flamelet model

International Journal of Multiphase Flow 125 (2020) 103216

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International Journal of Multiphase Flow journal homepage: www.elsevier.com/locate/ijmulflow

Large eddy simulation of a partially pre-vaporized ethanol reacting spray using the multiphase DTF/flamelet model Yong Hu a,∗, Reo Kai b, Ryoichi Kurose b, Eva Gutheil c, Hernan Olguin d a

School of Aeronautics and Astronautics, Sichuan University, Chengdu 610065, China Department of Mechanical Engineering and Science, Kyoto University, Kyoto daigaku-Katsura, Nishikyo-ku, Kyoto 615-8540, Japan c Interdisciplinary Center for Scientific Computing, Heidelberg University, Heidelberg, Germany d Department of Mechanical Engineering, Universidad Técnica Federico Santa María, Valparaíso, Chile b

a r t i c l e

i n f o

Article history: Received 8 October 2019 Revised 9 January 2020 Accepted 9 January 2020 Available online 10 January 2020 Keywords: Pre-vaporized spray jet Spray flamelet Multi-regime Evaporation model Spray-flame coupling

a b s t r a c t Spray reactive flow finds application in various technical devices, and due to the complex nature, their optimization is very challenging, requiring proper modeling of turbulence/chemistry interactions as well as of the contribution from spray evaporation. This work presents a study of sub-grid scale combustion models, where relevant assumptions on multiphase coupling and their effects are analyzed in detail. For this purpose, two different flamelet approaches, i.e. progress variable spray flamelet and multi-regime gas flamelet are examined in an implementation coupled with the dynamic thickened flame model, along with which the impact of inlet inhomogeneities condition and droplet evaporation taking into account internal temperature gradient is also investigated. The numerical evaluation is carried out in large eddy simulations of a benchmark ethanol spray flame with partial pre-vaporization, where an EulerianLagrangian numerical framework is adopted. The analysis demonstrated that the flame dynamics under consideration is governed by a close coupling between spray evaporation, turbulent dispersion and unsteady flame propagation at upstream shear layers. Results show that the spray flamelets built from counterflow partially-premixed spray flames achieved a better agreement with experiments, capturing the flame structure in terms of gas-phase temperature, OH mass fraction as well as spray statistics. © 2020 Elsevier Ltd. All rights reserved.

1. Introduction Facing the increasing concerns on energy shortage and environmental pollution, energy generators powered by (gas or liquid) fossil fuels combustion are currently receiving a prevailing attention on their optimization. Design solutions based on computational fluid dynamics (CFD), especially on Large Eddy Simulation (LES), to capture reliably the unsteady flow features, are gaining in relevance (Pope, 20 0 0). Moreover, the significance of the development of accurate and computational efficient models, describing the complex interplay between the sub-processes involved in multidimensional CFD of turbulent combustion, is widely recognized (Bilger et al., 2005; Bray, 1996). For multiphase reactive flows, this means that more efforts are needed for the modeling of both gas and dispersed liquid phases, taking into account the coupling between not only the gas dynamics and chemical reactions but also spray atomization, evaporation and droplet dispersion. Owing to such complexity and the wide range of their engi∗

Corresponding author. E-mail address: [email protected] (Y. Hu).

https://doi.org/10.1016/j.ijmultiphaseflow.2020.103216 0301-9322/© 2020 Elsevier Ltd. All rights reserved.

neering applications, two-phase reacting spray LES has been one of the major focus of model development in the past few years (Jiang et al., 2010; Xiao et al., 2016; Kitano et al., 2016). Turbulence-chemistry interactions are one of the fundamental processes influencing flame dynamics and other sensitive phenomena, such as autoignition and pollutant formation. In twophase flows these interactions show a distinct feature and challenge compared to their gaseous counterpart, which is mainly attributable to the presence of spray evaporation. A major issue is the partially-premixed burning considered as the dominant reaction mode produced in multiphase combustion (Réveillon and Vervisch, 2005), the proper modeling of which requires relaxing several restrictive assumptions introduced in traditional turbulence combustion models. For the large scale calculations, Flamelet models have been generally believed to be an efficient approach, which requires a priori determination of the type of flame regime and then describes the mixing and chemistry interaction following a flamelet-like diffusion or premixed flame structure (Peters, 1984; Pandal et al., 2018). This however, might be a problem because of the flame nature of multiple reaction regimes in a reactive twophase flow. Baba and Kurose (2008) compared the performance

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Y. Hu, R. Kai and R. Kurose et al. / International Journal of Multiphase Flow 125 (2020) 103216

Nomenclature

α αt ξ¯ χ δl0 E F

μt  τ˜

 u c W 2 C  C 2 Z  Z D E h Nu Pr Su0 Sc Sh Zst vd

ρ ρl ρu

md p R0 T Td v0

mass diffusivity, m2 /s eddy diffusivity, m2 /s flame index, dissipation rate of scalar , 1/s laminar flame thickness, m efficiency function, flame thickening factor, turbulent viscosity, kg/m/s flame sensor, stress tensor, N/m2 gas velocity, m/s. reaction source term of progress variable, 1/s variance of progress variable, reaction progress variable, variance of mixture fraction, mixture fraction, jet nozzle diameter, m global equivalence ratio, enthalpy, J/kg Nusselt number, Prandtl number, laminar flame speed, m/s Schmidt number, Sherwood number, stoichiometric mixture fraction, droplet velocity, m/s filter width, m gas density, kg/m3 liquid density, kg/m3 unburnt mixture density, kg/m3 droplet mass, kg pressure, Pa initial droplet radius, m gas temperature, K droplet temperature, K initial droplet velocity, m/s

of different flamelet tabulation methods in simulations of both gas and spray jet flames, and they found that single-regime burning flamelets can fail to predict the coexistence of diffusion and premixed combustion occurring in spray flames. In a DNS evaluation of flamelet libraries within a laminar counterflow spray flame, Luo et al. (2018) adopted the diffusion flamelet structure with a modification of temperature at boundary conditions. The study indicated that even though the mass fraction of species cannot be correctly predicted in all regimes, the flame temperature can generally be reproduced using this strategy. However, it is still unclear whether these observations obtained in laminar spray flames have applicability in LES of turbulent spray flames due to the turbulent modification in sub-scale structures. In a recent study with turbulent acetone spray flames, tabulated chemistry approaches based on single-burning regime assumptions traditionally employed in nonpremixed or premixed combustion modeling led to an apparent deviation capturing the main features of spray flames (Hu and Kurose, 2018). Hence, the extension of conventional flamelet models to multi-regime spray combustion was attempted in Refs. (Knudsen et al., 2015; Hu and Kurose, 2019; Franzelli et al., 2013). Knudsen et al. (2015) proposed a partially-premixed flamelet in multiphase LES by use of a flame index parameter combining locally gaseous diffusion and premixed flamelet properties. Different closure assumptions for liquid evaporation were tested and the

new model was demonstrated to be capable of accommodating the structures in different combustion regime. At the meantime, the influence of the droplet models was revealed to be important, highlighting the need of taking it into account in the validation of the combustion model. Hu and Kurose (2019) further validated a modified multi-regime flamelet approach in acetone lean and rich spray flames. By contrast with the single-regime flamelet simulation, the proposed multi-regime flamelet model showed a promising performance in predicting the major burning structures. Additionally to this flame index based multi-regime flamelet model, Gutheil and coworkers (Hu et al., 2017b; Honhar et al., 2017; Olguin and Gutheil, 2014) developed a spray flamelet model including the influence of the liquid phase explicitly in the tabulated structures. Hollmann and Gutheil (1998) generated a chemistry library by computing a priori laminar spray flame and using five parameters for their tabulation. The resulting spray flamelet was validated in the context by the prediction of turbulent spray flame characteristics within the zone where liquid evaporation is present. Further model development was carried out by Gutheil and coworkers of combining the spray flamelets with transported PDF methods (Honhar et al., 2017) and generating spray flamelet libraries directly considering fuel prevaporization for properly capturing partially premixed features (Hu et al., 2017a). This last approach is similar to the one proposed by Franzelli et al. (2013) for gas flame. In (Hu et al., 2017a), a progress variable was used instead of the scalar dissipation rate to parameterize the flamelet structures. Its successful application was reported in the context of spray flamelet/transported PDF calculation of a pre-vaporized ethanol spray flame. Essentially, the above mentioned attempts differ in the prototype structure devised to represent the composition space occurring in spray flames, and a thorough understanding of their performance is needed for the further improvement in reactive spray modelling. In addition, as the fuel-droplets evaporation and their turbulent dispersion cause a strong heterogeneities in local mixture composition, which would enable the flame front propagating in a stratified gas flow, a numerical strategy is then also required to account for the unresolved reaction zone and thereafter the accurate flame speed in sub-filter field of LES. A well-established solution includes the Dynamically Thickened Flame model (DTF) (Wang et al., 2011; Legier et al., 20 0 0), which shows its promising performance in its initial implementation using Arrhenius law closure for gaseous premixed flame (Colin et al., 20 0 0). Later on, to account for the finite-rate effect and meanwhile reduce the computational cost when implement detailed chemistry, in a coupled way with single-regime premixed flamelet model it is widely used in various partially or perfectly premixed gas flame simulation (Han et al., 2019; Kuenne et al., 2011). A few applications to spray flames were recently reported in Refs. (Felden et al., 2018; Rittler et al., 2015). In respect of the inherent complexity of mixed burning regimes interplayed with a dispersed phase, there are only few analysis in the literature of DTF concept in context of the extended flamelet model for spray combustion. In view of this background, a combined formulation of dynamically thickened flame model with different multi-regime flamelet models for two-phase reacting spray is presented in this work, and its capability accounting for two-phase interaction effects is studied together with use of different types of tabulated detailed chemistry based approaches. In particular, the spray and multiregime gas flamelet models are discussed in the LES of a prevaporized ethanol spray flame which is expected to present a complex multi-regime flame topology. To the author’s best knowledge, there is no study showing the impact of such a choice on the simulation of this kind of flame. As indicated in a previous work (Knudsen et al., 2015), the spray model for two-phase mass and heat transfer is expected to be a nontrivial factor affecting the

Y. Hu, R. Kai and R. Kurose et al. / International Journal of Multiphase Flow 125 (2020) 103216

multiphase LES. Thus, the sensitivity of the computation to the assumptions made for the liquid phase is also investigated in the present numerical framework. Specifically, the droplet heating with finite conductivity, inlet inhomogeneous gas and liquid distribution are examined. The paper is organized in the next section as follows. Firstly, the gas equations and model description will be presented in the next section, after which the target flame and the used numerical approach are described. The results and conclusions will be given in Sections 4 and 5, respectively. 2. Mathematical methods

with 2 ) = RI ( Z , Z 2 ) = RII ( Z , Z



1

0

d2Yceq (Z ) 2 )dZ β (Z ;  Z , Z (Z ) dZ 2

1 Yceq

0



3

1

dYceq (Z ) 2 )dZ β (Z ;  Z , Z (Z ) dZ

1 Yceq

(9)

where the filtered mean values are computed using a β -shaped 2 ), to integrate the mixture fraction. χ Z and χ Z,C PDF, β (Z ;  Z , Z are the scalar dissipation rate and the cross-dissipation rate. Apart ¯˙ c , the equilibrium value and its from the species reaction rate ω derivative related terms RI and RII are taken from the chemical table as well, which is also generated by flamelet methods.

2.1. LES equations 2.2. DTF model The LES of spray reactive flows include the solution of the Favre-filtered balance equations for gas-phase density ρ , velocity u, energy (gas enthalpy h) and species mass fraction Yi . As to the hydrocarbon fossil fuel, a detailed reaction mechanism could lead to hundreds or even thousands of equations to be solved. This is avoided by use of flamelet model that allows to represent the detailed chemical space in a low-dimensional manifold (Ihme and Pitsch, 2008; Fiorina et al., 2005). The thermo-chemical properties, like species mass fractions or reaction rates are described by a chemical database stored in a pre-generated flamelet library with a reduced set of controlling parameters, commonly used the mixture fraction Z and a reaction progress variable C. The intact set of two-phase LES equations for carrier gas flows with a dilute spray assumption is then given as (Kitano et al., 2016)

∂t ρ¯ + ∇ · (ρ¯  u ) = S˙ v ,

(1)

∂t (ρ¯  u ) + ∇ · (ρ¯  u u ) = −∇ p¯ + ∇ · (τ¯ + τ¯sgs ) + S˙ m ,

(2)

  ∂t (ρ¯ h ) + ∇ · (ρ¯  u h ) = ∇ · (ρ¯ α∇ h + J¯hsgs ) + S˙ e ,

(3)

  ∂t (ρ¯  Z ) + ∇ · (ρ¯  u Z ) = ∇ · (ρ¯ α∇ Z + J¯Zsgs ) + S˙ v ,

(4)

) = ∇ · (ρ¯ α∇  + J¯sgs ) + ρ¯ W c + S˙ c ,  C ∂t (ρ¯ C) + ∇ · (ρ¯  uC C

(5)

−  ), J = ρ¯ ( where p¯ is the pressure. τ¯sgs = ρ¯ ( u u − uu u   u ) ( = {h, Z, C } ) are the subgrid terms of stress tensor and SGS scalar convective flux (Peters, 20 0 0), and SGS diffusive contribution is neglected as adopted in general practice in LES. By use of the eddy viscosity approximation, their closed forms are ¯sgs

sgs , τ¯sgs = μt [(∇  u ) + (∇  u )T ], J¯ = ρα ¯ t∇

(6)

where μt and α t denote the turbulent viscosity and eddy diffusivity, respectively. In the jet-flame simulation, these are usually related in a formulation of αt = μt /(ρ¯ Sc ) with constant Schmidt number Sc = 0.4 (Kitano et al., 2016), where μt is determined by the dynamic Smagorinsky-Lilly model (Lilly, 1992). The mixture fraction Z describes the mixing state of fuel and oxidizer, and the reaction progress of mixture is denoted by the variable C, which is defined as the normalized mass fraction of species,

C=

Yc , Yceq (Z )

(7) eq

and Yc = YCO2 + YCO (Fiorina et al., 2005). Yc (Z ) represents the equilibrium state of reactant mixture given at each value Z. In c of C, taking into account the Eq. (5), the reaction source term W changes of the equilibrium state with respect to the mixture fraction, is modeled as (Hu and Kurose, 2018)

χ 2 ) + 2χ 2 ), c = ω ¯˙ c + C Z RI ( Z,C RII ( W Z , Z Z , Z

(8)

The characteristics of mixed combustion regimes in spray flames leads to the presence of a flame front propagation, separating the fresh and burnt gases, and in a spatially filtered LES, the subgrid thickened flame model is adopted to properly predict the flame propagation speed. The main concept on which this methodology is based, is that the flame front is artificially thickened by a factor to make it resolvable on the LES mesh. The general form of scalar  governing equations (referred to Eqs. (3)–(5)) in DTF framework are re-written as

    ∂t ρ¯  + ∇ · ρ¯  u   ˙  + (1 − )J¯sgs + E ρ¯ ω   = ∇ · EF ρ¯ α∇ φ ,g /F + S˙ φ ,d 

(10)

In the above equation, the additional parameters F, E, and  denote the thickening factor, the efficiency function and the flame sensor, respectively. The LES grid size and the laminar flame thickness δl0 are used to measure the dynamic thickening factor as



F = 1 + (Fmax − 1 ),

and

Fmax = max

n

δl0



,1

(11)

where the constant n represents the suggested grid-point number used to adequately resolve the reaction zone, and following the work of Charlette et al. (2002), it is taken as 5. δl0 is computed using the temperature profile from a 1-D premixed flame calculation and it is parameterized as a function of mixture fraction. To limit the thickening applied only inside intense reaction zone,  works to locate the flame region and is determined as (Kuenne et al., 2011)

 = 16[C (1 − C )]2

(12)

In the domain with C = 0.5 where flame front is detected, F approaches its maximum and as indicated in Eq. (10) only the molecular diffusion associated with the reaction zone is modified to invoke the thickening procedure. Proch et al. (2017) found that a DTF modeling without this limit imposed by a flame sensor could cause an excessive spreading of the burnout zone and damping of small flame structures. In addition, E accounts for the lost effect of flame wrinkling on the sub-filter scale level, for which the power-law closure model of E suggested by Charlette et al. (2002) is employed in this work. ˙ In Eq. (10), ω φ ,g is the reaction source term that functions for reaction progress C. Its evaluation is based on the flamelet tabulated chemistry introduced below. The closure of the spray source terms S˙ φ ,d will be detailed in Section 2.4.2 2.3. Flamelet tabulation technique Flamelet models allow the direct integration of detailed reaction mechanisms at a low computational cost, which is attractive for practical simulations. Two distinct tabulation strategies are introduced for the present multi-phase LES.

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2.3.1. Multi-regime gaseous flamelet This model takes into account the coexistence of diffusion- and premixed-type reaction structures accessed by spray combustion. The idea is to establish a numerical framework where the local chemical structure of different kind of flames can be modeled by the usage of the adequate flamelet chemistry in each flame region. By use of presumed a PDF and assuming statistical independence, the single-regime nonpremixed (ψ˜ non ) and premixed flamelet databases (ψ˜ pre ), determined by a pre-computation of 1-D counterflow diffusion flame and freely-propagating premixed flame, respectively, take the following form

ψ˜ (non,pre) =



1



0

1

0

, C 2 )β (C ; C 2 )dZdC, ψβ (Z ;  Z , Z

(13)

 2 and C 2 dewhere the beta PDF for both Z and C is applied. Z note the variance of mixture fraction and progress variable, respectively, and because of the small evaporation effect on fluctuation  2 = near droplets in present spray flame, they are computed by   |2 (Peters, 20 0 0), where β v is set as 0.15 (Chrigui et al., βv 2 |∇  2012). According to Knudsen et al. (2015), the thermo-chemical properties are then calculated locally by a weighting operation of contributions from both nonpremixed and premixed flame regime

ψ˜ = (1 − ξ¯ ) · ψ˜ non + ξ¯ · ψ˜ pre ,

(14)

and E. However, it is practically difficult to determine a priori the profile of S˙ v for the computation of flamelet library. For the PDF distribution of the spray quantities i.e. initial droplet radius, spray velocity and equivalence ratio, a dirac-delta function δ (R0 − R∗0 ), δ (v0 − v∗0 ) and δ (E − E ∗ ) is adopted as the first approximation due to the flame stability consideration of the laminar spray flame calculation in the counterflow configuration (Gutheil and Sirignano, 1998),

ψ˜ spray =



∞



∞



−∞

0

∞



0

1



0

1 0

, C 2 )β (C ; C 2 ) ψβ (Z ;  Z , Z

δ (R0 − R∗0 )δ (v0 − v∗0 )δ (E − E ∗ )dZdCdR0 dv0 dE R∗0 ,

v∗0

(17)

E∗

where and are determined in line with the specific conditions of the turbulent spray flame to be simulated. Then, the final spray flamelet library tabulating the chemical flame properties indicated by Eq. (16) is computed from the collection of 1-D laminar counterflowing spray flames under different level of stretch, which is achieved by varying the carrier flow velocity. The characteristics of the laminar spray flames are considerable different from those of their gaseous counterparts (Gutheil and Sirignano, 1998), and their influence on the turbulent flame prediction would be thoroughly investigated in this work.

with a definition of the flame index ξ¯ discriminating different combustion regime, as (Hu and Kurose, 2019)

2.4. Liquid phase model

| V  pre dV |

ξ¯ =

| V  pre dV | + | V non dV | + 

2.4.1. Droplet equations The liquid phase description is important to capture the droplet-fluid interaction through the mass, momentum and energy exchange, which is responsible for the close coupling in twophase flows. In this work, the spray model includes the Lagrangian evolution equations for the droplet motion, heating and evaporation (Kitano et al., 2014; 2016; Pillai and Kurose, 2019). The droplet is supposed to be spherical. Considering the interactive drag force, gravity and SGS perturbation due to random force, the individual droplets’ velocity evolves as



0 2Q  )] − ρ¯ χ˜C ∂  C  pre = ∂ ¯ α∇ C Qk [ρu Su |∇ C | − ∇ · (ρ C k 2Q non = −ρ¯ χ˜ Z ∂ Z k

(15)

In Eq. (15), V is the LES computational cell volume and  is a small number. ξ¯ approaches 0 and 1 in the limit condition of pure diffusion and premixed flames, respectively. pre and non represent the dominant transport terms relevant in diffusion and premixed flames, respectively, and are derived from a twodimensional flamelet species equation. ρ u and Su0 are the unburnt mixture density and flame speed, respectively. Here, a choice of Qk = YFuel + YCO is applied since the fuel consumption can indicate the reacting region. 2.3.2. Spray flamelet The spray flamelet model proposed by Hollmann and Gutheil (1998) is an extension of the traditional flamelet concept to the simulation of two-phase reacting flows. In this model, the flamelet chemistry is formulated relying on a prototypical laminar spray flame, which would explicitly take into account the evaporation effect that has been illustrated to be of major importance in the characterization of spray flames (Olguin and Gutheil, 2014). Differing from gas flamelets (Peters, 1984), spray flamelet structures cannot be uniquely parameterized by the mixture fraction and its scalar dissipation rate, and, as investigated by Hollmann and Gutheil (1998), the additional quantities relating to the liquid phase should be included to revise the formulation. Following the (Hu et al., 2017a), the spray flamelet model reads

ψ˜ spray =



∞ 0



∞ −∞



∞ 0



1 0



1 0

ψ P˜(Z, C, R0 , v0 , E )dZdCdR0 dv0 dE , (16)

where R0 and v0 denote the initial droplet radius and velocity of the mono-disperse spray, respectively, and E is the global equivalence ratio including vapor and droplet fuel at the spray inlet. Recently, Olguin and Gutheil (2014) proposed the use of spray evaporating rate S˙ v to correlate the spray influence instead of R0 , v0

dvd =

CD ρg Ad C ksgs |u˜ − vd |(u˜ − vd )dt + gdt + ( 0 )1/2 dW, 2md τc

(18)

Here, the droplet properties are denoted by the subscript ‘d’. ρ g ˜ the is the gas density, g is the gravitational acceleration, and u gas velocity. CD represents the drag coefficient and measures the resistance of droplet to the carrier flow. It is given by an empirical expression based on the slip and blowing velocities (Pillai and Kurose, 2019). ksgs is the subgrid kinetic energy and C0 = 1. dW denotes the increment of a stochastic Wiener process. τ c denotes the timescale characterizing the interactions between the particle and turbulence. For the droplet with radius rd , its mass md is md = 4/3π rd3 ρl and droplet cross-section area Ad = π rd2 . For the evaporating droplets, the determination of their vaporization rate requires the analysis of heat and mass transfer processes surrounding each droplet, the typical conservation equations for which are given as (Miller et al., 1998)

dmd = − dTd =

md

τdSt 

Nu 3P r





Sh ln(1 + BM )dt. 3Sc

c p,g c p,l



f2 (T˜g − Td )

τdSt

dt +

(19) LV c p,l



d md md

,

(20)

In the above equation, τdSt = 2ρl rd2 /9μ is the particle relaxation time in Stokes regime. BM is the Spalding mass transfer number and LV the latent heat of vaporization. Sh and Nu are the Sherwood number and Nusselt number given by Ranz and Marshall correlations,

Sh = 2 + 0.552Re1d/2 Sc1/3 ,

and Nu = 2 + 0.552Re1d/2 P r 1/3 , (21)

Y. Hu, R. Kai and R. Kurose et al. / International Journal of Multiphase Flow 125 (2020) 103216

accounting for the effect of forced convective mass and heat transfer at the droplet surface. Due to the small droplet slip velocity (0 ~ 5 m/s) indicated by Gounder et al. (2012), in the present ˜ − vd |/μ flame, most of droplets’ Reynolds number Red = 2ρg rd |u is less than 200. Other convective correlations for higher Reynolds number are provided by Michaelides (2006). Pr and Sc are the gasphase Prandtl and Schmidt numbers, cp,g and cp,l the specific heat capacities of the gas and liquid phase. f2 is the correction factor for the thermal transfer between phases around droplets (Miller et al., 1998). T˜g is the gas-phase temperature. In Eq. (20), a spatially uniform temperature distribution inside the droplets is supposed adopting the infinite conductivity assumption. This treatment however becomes improper since the temperature gradient inside droplet could be the controlling factor of droplet vaporization and ignition (Abramzon and Sirignano, 1989; Sazhin et al., 2005). The transient heating also needs to be taken into account with a reasonable amount of computer time. A parabolic function is suggested (Dombrovsky and Sazhin, 2003) for temperature profile inside the droplets. In this model, the key parameters needed are the average temperature Td,m and the surface temperature Td,s as computed by

dTd,m =

Nu 3P r +



c p,g c p,l



f2 (T˜g − Td,s )

τdSt 

dt +

LV c p,l



d md md



dmd  T − Td,m , md d,s

(22)

and,

Td,s =

Td,m + γ T˜g 0.5LV m˙ d γ + and 1+γ 2π rd λl (1 + γ )

γ = 0.1λgl Nu f2 ,

(23)

where λgl is the ratio of gas thermal conductivity to liquid conductivity as, λgl = λg /λl . Compared to the isothermal model, the internal variable heating is accounted for and the full solution in this parabolic model requires Eqs. (22) and (23). 2.4.2. Spray source terms The carrier phase quantities are interpolated using the droplets position, while the perturbations originated by the particles are introduced through source terms in each computational cell. Spray source terms play a role in the communication between phases and they ensure mass and energy conservation. For example, with the above droplet solutions, the spray source terms S˙ φ ,d are closed as

S˙ φ ,d = −

Nd 1  d nd,i δF φ (md,i ) V dt

(24)

i=1

Here, the summation is applied to all the particles i that traverse the local computational volume V. nd,i is the real number of droplets in each computational parcel, and δ is the Kronecker symbol and subscript F denotes species of fuel. In the DTF implementation for two-phase flow simulations, as pointed out by Filho et al. (2017), the mass imbalance and false evaporation could be induced by conventional Eulerian thickening procedure, and therefore a Lagrangian-type DTF transformation is employed on the liquid phase,

d md =

1 ˙ dt, M F d

and

dTd =

1 T˙ dt, F d

(25)

where the thickening factor F acts to reduce the timescale of dis˙ d and T˙ d reprepersed phase within the thickened flame front. M sent the variances at right-hand side of Eqs. (19), (20) and (22), respectively. Thus, the spray source terms calculated by Eq. (24) include the thickening effect implicitly.

5

Table 1 Boundary condition of spray EtF7 at jet exit (Gounder et al., 2012). Parameters

Value

Bulk velocity [m/s] Carrier air flow rate [g/min] Liquid fuel flow rate [g/min] Vapor fuel flow rate [g/min] Jet Reynolds number [-] Mixture fraction [-] Equivalence ratio [-]

60 376 70 5 45600 0.013 0.1

3. Configuration 3.1. Experimental setup The flame configuration under consideration is the Sydney piloted spray burner with ethanol fuel (Gounder et al., 2012). The spray generated by an ultrasonic nebulizer is initialized 215 mm away from the nozzle exit with a diameter of D = 10.5 mm. To stabilize this central spray jet, a pilot flame that is set to a stoichiometric mixture of hydrogen, acetylene and air is used and is injected coaxially with an outer diameter of 25 mm and an unburnt velocity 1.5 m/s. A co-flowing air stream of 4.5 m/s is supplied within a diameter of 104 mm. This burner has been well designed to be representative of reacting spray flows stabilized by recirculated hot products that are widely encountered in real engine combustion. Here, the experimental set B for EtF7 is used providing measurements for both axial and radial velocity profiles, for which the inlet condition is described in Table 1 and more information on the spray measurements can be found in (Gounder et al., 2012; Chen et al., 2006). 3.2. Numerical implementation The LES computation is carried out using an in-house semiimplicit solver FK3 (Kitano et al., 2016; Pillai and Kurose, 2019). The gas-phase conservation equations are discretized with the finite difference method adopting a staggered Cartesian grid. The fourthorder central differencing scheme approximates the nonlinear term in the momentum equation, while a WENO-scheme is used for the convective terms of scalar quantities. The marching in time is achieved using a third-order accurate TVD Runge-kutta scheme. A cuboid computational domain extending to 48D × 11D × 11D in stream-wise and span-wise directions, respectively, is employed and is discretized with grid points of 216 × 128 × 128. The validity of the present mesh has been illustrated in previous study of acetone spray flames (Hu and Kurose, 2019). The flamelet thermo-chemistry library is generated using the FlameMaster program (Pitsch, 1998) and an in-house 1-D spray flame code by integration of a detailed transport properties and chemical mechanism consisting of 38 species and 337 elementary reactions (Olguin et al., 2019). The tabulation of gaseous diffusion and premixed flame structures required by the multi-regime flamelet model relies on the pre-calculation of a 1D counterflow diffusion flame and a premixed freely propagating flame. The ethanol fuel and air at ambient temperature 300K are prescribed for the boundary condition at Z = 1 and Z = 0, respectively. For the spray flamelet library, an axisymmetric counterflow spray flame is solved, in which a spray stream carried by air is injected in opposite direction against an air-flow. This configuration representative for the basic structure in spray combustion has the advantages of being easily applied in the generation of the required flamelet databases (Gutheil and Sirignano, 1998). The choice of boundary conditions for this spray flame computation is motivated by the experimental conditions of the turbulent flame as shown in Table 1.

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Since spray EtF7 is partially pre-vaporized, the mixture of liquid and vapor fuel at the spray jet is taken into account by a choice of initial spray equivalence ratio E0∗ = 1.63 to match the experimental global value. The initial droplet size R∗0 at the spray inlet is set as 50 μm (in radius) because the disperse phase has negligible mass larger than this value based on the observation of the inlet experimental droplet size distribution. The measured mean data of gas and liquid phases at the first experimental cross-section (x = 0.3D) are used to determine the computational inlet conditions, and a pseudo turbulence generated by a digital filter technique is imposed on the prescribed inlet gas velocity profiles (Hu and Kurose, 2019). Due to the high volatility of ethanol and the vaporization of droplets traveling through the long nozzle, the gas-phase fuel concentration at the exit is expected to be inhomogeneous (Heye et al., 2014). For simplicity, the vaporized mixture fraction is often taken as uniform with value of 0.013 based on the air and vapor fuel mass flow-rate (Ukai et al., 2014), whereas the experiments do not have the necessary information on this scalar distribution to quantify its influence. For the disperse phase, the assignment of fuel droplets at the inlet follows a random approach, and the droplet velocity is determined with experimental velocity for different size class. In the experiment, the inlet droplet size distribution measured at different radial locations differs in a way that the size distribution shows bias towards small droplet when moving close to the jet wall (Gounder et al., 2012). The common strategy uses the profile at centerline since the large droplets are vital for the disperse phase properties and progress of combustion at later sampling sections (Rittler et al., 2015; Chrigui et al., 2012). Indicated by the experiments, the maximum droplet volume fraction is 3 × 10−4 and the spray is diluted without consideration of droplet-droplet interaction. The spray and flame statistics are obtained by parallel simulations with the usage of supercomputer at ACCMS, Kyoto University for approximately 48,0 0 0 CPU hours. This work will focus on the impact of the tabulated chemistry properties by using different flamelet models in the two-phase LES/DTF simulation, along with which the influence of inflow inhomogenieties and the droplet distribution on the flame behavior interplayed with sprays is also investigated. 4. Results and discussion 4.1. Effects of inflow conditions In this section, three computed cases are compared, which include 1) sim0: the baseline case considering the spray flamelet and an uniform mixture (Z = 0.013) at inlet; 2) sim0In: Differing from sim0, a designed mixture fraction Z profile to assess the inhomogeneity effect at fuel inlet is prescribed by an assigned linear function of Z = 2 · r/D · Zo + (1 − r/D · 2 ) · Zc , where Zo = 0.006 and Zc = 0.02 are adopted to simulate the wall effect on droplet evaporation inside the nozzle as well as to keep the mass flux the same as the uniform case; 3) sim0Sz: based on sim0, instead of the use of measured droplet size distribution at the centerline, the measured profile close to the jet wall is employed for the inlet droplets, which would bias the injection of smaller droplets (Gounder et al., 2012). Shown in Fig. 1 is the comparison of simulated and measured gas temperature at three different cross-sections. In general, all cases show a similar trend in agreement with experiments, while an apparent discrepancy is found at upstream close to the inlet plane, which, however, is also observed in LES available in the literature (De and Kim, 2013). At location x/D = 10, sim0In has the highest peak temperature, whereas sim0Sz leads to a slight shift of peak temperature away from the centerline. When moving downstream to x/D = 20, the difference between sim0 and sim0In be-

Fig. 1. Comparison of gas-phase temperature (dots: experiments and line: computed results) at axial locations x/D = 10, 20, 30.

Fig. 2. Comparison of droplet Arithmetic Mean Diameter (AMD) (dots: experiments and line: computed results) at axial locations x/D = 5, 10, 15, 20, 25, 30.

comes small in comparison to the one found between sim0 and sim0Sz. This indicates that the uncertainty on the gas phase mixture can induce the obvious changes of flame structure near the jet exit but its impact on downstream combustion behavior is relatively small in contrast to the inlet dispersed phase distribution. It can conclude to the fact that the lean-premixed nature of the current flame favors the changes on inlet fuel concentration, while this effect is smeared out at the region close to rich part of flame. In addition, a larger portion of small droplet is injected in case sim0Sz and this results in a small contribution to the fuel pool inside the cold spray jet but fuels the surrounding pilot flame. It can also be seen that the inlet uncertainty from both gas and liquid phase has a marginal effect on the gas temperature distribution at further downstream (x/D = 30), especially for the inner side of the main jet. Fig. 2 shows the predicted spray statistics of the droplet arithmetic mean diameter (AMD) for different inlet setups at six downstream cross-sections. It is seen that the AMD value computed from sim0Sz is smaller than the experiments and the two other approaches match the measured data in most cross-sections. Although at x/D = 5 sim0Sz predicted almost half of the experimental data near the central axis around 10 μm, its computed result increases slowly at downstream to reach the same level as the sim0 and sim0In. This can be attributed to the consumption of small droplets, after which the large droplets survive and become crucial in the determination of the far-field spray dynamics. In this sense, the downstream mean spray structure is insufficient to evaluate the inlet condition. It should be also noted that at the edge of the main jet sim0Sz attains a more reasonable prediction and the usage of size distribution at centerline for exit plane initialization might overestimates AMD. Meanwhile, the LES droplet size statistics show a lack of sensitivity to inlet gas-phase non-uniformities with equivalent results obtained for sim0 and sim0In. As pointed out by Pichard et al. (2002), when small droplets are injected, the flame may behave like a gaseous premixed flame

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rather than a two-phase flame since the small droplets tend to pre-vaporize completely before reaching reaction zone. Considering the evaluation of the transit time (τ tr ) of droplets traversing the stoichiometric flame front (at Zst = 0.1) based on the flame front thickness and laminar flame speed, and the evaporation time (τ evp ) estimated from the D2 law, for the present ethanol flame with droplet size AMD = 10 μm, τ tr is around 0.67 s and τ evap is 0.95 s, which indicates τ evap /τ tr > 1. Therefore, the combustion cases under consideration are mainly characterized by the twophase flame regime that requires a careful choice of the SGS combustion model (Wang et al., 2019), the influence of which would be discussed in Section 4.2. Figs. 3 and 4 present the prediction of radial profiles of mean and fluctuating droplet velocities. It is seen that all three cases show only small difference for the computation of mean particle velocity. After the axial location 5D away from the burner exit, the predicted velocities start to deviate from the experimental data and an under-prediction is found inside the main jet region. This deviation could be due to the over-prediction of the gas temperature, which causes the small droplets to evaporate faster and the inertia effects of the remaining large droplets contributes to these low particle mean velocities. Another possible reason may stems from the LDV measurement error because of the selective seeding (Gounder et al., 2012). On the contrary, the radial distribution of the droplet rms axial fluctuation velocity shows a higher value compared to the experimental measurements near the centerline.

However, a noticeable difference between the achieved results is observed when the initial droplet distribution that peaks towards the small droplet size is adopted. Due to the small Stokes number, small droplets tend to follow the carrier flow, attaining enhanced fluctuations in the dispersed phase, which also help explaining the lower computed r.m.s of Ud obtained at further downstream locations where small droplets evaporate completely. The figure also suggests that the inhomogeneous mixture conditions at the inlet may leads to the lower particle fluctuations. This may be attributed to the higher computed gas temperature (see Fig. 1). A reasonable agreement with experiments is also found for the radial particle velocity profiles depicted in Fig. 5. At x/D = 5 (close to the exit), the injected droplets experience a sudden expansion in geometry and consequently an apparent increase in their mean radial velocity at the edge of the jet around position r/D = 0.5. This trend is fairly well captured in the computations, although an overestimation is obtained partly because of the over-estimated momentum transfer from axial to radial direction. The evolution of the peak of Vd radial profile marks the downstream expansion of the main spray stream, and it is in a good agreement with experiments. The corresponding rms velocity is shown in Fig. 6. The fluctuation intensity is well captured at upstream locations, but a disagreement is observed far away from the nozzle exit. Worth noticing is that the case sim0In tends to predict a lower rms radial velocity. This can be related to the higher gas-phase temperature predicted and the resulting higher number of large droplets,

Fig. 3. Comparison of droplet axial mean (Ud ) velocity (dots: experiments and line: computed results) at axial locations x/D = 5, 10, 15, 20, 25, 30.

Fig. 5. Comparison of droplet radial mean (Vd ) velocity (dots: experiments and line: computed results) at axial locations x/D = 5, 10, 15, 20, 25, 30.

Fig. 4. Comparison of droplet axial fluctuating (Ud rms) velocity (dots: experiments and line: computed results) at axial locations x/D = 5, 10, 15, 20, 25, 30.

Fig. 6. Comparison of droplet radial fluctuating (Vd rms) velocity (dots: experiments and line: computed results) at axial locations x/D = 5, 10, 15, 20, 25, 30.

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Fig. 7. Comparison of gas-phase temperature (dots: experiments and line: computed results) at axial locations x/D = 10, 20, 30.

which retain their own momentum because of large Stokes number and impose a slow radial dispersion (Chen et al., 2006; Glaze and Frankel, 20 0 0). 4.2. Effects of flamelet chemistry and the liquid phase model In this section, the detailed flame structures generated by the different flamelet models under investigation (i.e, spray flamelet and multi-regime gas flamelet) and their implementation coupled with the dynamically thickened flame approach are discussed. In the context of the LES of spray flame, the model for the liquid phase evaporation considering the internal transient heating is also investigated. The nomenclature of the computed cases for comparison is: 1) sim0Pa adopts the parabolic droplet heating model of Eqs. (22) and (23) in a comparison to the isothermal model used in baseline case sim0; 2) Differing from the usage of the spray flamelet in case sim0Pa, the multi-regime gas flamelet is employed for the description of combustion chemistry in sim0PaGf. Fig. 7 presents the computed gas temperature and the experimental measurements at different axial locations. As shown in the plots, the use of the parabolic approximation for the inner temperature gradient of the droplets helps improving the gas-phase temperature prediction and leads to a computed value at x/D = 10 that is closer to experiments, which is in contrast with the lowest temperature achieved by case sim0PaGf. However, the underestimation of the gas temperature by sim0PaGf persists till the axial position x/D = 30, while the effects of using different liquid-phase models diminishes at this far away downstream location. By referring to the observation made for the temperature calculation obtained in the last section, it seems that, despite of the different inlet setups, chemistry and liquid-phase models, all simulated cases give a similar deviation on gas temperature computations at x/D = 10. This tendency to under-predict the peak gas temperature near nozzle exit also occurs in the simulations in Ref. (De and Kim, 2013). That work tested different closure methods for modeling the sub-grid scale scalar variance, and they attributed this discrepancy to the three-stream nature of the jet flame studied, where the mixing between fuel and air jets is interfered by the presence of a pilot flame, a phenomenon that is not explicitly taken into account by flamelet calculations. Another key factor affecting the performance of tabulated chemistry methods is the assumption of stochastic distributions of the control variables (Ihme and Pitsch, 2008). However, the assessment work conducted by Rittler et al. (2015) revealed that the predictions using a beta-PDF model is comparable to the results obtained using a top-hat function (TH) for Z and C, and the joint impact of Z and C is small for the present spray flame. Fig. 8 compares the LES droplet size statistics with experimental data. As shown in the figures, among all three cases, significant differences are found when the gas flamelet library is used for the SGS combustion modeling. The droplet evolution is affected by both gas-phase temperature and species concentration. As illustrated in Fig. 7 and the discussion below, the multi-regime gas flamelet presents a different profile on gas thermal properties, interplayed with which liquid-phase yields a different particle statis-

Fig. 8. Comparison of droplet Arithmetic Mean Diameter (AMD) (dots: experiments and line: computed results) at axial locations x/D = 5, 10, 15, 20, 25, 30.

Fig. 9. Comparison of droplet axial mean (Ud ) velocity (dots: experiments and line: computed results) at axial locations x/D = 5, 10, 15, 20, 25, 30.

tics. The flame width predicted by the gas flamelet library appears to be narrower (see Fig. 7) and the influence of flow expansion by the reacting gas dilatation is then not expected to be significant, which explains the small AMD profiles at outer radial positions in case sim0PaGf. Combining the assessment with Fig. 2, it can be concluded that the deviation of AMD profile found at downstream locations is mainly attributed to the overestimated gas temperature near the nozzle-exit and the resulting consumption of small droplets, which is also evidenced in other studies (Chrigui et al., 2012; De and Kim, 2013; Rittler et al., 2015). A more sophisticated combustion modeling could possibly cure this discrepancy. Fig. 9 illustrates the LES profiles of droplet mean velocity compared with measurements. Generally, all predictions provide comparable results and properly agree with the measured data. However same as for the AMD profile, the case sim0PaGf gives a higher droplet axial velocity due to the coupling between the radial spreading of droplet and the gas-phase thermal expansion. The comparison of statistical results for the particle fluctuating velocity is shown in Fig. 10. With sim0PaGf, few droplets escape from the fuel jet and cause a larger rms Ud around r/D = 1 at upstream cross-sections; the resulting large number of droplets leads to a higher rms Ud profiles in regions close to the centerline at downstream. Similarly, the particle radial velocity is affected by the different combustion models in a same way that particle dispersion is closely connected to the carrier gas temperature and its aforemen-

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Fig. 10. Comparison of droplet axial fluctuating (Ud rms) velocity (dots: experiments and line: computed results) at axial locations x/D = 5, 10, 15, 20, 25, 30.

Fig. 11. Snapshots of OH mass fraction. The predictions from cases sim0, sim0Pa and sim0PaGf are compared to the measured results (Exp.) at left-hand side.

tioned dilatation effect. For brevity, their profiles are not shown here. Fig. 11 shows a comparison of the computed instantaneous data of OH mass fraction with the laser induced fluorescence (LIF)

9

images collected experimentally at various downstream positions. The LIF planar imaging of OH species is usually adopted for the visualization of the heat release zone and examination of its complex structure interacting with the turbulent flow-field (O’Loughlin and Masri, 2012). Here, the representative snapshots from different computed cases show that the main features of this flame, i.e., flame wrinkling, flame width and the location of flame front, are qualitatively captured by the simulations, while the predictions with spray flamelet chemistry attain good agreement with measurements on the downstream evolution of flame dynamics in contrast to the case with gas flamelet. A distinct characteristics is observed in sim0PaGf, showing a thinner reaction zone with a corrugated structure at the inner jet. It is worthy to mention that in present multi-regime flamelet formulation, the cross-correlations between diffusion and premixed flame structures are not considered. As indicated by Knudsen and Pitsch (2012), this treatment may lead to the spurious prediction of unsteady effects at the initial flame evolution. To examine the influence of the different models on the sprayflame interaction, the representative distribution of fuel droplets overlaid with the contour plot of OH mass fraction and flame thickening factor F is illustrated in Fig. 12. As indicated by Eq. (11), the thickening factor F is determined along with the parameter  (Eq. (12)) which displays the location of the propagating flame front. Hence, the spatial distribution of F suggests the presence of a premixed mode of combustion near the burner exit and at the shear layer there is a strong direct interaction between the spray core and the reaction zone. This is consistent with the findings in (Hu and Kurose, 2018; Chrigui et al., 2012) where this burner was found to exhibit various regimes of upstream premixed-like reactions coupled with downstream nonpremixed combustion. Meanwhile, with the usage of the parabolic temperature model for the evaporating droplets in cases sim0Pa and sim0PaGf, the burning islands appear to be formed much closer to the exit plane where the dispersed phase shows a higher evaporation rate. As shown by Dombrovsky and Sazhin (2003), at the initial stage of droplet heating, the uniform temperature assumption adopted by the infinite thermal conductivity model leads to a lower droplet surface temperature when compared to the parabolic model. Consequently, the droplet evaporation is relatively lower when the infinite conductivity model is used (Eq. (20)). In addition, different flamelet modeling strategies in such two-phase LES induce a different pattern on both gas and spray structures. Concerning the importance of the flamelet chemistry on the fuel evaporation and consequently on the flame structure, further assessment is made with the definition of evaporation parameter

Fig. 12. Snapshots of OH mass fraction and flame thickening factor F from computations of sim0(left), sim0Pa(middle), and sim0PaGf (right). Filled dots are the representative distribution of fuel droplets colored by the evaporation rate m˙ d .

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Fig. 13. Joint normalized histogram of mixture fraction Z and evaporation parameter θ l computed by spray flamelet (1st row: sim0Pa) and multi-regime gas flamelet (2nd row: sim0PaGf) at four axial locations. The dash-line indicates the position of stoichiometric mixture fraction Zst = 0.1.

θl = S˙ v /ρ , the joint histogram of which together with the mixture fraction Z is illustrated in Fig. 13 for different axial locations. At the downstream location, away from the jet exit around x/D = 2, it can be seen that droplet evaporation occurs mainly at the lean side of the reactive mixture, which is due to the low equivalence ratio of the incoming carrier phase. Specifically, the gas flamelet calculation attains a doubly-conditioned profile and, as discussed in Fig. 9 of higher axial momentum in this case, fuel-droplets evaporate in a strong stratified flow condition that prevents a sufficient local mixing. For the farther downstream locations, the distribution shifts towards the stoichiometric mixture fraction Zst as the spray evaporation has a direct interplay with the flame zone. At x/D = 15, the appearance of the interaction is predicted by the spray flamelet model, which means that the mass release from the droplets into the carrier phase is promoted in this simulation, and thereafter the flame propagation from the hot-pilot stream to the inner core jet is well developed. It explains the computation of the gas temperature by case sim0Pa (shown in Fig. 7) and indicates the importance of the inclusion of spray-flame interactions in the flamelet library to properly reproduce the flame dynamics. Therefore, more droplets penetrate the high temperature mixture layers in the simulation based on gas flamelet library, which form another evaporation zone at the rich side shown at x/D = 20. 5. Conclusions In the present work, model assumptions relevant for the accurate description of complex spray combustion were investigated, and their impact in multi-phase LES of the Sydney partially prevaporized spray flame EtF7 was reported. The carrier gas flow and chemical reactions were simulated by LES and SGS combustion models based on tabulated detailed chemistry, while the dispersed phase is computed using a Lagrangian tracking method. Focusing on the proper modeling of the mixed combustion regime in spray flames, the different tabulation strategies, i.e., spray flamelet and multi-regime gas flamelet models were considered. Using the available experimental data, the quantitative comparisons of spray

and flame properties were made to study the influence of inlet uncertainty on gas and liquid distribution, the spray assumption used for droplet heating, as well as the tabulation techniques. The main conclusions are drawn as follows, •

For the computational inflow conditions of the present lean spray flame, the inhomogeneities of the gas mixture at the inlet provided an improved prediction of peak gas-phase temperature at upstream location, although its effect vanishes when moving to further downstream cross-sections. Comparatively, the computed results of both gas and liquid phases show a higher sensitivity to the different droplet size distributions. The computed peak temperature decreases with the initial size distribution that bias the small droplets, but considering a large number of small droplets contributes to the fuel supply for hotpilot and it shift the gas temperature profile slightly outwards.

•

The comparative study of different flamelet methods confirmed that the turbulence-chemistry coupling is the dominant factor affecting the overall flame structure. The use of gas flamelet libraries induced a thinner combustion zone with an underprediction of the gas temperature. On the contrary, the chemical flamelet database built from counterflow partially-premixed spray flames achieved a better agreement with experiments by capturing the flame structures in terms of gas temperature, OH mass fraction as well as spray statistics. The analysis of joint PDF of mixture fraction and evaporation parameter confirmed the importance of inclusion of spray-flame coupling in flamelet library in order to capture the flame evolution under the influence of spray evaporation effects. In addition, the consideration of droplet inner temperature gradients intensified the formation of combusting islands close to the inlet plane and led to an improved temperature prediction at upstream locations.

The present model formulation of a coupled spray or multiregime gas flamelet method with a dynamic thickened flame model showed its promising capability in reproducing the behavior of spray combustion with partial pre-vaporization. This fea-

Y. Hu, R. Kai and R. Kurose et al. / International Journal of Multiphase Flow 125 (2020) 103216

ture is attractive for applications to model modern combustion technologies where lean combustion and two-phase regimes with partially premixed-, non-premixed- and propagating premixed-like multiple-reactions are encountered, for which further investigations is required.

Declaration of Competing Interest None.

CRediT authorship contribution statement Yong Hu: Conceptualization, Methodology, Software, Validation, Investigation, Formal analysis, Data curation, Writing - original draft, Writing - review & editing, Visualization, Project administration, Funding acquisition. Reo Kai: Software, Formal analysis, Data curation. Ryoichi Kurose: Resources, Writing - review & editing, Supervision, Project administration, Funding acquisition. Eva Gutheil: Resources, Writing - review & editing, Project administration, Funding acquisition. Hernan Olguin: Resources, Software, Writing - review & editing, Project administration, Funding acquisition. Acknowledgements This research has received funding support from “the Fundamental Research Funds for the Central Universities” under the Project YJ201943. The author also would like to thank for the support by MEXT (Ministry of Education, Culture, Sports, Science, and Technology) as “Priority issue on Post-K computer” (Accelerated Development of Innovative Clean Energy Systems). EG acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Projektnummer 374463455 and through HGS MathComp. HO thanks CONICYT (Chile) for financial support through FONDECYT Grant 11171020. References Abramzon, B., Sirignano, W.A., 1989. Evaluation of equilibrium and non-equilibrium evaporation models for many-droplet gas-liquid flow simulations. Int. J. Heat Mass Transf. 32, 1605–1618. Baba, Y., Kurose, R., 2008. Analysis and flamelet modeling for spray combustion. J. Fluid Mech. 612, 45–79. Bilger, R.W., Pope, S.B., Bray, K.N.C., Driscoll, J.F., 2005. Paradigms in turbulent combustion research. Proc. Combust. Inst. 30, 21–42. Bray, K.N.C., 1996. The challenge of turbulent combustion. Symp. (Int.) Combust. 1–26. Charlette, F., Meneveau, C., Veynante, D., 2002. A power-law flame wrinkling model for LES of premixed turbulent combustion part i: non-dynamic formulation and initial tests. Combust. Flame 131, 159–180. Chen, Y.C., Starner, S.H., Masri, A.R., 2006. A detailed experimental investigation of well-defined, turbulent evaporating spray jets of acetone. Int. J. Multiphase Flow 32, 389–412. Chrigui, M., Gounder, J., Sadiki, A., Masri, A.R., Janicka, J., 2012. Partially premixed reacting acetone spray using LES and FGM tabulated chemistry. Combust. Flame 159, 2718–2741. Colin, O., Ducros, F., Veynante, D., Poinsot, T., 20 0 0. A thickened flame model for large eddy simulations of turbulent premixed combustion. Phys. Fluids 12, 1843. De, S., Kim, S.H., 2013. Large eddy simulation of dilute reacting sprays: droplet evaporation and scalar mixing. Combust. Flame 160, 2048–2066. Dombrovsky, L.A., Sazhin, S.S., 2003. A parabolic temperature profile model for heating of droplets. J. Heat Transf. 125, 535. Felden, A., Esclapez, L., Riber, E., Cuenot, B., Wang, H., 2018. Including real fuel chemistry in LES of turbulent spray combustion. Combust. Flame 193, 397–416. Filho, F.L.S., Kuenne, G., Chrigui, M., Sadiki, A., Janicka, J., 2017. A consistent artificially thickened flame approach for spray combustion using LES and the FGM chemistry reduction method: validation in lean partially pre-vaporized flames. Combust. Flame 184, 68–89. Fiorina, B., Gicquel, O., Vervisch, L., Carpentier, S., Darabiha, N., 2005. Approximating the chemical structure of partially premixed and diffusion counterflow flames using FPI flamelet tabulation. Combust. Flame 140, 147–160.

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Franzelli, B., Fiorina, B., Darabiha, N., 2013. A tabulated chemistry method for spray combustion. Proc. Combust. Inst. 34, 1659–1666. Glaze, D.J., Frankel, S.H., 20 0 0. Effect of dispersion characteristics on particle temperature in an idealized nonpremixed reacting jet. Int. J. Multiphase Flow 26, 609–633. Gounder, J.D., Kourmatzis, A., Masri, A.R., 2012. Turbulent piloted dilute spray flames: flow fields and droplet dynamics. Combust. Flame 159, 3372– 3397. Gutheil, E., Sirignano, W.A., 1998. Counterflow spray combustion modeling including detailed transport and detailed chemistry. Combust. Flame 113, 92–105. Han, W., Wang, H., Kuenne, G., Hawkes, E.R., Chen, J.H., Janicka, J., Hasse, C., 2019. Large eddy simulations/dynamic thickened flame modeling of a high Karlovitz number turbulent premixed jet flame. Proc. Combust. Inst. 37, 2555– 2563. Heye, C., Kourmatzis, A., Raman, V., Masri, A.R., 2014. A comparative study of the simulation of turbulent ethanol spray flames. In: Gutheil, E., Merci, B. (Eds.), Experiments and Numerical Simulations of Turbulent Combustion of Diluted Sprays, vol. 19. Springer, Cham, Switzerland, pp. 31–54. ERCOFTAC Series Hollmann, C., Gutheil, E., 1998. Diffusion flames based on a laminar spray flame library. Combust. Sci. Technol. 1, 175–192. Honhar, P., Hu, Y., Gutheil, E., 2017. Analysis of mixing models for use in simulations of turbulent spray combustion. Flow Turbul. Combust. 99, 511–530. Hu, Y., Kurose, R., 2018. Nonpremixed and premixed flamelets LES of partially premixed spray flames using a two-phase transport equation of progress variable. Combust. Flame 188, 227–242. Hu, Y., Kurose, R., 2019. Partially premixed flamelet in LES of acetone spray flames. Proc. Combust. Inst. 37, 3327–3334. Hu, Y., Olguin, H., Gutheil, E., 2017. A spray flamelet/progress variable approach combined with a transported joint PDF model for turbulent spray flames. Combust. Theory Model. 21, 575–602. Hu, Y., Olguin, H., Gutheil, E., 2017. Transported joint probability density function simulation of turbulent spray flames combined with a spray flamelet model using a transported scalar dissipation rate. Combust. Sci. Technol. 189, 322– 339. Ihme, M., Pitsch, H., 2008. Prediction of extinction and reignition in nonpremixed turbulent flames using a flamelet/progress variable model. 2. Application in LES of Sandia flames D and E. Combust. Flame 155, 90–107. Jiang, X., Siamas, G., Jagus, K., Karayiannis, T., 2010. Physical modelling and advanced simulations of gas-liquid two-phase jet flows in atomization and sprays. Prog. Energy Combust. Sci. 36, 131–167. Kitano, T., Kaneko, K., Kurose, R., Komori, S., 2016. Large-eddy simulations of gasand liquid-fueled combustion instabilities in back-step flows. Combust. Flame 170, 63–78. Kitano, T., Nishio, J., Kurose, R., Komori, S., 2014. Effects of ambient pressure, gas temperature and combustion reaction on droplet evaporation. Combust. Flame 161, 551–564. Knudsen, E., Pitsch, H., 2012. Capabilities and limitations of multi-regime flamelet combustion models. Combust. Flame 159, 242–264. Knudsen, E., Shashank, Pitsch, H., 2015. Modeling partially premixed combustion behavior in multiphase LES. Combust. Flame 162, 159–180. Kuenne, G., Ketelheun, A., Janicka, J., 2011. LES modeling of premixed combustion using a thickened flame approach coupled with FGM tabulated chemistry. Combust. Flame 158, 1750–1767. Legier, J. P., Poinsot, T., Veynante, D., 20 0 0. Proceedings of the Summer Program 20 0 0, Center for Turbulence Research, 157–168 Lilly, D.K., 1992. A proposed modification of the Germano subgrid-scale closure method. Phys. Fluids A 4, 633–635. Luo, Y., Wen, X., Wang, H., Luo, K., Fan, J., 2018. Evaluation of different flamelet tabulation methods for laminar spray combustion. Phys. Fluids 20, 053603. Michaelides, E.E., 2006. Particles, bubbles, and drops: their motion, heat and mass transfer. World Scientific. Singapore, Hackensack, NJ Miller, R.S., Harstad, K., Bellan, J., 1998. Droplet vaporization model for spray combustion calculations. Int. J. Multiphase Flow 24, 1025–1055. Olguin, H., Gutheil, E., 2014. Influence of evaporation on spray flamelet structures. Combust. Flame 161, 987–996. Olguin, H., Scholtissek, A., Gonzalez, S., Gonzalez, F., Ihme, M., Hasse, C., Gutheil, E., 2019. Closure of the scalar dissipation rate in the spray flamelet equations through a transport equation for the gradient of the mixture fraction. Combust. Flame 208, 330–350. O’Loughlin, W., Masri, A.R., 2012. The structure of the auto-ignition region of turbulent dilute methanol sprays issuing in a vitiated co-flow. Flow Turbul. Combust. 89, 13–35. Pandal, A., García-Oliver, J.M., Novella, R., Pastor, J.M., 2018. A computational analysis of local flow for reacting diesel sprays by means of an Eulerian CFD model. Int. J. Multiphase Flow 99, 257–272. Peters, N., 1984. Laminar diffusion flamelet models in non-premixed turbulent combustion. Prog. Energy Combust. Sci. 10, 319–339. Peters, N., 20 0 0. Turbulent Combustion. Cambridge University Press, Cambridge, UK. Pichard, C., Michou, Y., Chauveau, C., Gökalp, I., 2002. Average droplet vaporization rates in partially prevaporized turbulent spray flames. Proc. Combust. Inst. 29, 527–533. Pillai, A.L., Kurose, R., 2019. Combustion noise analysis of a turbulent spray flame using a hybrid DNS/APE-RF approach. Combust. Flame 200, 168–191. Pitsch, H., 1998. Flamemaster v3.1: A c++ computer program for 0d combustion and 1d laminar flame calculations. Pope, S.B., 20 0 0. Turbulent Flows. Cambridge University Press, Cambridge, UK.

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Y. Hu, R. Kai and R. Kurose et al. / International Journal of Multiphase Flow 125 (2020) 103216

Proch, F., Domingo, P., Vervisch, L., Kempf, A.M., 2017. Flame resolved simulation of a turbulent premixed bluff-body burner experiment. Part II: a-priori and a– posteriori investigation of sub-grid scale wrinkling closures in the context of artificially thickened flame modeling. Combust. Flame 180, 340–350. Réveillon, J., Vervisch, L., 2005. Analysis of weakly turbulent dilute-spray flames and spray combustion regimes. J. Fluid Mech. 537, 317–347. Rittler, A., Proch, F., Kempf, A.M., 2015. LES of the sydney piloted spray flame series with the PFGM/ATF approach and different sub-filter models. Combust. Flame 162, 1575–1598. Sazhin, S.S., Abdelghaffar, W.A., Sazhina, E.M., Heikal, M.R., 2005. Models for droplet transient heating: effects on droplet evaporation, ignition, and break-up. Int. J. Therm. Sci. 44, 610–622.

Ukai, S., Kronenburg, A., Stein, O.T., 2014. Simulation of dilute acetone spray flames with LES-CMC using two conditional moments. Flow Turbul. Combust. 93, 405–423. Wang, G., Boileau, M., Veynante, D., 2011. Implementation of a dynamic thickened flame model for large eddy simulations of turbulent premixed combustion. Combust. Flame 158, 2199–2213. Wang, Q., Jaravel, T., Ihme, M., 2019. Assessment of spray combustion models in large-eddy simulations of a polydispersed acetone spray flame. Proc. Combust. Inst. 37, 3335–3344. Xiao, G., Jia, M., Wang, T., 2016. Large eddy simulation of n-heptane spray combustion in partially premixed combustion regime with linear eddy model. Energy 97, 20–35.