two different cases involving an open and a confined
and seldom occurs in a single mode. swirl-stabilized spray flames/
There are several approaches used in the study It is well known that considerable effort usually
of turbulent spray flames. 1-1° E1 Banhawy and goes into generating structured-grid meshes for grid-
Whitelaw 2 in their study on the prediction of the ding up practical combustor geometries which tend
local flow properties in a spray flame employed an to be very complex in shape and configuration. The
assumed shape PDF model with values for its mean grid generation time could be reduced considerably
and variance obtained from the solution of additional by making use of existing, automated unstructured
transport equations. Raju and Sirignano 4 in their grid generators. 1_-1s With the aim of advancing the
study on multicomponent spray computations in a current multi-dimensional computational tools used
swirl-stabilized center-body combustor employed the in the design of advanced technology combustors, two
eddy break-up model of Spalding 13 to take some ef- new computer codes - LSPRAY s and EUPDF 19 -
fect of turbuler/ce on combustion. The eddy break-up were developed here, thereby extending our previ-
model might be applicable in cases where chemical ous work 7 on the Monte Carlo PDF and sprays to
reaction rates are either very fast or slow compared unstructured grids as a part of the National Com-
with the turbulent time scales. bustion Code (NCC) activity. NCC is being devel-
Most of the turbulence closure models for reac- oped in the form of a collaborative effort between
tive flows have difficulty in treating nonlinear reaction NASA GRC, aircraft engine manufacturers, and sev-
rates. 11-12 The use of assumed shape PDF methods eral other government agencies and contractors. 17
was found to provide reasonable predictions for pat- The unstructured 3D solver is designed to be mas-
tern factors and NOx emissions at the combustor sively parallel and accommodates the use of an un-
exit. 14However, their extension to multi-scalar chem- structured mesh with mixed elements comprising of
istry becomes quite intractable. The solution proce- either triangular, quadrilateral, and/or tetrahedral
dure based on the modeled joint composition PDF type: The ability to perform the computations on
transport equation has an advantage in that it treats unstructured meshes allows representation of com-
the nonlinear reaction rates without any approxima- plex geometries with relative ease. The application of
tion. This approach holds the promise of modeling the unstructured grid extension to a confined swirl-
various important combustion phenomena relevant stabilized spray flame provided reasonable agreement
to practical combustion devices such as flame extinc- with the available droplet velocity measurements. 9
tion and blow-off limits, and unburnt hydrocarbons A current status of the the use of the paral-
(UHC), CO, and NOx predictions. 11-12'14 lel computing in turbulent reacting flows involving
The success of any numerical tools used in the sprays, scalar Monte Carlo PDF and unstructured
multidimensional combustor modeling depends not grids was described in Ref. 10. It also outlines sev-
only on the modeling and numerical accuracy con- eral numerical techniques developed for overcoming
someofthehighcomputetrime-and-storalgimeita-
the gas-phase in Eulerian coordinates derived for the
tionsplacedbytheuseofMonteCarlosolutionmeth-
multicontinua approach. 21 This is done for the pur-
ods.Theparallepl erformancoefboththePDFand
pose of identifying the interphase source terms arising
CFDcomputationwsasfoundtobeexcellenbtutthe
from the exchanges of mass, momentum, and energy
resultsweremixedforthespraymoduleshowingrea-
with the liquid-phase.
sonablpeerformancoenmassivelpyaralleclomputers
likeCrayT3D;butitsperformancweaspooronthe The conservation of the mass leads to:
workstationclusters.In ordertoimprovethepar-
allel performance of the spray module, two different
b yo],,+ = s ,o = nk (1)
domain decomposition strategies were developed and
k
the results from both strategies were summarized. I°
The main objective of our present work is to For the conservation of the jth species, we have:
to investigate the importance of considering chem-
istry/turbulence interactions in the calculation of
a reacting spray. This was done by making de-
tailed comparisons for the case of a reacting non-
swirling spray for which experimental data was re-
sm ,= (2)
ported by McDonell and Samuelson. 2° The compar-
k
isons involved predictions from two different sets of
where
computations, one in which the solution for the tem-
perature and species fields is obtained from the use of
the scalar Monte Carlo PDF method and in the other
they are obtained from the solution of a conventional Z (vj = 0 and Z ej = 1
CFD solution. The second objective is to demon- J J
strate the computational viability of the present so-
For the momentum conservation, we have:
lution procedure for its use in 3D computations. This
was done by summarizing the results of a 3D test case
which was designed primarily to examine the appli-
+ + [pv4,.,-
cability of the spray particle search algorithm in 3D
computations and of the newly-implemented periodic
[OV_rij],=j - [(1 - O)Vc_ij],=j = s,,_trn=
boundary conditions. For clarity, the overall solution
procedure from Ref. 10 is repeated here. However, 4_
for a detailed account of the parallel performance to-
k k
gether with the development and implementation of
the parallel method, the interested reader is referred where _ = the void fraction of the gas which is defined
to Ref. 10. as the ratio of the equivalent volume of gas to a given
First, complete details of the overall solution volume of a gas and liquid mixture. For dilute sprays,
procedure with a particular emphasis on the PDF the void fraction is assumed to be equal to one. The
and spray algorithms are presented along with sev- shear stress rq in Eq. (3) is given by:
eral other numerical issues related to the coupling
between the CFD, spray, and PDF solvers. It is fol-
lowed by the results-and-discussion section where the 2
vii = I.t[ui,rj + u.i,x,] -- "_6ijui,x_
application of the method to predict the local prop-
erties of two different cases are presented. Finally, we
For the energy conservation, we have:
conclude the paper by presenting a brief summary of
important results along with the parallel performance
of the three solvers (CFD, PDF, and spray).
[_Vch],,+_SVcuih],x,-[OVc)_T,_,],_,-[(1-8)VcAtT,:,],_,
4 GOVERNING EQUATIONS FOR THE
GAS PHASE = (4)
k
Here, we summarize the conservation equations for
5 GAS-PHASE SCALAR JOINT PDF The second term on the right-hand side repre-
EQUATION sents transport in the scalar space due to molecu-
lar mixing. A mathematical description of the mix-
The transport equation for the density-weighted joint ing process is rather complicated, and the interested
PDF of the compositions, P, is: reader is referred to Ref. 12. Molecular mixing is
accounted for by making use of the relaxation to the
ensemble mean submodel. 11
_],, + [paip'],.,
{Transient} {Mean convection}
< pjai,., I¢_ >= -c#_(¢_ - $_) (7)
+ [p_(c__)p'],¢o= -[p < u71¢_> _,.,
{Chemical reactions} {Turbulent convection} where w = e/k, and C¢ is a constant. For a
conserved scalar in a homogeneous turbulence, this
I
-[P<_ 1jai,=,l¢->- p-],,o-[_< ps_lC->p-],¢. (5) model preserves the PDF shape during its decay, but
there is no relaxation to a Gaussian distribution} 2
{Molecular mixing} {Liquid-phase contribution} However, the results of Ref. 14 indicate that the
choice between the different widely-used mixing mod-
where
els is not critical in the distributed reaction regime of
= chemical source term
Wa premixed combustion as long as the turbulent mixing
for the c_-th
frequencies are above I000 Hz. Most of the practical
composition variable,
combustors seem to operate at in-flame mixing fre-
< u_II ]¢_.> = conditional average of
quencies of 1000 Hz and above. The application of
Favre velocity
this mixing model seemed to provide some satisfac-
fluctuations,
tory results when applied to flows representative of
1 ct
< ;J_,_, [¢ > = conditional average those encountered in the gas-turbine combustion} 4
of scalar dissipation,
The third term on the right-hand side represents
and
the contribution from the spray source terms:
1
< ;s_ ]¢ > = conditional average of
spray source terms.
The terms on the left-hand side of the above 1 1
< -so 1¢ >= -- _'_-kmk(¢o. - ¢_) (s)
equation could be evaluated without any approxima- p - pAY
tion, but the terms on the right-hand side of the equa-
where ¢o = y_,a= 1,2,...,s= o'- 1
tion require modeling. The first term on the right
represents transport in physical space due to turbu-
1 1
lent convection. 12 Since the joint PDF, /3, contains < -s_ I¢ >= _ _ n_m_(-l_,_jj + h_. - ¢_)
no information on velocity, the conditional expecta- p -- pAy
(9)
tion of < uIiI J¢ > needs to be modeled. It is mod-
eled based on a gradient-diffusion model with infor- where ¢_ = h and is defined by:
mation supplied on the turbulent flow field from the
a-1
flow solver} 2 h = _,ihi (10)
i=l
- < u7I¢ > P= r_p,_, (6)
where
The fact that the turbulent convection is modeled
as a gradient-diffusion makes the turbulent model no
better than the k - e model. The uncertainties asso- hi = h°li + CpiyidT,
rtJ
ciated the use of a standard k- cturbulence model to
swirling flows are well known32 Some of the modeling
}:_tt
uncertainties associated with the use of the standard
Cpi = "_i(Ati + A2iT + A3iT 2+ A4iT 3+ AhiT4),
k - cmodel would be addressed in our future studies
with the implementation of a non-linear k - ¢ devel- h°li is the heat of formation of ith species, R_ is the
oped for the modeling of swirling flows} 2 universal gas constant, cas is a mass fraction of the
evaporatinsgpecieastthedroplet
surface, and lk,,]:
is the effective latent heat of vaporization as modified c9 = _ 1+ (15)
by the heat loss to the droplet interior:
For droplet size, the droplet regression rate is
(11) determined from one of three different correlations
depending upon the droplet-Reynolds-number range.
When Rek > 20, the regression rate is determined
Here we assumed that the spray source terms
based on a gas-phase boundary-layer analysis valid
could be evaluated independent of the fluctuations in
for Reynolds numbers in the intermediate range. 23
the gas-phase compositions of species and enthalpy.
The other two correlations, valid when Rek < 20, are
Eqs. (8)-(10) represent the modeled representation taken from Clift et al. _4
for the conditional averages of the spray contribution
to the PDF transport equation.
ds_ -2 I__[[[2;ne,]j1/2 f(B,) ;fR. >20
-E=
Pk
6 LIQUID-PHASE EQUATIONS
ds___£
The spray model is based on the multicontinua ap- dt = _Pp._kL [1+ (1+ Rek) '13] Re°£°771n(1 + Bk)
proach which allows for resolution on a scale greater
if 1 < Rek <_20 (16)
than the average spacing between two neighboring
droplets. A Lagrangian scheme is used for the liquid-
-d-sk = ---#_ [1+ (1+ Re_) 1/3] ln(l+Bk) if Rek <
phase equations as it eliminates errors associated dt pk
with numerical diffusion. The vaporization modelofa
where Bk is the Spalding transfer number defined in
polydisperse spray takes into account the transient ef-
Eq. (22). The function f(Bk) is obtained from the
fects associated with the droplet internal heating, the
solution of Emmon's problem. 2s The range of validity
forced convection effects associated with droplet in-
of this function was extended in l_ju and Sirignano 4
ternal circulation and the phenomena associated with
to consider the effects of droplet condensation.
boundary layers and wakes formed in the intermedi-
The internal droplet temperature is determined
ate droplet Reynolds number range. 4The present for-
based on a vortex model. 23 The governing equation
mulation is based on a deterministic particle-tracking
for the internal droplet temperature is given by:
method and on a dilute spray approximation which
is applicable for flows where the droplet loading is
>,,
low. Not considered in the present formulation are +
the effects associated with the droplet breakup, the "3F = 17_c--,-,=_--p__;_,-,[_ _ (1 + C(Oa ) TEJ
droplet/shock interaction, the multi-component na- (17)
ture of liquid spray and the phenomena associated
with dense spray effects and super-criticM conditions. where
The spray method provided some favorable results
(18)
when applied to several unsteady and steady-state rk-_--
calculations.4- lo
For the particle position of the kth droplet where a represents the coordinate normal to the
group, we have: streamsurface of a Hill's Vortex in the circulating
fluid, and C(t) represents a nondimensional form of
dxik the droplet regression rate. The initial and boundary
= uik (12)
dt conditions for Eq. (17) are given by:
For the droplet velocity:
"-- tinjection, T;, = T,Lo (19)
dui_ 3 CDpa,Rek
(13) 1 [c,,p,]
dt - 16 pkr_ [ui9 -- Uik] 0__ 0_ = l"ff [--_--_ J r_ at
where OT_
O:=1,
O_
Rek =2_--_-p[(u9- _,k).(u9- uk)]1/2 (14)
(21)
I-tas
where a = 0 refers to the vortex center, and c_ = 1 grid spacing and average droplet spacing and velocity.
refers to the droplet surface. However, our experience showed that for the case of a
The Spalding transfer number is given by: steady-state solution, a time step based on the aver-
age droplet lifetime yields better convergence. 4-s Its
- - (Y:"- Y:) (22) value typically ranges between 1 and 2 milli-seconds
Ik,,:: (1- y:,)
for the case of reacting flows.
The spray computations facilitate fuel injection
y-:: ,= l+Ma( x:-1,- :) through the use of a single fuel injector comprised of
different holes, s-s However, multiple fuel injection in
where Ma is the molecular weight of the gas excluding
a steady-state calculation could be simulated by sim-
fuel vapor.
ply assigning different initial conditions for the spatial
Based on the assumption that phase equilibrium
locations of the droplet groups associated with each
exists at the droplet surface, the Clausius-Clapeyron
one of the different holes. For a polydisperse spray,
relationship yields
the spray computations require inputs for the num-
ber of droplet groups in a given stream and for the
Pn :h initial droplet locations and velocities. However, the
number of droplets in a given group and their sizes
In Eq. (14), the molecular viscosity is evaluated at a could be either input directly or computed from a
reference temperature using Sutherland's equation properly chosen function for the droplet size distribu-
tion. The specified initial inputs should be represen-
tative of the integrated averages of the experimental
T3/2 conditions:- :0
pg,(T_]) = 1.4637 10-6 "'_1 (25)
T,_: + 120 One correlation typical of those used for the
droplet size distribution is taken from Ref. 2:
where
T,,:= [Tg+ _T_, (26)
The droplets may evaporate, move along the [ d ]3.5 , _ ,o.4dd
wall surfaces, and/or reflect with reduced momen- dnn = 4.21 IOs [d-'_32J e-:6"gs(s_7) --d32 (27)
tum upon droplet impingement with the combustor
walls. In our present computations, subsequent to
the droplet impingement with the walls, the droplets where n is the total number of droplets and dn is
are assumed to flow along the wall surfaces with a the number of droplets in the size range between d
velocity equal to that of the surrounding gas. and d-4-dd. The Santer mean diameter, d32, could
be either specified or estimated from the following
7 DETAILS OF DROPLET FUEL correlation: 26
INJECTION
The success of any spray model depends a great
deal on the specification of the appropriate injector 2_roz_. (28)
d32 -" Bd pg-'-_ m
exit conditions. However, a discussion involving the
physics of liquid atomization is beyond the scope of
this subject matter. In our present computations, the
where Bu is a constant, VT is the average relative
liquid fuel injection is simulated by introducing a dis-
velocity between the liquid interface and the ambi-
cretized parcel of liquid mass in the form of spherical
ent gas, and _* is a function of the Taylor number,
droplets at the beginning of every fuel-injection time
step.
For certain cases, the fuel-injection time step, A typical droplet size distribution obtained
Atiz, needs to be determined based on the resolution from the above correlation in terms of the cumulative
permitted by the length and time scales associated percentage of droplet number and mass as a function
with several governing parameters such as average of the droplet diameter is shown in Fig. 1.7
22 where subscript n refers to the nth-face of the com-
1.e
putational cell. The coefficient c,_ represents the
_g _ _ Number / c
.9 o transport by convection and diffusion through the
18 nth-face of the computational cell, p. The convec-
_J
16 .o tion/diffusion coefficients in the above equation are
.7 '[ determined by one of the following two expressions:
"- 12
2a n ._
.5 _
E In
-o B .4 E
2a_n.an .
6 .3
c, = rnaz[10.5fia_ntU_n I,I'¢(A_-p _V ) ] - 0.5_an.U_a
4 .2 O"
and
,-1
2 .1
J Cp "-- Z Cn
J
e
|
In both the above expressions for cn, a cell-centered
finite-volume derivative is used to describe the vis-
Drop|et diameter, mlc_-ons
cous fluxes; but an upwind differencing scheme is used
for the convective fluxes in the first expression and a
hybrid differencing scheme in the second.
Figure 1 Droplet-size distribution.
9.1 Numerical Method Based on
8 CFD SOLUTION ALGORITHM
Approximate Factorization
The gas-phase mass and momentum conservation The transport equation is solved by making use of an
equations together with the standard k-e turbulence approximate factorization scheme) 2 Eq. (29) can be
equations with wall functions are solved by making recast as:
of a finite-volume solver with an explicit fourth-stage
Runge-Kutta scheme. Further details of the code can
_,,(¢__,+ At)=
be found in Refs. 16 & 18.
(I+AtR)( (I+AtS)( I+AtM)( I+AtT)_p (¢, _)+O(At 2)
9 PDF SOLUTION ALGORITHM
(30)
In order to facilitate the integration of the Monte where I represents the unity operator and T, M, S,
Carlo PDF method in a finite-volume context, the and R denote the operators associated with spatial
volume integrals of convection and diffusion in Eq. transport, molecular mixing, spray, and chemical re-
(5) were first recast into surface integrals by means of actions, respectively. The operator is further split
a Gauss's theorem. -_7Partial integration of the PDF into a sequence of intermediate steps:
transport equation would yield:
_(¢_,t) = (I + AtT)fiv(C_.,t ) (31)
cpAt
ifip(__, t + At) = (I - #--_)ifip (__,t)
_v (C__,t) ----(I + AtM)_(C_,t) (32)
c_ At
+ _ _(_¢,_)- A@o(c_)v'],¢o
Tt
p_(¢_., t) = (I + AtS)W (¢__,t) (33)
- A_[<1S,_,=#,1_¢> p-],_:- A_[<}_ I_¢> p-3,¢_
(29) ,_p(¢_., t + At) = (I + AtR)p"_(¢_., t) (34)