Issue 
Sci. Tech. Energ. Transition
Volume 77, 2022
Dossier LES4ECE’21: LES for Energy Conversion in Electric and Combustion Engines, 2021



Article Number  5  
Number of page(s)  12  
DOI  https://doi.org/10.2516/stet/2022004  
Published online  22 April 2022 
Regular Article
Evaluation of the unsteady flamelet progress variable approach in Large Eddy Simulations of the ECN Spray A
Department of Mechanical Engineering, Simulation of reactive ThermoFluid Systems, Technical University of Darmstadt, OttoBerndtStr. 2, 64287 Darmstadt, Germany
^{*} Corresponding author: scholtissek@stfs.tudarmstadt.de
Received:
15
October
2021
Accepted:
3
March
2022
Within the Unsteady Flamelet Progress Variable – Large Eddy Simulation (UFPVLES) approach the local scalar dissipation rate represents one key parameter, significantly affecting the ignition behaviour. In this study, the UFPVLES approach is evaluated for ECN Spray A baseline conditions, relevant for diesel engines. After confirming its general applicability, using experimental data under nonreacting and reacting conditions, special attention is paid to the distribution of the local scalar dissipation rate. Based on the findings of this analysis, a reduced modeling approach, considering only igniting flamelets starting from the adiabatic mixing line between the fuel and oxidizer, is investigated. The performance of this reduced approach is assessed systematically, using the UFPVLES results as a reference. Based on an apriori analysis, regions affected by the model reduction are identified and evaluated. A subsequent evaluation in an aposteriori analysis, i.e. a coupled LES, reveals similar results in terms of local flame structure as well as global ignition characteristics and confirms the applicability of the reduced model under the ECN Spray A baseline conditions.
Key words: ECN Spray A / LES / Unsteady flamelet progress variable approach
© The Author(s), published by EDP Sciences, 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 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 heavyduty 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 efuels) represents a promising way to achieve carbonneutral combustion in the shortterm. 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 hightemperature/highpressure environment under diesel enginerelevant conditions remains of practical interest. The Engine Combustion Network (ECN) [3] provides a comprehensive experimental database for the validation of numerical models for highpressure sprays. In recent years, different diesel surrogate fuels have been investigated in singlehole injectors under conditions relevant for diesel engines. Based on the ECN Spray H, operating with nheptane, the research focus has evolved towards more diesellike fuels, e.g., ndodecane in the ECN Spray A. However, longchain 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 dieselenginerelevant conditions. The fundamental assumption of flameletbased 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 onedimensional representation of the flame structure denoted by socalled flamelets. A turbulent flame is considered to be an ensemble of such 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 socalled flamelet equations is the scalar dissipation rate χ = 2D(∇Z · ∇Z), 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 χ_{st}. Using a definition for χ_{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 [11–13] which was recently applied in LES of the ECN Spray A in [14].
One alternative to the interactive flamelet simulation is the pretabulation 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 LiftOff Lengths (LOLs) and Ignition Delay Times (IDTs) were predicted within an error of 25%. The influence of χ_{st} on the IDT was connected to the time needed to reach the value of χ_{st} at which ignition can occur. Furthermore, the LOL was found to be connected to the axial location where χ_{st} equals its value, limiting the ignition process. It was further shown that the χ_{st} distribution upstream of the LOL is similar for nonreacting and reacting sprays [16]. Subsequently, Ameen [17] utilized the UFPV approach to investigate the ignition behavior of nheptane jets under dieselrelevant conditions by means of RANS and LES. In their study, they confirmed the interplay between LOL and χ_{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 χ_{st} values included in the lookup 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 autoigniting flamelet solutions are utilized in the tabulation of socalled Approximated Diffusion Flamelets (ADF) [21] in spray LES [22–29]. 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 GarcíaOliver et al. [27] who compared the ECN Spray A and Spray D sprays based on RANS simulations and Desantes et al. [28] and PérezSánchez 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 LESUFPV 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.
2 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. ndodecane at a pressure of 1500 bar is injected through a singlehole injector into a constantvolume preburn 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 () is additionally investigated. The ECN Spray A injector, serial number 210677, is simulated for nonreactive 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.
3 Modeling approach
The twophase 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.
3.1 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 semiempirical 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(1)where denotes the mass fraction of the fuel species in the gas phase in the vicinity of the droplet and ∞ corresponds to the environmental conditions. The breakup of the liquid phase is modelled utilizing the KHRT model [35] with model parameters set to B_{0} = 0.61, B_{1} = 5, C_{τ} = 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].
3.2 Gas phase
The gas phase model is formulated for a LES framework. Transport equations are solved for the Favrefiltered conservation equations for mass, momentum and total enthalpy given by(2) (3) (4)where ρ denotes the density, u_{i} the velocity in direction i, p the pressure, g_{i} the gravitational acceleration in direction i and , , and the mass, momentum and total enthalpy source terms arising from the coupling with the liquid phase. The operators and denote spatial and densityweighted (Favre) spatial filtering, respectively. The filtered viscous stress tensor and heat flux are given by(5) (6)with μ denoting the dynamic viscosity, λ 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 and denote the unclosed terms due to the application of the filter operator. Applying the Boussinesq hypothesis [36], the anisotropic part of is modeled in analogy to the viscous stress tensor as(7)
The isotropic part equals twice the subgrid scale kinetic energy and is absorbed in the filtered pressure , as usually done within the limit of low Mach number flows [37]. The subgrid scale viscosity ν_{SGS} is modeled by means of the model [38](8)where the differential operator is based on the singular values of the velocity gradient tensor σ_{i} and is defined as(9)
A value for the model constant of C_{σ} = 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(10)
The unresolved enthalpy flux is modelled by means of a gradientflux assumption leading to(11)where ν_{SGS} is obtained from the subgridscale 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(12)is utilized.
3.3 Combustion model
3.3.1 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](13) (14)
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(15)follows from the coordinate transformation. The model derived for the steady laminar counterflow diffusion flame with constant density and diffusion coefficients by Peters [7](16)is used in this study to close this set of equations. It is noted that the function is parameterized by the scalar dissipation rate at stoichiometry χ(Z = Z_{st}) = χ_{st}. The boundary conditions corresponding to the experimental setup, see Section 2, are given in Table 1. The steady solutions of the flamelet equations consist of stable and unstable solutions forming an Sshaped curve, which is abbreviated by Scurve in the following and visualized in Figure 2. Unsteady igniting flamelet solutions, starting from adiabatic mixing between fuel (Z = 1) and oxidizer (Z = 0) until a steady state is obtained, are simulated for different scalar dissipation rates at stoichiometry and stored in a Flamelet LookUp Table (FLUT). Thus, the influence of the scalar dissipation rate on the ignition delay time is considered. In addition, unsteady igniting and extinguishing solutions starting from the unstable branch of the characteristic Scurve are incorporated, which is further discussed in Section 3.3.3. The thermochemical state vector ϕ = [p, T, Y_{k}]^{T} is parameterized by Z, χ_{st} and the Lagrangianlike flamelet time τ, i.e., ϕ = ϕ(Z, χ_{st}, τ). As proposed by Pitsch and Ihme [15] and Ihme and See [8], the Lagrangianlike time is mapped to a progress variable Y_{C}, defined as a weighted sum of species mass fractions. A suitable definition must ensure a unique mapping between τ and Y_{C}. The definition applied in this study is derived and assessed in a onedimensional transient counterflow configuration under the operating conditions under investigation in [41] and given by(17)
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.
Furthermore, Y_{C} is normalized by its corresponding minimum and maximum values, reading(18)
This variable is statistically independent of Z [42] and leads to a cubically shaped lookup table, which permits an efficient table access [43, 44]. Note, that a twostage request strategy is applied to evaluate equation (18). To accurately capture the onset of ignition, the progress variable source term for C ∈ [0; C_{init} = 10^{−6}] is set to a constant and finite value preserving the mass of Y_{C} produced by chemical reactions [41]. To consider the nonresolved fluctuations of the request variables, a presumed β PDF is applied to model the marginal PDF of Z parameterized by the normalized variance(19)while a δ distribution is chosen for the marginal PDFs of C and χ_{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 CPDF, see [8]. The evaluation of such models is interesting for future studies of highpressure spray flames. The resulting parameterization of the Favrefiltered thermochemical state reads .
3.3.2 UFPVLES coupling for spray combustion
The variables needed for the table lookup are obtained by solving additional transport equations for and given by(20) (21)
The variance of the mixture fraction is obtained following [45] by (22)with [46, 47] and Sc_{SGS} = 0.4 in accordance with the Prandtl number in equation (11). The Favrefiltered scalar dissipation rate is obtained by an algebraic relation given by [48] (23)
From the local Favrefiltered scalar dissipation rate, the scalar dissipation rate under stoichiometric conditions used to parameterize the FLUT can be calculated using [49](24)
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 thermophysical properties in the LES. With this, the cooling effect due to evaporation is accounted for. This UFPVLES coupling is summarized in Figure 1.
Fig. 1
Coupling scheme of the UFPVLES approach for spray combustion. 
3.3.3 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 Scurve 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 lookup table [24]. Such an approach requires the ignition process to be handled in a specific way, since fluid elements potentially enter regions of higher χ_{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 χ_{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 χ_{st} > χ_{st,ign} and hence in the estimated temperature and thermophysical properties, respectively. To overcome this problem, the maximum value of χ_{st}, used for the table lookup after the onset of ignition, is limited to χ_{st,ign}. This is expressed by the relation(25)where denotes the value of χ_{st} obtained in the coupled simulation based on equation (24) and the maximum Y_{C} value for the local combination on the lower stable branch of the Scurve for . A FLUT request based on the reduced state space by applying equation (25) is denoted by RUFPV approach in the remainder of this study. The corresponding trajectory in the T_{st} − χ_{st} state space is indicated by the blue line in Figure 2. By this means, strong local gradients in the thermophysical 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.
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 Table 1. 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 χ_{st} until which hightemperature ignition is possible is marked by a vertical gray dashed line and denoted by χ_{st,ign}. Possible T − χ_{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. 
3.4 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 × 110 × 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 μm, 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 secondorder scheme is utilized for time discretization with a maximum Courant number set to 0.5, resulting in a minimum time step size of Δt = 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°.
Fig. 3
Computational mesh discretizing the gas phase colored by the solution of the Favrefiltered mixture fraction field at 0.40 ms after start of injection. Black dots denote parcel discretizing the liquid phase. 
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 nonreactive 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 × 10^{−6} mg/parcel following the recommendation given in [53, 54]. Note that the simulations are performed until 2 ms after the start of injection.
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]. 
4 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 UFPVLES 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.
4.1 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.
Fig. 5
Temporal evolution of Vapor Penetration (VP) (dashed lines) and Liquid Penetration (LP) (solid lines) for the ECN Spray A under nonreactive 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. 
The mixture formation is validated by means of the experimentally determined radial distribution of the ensembleaveraged mixture fraction field. This information is obtained from Raman measurements [52]. The singleshot LES results are temporally averaged over a period of 1.5–2.0 ms After Start Of Injection (ASOI), where the spray reached a quasisteady 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 UFPVLES approach for reactive sprays cases is evaluated.
Fig. 6
Averaged radial mixture fraction distribution obtained from Raman measurements [52] (black curve) and LES (blue curve) at 17.5 mm (a) and 30 mm (b) downstream the nozzle orifice. 
4.2 Validation of the UFPVLES approach under reactive conditions
In the following section, the UFPVLES 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 UFPVLES approach correctly reproduces the well known characteristics of this spray flame [55]. The ignition delay time and the steady liftoff 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 liftoff 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.
Fig. 7
Distribution of the temperature obtained using the original UFPVLES approach for 0.30 ms (left column), 0.40 ms (middle column) and 1.00 ms ASOI (right column). 
A subsequent validation of the spatial spray flame structure during the different stages of ignition is performed in the following based on experimental 355 nmPLIF 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 UFPVLES 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 liftoff length. This can be clearly seen at 190 μs 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 μs ASOI, the signal intensity shows a maximum near the spray centerline slightly downstream of the steady liftoff length. The spatial extent of the region including noticeable signal intensity increases during the subsequent spray development and reaches a maximum at 390 μs ASOI, close to the experimentally determined ignition delay time. At 490 μs ASOI, in particular after the onset of secondstage 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 UFPVLES. 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 μs 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 μs ASOI corresponding to the underestimation of the ignition delay time of around 0.04 ms.
Fig. 8
Measured CH_{2}O PLIF signal [58] (left column) in comparison with the CH_{2}O mass fraction obtained by the UFPVLES approach (right column). Vertical white line denotes measured steady liftoff length of 16.10 mm [56]. Experimental data reprinted from Skeen et al. [58], with permission from Elsevier. 
In summary, the main characteristics observed in the experimental PLIF signal are well reproduced by the UFPVLES 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.
4.3 Distribution of the scalar dissipation rate
After the validation of the UFPVLES approach, the spatial distribution of the scalar dissipation rate 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 are clearly present upstream of the liquid penetration length, and decrease downstream. Overall, small scalar dissipation rate values are observed after the experimentally determined liftoff length is passed. Furthermore, upstream of the liftoff length, large values of are found near the spray centerline. With increasing distance to the nozzle, this is not the case and 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 values. Hence, is potentially acting as ignition inhibition upstream the liftoff 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 liftoff length to the scalar dissipation rate limiting the ignition process χ_{st,ign} or the extinction scalar dissipation rate, respectively.
Fig. 9
Distribution of the scalar dissipation rates at stoichiometry used for table lookup (top row) and temperature overlaid with the isoline of colored according to the (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 liftoff length LOL of 16.1 mm [56], respectively. 
In the following, the influence of the region between the stable and unstable branches of the Scurve 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 v alues. A slice through the corresponding data is visualized in the lower row of Figure 9 based on the isoline of colored by used for the table lookup. One fact, which is more obvious in this visualization than in the discussion above is the presence of large values at the spray tip at 0.40 ms ASOI, where second stageignition is still ongoing in the simulation.
The question remains of whether such combinations only slow down the ignition process or lead to extinction. To address this issue, the temperature under stoichiometric conditions is plotted in space in conjunction with the Scurve 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 lookup 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 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 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.
Fig. 10
Scatter plot of the temperature under stoichiometric conditions over the corresponding scalar dissipation rate 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. 
4.4 Evaluation of the reduced UFPV approach
Based on the previous analysis of the spatial 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 apriori analysis. Second, the influence in a coupled spray simulation is investigated, denoted by aposteriori analysis.
4.4.1 Apriori analysis
Within the apriori analysis, the RUFPV solution is compared to the UFPV reference solution at the same conditions. Therefore, the FLUT request parameter vector obtained by the UFPVLES results is used to perform a lookup from the RUFPV FLUT. To estimate the difference in the ignition prediction, the progress variable source term obtained by the RUFPV approach is compared to the UFPVLES reference solution . The deviation of the progress variable source term between both approaches is quantified by . In Figure 11, is shown along with ϵ for t = 0.40 ms ASOI. At this representative time, secondstage ignition is still established in the simulation. In line with the distribution presented in Section 4.3, the part of the spray flame that is affected by the reduction in the state space is limited to a narrow region at the edges of the vaporized fuel jet. Furthermore, the order of magnitude of ϵ is low compared to 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.
Fig. 11
Distribution of the progress variable source term obtained using the original UFPVLES approach together with the differences between the source term obtained from the RUFPV FLUT in the no analysis and that of UFPVLES approach ϵ for 0.40 ms ASOI. 
4.4.2 Aposteriori 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 UFPVLES 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.
Fig. 12
Evolution of the maximum temperature (solid lines) and OH mass fraction (dashed lines) during the simulation of the ECN Spray A for the UFPV (black lines) and RUFPV approach (blue lines). 
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 secondstage 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 fuelrich mixtures [61] after the secondstage ignition. It can be concluded, that neglecting the unstable branch of the Scurve is a suitable assumption for the ECN Spray A baseline conditions.
Fig. 13
Distribution of the CH_{2}O mass fraction and the temperature 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 isoline of is shown as gray contour. 
5 Summary and conclusion
In this work, the UFPVLES 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 UFPVLES results for reactive conditions show an underestimation of the ignition delay time by 8% and an overestimated steady liftoff 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 nmPLIF 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 apriori 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 UFPVLES approach.
In summary, the UFPVLES 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 liftoff length after the onset of ignition under the investigated operating conditions.
Acknowledgments
Calculations for this research were conducted on the Lichtenberg II high performance computer of the TU Darmstadt. The authors further thank Alessandro Stagni from Politecnico di Milano for kindly providing the mechanism for the ignition of higher hydrocarbons.
References
 Reitz R.D., Ogawa H., Payri R., Fansler T., Kokjohn S., Moriyoshi Y., Agarwal A.K., Arcoumanis D., Assanis D., Bae C., Boulouchos K., Canakci M., Curran S., Denbratt I., Gavaises M., Guenthner M., Hasse C., Huang Z., Ishiyama T., Johansson B., Johnson T.V., Kalghatgi G., Koike M., Kong S.C., Leipertz A., Miles P., Novella R., Onorati A., Richter M., Shuai S., Siebers D., Su W., Trujillo M., Uchida N., Vaglieco B.M., Wagner R.M., Zhao H. (2020) IJER editorial: The future of the internal combustion engine, Int. J. Engine Res. 21, 1, 3–10. [CrossRef] [Google Scholar]
 Onarati A., Raúl Payri B.M., Vaglieco A.K., Agarwal C., Bae G., Bruneaux M., Canakci M., Gavaises M., Günthner C., Hasse S., Kokjohn S.C., Kong Y., Moriyoshi R., Novella A., Pesyridis R., Reitz T., Ryan R.Wagner, Zhao H. (2022) The role of hydrogen for future internal combustion engines, Int. J. Engine Res.23, 4, 529–540. [CrossRef] [Google Scholar]
 Pickett L.M. (2013) Engine Combustion Network, Sandia National Laboratory. https://ecn.sandia.gov. [Google Scholar]
 Kahila H., Wehrfritz A., Kaario O., Masouleh M.G., Maes N., Somers B., Vuorinen V. (2018) LargeEddy simulation on the influence of injection pressure in reacting Spray A, Combust. Flame 191, 142–159. [CrossRef] [Google Scholar]
 Kundu P., Scroggins J., Ameen M.M. (2020) A novel in situ flamelet tabulation methodology for the representative interactive flamelet model, Combust. Sci. Technol. 192, 1, 1–25. [CrossRef] [Google Scholar]
 Peters N. (1988) Laminar flamelet concepts in turbulent combustion, in: Symposium (International) on Combustion, Vol. 21, 1, Elsevier, pp. 1231–1250. [CrossRef] [Google Scholar]
 Peters N. (1984) Laminar diffusion flamelet models in nonpremixed turbulent combustion, Prog. Ener. Combust. Sci. 10, 3, 319–339. [CrossRef] [Google Scholar]
 Ihme M., See Y.C. (2010) Prediction of autoignition in a lifted methane/air flame using an unsteady flamelet/progress variable model, Combust. Flame 157, 10, 1850–1862. [CrossRef] [Google Scholar]
 Pitsch H., Wan Y.P., Peters N. (1995) Numerical investigation of soot formation and oxidation under diesel engine conditions, SAE Technical Papers 412. [Google Scholar]
 Pitsch H., Barths H., Peters N. (1996) Threedimensional modeling of NOx and soot formation in DIdiesel engines using detailed chemistry based on the interactive flamelet approach, SAE Trans. 2010–2024. [Google Scholar]
 Barths H., Antoni C., Peters N. (1998) Threedimensional simulation of pollutant formation in a DI Diesel engine using multiple interactive flamelets, SAE Trans. 987–997. [Google Scholar]
 Barths H., Hasse C., Bikas G., Peters N. (2000) Simulation of combustion in direct injection diesel engines using a Eulerian particle flamelet model, Proc. Combust. Inst. 28, 1, 1161–1168. [CrossRef] [Google Scholar]
 Barths H., Hasse C., Peters N. (2000) Computational fluid dynamics modelling of nonpremixed combustion in direct injection diesel engines, Int. J. Engine Res. 1, 3, 249–267. [CrossRef] [Google Scholar]
 Davidovic M., Falkenstein T., Bode M., Cai L., Kang S., Hinrichs J., Pitsch H. (2017) LES of nDodecane spray combustion using a multiple representative interactive flamelets model, Oil Gas Sci. Technol.  Rev. IFP Energies nouvelles 72, 5, 29. [CrossRef] [Google Scholar]
 Pitsch H., Ihme M. (2005) An unsteady/flamelet progress variable method for les of nonpremixed turbulent combustion, in: 43rd AIAA Aerospace Sciences Meeting and Exhibit – Meeting Papers, pp. 2593–2606. [Google Scholar]
 Bajaj C., Ameen M., Abraham J. (2013) Evaluation of an unsteady flamelet progress variable model for autoignition and flame liftoff in diesel jets, Combust. Sci. Technol. 185, 3, 454–472. [CrossRef] [Google Scholar]
 Ameen M.M. (2014) Unsteady flamelet progress variable modeling of reacting diesel jets, Dissertation, Purdue University. [Google Scholar]
 Lucchini T., D’Errico G., Onorati A., Frassoldati A., Stagni A., Hardy G. (2017) Modeling nonpremixed combustion using tabulated kinetics and different fame structure assumptions, SAE Int. J. Engines 10, 2, 593–607. [CrossRef] [Google Scholar]
 Bekdemir C., Somers L.M.T., De Goey L.P.H. (2011) Modeling diesel engine combustion using pressure dependent Flamelet Generated Manifolds, Proc. Combust. Inst. 33, 2, 2887–2894. [CrossRef] [Google Scholar]
 Wehrfritz A., Kaario O., Vuorinen V., Somers B. (2016) Large Eddy Simulation of ndodecane spray flames using Flamelet Generated Manifolds, Combust. Flame 167, 113–131. [Google Scholar]
 Michel J.B, Colin O., Veynante D. (2008) Modeling ignition and chemical structure of partially premixed turbulent flames using tabulated chemistry, Combust. Flame 152, 1–2, 80–99. [CrossRef] [Google Scholar]
 Tillou J., Michel J.B., Angelberger C., Veynante D. (2014) Assessing LES models based on tabulated chemistry for the simulation of Diesel spray combustion, Combust. Flame 161, 2, 525–540. [CrossRef] [Google Scholar]
 AubagnacKarkar D., Michel J.B., Colin O., Darabiha N. (2018) Combustion and soot modelling of a highpressure and hightemperature Dodecane spray, Int. J. Engine Res. 19, 4, 434–448. [CrossRef] [Google Scholar]
 Winklinger J.F. (2014) Implementation of a combustion model based on the flamelet concept and its application to turbulent reactive Sprays, PhD Thesis, Universitat Politecnica de Valencia. [Google Scholar]
 Desantes J.M., GarcíaOliver J.M., Novella R., PérezSánchez E.J. (2017) Application of an unsteady flamelet model in a RANS framework for spray A simulation, Appl. Therm. Eng. 117, 50–64. [CrossRef] [Google Scholar]
 Payri F., GarcíaOliver J.M., Novella R., PérezSánchez E.J. (2019) Influence of the ndodecane chemical mechanism on the CFD modelling of the diesellike ECN Spray A flame structure at different ambient conditions, Combust. Flame 208, 198–218. [CrossRef] [Google Scholar]
 GarcíaOliver J.M., Novella R., Pastor J.M., Pachano L. (2020) Computational study of ECN Spray A and Spray D combustion at different ambient temperature conditions, Transport. Eng. 2, 100027. [CrossRef] [Google Scholar]
 Desantes J.M., GarcíaOliver J.M., Novella R., PérezSánchez E.J. (2020) Application of a flameletbased CFD combustion model to the LES simulation of a diesellike reacting spray, Comput. Fluids 200, 104419. [CrossRef] [MathSciNet] [Google Scholar]
 PérezSánchez E.J., GarcíaOliver J.M., Novella R., Pastor J.M. (2020) Understanding the diesellike spray characteristics applying a flameletbased combustion model and detailed Large Eddy Simulations, Int. J. Engine Res. 21, 1, 134–150. [CrossRef] [Google Scholar]
 Michel J.B., Colin O., Veynante D. (2009) Comparison of differing formulations of the PCM model by their application to the simulation of an autoigniting H_{2}/air Jet, Flow Turbul. Combust. 83, 1, 33–60. [CrossRef] [Google Scholar]
 Dukowicz J.K. (1980) A particlefluid numerical model for liquid sprays, J. Comput. Phys. 35, 2, 229–253. [CrossRef] [MathSciNet] [Google Scholar]
 Stiesch G. (2003) Modeling engine spray and combustion processes, Springer, Berlin Heidelberg, Berlin, Heidelberg. [CrossRef] [Google Scholar]
 Baumgarten Carsten (2006) Mixture formation in internal combustion engine, SpringerVerlag, Berlin/Heidelberg. [Google Scholar]
 Wehrfritz A., Vuorinen V., Kaario O., Larmi M. (2013) Large eddy simulation of highvelocity fuel sprays: Studying mesh resolution and breakup model effects for spray A, Atom. Sprays 23, 5, 419–442. [CrossRef] [Google Scholar]
 Reitz R.D. (1987) Modeling atomization processes in highpressure vaporizing sprays, Atom. Spray Technol. 3, 309–337. [Google Scholar]
 Boussinesq J. (1877) Théorie de l’écoulement tourbillant, Mémoires présentés par divers savants à l’Académie des Sciences 23, 46–50. [Google Scholar]
 Poinsot T., Veynante D. (2012) Theoretical and numerical combustion, 3rd edn., RT Edwards, Inc. [Google Scholar]
 Nicoud F., Toda H.B., Cabrit O., Bose S., Lee J. (2011) Using singular values to build a subgridscale model for Large Eddy Simulations, Phys. Fluids 23, 8, 085106. [CrossRef] [Google Scholar]
 Pitsch H., Steiner H. (2000) LargeEddy Simulation of a turbulent piloted methane/air diffusion flame (Sandia flame D), Phys. Fluids 12, 10, 2541–2554. [CrossRef] [Google Scholar]
 Kempf A., Flemming F., Janicka J. (2005) Investigation of lengthscales, scalar dissipation, and flame orientation in a piloted diffusion flame by LES, Proc. Combust. Inst. 30, 1, 557–565. [CrossRef] [Google Scholar]
 Sun Z., Gierth S., Pollack M., Hasse C., Scholtissek A. (2021) Ignition under strained conditions: unsteady flamelet progress variable modeling for diesel engine conditions in the transient counterflow configuration, Combust. Flame 240, 111841. [Google Scholar]
 Domingo P., Vervisch L., Veynante D. (2008) Largeeddy simulation of a lifted methane jet flame in a vitiated coflow, Combust. Flame 152, 3, 415–432. [CrossRef] [Google Scholar]
 Weise S., Messig D., Meyer B., Hasse C. (2013) An abstraction layer for efficient memory management of tabulated chemistry and flamelet solutions, Combust. Theor. Model. 17, 411–430. [CrossRef] [Google Scholar]
 Weise S., Hasse C. (2015) Reducing the memory footprint in Large Eddy Simulations of reactive flows, Parallel Comput. 49, 50–65. [CrossRef] [MathSciNet] [Google Scholar]
 Pierce C.D., Moin P. (2004) Progressvariable approach for largeeddy simulation of nonpremixed turbulent combustion, J. Fluid Mech. 504, 73–97. [CrossRef] [MathSciNet] [Google Scholar]
 Popp S., Hunger F., Hartl S., Messig D., Coriton B., Frank J.H., Fuest F., Hasse C. (2015) LES flameletprogress variable modeling and measurements of a turbulent partiallypremixed dimethyl ether jet flame, Combust. Flame 162, 8, 3016–3029. [CrossRef] [Google Scholar]
 Hunger F., Zulkifli M.F., Williams B.A.O., Beyrau F., Hasse C. (2016) A combined experimental and numerical study of laminar and turbulent nonpiloted oxyfuel jet flames using a direct comparison of the Rayleigh signal, Flow Turbulence Combust. 97, 1, 231–262. [CrossRef] [Google Scholar]
 Steiner H., Pitsch H. (2000) Scalar mixing and dissipation rate in largeeddy simulations of nonpremixed turbulent combustion, Proc. Combust. Inst. 28, 1–9. [Google Scholar]
 Hellström T. (1997) RIF Implementation and Testing, Technical Report. [Google Scholar]
 Sweby P.K. (1984) High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal. 21, 5, 995–1011. [CrossRef] [MathSciNet] [Google Scholar]
 Payri R., Salvador F.J., Gimeno J., Bracho G. (2013) Virtual Injection Rate Generator. https://www.cmt.upv.es/ECN03.aspx. [Google Scholar]
 Pickett L.M., Abraham J.P., Bajaj C. (2009) ECN Diesel Database, SprayA nominal conditions (nonreactive), dataset bkldaAL4. [Google Scholar]
 Wehrfritz A. (2016) Large Eddy Simulation of fuel spray combustion, PhD Thesis, Aalto University. [Google Scholar]
 Gadalla M., Kannan J., Tekgül B., Karimkashi S., Kaario O., Vuorinen V. (2020) Largeeddy simulation of ECN Spray A: sensitivity study on modeling assumptions, Energies 13, 13, 3360. [CrossRef] [Google Scholar]
 Maes N., Meijer M., Dam N., Somers B., Toda H.B., Bruneaux G., Skeen S.A., Pickett L.M., Manin J. (2016) Characterization of Spray A flame structure for parametric variations in ECN constantvolume vessels using chemiluminescence and laserinduced fluorescence, Combust. Flame 174, 138–151. [CrossRef] [Google Scholar]
 Pickett Lyle M., Skeen S.A. (2012) ECN diesel database, Spray A nominal condtions, dataset jkldnAL4. https://ecn.sandia.gov/ecndatasearch/ . [Google Scholar]
 Dahms R.N., Paczko G.A., Skeen S.A., Pickett L.M. (2017) Understanding the ignition mechanism of highpressure spray flames, Proc. Combust. Inst. 36, 2, 2615–2623. [CrossRef] [Google Scholar]
 Skeen S.A., Manin J., Pickett L.M. (2015) Simultaneous formaldehyde PLIF and highspeed schlieren imaging for ignition visualization in highpressure spray flames, Proc. Combust. Inst. 35, 3, 3167–3174. [CrossRef] [Google Scholar]
 Yen M., Abraham J. (2013) Modeling lifted diesel jets: Insights into the correlation between flame liftoff height and soot formation, in: 8th US National Combustion Meeting, Vol. 1, pp. 579–591. [Google Scholar]
 Pickett L.M. ECN diesel database, Spray A nominal condtions, dataset AJKLDNLASRFA, 2017. https://ecn.sandia.gov/ecndatasearch/. [Google Scholar]
 Sim H.S., Maes N., Weiss L., Pickett L.M., Skeen S.A. (2020) Detailed measurements of transient twostage ignition and combustion processes in highpressure spray flames using simultaneous highspeed formaldehyde PLIF and schlieren imaging, Proc. Combust. Inst. 38, 5713–5721. [Google Scholar]
All Tables
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.
All Figures
Fig. 1
Coupling scheme of the UFPVLES approach for spray combustion. 

In the text 
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 Table 1. 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 χ_{st} until which hightemperature ignition is possible is marked by a vertical gray dashed line and denoted by χ_{st,ign}. Possible T − χ_{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. 

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

In the text 
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]. 

In the text 
Fig. 5
Temporal evolution of Vapor Penetration (VP) (dashed lines) and Liquid Penetration (LP) (solid lines) for the ECN Spray A under nonreactive 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. 

In the text 
Fig. 6
Averaged radial mixture fraction distribution obtained from Raman measurements [52] (black curve) and LES (blue curve) at 17.5 mm (a) and 30 mm (b) downstream the nozzle orifice. 

In the text 
Fig. 7
Distribution of the temperature obtained using the original UFPVLES approach for 0.30 ms (left column), 0.40 ms (middle column) and 1.00 ms ASOI (right column). 

In the text 
Fig. 8
Measured CH_{2}O PLIF signal [58] (left column) in comparison with the CH_{2}O mass fraction obtained by the UFPVLES approach (right column). Vertical white line denotes measured steady liftoff length of 16.10 mm [56]. Experimental data reprinted from Skeen et al. [58], with permission from Elsevier. 

In the text 
Fig. 9
Distribution of the scalar dissipation rates at stoichiometry used for table lookup (top row) and temperature overlaid with the isoline of colored according to the (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 liftoff length LOL of 16.1 mm [56], respectively. 

In the text 
Fig. 10
Scatter plot of the temperature under stoichiometric conditions over the corresponding scalar dissipation rate 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. 

In the text 
Fig. 11
Distribution of the progress variable source term obtained using the original UFPVLES approach together with the differences between the source term obtained from the RUFPV FLUT in the no analysis and that of UFPVLES approach ϵ for 0.40 ms ASOI. 

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

In the text 
Fig. 13
Distribution of the CH_{2}O mass fraction and the temperature 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 isoline of is shown as gray contour. 

In the text 
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