An Application of the Probability Density Function Model to Diesel

An Application of the Probability Density Function Model to Diesel Engine Combustion P.DURAND M.GOROKHOVSKI1 : CORIA / LTH - UMR 6614, CNRS-Universite de Rouen

Place Emile Blondel, 76821 Mt-St-Aignan Cedex, FRANCE

R.BORGHI : UMR 6594-CNRS, IRPHE-IMT/Universite Aix Marseille 13451 Marseille Cedex, FRANCE

Abstract A turbulent combustion model based on the probability density function (PDF) approach has been extended for the spray combustion computations under simulated diesel engine conditions. This approach accounts for the eects of turbulence and of random dynamics of evaporating droplets on the mean rate of chemistry. The paper begins with formulation of the equation for the joint PDF for the gaseous mixture variables where evaporating droplets are viewed as point sources. Then the modi ed micromixing model that involves evaporation process is described and details of a Monte-Carlo modeling of PDF-equation are given. After that, the results of numerical studies dealing with diesel spray combustion are discussed. Three dierent examples are considered. First, the computations of the evaporating spray injected into heated nitrogen atmosphere are carried out. The contribution of evaporating droplets dynamics to the mean and variance distributions of temperature and vapor concentration is demonstrated. Next, the spray combustion under light-duty and heavy-duty diesel conditions is simulated, and the results are compared with calculations using the Eddy-Break-Up combustion model, and with experimental data. It is shown that the PDF equation model is able to predict experimental data signi cantly better than the Eddy-BreakUp model. The last part of computations concerns the diesel spray autoignition governed by the strong turbulence eects. Spatial probability distributions of self-ignition sites are displayed and compared with experimental observation. It is shown that the region with high probability of ignition sites occurs at the level of dense spray and is displaced towards the nozzle hole if the inlet air temperature and pressure increase. Keywords : Spray combustion, turbulent mixing, PDF-equation method.

1 Introduction Motivation for modeling the combustion process in sprays comes, in particular, from applications related to diesel engines. In a diesel engine, the liquid fuel vaporizes, vapor is then dispersed into surrounding gas forming a mixture that can be ignited. Further, the ignition sites evolve a combustion zone close to the liquid spray that vaporizes, mixes with air, and burns. At some points of

ame the droplets disappear due to evaporation; this enhances temperature/concentrations mixing that eventually leads to the turbulent combustion. To calculate the mean rate of combustion, one needs the statistical information about the uctuations of temperature and concentrations in the gaseous mixture. At the other points, where droplets are not yet vaporized, the random positions of droplets and their turbulent dispersion are determined by the general mixing processes in the gas. Corresponding author : CORIA / LTH - UMR 6614, CNRS-Universite de Rouen, Place Emile Blondel, 76821 Mt-St-Aignan Cedex, FRANCE 1

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At these points the irregular behavior of droplets, viewed as random carriers of pure vapor, modi es the local distribution of concentration/temperature probability density and the mean local rate of chemical reactions, consequently. Here again, to compute the mean combustion rate, one needs to know the scalar probability distributions in gas but with accounting for the random mobility and properties of vaporizing droplets. In this situation, the coupling between dynamics of vaporization, mixing and chemistry are of primary importance in the spray combustion computations. It controls auto-ignition process, burning rate, mean thickness and con guration of the ame. Most of the current computer models of spray combustion suppose that the vapor issued from a drop or from a group of drops is immediately mixed up to the local mean composition, and combustion is governed either by the rate of Arrhenius kinetics with homogeneous mixture conditions (Amsden, O'Rourke and Butler ,1989), or by the rate of decay of large-scale turbulence, Spalding (1971). Borghi (1995) shows that this is valid only when the chemical reactions are slow, in such a way that a characteristic thickness of amelets within the burning spray is larger than the mean spacing between the droplets. When the chemical reactions are fast, this condition is likely to be violated. Several papers have addressed the turbulence-chemistry coupling in the diesel spray computation alongside with amelet concept. In the paper of Musculus and Rutland (1994,1995) the coherent

amelet model (Bray and Moss, 1977; Marble and Broadwell, 1977; Candel et al, 1990) was extended to diesel combustion. Here the mean combustion rate is calculated from the transport equation for density of the ame area, where eects of turbulence are involved in the formation of ame area. In the paper of Gill et al (1996) the classical amelet model with assumed -function shape of the PDF (Peters, 1984), was employed for spray combustion modeling in engines. Similar approach was used by Chang et al (1996) in the spray auto-ignition modeling. Another modern treatment of chemically reacting gaseous turbulent ows, that accounts for turbulencechemistry interaction, is often based on the probability density function (PDF) method (see rsts works of Frost, 1971; Kuznetsov, 1977; O'Brien, 1980; Pope, 1981). The main appealing feature of such application is the possibility to provide the non-linear chemical reaction terms in the closed form. However, in the PDF method, the small-scale mixing has to be modeled. Moreover, in the case of spray combustion, the micro-mixing modeling requires to take into account the distribution of vaporizing droplets. In this paper the Monte Carlo modeling of joint PDF equation for mixture species is implemented using the multidimensional computational computer code KIVA II. Evaporation is included into stochastic processes. Micromixing term itself is treated here within the modi ed coalescencedispersion model (Curl, 1963). Our previous results have been reported in Durand, Gorokhovski and Borghi (1995,1996,1997). The general objective in this work is to assess the PDF approach in the diesel-like conditions. The rst objective is to examine the PDF approach in the computations of vaporizing chemically inert spray. Experimental imaging data have been used for that purpose. The second objective in this work is to assess the developed model for burning spray. Predictions of combined KIVA II-PDF equation model will be compared with the experimental visualization of burning spray and with KIVA II results using Eddy-Break-Up model. Our third objective in this work is to employ the PDF approach in the simulation of diesel spray self-ignition governed by the strong turbulence eects. As it was shown by Baritaud et al (1994), the self-ignition in diesels does not occur at the same location for each cycle. Multiple ignition sites can be seen at the same time resulting from the local concentration and temperature mixing history. On another hand, the statistical approach in the self-ignition simulation allows to predict the spatial probability distribution of self-ignition sites at the moment when the local mean temperature is greater than a given critical

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temperature criteria. As a cumulative result, one can display the most probable ignition region. Does it match the experimental observations of self-ignition sites region ? Does it follow the same tendency which was observed in the experimental imaging ? How this modeled region is sensitive to the fuel injection velocity, turbulence level, initial temperature and pressure of air in chamber ? These questions were subject of our self-ignition simulations.

2 Joint PDF equation for reacting mixture variables in the gas Consider a reacting vapor-air ow (Re >> 1), seeded by the moving droplets. Equations that govern the trajectory and vaporizing history of each single droplet comprise the random coecients due to the random eld of velocity and temperature in the gas. It implies that the motion of droplets is random. The random behavior of evaporating droplets alter randomly the continuous phase ow eld, at least locally, if the considered point is visited at the given instant by a droplet or group of droplets. Then, each dependent variable in the gas mixture is locally uctuating due to the gas turbulence and, additionally, due to the random positions of evaporating droplets. Fluctuations are ampli ed by the chemical reactions occurring in the mixture. Because of all random processes one can represent an ensemble of local values of dependent variables at a given instant as an event in the sample space. The temporal sequence of such events can be treated as random and expressed stochastically. In principle, the latter implies an existence of joint PDF of all variables that describes statistical characteristics of smoothly varying mixture. Then a probability distribution of gas variables can be introduced and dierential equation for PDF can be stated from instantaneous governing equations in the gas mixture with vaporizing droplets. Let us demonstrate this for one random variable, such as vapor mass fraction Yv , and for droplets with neglected void. The instantaneous continuity equation for vapor mass fraction satis es : v ~ (1)  @Y @t + V
rYv = r (DrYv ) + !_ lg (1 , Yv ) + !chem where V~ is the local gas velocity, D is the mass diusion coecient, !lg is the intensity of liquid-gas mass transfer, !chem is the chemical reaction rate and  is gas mixture density. The last one is the sum of densities of the gaseous species including the vapor density. Then the continuity equation for gas mixture density supplements equation (1) : @ + r
V~  = !_ ; (2) lg @t Equation (1) describes the evolution of the vapor mass fraction in a random medium. Two types of randomness enter here : a random velocity eld and a random rate of vaporization/condensation. From this equation one can construct the unconditional PDF balance equation by the usual means (Kuznetsov and Sabel'nikov, 1990):

@ hjY = Y^ iP(Y^ ) + r hhV~ jY = Y^ iP(Y^ )i = h!_ jY = Y^ iP(Y^ ) v v v v v v lg v v v @t h i , @^ hr (DrYv ) jYv = Y^v iP(Y^v ) @ Yv

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h  i , @^ P(Y^v )h!_ lg jYv = Y^v i 1 , Y^v

@ Yv h i (3) , @^ P(Y^v )h!chem jYv = Y^v i @ Yv In (3) ^ stands for independence of random variables on space and time and brackets hi denote expected values conditioned upon the mass fraction Yv = Y^v . Note that the integration of (3) over all spectrum of Yv leads to the averaged continuity equation : @ + r
V~  = !_ lg @t The term on the left hand side of (3) represents convective transport of PDF due to conditionally averaged convection of vapor. First term on the right hand side stands for the source of probability due to events when the considered point in physical space is visited by the vaporizing droplets. Another three terms on the right hand side are associated with the evolution of PDF in phase space due to the viscous decay of vapor uctuations, vaporization and chemistry. Chemistry rate in (3) is presented in exact way. Note, that vaporization, as well, does not give any problem for computation. It can be computed from the mass balance at the surface of the droplets visiting a given control volume. Motion of droplets obeys usually the spray equation (Williams, 1958; O'Rourke, 1981) modeled by the Lagrangian Monte Carlo procedure (Dukowicz, 1980) with eects of droplets collision (O'Rourke, 1981) and their breakup (O'Rourke and Amsden, 1987; Reitz, 1987; Ibrahim et al, 1993). At the same time, the micromixing must be modeled. For the random mixture variable, being composed from the gaseous species mass fractions Y1;


; YM and the gas enthalpy values h, one can introduce the unconditional probability distribution. PDFequation is written similarly to (3) :

@ hiP Y^ ; :::; Y^ ; h^  + r hhV~ iP Y^ ; :::; Y^ ; ^hi = h!_ iP Y^ ; :::; Y^ ; h^  1 M 1 M lg 1 M @t M @ h  i X ^1; :::; Y^M ; ^h , ( hr ( D r Y ) i , h  ! _ i Y + h  ! _ i  ) P Y i lg i lg iv i=1 @ Y^i h  i , @^ (hr (Drh)i + hq_lg i) P Y^1; :::; Y^M ; ^h @h M @  X ^1; :::; Y^M ; ^h , h ! i P Y chem i=1 @ Y^i Here iv is the Kronecker symbol (equal to one only for vapor specie, i = v), q_lg is the enthalpy source (or sink) due to the droplets. It is zero if the heat exchange from the gas to the droplet serves only vaporization of a given mass of liquid and it is negative when the gas heats the droplet. Similarly to the vaporization/condensation rate, it can be calculated, with usual assumptions, as a function of the droplet diameter, temperature and velocity, and it is zero only if T = Ts, where Ts, the surface temperature, has to be calculated from the droplet energy balance or h = hs (Ts) (the enthalpy of the gaseous mixture at the surface of the droplet). Note that in the equation for joint PDF the brakets h i denote again the conditional mean hjY^1 ;


; Y^M ; h^ i.

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3 Micromixing modeling Numerous models have been proposed, Curl (1963); Dopazo and O'Brien (1974); Pope (1982); Kerstein (1988); Chen et al (1989); Vali}no and Dopazo (1994); Fox (1994,1995); Burluka et al (1997) to describe the scalar mixing process in the presence of the small-scale turbulent velocity eld. One of the well-known micromixing model is the classical Curl model (Curl, 1963) that is employed in this paper. This model does not correspond to the results of direct numerical simulation of micromixing (Erswaran and Pope, 1988), and it lacks the physical basis, but it is relatively simple model representing the viscous decay of scalar uctuations as a pairwise interaction between statistical particles (two randomly chosen particles mix with a frequency given by turbulent time scale, their scalar values approaching the mean scalar value of the pair). In the case of the micromixing in sprays, the more insight into the micromixing can be obtained if the Curl model is modi ed in the "quasilaminar" vicinity close to the vaporizing droplets. Indeed, the mixing between neighbouring isosurfaces in this vicinity is more probable than their random coalescence-redispersion rearrangement within all spectrum of concentrations. Consider the motion of vapor issued from a group of droplets in the turbulent eld. Assume that this droplet cluster is reproduced by one drop releasing the same quantity of vapor as the group of droplets with eective surface vapor mass fraction Yvs that varies in time randomly to represent the collective eect of droplet interaction. Let a laminar air/vapor mixing be performed at the given instant at the very small scales around the drop. At the length scales of Kolmogorov size  the vapor isolines are distorted due to small eddies of order of . Further, greater gas turbulent scales are involved in the turbulent mixing with increasing distance from the drop. Therefore, in this paper, the Curl model ("turbulent" part of micromixing) is used for the vapor mass fractions that dier from one of saturated vapor Y^v < Yvs , while a "quasilaminar" part of micromixing is postulated for accounting of the laminar mass transfer for the range of concentrations close to Y^v = Yvs . It was shown by Annamalai and Ryan (1992) that the time required for the heat wave to penetrate the cloud of droplets (here they are supposed to ll a volume of length scale of Kolmogorov size ) is larger than the cloud evaporation time scale. Then the quasi-steady evaporation can be assumed around the drops in the micromixing modeling. The "quasilaminar" part of micromixing is proposed as follows. At a given moment, the instantaneous vapor distribution around the drop obeys the laminar steady-state solution known from Spalding (1955). At the next moment, the vapor distribution is modi ed by the random rearrangement of the drop, but it is still obeying the laminar steady-state solution and etc . . . Then, in statistical sense, we can present this rearrangement of vapor distributions by the PDF constructed directly from the laminar steady-state solution. Statistical particles sampled from this PDF (with mass conservation satis ed) will introduce the result of the "laminar" part of mixing around the drop. Further, ensemble of all statistical particles is involved in the usual Curl interaction to represent the "turbulent" part of mixing. Note that similar idea to introduce the "quasilaminar" part in the micromixing modeling was used already by Frost (1973) and Kuznetsov (1992) for the vicinity of laminar ame front in the turbulent reacting ow. The steady-state solution for evaporating droplet is (Spalding, 1955) : ! ! lg ; (4) Yv = 1 , (1 , Yv )exp , 4 Dr g where Yv is the vapor mass fraction in surrounding gas mixture and r is radial distance. For small values of r we can modify solution (4) with replacing r by r0 +
r, where r0 is radius of eective drop and
r is small : 1

1

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!

!

(r , r0) ; Yv = 1 , (1 , Yv1 ) exp , 4!lgDr exp !4lgDr (5) 2 g 0 0 The derivation from (5) gives : dYv = !lg (Yv , 1) (6) dr 4Dr02 v One can see that in the rst approximation dY dr is linearly dependent on Yv in the considered small range r 2 [r0; r0 +
r]. From the geometric interpretation of the scalar PDF (Kuznetsov and Sabel'nikov, 1981), it is known that   ^v !,1 d Y ^ P Yv  dr or with (6) it takes a form :

  (7) P Yvs  Y^v  Yv  ^1 Yv The question now is to de ne the lower boundary in the range of concentration Yv  Y^v  Yvs where the probability distribution (7) can be applied. This value can be calculated from (5) at the distance of order of Kolmogorov size , estimated from the theory of homogeneous turbulence :

 =  3=4",1=4 ; where " comes from k-" model.

4 KIVA II computational model The KIVA II code (Amsden, O'Rourke and Butler, 1989), is written for the numerical computation of transient two and three dimensional chemically reactive uid turbulent ows with sprays. This model incorporates the governing averaged equations for the gas phase mixture and the stochastic particle method for computation of evaporating liquid sprays, including the eects of droplet dispersion, collision and aerodynamic breakup. The interactions between the spray droplets and the gas phase are also accounted for. Turbulence is modeled using the standard k-" turbulence model equations. One step global reaction was applied in all computations with kinetic constants given by Westbrook and Dryer (1981). KIVA II general model and numerical solution procedure are discussed in detail by Amsden, O'Rourke and Butler (1989); it is often used in diesel engines computations and we will not discuss it here. It is only worth to note that alongside with the numerical integration of KIVA II averaged equations, the PDF-equation modeling supplies each cell of nite dierence mesh with the probability density distributions of scalar values that allows us to calculate the local mean rate of chemical reactions.

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5 Monte Carlo modeling of PDF equation Before the practical use of PDF equation, all conditional means in this equation have to be evaluated. The method of evaluation is well-known and has been found suciently accurate. This approach was proposed rst by Pope (1981,1982). Let us demonstrate it for a random variable Xv , composed from pair of random values Yv and h. In each nite-dierence cell the PDF for turbulent mixture is considered to be composed of N statistical particles. The ensemble average probability corresponding to the variable Xv (X^v < Xv < X^v + dX^v ) is de ned as a ratio of ensemble averaged particle nXv associated with value Xv to the number N of the particles (N is constant for each cell) : P (Xv )
Xv = nNXv Averaged and variance values of Xv are de ned as : Z1

Xv = Xv P (Xv ) dXv = 0

class X

N X Xvi nNi = N1 Xvi i=1 i=1

Z1

N  2 X Xv , Xv i Xv = Xv2P (Xv ) dXv = N1 0

2

i=1

0

Ensembles of statistical particles are modi ed at each time step as a result of the stochastic simulation of nite-dierence schemes associated with PDF convection-diusion transport computations and micromixing-vaporization-chemistry terms.

5.1 Monte Carlo computation of PDF transport

Incorporation of Monte Carlo modeling of PDF convection and diusion in the KIVA II codes follows ideas of Pope (1981) . Convective transport is caused in KIVA code by the motion of the lagrangian cell of mesh relative to the state of uid computed at the previous time moment. This is done by calculating the volume V swept out by each cell face as it rezones from its lagrangian position to its position given by the nodes distribution in eulerian mesh. The product of V and the volume density of transport variable then gives the amount of this variable transported between two cells common to face . The pure donor cell dierencing gives for PDF convective transport over the face  : Pea?(Xv ) = Pea(Xv ) + VV  Ped(Xv ) where Pea(Xv ) and Ped (Xv ) are the values of Favre PDF in acceptor and donor cells respectively, V is the volume of cell,  denotes the face of acceptor cell common with donor cell, and ? denotes an intermediate value. The coecient = VV  represents an eective Courant number based on uid speed relative to the mesh. Monte-Carlo simulation of pure donor cell dierencing consists in the selection at random of Nc = int( N ) statistical particles from the donor cell to replace the same number of particles in the

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acceptor cell (selected at random as well). The total contribution of convection is obtained when the sampling is made over all faces of the acceptor cell. Schematic description of convective transport of statistical particles is given below. Sch1 The spatial nite-dierencing for diusion based in KIVA codes on the geometrical calculation of the surface A~  of the face , which is the common face of two neighbouring cells. It is applied then to the diusion ux crossing face , where A~  is expressed in its parametrical form A~  = a(~r , r~). The diusion operator for PDF diusion can be written as:  ?  n (Dturb )n a
t  Pe = Pe + rP V or Pea? = Pean (1 , F) + F Pedn ; where Pea (Xv ) and Ped (Xv ) are acceptor and donor cells respectively, and F = (DturbVa
t) represents an eective Fourier number,  is the mean gas density. This is simulated by random selection of Nd statistical particles (Nd = int(FN )) from the donor cell, and random replacing of the same number of particles in the acceptor cell. Schematic description of PDF diusion transport is shown below. Sch2 Explicit Monte-Carlo diusion simulation is adapted to the implicit KIVA diusion numerical scheme by subcycling of sampling procedure. Namely, the sampling is made nsub times, each one for the time step
tsub : # " ( D turb a
tsub ) :N Nd = int V  
t where the number of diusive subcycles is nsub = int
t and
tsub is calculated from the overall sub diusive stability restriction : !,1 1 1 1 1
tsub  2D + + turb
x2
y 2
z 2

5.2 Monte Carlo representation of micromixing and vaporization terms

The evolution of scalars due to evaporation and micromixing is modeled here as follows : consider an ensemble of N particles that represents statistical composition of gas mixture in the mesh control volume. For every evaporation step, an additional ensemble of Ninj particles is injected such that they possess in sum the mass evaporated during the time step !lg
t in the control volume. To each injected statistical particles we ascribe a pair of Yvs and hs values calculated from the mass-and-heat balance relations at the surface of drops. Further (N + Ninj )-ensemble of particles participate in pairwise Curl interaction with mixing frequency calculated by k-" model. Finally, a new obtained vapor distribution function for (N + Ninj )-ensemble is reconstructed for N -ensemble of particles to identify the statistical properties of gas phase for each mesh cell. Schematic description of micromixing modeling is shown below. Sch3 Examples of distributions resulting from pure micromixing modeling are demonstrated in Fig.1-3 for normalized variable Yv = YYv and for initial mean equivalence ratio taken as 0.2, 1 and 5. For vs

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that, an isolated zero-dimentional box with evaporating drops in turbulent media is considered. The mixing time is taken as four times of evaporation time scale te calculated from d2 law. Fig1 As can be seen, the distributions are very dierent from the Dirac -function that would corre- Fig2 spond to the perfectly mixed state. Starting from one -function at Yv = 0 (extremities correspond Fig3 to the pure air side Yv = 0 and the pure vapor at the surface side Yv = 1 ), the form of PDF goes to the cliped at the extremities distributed shape with displacement of mean value Yv according to the evaporated mass and the initial equivalence ratio.

5.3 Modi ed micromixing modeling

Modi cation of micromixing model concerns the statistical contribution of mixing appearing in the near eld around the drops. We prescribe the value sampled from the PDF distribution (7) to each statistical particle that compose the Ninj -ensemble. Further all statistical particles participate in the pair-exchange Curl model. Technique of sampling from (7) is involved as follows : let be Y^v 2 [Yvs ; Yva] a random variable corresponding to the given distribution function. Then the ratio YRv   P Y^v dY^v Yvs YRva   P Y^v dY^v Yvs

= rnd 2 [0; 1]

(8)

is uniformly distributed in the range of random numbers from zero to one. Expression (8) can be rewritten using (7) : !rnd ^ Y vs Y^v = Y^va ^ (9) Yva The shapes of P (Y^v) in the homogeneous spray using modi ed micromixing modeling are presented on Fig. 4-6. Fig4 It is seen that the mixed scalar eld lies in the vicinity of Yvs -values, as opposed to the computed Fig5 -peak distributions presented on Fig 1-3. This implies a more adequate modeling in the case of fast Fig6 reactions close to the drops surface.

6 Results and discussion

6.1 Evaporating spray computation

A numerical experiment was conducted with n-tridecane spray that is injected into nitrogen gas at 830 K and 3.4 MPa. The initial conditions were selected to be quiescent to remove any uncertainties associated with uid motion. Computations were performed to simulate constant volume bomb experiments of Hiroyasu and Kadota (1974). Sch4 , 1 The injection velocity was held constant at 200ms . The nozzle hole diameter was 160m and the fuel amount to be injected during 2 ms was 12 mg. Ensemble of 1000 statistical particles in each cell was used for PDF simulations. Augmentation of number of nodes in the computational mesh did not change the PDF-distributions. Several meshes were used to analyse consequences of the mesh eect known for Diesel spray computation (Gosman and Clerides, 1997). Finally computations were

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done with 30x50 nodes. Numerical testing with dierent number of statistical particles showed that 1000 particles is sucient number for PDF computations. In the calculation with more than 1000 particles the form of PDF does not change much. Two examples of the mean and the variance of vapor mass fraction distributions are shown in Fig.7 and Fig.8 at 2ms and 3.6ms. The islands (of order of several millimeters) of intermittent intensity of the vapor mass fraction variance can be seen in the region where turbulent evaporation occurs. Distributions show high values of the vapor mean and variance at the spray tip, with steep gradients in upstream direction. Such an inhomogeneous structure of fuel vapor eld was observed by Carabell and Farrell (1994); Hosaka and Kamimoto (1993); Yeh it et al (1993); Kamimoto (1994). The evolution in time of typical probabilistic distributions of vapor are shown in Fig.9 at the given axial point, located in 2 cm below the injector ori ce. These PDF's display a distribution during the time when the vaporizing droplets ll up the control volume and turbulent vapor-air mixing occurs. Further, the cloud of droplets goes down to the bottom side, hot gases enter the control volume and PDF's show a sharp shape, with the low amount of vapor and high temperature. In Fig.10 and Fig.11 the mean and variance temperature distribution are shown at 2.0ms and 3.6ms. Zones containing the evaporating droplets display the region of temperature uctuations. Note, that the level of uctuations caused by turbulent evaporation is computed of maximum 10% from the mean value. Two next gures Fig 12 and Fig 13 concern the comparisons with experimental imaging of Kosaka and Kamimoto (1997). Distributions of mean and variance of vapor concentration are given for two dierent injection pressures : 55 MPa with injection time 3.7 ms and 110 MPa with injection time 2.2 ms. The pressure in the bomb was 2.9 MPa. It was found in the experiment that the maximal ratio of concentration uctuation to its mean value Cf0 =Cf that can be achieved is 0.97 for the case of 55 MPa and 0.88 for the case of 110 MPa. The comparison shows satisfactory agreement. Maximal computed value Cf0 =Cf is 0.88 for 55 MPa and 0.82 for 110 MPa.

6.2 Spray combustion computations

Numerical simulations concerning the evolution of diesel spray in high-temperature, high-pressure, nearly quiescent air environment were performed with initial conditions taken from the model experiment of Ragucci et al (1994). In particular, experimental data of Ragucci et al (1994) have been obtained for a diesel spray injected into air at 4 MPa and 900 K with two dierent injection conditions. The injection periods were 0.65 and 1.2 ms corresponding to the injected fuel quantities of 1.6 an 6.0 mg. The experimental injection pressure pro les are taken from Cavaliere and Ragucci (1993) where experiment of Ragucci et al (1994), for light-duty conditions is discussed in details. Patterns of fuel-like particles scattering (starting from injection and up to 3.5 ms) are shown on the upper picture of Fig.14 and Fig.15 for both (lower and higher) fuel injection rate conditions. The solid and dashed lines on these pictures represent ensemble-averaged contours of fuel-like particles scattering and OH-emissions patterns, respectively.

Fig7 Fig8

Fig9 Fig10 Fig11

Fig12 Fig13

Fig14 The middle part of Fig.14 and Fig.15 display the calculated liquid droplets positions, the most Fig15 probable ignition region, and a sequence of mean temperature distributions, with a time step of 0.5 ms as it was done in the experiment. Presence of fuel is showed by solid line taken from the calculated fuel mean distributions (the lower boundary is supposed to be de ned by vapor mass fraction isoline corresponding to Yv = 0:01). Besides, the contour of temperatures higher than 1800K is given on the

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gures (dotted line). It was assumed here that this zone can be compared with the mean contour of experimental OH-emission patterns. Hot ignition was de ned when the temperature exceeded 1100K or had a rise rate greater than 107 K/s. At this moment the spatial probability distribution of ignited statistical particles was calculated to supplement the picture of auto-ignition. These probabilistic distributions (identi ed here as a probability distribution of self-ignition sites) are demonstrated on Fig.14 and Fig.15 for the above injection conditions. As can be seen, the numerical results represent qualitatively the features observed in the experiment. The auto-ignition spots are attached to the lower part of liquid fuel, not far from the nozzle hole. Ignition occurs at the end of injection period, or shortly thereafter, when a part of liquid fuel is already evaporated and mixed. Auto-ignition sites radiply fuse in a combustion zone that lls entirely the fuel region and removes slightly from the nozzle ori ce. Ignition delay, and contours that de ne averaged domains of presence of the fuel and of the high temperature zone due to combustion, are predicted in qualitative agreement with experimental data. It is mentioned by Ragucci et al (1994), and it can be seen from calculations, that the domain of signi cant fuel compsumption is situated at the bottom part of spray. The bottom parts of Fig.14 and Fig.15 correspond to numerical KIVA II results with the Eddy-Break-Up model modi ed by Magnussen and Hyertager (1977). As previously, the fuel-rich domain and the hot zone contour are noted by solid and dotted lines, respectively. Kinetic coecients and activation temperatures are taken the same as in computations with PDF. The A and B coecients in the Magnussen model are taken as 20 and 2.5 correspondingly, Pinchon (1989). The use of A and B values that were recommended by Magnussen and Hyertager (1977) did not lead to the autoignition and combustion in these spray computations. It is seen that, while giving longer ignition delay, the Eddy-Break-Up model yields a considerably overestimated turbulent heat release, and the spatial distributions dier from experimental visualization. Simulations presented on Fig.14 and Fig.15 are examined in the case reere to above as light-duty diesel conditions. The combustion zone covers the fuel-rich domain without visible ame front, which is typical for turbulent diusion ames. Example of computations related to the heavy-duty disel conditions are given on Fig.16. Parameters of the constant volume bomb as well as injection conditions have been taken from Baritaud (1993). In this work 12 mg of dodecane were injected during 3.0 ms at the injection velocity associated with the injection pressure of 35 MPa in the air environment at 1000 K and 2.9 MPa. Three pictures of temperatures corresponding to the time moment 5.0 ms after the start of injection (2.4 ms after the auto-ignition) are given on Fig.16 : mean temperature distribution obtained with the Eddy-Break-Up model, the mean and the variance temperature distributions computed by PDF method. Fig16 It is seen that distributions of mean temperature, calculated from the Eddy-Break-Up model, demonstrate a thick high-temperature zone. Instead, the PDF simulation yields more narrow domain, that is occuped by the high mean and variance values of temperature. It can indicate the position and the thickness of ame front.

6.3 Auto-ignition sites simulation

A set of numerical calculations have been carried out in the connection with experimental visualisation (Baritaud et al, 1994; Baritaud, 1993) of auto-ignition sites in the spray of n-heptane, injected from the single-hole nozzle into the combustion chamber. Experimental conditions involved changes of air temperature, the pressure in the cylinder, and the injected quantity. Due to space limitations in the present volume, we focus our attention on the self-ignition sites location. On the upper part of Fig.17, the experimental patterns of self-ignition sites collected from each cycle are presented for

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three dierent air inlet temperatures - 713, 800 and 900 K- and air pressure of 4.6, 5.0 and 5.6 MPa, respectively. On the bottom picture in Fig.17 the calculated spatial probability distribution of ignited particles are given at the moment of mean auto-ignition.

Fig17

One can see the rather good qualitative agreement between numerical results and experimental observations. Calculated most probable self-ignition region occurs at the level of dense liquid spray, its location is not far from the experimental one; ignition sites are more concentrated in space if the inlet air temperature increases with displacement of auto-ignition zone towards the nozzle hole. Fig.18 demonstrates the in uence of the inlet bomb pressure on the auto-ignition sites distributions. The inlet temperature 800 K and the inlet bomb pressures 3.8, 5.0 and 6.4 MPa are taken from Baritaud et al (1994). Here again, the upper part of gure corresponds to the experimental results, while the lower part shows numerical predictions. It is seen that the ignition sites are more concentrated in space as inlet pressure increases and auto-ignition zone draws near nozzle hole. Fig18

7 Conclusion A turbulent combustion model based on the probability density function (PDF) approach has been extented for the spray combustion computations under simulated diesel engine conditions. This approach accounts for the eects of turbulence and of the random dynamics of vaporizing liquid droplets on the mean rate of chemistry. An equation for the joint PDF for a species in the gas mixture is formulated where vaporizing droplets are viewed as point sources. New modi cation of micromixing is proposed to account for an additional mixing eect occurring in the close vicinity of vaporizing droplets. This model was, rst, demonstrated in the case of one isolated zero-dimensional box with evaporation of droplets in turbulent media. The combustion process including spray dynamics, evaporation, mixing and chemistry, as well as local PDF distributions, have been treated using the combined KIVA II-PDF equation model. The Monte Carlo method is employed to model the evolution of the joint PDF equation. Results of numerical studies dealing with diesel spray combustion are discussed. Three dierent examples are considered. First, the computation of the vaporizing spray injected into heated nitrogen atmosphere is carried out. The contribution of vaporizing droplets to the mean and variance distributions of temperature and vapor concentration is demonstrated. The calculated inhomogeneous fuel vapor eld showed the islands (of order of several millimeters) of intermittent intensity of vapor mass fraction variance in the region where turbulent evaporation occurs. The computed maximal ratio of uctuation of vapor concentration to its mean value was close to the mesured one. Then, the spray combustion under light-duty and heavy-duty diesel conditions is simulated, and the results are compared with calculations using the Eddy-Break-Up combustion model, and with the experimental data. It is shown that the PDF equation model is able to predict experimental data signi cantly better than the Eddy-Break-Up model. In fact, giving longer ignition delay, the Eddy-Break-Up model leads to the considerably faster rate of fuel compsumption ; instead, the PDF simulations yield improved combustion computations : contours that de ne averaged domains occuped by the fuel-like particles and by the high-temperature zone are predicted in a qualitative agreement with experimental data. The last part of computations concern the diesel spray auto-ignition governed by the strong turbulence eects. The spatial probability distributions of self-ignited particles were displayed when the mean temperature exceeded 1100 K or had a rise rate greater than 107 K/s. These distributions are to be identi ed with the probability distribution of self-ignition sites. They are compared with

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experimental observations from each cycle. Comparisons are given for dierent air inlet temperatures and pressures and show a good qualitative agreement between numerical and experimental results. The calculated most probable self-ignition region occurs at the level of the dense liquid spray, its location is not far from experimental one; the ignition sites are more concentrated in space if the inlet air temperature and pressure increase with displacement of the auto-ignition zone towards the nozzle hole. The model proposed has at least two weak points. Corrective factors accounting for the convective eect on heat and mass transfer in the close vicinity of vaporizing droplets must be introduced as an extra parameter in the micromixing modeling. That warrants an additional examination of the micromixing model. Another weak point is that all computations are performed with one-step global reaction, and the comparison of the computed high-temperature zone (T>1800 K) with the measured OH-emission domain is only tentative. Future computations can be supplemented by more detailled chemical mechanism.

Acknoledgements

The present study was supported by the French National Center of Spatial Study CNES and by the European Society of Propulsion SEP within the GdR "Combustion in Rocket Engine". It was also supported by the Region Haute-Normandie. The authors thank Prof. V.A. Sabelnikov and Prof. G. Ryskin for their helpful remarks made after reading of the manuscript.

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