Simulations of spark ignition of a swirling n-heptane spray flame with conditional moment closure

48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition 4-7 January 2010, Orlando, Florida AIAA 2010-614 Simulations of spark ignition of a swirling n-heptane spray flame with conditional moment closure P. Schroll, E. Mastorakos Engineering Department, University of Cambridge, Cambridge, CB2 1PZ, UK R. W. Bilger School of Aerospace, Mechanical and Mechatronic Engineering, University of Sydney, Australia Conditional Moment Closure (CMC) extended to two-phase flows and interfaced with a RANS two-phase flow solver has been used to study the ignition process of an n-heptane spray flame in a flow typical of a liquid-fueled burner. The aim was to investigate the influence of the additional source terms due to the droplet evaporation process in the conditional species and energy equation. The droplet source terms in the mean and variance of the mixture fraction equation were also modelled. For comparison, simulations without droplet source terms in the CMC equations were also carried out. The air velocity field and droplet size were compared with experimental data and good agreement was found. The flame expansion process following ignition compared qualitatively with fast-camera images from the experiment and the overall flame shape and speed were captured satisfactorily. Comparisons of conditional temperature at different locations in the flow field gives further insight on how the additional droplet evaporation source terms affect the flow field. I. Introduction Many industrial applications, such as conventional diesel or homogeneous charge compression ignition (HCCI) engines, aircraft gas turbines and liquid fuelled furnaces, involve spray combustion. The interactions between droplet evaporation, turbulence and chemical reactions make spray combustion a complex field for simulations. For single-phase flows, advanced modeling based on conditional moment closure, developed by Bilger and Klimenko, 1 has showed very good results. CMC was successfully used for autoignition problems, 2 4 attached turbulent flames 5, 6 and lifted flames. 7 9 The extension of the CMC model to two-phase flow problems allows a very effective simulation tool for spray combustion with complex geometries including complex chemistry. There are several applications of the CMC method to spray combustion in the literature. For Diesel engine spray combustion the CMC method was used 10 with the approach of neglecting all source terms due to evaporation except in the mean mixture fraction equation. In an attempt to account for all droplet source terms, various studies derived the conditional species and temperature equations, 11 13 but leading to different versions that are in certain details inconsistent to each other. For the droplet source terms in the mixture fraction variance equation different closure models are available in literature. 14 17 In this work, the rigorous approach by Mortensen and Bilger 18 for the conditional moment closure equations extended to twophase flows is used for spray flame ignition, which is a situation of great relevance to aviation engines. The configuration studied is typical of a liquid-fueled burner, with a spray from a hollow-cone atomiser placed in the centre of a bluff body in swirling air flow. Ignition is achieved by an electrical spark and the experimental work focuses on the ignition process. 19 This configuration has been studied very extensively 20, 21 in regard to the interaction of evaporation, turbulence and ignition effects and is very relevant to gas turbine combustion. Spark ignition phenomena in gaseous and spray turbulent flows have been reviewed recently by Mastorakos. 22 PhD student. Professor of Energy Technologies. Senior Member AIAA. Email: em275@eng.cam.ac.uk. Emeritus Professor. 1 of 12 Copyright 2010 by the, Inc. All rights reserved.

The objectives of this paper are (i) to investigate the impact of the additional evaporation source terms in the CMC equations; (ii) to compare the simulations with experimental data. II. Formulation II.A. CMC governing equations The conditional mean of the mass fraction Y α of species α and the temperature T, are defined by Q α (η, x, t) = Y α (x, t) ξ(x, t) = η (1) Q T (η, x, t) = T (x, t) ξ(x, t) = η (2) where η is the sample space variable for the mixture fraction ξ. The expression f a = b denotes an ensemble averaging over all cases that fulfill the condition on the right of the vertical bar. The derivation of the CMC 1, 23 equations for two-phase flows can be derived in a similar manner as for single-phase flows. Recently Mortensen and Bilger 18 presented a derivation of the CMC equations for two-phase flows based on a twofluid formulation. In the present paper, their CMC equations are used under the following assumptions: (i) dilute spray (i.e. volume fraction of liquid phase 0); (ii) Le = 1 for all species; (iii) Y S η = 0. The validity of these assumption for the present problem will be discussed in the full paper. Then, the simplified equations become: Q α t + u j η Q α = 1 x j ρ P (η) [ ρ x P ] (η) u j Y α η j + N η 2 Q α η 2 + W α η + S η S η Q α (1 η) S η Q α η (3) and Q T t + u j η Q T = 1 x j ρ P (η) [ + N η + 1 c pη [ ρ x P ] (η) u j T η j ( 1 n cpη c pη η + α=1 + 1 W H η c pη 1 p ρ t η c p,αη Q α η ) Q T η + 2 Q T η 2 ] h fg S η Q T S η S η (1 η) Q T c pη η (4) for the conditional mass fraction and temperature, respectively. In Eq. (3) ρ is the mean density, p the pressure, u j the velocity in the j direction and W α the production rate of species α due to chemical reaction. In equation (4), W H is the heat release rate, h fg is the latent heat of vaporization and c pη denotes the conditional heat capacity. In both equations the last three terms on the r.h.s. are due to the droplet evaporation process emerging in the derivation for equation (3) from the continuity, the mass fraction and mixture fraction equation and, for Eq. (4), from the continuity, temperature and mixture fraction equation. The pressure work term will be important for internal combustion engines where compression or expansion occur and it may not be negligible during very quick ignition transients in ignition problems. Here, this term will be neglected as the ignition process we are dealing with is relatively slow (compared to the speed of sound) and occurs in an burner open to the atmosphere. The conditional turbulent fluxes can be modelled by analogy with the diffusion approximation used for the unconditional scalar flux u j Y α Q α η = D t (5) x j 2 of 12

where D t = µ t /Sc t is the turbulent diffusivity with µ t and Sc t the turbulent viscosity and turbulent Schmidt number, respectively. The conditional expectation of the velocity can be closed by a linear approximation 1 with u j η = ũ j + u j ξ u j ξ ξ 2 (η ξ) (6) = D t ξ x j (7) The conditional scalar dissipation rate is modelled by the Amplitude Mapping Closure (AMC), 24 giving with N η = N 0 G(η) (8) G(η) = exp ( 2[erf 1 (2η 1)] 2) (9) N 0 = χ 2 1 0 G(η) P (η)dη where χ is the mean scalar dissipation rate computed from the mean turbulence quantities of the flow field. First order closure is performed for the chemical source terms (10) W α η = W α (Q α, Q T, p) (11) W H η = n h α W α η (12) α=1 In Eqs. (3) and (4) P (η) is the probability density function of η. To find P (η), a β function pdf is presumed, which needs the mean and the variance of mixture fraction, ξ and ξ 2 respectively. Both variables are influenced by the evaporation of the droplets in two-phase flows. The mean mixture fraction transport equation can be written in conserved form as ρ ξ t + ρũ j ξ x j = x j [ ρ(d + D t ) ξ ] + ρ x S (13) j The transport equation for ξ 2 has been derived by various authors. 15, 16, 18, 25 Using the transport equations for ξ 2 and ξ 2 and using ρ ξ 2 ρ ξ 2 = ρ ξ 2 (14) t t t leads to [ ] ρ ξ 2 + ρũ j ξ 2 = ξ 2 ρ(d + D t ) ρ χ + 2ρD ξ ξ t + ρs ξ (15) t x j x j x j x j x j with ) S ξ = 2 ( ξs ξ S ( ξ2 S ξ ) 2 S (16) The last terms on the r.h.s. in Eqs. (13) and (15) are due to the droplet evaporation process. The mean scalar dissipation rate χ that is needed in Eqs. (15) and (10) is modelled by χ = C d ɛ k ξ 2 (17) with the constant C d given the usual value of 2. The turbulent kinetic energy k and its dissipation rate ɛ come from the turbulence model, discussed later. 3 of 12

II.B. Spray model In the above equations, S is the evaporation source term. In the present paper, the spray is modelled using a conventional Lagrangian approach, in which representative droplet parcels are tracked in the flow using a stochastic component of the drag force in order to simulate turbulent dispersion. The RANS CFD solver STAR-CD 26 has been used with standard Lagrangian tracking for the position, velocity, temperature and size of the droplet. In the context of the Lagrangian spray approach, the mean evaporation rate at a point can be described by summing the evaporation rates of each droplet present in a particular computational cell: ρ S = 1 N ṁ i (18) V where V and ṁ i denote the cell volume where the droplets are located and the evaporation rate of the i-th droplet, respectively. The present spray model implies that the droplet surface is at the saturation fuel mass fraction corresponding to the droplet temperature. Neglecting, for the time being, the difference between mixture fraction and fuel mass fraction, the mixture fraction at the droplet surface will be ξ s, where the subscript s denotes saturation. It is conceivable that every droplet in volume V will have a different ξ s. The 16, 25 correlations between ξ and S in Eq. (17) can hence be expressed as which in the usual context of the spray trajectory approach become: i=1 ξs = ξ s S (19) ξ 2 S = ξ 2 ss (20) ρ ξ S S = 1 V N ξ s,i ṁ i (21) i=1 ρ ξ 2 S S = 1 V N ξs,iṁ 2 i (22) This physical picture of the evaporation process is in contrast to models 15, 17 that suggest a conditional source S η to be proportional to the value of ξ cell, which is the mixture fraction value of the computational cell where the droplet is located. Recent findings from Direct Numerical Simulations 27 showed that ξ s, conditionally-averaged over ξ cell was virtually independent from ξ cell, leading to the conclusion that a given value of ξ cell does not imply a corresponding evaporation rate (such a dependence may come from the fact that the evaporation rate is affected by the vapour present already in the droplet surroundings but this dependence is of opposite sign to the previously suggested models for S η 15, 17 ). As suggested by Mortensen and Bilger, 18 DNS studies with high resolution close to the droplet surface are needed for further investigations of the mixture fraction distributions in the inter-droplet space. Setting the fuel mass fraction at the droplet surface (YF s ), which is used in the usual Lagrangian trajectory approach and calculated from the Clausius-Clapeyron equation for each droplet, equal to the mixture fraction at the droplet surface (ξ s ), which appears in Eqs. (19), is identically correct for the case of inert flow (i.e. before spark ignition) and approximately correct for autoignition problems in the pre-ignition phase (due to the negligible consumption of reactants before ignition). For flames, the assumption that YF s ξ s will be quite bad around stoichiometry (ξ st = 0.062 for heptane), but the difference between the two is diminishing as the mixture fraction increases. YF s tends to be large for volatile fuels at high temperatures: for heptane at ambient conditions, YF s = 0.18 and Y F s will be increasing as the droplets are exposed to flame temperatures and hence their temperature approach boiling point. Therefore, we expect the models provided here for the correlations between ξ and S to be approximately valid for flame problems, with the degree of approximation increasing for ignition problems. II.C. Numerical methods A schematic of the coupling between the CFD and the CMC solver can be seen in Figure 1. The flow is solved by STAR-CD using a block structured grid, using a RANS formulation and the k ɛ model for low Reynolds numbers. The transport equations for the mean and variance of mixture fraction (Eqs. 13, 15) are calculated i=1 4 of 12

Figure 1. Schematic of the STAR-CD CMC coupling and the droplet evaporation subroutine. in STAR-CD with the droplet source terms supplied by a user subroutine. The built-in Lagrangian-Eulerian formulation between liquid and gas phase is used with the Reitz-Diwakar model 28 for droplet atomization. The thermo-physical droplet properties are supplied by a user subroutine. The species mass fraction and temperature equations are not solved in STAR-CD, as they are computed by the CMC solver. The species mass fractions and temperature returned to STAR-CD are obtained in the CMC solver by integrating the conditional averages over P (η). Therefore transport equations for ξ and ξ 2 (Eqs. 13,15) have to be solved by STAR-CD to establish a presumed PDF of mixture fraction. The droplet evaporation source terms in Eqs. (13 and 15) are supplied by a user subroutine which loops for each time step over all droplet parcels and creates the droplet source terms, using the evaporation model as described above, and returns them to STAR-CD for the next iteration step. The conditional droplet source term S η is established by this subroutine as well and is supplied to the CMC solver. Figure 2. Schematic of burner 19 (left) and block structured axi-symmetric grid (right). The CMC equations are discretized using finite differences with 81 nodes in conserved scalar space, clustered around the stoichiometric value to enhance local resolution. To reduce the number of simultaneously solved ODEs, operator splitting (OS) was used, separating the chemical part of the system from the convection and diffusion part. The evaporation source terms are computed together with the reaction and diffusion in mixture fraction space, after the computational step of the physical space convection and diffusion. Integration of the ODEs is done with the VODPK solver. Details of the CMC code have been reported 29, 30 10, 31 previously. The 2D axisymmetric block-structured CFD grid is shown on the r.h.s. of Figure 2 and consists of 22820 cells. The overlaying CMC mesh consists of 40 cells in radial and 80 cells in axial direction, ensuring that at least one CFD cell is located in each CMC cell. For both CFD and CMC mesh, the grid size near the bluff body is refined. For all simulations the timestep is t = 5µs. II.D. Chemical mechanism and spark representation Reaction rates are calculated by a modified one-step chemistry for heptane, 32 developed by the method of Tarrazo-Fernandez et al. 33 for partially premixed combustion. This method is based on a tuning of the heat release rate as a function of the local equivalence ratio in order to produce the correct adiabatic flame temperature, which results in a substantial improvement in the prediction of the laminar burning velocity of premixed flames at all flammable values of equivalence ratio. For non-premixed combustion, the methods 5 of 12

< T η > 2500 2000 1500 1000 N 0 = 100 s -1 N 0 = 150 s -1 N 0 = 190 s -1 N 0 = 200 s -1 < Y η > 1 0.8 0.6 0.4 Y C7 at N H 16 0 = 100 s -1 Y C7 at N H 16 0 = 150 s -1 Y C7 at N H 16 0 = 190 s -1 Y C7 at N H 16 0 = 200 s -1 Y O2 at N 0 = 100 s -1 Y O2 at N 0 = 150 s -1 Y O2 at N 0 = 190 s -1 Y O2 at N 0 = 200 s -1 500 0.2 0 0 0.2 0.4 0.6 0.8 1 η 0 0 0.2 0.4 0.6 0.8 1 η (a) (b) Figure 3. Conditional temperature (a) and mass fraction (b) profiles for different N 0 from the 1-step n-heptane mechanism. 32 Table 1. Cases simulated. Case Spark location CMC evaporation source terms A r=0mm z=23mm ON B r=0mm z=23mm OFF results in flame temperatures and extinction strain rates that are comparable to those from the detailed mechanism. Figure 3 shows the temperature and mass fraction profiles for fuel and oxygen for different peak values of scalar dissipation given by Eq. (10) obtained with this modified one-step chemistry. The energy deposition of the spark is modelled by placing this burnt gas solution, for a low scalar dissipation, in a few CMC cells. Ignition occurs by physical transport of heat and species from this region to the surrounding regions through the convection and turbulent diffusion terms in the CMC equation. II.E. Experimental case The configuration studied here is based on the experimental study of spark ignition of a turbulent swirling n-heptane spray flame by Marchione et al. 19 The schematic of the burner can be seen in Figure 2. Fuel is supplied through a 60 degree hollow cone fuel nozzle positioned centrally in the bluff body, with a nozzle exit diameter of 0.15 mm. The air enters from the annular open area between the bluff body and the outer duct wall after passing a swirler. The air mass low ṁ air and fuel mass flow ṁ fuel are 0.42 kg/min and 0.025 kg/min, respectively, resulting to a global equivalence ratio of 0.9. The fuel spray initial droplet size distribution p(d) follows a Rossin Ramler distribution p(d) = 1 exp (d/d) q (23) where D = 30 µm and q = 2.55 and p(d) is approximated by 10 droplet size classes. All droplet parcels have the same initial velocity magnitude of 34.5 m/s, given by the nozzle diameter (0.15 mm) and the volume flow rate ṁ fuel. However due to the size distribution the differing drag forces will quickly create a nonuniform velocity distribution. Table 1 shows the different simulation set ups investigated in this paper. For comparison purposes two different simulation set-ups are used to investigate the impact of the extra droplet source terms in the CMC equations. III. Results III.A. Cold Flow Contour plots of the mean axial and swirl velocity before spray injection are shown in Fig. 4. The axial velocity shows a recirculation zone above the bluff body as expected. Predicted and measured radial profiles of the mean of the axial and swirl velocity are shown in Fig. 4 at several axial positions. In general both the 6 of 12

(a) (b) Figure 4. (a) Contour plot of swirl (left) and axial (right) velocity of the cold flow in absence of spray. (b) Predicted and measured radial profile of mean swirl (top) and axial (bottom) velocity for air flow in absence of spray at the indicated axial positions. Experimental data from Marchione et al. 19 predicted axial and swirl velocities are in good agreement with the experiment. Radial profile predictions for both velocities are slightly more overpredicted further upstream than in the proximity of the bluff body. With the initiation of the fuel spray in the cold flow, Figure 5 shows the radial profile of the mean Sauter Mean Diameter (SMD) at different axial positions. A good agreement with the experimental data 19 can be found. Figure 6 shows ξ and ξ 2 for the cold spray flow prior to spark ignition. The maximum of ξ can be found near the core spray up to z = 20 mm. High ξ 2 is found in regions with high gradients of ξ. 5e-05 4e-05 SMD, m 3e-05 2e-05 1e-05 0 0 0.005 0.01 0.015 0.02 0.025 0.03 Radial position, r m 12mm CFD 12mm Exp. 20mm CFD 20mm Exp. 35mm CFD 35 mm Exp. Figure 5. Radial profile of the Sauter Mean Diameter (SMD) at the indicated axial position for cold spray flow. III.B. Spark ignition Figure 7 shows a comparison of Favre-averaged temperature contours of case A with high-speed images from Ref. 19 at four different time steps after spark ignition. The bluff body is at the bottom of these images 7 of 12

Figure 6. Contour plot of mean mixture fraction (left) and variance of mixture fraction (right) of cold spray flow. On the left side, the white line denotes the isoline ξ = ξ st. in the center. A good qualitative agreement in the time evolution of the flame can be observed between simulation and experiment. As the flame expands and the flow gets hotter, the gaseous mixture fraction increases as the droplets evaporate quickly. Figures 8 and 9 show the mean and variance of the mixture fraction respectively at various times after ignition. In the immediate vicinity of the spark, the mixture fraction increases quickly to about 0.6, while later the whole recirculation zone has become rich. The inert and the frozen distributions of the mixture fraction are very different. The variance of the mixture fraction peaks at the locations of the steepest gradients of the mean. To get a better insight of the droplet evaporation effect between case A and B, Figure 10 shows contour plots of the Favre-mean temperature difference between the two solutions, T = T A T B, together with ξ, the corresponding difference of the mean mixture fraction. Near the bluff body, where the spray is introduced, the cooling effect of the extra droplet source terms in case A can be seen. In the location of the spark, case A shows slightly higher temperatures in the beginning of the flame establishment while upstream of the spark location a region is visible with a strong cooling effect in case A, since a large amount of droplets is found there. In the downstream development of the flame, it can be seen that case A is growing faster than case B, visualized by the higher temperature of case A. The comparison of ξ between case A and B shows that near the bluff body at the spray injection, ξ is lower in case A due to the cooling effect which lowers the evaporation rate. In the spark location and downstream, case A shows higher mixture fractions since here fewer droplets are available and so the cooling effect is weaker than the effect of the additional mass fraction source term in mass fraction CMC equation. Finally, Figure 11 shows the conditional temperature T η of the two solutions at the two locations marked on Fig. 10 and for a few times from spark initiation. In the first time instant (0.238 ms) the cooling effect in case A can be seen in both locations and results in a decrease in conditional temperature at the saturation mixture fraction. In location 1, nearer to the bluff body, the effect is larger since more droplets are present. At the second time instant, T η is also lower for case A at both locations due to the heat needed for droplet evaporation. At 15.9 ms and in location 1, both temperature profiles for case A and B have become almost identical since the lower number of droplets in this location means that all droplets are evaporated and the cooling effect is visible any more. In location 2, near to the spray injection, droplets are still available for evaporation leading to a lower conditional temperature for case A in the saturation mixture fraction region. Note also that the distributions of T η in mixture fraction space show large spatial and temporal variations throughout the ignition transient and that it takes significant time for the T η to approach the fully-burning solution as in Fig. 3. More details on this transient for gaseous burner spark ignition with large-eddy Simulations and the CMC model can be found 8 of 12

Figure 7. Comparison between contour plots of mean temperature for case A and high-speed images 19 at different time instants. Case A, initial spark kernel location at r = 0 and z = 23mm. White line denotes the isoline ξ = ξ st. The flow comes from below. in Triantafyllidis et al. 31 IV. Conclusions The CMC model for sprays works well for the flame expansion following spark ignition in a turbulent swirling recirculating n-heptane spray flame, suggesting that the model is capable of handling ignition events in realistic gas turbine combustors. The extra droplet source terms in the conditional mass fraction and temperature equations cause some differences in the simulations. Further validation and comparison with experimental data is needed. Acknowledgments This work received funding from the European Community through the project TIMECOP-AE (Project number AST5-CT-2006-030828). It reflects only the authors views and the Community is not liable for any use that may be made of the information contained therein. R.W. Bilger has been supported by a UK Royal Academy of Engineering Distinguished Visitor Fellowship that enabled him to visit the University of Cambridge. References 1 Klimenko, A. Y. and Bilger, R. W., Conditional Moment Closure for turbulent combustion, Prog. Energ. Combust. Sci., Vol. 25, 1999, pp. 595 687. 2 Kim, S. H., Huh, K. Y., and Tao, L., Application of the Elliptic Conditional Moment Closure Model to a Two- Dimensional Nonpremixed Methanol Bluff-Body Flame, Combust. Flame, Vol. 120, 2000, pp. 75 90. 3 Mastorakos, E. and Bilger, R. W., Second-order conditional moment closure for the autoignition of turbulent flows, Phys. Fluids, Vol. 10, 1998, pp. 1246 1248. 4 Sayed, A. E., Milford, A., and Devaud, C. B., Modelling of autoignition for methane-based fuel blends using Conditional Moment Closure, Proc. Combust. Inst., Vol. 32, 2009, pp. 1621 1628. 5 Fairweather, M. and Woolley, R. M., First-order conditional moment closure modeling of turbulent, nonpremixed hy- 9 of 12

Figure 8. Contour plot of mean mixture fraction at various times after ignition. Case A, initial spark kernel location at r = 0 and z = 23mm. (a) t = 0.3ms, (b) t = 0.7ms, (c) t = 1.9ms, (d) t = 7.1ms, (e) t = 13.5ms and (f) t = 16.5ms. The white line denotes the isoline ξ = ξ st. drogen flames, Combust. Flame, Vol. 133, 2003, pp. 393 405. 6 Roomina, M. R. and Bilger, R. W., Conditional moment closure (CMC) predictions of a turbulent methane-air jet flame, Combust. Flame, Vol. 125, 2001, pp. 1176 1195. 7 Devaud, C. B. and Bray, K. N. C., Assessment of the applicability of conditional moment closure to a lifted turbulent flame: first order model, Combust. Flame, Vol. 132, 2003, pp. 102 114. 8 Kim, I. S. and Mastorakos, E., Simulations of turbulent lifted jet flames with two-dimensional conditional moment closure, Proc. Combust. Inst., Vol. 30, 2005, pp. 911 918. 9 Patwardhan, S. S., De, S., Lakshmisha, K. N., and Raghunandan, B. N., CMC simulations of lifted turbulent jet flame in a vitiated coflow, Proc. Combust. Inst., Vol. 32, 2009, pp. 1705 1712. 10 Wright, Y. M., de Paola, G., Boulouchos, K., and Mastorakos, E., Simulations of spray autoignition and flame establishment with two-dimensional CMC, Combust. Flame, Vol. 143, 2005, pp. 402 419. 11 Rogerson, J. W., Kent, J. H., and Bilger, R. W., Conditional moment closure in a bagasse-fired boiler, Proc. Combust. Inst., Vol. 31, 2007, pp. 2805 2811. 12 Smith, N. S. A., Cha, C. M., Pitsch, H., and Oefelein, J. C., Simulation and modeling of the behaviour of conditional scalar moments in turbulent spray combustion, Proc. Summer Prog., 2000, pp. 207 218. 13 Kim, S. H. and Huh, K. Y., Application of the elliptic conditional moment closure model to a two-dimensional nonpremixed methanol bluff-body flame, Proc. Combust. Inst., Vol. 29, 2002, pp. 273 279. 14 Hollmann, C. and Gutheil, E., Modeling of turbulent spray diffusion flames including detailed chemistry, Proc. Combust. Inst., Vol. 26, 1996, pp. 1731 1738. 15 Réveillon, J. and Vervisch, L., Spray vaporization in nonpremixed turbulent combustion modeilng: A single droplet model, Combust. Flame, Vol. 121, 2000, pp. 75 90. 16 Demoulin, F. X. and Borghi, R., Assumed PDF Modeling of Turbulent Spray Combustion, Combust. Sci. Technol., Vol. 158, 2000, pp. 249 271. 17 Sreedhara, S. and Huh, K. Y., Conditional statistics of nonreacting and reacting sprays in turbulent flows by direct numerical simulation, Proc. Combust. Inst., Vol. 31, 2007, pp. 2335 2342. 10 of 12

Figure 9. Contour plot of mixture fraction variance at various times after ignition. Case A, initial spark kernel location at r = 0mm and z = 23mm. (a) t = 0.3ms, (b) t = 0.7ms, (c) t = 1.9ms, (d) t = 7.1ms, (e) t = 13.5ms and (f) t = 16.5ms. The white line denotes the isoline ξ = ξ st. 18 Mortensen, M. and Bilger, R. W., Derivation of the conditional moment closure equations for spray combustion, Combust. Flame, Vol. 156, 2009, pp. 62 72. 19 Marchione, T., Ahmed, S. F., and Mastorakos, E., Ignition of turbulent swirling n-heptane spray flames using single and multiple sparks, Combust. Flame, Vol. 156, 2009, pp. 166 180. 20 Spalding, D. B., Combustion and Mass Transfer, Pergamon Press, 1979. 21 Lefebvre, A. H., Gas Turbine Combustion, Taylor and Francis, 1998. 22 Mastorakos, E., Ignition of turbulent non-premixed flames, Prog. Energy Combust. Sci., Vol. 35, 2009, pp. 57 97. 23 Bilger, R., Conditional moment closure for turbulent reacting flow, Phys. Fluids, Vol. 5, 1993, pp. 436 444. 24 O Brien, E. E. and Jiang, T.-L., The conditional dissipation rate of an initially binary scalar in homogeneous turbulence, Phys. Fluids, Vol. 3, No. 3121-3123, 1991. 25 Demoulin, F. X. and Borghi, R., Modeling of turbulent spray combustion with application to diesel like experiment, Combust. Flame, Vol. 129, 2002, pp. 281 293. 26 CD-adapco Group, STAR-CD v3.26 User s Manual, 2004. 27 Schroll, P., Wandel, A. P., Cant, R. S., and Mastorakos, E., Direct numerical simulations of autoignition in turbulent two-phase flows, Proc. Combust. Inst., Vol. 32, 2009, pp. 2275 2282. 28 Reitz, R. D. and Diwakar, R., Effect of Drop Breakup on Fuel Sprays, SAE950283, 1986. 29 Brown, P. N., Byrne, G. D., and Hindmarsh, A. C., VODE, a Variable-Coefficient ODE Solver, SIAM J. Sci. Statist. Comput., Vol. 10, 1989, pp. 1038 1051. 30 Brown, P. N. and Hindmarsh, A. C., Reduced Storage Matrix Methods in Stiff ODE Systems, J. Appl. Math. Comput., Vol. 31, 1989, pp. 49 91. 31 Triantafyllidis, A., Mastorakos, E., and Eggels, R. L. G. M., Large Eddy Simulations of forced ignition of a non-premixed bluff-body methane flame with Conditional Moment Closure, Combust. Flame, Vol. 156, 2009, pp. 2328 2345. 32 Richardson, E. S., Ignition Modelling for Turbulent Non-Premixed Flows, PhD Thesis, University of Cambridge, 2007. 33 Fernandez-Tarrazo, E., Sanchez, A., Linan, A., and Williams, F. A., A simple one-step chemistry model for partially premixed hydrocarbon combustion, Combust. Flame, Vol. 147, 2006, pp. 32 38. 11 of 12

Figure 10. Upper row: Contour plots of mean temperature difference T between case A (with) and case B (without extra droplet source terms in the CMC equations). Lower row: Contour plots of mean mixture fraction difference ξ between case A and case B. Figure 11. T η for case A (red dotted) and B (black) at two locations: location 1 is at r = 8.8 mm and z = 5.7 mm, location 2 at r = 15 mm and z = 32.5 mm. 12 of 12