LES-CMC of a dilute acetone spray flame with pre-vapor using two conditional moments

LES-CMC of a dilute acetone spray flame with pre-vapor using two conditional moments S. Ukai, A. Kronenburg, O.T. Stein Institut für Technische Verbrennung, Universität Stuttgart, 774 Stuttgart, Germany Abstract Turbulent spray combustion is a complex phenomenon, involving interactions of multi-scale physics which complicate the modelling. An earlier study by the authors (Ukai et al. 23) gave reasonable results for the prediction of a turbulent spray flame by utilizing large-eddy simulation (LES) and conditional moment closure (CMC), but also some limitations of the method became apparent especially those related to the upper limit in mixture fraction space. The scope of this paper is to propose a two-conditional moment approach that solves two sets of conditional moments with different mixture fractions. The solution of each CFD cell is found by interpolating between two conditional moments weighted by the amount of the vapor emitted within the domain, and the cell filtered value is given by integration of the conditional moment across mixture fraction space using a bounded β-pdf for the distribution of the scalar. The new two-conditional moment approach results show distinct improvements in the predictions of temperature along the centerline. Introduction Many industrial combustion systems involve turbulent spray flames, a complex physio-chemical problem the details of which are yet to be fully understood. A crucial aspect of this problem is the effect of turbulence on the spray and combustion chemistry, and such turbulencespray and turbulence-chemistry interactions must be modelled appropriately. We apply large-eddy simulation (LES) to represent the largest scales of turbulent fluid motion and model small scale eddies. Spray evaporation and combustion are sub-grid processes in LES, and we use a Lagrangian approach for the droplets and the conditional moment closure (CMC) framework to account for these sub-grid phenomena. CMC is a sub-grid combustion model originally developed for single phase reacting turbulent flow, and it has been validated against different types of flames. CMC has been extended to two-phase turbulent flames with extra spray source terms []. Recently, Ukai et al. [2] applied two-phase CMC- LES to a dilute acetone spray jet flame and discussed the impact of the additional spray terms. The study also pointed out difficulties of selecting an appropriate upper limit of mixture fraction, ξ UL, due to the effect of evaporation. In general, a single-phase CMC implementation (or non-premixed approaches in general) assumes ξ UL to be fixed at the fuel condition. However, the maximum mixture fraction varies in space and time due to evaporation, and it is difficult to define a single ξ UL applicable throughout the entire domain. One compromise applied in the past study [2] is to fix ξ UL at the fuel jet mixture fraction ξ jet originating from pre-vapor inside the jet nozzle. This approach led to a good match of the predictions with experiments near the jet inlet where most of the evaporation occurs without any mixing with the surrounding streams. However, the assumption is not fully valid where droplet vapor and oxidizer are mixed more homogeneously, and, as a result, low temperature Corresponding author: kronenburg@itv.uni-stuttgart.de Proceedings of the European Combustion Meeting 23 regimes were observed along the centerline further downstream. An alternative approach is to adjust ξ UL dynamically based on the mixing fields, and better predictions can be expected downstream. However, if only the dynamically moving ξ UL is used, the scalar dissipation can cause unphysical mixing in the jet core where mixing with the surrounding flow should not occur. Thus, a new method must be developed to treat evaporating sprays more appropriately. The scope of the current study is to develop a new two-conditional moments approach that can deal with a shift of ξ UL dynamically. This paper explains the difficulties of a classical CMC method with one set of conditional moments. Then, the new two-conditional moment approach is discussed. This method is applied to simulate a dilute acetone spray flame, and numerical results are compared against well-defined experimental data [3]. Gas and Liquid Phase Formulations Large-eddy simulation resolves turbulent eddies down to the size of the computational grid, whereas the small scale motion is modelled. The Favre-filtered continuity and momentum equations read t p ρ ru iq ρ t p ρ ru i ru p j q x j x i x j p ρ ru j q 9ρ, () τ ij τ sgs ij x j x j 9 F i, (2) where ρ is density, u i are the velocities in i direction, p is 9 pressure and τ denotes the viscous stress tensor, 9ρ, F i are the source terms associated with the liquid phase in the mass, momentum, τ sgs ij is the subgrid stress tensor. We have used standard Smagorinsky type closures for τ sgs ij with a dynamic model with the least square method [4] to locally compute the Smagorinsky constant. Mixture fraction is a useful concept to analyze nonpremixed flames. The mixture fraction is usually set to

unity in the fuel stream, and the maximum mixture fraction within a flow is kept less than.. However, in turbulent spray flames, the maximum mixture fraction can exceed the value associated with the gaseous fuel jet at the inlet. Thus, here two mixture fractions are solved simultaneously: the total mixture fraction,, and the conserved mixture fraction, ξ cons. represents the mixture fraction associated with the inlet conditions (a pilot flame and fuel originating from pre-evaporated droplets that lead to fuel vapor at the jet exit) and fuel originating from droplet evaporation within the domain. ξ cons originates from pre-evaporation within the nozzle and a pilot flame. Solving these two mixture fractions, the mixture fraction evaporated from the droplets after exiting the nozzle emitted can be computed as ξ cons, and it plays an important role in the following modelling approach. The maximum value of changes dynamically and is defined to be unity when the mixture is pure fuel. The maximum value of ξ cons corresponds to the maximum inlet mixture fraction. The Favre-filtered scalar transport equations for these mixture fractions are t p ρ q p ρ ru J i,tot J sgs i,tot j ξtot q 9ξ, (3) x j x j x j t p ρ ξ cons q p ρ ru J i,ξ cons J sgs i,ξ cons j ξcons q, x j x j x j (4) where J i is the diffusion flux of the mixture fraction, ξ 9 is the source term of the mixture fraction due to evaporation, and J sgs i is the subgrid diffusion term. The governing equations are solved with a SIMPLE-type predictorcorrector procedure with pressure smoothing as described in [5]. We use a Lagrangian scheme to track the evolution of the droplets. Every droplet is tracked individually, and the unresolved turbulent fluctuation is coupled to the Lagrangian scheme with stochastic terms. The detailed formulation can be found elsewhere [6]. Conditional Moment Closure The conditional moment closure method [7] is used to model the transport of the reacting species. The turbulent fluctuations of the reacting species are thought to correlate well with fluctuations of mixture fraction and thus, a first order closure of the chemical source term suffices. The conditional moment Q α of species α is defined as Q α pξ, x, tq Y α px, tq ξpx, tq η where angular brackets denote the average and the condition is given by the expression on the right of the vertical bar. However, the conditional fluctuation can be very large for spray flames, and it might be impractical to represent the properties by only one conditional moment. Therefore, two sets of conditional moments conditioned on ξ cons and are solved simultaneously, and the solution is obtained by interpolation between the two conditional moments as presented in the later section. The CMC equation for the conditional moment on ξ cons is t Qα,cons ρ r η,cons P η,cons ρ η,cons r P η,cons puη,consq α,cons Dt,η,cons Q α,consq rω 2 η,α,cons Nη,cons η 2 Q α,cons Q α,cons ρ η,cons P r η,cons P η,consu η,cons, η,cons (5) where subscript cons indicates the conditioning on the random variable, ξ cons, N is the conditional scalar dissipation rate, rω η,α is the chemical source term of species α that can be closed by a first order approximation, and subscript η denotes conditionally averaged quantities. The CMC equation for the moments conditionally averaged on includes additional spray source terms [, 8] as t Qα,tot ρ r η,tot P η,tot ρ η,tot r P η,tot puη,totq α,tot Dt,η,tot Q α,totq rω 2 η,α,tot Nη,tot η 2 Q α,tot (6) Q α,tot ρ η,tot P r η,tot P η,totu η,tot η,tot Q,α Q α,tot p ηq η Qα,tot Π η, where the subscript tot indicates the conditioning on the random variable, Q,α denotes the composition of the liquid fuel and Π is the conditional volumetric fuel evaporation rate. Similarly, the conditionally averaged enthalpy equations are given given by t Q h,cons ρ η,cons r P η,cons ρ η,cons r P η,cons U η,consq h,cons D t,η,cons Q h,cons N 2 η,cons η 2 Q h,cons Q h,cons η,cons P η,cons t Q h,tot ρ η,tot r P η,tot ρ η,cons r P η,consu η,cons ρ η,tot r P η,tot U η,totq h,tot D t,η,tot Q h,tot N 2 η,tot η 2 Q h,tot e rad,η,tot e rad,η,cons, Q h,tot ρ η,tot P r η,tot P η,totu η,tot η,tot Q,h Q h,tot p ηq η Q h,tot Π η (7) Ψη, (8) where Q h is the conditionally averaged enthalpy, e rad is the radiation heat loss that is modelled assuming an optically thin flame, and Ψ is the heat transfer term between spray and gas phases. Unity Lewis numbers has been assumed. The modelling approaches of the conditionally averaged quantities (Π η, Ψ η, N η, U η and D t,η ) are explained and discussed in our previous work [2], and the current study considers that these terms are identical for both the formulations of Q tot and Q cons. rω η,α and e rad,η are functions of each conditional moment, (e.g. rω η,α,tot f pq α,η,tot, T η,tot q), so that rω η,α,tot rω η,α,in and e rad,η,tot e rad,η,cons. Computational Configuration The present study is based on the series of experiments on acetone spray flames conducted at Sydney University [3]. The nozzle diameter, D, is.5 mm, the outer annulus diameter of the piloted flow is 25 mm, and the co-flow diameter is 4 mm. Parametric studies of bulk velocities and spray mass flow rate were conducted in the 2

experiment. For the case considered here, the bulk velocity of the pilot is.9 m{s and of the co-flow is 4.5 m{s, the jet bulk velocity 24 m{s. The numerical domain is 4D in axial direction, the cross section is D D at the inlet and diverges towards the outlet. 9 9 24 LES cells are used with grid clustering close to the nozzle. In shear-layer type flow, the conditional moments do not change much with radial position and the CMC grid can be relatively coarse compared to the LES mesh. The current study uses 5 5 4 CMC cells in the computational domain, so that one CMC cell contains 6 6 6 LES cells. The inflow velocity profiles for the LES are based on measured mean and variance of the axial velocity at the nozzle exit. The digital filtering technique of Klein et al. [9] is used to generate the inflow turbulence. The temperature of the jet was not measured in the experiments, so that 293K is used as proposed in [3]. Acetone spray is generated upstream of the nozzle exit, so that a certain portion of the spray evaporates before reaching the nozzle exit. The total acetone mass flow rate injected into the domain is 45. g{min, with 29. g{min evaporated and 5.9 g{min liquid fuel. Since the particle statistics are measured from z{d.3, the particles are inserted into the numerical domain at this location. Since the maximum mixture fraction within the domain dynamically changes due to the evaporation, the mixture fraction is scaled such that it is unity for pure fuel mixtures. Thus, the mixture fraction of the jet fuel is taken to be ξ jet.62, and the mixture fraction of the pilot is ξ pilot.858. Inflow condition with the co-flow velocity is used at the side boundary, and zero-gradient Cs are applied at the outlet. Several chemical mechanisms were tested, but their influence on quantities that could be used for validation was minor. Only results from an acetone mechanism based on GRI-Mech 3. with seven additional acetone sub-steps [] are reported here. the CMC solution. LES Representations in Mixture Fraction Space The evaporation within the domain can increase beyond ξ jet. A base mixture fraction ξ base is defined as the minimum amount of vapor emitted from droplets,, i.e., # ξbase if ξ jet. (9) ξ base ξtot ξjet ξ jet if ξ jet In other words, for ξ jet, only fluid originating from the jet is found along the base line (green line in Fig. ) that has not yet mixed with the surrounding fluid (e.g. from the pilot or co-flow). Figure shows the mixture fraction originating from droplets in the domain at different CMC cell positions. Since there is no mixing with the surrounding materials, within the jet core clusters near ξ base, and the conditional fluctuation is relatively small. However, the samples from the CMC cell containing the jet core and the pilot show a wide distribution of around.6. It is obvious that a simple first order closure is not forthcoming with only one conditional moment since realizations along the base line must represent cold solutions but those with large can be hot and thus, conditional fluctuations will not be negligible..3.2. Analysis of the Two-Conditional Moment Approach The past study [2] has discussed limitations of the upper mixture fraction boundary limit. One example is that the upper boundary condition had to be fixed at the jet condition, but in reality, the mixture fraction changes due to the evaporation process, and the nominal upper mixture fraction boundary changes with time and position. Therefore, a new definition of ξ UL should be given. Also, the evaporation process can violate the assumption of small conditional fluctuations in mixture fraction space. Therefore, not one but two conditional moments are computed that represent the extrema, and an interpolation routine is implemented to determine a suitable average of the two. An example study is made here by performing a non-reacting simulation with an enhanced evaporation rate for the same configuration as presented in the previous section. The non-reacting case allows for the computation of the LES-filtered values of acetone on the LES cell that can then be used for comparison with 3..2.4.6.8 Figure : Realization of within a CMC cell that lies within the jet core (black) and a CMC cell at the interface of the jet core and the pilot (red). The green line is ξ base (Eqn. 9). Selection of Upper Mixture Fraction oundary An upper and a lower mixture fraction boundary is necessary to solve the scalar dissipation term in Eqns 6 and 7. An upper mixture fraction boundary ξ UL in single phase flow is taken to be the fuel jet condition, ξ jet, which is unmixed with the surrounding materials (e.g. pilot or co-flow). However, it is not very straightforward to determine the upper limit, ξ UL, in two-phase flow since the mixture fraction is not conserved due to droplet evaporation. In the microscopic view, the highest mixture fraction within the domain is at a droplet surface, ξ surf. However, this would be an impractical choice unless a fully-resolved simulation is performed, since the surface

values tend to be much larger than the LES filtered quantities, and adequate conditional scalar dissipation modelling for mixture fraction values ξ jet ξ ξ surf is unknown. Thus, we choose ξ UL based on the LES filtered quantities. In a single phase flow, a non-reacting scalar mixes linearly between fuel and oxidizer in the mixture fraction space. Thus, if a quantity of the conserved scalar, φ, at a certain mixture fraction is known, φ at the upper boundary φ UL can be given as φ ξ UL UL φ, () ξ and we denote this slope as a mixing line. Using a single phase flow as an analogy, ξ UL must be unmixed with the surroundings. In other words, the solution must lie on the unmixed limit (Eqn. 9). Thus, here we propose a new selection method of ξ UL with the combination of the base mixture fraction and the mixing line as shown in Fig. 2. From the base mixture fraction (Eqn. 9), an amount of at ξ UL can be determined as,@ξul ξ UL ξ jet ξ jet, () and the linear mixing line as in Eqn., (φ UL,@ξUL and φ ),@ξul ξ UL. (2) Thus, ξ UL as a pseudo-unmixed material is obtained as..8.6.4.2 ξ UL Mixing line ξ ξ jet Corresponding unmixed property LES value. (3) ξtot ξ jet ase mixture fraction However, the current study requires a modification to the boundary of the β-function due to the dynamic change of ξ UL. Thus, the bounded β-pdf is implemented as [] f px; p, q, ξ LL, ξ UL q Γpp, qq px ξ LL q p pξ UL xq q pξ UL ξ LL q p q, (5) where ξ LL and ξ UL are the lower and upper bound limit of the PDF, x is bounded within ξ LL x ξ UL, and p and q are shape parameters constructed as ξ ξ LL p ξ ξ LL qpξ UL xq p, (6) ξ UL ξ LL ξ 22 q ξ UL ξ p ξ ξ LL qpξ UL xq, (7) ξ UL ξ LL ξ 22 where ξ 22 is the mixture fraction variance modelled as ξ 22 C 2 ξ ξ, (8) x i x i and C ξ is a coefficient set to be., and is the LES cell size. In this study, ξ LL is always set to.. Interpolation Method The LES quantities are computed using Eqn. 4 with Q α,η that is obtained by interpolation between Q tot and Q cons depending on the amount of. It can be said that Q cons is the solution along ξ base, and Q tot is approximately the solution over η. Thus, when is close to ξ base, the solution must be close to Q ξcons, and the large amount of should weight the solution towards Q tot as shown in Fig. 3. The weighting factor θ is given by Q η θq η,tot p θqq η,cons, (9) with " θ ξ ξ base η ξ base if ξ base η θ. if η.3. (2).5..5.2.25 ξ Figure 2: Estimation of ξ UL. Cross - actual quantity in a LES cell (, ), circle - (imaginary) unmixed property corresponding to the LES cell (ξ UL,,@ξUL ), solid line - mixing line, dashed line - ξ base. ounded β-function A β-function is often utilized to construct the PDF of the mixing field, and used to compute the LES filtered quantities as ry α» Q α,η P pηqdη. (4).2. Apply Q tot here Apply Q cons here..2.4.6.8 Figure 3: Realization of in the CMC cell over the core and the pilot (red square) with η (blue line). The green line is the base mixture fraction (Eqn. 9). 4

Comparisons of CMC and LES solutions This two-conditional moment approach is validated by comparing CMC solutions with the LES results for a non-reacting case. The mass fraction of acetone fuel, Y ace, is chosen as a tracer, and the LES solution is obtained by solving another Favre filtered transport equation, t p ρỹaceq p ρ ru j Ỹ J i,ace J sgs i,ace ace q 9Y ace, x j x j x j (2) where the source term is Y 9 9 ace ξ. The CMC solution is given by integrating the conditional moments as in Eqn. 4. Even though LES and CMC solutions are obtained by different methods, Fig. 4 shows that the distributions of Y ace within a CMC cell seem to agree very well. It indicates that the two-conditional moment approach is capable of modelling an evaporating spray with pre-evaporated fuel. If only one set of conditional moments is used, the LES solution cannot be reproduced. Y ace.2.5..5 LES CMC Q tot Q cons..2.4.6.8 Figure 4: Realization Y ace of LES (black plus symbol) and CMC (red X symbol) solutions over profiles of Q ace,tot (green line) and Q ace,cons (blue line). Results and Discussions Conditional Profiles The conditional temperature profiles with the LES realizations at different cross sections along the centerline are shown in Fig. 5. At z{d, there are some cells along Q cons near ξ.62. This indicates a predominantly cold solution for this position on the centerline. More mixing occurs downstream, and consequently no cold solution is found at z{d 2 and z{d 3. Due to the evaporation, the conditional moments require a higher upper boundary limit. The conditional temperature at z{d shows a local minimum around ξ.65, and that is caused by the convective flux. A large portion of non-reacting gases exists around ξ jet from the inlet condition as seen in LES solution, and since the convective flux is not split into Q tot and Q cons, Q tot can be influenced by the low temperature gases. The effect of the inlet condition eventually becomes small at the downstream positions due to the dissipation term. 5 Temperature (K) 4 2 8 6 4.4.6.8.2.22.24 Figure 5: Instantaneous temperature profiles within CMC cells along the centerline. Dots - LES realization, solid line - Q tot, dashed line - Q cons. lack - z/d=, red - z/d=2, blue - z/d=3. Unconditional Profiles Figure 6 compares the two-conditional moments approach, the conventional approach only with Q cons and the experimental results. Very low temperature is observed near the centerline by using only Q cons, since the upper mixture fraction is fixed and underestimates the temperature rise in rich mixtures. The new two-conditional moment approach predicts the centerline temperature successfully. Spray behavior The spray axial velocity statistics are compared at z/d= in Fig. 7. The mean profile shows good agreement with the experimental results. The two-conditional moment approach causes a slightly higher mean axial velocity since temperature rises along the centerline and causes thermal expansion. The variance matches with the experimental values around the centerline, but a large difference is found towards outer radii. Conclusions and Future Work The main goal of the paper is to amend the shortcomings of modelling an evaporating spray flame as reported in our previous study [2], such as the dynamically moving upper mixture fraction boundary, and the large conditional fluctuations in mixture fraction space. This assumption resulted in unphysically low temperatures along the centerline. Thus, this study introduces a new two-conditional moment approach, which models the conditional moments based on the total mixture fraction and the conserved mixture fraction. Due to the shift of the mixture fraction caused by evaporation, a new method to select the upper boundary is also proposed. The solution is computed by interpolating two conditional moments weighted by the mixture fraction from the evaporation, and the new approach is validated by comparing LES and CMC solutions of a non-reacting evaporating spray jet. The analysis of the turbulent spray flame case demonstrates that the new two-conditional moment

Temperature (K) Temperature (K) Temperature (K) 2 5 5 2 5 5 2 5 5 2 3 (a) z/d= 2 3 (b) z/d=2 2 3 (c) z/d=3 Figure 6: Radial profiles of mean temperature. Crosses - experiments [3], solid line - the two-conditional moment approach dashed line - Only with Q cons. rms W (m/s) 4 3.5 3 2.5 2.5.5 - -.8 -.6 -.4 -.2.2.4.6.8 Figure 7: Mean (right) and rms (left) axial droplet velocity profiles for the diameter range 2-3 µm at z/d=. Crosses - experiments [3], scatter points - single realization of a simulated droplet, solid line - the twoconditional moment approach, dashed line - only with Q cons. 4 35 3 25 2 5 W (m/s) 6 approach leads to a significant improvement of the radial temperature profiles, particularly near the centerline. The spray statistics are also slightly improved. The future work will apply the two-conditional moment approach to wider ranges of flow configurations. Acknowledgements The authors would like to acknowledge the funding support by DFG (grant no. KR3684/-2) and the computational resources provided by bwgrid. We would also like to thank Profs. A. Masri and W. P. Jones for providing the experimental data sets and the original CFD routines, respectively. References [] M. Mortensen, R. ilger, Derivation of the conditional moment closure equations for spray combustion, Combust. Flame 56 () (29) 62 72. [2] S. Ukai, A. Kronenburg, O. Stein, LES-CMC of a dilute acetone spray flame, Proceedings of the Combustion Institute 34 () (23) 643 65. [3] A. R. Masri, J. D. Gounder, Turbulent spray flames of acetone and ethanol approaching extinction, Combust. Sci. Techn. 82 (4) (2) 72 75. [4] M. Germano, U. Piomelli, P. Moin, W. H. Cabot, A dynamic subgrid-scale eddy viscosity model, Phys. Fluids A 3 (7) (99) 76 765. [5] N. ranley, W. P. Jones, Large eddy simulation of a turbulent non-premixed flame, Combust. Flame 27 (2) 94 934. [6] M. ini, W. P. Jones, Large eddy simulation of an evaporating acetone spray, Int. J. Heat Fluid Flow 3 (29) 47 48. [7] A. Y. Klimenko, R. W. ilger, Conditional moment closure for turbulent combustion, Prog. Energy Combust. Sci. 25 (999) 595 687. [8] G. orghesi, E. Mastorakos, C.. Devaud, R. W. ilger., Modeling evaporation effects in conditional moment closure for spray autoignition, Combust. Theo. Model. 5 (5) (2) 725 752. [9] M. Klein, A. Sadiki, J. Janicka, A digital filter based generation of inflow data for spatially developing direct numerical or large eddy simulations, J. Comput. Phys. 86 (2) (23) 652 665. [] C. Chong, S. Hochgreb, Measurements of laminar flame speeds of acetone/methane/air mixtures, Combust. Flame 58 (8) (2) 49 5. [] H.-W-Ge, E. Gutheil, Probability density function (PDF) simulation of turbulent spray flows, Atomiz. Sprays 6 (5) (26) 53 542.