{'fraction_non_alphanumeric': 0.0255839549957197, 'fraction_numerical': 0.010028127675186498, 'mean_word_length': 4.727934995797142, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Numerical analysis of dilute methanol spray flames in vitiated coflow using extended Flamelet Generated Manifold model

The present work focuses on the large eddy simulation (LES) and the study of turbulent dilute methanol spray flames in vitiated coflow using the secondary oxidizer Flamelet Generated Model (FGM). The modified FGM model uses an additional secondary oxidizer parameter in addition to the three other parameters previously used for spray flames progress variable, mixture fraction, and enthalpy. The results for gas phase and droplet properties are validated against the dilute methanol spray flame database for varying fuel injection amounts. The droplet statistics and the lift-off flame heights are accurately captured for all the cases. A proper orthogonal decomposition (POD) of the scalar fluctuating hydroxyl radical (OH) field and velocity-temperature field capture the flame structures in the downstream region of ignition kernels. The detailed POD analysis reveals that the base frequency of the dominant OH field equals that of the dominant vortical structure of 67.3 Hz. The flame propagation happens around these dominant vortical structures because of the less-strained fluid mixing.

3 burners where the main flame and the secondary burner employ the same fuel. In the dilute methanol auto-igniting flame used in the present study, the hot coflow is generated from a H2/air flame and does not contain the product of the main flame, thus rendering the DAFGM 27 inapplicable. Instead, in the current study, the Flamelet Generated model (FGM) is extended with an additional parameter to distinguish between the two oxidizers with different compositions and temperatures. The FGM structure is similar to that developed by Sarras et al. 26 . The model is validated against the auto-igniting methanol flame as a test case.
The availability of extensive data related to droplet statistics and flame structure allows for a complete test. However, the relation between flow properties and chemical reaction is not very clear from single-point statistics. To provide deeper insight into the reacting flow, we perform a proper orthogonal decomposition (POD) on the velocity-temperature field and the OH field of the flame to identify the dominant fluctuating structures. These structures help us find the ignition kernels and the flame growth locations that, otherwise, are difficult to locate in an instantaneous field. The procedure allows seeing relations between mixture fraction fields, vortical structures (fluid strain), and the creation of a flame, using the OH species as a marker for flame structure [28][29][30] .

II. NUMERICAL METHODOLOGY
The multiphase simulations are carried out using an Eulerian-Lagrangian approach. The dispersed (liquid) phase is solved with the Lagrangian approach, while the continuous phase is solved with the Eulerian framework. The gas-phase reaction of evaporated fuel and gaseous oxidizer is modeled with the FGM (Flamelet Generated Manifold) model. Eulerian equations, Lagrangian equations, and FGM are explained in the following.

A. Basic Governing Equations
Filtered transport equations are solved for the Eulerian (gas) phase, continuity equation and momentum equation, energy equation, and scalar equations related to the FGM model. Continuity and momentum equations, including sources describing transfer between the two phases, are given by: The term t ij in Eq. 2 denotes the stress tensor, including laminar and subgrid-scale effects. The unresolved stress is closed by the dynamic k-equation model, which is a one-equation eddy viscosity SGS model. This model was primarily developed by Kim & Menon 31 based on the previous works of Germano et al. 32 . In recent years, many improvements to the models have been proposed 33 (4) and eff  is the effective kinematic viscosity (both molecular and sub-grid viscosity). The turbulent kinetic energy is obtained from (5) where sgs k is the sub-grid scale kinetic energy. The k C , e C constants are computed based on the dynamic formulation from Germano et al. 32 Mean scalar equations (enthalpy and the scalars of the FGM model) are closed using the gradient diffusion assumption for the unresolved flux. The partial differential equations (PDEs) are discretized using a Finite Volume Method (FVM) code implemented in OpenFOAM-v19.0 35 . The convective flux discretization uses a second-order TVD (Total Variation Diminishing) scheme. In contrast, viscous flux discretization involves a second-order central scheme. The temporal term deploys the second-order Crank-Nicolson scheme with sufficiently small time steps to maintain stability and reduce numerical diffusion.

Lagrangian Particle Tracking (LPT)
The Lagrangian particles represent the liquid droplets, which act as a point source of mass, momentum, and energy for the above equations. In the numerical approach, the full spray is represented by parcels, and computational particles represent a set of physical droplets with identical properties. In the following discussion, they will be denoted as particles or droplets. The 5 equations for particles contain sub-models for dispersion, collision, atomization, and heat transfer. The Basset-Boussinesq-Ossen (BBO) equation is solved for momentum conservation of the Lagrangian particles. The equation is given by, p p d dt = x u (7) DG p p T d m dt = + + u F F F (8) where p x , mp, p u is each particle's position, mass, and velocity, respectively.  (12) where Re p is the particle Reynolds number and m g is the gas phase dynamic viscosity. The KHRT (Kelvin-Helmholtz and Rayleigh-Taylor) model describes the aerodynamic break-up for secondary atomization. Su et al. 37 showed that the KHRT model predicts a better drop size distribution, closer to the experimental results than the KH model alone. Later, Ricart et al. 38 used the KHRT model with a break length concept in which the KH instability dominates the breakup in the area within breakup 6 length. Outside the breakup length in the region of secondary atomization, RT instability starts to compete with the KH instability. In other words, the breakup length concept, along with the KH-RT model, considers both the primary atomization and the secondary atomization. Since the present case simulates the dilute sprays, only the secondary atomization is dominant.
Thus, the KH-RT breakup of the droplets occurs without considering the breakup length concept. Hence, the model by Su et al. 37 is used in the present study. The wavelength of the KH waves on the droplet surface and the characteristic breakup time is calculated as, The acceleration t g is in the direction of the droplet trajectory, C  is a constant assumed equal to unity, and The blowing effect becomes significant at higher temperatures. In the case of temperatures reaching above boiling point, the flashing of liquid is also evident. Thus, a combination of evaporation and the flashing process is based on the model proposed by Zuo et al. 43 .
. (19) Here is the evaporation rate, which depends on the heat conductivity k , heat capacity Standards and Technology 44 is used to calculate the thermodynamic properties for the two phases at different pressures and temperatures.

C. Flamelet generation
In the FGM method, the local states in a turbulent flame are described using a few controlling variables, either representing the mixing (mixture fraction) or the chemical evolution (progress variables). The other composition variables are related to them according to relations found in a selected set of one-dimensional laminar flames known as flamelet 45 . In the first step, a group of flamelets is pre-computed, and the structure information is stored in a library. Assuming that the flamelet relations hold in the turbulent flame, only the dynamics of the controlling variables have to be solved. But in RANS and LES, it has to be taken into account that the controlling variable has turbulent fluctuations. In the present study, the code CHEM1D 46 is employed for the flamelets' calculation at different conditions. It solves one-dimensional temperature, species, and flow equations with an equation of state (EOS) using adaptive grid refinement to define the one-dimensional flame structure completely 47 . For the present atmospheric flames, the ideal gas EOS has been used. The states closest to the chemical equilibrium state are reached when the scalar dissipation value (or strain rate) approaches zero. These are the flamelets with the maximum temperature near the adiabatic mean temperature depicted in Fig. 1. On the other hand, the mixing solution is obtained without reaction.  Alternatively, when at a relevant scalar dissipation rate, the mixing solution auto-ignites the transient states reached during ignition and can be used to construct the manifold. The corresponding FGM is called Igniting FGM (IFGM). IFGM can be 9 used in the case of a hot vitiated coflow (hot oxidizer). In the system studied here, cold air acts as an oxidizer, and a treatment using only IFGM is impossible, and EFGM is used.

D. Flamelet library generation
The flamelets described in the previous section correspond to an oxidizer boundary condition of hot vitiated coflow and do not consider heat loss. To obtain a complete library required for the turbulent flame generation, the flamelet simulations have to be repeated, varying the oxidizer composition while also including the heat loss. This leads to a library with four independent degrees of freedom, as explained in the following. The methanol spray flame case has two oxidizer streams: an airflow from the central jet core maintained at room temperature and hot coflow produced by lean H2/Air flames. Both streams have a constant but different temperature. This system can be handled straightforwardly by introducing an extra mixing parameter (mixture fraction) to define the relative fraction of the two oxidizer streams in the three-stream mixture with fuel. It should be noted that the system does not fit the scope of models considering a product stream as a third stream, either diluted-air-based FGM models 27 or flamelet progress variable models 49 . Indeed, the vitiated coflow is different in composition from the product stream of the main flow.

Mixture Fraction (Z1)
The primary mixture fraction (Z1) represents the mass fraction of the fuel in a mixture of fuel-oxidizer. By definition in a twostream problem the composition of the oxidizer stream and the fuel stream respectively correspond to mixture fraction values 0 and 1 at progress variable 0. With  denoting any relevant thermochemical variable needed to describe the state of the mixture, the boundary condition is

Oxidizer mixture fraction (Z2)
The central carrier jet consists of cold air at 283-288K, and some evaporated fuel. The hot outer coflow is the product of hydrogen/air flame burnt in lean conditions at an approximately constant temperature of 1430K. Further, the product of lean combustion of premixed hydrogen/air flame majorly consists of H2O, N2, and O2 only. The difference between composition and temperature demands a separate parameter that tracks both the oxidizers' mixing and helps in better definition of thermochemical quantities. Sarras et al. 26 used a similar parameter to define and distinguish the air from vitiated coflow of varying oxygen content in Delft Jet-in-Hot-Coflow (DJHC). This methodology also allows for flamelet generation for oxidizers at intermediate states of oxygen content. It should be noticed that the current case cannot be treated as MILD combustion with exhaust gas recirculation (EGR). In EGR, fuel combustion products with the oxidizer act as an inert diluent to the oxidizer.
There are various studies on EGR-based flames like furnace flames 27,49 . Furthermore, the diluent temperature will be the adiabatic flame temperature from the complete combustion of fuel and oxidizer, provided the enthalpy loss is neglected. If air is considered the oxidizer in the present case, the combustion product will contain a carbon-based product like CO2, which is absent in the hot coflow. In other words, the temperature of 1430K in hot coflow will require a completely different hot-coflow composition if the coflow was to be made by diluting the air with exhaust gas. This justifies the treatment of cold air and hot coflow as two separate oxidizers for the present case. Moreover, the oxidizer mixture fraction helps to accurately define the amount of fuel evaporation in the carrier jet. The oxidizer mixture fraction (Z2) is a normalized scalar representing a mixture of two oxidizerscold air and hot coflow. By definition, the composition of the hot coflow stream and the air respectively correspond to second mixture fraction values 0 and 1.
Flamelets can be generated for both the oxidizer streams or any mixture of the two oxidizer streams by varying the second mixture fraction.

Scaled Progress Variable (C)
The set of computed flamelets can be tabulated as a function of the two mixture fractions, a progress variable tracking the conversion progress, and a variable characterizing the enthalpy deficit relative to an adiabatic reference state. The progress variable is scaled to a range [0,1] for tabulation convenience. The value of C equals zero would define either a pure state of fuel or oxidizer (as shown in Eq. 20,21) or a state of mixing between them 50 . Figure 2 shows the temperature as a function of scaled progress variable and mixture fraction at different values of enthalpy deficit ( ) and Z2. The brighter white region indicating high temperature lies near the top, where the value of C is close to 1. At the same time, C equals zero means the temperature of a nonreactive mixing of oxidizer and fuel. The high flame temperature zone decreases as the enthalpy deficit or second mixture fraction increases. Reducing the second mixture fraction represents a mixture containing more hot-coflow and a higher oxidizer temperature. This is why a broader high-temperature zone arises for Z2 equals 0, and it results in quicker autoignition of the fuel-oxidizer mixture even in lean conditions at lower mixture fraction values.

Enthalpy deficit ( ) or non-adiabatic FGM
The multiphase reactive flow, considered here, involves liquid fuel evaporation before mixing and combustion. The evaporation process reduces the local temperature and enthalpy of the surrounding fluid, affecting the reaction zone and flame structure.
Higher enthalpy losses due to evaporation may also lead to local flame extinction. It becomes more critical for the auto-igniting flames where the formation of ignition kernel is decisive for flame lift-off height. The region near to fuel injection nozzle has a high density of spray droplets, and the liquid fuel evaporation delays the auto-ignition, thus, leading to an increase in the liftoff height. Hence, the enthalpy deficit represents the difference between the actual condition and the adiabatic case. Similarly, a cold wall may also lead to enthalpy loss, local quenching, and detachment of a flame. Different methods for including heat loss in a flamelet library have been used in earlier studies. The first would be to calculate non-adiabatic flamelets by including a heat loss term in the energy equation 51 . Since the phenomena of heat loss in a single flamelet and a spray flame environment are quite different, the inclusion of enthalpy loss due to evaporation in the flamelet library will not cover a correct range of conditions 52 . Another method to achieve enthalpy deficits is to generate flamelets for reduced temperature boundary conditions.
Marracino & Lentini 53 assumed an equal enthalpy loss for both the fuel and oxidizer in creating flamelets. In contrast, Ma et al. [24][25] , and Huang et al. 27 assumed the enthalpy loss for the oxidizer only. The former examines a methane/air flame for radiation effects, thus giving rise to source/sink terms in the enthalpy equation. The latter is concerned with the multiphase reactive system where a part of a liquid fuel droplet is assumed to absorb the latent energy from its surrounding (oxidizer) during the evaporation. It results in a more pronounced effect on the oxidizer only. The latter approximation is adopted for enthalpy loss effects in the present study. Hence, the enthalpy deficit ( ) condition would apply to the oxidizer only, and the flamelet state at the boundary is given by: The value of the maximum enthalpy deficit must be specified depending on the insight of the studied flame. The magnitude of the considered range of enthalpy deficit is temperature and composition-dependent and therefore depends on Z2. But for every case, the range is described by a parameter  with range [0,1].

Construction of a lookup FGM table
The thermochemical states of flamelets calculated in Section II(C) can be written as functions of the scaled independent variables ( ) 12 , , , Z C Z  . The range of tabulated conditions depends on the set of flamelets considered and should be sufficiently large. The definition of two mixture fractions defines all possible mixing states. The set of flamelets is well complete to cover the entire conversion process from inert to fully burnt state. The heat loss range is also sufficient to cover the actual heat loss due to evaporation. In a turbulent flame, the independent variables are fluctuation. In LES, subgrid-scale fluctuations are not resolved and, instead, they are modeled using a presumed joint PDF of the independent variables. The density-weighted filtered thermochemical quantities are obtained from the flamelet data by integrating over the PDF 50 : It is assumed that all the control variables are independent of each other. Furthermore, the mixture fraction Z1 and scaled progress variable C are assumed to vary according to the -function PDF 54 The independent variables of the

Calculation of control variables
More detailed specification on the definition, range, and scaling of the control variable is needed. The details of the definition of control variables in physical space and their dependency on each other are provided here.

a. Mixture Fraction
A mixture fraction is defined here following the work of Bilger et al. 55 : , , , 14 Here, Ye and Mw are the element mass fraction and molecular weight that need to be calculated for carbon, hydrogen and oxygen atoms. These values are required to be calculated for fuel and oxidizer for a given boundary condition. This definition of mixture fraction is used to convert the flamelet results in physical space obtained using CHEM1D to flamelet profiles in mixture fraction space needed in the tabulation.

b. Progress Variable
A (unscaled) progress variable ( c Y ) may be defined as a sum of species whose formation indicates the completion of a combustion process. For the present work, species -CO2, H2O, and H2 are chosen to resolve the reaction zone sufficiently.
Following the previous works 16,41,45 , the control variable c Y is defined as Here, the reciprocal of molar mass M acts as a weighting factor for the mass fraction Y of species CO2, H2O, and H2. The unscaled progress variable is scaled based on the minimum and maximum values of the progress variable. In this case of four control variables, the extreme value of the progress variable will depend on the mixture fraction, enthalpy loss, and the second mixture fraction describing the mixing of the two oxidizer streams: Here, 'u' and 'b' indicate c Y values in unburnt and burnt states.

c. Enthalpy loss
The adiabatic enthalpy had of a fuel-oxidizer mixture can be expressed as: 11 (1 ) 15 Here, f h and Ox h are the adiabatic enthalpy of fuel and oxidizer, respectively. In our case the oxidizer is a mixture of two oxidizer streams, as described by the second mixture fraction Z2. Then the adiabatic enthalpy considering the oxidizer composition is given by: Here, 1 Ox h and 2 Ox h are the adiabatic enthalpy of air and hot-coflow, respectively. The normalized enthalpy loss is calculated as:

d. Oxidizer mixture fraction
An oxidizer mixture fraction is introduced to define the mass fraction of the two oxidizers in the mix of oxidizers. In the current case, two oxidizers are air and hot-coflow. It is defined in the present case as: 22 22 , 2 ,, Every flamelet has a fixed value of Z2 at its oxidizer boundary. It is a non-reacting scalar. Together with Z1, it fully defines the mass present locally that is originating from each oxidizer stream.

e. Variances of mixture fraction and progress variable
The scaled variance in the lookup table varies between zero to one and are related to the variance by: .
16 Similarly, the scaled variance of progress variable is calculated as:

Calculation of resolved independent scalars and their variances
The control variables are calculated in the turbulent flow simulation either using the transport equations or through the algebraic equations. The equations of the Favre-averaged control variablesmixture fraction , oxidizer mixture fraction , unscaled progress variable , total absolute enthalpy and variances of mixture fraction and progress variable are provided below, , , , , , Here,  is the mixture density, is the mixture velocity, k D is the total diffusivity equal to t DD + , while t D is the turbulent diffusivity. The turbulent scalar dissipation rate for variance of mixture fraction is modelled as: , Here, Zv C is assumed as 2. Following previous work 25,52 , the variance of progress variable is algebraically calculated assuming the equilibrium of the generation and dissipation of scalar variance at sub-grid scale, .
The second term accounts for the progress variable source term. The constant Next the eigenvalue problem of A is solved, providing eigenvalues l k and eigenfunctions j k , 1£ k £ N : The eigenfunctions are ordered from high to low eigenvalue, expressing the energy content of the modes.
The POD modes are the projections of X on the eigen vectors: The first P<N modes are sufficient to reconstruct the original spatio-temporal data fields, maintaining the spectral energy contained in the first P modes.

   and
x the fluctuating field of the covariance matrix X for a k th snapshot.
This method is used here to study the development of the ignition kernel and the flame growth. If X is the velocity ' i u , the trace of the covariance matrix represents the averaged energy of the fluctuating LES velocity field, Here,  is the time-averaging operator. Moreover, the analysis has also been performed on velocity-temperature field ( ' The composition of the fuel, air, and hot coflow is mentioned earlier in Table I.

Eulerian boundary condition
The details of the three studied methanol flames -Mt2A, Mt2B and Mt2C are mentioned in case of Table I Table I for all the simulated cases. The case Mt2C-NG is identical to case Mt2C, except for the fact that the gas phase of the central injector stream does not contain fuel vapor.

Spray boundary condition
An essential factor for a dispersed flow is the boundary condition applied to the particles, especially where definite particle statistics and the subsequent dependent processes are required. The auto-igniting dilute spray flames 15 provide extensive data on spray droplet size, mean and fluctuation velocity, temperature, etc., at the inlet. The radial profiles of particle data are provided at the inlet, just above the burner, and at various downstream distances. The spray inlet is divided into multiple injection patches with center points uniformly distributed in the inlet jet-exit plane by dividing its radius into ten intervals and the azimuthal direction into sixteen intervals. The radial divisions are based on the location of experimental measurements. The 21 azimuthal distribution is uniform and sufficiently fine to represent the circumferential uniformity of droplet injection but not too fine to allow the injection of parcels from every patch at each time step.
Further, any patch consists of five overlapping sub-patches to meter the input mass flux of the droplets corresponding to five different size bins. These bins are taken from the experiments -0 -10 , 10 -20 , 20 -3re0 , 30 -40 and 40 -50 .
In addition to the mass flux, the mean and fluctuating r.m.s. velocity is also fed to injector patches. The data of various radial profiles are provided in a tabulated form for each patch. Hence, multiple droplet parameters are set at the injection. Around 2 million parcels are injected per second from the complete injector at an initial temperature of 283K for Mt2A and 288K for Mt2B and Mt2C cases.

IV. RESULTS AND DISCUSSION
A. VALIDATION

Grid Independence
The coarse mesh out of the two chosen grid sizes has been refined enough to capture essential features of the flow. As  Fig. 4 reports axial velocity and temperature radial profiles at two downstream locations. The agreement with the experimental data is satisfactory. The grid is sufficiently refined to capture the mixing and reaction zones.
Therefore, the coarse mesh is chosen for the rest of the simulations in this study.
Additionally, the grid resolution is also found sufficient according to the LES Index of Resolution Quality (LES_IQ), given by Celik et al. 71 . The recommended value of LES_IQ of more than 75%-80% is acceptable for most cases. The minimum value of LES_IQ is found to be near 95% for both the coarse and fine grid sizes.

Temperature Variation
The effect of fuel mass loading is noticeable in radial temperature distribution at downstream distances, as shown in Fig. 5. The discrepancy in numerical predictions may be explained in two ways: first, early ignition is predicted at a distance less than 10D in all the three flames, and second, faster mixing of the free shear layer from either side of the jet. As discussed in the following sub-sections, the first is not noticed from the OH distribution, and the ignition locations change with the fuel loading.
In contrast, the shear layer development is sensitive to turbulence characteristics. In the present simulations, a single turbulent kinetic energy equation-based model (dynamic k-equation) is used, as shown by Eq. 5. The inlet mean velocity and k profile is taken from the experiments 15 . Studying the influence of turbulent inlet modeling or the turbulence model on the flame characteristics near the jet exit may provide insight into this discrepancy. simulation results in several cases over predict the experimental data, namely for size bin D [30,40] in flame Mt2A and all the size bins of Mt2C as shown in Fig. 6. The droplet data of the Mt2A flame at 30D are under-predicted, but the rest is in good agreement for all three flames. It should be noted that there is some asymmetry between the data from the two sides of the central axis, seen in both the experimental and the simulation results. Such asymmetry can result from insufficient statistics in 25 the simulation, but it may have another reason in the experiments. For this reason, a perfect agreement between model and experiment, therefore, cannot be expected. In the flame Mt2A, the biggest droplets have lesser velocity than the small droplets at 10D. But this trend is reversed at a 30D downstream location for all the flames. At 30D, however, the droplet velocity is higher for larger droplets than the smaller ones. This is a consequence of the difference in inertia between small and large droplets. The rapid relaxation of the velocity of small droplets (size less than 10 m) to gas velocity allows them to be considered as seeding particles for gas velocity measurement 15 . On the hand, the higher inertia of larger droplets delays their response to the velocity changes in the carrier medium. Therefore, at 30D of Mt2C flame, the velocity of droplets with a size less than 20 m in Fig.8 is closer to the gas velocity in Fig. 4(a) when compared to the droplets of larger size. This also explains a near parabolic profile at 10D compared to the flatter velocity profiles of droplets at 30D downstream from the jet exit.

26
The droplet mean size is shown in Fig. 8 at various downstream distances for all three flames. It is to be noticed that the droplet size remains approximately constant across all flames at a given downstream location. The droplet size is observed to increase with the downstream location away from the jet exit. The rough portion is observed mainly at 40D, and a smoother curve is obtained at near-jet exit locations. This is because significantly fewer large droplets are available as a sample at 40D. The high temperature of coflow surrounding the center cold-carrier jet evaporates a large number of smaller droplets leaving behind the bigger droplets. Hence, at locations far away from the jet, the region is dominated by fewer but bigger droplets. Except for a few locations, the simulation can adequately capture the details of all three flames.

Lift-off Height and Flame Structure
As discussed in the previous sub-sections, the flame lift-off height (L) increases on decreasing the fuel loading, as shown in Fig. 9(a). The OH radical is a necessary species for ignition, and its concentration is a reasonable indicator of the auto-ignition as has been used previously 15,28 . Thus the OH concentration is shown in Fig. 9(b). The lift-off height would vary with the chosen minimum threshold value of the OH mass fraction. It was assumed as 2x10 -4 in Refs. 20,72 and 6x10 -4 in Refs. 73-74. Also, the experimental lift-off height is based on visual observation using the still images from a regular digital camera, making it challenging to assume an appropriate OH species mass fraction value for precise validation. Hence, two data sets are considered based on Favre-averaged mean mass fraction of OH as 2x10 -4 and 6x10 -4 , as shown in Fig. 9(a)  for the progress variable better represents the presence of the reactions. As shown in Fig. 10(b), the prominent annular structure is visible for Mt2C and Mt2B flame, whereas the annular structure is almost absent in Mt2A flame. Another feature to note is that the white-dashed line representing the lift-off height from OH concentration lies downstream of the chemical source term.
It indicates an early start of reactions than predicted by the flame visualization from OH concentration. For flames Mt2B and Mt2C, the annular structure extends well beyond the lift-off height, which is not the case for Mt2A flame. The higher fuel loading results in a near-stoichiometric mixture for flames like Mt2C in the shear layer, initiating the early reactions.

Effect of evaporated fuel in the carrier jet
In the additional case of Mt2C-NG, a simulation is carried out without evaporated fuel in the carrier jet. It shows the effect of an already evaporated fuel on the lift-off height of the Mt2C flame, which is around 8 percent of the carrier jet air flow rate or almost one-third of the liquid fuel injected through the nebulizer (refer to Table I

B. IGNITION KERNEL FORMATION AND GROWTH
The previous section used OH species as the flame and ignition location indicator. To further study the spread of flame within the shear layer, it is crucial to track the development of OH species in the flow. It is difficult to identify the critical processes in flame evolution and its interaction with flow structures directly from the instantaneous field, especially flame propagation. At any instant, fields like OH species can be split into their mean and fluctuating components. A mean component may provide the lift-off height, as shown in Fig. 9(b), but the fluctuating data helps in the study of flame propagations and ignition kernels. This is addressed using POD.

Proper Orthogonal Decomposition setup
Proper Orthogonal Decomposition (POD) of a flow identifies the dominant structures, explaining most of the variance of the fluctuations. It allows looking beyond the mean and the insignificant flow features. In Fig. 9(b), the mean distribution of OH primarily lies in the mean shear boundary region. Mixing fuel and oxidizer is vital to sustaining a continuous heat release in the given region. The liquid droplets that evaporate in the neighborhood of hot coflow and the already evaporated fuel from the upstream of a pipe form a premix of fuel and oxidizer in the carrier jet (with very low mixture fraction values). The vortex structures formed in the shear layer between carrier jet and coflow and their transition into turbulence allows the hightemperature oxidizer from vitiated coflow to mix with the fuel in the jet. As discussed in Section II (and also shown in Fig. 2), the flamelets achieve high temperatures at the location near stoichiometric mixture fraction values. The vortices play a significant role in reaching the required mixture fraction value. It may also be pointed out that the value of scalar dissipation rates higher than the critical value prevents the early ignition of fuel with oxidizer from carrier jet or hot vitiated co-flow.
The POD is performed on the fluctuating LES fields of OH and the combined LES fields of velocity and temperature. Around as shown in Fig. 12(a). The first few most energetic modes are resolved for each set, but differences arise in higher modes. The eigenvalue curves of sets B and C nearly overlap, showing that it is sufficient to consider set B for reconstruction and further analysis at a lower computational cost than set C. The fast Fourier transform (FFT) is performed on the POD time coefficients k b , allowing the identification of the frequencies of field fluctuations corresponding to each mode (see Fig. 12(b)) and taking this into account in the analysis of POD results.

Proper Orthogonal Decomposition results
The FFT of the first six modes, which constitute 60 percent of the energy (co-variance), shows a distinct peak at around 67.3 Hz and another peak of 269.2 Hz for structures corresponding to the first mode. The second mode also shows a peak at 269.2 Hz, whereas the third mode is again similar to the first mode with the same peaks. The fourth mode shows a peak at 134.4 Hz.
It may be noticed that all these frequencies are integer multiples of the base frequency of 67. The reconstructed field lies within the instantaneous field in most regions for all three snapshots. The iso-contour value of the mass fraction is chosen to be 0.0002, a smaller value for the better visualization of small patches of the field variable associated with ignition kernels. The structures like ignition kernels are verified to be independent and detached from the rest of the OH field by further reducing the value of mass fraction for iso-contour (it is shown here for only 0.0002). The ignition kernels are 32 visible in the instantaneous field, not the reconstructed field. As the flow progresses, these kernels get attached to the rest of the flame or each other through flame propagation (as can be seen in snapshot 100 of Fig. 13). The attachment of ignition kernels happens by forming finger-like streamwise elongated, low-frequency structures. As shown in Fig.13 (and better visualized in the supplementary video file), the ignition kernels are more stable near the shear-layer region and show better flame propagation. On the other hand, the ignition kernels formed near the jet centerline are unstable, and a few of them tend to extinguish as they are convected downstream.
FIG. 13. Iso-contour of the instantaneous field of OH and reconstructed field of six POD modes of fluctuating OH species with the value of 2x10 -4 at three snapshots = 35, 50, and 100. Ignition kernels are visible in the "instantaneous OH" field, whereas the "reconstructed fluctuating OH" field shows the flame propagation. Watch the animation for more details. ( Multimedia video) On the other hand, flame propagation is analyzed through a reconstructed POD-based fluctuating OH field. A positive isocontour value of 0.0002 for the fluctuation field represents the flame propagation in the region that favors reactions. Another important observation is that most of the instantaneous field region is occupied by the reconstructed OH field. In other words, the six POD modes are sufficient to resolve the OH field and, thus, capture the flame dynamics. Together, these six POD modes control the flame evolution, whose frequency corresponds to the harmonics of 67.3 Hz (Fig 12(b)). Hence, it can be said that The comparison of the location of the instantaneous OH field from Fig. 13 with the reconstruction of only the first dominant mode of velocity-temperature-based POD and the reconstruction of six OH field-based POD modes shows that the ignition kernels develop in the region (around 5D to 10 D) where the larger streamwise or the broken ring vortical structures are still present (refer to Fig. 14(c)). The OH engulfs more regions around the vortices as the structures evolve further downstream.
The larger OH-based structures at downstream locations can also form due to the flame propagation around the "reactive mixture" in the surrounding region of vortices. As discussed in Ref. 15, patches of local extinction and closures in the flame may also give rise to the OH fluctuation field. The zoomed-in image in Fig. 14(c) shows that the two regionsflame development and vortices overlap. The vortex structures allow easy fluid mixing at a lesser strain, as witnessed in the vortical ring structures and the streamwise vortex pairs in the braid region of free-shear jets 62,75 . It enhances the chances of formation of the reaction zones, thus promoting the flame propagation (or maybe ignition kernels) around vortices. Thus, the streamwise 35 braid vortices formed in the region between 5D to 10D (Fig. 14) help create the elongated, tubular streamwise OH-based structures (Fig. 13).

V. CONCLUSION
LES simulations of methanol spray flames have been carried out using an extended FGM model. The extended FGM model contains an additional parameter (second mixture fraction) as a key feature, allowing the distinction between the two oxidizers (air and hot coflow) in the studied flames. The model requires different laminar flamelet generation for methanol vapor burner in any mixture of the two oxidizers in the presence of heat loss due to evaporation effects. The solver is validated against an extensive auto-igniting dilute methanol flame database in a vitiated coflow 15 . The spray droplets are tracked using Lagrangian particles while the combustion model and the gaseous flow are solved in the Eulerian phase. The pre-evaporated gaseous fuel present in the carrier jet is also considered.
The droplet statistics, including the particle velocity, size, and mean-field temperature, are verified for all three flames with different fuel mass flow rates. The predicted flame structure is in good agreement with the experimental results. The liftoff height is accurately predicted using OH mean-field data, where it is shown that the height decreases with an increase in fuel loading. This flame height is also compared to the flame without the presence of evaporated fuel in the carrier jet for the case with the highest fuel mass flow rate (Mt2C). It is found that not taking into account the pre-evaporated fuel at the jet exit increases the lift-off height by a factor of five.
Further, the Mt2C flame is investigated using proper orthogonal decomposition (POD), performed on the fluctuating OH field and the velocity-temperature fields. The study of the instantaneous OH field is found to be best for the prediction and for revealing ignition kernels. It includes all relevant structures with different frequencies and energy. The POD reconstruction based on the OH field is best suited for flame propagation studies. The ignition kernels formed in the peripheries of the shear layer are observed to be more stable than those near the jet centerline. The tubular, streamwise elongated structure formation precedes the ignition kernel, especially near the shear layer region. These structures seem to self-sustain through flamepropagation without forming any upstream ignition kernels. It is found that these dominant structures corresponding to both the OH and velocity-temperature data fields have an approximately same base frequency of 67.3Hz. The presence of the harmonics of this base frequency also indicates that these structures have periodic motion. The streamwise, braid vortices captured in the first, energetic POD mode seem to assist the formation of tubular-shaped, elongated flame structures. Based on 36 the POD and instantaneous OH-field analysis, it is proposed that the ignition kernels development and flame propagation are dominant features responsible for sustaining the lift-off height of the annular-shaped Mt2C flame.

DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.