Evaluation of the unsteady ﬂ amelet progress variable approach in Large Eddy Simulations of the ECN Spray A

. Within the Unsteady Flamelet Progress Variable – Large Eddy Simulation (UFPV-LES) approach the local scalar dissipation rate represents one key parameter, signi ﬁ cantly affecting the ignition behaviour. In this study, the UFPV-LES approach is evaluated for ECN Spray A baseline conditions, relevant for diesel engines. After con ﬁ rming its general applicability, using experimental data under non-reacting and reacting conditions, special attention is paid to the distribution of the local scalar dissipation rate. Based on the ﬁ ndings of this analysis, a reduced modeling approach, considering only igniting ﬂ amelets starting from the adiabatic mixing line between the fuel and oxidizer, is investigated. The performance of this reduced approach is assessed systematically, using the UFPV-LES results as a reference. Based on an a-priori analysis, regions affected by the model reduction are identi ﬁ ed and evaluated. A subsequent evaluation in an a-posteriori analysis, i.e. a coupled LES, reveals similar results in terms of local ﬂ ame structure as well as global ignition characteristics and con ﬁ rms the applicability of the reduced model under the ECN Spray A baseline conditions.


Introduction
Even though the electrification of the transportation sector is an ongoing process, there are obstacles that prevent a complete replacement of Internal Combustion Engines (ICE).As pointed out by [1], the limited specific capacities of current battery systems are deferring the adoption of Battery Electric Vehicles (BEV) in certain economic sectors, especially for heavy-duty applications.Direct Injection (DI) diesel engines account for most of the market share in this segment since they are robust and highly efficient.A major challenge is the reduction of greenhouse gases, especially CO 2 .In that regard, the utilization of renewable diesel fuels (e.g., biogenic or e-fuels) represents a promising way to achieve carbon-neutral combustion in the short-term.Investment costs for this transition are low, since infrastructure (refueling systems, etc.) and existing technology can be used, which is a great advantage.In addition, (liquefied) H 2 is a promising supplement of renewable fuels in diesel engines which will allow to further reduce carbonaceous emissions in the future [2].However, combustion characteristics of renewable diesel fuels can strongly differ from that of conventional hydrocarbon fuels.Hence, the investigation of ignition of liquid fuels in a high-temperature/high-pressure environment under diesel engine-relevant conditions remains of practical interest.The Engine Combustion Network (ECN) [3] provides a comprehensive experimental database for the validation of numerical models for high-pressure sprays.In recent years, different diesel surrogate fuels have been investigated in single-hole injectors under conditions relevant for diesel engines.Based on the ECN Spray H, operating with n-heptane, the research focus has evolved towards more diesel-like fuels, e.g., n-dodecane in the ECN Spray A. However, long-chain hydrocarbons lead to more complex chemical processes during ignition and combustion.Their description requires chemical kinetic mechanisms incorporating a large number of species and reactions [4].Detailed kinetic mechanisms are numerically very expensive in the coupled simulation [5].Therefore, reduced approaches, e.g., based on the flamelet concept [6], are frequently applied in spray combustion simulations under dieselengine-relevant conditions.The fundamental assumption of flamelet-based approaches are large Damköhler numbers, where chemical reactions occur in asymptotically thin layers [7].The alignment of the gradients of the mixture fraction Z, the temperature and the species mass fractions normal to these layers leads to a one-dimensional representation of the flame structure denoted by so-called flamelets.A turbulent flame is considered to be an ensemble of such Dossier LES4ECE'21: LES for Energy Conversion in Electric and Combustion Engines, 2021 Edited by Christian Angelberger, Stéphane Jay laminar diffusion flamelets for which the influence of the surrounding turbulent flow field is limited to their deformation and strain [8].The corresponding set of governing equations were initially derived by Peters [6,7].The diffusion coefficient present in these so-called flamelet equations is the scalar dissipation rate v = 2D(rZ Á rZ), denoting the connection between the mixture fraction and physical space.The functional form of this parameter is usually parameterized by its value in stoichiometric conditions v st .Using a definition for v st and other boundary conditions from the CFD (e.g., pressure), these equations can be solved interactively.This is done in the Representative Interactive Flamelet (RIF) approach [9,10] using one flamelet representative for the overall domain or by utilizing multiple representative flamelets in the MRIF approach [11][12][13] which was recently applied in LES of the ECN Spray A in [14].
One alternative to the interactive flamelet simulation is the pre-tabulation of the manifold.In this context, the Unsteady Flamelet Progress Variable (UFPV) approach was proposed by Pitsch and Ihme [15] and further extended by Ihme and See [8] to describe ignition processes.In this approach, unsteady flamelet solutions, initialized by an adiabatic mixing of fuel and oxidizer, are incorporated as well as igniting and extinguishing solutions with unstable steady flamelet solutions as the initial condition.The first application of this approach to diesel spray combustion was presented by Bajaj et al. [16] in RANS simulations of the ECN Spray H and A over a wide range of operating conditions.By this means, Steady Lift-Off Lengths (LOLs) and Ignition Delay Times (IDTs) were predicted within an error of 25%.The influence of v st on the IDT was connected to the time needed to reach the value of v st at which ignition can occur.Furthermore, the LOL was found to be connected to the axial location where v st equals its value, limiting the ignition process.It was further shown that the v st distribution upstream of the LOL is similar for non-reacting and reacting sprays [16].Subsequently, Ameen [17] utilized the UFPV approach to investigate the ignition behavior of n-heptane jets under diesel-relevant conditions by means of RANS and LES.In their study, they confirmed the interplay between LOL and v st distribution as presented by Bajaj et al. [16] in LES.The UFPV approach was further applied by Lucchini et al. [18] in RANS simulations of diesel engine combustion.
Beside the UFPV approach proposed by Pitsch and Ihme [15], reduced versions have been used in diesel spray combustion simulations with respect to, among other things, the range of v st values included in the look-up table generation.In this context, only one igniting flamelet at a constant strain rate is incorporated in the Flamelet Generated Manifold (FGM) approach for diesel engine combustion [4,19,20] and, usually, only auto-igniting flamelet solutions are utilized in the tabulation of so-called Approximated Diffusion Flamelets (ADF) [21] in spray LES [22][23][24][25][26][27][28][29].The latter approach was systematically evaluated in [24,30] based on a lifted turbulent H 2 /N 2 jet flame.Furthermore, Desantes et al. [25] assessed the approach within RANS simulations of the ECN Spray A for different ambient temperatures and oxygen contents, effectively capturing the trend of increasing LOL with increasing IDT, while quantitative differences stayed within acceptable range.Further applications are presented by Payri et al. [26], who investigated the influence of the chemical mechanism in RANS simulations of the ECN Spray A, by García-Oliver et al. [27] who compared the ECN Spray A and Spray D sprays based on RANS simulations and Desantes et al. [28] and Pérez-Sánchez et al. [29] investigating the ECN Spray A by means of the LES framework.However, a systematic comparison of this reduced model with the original UFPV approach [15] in LES of the ECN Spray A has not been performed so far.
Thus, the objective of this work is twofold: to assess the LES-UFPV approach as proposed by Pitsch and Ihme [15], including a detailed analysis of the scalar dissipation rate distribution and its effect on the ignition behavior under the ECN Spray A baseline conditions (1), to evaluate the tabulation approach, accounting only for igniting flamelets starting from the adiabatic mixing line (2).
The remainder of this paper is structured as follows: In Section 2, a brief description of the experimental setup is given.In Section 3, the modeling approach is described.In Section 4, the results are presented.The work closes with a summary and conclusions in Section 5.

Experimental setup
The ECN Spray A conditions are described in detail in various publications with corresponding measurement data collected on the ECN webpage [3].In this study, the ECN Spray A base line conditions are investigated.n-dodecane at a pressure of 1500 bar is injected through a single-hole injector into a constant-volume pre-burn combustion chamber under nominal conditions of 900 K and 60 bar with a molar oxygen concentration of 15%.For the purpose of validation, the spray under inert conditions (X O 2 ¼ 0) is additionally investigated.The ECN Spray A injector, serial number 210677, is simulated for non-reactive conditions while the injector, serial number 210370, is investigated under reactive conditions due to the availability of more recent experimental data.The two injectors differ slightly in terms of nozzle diameter but otherwise share similar characteristics.

Modeling approach
The two-phase flow considered in this work is modeled as a continuous gas and dispersed liquid phase.The gas phase is described by a set of differential equations in an Eulerian frame of reference, while the dispersed phase of liquid droplets is described in a statistical sense by the distribution function with a characteristic set of internal coordinates.This is known as the Eulerian-Lagrangian method [31], with further explanation given in [32,33].The suitability of this approach for the experimental setup under investigation has been shown in the literature e.g., see [14,20,34].The description of the liquid phase is given in Section 3.1 while the gas phase model is described in Section 3.2.The combustion model is introduced in Section 3.3.

Liquid phase
As mentioned above, the liquid phase is modeled with a Lagrangian approach where numerical parcels represent a number of droplets sharing the same properties.The description of their evolution is based on semi-empirical laws for drag, heat and mass transfer correlations following [20,34].The Spalding mass number B m,f of the fuel species f is utilized in this work in the evaluation of the evaporation model given by where Y s f denotes the mass fraction of the fuel species in the gas phase in the vicinity of the droplet and 1 corresponds to the environmental conditions.The break-up of the liquid phase is modelled utilizing the KH-RT model [35] with model parameters set to B 0 = 0.61, B 1 = 5, C s = 1, C RT = 0.1, ms Limit = 0.05 and We Limit = 6.These are close to the values suggested in the literature for this spray and simulation strategy [34].

Gas phase
The gas phase model is formulated for a LES framework.Transport equations are solved for the Favre-filtered conservation equations for mass, momentum and total enthalpy given by where q denotes the density, u i the velocity in direction i, p the pressure, g i the gravitational acceleration in direction i and _ S m , _ S u;i , and _ S ht the mass, momentum and total enthalpy source terms arising from the coupling with the liquid phase.The operators ðÁÞ and ðÁÞ denote spatial and density-weighted (Favre) spatial filtering, respectively.The filtered viscous stress tensor s ij and heat flux q j are given by with l denoting the dynamic viscosity, k the thermal conductivity and c p the specific heat capacity at constant pressure.Note that the mechanical source in equation ( 4) and the enthalpy flux due to mass diffusion are neglected here following [20].The terms s SGS ij ¼ q g u i u j À ũi ũj À Á and q SGS j ¼ q g u j h t À ũj ht denote the unclosed terms due to the application of the filter operator.Applying the Boussinesq hypothesis [36], the anisotropic part of s SGS ij is modeled in analogy to the viscous stress tensor as The isotropic part 1 3 s SGS kk equals twice the subgrid scale kinetic energy and is absorbed in the filtered pressure p, as usually done within the limit of low Mach number flows [37].The subgrid scale viscosity m SGS is modeled by means of the r-model [38] where the differential operator D r is based on the singular values of the velocity gradient tensor r i and is defined as A value for the model constant of C r = 2.0 is chosen in this work.In the numerical solution procedure, the filter width is modeled as a function of the cell volume V cell as The unresolved enthalpy flux is modelled by means of a gradient-flux assumption leading to where m SGS is obtained from the subgrid-scale model described above and Pr SGS is a model constant set to 0.4 in this work [39,40].To close the system of equations given above, the filtered equation of state is utilized.

UFPV approach
The present study focuses on the evaluation of combustion modeling using the UFPV approach [8,15].This approach uses the flamelet equations for the species mass fractions Y k and the temperature T derived by Peters [7] q The Author(s): Science and Technology for Energy Transition 77, 5 The Lewis number is assumed to be unity for each species (i.e., Le k = 1) and the pressure to be constant.The scalar dissipation rate of the mixture fraction follows from the coordinate transformation.The model derived for the steady laminar counterflow diffusion flame with constant density and diffusion coefficients by Peters [7] vðZ Furthermore, Y C is normalized by its corresponding minimum and maximum values, reading This variable is statistically independent of Z [42] and leads to a cubically shaped look-up table, which permits an efficient table access [43,44].Note, that a two-stage request strategy is applied to evaluate equation (18).To accurately capture the onset of ignition, the progress variable source term _ x Y C for C 2 [0; C init = 10 À6 ] is set to a constant and finite value preserving the mass of Y C produced by chemical reactions [41].To consider the non-resolved fluctuations of the request variables, a presumed b PDF is applied to model the marginal PDF of Z parameterized by the normalized variance while a d distribution is chosen for the marginal PDFs of C and v st [15].This approach is consistent with previous studies on the ECN Spray A using LES, see e.g., [20,28].Note that there are more complex models for the C-PDF, see [8].The evaluation of such models is interesting for future studies of high-pressure spray flames.The resulting parameterization of the Favre-filtered thermochemical state reads e / ¼ e / e Z ; g Z 00 2 N ; e v st ; e C .

UFPV-LES coupling for spray combustion
The variables needed for the table look-up are obtained by solving additional transport equations for e Z and e Y C given by Table 1.Boundary conditions for the flamelet model under the ECN Spray A baseline conditions (Z = 0: oxidizer; Z = 1: fuel; Z = Z st = 0.045: stoichiometry).The pressure is p = 60 bar.
0 900 0 0.15 0.7515 0.0622 0.0362 The variance of the mixture fraction is obtained following [45] by with C f Z 00 2 ¼ 0:13 [46,47] and Sc SGS = 0.4 in accordance with the Prandtl number in equation (11).The Favre-filtered scalar dissipation rate is obtained by an algebraic relation given by [48] From the local Favre-filtered scalar dissipation rate, the scalar dissipation rate under stoichiometric conditions used to parameterize the FLUT can be calculated using [49] e With the request parameters determined in this way, the progress variable source term is obtained from the UFPV FLUT along with the species mass fractions.The temperature is then retrieved from the transported enthalpy obtained from equation ( 4) and the species mass fractions retrieved from the FLUT and is used to obtain the thermo-physical properties in the LES.With this, the cooling effect due to evaporation is accounted for.This UFPV-LES coupling is summarized in Figure 1.

Reduced UFPV approach
In this study, the original UFPV approach incorporating the state space as described in Section 3.3.1 is utilized.The inclusion of unsteady flamelet solutions originating from the unstable branch of the S-curve necessitates its calculation prior to table generation.To circumvent this additional effort and thus reduce the computational cost of the table generation, these results are often not incorporated in the look-up table [24].Such an approach requires the ignition process to be handled in a specific way, since fluid elements potentially enter regions of higher v st values after the onset of ignition.This is marked by the red trajectory in Figure 2.An interpolation between the states of the neighboring v st values (without considering the solutions starting from the unstable branch) would lead to a significant change in the species mass fractions when entering regions with v st > v st,ign and hence in the estimated temperature and thermo-physical properties, respectively.To overcome this problem, the maximum value of v st , used for the table look-up after the onset of ignition, is limited to v st,ign .This is expressed by the relation  25) is denoted by RUFPV approach in the remainder of this study.The corresponding trajectory in the T st Àv st state space is indicated by the blue line in Figure 2. By this means, strong local gradients in the thermo-physical properties during the coupled simulation are avoided.However, by neglecting the solutions between the stable and unstable branches, the slow down in ignition and possible extinction during the ignition process is, in contrast to the original UFPV approach, not incorporated by the RUFPV approach.The influence of this reduction of the state space is discussed in Section 4.

Numerical framework and discretization
The mesh topology used to discretize the domain is based on the topology presented in [34] and shown in Figure 3.The mesh has the form of a cuboid with dimensions of 60 Â 110 Â 60 mm and a base cell size of 1 mm.Based on the recommendation given in [34], the near nozzle resolution is set to 64.5 lm, which is approximately 2/3 of the nozzle diameter, via an embedded grid refinement with 4 levels, each halving the grid size in each direction, resulting in an overall number of approximately 12.8 million cells.An implicit second-order scheme is utilized for time discretization with a maximum Courant number set to 0.5, resulting in a minimum time step size of Dt = 33 ns during the reactive simulation.For spatial discretization, CDS is used for the convective flux in the momentum equation and a TVD scheme utilizing the Sweby limiter [50] for the convective scalar fluxes.The Lagrangian parcels, representing the liquid phase are injected from a disc source inside the domain with the center at y = 1.5 mm and a diameter equal to that of the nozzle.The half cone opening angle is 10.75°.
The Rate Of Injection (ROI) is obtained using the virtual injection rate generator [51] and is shown in Figure 4.Note that the ROI for the Spray A injector 210677 used for the non-reactive simulation is scaled to the given experimental overall injected mass of 13.77 mg for the injection time of 6 ms [52].About 0.76 million parcels/ms are injected for the injector 210370 resulting in an averaged parcel mass of 3.4 Â 10 À6 mg/parcel following the recommendation given in [53,54].Note that the simulations are performed until 2 ms after the start of injection.

Results and discussion
To validate the LES setup for spray simulations, the ECN Spray A under inert conditions is investigated first.The corresponding results are presented in Section 4.1.Based on the validated LES spray setup, the original UFPV-LES approach for spray flame simulations is assessed, and the results are shown in Section 4.2.The distribution of the scalar dissipation rate is analyzed in Section 4.3.The RUFPV approach is evaluated in Section 4.4.

Validation of the LES setup under inert conditions
The validation of the LES spray setup under inert conditions is based on a comparison with experimental data for the predicted liquid and (fuel) vapor penetration lengths, as well as the mixture fraction distribution.From the simulation results, the liquid penetration is evaluated as the axial distance to the nozzle orifice, where 99% of the liquid mass is incorporated and the vapor penetration as the minimum distance to the nozzle, where the threshold of Z = 0.001 is reached.The corresponding results are shown in Figure 5.The experimental data is obtained from Mie scattering and Schlieren measurements [52].An overall good agreement between the experimental and numerical data is observed.
The mixture formation is validated by means of the experimentally determined radial distribution of the ensemble-averaged mixture fraction field.This information is obtained from Raman measurements [52].The single-shot   [51] with the overall injected mass evaluated as the integral of the given ROI profiles.Note, that the ROI for the injector 201677 is scaled to ensure the experimentally given value for the injected mass of 13.77 mg [52].Fig. 5. Temporal evolution of vapor penetration VP (dashed lines) and liquid penetration LP (solid lines) for the ECN Spray A under non-reactive conditions from the LES (blue curves) and experiments obtained by Schlieren measurements [52] and Mie scattering [52] (black curves).The experimental standard deviation is marked by gray shading.
LES results are temporally averaged over a period of 1.5-2.0ms After Start Of Injection (ASOI), where the spray reached a quasi-steady state in the regions of interest and additionally in circumferential direction, e.g., see [14,20].The comparison suggests that the mixture fraction is underpredicted at the spray centerline (Fig. 6).This deviation is, however, in the range of the data previously shown in literature for similar simulation strategies [14,20].Overall, the presented numerical LES framework delivers good agreement under inert conditions.In the following section the UFPV-LES approach for reactive sprays cases is evaluated.

Validation of the UFPV-LES approach under reactive conditions
In the following section, the UFPV-LES approach is utilized to simulate the ECN Spray A under reactive conditions.In Figure 7, the ignition process is visualized based on the temperature distribution at selected moments in time, in particular 0.30, 0.40 and 1.00 ms ASOI.Ignition is seen to start at the periphery of the spray head and then subsequently develop towards the spray flanks due to increasing residence times in the outer shear layer of the evolving jet.With this, the UFPV-LES approach correctly reproduces the well known characteristics of this spray flame [55].
The ignition delay time and the steady lift-off length are evaluated following [20] based on the 2% criterion (of the maximum OH mass fraction occurring during the simulation).With an absolute value of 0.368 ms, an underprediction of the ignition delay time of 8% is found compared to the experimental value of 0.41 ms [56] and the steady lift-off length is slightly overestimated with 16.84 mm by 4.3% compared to the experimental reference value of 16.10 mm [56].A study based on a Lagrangian flamelet simulation [57] performed during this work suggests that these results can be further improved by applying a more comprehensive version of the chemical reaction mechanism incorporated in this work.
A subsequent validation of the spatial spray flame structure during the different stages of ignition is performed in the following based on experimental 355 nm-PLIF data presented by Skeen et al. [58].The PLIF signal correlates with the CH 2 O concentration and is compared to the CH 2 O mass fraction obtained using the UFPV-LES approach shown in Figure 8.The experimental signal suggests that CH 2 O builds up at the periphery of the vaporized fuel jet upstream the steady lift-off length.This can be clearly seen at 190 ls ASOI, where a first noticeable PLIF signal intensity is detected in the experiment.This signal intensity then increases with increasing distance to the nozzle.Starting from 240 ls ASOI, the signal intensity shows a maximum near the spray centerline slightly downstream of the steady lift-off length.The spatial extent of the region including noticeable signal intensity increases during the subsequent spray development and reaches a maximum at 390 ls ASOI, close to the experimentally determined ignition delay time.At 490 ls ASOI, in particular after the onset of second-stage ignition, the region of noticeable signal intensity is similar compared to the previous times, but with very low intensities at the spray head.The same qualitative behavior is observed for the CH 2 O mass fraction determined by the UFPV-LES.CH 2 O builds up at the spray flanks, develops towards the spray axis with increasing value and is latter consumed mainly on the spray head.These processes occur slightly too early, however, which is linked to the underprediction of the ignition delay time.In particular, CH 2 O can be detected in the simulation at 140 ls ASOI, where no PLIF signal intensity is detected in the experiment.At later instants in time, the first axial distance where CH 2 O is observed in the simulation is also smaller than this length for the PLIF signal.The predicted CH 2 O consumption at the spray tip occurs between 340 and 390 ls ASOI corresponding to the underestimation of the ignition delay time of around 0.04 ms.
In summary, the main characteristics observed in the experimental PLIF signal are well reproduced by the UFPV-LES approach with a temporal offset due to an underestimation in the ignition delay time in acceptable range remaining.As a result, the approach is a suitable means of analyzing the spatial scalar dissipation rate distribution performed in the next section.

Distribution of the scalar dissipation rate
After the validation of the UFPV-LES approach, the spatial distribution of the scalar dissipation rate e v st and its influence on the ignition process is discussed next.In Figure 9, the spatial scalar dissipation rate distribution is shown for 0.40 ms and 1.00 ms ASOI in the top row together with the corresponding temperature field in the bottom row.High values of e v st are clearly present upstream of the liquid penetration length, and decrease downstream.Overall, small scalar dissipation rate values are observed after the experimentally determined lift-off length is passed.Furthermore, upstream of the lift-off length, large values of e v st are found near the spray centerline.With increasing distance to the nozzle, this is not the case and e v st is larger at the border of the vaporized fuel jet than at the axis for both moments in time shown.The temperature distribution presented in the lower row of Figure 9 shows the stabilization of the flame downstream the region with high e v st values.Hence, e v st is potentially acting as ignition inhibition upstream the lift-off length.This observation is in line with findings from other groups [16,59].It leads notably to the idea presented in [16] of connecting the lift-off length to the scalar dissipation rate limiting the ignition process v st,ign or the extinction scalar dissipation rate, respectively.
In the following, the influence of the region between the stable and unstable branches of the S-curve is investigated looking at the stoichiometric conditions representative of the ignition process.In particular, the temperature under these conditions is analyzed in conjunction with the respective e v st values.A slice through the corresponding data is visualized in the lower row of Figure 9 based on the iso-line of e Z ¼ e Z st colored by e v st used for the table look-up.One fact, which is more obvious in this visualization than in the discussion above is the presence of large e v st values at the spray tip at 0.40 ms ASOI, where second stage-ignition is still ongoing in the simulation.
The question remains of whether such e T À e v st combinations only slow down the ignition process or lead to extinction.To address this issue, the temperature under stoichiometric conditions e T st ¼ e T ð e Z st Þ is plotted in e v st space in conjunction with the S-curve in Figure 10.At 0.40 ms ASOI, fluid elements with elevated temperatures and below the unstable branch are observed.These fluid elements undergo extinction to a certain extent.This finding suggests that this effect needs to be taken into consideration in the look-up table.However, only a few occurrences can be detected.This is further quantified in the lower row of Figure 10 where the mass distribution in the e T st À e v st space is shown.The mass that undergoes extinction is clearly relatively small.The same holds true for fluid elements which undergo a slowdown in ignition when passing e v st;ign above the unstable branch at 1.00 ms ASOI shown in the right column of Figure 10.Hence, states between the stable and unstable branches are present in the coupled spray simulation, but only for small amount of mass.This suggests that the error introduced by neglecting this region during the FLUT generation will not influence the spray flame predictions significantly.This is analyzed in more detail in the next section.

Evaluation of the reduced UFPV approach
Based on the previous analysis of the spatial e v st distribution, the RUFPV approach, presented in Section 3.3.3, is evaluated and compared to the original UFPV approach.First, the assessment of the underlying tabulated manifold  is conducted by means of an a-priori analysis.Second, the influence in a coupled spray simulation is investigated, denoted by a-posteriori analysis.

A-priori analysis
Within the a-priori analysis, the RUFPV solution is compared to the UFPV reference solution at the same conditions.Therefore, the FLUT request parameter vector obtained by the UFPV-LES results e w UFPV ¼ ½ e Z UFPV ; g Z    together with the differences between the source term obtained from the RUFPV FLUT in the a-priori analysis and that of UFPV-LES approach e for 0.40 ms ASOI.
is shown along with for t = 0.40 ms ASOI.At this representative time, second-stage ignition is still established in the simulation.In line with the e v st distribution presented in Section 4.3, the part of the spray flame that is affected by the reduction in the e v st state space is limited to a narrow region at the edges of the vaporized fuel jet.Furthermore, the order of magnitude of is low compared to _ x UFPV Y C in the region of main ignition, suggesting a minor difference in the ignition prediction between the UFPV and the RUFPV approach.This is investigated based on the coupled simulation in the next section.

A-posteriori analysis
In the following section, the RUFPV approach is used in a coupled spray LES and the corresponding results are compared with those presented in Section 4.2 based on the original UFPV-LES approach.To investigate the difference in the global ignition behavior, the evolution of the maximum temperature and OH mass fraction over time is shown in Figure 12.Only slight differences are seen.In particular, the ignition delay time is slightly lower when using the RUFPV compared to the original UFPV approach.
The results for the local CH 2 O mass fraction and temperature distribution are shown in Figure 13 for the same moments in time used in Figure 7.At 0.30 ms ASOI, both simulations show a similar CH 2 O distribution without noticeable temperature increase in the shown range.At 0.40 ms ASOI, the consumption of CH 2 O at the spray tip connected with an increase in temperature is predicted by both approaches.The region, where this second-stage ignition takes place, seems to extend a bit more upstream for the RUFPV approach compared to the UFPV approach.This observation is in line with the smaller ignition delay time for the RUFPV approach discussed above.However, the differences are relatively small.The flame structures obtained by the two approaches show an overall close agreement which is also preserved at 1.00 ms ASOI, where parts of the flame are in a steady state.A slight difference in the length of moderate temperature along the spray axis is observed at this time which is potentially caused by a periodic detachment of fuel-rich mixtures [61] after the second-stage ignition.It can be concluded, that neglecting the unstable branch of the S-curve is a suitable assumption for the ECN Spray A baseline conditions.

Summary and conclusion
In this work, the UFPV-LES approach [15] is evaluated in the simulation of the ECN Spray A baseline conditions.The LES setup is first assessed under inert conditions showing a very good agreement in terms of penetration lengths with the experimental data.Only a slight underestimation is observed in the mixture fraction distribution, similar to computational results previously reported in the literature [14,20].The UFPV-LES results for reactive conditions show an underestimation of the ignition delay time by 8% and an overestimated steady lift-off length by 4%.In addition to these global ignition quantities, the local flame structure is investigated in terms of CH 2 O mass fraction with experimental 355 nm-PLIF data [58] showing an overall good agreement with remaining deviations connected to the difference in the ignition delay time.The results reveal the presence of increasing scalar dissipation rates downstream the ignition location for distinct fluid elements, leading to delayed ignition and local extinction.They are, however, limited to a relatively small amount of mass near the jet border.The reduced tabulation (RUFPV) approach, solely including flamelet solutions starting from the adiabatic mixing line between fuel and oxidizer, is subsequently assessed in the direct comparison with the original UFPV approach.First, an a-priori analysis is performed to identify regions which are affected by the state space reduction in terms of deviations in the progress variable source term.It is found, that this difference is only noticeable in a narrow region at the edges of the fuel jet which is attributed to the scalar dissipation rate distribution.The deviation is in particular small in regions of large source term values where the spray flame stabilizes.A coupled LES using the reduced approach hence leads to a close agreement in terms of global ignition characteristics and spatial CH 2 O and temperature distribution compared to the original UFPV-LES approach.
In summary, the UFPV-LES approach is well suited for simulating the ECN Spray A baseline conditions, and the approach, solely incorporating igniting flamelet solutions is justified by the limited influence of high scalar dissipation rate values downstream of the lift-off length after the onset of ignition under the investigated operating conditions.
max ¼ Y C;max Z CFD ; v CFD st À Á the maximum Y C value for the local Z CFD ; v CFD st À Á combination on the lower stable branch of the S-curve for v CFD st > v st;ign .A FLUT request based on the reduced state space by applying equation (

Fig. 2 .
Fig. 2. Excerpt of the FLUT obtained from unsteady flamelet solutions (dots, colored by their respective Y C value) under stoichiometric conditions.The boundary conditions are given in Table1.Solid black lines are built by stable steady flamelet solutions and the dashed black line denotes unstable steady solutions.Black arrows indicate the direction of evolution of the unsteady solutions depending on their initial condition.The value of v st until which high-temperature ignition is possible is marked by a vertical gray dashed line and denoted by v st;ign .Possible T À v st trajectories for the original UFPV approach are indicated by a red solid line while the corresponding trajectory for the RUFPV approach following equation (25) is shown by a blue dashed arrow.

Fig. 3 .
Fig. 3. Computational mesh discretizing the gas phase colored by the solution of the Favre-filtered mixture fraction field at 0.40 ms after start of injection.Black dots denote parcel discretizing the liquid phase.

Fig. 4 .
Fig.4.Rate Of Injection (ROI) profiles for the ECN Spray A injectors.ROI shapes are obtained from the virtual injection rate generator[51] with the overall injected mass evaluated as the integral of the given ROI profiles.Note, that the ROI for the injector 201677 is scaled to ensure the experimentally given value for the injected mass of 13.77 mg[52].

Fig. 7 .
Fig. 7. Distribution of the temperature T obtained using the original UFPV-LES approach for 0.30 ms (left column), 0.40 ms (middle column) and 1.00 ms ASOI (right column).

Fig. 8 .
Fig. 8. Measured CH 2 O PLIF signal [58] (left column) in comparison with the CH 2 O mass fraction obtained by the UFPV-LES approach (right column).Vertical white line denotes measured steady lift-off length of 16.10 mm [56].Experimental data reprinted from Skeen et al. [58], with permission from Elsevier.

00 2 N
;UFPV ; e v UFPV st ; e C UFPV T is used to perform a look-up from the RUFPV FLUT.To estimate the difference in the ignition prediction, the progress variable source term obtained by the RUFPV approach _ x RUFPV Y C e w UFPV is compared to the UFPV-LES reference solution _ x UFPV Y C .The deviation of the progress variable source term between

Fig. 9 .
Fig.9.Distribution of the scalar dissipation rates at stoichiometry ṽst used for table look-up (top row) and T overlaid with the iso-line of Z ¼ Z st colored according to the ṽst (bottom row) for 0.40 ms (left column) and 1.00 ms ASOI (right column).The black horizontal lines denote the experimentally obtained liquid penetration LP of 9.6 mm[60] and the steady lift-off length LOL of 16.1 mm[56], respectively.

Fig. 10 .
Fig. 10.Scatter plot of the temperature under stoichiometric conditions T st ¼ T ð Zst Þ over the corresponding scalar dissipation rate ṽst colored by the axial distance to the nozzle (top row) and corresponding mass distribution (bottom row) for 0.40 ms (left column) and 1.00 ms (right column) ASOI.

Fig. 11 .
Fig. 11.Distribution of the progress variable source term obtained using the original UFPV-LES approach _ x UFPV Y C

Fig. 12 .
Fig.12.Evolution of the maximum temperature T max (solid lines) and OH mass fraction Ỹ OH ;max (dashed lines) during the simulation of the ECN Spray A for the UFPV (black lines) and RUFPV approach (blue lines).

Fig. 13 .
Fig.13.Distribution of the CH 2 O mass fraction Ỹ CH2O and the temperature T obtained using the original UFPV approach (top row) and the RUFPV approach (bottom row) for 0.30 ms (left column), 0.40 ms (middle column) and 1.00 ms ASOI (right column).The iso-line of Z ¼ Z st is shown as gray contour.