www.elsevier.com/locate/compfluid

Computers & Fluids 36 (2007) 601–610

System decomposition technique for spray modelling in CFD codes

V. Bykov a, I. Goldfarb a, V. Gol’dshtein a, S. Sazhin b,*, E. Sazhina b

a Department of Mathematics and Computer Sciences, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israelb School of Engineering, Faculty of Science and Engineering, University of Brighton, Cockcroft Building, Brighton BN2 4GJ, UK

Received 1 August 2005; received in revised form 13 December 2005; accepted 10 February 2006Available online 21 July 2006

Abstract

A new decomposition technique for a system of ordinary differential equations is suggested, based on the geometrical version of theintegral manifold method. This is based on comparing the values of the right hand sides of these equations, leading to the separation ofthe equations into ‘fast’ and ‘slow’ variables. The hierarchy of the decomposition is allowed to vary with time. Equations for fast vari-ables are solved by a stiff ODE system solver with the slow variables taken at the beginning of the time step. The solution of the equationsfor the slow variables is presented in a simplified form, assuming linearised variation of these variables for the known time evolution ofthe fast variables. This can be considered as the first order approximation for the fast manifold. This technique is applied to analyse theexplosion of a polydisperse spray of diesel fuel. Clear advantages are demonstrated from the point of view of accuracy and CPU effi-ciency when compared with the conventional approach widely used in CFD codes. The difference between the solution of the full systemof equations and the solution of the decomposed system of equations is shown to be negligibly small for practical applications. It isshown that in some cases the system of fast equations is reduced to a single equation.� 2006 Elsevier Ltd. All rights reserved.

1. Introduction

Decomposition of complex systems into simpler subsys-tems is de facto almost universally used in engineering andphysics applications. It allows the numerical simulation tofocus on the subsystems, thus avoiding substantial difficul-ties and instabilities related to numerical simulation of theoriginal systems. Special rules are introduced to incorpo-rate the results of numerical simulation of the subsystemsinto the general scheme of the simulation of the whole sys-tem. To the best of our knowledge, the hierarchy of thedecomposition process has been so far fixed for the dura-tion of a process.

As an example of such decomposition we can mentionthe solutions of ordinary and partial differential equations(ODEs and PDEs) describing spray dynamics in computa-tional fluid dynamics (CFD) codes. Numerical spray mod-elling is traditionally based on the Lagrangian approach

0045-7930/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compfluid.2006.02.003

* Corresponding author.E-mail address: S.Sazhin@brighton.ac.uk (S. Sazhin).

coupled with the Eulerian representation of the gasphase. This permits the decomposition of complicatedand highly nonlinear systems of PDEs, describing interac-tions between computational cells, and the systems ofODEs that govern processes in individual computationalcells, including liquid/gas phase exchange and chemicalkinetics. The systems of ODEs are usually integrated usingmuch shorter time steps dt (typically 10�6 s) than the globaltime steps used for calculating the gas phase Dt (typically10�4 s). Thus the decomposition of ODEs and PDEs is de

facto used although its basis has not been rigorously inves-tigated to the best of our knowledge [1,2].

Further decomposition of the system of ODEs, describ-ing droplets dynamics inside individual computationalcells, is widely used. The simplest decomposition of thissystem of ODEs is based on the sequential solution of indi-vidual subsystems comprising this system (split operatorapproach). In this approach, the solution of each individ-ual subsystem for a given subset of variables is based onthe assumption that all the other variables are fixed. Thesequence of the solution of individual subsystems is often

602 V. Bykov et al. / Computers & Fluids 36 (2007) 601–610

chosen rather arbitrarily and the results sometimes varysubstantially depending on the order in which these subsys-tems are solved. In the case of a multiscale system, the reli-ability of this approach becomes questionable altogether,as shown later in Section 4.

We believe that to overcome these problems the multi-scale nature of ODEs needs to be investigated before anyattempt to solve them is made. This idea could beprompted by the approaches used in [3] for the analysisof the processes in CO2 lasers and the one used in [4] forthe analysis of equations describing the autoignition of die-sel fuel (the Shell model). Before solving a system of fivestiff ODEs describing five temperatures in these lasers,the characteristic time-scales of these equations wereanalysed [3]. It was shown that two of these equationsdescribe rather rapid relaxation of two temperatures tothe third one. This allowed the authors of [3] to replacethe solution of five stiff equations by the system of justthree non-stiff equations without any significant loss ofaccuracy. The approach used in [4] was different from theone used in [3], but the ultimate result of reduction of thenumber of ODEs to be solved, and elimination of the stiff-ness of the system of equations, remained the same. Inmathematical terms in both papers the dimension of theODE system was reduced. In other words, the systemwas decomposed into lower dimension subsystems.

A similar system decomposition into lower dimensionsubsystems have been used in constructing reduced chemi-cal mechanisms based on intrinsic low-dimensional mani-folds (ILDM) (e.g. [5–7]) and computational singularperturbation (CSP) (e.g. [8–12]). There are many similari-ties between these methods. They are based on a rigorousscale separation such that ‘fast’ and ‘slow’ subspaces ofthe chemical source term are defined and mechanisms ofmuch reduced stiffness are constructed. These approaches,however, were developed with a view of application tomodelling chemical kinetics. Their generalisation to generalCFD codes has not been considered to the best of ourknowledge.

A useful analytical tool for the analysis of stiff systemsof ODEs, used for modelling of spray heating, evaporationand ignition, could be based on the geometrical asymptoticapproach to singularly perturbed systems (integral mani-fold method) as developed by Gol’dshtein and Sobolev[13,14] for combustion applications (see also [15]). Thisapproach overcomes some of the earlier mentioned prob-lems and is, essentially, focused on systems of ordinary dif-ferential equations of the form:

dx

dt¼ F ðx; yÞ; e

dy

dt¼ Gðx; yÞ;

where x and y are n and m-dimensional vector variables,and e is a small positive parameter (e� 1). The first subsys-tem dx

dt ¼ F ðx; yÞ is called a slow subsystem and the secondone e dy

dt ¼ Gðx; yÞ is called a fast subsystem. In practicalimplementations of the integral manifold method a numberof simplifying assumptions have been made. These include

the assumption that the slow variable is constant during thefast processes. This assumption opened the way to analyt-ical study of the processes [16–19].

These approaches to decomposing systems of ODEswere developed and investigated with a view to applicationto rather special problems, and were based on a number ofassumptions. These include fixing of the decompositionover the whole period of the process, and not allowing itshierarchy to change with time. The underlying philosophyof these approaches, however, seems to be attractive forapplication to the analysis in a wide range of physicaland engineering problems including spray modelling ingeneral computational fluid dynamics (CFD) codes. Thedevelopment of a rather universal new method of decom-position of the system of ODEs, allowing the change ofthe nature of decomposition with time (dynamic decompo-sition), will be the main objective of this paper.

As in the original integral manifold method, our formalapproach to decomposition of the system of ODEs is basedon the division of system variables into ‘fast’ and ‘slow’.This leads to the division of this system onto ‘fast’ and‘slow’ subsystems. In contrast to the original version ofthe integral manifold method, however, linearised varia-tions of slow variables during the time evolution of the fastvariables will be taken into account as the first orderapproximation to the fast manifold. The usefulness of thisdivision depends on whether the ‘fast’ subsystem has lowerdimension compared with the ‘slow’ subsystem. The proce-dure can be iterative and result in a hierarchical division ofthe original system. For example the ‘slow’ subsystem can,in its turn, be subdivided into ‘slow’ and ‘very slow’subsystems.

The proposed procedure will be initially focused on thesimplest possible subdivision of the original system intotwo subsystems, and applied to spray combustion model-ling. Note that ‘fast’–‘slow’ decomposition in this casecan be different for different phase space regions [16,17]and for different time intervals. Wider range of applicationof this method is anticipated.

The preliminary description of this method and its appli-cations was given in [20–22].

The general mathematical description of the method andthe assumptions on which it is based are presented inSection 2. The rest of the paper discusses application of thismethod to a specific problem of modelling of sprays. Basicequations describing these processes are presented inSection 3. In Section 4 the results of applying the newmethod to numerical modelling of these processes aredescribed and discussed. The main results of the paperare summarised in Section 5.

2. Decomposition of the system of equations

Let us consider the system, the state of which is charac-terized by n dimensionless parameters, which are denotedas Zn (n = 1,2, . . . ,n). The value of each of these parame-ters for a given place in space depends on time t, i.e.

V. Bykov et al. / Computers & Fluids 36 (2007) 601–610 603

Zn = Zn(t). This dependence can be found from the solu-tion of the system of n equations, which can be presentedin a vector form:

dZ

dt¼ UðZÞ; ð1Þ

where

Z ¼ ðZ1; Z2; . . . ; ZnÞ; U ¼ ðU1;U2; . . . ;UnÞ:In the general case, a rigorous coupled numerical solu-

tion could be found. This may be not practical, when toomany equations are involved. We believe that a more prac-tical approach to this problem could be based on reducingthe dimensions of this system as discussed in Section 1.This could be based on organising Eq. (1) in terms ofdecreasing parameter Yi defined as

Y i ¼UiðtkÞZiðtkÞ

��������; ð2Þ

where Ui(tk) � Uik � Ui(Z1(tk),Z2(tk), . . . ,Zi(tk), . . . ,Zn(tk))and Zi(tk) � Zik are the right hand sides of Eq. (1) andthe values of Zi taken at the time tk for the time step: Dt:tk! tk+1, i = 1,2, . . . ,n. For Zik close to zero we have aspecial case which requires additional investigation. Inpractice, in most cases we can just assume that Zik is afinite number.

Note that the Taylor expansion of the right hand side ofEq. (1) is

UðZðtkþ1ÞÞ � UðZkþ1Þ¼ UðZkÞ þDUjZ¼Zk

ðZkþ1 � ZkÞþ oðZkþ1 � ZkÞ; ð3Þ

where DU is the Jacobian matrix of the vector field U.In the general case the value of U(Zk+1) is controlled

mainly by U(Zk). However, in the special case whereU(Zk) = 0, the second term on the right hand side of Eq.(3) becomes dominant. Only in this or similar cases, undercertain conditions, Eq. (1) can be simplified to

dZiðtkÞdt

¼ kiðtkÞZiðtkÞ ð4Þ

and the values of Yi coincide with ki. Our analysis, how-ever, will be focused not on this special case, but on thegeneral case when Yi is not directly related to ki.

If Yi is greater than a certain a priori chosen positivenumber a < 1, then the corresponding equation can be con-sidered fast and this should be solved rigorously. If thenumber of ‘fast’ equations is f 5 0, then the system iscalled multiscale and this procedure effectively reducesthe dimension of the system of Eq. (1) from n to f. Thisdimension reduction is particularly attractive when f issmall (1 or 2).

In an alternative approach, which turned out to be morepractical, we reorganize Yi in descending order as

Y P Y P � � �P Y P � � �P Y : ð5Þ

i1 i2 ij in

If we are able to find j = f such that

Y ifþ1

Y if

< �; ð6Þ

where � is another a priori chosen small parameter, then wecan conclude that the system can be decomposed locally(Dt: tk! tk+1) into two subsystems: ‘fast’ and ‘slow’. Notethat the subscript of ij just indicates the order in which theparameters are organized (no summation over f in the righthand side of (6) takes place).

Equations for these subsystems can be presented in vec-tor form as

dU

dt¼ Uf

U

V

� �; ð7Þ

dV

dt¼ Us

U

V

� �; ð8Þ

where

Uf ¼ ðUi1 ; . . . ;Uif Þ; Us ¼ ðUifþ1; . . . ;UinÞ;

U ¼ ðZi1 ; . . . ; Zif Þ; V ¼ ðZifþ1; . . . ; ZinÞ:

The transformation from the original order of variables

Z ¼ ðZ1; Z2; . . . ; ZnÞ

to the new order of the same variables

Z0 ¼ ðZi1 ; . . . ; Zif ; Zifþ1; . . . ; ZinÞ

was performed with the help of the transformation matrixQ ¼ Qi;ij , where Qi;ij ¼ 1 when i corresponds to the originalposition of the variable, ij is the final position of the vari-able, and Qi;ij ¼ 0 for all other i and ij. In this case wecan formally write:

Z ¼ QðZ0ÞZ0 ¼ QðZ0ÞU

V

� �¼ QfðZ0Þ QsðZ0Þð Þ

U

V

� �;

ð9Þwhere Q is calculated for the values of Z at the initial timet0 or the beginning of the time step (Z0). The first f columnsof the matrix Q refer to the fast subsystem, while theremaining n � f columns refer to the slow subsystem. Thisis indicated by introducing the additional matrices Qf andQs.

Note that so far we considered the simplest form ofmatrices Q and Q�1 performing the change of the orderof variables. More complex forms of these matrices couldpotentially perform the decomposition of the originallynon-multiscale system into the multiscale one. Analysis ofthe latter decomposition, however, is beyond the scope ofthis paper.

Having introduced a new small positive parameter e� 1and remembering the definitions of Qf and Qs we canrewrite the system of equations (7) and (8) in a form similarto the one used in Integral Manifold Method:

604 V. Bykov et al. / Computers & Fluids 36 (2007) 601–610

edU

dt¼ eQ�1

f ðZ0ÞU QðZ0ÞU

V

� �� �� Ufe

U

V

� �; ð10Þ

dV

dt¼ Q�1

s ðZ0ÞU QðZ0ÞU

V

� �� �� Us

U

V

� �; ð11Þ

where Ufe = eUf. In this presentation the right hand sidesof Eqs. (10) and (11) are expected to be of the same orderof magnitude over the specified period (time step).

Eqs. (10) and (11) will be integrated over the time periodDt: tk! tk+1. The zeroth order solution of Eq. (11) is justa constant value of the slow variable: V0

kþ1 ¼ Vk ¼ðZifþ1k; . . . ; ZinkÞ, where the superscript 0 indicates the zer-oth approximation, while the subscripts k and k+1 indicatethe point in time. The zeroth order for the fast variable isfound from Eq. (10) with V = Vk. This could be interpretedas the equation for the slow variable on the fast manifold.This means that Eq. (10) (or (7)) is approximated by thefollowing system:

dU

dt¼ Uf

U

Vk

� �: ð12Þ

The solution of Eq. (12) at t = tk+1 (U0kþ1) is the zeroth

order approximation of the fast motion on the fast mani-fold at t = tk. Note that the system of equations (12) canbe stiff in the general case, but with a reduced level of stiff-ness, compared with the original system (1). Hence, thesuggested method is expected to reduce the level of stiffnessof the system and not to eliminate the stiffness altogether.

Under the same zeroth order approximation the slowvariables would remain constant over the same time step.This assumption was used in the original formulation ofthe Method of Integral Manifolds [14]. This, however,might lead to an unphysical result when slow variableswould remain constant for any time t > t0. Hence, the needto calculate slow variables using at least the first orderapproximation. In the case when e is not asymptoticallysmall, further, or higher order, approximations need tobe considered. In this case we introduce the new time scales = t/e, and formally present the slow and fast variables as

V ðsÞ ¼ V ð0Þ þ eV ð1ÞðsÞ þ e2V ð2ÞðsÞ þ � � �UðsÞ ¼ U ð0ÞðsÞ þ eU ð1ÞðsÞ þ e2U ð2ÞðsÞ þ � � �

): ð13Þ

Having substituted expressions (13) into Eq. (11) weobtain:

dðV ð0Þ þ eV ð1ÞðsÞ þ e2V ð2ÞðsÞ þ � � �Þds

¼ eUsU ð0ÞðsÞ þ eU ð1ÞðsÞ þ e2U ð2ÞðsÞ þ � � �V ð0Þ þ eV ð1ÞðsÞ þ e2V ð2ÞðsÞ þ � � �

!:

ð14Þ

Eq. (14) allows us to obtain the first order solution forthe slow variable in the form:

V kþ1 ¼ V ð0Þkþ1 þ eV ð1Þkþ1 ¼ V ð0Þk þ eUs

U ð0Þkþ1

V ð0Þk

!Ds: ð15Þ

Returning to the original variables we can write the expres-sion for the value of V(tk+1) � Vk+1 in the form:

Vkþ1 ¼ Vð0Þk þUs

Uð0Þkþ1

Vð0Þk

!Dt

¼ Zifþ1k þ Uifþ1Uð0Þkþ1;V

ð0Þk

� �Dt; . . . Zink

�þUin U

ð0Þkþ1;V

ð0Þk

� �Dt�: ð16Þ

To increase accuracy of calculations one could continuethe process to take into account the first order solution forthe fast motion. Then the second order solution for theslow motion could be obtained etc.

Since the abovementioned decomposition of the originalsystem of equations is allowed to vary with time, we sug-gest to call this dynamic decomposition approach.

3. Basic equations for spray modelling

In this section basic equations used for modelling drop-lets heating, evaporation and combustion will be summa-rised. These equations will be presented in the generalform, following [23,24], and in the simplified form (partlyfollowing [25]). The latter form of these equations will allowus to perform a direct comparison between the predictionsof conventional CFD approach, and the approachdescribed in this paper. A number of processes, includingdroplet dynamics, break-up and coalescence, and the effectsof temperature gradient inside droplets will be ignored atthis stage (see [26–32]). This can be justified by the fact thatthe main emphasis of this paper will be on the investigationof the new method of the solution of the systems of ODEsrelevant to spray combustion modelling rather than provid-ing a detailed analysis of the processes involved.

3.1. Droplet mass

As shown in [23,32], the equation for stationary dropletmass md can be presented as

_md ¼ 4pkgRd

cpF

lnð1þ BT Þ; ð17Þ

where

BT ¼cpFðT g � T sÞ

Leff

is the temperature Spalding number,

Leff ¼ LðT sÞ þQL

_md

;

L is the specific latent heat of vaporization, QL is the heatspent on droplet heating, kg is the average gas thermal con-ductivity, Rd is droplet radius, cpF is specific heat capacityof fuel vapour.

Analysis of Eq. (17) is not trivial and requires iterations[23]. To test the new system dimension reduction techniquewe will use a much simpler form of this equation [25]:

V. Bykov et al. / Computers & Fluids 36 (2007) 601–610 605

_md ¼ 4pkgRd

cpF

lnð1þ BMÞ; ð18Þ

where BM = (Yfs � Yf1)/(1 � Yfs) is the Spalding massnumber, Yfs and Yf1 are the mass fractions of fuel vapournear the droplet surface and in the ambient gas respec-tively. We present the expression for Yfs as

Y fs ¼ 1þ ppfs

� 1

� �Ma

M f

� ��1

; ð19Þ

p and pfs are ambient pressure and the pressure of saturatedfuel vapour near the surface of droplets respectively, Ma

and Mf are molar masses of air and fuel; pfs can be calcu-lated from the Clausius–Clapeyron equation presented inthe form [25]:

pfs ¼ exp af �bf

T s � 43

� �; ð20Þ

af and bf are constants to be specified for specific fuels, Ts isthe surface temperature of fuel droplets in K; pfs predictedby Eq. (20) is in kPa. We assumed that Le = 1, and ignorethe temperature dependence of liquid density, gas thermalconductivity and viscosity. In the calculations we tookthe values af = 15.5274 and bf = 5383.59 recommendedfor diesel fuel [25].

3.2. Droplet temperature

Following [23,33,34] we can present the equation for sta-tionary droplet temperature as

mdcl

dT d

dt¼ 4pR2

dhðT g � T dÞ � _mdLþ 4pR2drQah

4R; ð21Þ

where

Nu ¼ 2hRd

kg

¼ 2lnð1þ BT Þ

BT; ð22Þ

hR is the radiative temperature, as calculated from P-1model (as the starting point we assume that hR = Text

(external temperature)), Qa is the average absorption effi-ciency factor, which can be calculated from the equation:

Qa ¼ arRbrd ; ð23Þ

ar and br are polynomials of hR, cl is liquid specific heatcapacity.

To test the new system dimension reduction techniquethe effect of thermal radiation will be ignored and we usea simplified form of the temperature dependence of L [25]:

L ¼ LT bn

T cr � T s

T cr � T bn

� �

where LT bnis the value of L at the droplet boiling temper-

ature Tbn, Tcr is the critical temperature. Following [25]we assume that Tbn = 536.4 K, Tcr = 725.9 K, and LT bn

=254,000 J/kg. Also, we assume that BT = BM and liquiddensity is constant.

Eqs. (18) and (21), written for individual droplets, willbe applied to describe droplet parcels, following the con-ventional approach widely used in computational fluiddynamics codes.

3.3. Fuel vapour density

This equation follows directly from the conservation offuel vapour:

ag

dqfv

dt¼ �agCTþ

Xi

_mdi=V

" #; ð24Þ

where qfv is the fuel vapour density, ag is the volume frac-tion of gas assumed equal to 1 in our calculations, the sum-mation is assumed over all droplets in volume V, Qf is theheat released per unit mass of burnt fuel vapour (in J/kg),CT is the chemical term describing fuel depletion (in kg/(m3 s)).

Following [35] we use the expression of the rate of reac-tion in the form:

kcr ¼ A½fuel�a½O2�b exp½�E=ðBT Þ�; ð25Þ

where kcr has units of mole/(cm3 s), while the concentra-tions of fuel [fuel] and oxygen [O2] has units of mole/cm3.The values of these coefficients given for C10H22 will beused. These are the closest to n-dodecane (C12H26) (theclosest approximation for diesel fuel) [35]:

A ¼ 3:8� 1011 1

s

mole

cm3

� �1�a�b

¼ 2:137� 109 1

s

kmole

m3

� �1�a�b

;

E ¼ 30kcal

mole¼ 1:255� 108 J

kmole; a ¼ 0:25; b ¼ 1:5:

Using A in 1sðmole

cm3 Þ1�a�b and E in Jkmole

, we can write:

CT ¼ AM�1:5O2

M0:75f q0:25

fv q1:5O2

exp½�E=ðBT Þ�; ð26Þ

where MO2¼ 32 kg/kmole, and Mf = 170 kg/kmole are

molar masses of oxygen and fuel respectively qO2is the

density of oxygen.In the case of diesel engines, one of the most widely used

autoignition chemical models is the so called Shell model[1,4,36].

3.4. Density of oxygen

A single step global reaction for n-dodecane combustioncan be written as

C12H26 þ 18:5O2 ) 12CO2 þ 13H2O:

Hence the equation for density of oxygen can be presentedas

dqO2

dt¼ �18:5

MO2

M f

CT ¼ �3:48235CT: ð27Þ

606 V. Bykov et al. / Computers & Fluids 36 (2007) 601–610

A useful characteristic, widely used as a measure of reac-tivity of the fuel vapour/air mixture is the equivalenceratio:

u ¼ Fuel=Air

Fuel=Airð Þstoich

¼ Fuel=Oxygen

Fuel=Oxygenð Þstoich

¼ 18:5� 32

170

qfv

qO2

¼ 3:48qfv

qO2

;

where (Fuel/Air)stoich is the stoichiometric ratio of the den-sities of fuel and air.

3.5. Gas temperature

The condition of energy balance leads us to the follow-ing equation for gas temperature:

cmixqmix

dT g

dt¼ agQfCT�

Xi

mdicl

dT di

dtþX

i

_mdiL

"

þX

i

_mdicpFðT g � T diÞ#,

V : ð28Þ

4. Application

The method described in Section 2 will be applied tosimulate polydisperse spray heating, evaporation and igni-tion. The model on which the analysis is based is chosen tobe rather simple, but capable nevertheless of capturing theessential features of the process. We consider three dropletswith initial radii 5 lm, 9 lm and 13 lm, respectively. Theinitial temperatures of all droplets is taken to be equal to400 K. The gas temperature is taken to be equal to 880 K[1]. The gas volume is chosen such that if the droplets arefully evaporated without combustion then the equivalenceratio of fuel vapour/air mixture is equal to 4. This is the sit-uation typical for diesel engines in the vicinity of the noz-zle. The initial density of oxygen is taken to be equal to2.73 kg/m3 (this corresponds to air pressure equal to3 MPa). The initial mass fraction of fuel is taken to beequal to zero. These values of the parameters can beconsidered as an approximation of the actual conditionsin diesel engines [1].

The calculations were based on Eqs. (18), (21), (24)–(28)and the equation for the density of the mixture qmix of fuelvapour and air:

dqmix

dt¼X

i

dqi

dt; ð29Þ

where qi are the densities of individual components. Thecalculations were performed until gas temperature reached1100 K. At this temperature the autoignition process wasassumed to be completed [1]. Since Eqs. (18) and (21) aresolved for three droplets, the maximal number of equationsto be solved was equal to 10. Note that the density of the

fuel vapour/air mixture could be derived algebraically frommass conservation. It was preferred, however, to solve theODE for it to enable us to monitor the mass conservationin the system as a validity check.

Once the smaller droplets have evaporated, the numberof equations was reduced. These coupled equations weresolved using three approaches.

Firstly, following widely used practice in CFD codes,the system of equations was divided into subsystems whichwere solved sequentially. This approach is widely referredto as the operator splitting technique (see [37]). These sub-systems include equations for mass and temperature ofeach droplet (three subsystems) and equations for gas tem-perature, fuel, oxygen and mixture density (additionalfourth subsystem). When each of these subsystems wassolved, the remaining variables were assumed to have con-stant values over the time step. This paper refers to thisapproach as ‘fixed decomposition approach’ to distinguishit from the ‘dynamic decomposition approach’ discussed inSection 2. More specifically, at the first step equations forthe mass and the temperature of each droplet (Eqs. (18)and (21)) were solved simultaneously (three systems ofequations). Then the results were used to calculate the den-sity of fuel and mixture, and temperature of gas withouttaking into account the chemical term (see Eqs. (24), (29)and (28)). Next, the chemical term was calculated basedon Eq. (26), and the result was used to calculate the densityof oxygen (Eq. (27)). Finally, the values of the density offuel and mixture, and temperature of gas were updatedusing the chemical term. This approach is effectively equiv-alent to the simplest form of (A–B) splitting as described in[37].

Secondly, these equations were solved rigorously usingDLSODAR stiff solver from ODEPACK developed inLLNL laboratory. This means that all equations weresolved simultaneously in a coupled way.

The third approach is based on decomposing of the ori-ginal system following the procedure described in Section 2with � = 0.25. Note that this parameter is not related to theparameter e used in Eqs. (10) and (13). The total number ofequations solved, and the number of equations for fastvariables could change with time as expected. The corre-sponding plots of the numbers of these equations areshown in Fig. 1. As follows from this figure, initially all10 equations were solved, when the first or secondapproaches were used. Then this number was reduced to8 when the smallest droplet evaporated, and to 6 whentwo smallest droplets evaporated. Initially, the number ofequations for fast variables to be solved was equal to 4,then it dropped to just one equation describing fuel vapourdensity. Between about 0.25 ms and 0.5 ms the number ofequations for fast variables was equal to two (equationsfor fuel vapour density and the radius of the smallest drop-let). Then again just the equation for fuel density wassolved. Between about 0.6 ms and 0.8 ms the number offast equations to be solved is comparable with the totalnumber of equations solved. During this period the decom-

Fig. 3. The same as Fig. 2 but for equivalence ratio.

Fig. 4. The largest droplet radius versus time, calculated using the firstapproach (fixed decomposition) (dashed), second approach (coupledsolution of the full system of equations) (solid), and the third approach(dynamic decomposition) (dotted). The time step of calculations is takenequal to 10�5 s.

Fig. 1. Plots of the total number of equations solved (dashed) and thenumber of equations for fast variables (solid) for the values of parametersas specified in the text. The calculations continued until the autoignitionprocess took place. The values of parameters are described in Section 4.

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position of the system is not expected to be useful. Afterabout 0.8 ms and until about 1.8 ms only one equation(fuel density) or two equations (fuel density and the radiusof the second droplet) were used. In this case the decompo-sition technique described in Section 2 is expected to beparticularly important.

The time evolution of gas temperature, equivalence ratioand the radius of the largest droplet, calculated using theabovementioned three approaches, are shown in Figs. 2–4 respectively. Fig. 2 appears to be the most informativeand practically important, as it indicates the total ignitiondelay time, i.e. the time required for the gas to reach1100 K. As follows from Fig. 2, the first approach appearsto be very sensitive towards the time step chosen. If thetime step 10�4 s is chosen then the predicted total ignitiondelay is almost four times longer than the one predictedbased on the second approach (coupled solution of thewhole system). If the time step is decreased to 5 · 10�5 sand 10�5 s then calculations using the first method appearto be more accurate than in the case when this time step isequal to 10�4, but still the accuracy of computations is

Fig. 2. Plots of gas temperature versus time, calculated using the firstapproach (fixed decomposition) (dashed), second approach (coupledsolution of the full system of equations) (solid), and the third approach(dynamic decomposition) (dotted). Plots ‘1’, ‘2’ and ‘3’ refer to calcula-tions based on the time steps 10�4 s, 5 · 10�5 s and 10�5 s, respectively.The gas volume is chosen such that if the droplets are fully evaporatedwithout combustion then the equivalence ratio of fuel vapour/air mixtureis equal to 4.

hardly acceptable for practical applications. Even for arather small time step, 10�5 s, the predicted total ignitiondelay is more than 20% greater than predicted by the rigor-ous coupled solution of this system of equations (secondapproach).

The application of the third approach to the solution ofthis system gives a rather different picture. Even in the caseof the largest time step (10�4 s) the error of calculations ofthe total ignition time delay was just 13%. In the case ofsmaller time steps, the time delay predicted by solving thedecomposed system almost coincides with the one obtainedby rigorously solving the whole system with possible errorsnot exceeding 2%. Essentially the same conclusion regard-ing the benefits of the third approach based on the decom-position of the original system of equations follows fromFigs. 3 and 4, showing the time evolution of the instanta-neous equivalence ratio and the radius of the largest drop-lets. Also, the difference of the values of other variables,predicted by the solutions of the system, using the abovementioned three approaches, was observed.

At the next step the solution of equations was performedfor a different set of parameters, typical for the peripheralregion of fuel sprays in diesel engines [1]. We consider threedroplets with initial radii 5 lm, 9 lm and 17 lm, respec-tively. The gas temperature is taken equal to 780 K [1].

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The gas volume is chosen such that if the droplets are fullyevaporated without combustion then the equivalence ratioof fuel vapour/air mixture is equal to 1. As in the previouscase, the initial density of oxygen and the initial droplettemperatures are taken equal to 2.73 kg/m3 and 400 K,respectively. The initial mass fraction of fuel is taken equalto zero. The plots of the number of equations solved usingthe abovementioned three approaches are shown in Fig. 5.Between about 0.1 ms and 0.4 ms only one equation for thetemperature of the smallest droplet was solved when thethird approach was used. In general, the decompositiontechnique described in Section 2 is expected to be particu-larly useful in the ranges 0.1–0.7 ms and 1–2.4 ms when notmore than two equations for fast variables were solved.

The plots of time evolution of gas temperature, calcu-lated using the abovementioned three approaches, areshown in Fig. 6. The calculations were performed for thesame Dt s as in the case shown in Fig. 2, but the results

Fig. 5. The same as Fig. 1 but for the radius of the largest droplet equal to17 lm, gas temperature equal to 780 K, and the gas volume is chosen suchthat if the droplets are fully evaporated without combustion then theequivalence ratio of fuel vapour/air mixture is equal to 1 (stoichiometricmixture).

Fig. 6. Plots of gas temperature versus time, calculated using the firstapproach (fixed decomposition) (dashed), second approach (coupledsolution of the full system of equations) (solid), and the third approach(dynamic decomposition) (dotted). Plots refer to calculations based on thetime step 10�5 s. The gas volume is chosen such that if the droplets arefully evaporated without combustion then the equivalence ratio of fuelvapour/air mixture is equal to 1 (stoichiometric mixture). Values of otherparameters are described in Section 4.

are shown just for Dt = 10�5 s. The dependence of theresults on Dt in the range of values of this parameter underconsideration (Dt = 10�5–10�4 s) turned out to be weakand the plots almost coincided. The comparison of thethree curves in this figure shows that the solution of thedecomposed system in this case turned out to be far moreaccurate than the solution used in CFD codes (firstapproach).

To compare the CPU efficiency of the new (dynamicdecomposition) and conventional CFD (fixed decomposi-tion) approaches, a series of runs for various time steps wereperformed for both cases considered in this section. A poly-disperse spray including three droplet parcels, 10,000 drop-lets each, was injected at the start of the calculation. Forfixed time steps the CPU requirements of both approacheswere about the same. As shown above, however, the accu-racy of the new approach was always higher than that ofthe conventional approach. Thus the comparison of CPUrequirements of both methods for fixed time steps wouldbe misleading. An alternative approach needs to take intoaccount the accuracy of calculations. As followed fromour calculations, the autoignition delays predicted by thedynamic decomposition approach coincided with those pre-dicted by the full coupled solution of the system of equa-tions for the time step of 10�6 s. This value of theautoignition delay was considered as the true value for fur-ther comparisons. For example, for the second set ofparameters errors less than 1.5% were achieved for time stepof 1.3 · 10�5 s for the conventional CFD approach, and fortime step of 2.4 · 10�5 s for the new approach. When thiseffect was taken into account then in all cases under consid-eration, the CPU time for the new method was always smal-ler than that for the conventional approach. In some cases,the CPU reduction for the new approach was as high asfactor of 3. The CPU time was estimated based on the cus-tomised function DATE_AND_TIME. The standard func-tion GETTIME did not give consistent results for smallCPU times.

A number of additional tests has been conducted tocompare the performances of the standard fix decomposi-tion approach (used in CFD codes) and the new dynamicdecomposition method. Both programs were run sequen-tially on two workstations (Silicon Graphics, Intel 64bitprocessor) and using two Fortran Compilers (Intel – For-tran 95, GNU – Fortran 77) with the additional optionof code optimisation. The abovementioned customisedfunction DATE_AND_TIME was used to estimate theCPU time required for system integration. It has beenfound that, for a given time step and for a low level of codeoptimisation, the CPU time is less for the standard fixdecomposition approach, whereas an increase in the levelof optimisation leads to comparable times required for pro-gram execution. For example, for the case shown in Figs.1–4 for the time step Dt = 5 · 10�5, the average time ofthe program execution with the code optimisation levelset to O0 (i.e. no code optimisation) was approximatelyequal to 0.12 s for the fixed decomposition approach and

V. Bykov et al. / Computers & Fluids 36 (2007) 601–610 609

0.21 s for the dynamic decomposition approach. For theoptimisation level set to O2 (this option is the defaultone, it enables optimisations for speed, including globalcode scheduling, software pipelining, prediction, specula-tion, etc.) these times become 0.15 s and 0.13 s, respec-tively. Therefore further improvements of realisation ofthe dynamic decomposition method are possible and mightbe implemented, leading to optimising the realisation struc-ture of the code and numerics.

Note that the coupled solution of the system of equa-tions using the stiff solver is always more accurate thanthe solutions based on dynamic and fixed decompositions,and is usually more CPU efficient. Hence, in the case whenthe number of equations is relatively small (as in the casesconsidered in this section) there is no need to develop anydecomposition technique at the first place. However, inrealistic engineering calculations, when the number ofdroplet parcels could be tens of thousands [1], no stiff sol-ver can cope with the full coupled system of ODEs describ-ing them. The real competition in this case is between fixedand dynamic decomposition approaches as describedabove. In our paper the coupled solution of the full systemof equations was used for estimation of the accuracy ofthese two approaches.

5. Conclusions

A new method of numerical solution of multiscale sys-tems of ordinary differential equations (ODEs) is sug-gested. This is based on a decomposition technique forsystems of ordinary differential equations, using the geo-metrical version of the Integral Manifold Method. Thecomparative analysis of the values of the right hand sidesof these equations, can result in the separation of the equa-tions for ‘fast’ and ‘slow’ variables. The hierarchy of thedecomposition is allowed to vary with time. Hence, thisdecomposition is called dynamic. Equations for fast vari-ables are solved by a stiff ODE system solver with the slowvariables taken at the beginning of the time step. This isconsidered as a zeroth order solution for these variables.The solution of equations for slow variables is presentedin a simplified form, assuming linearised variations of thesevariables during the time evolution of the fast variables.This is considered as the first order approximation forthe solution for these variables or the first approximationfor the fast manifold. This approach is applied to the anal-ysis of the problem of explosion of a polydisperse spray ofdiesel fuel for the parameters typical for the vicinity of thenozzle and the periphery of the spray. Our results showclear advantages of the new approach from the point ofview of accuracy and CPU efficiency when compared withthe conventional approach widely used in CFD codes. Thelatter is called the fixed decomposition approach. The dif-ference in the solutions of dynamically decomposed andfull systems of equations is shown to be negligibly smallfor practical applications.

Acknowledgements

The authors are grateful to the EPSRC (Grant GR/S98368/01), German-Israeli Foundation (Grant G-695-15.10/2001) and the Academic Study Group (UK) for thefinancial support of this project. This paper was preparedduring the visits of VB and VG to the University of Brigh-ton (UK), and the visits of SS and ES to Ben Gurion Uni-versity of the Negev (Israel). The authors are grateful toboth universities for providing hospitality. Our specialthanks are to Mrs. Yu. Shramkova for useful discussionsof technical details and numerics.

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