# Large-Eddy Simulations of kerosene spray combustion in a supersonic jet flow

Florian Kissel, Guillaume Ribert, Pascale Domingo

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Florian Kissel, Guillaume Ribert, Pascale Domingo. Large-Eddy Simulations of kerosene spray combustion in a supersonic jet flow. Aerospace Science and Technology, 2025, 161, 10.1016/j.ast.2025.110164. hal-05326650

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# Large-Eddy Simulations of kerosene spray combustion in a supersonic jet flow

Florian Kissel, Guillaume Ribert ID*, Pascale Domingo ID

CORIA-CNRS, Normandie Univ., INSA de Rouen Normandie, Rouen, 76000, France

<figure><img src="image_2.png" /></figure>

## ARTICLE INFO

Communicated by Suresh Menon

**Keywords:**

Kerosene

Supersonic

Spray

Reduced chemistry

LES

Combustion

## ABSTRACT

High-speed reactive two-phase flows, relevant for the development of future scramjet engines, are studied by Large-Eddy Simulations. The present configuration is inspired by the supersonic Cheng's burner replacing hydrogen with kerosene in the gaseous or liquid phase. The sonic kerosene injection is surrounded by a supersonic co-flow of hot vitiated air, ensuring a jet flame's stabilization. The global equivalence of the burner is set to 0.5. A 5-species global mechanism and a more detailed mechanism from the literature involving 19 species and 54 reactions are compared with a new optimized chemistry involving 18 species and 29 reactions. The impact of ignition delay and droplet size on flame stabilization dynamics and combustion modes is discussed. The necessity of a sub-grid closure for the source terms is assessed. Whatever the kerosene phase, the flame structure is dominated by the non-premixed combustion regime, even if a non-negligible contribution of the rich premixed regime is detected.

## 1. Introduction

The development of hypersonic propulsion technologies such as scramjets requires a better understanding of the physical mechanisms involved in supersonic combustion [1]. The main difficulties of these technologies lie in the high velocity of the airflow in the combustion chamber, limiting the residence time required for the fuel-oxidant mixture, ignition, and flame stabilization [2]. While the first scramjet studies were conducted by injecting hydrogen or gaseous ethylene [3–7], liquid hydrocarbon fuels are more likely to be used (up to Mach 8 [1]) thanks to their high densities and their easier storage. Consequently, the use of liquid fuel requires an advanced understanding of the interactions between supersonic flow, turbulence, combustion, and liquid vaporization [8].

For subsonic dilute spray jets and flames, extensive experimental databases exist for selected configurations [9] even if detailed measurements in turbulent spray flames remain a difficult task [10]. However, performing experiments with high-enthalpy conditions in ground-based combustion facilities is much more challenging [11]. For gaseous injections, the supersonic turbulent burner of Cheng et al. [12] that yields a hydrogen/air jet flame, appears as a reliable experiment for numerical code validation [13–17]. However, there is no equivalent experimental setup with spray injection, and most available configurations in the

literature are mainly related to scramjets designed with a cavity flame holder. This arrangement enhances mixing by trapping fuel and air in recirculation zones to provide a stable flame. For example, in [18], liquid kerosene is injected with an air barbotage atomizer system into the supersonic airflow either upstream of the cavity or at the cavity floor. Such a strategy increases the jet penetration and provides better atomization. A pilot hydrogen flame may be added upstream of the cavity to trigger ignition and sustain combustion that occurs inside the cavity. For such a transverse injection in a supersonic flow [19], the spray penetration and fuel-oxidizer mixing are enhanced by a high injection pressure, characterized by a faster atomization process and the formation of a bow shock [19]. The liquid is often pressurized with a piston [20], and the subsequent liquid jet disintegration is driven by the physical processes of primary and secondary atomization and the relative importance of Kelvin-Helmholtz and Rayleigh-Taylor instabilities [8]. A significant improvement in combustion efficiency is observed when turbulence promotes mixing between small drops of fuel and the oxidizer or when a gas barbotage injection technique is used, i.e. when a gas like air or hydrogen [18,21] is added to the liquid to inject.

Since experimental studies provide a limited amount of information, numerical simulations are often performed to characterize supersonic spray combustion [8]. However, in the case of heavy hydrocarbon combustion, like kerosene, the cost of detailed chemistry, coupled with the

<footnote>* Corresponding author.
E-mail address: guillaume.ribert@insa-rouen.fr (G. Ribert).</footnote>
stiffness of the chemical mechanisms, is a major issue to consider. Indeed, since the kerosene oxidation yields many radical species [22], solving for the mass fraction of each species leads to two main difficulties [23]: the number of computing operations increases dramatically with the number of species, and the stiffness of the chemical system imposes fine mesh and time resolutions [24]. As both highlighted in [25,26] for a two-phase supersonic reacting mixing layer, the use of global mechanisms does not allow capturing the local flame blowout as well as instabilities phenomena. In the literature, global reaction mechanisms [27,28] are nevertheless used to drastically reduce the computational time [29,25,26,8] at the expense of a good description of the combustion phenomenon. Accordingly, skeletal mechanisms have also been constructed [30–32] to more accurately simulate the gaseous or spray kerosene combustion in scramjet configurations. For instance, Yao et al. [33] developed several skeletal mechanisms that perform well in autoignition delays and laminar flame speed predictions for the most detailed one.

In the present study, the combustion of a kerosene spray is addressed with well-refined large-eddy simulations in the context of highly compressible flows. The configuration of Cheng’s burner [12] (supersonic hydrogen combustion) is adapted by replacing hydrogen with kerosene. Two kinds of kerosene injection are envisaged, one using a supersonic gaseous injection of kerosene, and the other one using a dilute spray of drops of kerosene within a nitrogen carrier phase. A new reduced and optimized mechanism derived from [33] is proposed and compared with kinetics from the literature. The need for a model for the filtered source terms is discussed and the resulting flame structure is detailed.

## 2. Numerical setup

### 2.1. Numerical solver

The coupled gas flow transport equations and Lagrangian equations for droplets are solved within the numerical code SiTCom-B, already fully detailed in [5,16,23] and validated for supersonic gaseous flows. SiTCom-B solves the conservation equations of species mass fractions, with complex molecular transport properties, momentum, and total sensible energy in their fully compressible form over a structured mesh in a finite volume formulation. Large-Eddy Simulations (LES) are performed with the Smagorinsky approach to model the subgrid scales of turbulence, with the model constant $C_s = 0.15$. Turbulent Prandtl and Lewis numbers are set to 0.8 and 0.7, respectively. The laminar model is used for the filtered chemical source terms, as previously done in [16] with $\tilde{\omega}_k(\rho, Y_k, T) = \dot{\omega}_k(\bar{\rho}, \tilde{Y}_k, \bar{T})$. $\rho$ stands for density, $T$ for temperature, and $Y_k$ is the mass fraction of the $k^{th}$ species. This modeling approach is questioned in section 3.2 through the subgrid Damköhler number, and the impact of the LES-PaSR model [34] on the solution is compared to the laminar approach on the target spray configuration (next section). Both strategies are often used when dealing with reacting supersonic flows [35]. With the LES-PaSR modeling, $\tilde{\omega}_k(\rho, Y_k, T) = \kappa \dot{\omega}_k(\bar{\rho}, \tilde{Y}_k, \bar{T})$, where the volume fraction of the fine structures is estimated as $\kappa = \tau_c/(\tau_c + \tau_m)$, with $\tau_c$ and $\tau_m$, the chemical time scale and subgrid mixing time scale, respectively.

A two-way coupling is used with a Lagrangian spray solver to address the two-phase flow problem, in which each drop ($d$) is considered like a source point and where fluctuations at the subgrid-scale are neglected. To accurately capture the momentum coupling between the two phases [36], through the Basset-Boussinesq-Oseen equations [37], it is crucial to rigorously model the force acting on each individual particle due to fluid-particle interactions. The total force exerted on a spherical particle undergoing arbitrary time-dependent motion in an unsteady, inhomogeneous ambient flow, at finite Reynolds and Mach numbers, is typically decomposed into several physically meaningful components: the quasi-steady (or drag) force, the stress-gradient force, the added-mass force, and the viscous-unsteady force. The latter three forces are collectively referred to as “unsteady forces”, as they only arise when either the fluid

<figure><img src="image_3.png" /><figcaption>Fig. 1. Drag coefficient as a function of the Reynolds number for two Mach numbers: ⊙ $Ma_1 = 0.141$; □ $Ma_2 = 0.594$. Symbols: experimental data from [42,43]. Lines plus symbols: Formulation of Henderson [40]. Solid line: subsonic correlation of Schiller and Naumann [41].</figcaption></figure>

or the particle experiences acceleration. For shock-particle interactions, unsteady forces can be an order of magnitude larger than the quasi-steady force in the flow field behind a shock wave [36]. These unsteady effects are active when the shock interacts with the particle but have little contribution to long-term particle motion if the droplet-to-fluid density ratio $\rho_d/\rho_g$ is large, as presently. The resulting set of equations is written as follows for the position ($x_d$), velocity ($u_d$), mass ($m_d$), and temperature ($T_d$) of the droplet, keeping only the drag force in $F_d$:

$$ \frac{dx_d}{dt} = u_d, \qquad (1) $$

$$ m_d \frac{du_d}{dt} = F_d, \qquad (2) $$

$$ \frac{dm_d}{dt} = \dot{m}_d = -\frac{S_h}{3S_c} \left( \frac{m_d}{\tau_d} \right) \ln(1 + B_M), \qquad (3) $$

$$ C_p \frac{dT_d}{dt} = \frac{1}{m_d} (\dot{m}_d L_v - \Phi_e). \qquad (4) $$

The drag force is expressed as:

$$ F_d = \frac{1}{2} C_d \rho_g (u_g - u_d) |u_g - u_d| A_d, \qquad (5) $$

where $A_d$ is the frontal area of the particle, and $u_g$ the gas-phase velocity in the vicinity of the particle. The droplet evaporation, having a characteristic time $\tau_d$, is modeled with the Langmuir-Knudsen approach to express the relations linking Sherwood ($S_h$), Schmidt ($S_c$), and Spalding ($B_M$) numbers [38]. Such an approach has already been used in supersonic configurations [9,11]. $L_v$ is the latent heat of vaporization and $\Phi_e$ corresponds to the heat flux applied to the surface of the droplet, modeled with Abramzon-Sirignano’s approach [39].

In Eq. (5), the drag force or coefficient ($C_d$) must be adapted to compressible flows (see the formulation of Henderson [40]), and cannot be only expressed as a function of the Reynolds number of the drop as in Schiller and Naumann [41] correlations. Such a behavior is illustrated in Fig. 1 for two subsonic Mach numbers, $Ma_1 < 0.3$ to be in the low-Mach region, and $Ma_2 > 0.3$ to reach compressible flows. The comparison with the experimental data [42,43] (Fig. 1) shows that the modified drag coming from Henderson [40] provides better results in the case $Ma_2$ than with classical subsonic correlations, as for $Ma_1$ with a Reynolds number above $10^3$. Results are of the same quality otherwise. Indeed, in [40], the wave drag is now dependent on the relative Mach number ($Ma_r$), the Mach number of the supersonic airflow along with the temperature of the drop ($T_d$) and gas flow ($T_g$). $Ma_r = |u_d - u_f|/c_f$ with $u_f$ and $c_f$, the surrounding fluid velocity and speed of sound, respectively. As in the work done in [44] for supersonic flows, $C_d$ is now plotted in Fig. 2 as a function of the Reynolds number with the charac-
<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Drag coefficient versus Reynolds number for Ma = 2. T<sub>d</sub> and T<sub>g</sub> are drop and gas flow temperatures, respectively: ⊙ T<sub>d</sub>/T<sub>g</sub> = 0.26, □ T<sub>d</sub>/T<sub>g</sub> = 1.0. Symbols (⊗, □): experimental data from [45]; symbols (×): exp. data from [43]. Lines plus symbols: Formulation of Henderson [40]. Solid line: subsonic correlation of Schiller and Naumann [41].</figcaption></figure>

teristics of flow and drop coming from the experimental data of [43,45]. For Ma = 2, when C<sub>d</sub> is based on Henderson's formulation, results are in good agreement with experimental data, while the subsonic correlations of Schiller and Naumann [41] fail in the prediction. The model proposed by Henderson [40] for calculating the drag coefficient is then kept for the rest of the study.

Overall, the modeling strategy proposed in this section introduces modeling uncertainties — such as breakup possibility, pressure drag, and evaporation — that require advanced validations, which are however currently unavailable in the literature. For example, in [46], the effect of Mach number on droplet aerobreakup is examined experimentally, but no modeling approach is proposed. Similarly, in [47], the authors suggest that droplet superheating may influence the extent and rate of vaporization under locally supersonic conditions, but without presenting a specific model. The present work uses a state-of-the-art approach [48] for spray-flame interactions with droplets around ten microns in size, recognizing that further modeling developments will be needed in the future.

## 2.2. Configuration setup

To study the kerosene spray combustion in supersonic flows, a 3D configuration adapted from Cheng's configuration [12] is simulated to get a supersonic spray jet flame on a rounded slot burner. No-slip adiabatic solid structures are described with the immersed boundary method as in [5,16].

The case under study is detailed in Fig. 3 where the kerosene fuel (NC<sub>10</sub>H<sub>22</sub>) is injected in the gaseous or liquid phase, with drops having a diameter of 10 or 20 µm for the latter case, as in [29]. In this configuration, 10-micron (20-micron) droplets evaporate in approximately 1.1 ms (4.4 ms), while they exit the injector tube in just 31 µs. As a result, evaporation within the injector tube is negligible in this numerical setup. As in Cheng's burner, the fuel injection is surrounded by vitiated air with an excess of oxygen to obtain a lifted kerosene flame. These two conditions lead to a fuel-air ratio of 0.5. The characteristics of the meshes used in this study are detailed in Table 1 along with the number of cells in the injector lips. The grid size field is given in Appendix A (Figs. A.25 and A.26) along with the ratio between the subgrid eddy viscosity and the laminar viscosity (v<sub>l</sub>/v), which is found close to one in flame fronts and above 10 at shock locations. In Fig. 3, classical diamond shock structures are observed close to the injector on the pressure gradient field. In the case of the spray injection, the internal structure near the symmetry axis is characterized by less density or pressure gradients compared to the gaseous injection. Such a modification can affect mixing, ignition, and jet flame opening.

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** 2D cut of the pressure gradient field for the slot burner inspired from supersonic Cheng's burner. Kerosene fuel (NC<sub>10</sub>H<sub>22</sub>) is injected liquid (left) or gaseous (right). The vitiated air composition is N<sub>2</sub> (66%), O<sub>2</sub> (21%) and H<sub>2</sub>O (13%). Ma: Mach number. The drop diameter (d<sub>d</sub>) is set to 10 or 20 µm for the liquid case.</figcaption></figure>

<table><caption>**Table 1**
Mesh properties.</caption><thead><tr><th># of cells</th><th>Mesh 1</th><th>Mesh 2</th><th>Mesh 3</th></tr></thead><tbody><tr><td>N<sub>x</sub></td><td>560</td><td>814</td><td>1280</td></tr><tr><td>N<sub>y</sub></td><td>140</td><td>200</td><td>320</td></tr><tr><td>N<sub>z</sub></td><td>140</td><td>200</td><td>320</td></tr><tr><td>Inj. Lips</td><td>3</td><td>5</td><td>8</td></tr></tbody></table>

## 2.3. Optimized chemistry for supersonic combustion

In the present study, kerosene is simply modeled by N-Decane (NC<sub>10</sub>H<sub>22</sub>). Small-size kerosene mechanisms suitable for supersonic combustion modeling are scarce in the literature and in many cases, compromises have been made between the accuracy of the results and CPU costs. Two mechanisms taken from the literature [27,30] are compared to a novel reduced and optimized mechanism developed with the in-house tool ORCh (Optimised and Reduced Chemistry) which is based on the DRGEP method combined with a genetic algorithm [49,50]. Historically, the most detailed mechanism for kerosene combustion was developed by Dagaut et al. [51,52], containing 2185 species and 8217 reactions. This mechanism served as a reference for the construction of skeletal mechanisms, notably in the work of Yao et al. [33], who proposed five skeletal schemes with five levels of reductions (89, 48, 39, 28, and 19 chemical species). These mechanisms are developed for three
<figure><img src="image_6.png" /><figcaption>Fig. 4. Ignition delay of kerosene in vitiated air depending on temperature for $\phi = 1$.</figcaption></figure>

<caption>**Table 2**</caption>

Characteristics of mechanisms with species and reaction numbers. The new reduced and optimized chemistry is called FK_18S_29R. S: species #. R: reaction #.

<table><thead><tr><th>Mechanisms</th><th>S</th><th>R</th></tr></thead><tbody><tr><td>Westbrook_5S_1R [27]</td><td>WB<sub>χ</sub></td><td>5 1</td></tr><tr><td>FK_18S_29R</td><td>FK<sub>χ</sub></td><td>18 29</td></tr><tr><td>Yao_19S_54R [30]</td><td>Yao<sub>χ</sub></td><td>19 54</td></tr><tr><td>Luche_91S_991R [53]</td><td>Luc<sub>χ</sub></td><td>91 991</td></tr></tbody></table>

surrogate kerosene (including N-Decane). In parallel, Luche et al. [53] proposed a reduced mechanism with 91 species, which fits perfectly with Dagaut's mechanism for canonical 1D flame configurations (pre-mixed, non-premixed) and auto-ignition delays. This mechanism will, therefore, be used as a reference in this work. Among the mechanisms developed by Yao et al. [33], the 19-species mechanism is of interest for CPU cost reasons even if it significantly overestimates the auto-ignition delay for temperatures above 1500 K (Fig. 4). Accordingly, a new reduced mechanism is proposed and optimized based on the existing 28-species mechanism [33] but using the results obtained with the kinetic of Luche et al. [53] on stochastic simulations [49] as a reference. This reduced mechanism for N-decane optimized with ORCh has 18 chemical species including C₂H₅ which is essential for a reliable prediction of the auto-ignition delay. The mechanisms compared in this study are summarised in Table 2, and validated hereafter on canonical flame configurations like auto-ignition, laminar premixed and non-premixed one-dimensional flames. Fig. 4 shows the evolution of auto-ignition delays for a temperature of the fresh mixture, $T_{in} \in [1250; 2000]$ K. For the operating temperature of the configuration sketched in Fig. 3, i.e. near 1750 K at stoichiometry, WB<sub>χ</sub> and Yao<sub>χ</sub> are inaccurate in predicting ignition delay compared to Luc<sub>χ</sub>, with an ignition 2.7 times too fast with WB<sub>χ</sub> and 3.4 times too slow for Yao<sub>χ</sub>. The optimized chemistry FK<sub>χ</sub> quasi matches the results of the Luche mechanism for $T > 1500$ K. Laminar flame speeds ($S_L$) are plotted in Fig. 5 for $\phi \in [0.6; 2]$. Whereas the global mechanism does not fit the reference, both skeletal mechanisms recover the trend for $S_L(\phi)$, with FK<sub>χ</sub> matching closely Luc<sub>χ</sub>. Diffusion flame main properties are displayed in Figs. 6 and 7 for temperature and major combustion products, with operating conditions for vitiated air and kerosene corresponding to what will be encountered in the numerical set-up of the modified Cheng's burner. Once again, the global chemistry WB<sub>χ</sub> fails to reproduce the flame structure by overestimating the temperature and production of major combustion products. Yao<sub>χ</sub> and FK<sub>χ</sub> provide similar results, the new mechanism being nevertheless more accurate in pre-heating zones. The chemical time step estimated

<figure><img src="image_7.png" /><figcaption>Fig. 5. Laminar flame speed against $\phi$ for a temperature of fresh gas at $T = 473$ K.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 6. Profile of temperature for a 1D-diffusion flame with pure kerosene at $T = 473$ K against vitiated air at $T = 1900$ K.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 7. Profile of species CO₂ for a 1D-diffusion flame with pure kerosene at $T = 473$ K against vitiated air at $T = 1900$ K.</figcaption></figure>
<caption>Table 3</caption>

Kerosene-air reduced and optimized kinetic scheme obtained with the tool ORCh. Units are in cal/mole, mole, cm⁻³, s, and K. The efficiencies of 3-body reactions are: H₂O = 16.25, CO = 1.875, CO₂ = 3.75, and H₂ = 2.5.

<table><thead><tr><th></th><th>FK_18S_29R</th><th>A</th><th>n</th><th>E<sub>a</sub></th></tr></thead><tbody><tr><td>1</td><td>NC<sub>10</sub>H<sub>22</sub> → 2C<sub>3</sub>H<sub>6</sub> + 2CH<sub>3</sub> + C<sub>2</sub>H<sub>4</sub></td><td>2.47E+17</td><td>0</td><td>83296</td></tr><tr><td>2</td><td>C<sub>3</sub>H<sub>3</sub> + O<sub>2</sub> ⇌ CH<sub>3</sub>O + HCO</td><td>1.55E+26</td><td>-3.92</td><td>7047</td></tr><tr><td>3</td><td>H + O<sub>2</sub> ⇌ OH + O</td><td>1.48E+14</td><td>0</td><td>16934</td></tr><tr><td>4</td><td>H + O<sub>2</sub> + (M) ⇌ HO<sub>2</sub> + (M)</td><td>7.56E+17</td><td>-0.79</td><td>0</td></tr><tr><td>5</td><td>H + OH + (M) ⇌ H<sub>2</sub>O + (M)</td><td>2.57E+22</td><td>-1.99</td><td>0</td></tr><tr><td>6</td><td>O + H<sub>2</sub> ⇌ OH + H</td><td>1.3E+04</td><td>2.8</td><td>5922</td></tr><tr><td>7</td><td>H<sub>2</sub> + OH ⇌ H<sub>2</sub>O + H</td><td>2.29E+08</td><td>1.49</td><td>3456</td></tr><tr><td>8</td><td>H<sub>2</sub> + O<sub>2</sub> ⇌ OH + OH</td><td>1.7E+13</td><td>0.0</td><td>41780</td></tr><tr><td>9</td><td>HO<sub>2</sub> + OH ⇌ H<sub>2</sub>O + O<sub>2</sub></td><td>1.8E+13</td><td>0</td><td>-497</td></tr><tr><td>10</td><td>H + HO<sub>2</sub> ⇌ H<sub>2</sub> + O<sub>2</sub></td><td>2.55E+13</td><td>0</td><td>1419</td></tr><tr><td>11</td><td>H + HO<sub>2</sub> ⇌ OH + OH</td><td>8.4E+13</td><td>0</td><td>875</td></tr><tr><td>12</td><td>CO + OH ⇌ CO<sub>2</sub> + H</td><td>4.47E+06</td><td>1.51</td><td>-740</td></tr><tr><td>13</td><td>CO + O + (M) ⇌ CO<sub>2</sub> + (M)</td><td>2.79E+13</td><td>0</td><td>-4540</td></tr><tr><td>14</td><td>HCO + (M) ⇌ H + CO + (M)</td><td>3.08E+17</td><td>-0.99</td><td>16941</td></tr><tr><td>15</td><td>HCO + OH ⇌ CO + H<sub>2</sub>O</td><td>1.28E+14</td><td>0</td><td>0</td></tr><tr><td>16</td><td>HCO + O<sub>2</sub> ⇌ CO + HO<sub>2</sub></td><td>6.25E+12</td><td>0</td><td>407</td></tr><tr><td>17</td><td>CH<sub>3</sub> + O ⇌ H + CH<sub>3</sub>O</td><td>1.02E+14</td><td>0</td><td>0</td></tr><tr><td>18</td><td>CH<sub>3</sub> + O ⇌ H + H<sub>2</sub> + CO</td><td>3.93E+13</td><td>0</td><td>0</td></tr><tr><td>19</td><td>CH<sub>3</sub> + O<sub>2</sub> ⇌ CH<sub>2</sub>O + OH</td><td>6.38E+11</td><td>0</td><td>14176</td></tr><tr><td>20</td><td>CH<sub>3</sub> + CH<sub>3</sub> ⇌ C<sub>3</sub>H<sub>5</sub> + H</td><td>2.44E+13</td><td>0</td><td>13456</td></tr><tr><td>21</td><td>CH<sub>2</sub>O + OH ⇌ HCO + H<sub>2</sub>O</td><td>1.53E+09</td><td>1.18</td><td>-447</td></tr><tr><td>22</td><td>CH<sub>2</sub>O + H ⇌ HCO + H<sub>2</sub></td><td>1.17E+08</td><td>1.81</td><td>3008</td></tr><tr><td>23</td><td>C<sub>2</sub>H<sub>4</sub> + H + (M) ⇌ C<sub>2</sub>H<sub>5</sub> + (M)</td><td>3.46E+09</td><td>1.27</td><td>1291</td></tr><tr><td>24</td><td>C<sub>2</sub>H<sub>4</sub> + (M) ⇌ C<sub>2</sub>H<sub>3</sub> + (M)</td><td>3.49E+17</td><td>0</td><td>95721</td></tr><tr><td>25</td><td>C<sub>2</sub>H<sub>4</sub> + OH ⇌ C<sub>2</sub>H<sub>3</sub> + H<sub>2</sub>O</td><td>2.49E+13</td><td>0</td><td>5939</td></tr><tr><td>26</td><td>C<sub>3</sub>H<sub>4</sub> + H ⇌ C<sub>3</sub>H<sub>3</sub> + H<sub>2</sub></td><td>1.28E+14</td><td>0</td><td>14964</td></tr><tr><td>27</td><td>C<sub>2</sub>H<sub>3</sub> + O ⇌ CH<sub>3</sub> + CO</td><td>1.88E+13</td><td>0</td><td>0</td></tr><tr><td>28</td><td>C<sub>3</sub>H<sub>6</sub> ⇌ C<sub>2</sub>H<sub>3</sub> + CH<sub>3</sub></td><td>1.65E+21</td><td>-1.19</td><td>97058</td></tr><tr><td>29</td><td>C<sub>3</sub>H<sub>6</sub> + H ⇌ C<sub>2</sub>H<sub>4</sub> + CH<sub>3</sub></td><td>7.33E+12</td><td>0</td><td>1300</td></tr></tbody></table>

in the above simulations is around 40 ns for WBₓ, 3 ns for Yaoₓ, and 1 to 1.5 ns for FKₓ. Emphasis should be laid on the fact that proposing an accurate mechanism with reasonable CPU times for kerosene chemistry remains a major challenge. The mechanism FK_18S_29R is provided hereafter (Table 3) Additional comparisons are provided in Appendix B (Figs. B.27 and B.28).

# 3. LES of gaseous kerosene flame

## 3.1. Main characteristics of the jet flame

The configuration of Fig. 3 (right) is first simulated with the laminar model using WBₓ. Fig. 8 shows the instantaneous fields of pressure gradient and heat release on the left, with the instantaneous field of kerosene mass fraction and temperature on the right. The stoichiometric isoline ($Z_{st} = 0.0567$) is displayed in white, using Bilger's formulation. A 3D contour is also provided. In this configuration, the gaseous injection of kerosene is under-expanded and sonic, inducing a characteristic diamond shock structure, reflected in the supersonic co-flow of vitiated air, leading to compressibility effects and enhancing ignition close to the injector. Because of the higher-pressure conditions in the diamond shock at the injector exit, kerosene ignites around $x/D = 4$ in the mixing layer between kerosene and vitiated air. A strong heat release and auto-ignition around the stoichiometric line is observed in Fig. 8 on the 3D contour. The flame then develops following the mixing layer evolution. Supersonic combustion enhances compressible effects. For instance, at $x/D = 30$, a strong compression wave appears at the base of the flame as already observed in [16] for H₂/air combustion. This feature is evidenced in Fig. 9 by comparing the present reactive field to the same case without activating chemistry, i.e. $\omega_k = 0$. The position of the resulting bow shock oscillates according to the position of the flame while inducing a strong release of heat inside the stoichiometric isoline. Fig. 10

shows the scatter plots of the temperature, scaled by the temperature of vitiated air, as a function of the mixture fraction for a given instant. A zoom is performed for $Z \in [0.01; 0.1]$ with the black solid line corresponding to the adiabatic temperature calculated at 1 bar, knowing that the pressure in the domain varies from 0.6 to 1.4 bar. Before kerosene ignition, i.e. for $0 < x/D < 3$, the fuel and vitiated air mixture increases temperature, and the stoichiometric mixture is found around 1700 K. At the first diamond shock (between $x/D = 4$ and $x/D = 10$), the temperature rise becomes significant, particularly around stoichiometry due to the combustion activation. Moving upstream, i.e. $x/D > 10$, the flame expands radially (Fig. 8). Mixing therefore occurs with the cold injected air which leads to values of the reduced temperature below one found over all the values of mixture fraction Z (Fig. 10).

Statistics have been recorded for 3 ms, which roughly corresponds to seven convective times based on the kerosene injection velocity. The mean temperature and species mass fractions of NC₁₀H₂₂, H₂O and CO₂ are plotted at three downstream locations ($x/D = 5, 25, 45$) in Fig. 11 for the three mesh resolutions of Table 1. Near the injector lips ($x/D = 5$), mixing starts between the injected kerosene and the vitiated air and between the vitiated air and the ambient cold air. The CO₂ species, which was not present in the injected vitiated air, begins to be created, meaning that combustion has already started. This can also be observed on the H₂O mass fraction and temperature profiles, which have slightly increased compared to their injection values (Fig. 3). A distinction between the three meshes is visible on the profiles of temperature and YCO₂. The mesh resolution of mesh 1 appears too coarse to capture this mixing. At $x/D = 25$, the kerosene is consumed, yielding a significant increase of CO₂ concentration, and the flame development leads to a deeper expansion of the vitiated air in the cold surrounding air. The difference between the three meshes is increased as the resolution impacts the maxima on the axis and the jet opening. Finally, at $x/D = 45$, the temperature in the core of the jet has increased up to 2500 K. A
<figure><img src="image_10.png" /><figcaption>Fig. 8. Gaseous kerosene jet flame (mesh 3, WB<sub>χ</sub>). Left: 2D cut of instantaneous pressure gradient with heat release rate (HRR) field. Right: 2D cut of instantaneous mass fraction of fuel with temperature field (sides) and 3D iso-contour of stoichiometry colored by temperature. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 9. Comparison of reacting (left) and non-reacting (right) instantaneous field of the pressure gradient (mesh 1, WB<sub>χ</sub>).</figcaption></figure>

Gaussian-type profile appears for each variable as a consequence of the strong mixing between the central and the two annular jets. The three meshes give roughly similar results in the core of the jet, although the mixing layer thickens slightly with the mesh size, certainly because of the turbulence that is better captured in the case of mesh 3. Overall, the difference between the three meshes is small, and the coarsest mesh will be used in what follows in combination with the skeletal mechanisms.

## 3.2. Impact of chemistry on flame structure

The configuration of Fig. 3 (right) is now performed with the reduced mechanism FK<sub>χ</sub> on mesh 1. Although the overall flow remains the same as described above with the global chemistry WB<sub>χ</sub>, a more detailed chemistry will provide a better understanding of the flame ignition and extinction phenomena. Fig. 12 shows an instantaneous field of temperature, heat release, and mass fraction of a pyrolysis species (C<sub>3</sub>H<sub>6</sub>). Given the construction of the chemical scheme, kerosene pyrolyzes directly into less heavy species (C<sub>3</sub>H<sub>6</sub>, C<sub>2</sub>H<sub>4</sub>, CH<sub>3</sub>), which are then oxidized. Pyrolysis occurs in the mixing layer between the injected kerosene and vitiated air. Unlike the simulation with WB<sub>χ</sub> where the

<caption>**Table 4**</caption>

Average ignition position (AIP, $x/D \pm 2 \times \text{standard deviation}$) for the injection case with gaseous kerosene.

<table><thead><tr><th>Kinetic</th><th>Mesh</th><th>$x/D \pm 2 \times \text{std}$</th></tr></thead><tbody><tr><td>WB<sub>χ</sub></td><td>1</td><td>03.64 ± 0.99</td></tr><tr><td>WB<sub>χ</sub></td><td>2</td><td>03.86 ± 1.31</td></tr><tr><td>WB<sub>χ</sub></td><td>3</td><td>04.01 ± 1.31</td></tr><tr><td>FK<sub>χ</sub></td><td>1</td><td>12.64 ± 2.35</td></tr><tr><td>Yao<sub>χ</sub></td><td>1</td><td>22.76 ± 1.37</td></tr></tbody></table>

flame ignites and stabilizes in the first diamond shock, a succession of ignition and extinction pockets is observed in the first two diamond shocks. A more complex scenario is then found when activating realistic chemistry as confirmed through Table 4 in which the average ignition position (AIP) is reported for the three reduced kinetics WB<sub>χ</sub>, FK<sub>χ</sub>, and Yao<sub>χ</sub>. Such a position is defined as the average of the axial locations cumulated over six convective times, where the temperature in the field is above the co-flow temperature +500 K and the heat release is higher than 10 GW·m<sup>-3</sup>. The results of Table 4 highlight the major role of chemistry, which, in our case, must precisely reproduce the autoignition delays (τ). Indeed, from Fig. 4, $\tau_{WB} < \tau_{FK} < \tau_{Yao}$. Consequently, the global mechanism (WB<sub>χ</sub>) predicts the AIP 3.5 times earlier than with FK<sub>χ</sub>, and the flame stabilization appears very far from the injector with Yao<sub>χ</sub>. From Table 4, the mesh refinement is found to have little impact on flame position with WB<sub>χ</sub> (see also Fig. 11), which suggests that the no-model assumption is presently justified.

To better quantify the impact of the mesh resolution, the subgrid Damköhler number ($Da_{SGS}$, [54]) is now estimated as,

$$Da_{SGS} = \frac{\tau_{SGS}}{\tau_C} \quad \text{with} \quad \tau_{SGS} = \frac{1}{\sqrt{2\tilde{S}_{ij}\tilde{S}_{ij}}}$$

$\tilde{S}_{ij}$ being the resolved deformation tensor and

$$\tau_C = \min\left(\frac{\bar{\rho}_{H_2O}}{\dot{\omega}_{H_2O}}, \frac{\bar{\rho}_{CO_2}}{\dot{\omega}_{CO_2}}\right).$$
<figure><img src="image_12.png" /><figcaption>Fig. 10. Scatter plot of reduced temperature ($T/T_{\text{coflow}}$) versus mixture fraction ($Z$) for an instantaneous field at different heights ($x/D$). Mesh 1. $Z_{\text{st}}$ = 0.0567 and $T_{\text{coflow}}$ = 1900 K.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 11. Mean transverse profiles of temperature (1st line), $Y_{\text{CO}_2}$ (2nd line), $Y_{\text{H}_2\text{O}}$ (3rd line) and $Y_{\text{NC}_{10}\text{H}_{22}}$ (4th line) at several axial positions, $x/D$ = 5 (left), 25 (middle), and 45 (right), with $\text{WB}_x$.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Fig. 12. Instantaneous field of temperature, heat release and pyrolysis C₃H₆- species mass fraction (FKχ).</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 13. Subgrid Damköhler distribution in the flame for different mesh refinements (mesh 1 or 3) and kinetic schemes.</figcaption></figure>

If DaSGS ≪ 1, subgrid segregations may be neglected and a no-model strategy can be applied. The distribution of DaSGS is shown in Fig. 13 for reactive cells having a heat release rate greater than 1% of the maximum found in the domain of computation. DaSGS < 0.2 for 80% of reactive cells with mesh 3 using WBχ meaning that the no-model assumption is then valid. However, this percentage drops to 60% with mesh 1, whatever the kinetics, suggesting that subgrid-scale modeling for chemical source terms could be required for a better description of the flame behavior.

<figure><img src="image_16.png" /><figcaption>Fig. 14. Instantaneous field of flame index considering reactive cells (FI = −1: lean premixed, FI = 0: non-premixed, FI = +1: rich premixed), for WBχ (up), FKχ (middle), and Yaoχ (down). Mesh 1. Gradation: x/D = 10.</figcaption></figure>

The flame structure is addressed with the flame index (FI), only calculated for HRR > 5 GW·m⁻³, and defined as [55,56]

$$ FI = \frac{1}{2} \frac{Z - Z_{st}}{|Z - Z_{st}|} \left( 1 + \frac{\nabla Y_F \cdot \nabla Y_O}{|\nabla Y_F \cdot \nabla Y_O|} \right), \quad (6) $$

with the mass fraction of fuel YF = YNC10H22 for the global chemistry and YF = ΣYcnHm + YH2, for the reduced mechanisms. Premixed mode is characterized by FI equal to −1 (lean mixture) or 1 (rich mixture). Diffusion mode is characterized by FI = 0. An instantaneous field of the flame index distribution is shown in Fig. 14 using WBχ or FKχ on mesh 1. Rich-premixed and non-premixed combustion regimes are predominant. Up to x/D = 20, combustion mainly occurs around the stoichiometric line in the mixing layer. For 20 < x/D < 55, due to the mixing and the compressibility effects, combustion occurs both in the jet center in a rich premixed regime and in the surrounding mixing layer. Finally, the non-premixed regime becomes predominant again for x/D > 55 with just a few pockets of lean premixed combustion emerging. Although the overall dynamics are similar between the mechanisms, differences exist, notably related to the estimation of the auto-ignition delay of each mechanism. For the WBχ, rich-premixed zones are found inside the jet at positions x/D = 4. The same observation is made for the FKχ, at approximately x/D = 8. In contrast, the rich premixing zones appear much later for the Yaoχ, at x/D = 22, since the flame ignites much higher.

To better quantify these observations, the normalized conditional mean of the heat release rate on the flame index (RFI) is defined as

$$ R_{FI} = \frac{\langle \dot{\omega}_E | FI \rangle}{\langle \dot{\omega}_E \rangle} \quad \text{with } FI = \{-1, 0, 1\}. \quad (7) $$

The contribution of each combustion mode to the total heat release is then expressed as

$$ C_{FI} = R_{FI} \frac{N_{FI}}{N_T}, \quad (8) $$

with NFI the number of points in the interval of each value of the flame index considered, and NT the number of reactive cells considered. RFI (dots) and CFI (bar plots) are given in Fig. 15. Combustion in the jet flame is dominated by diffusion with CFI > 70% and negligible lean
<figure><img src="image_17.png" /><figcaption>**Fig. 15.** Dimensionless conditional mean of heat release by flame index ($R_{FI}$ in dots) and the contribution of each combustion mode to the total heat release ($C_{FI}$ in histograms) for gaseous injection.</figcaption></figure>

premixed combustion, whatever the kinetics used. The role of the rich premixed regime is correlated with the flame stabilization height, of the order of 25% with $Yao_\chi$, which has the longest ignition delay and is of the order of 5 to 10% with $WB_\chi$ with the shortest ignition delay. This can be easily explained by the fact that the longer the auto-ignition delay is, the more time the fuel and vitiated air have to mix with each other. Regarding $R_{FI}$, none of the three combustion regimes reveals weak or strong combustion, and the lean premixed regime contributes less to heat release regardless of the skeletal mechanism used.

Finally, the mean temperature field is provided in Fig. 16 for the three kinetic schemes. As expected from the average ignition position (Table 4), the average flame base position on the central axis is found closer to the injector with $WB_\chi$. The kerosene penetration is thus limited, like the jet flame opening. Using a more detailed kinetic scheme, with an increased ignition delay, allows the kerosene to penetrate further downstream, with consequences for the flame development. $FK_\chi$ having a shorter ignition delay than $Yao_\chi$, most of the reacting area is found at the end of the kerosene central jet as for the case with $WB_\chi$ but with a larger jet opening. This behavior is exacerbated with $Yao_\chi$, the axial flame position being pushed downstream.

## 4. LES of kerosene spray flame

The gaseous kerosene is now replaced by a kerosene spray containing droplets of 10 microns in diameter in a nitrogen-carrying phase. The injection of droplets is performed through a 14° angle cone located 3.2 mm from the lips of the injector inside the injection tube. The mass ratio and global equivalence ratio are roughly the same as in the gaseous injection ($\phi = 0.51$ and $\dot{m}_f = 1.6 \text{ g}\cdot\text{s}^{-1}$), considering the dilution by the nitrogen carrier phase. The dilute spray hypothesis is verified with a volume fraction of liquid $\alpha_\ell < 1\%$ almost everywhere, a maximum of 4% and 7% being found inside the injector for mesh 1 and 3, respectively. $\alpha_\ell$ is defined as

$$ \alpha_{\ell} = \frac{V_{\ell}}{V_{\text{cell}}}, \quad \text{with} \quad V_{\ell} = \sum_{d} \rho_{\ell} \frac{\pi d^{3}}{6}, \qquad (9) $$

$N_d$ being the number of drops in the volume of the cell considered, $V_{cell}$. The additional cost of the Lagrangian approach remains reasonable, multiplying the CPU cost by 3 in the case of mesh 1 and by 2 in the case of mesh 3.

<caption>**Table 5**</caption>

Average ignition position ($x/D \pm 2 \times \text{standard deviation}$) for the injection case with spray kerosene, $d_{10}$.

<table><thead><tr><th>Kinetic</th><th>Mesh</th><th>x/D ± 2 × std</th></tr></thead><tbody><tr><td>WB<sub>χ</sub></td><td>1</td><td>04.77 ± 1.52</td></tr><tr><td>WB<sub>χ</sub></td><td>2</td><td>04.97 ± 1.54</td></tr><tr><td>WB<sub>χ</sub></td><td>3</td><td>04.98 ± 1.79</td></tr><tr><td>FK<sub>χ</sub></td><td>1</td><td>22.16 ± 3.53</td></tr></tbody></table>

$WB_\chi$ and $FK_\chi$ kinetics are retained in what follows but only mesh 1 is used with $FK_\chi$ because of the CPU cost.

### 4.1. Main characteristics of the jet flame

The evolution of the spray flame is given in Fig. 17 with an overall flame structure similar to the gaseous flame. The kerosene spray is injected with nitrogen as a carrier phase at a temperature of 473 K (see Fig. 3 (left) for inlet characteristics), then evaporates and mixes with surrounding species. A maximum mass fraction is found between $x/D = 10$ and $x/D = 30$. The time required for the evaporation process, coupled with the cooling of the drops, induces a shift in the flame stabilization position compared to the gaseous case. In addition, the injection of drops modifies the flow velocity and therefore the compressibility of the flow i.e. the diamond shock structure. Fig. 18 compares the diamond shock structures at the injector outlet between the spray and gaseous cases. In Fig. 18, the gaseous injection being supersonic, the diamond shocks are reflected right into the core of the jet, whereas in the liquid case, the diamond shocks are reflected on the spray envelope. Nearly 40% of the drops have a relative Mach number greater than 0.3, justifying the use of wave drag correlations developed for compressible flows [42,44], especially for the description of the drops/vitiated air interaction at the injector outlet. Injecting drops has therefore consequences for the evolution of the ignition position (AIP), which is provided in Table 5 depending on the mesh size and the kinetic mechanisms considered. With $WB_\chi$, ignition happens at the first diamond shock (compression wave) as for the gaseous case due to the very short ignition delay of the kinetic scheme. The first ignition pockets appear around $x/D = 5$, and the impact of the mesh resolution is almost negligible. In the case of $FK_\chi$, ignition appears, as for the gaseous case, much later around $x/D = 22$, with greater fluctuations in position. This is 4.5 times higher than with $WB_\chi$ because ignition with $FK_\chi$ starts slowly in the first diamond shock. A difference is then observed between gaseous (Table 4) and spray (Table 5) flames, which is small with $WB_\chi$ but important for $FK_\chi$. Indeed, the endothermic process of evaporation coupled with the pyrolysis included in the kinetic scheme leads to a decrease in the temperature at the stoichiometric surface of the order of 75 K. As for the gaseous flame, the first diamond shock increases the heat release in the mixing layer, resulting in the pyrolysis of kerosene into lighter hydrocarbons, but thermodynamic conditions are not yet favorable for an efficient autoignition of the stoichiometric mixture, and the flame is pushed away compared to the gaseous case. Fig. 19 displays the heat release rate, averaged over four convective times, for the gaseous and spray configurations. The maximum heat release is reached at the first diamond shock in the gaseous case but at the second diamond shock for the spray flame. A strong interaction thus occurs between flow compressibility and flame ignition impacting the mixing and flame development.

$R_{FI}$ and $C_{FI}$ are plotted in Fig. 20. As in the gaseous case, the predominant regime is diffusion, with $C_{FI}$ of the order of 85-90% for $WB_\chi$ and 70% for $FK_\chi$. The rich premixed combustion regime is much more important with $FK_\chi$ than with $WB_\chi$ because the flame is located further downstream. Therefore, Kerosene and air have time to mix and not do only burn as a pure diffusion flame. It is also interesting to note that a similar behavior is observed when using $Yao_\chi$ on the gas configuration. Similar proportions are found for the combustion regimes with 70% for
<figure><img src="image_18.png" /><figcaption>Fig. 16. Mean temperature field for WB<sub>χ</sub> (left), FK<sub>χ</sub> (middle), and Yao<sub>χ</sub> (right).</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Fig. 17. Spray flame evolution with the instantaneous field of kerosene mass fraction (left), droplet spray colored by the temperature of the carrier phase (middle) and temperature field (right). Stoichiometry is defined using the gaseous mixture fraction (white contour). WB<sub>χ</sub>, mesh 3, d<sub>10</sub>. Gradation: x/D = 10.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Fig. 18. Comparison of instantaneous pressure gradient showing the diamond shock structure for both gaseous and spray injections (WB<sub>χ</sub>, mesh 3, d<sub>10</sub>).</figcaption></figure>

diffusion and 25% for the rich premixed regime. Finally, no combustion regime can be neglected since R<sub>FI</sub> is of the order of 0.8-1 for each combustion regime.

The kerosene penetration length is thus naturally found longer with FK<sub>χ</sub>.

<figure><img src="image_21.png" /><figcaption>Fig. 19. Comparison of heat release rate averaged over four convective times for gaseous case (left side) and spray case (right side), FK<sub>χ</sub>, mesh 1, d<sub>10</sub>.</figcaption></figure>

than with WB<sub>χ</sub> because of the evaporation process. The maximum position of the droplets reached before complete evaporation is given in Table 6. Results are in agreement with the above observations.

The impact of the kerosene droplet size is provided in Fig. 21, where the time evolutions of Lagrangian particles and of the Eulerian temperature field give insight into the physical processes at play for the kerosene spray flame in a supersonic flow. The droplet diameter (d<sub>d</sub>) is increased from 10 to 20 µm with an unchanged mass flow rate. On mesh 3, WB<sub>χ</sub> is used instead of FK<sub>χ</sub> due to CPU cost. However, the trend observed when switching from 10 to 20 µm should remain unchanged even with a more detailed chemical model. Cases will be denoted d<sub>10</sub> and d<sub>20</sub>, respectively.
<figure><img src="image_22.png" /><figcaption>**Fig. 20.** Dimensionless conditional mean of heat release by flame index ($R_{FI}$ in dots) and the contribution of each combustion mode to the total heat release ($C_{FI}$ in histograms) for spray injection, $d_{10}$.</figcaption></figure>

<caption>**Table 6**</caption>

Average spray length penetration ($x/D$ ± standard deviation) for the injection case with spray kerosene, $d_{10}$.

<table><thead><tr><td>Kinetics</td><td>Mesh</td><td>x/D ± std</td></tr></thead><tbody><tr><td>WB<sub>x</sub></td><td>1</td><td>37.02 ± 2.62</td></tr><tr><td>WB<sub>x</sub></td><td>2</td><td>36.04 ± 2.97</td></tr><tr><td>WB<sub>x</sub></td><td>3</td><td>36.63 ± 1.64</td></tr><tr><td>FK<sub>x</sub></td><td>1</td><td>45.46 ± 4.02</td></tr></tbody></table>

<figure><img src="image_23.png" /><figcaption>**Fig. 21.** Instantaneous picture of Lagrangian particles added to a 2D-cut of the Eulerian temperature field for two droplet sizes at inlet: 10 µm (left) and 20 µm (right). WB<sub>x</sub>, mesh 3. Gradation: $x/D = 10$.</figcaption></figure>

From a macroscopic point of view, both reacting flows exhibit similar structures except for the droplet penetration, which is longer with $d_d = 20$ µm. Indeed, such drops have a longer evaporation time, i.e., a longer residence time than 10-micron drops. Therefore, the 20-micron diameter drops have a greater thermal inertia. Evaporation is thus slower, resulting in a weaker endothermic process. In parallel, the 20-micron

<figure><img src="image_24.png" /><figcaption>**Fig. 22.** Eulerian fields of kerosene (NC<sub>10</sub>H<sub>22</sub>) and CO<sub>2</sub> mass fraction for two droplet sizes (10 µm (left) and 20 µm (right)). White line ≡ stoichiometric isoline. WB<sub>x</sub>, mesh 3. Gradation: $x/D = 10$.</figcaption></figure>

drops interact more intensely with the flow of vitiated air, promoting the fuel/oxidant mixture. The temperature conditions are, therefore, more favorable to the start of combustion closer to the injector in the case $d_{20}$ than in the case $d_{10}$. Drops thus exit the central flow with a 14° angle cone and then interact with the surrounding vitiated air. Drops near the injector lips ($x/D = 5$) are spread over a height close to twice the injector diameter for both cases. This expansion increases with $d_{20}$ but is stopped for $d_{10}$, as visible at $x/D = 20$. Finally, the average ignition position ($x/D$) is only slightly modified with the flame located at $x/D = 4.98 \pm 1.79$ for the case $d_{10}$ and $x/D = 4.31 \pm 1$ for $d_{20}$. The drop size continuously decreases because of the evaporation process up to a diameter close to 10 µm in the 20-micron case. The remaining drops found for $x/D > 25$ are much more diluted, now in a mixture of nitrogen, kerosene vapor, and burnt gases, and are strongly impacted by the surrounding flow. Finally, the central jet destabilizes. For the 10-micron case, $d_d$ similarly decreases to tiny sizes, but larger drops (7-8 µm), coming from the inner part of the injection cone, are found around $x/D = 20$. The impact of the kerosene droplet size is also shown on the Eulerian fields of kerosene and CO<sub>2</sub> mass fractions, in Fig. 22. In both cases, a flame develops following the stoichiometric isoline in the mixing layer, especially in the first part of the flow, where the interaction with the vitiated air is more intense. For $d_{20}$, such an interaction intensifies, leading to a thicker flame than with $d_{10}$. Combustion is thus significant in the
<figure><img src="image_25.png" /><figcaption>Fig. 23. Comparison of the no-model (right side) and PaSR (left side) approaches on averaged temperature (left), CO₂ (middle), and CO (right) mass fractions. FK_χ, mesh 1 (spray case). Gradation: x/D = 10.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>Fig. 24. Instantaneous field of κ (left) used with PaSR, and comparison of the averaged temperature at x/D = 60 for the no-model and PaSR modeling. Gradation: x/D = 10.</figcaption></figure>

first two diamond shock structures, particularly for $d_{20}$. The resulting va- por of fuel changes according to the case considered with a maximum closer to the injector with $d_{10}$ than with $d_{20}$. Indeed, the longer evapora- tion time of larger drops shifts such a peak of concentration downstream, yielding a longer central jet. Having a polydisperse distribution would slightly change such penetration length. A lower temperature of 60 K is then found with $d_{20}$ compared to $d_{10}$ for x/D ∈ [40; 60]. Finally, CO₂ is created on the stoichiometric surface before mixing with the surround- ing flow. H₂O has the same behavior but also mixes with some injected H₂O.

## 4.2. LES-PaSR impact

Using the laminar chemistry assumption in simulations can lead to overestimating the reaction rate, resulting in an overestimation of the

temperature. Accordingly, many models have been proposed in the lit- erature (as in [35]) for supersonic combustion, including the partially stirred reactor (PaSR) approach [57,58]. The LES-PaSR modeling used in this study is classically founded on the chemical time scale ($\tau_C$) and the subgrid mixing time scale ($\tau_{SGS}$), to estimate $\kappa = \tau_C/(\tau_C + \tau_{SGS})$ that is used to compute the filtered species source term $\tilde{\omega}_k(\rho, Y_k, T) = \kappa \dot{\omega}_k(\bar{\rho}, \bar{Y}_k, \bar{T})$:

$$ \tau_C = \min \left( \frac{\bar{\rho}_{H_2O}}{\dot{\omega}_{H_2O}}, \frac{\bar{\rho}_{CO_2}}{\dot{\omega}_{CO_2}} \right) \quad \text{and} \quad \tau_{SGS} = \frac{1}{\sqrt{2\bar{S}_{ij}\bar{S}_{ij}}} \quad (10) $$

FK_χ is used on mesh 1 (spray case).

Results are provided in Fig. 23 for the averaged temperature and species mass fractions estimated over three convective times. The temperature fields are very close, with a mean dense core ending around x/D = 35.
Most of the flame is above $x/D = 35$, even if CO production appears in the mixing layer between the kerosene spray and the vitiated air from $x/D = 20$. For $x/D < 40$, $κ$ is close to 1, meaning that the PaSR model barely activates (Fig. 24).

For $x/D > 40$, 90% of $κ \in [0.8; 1.0]$ with the lowest values ($κ \approx 0.6$) found in vortices. This explains why the addition of a subgrid model for the filtered source terms modifies only slightly the results compared to the no-model assumption.

## 5. Conclusion

A numerical test case has been proposed to study the injection and combustion of kerosene with thermodynamic conditions similar to those encountered in a scramjet. Inspired by Cheng's burner, kerosene is injected as a gas or a dilute spray surrounded by hot vitiated air to lead to spontaneous combustion. The resulting flame is found more or less lifted according to the chemistry used and the injection modes (spray or gaseous).

The flow structure features successive diamond shockwaves. In the case of one-step chemistry, the flame ignites in the first shock diamond, whatever the kerosene injection modality. Using a more detailed chemistry leads to an ignition in the second shock diamond. A strong coupling thus exists between compressible aerodynamics, spray evaporation, and auto-ignition delays provided by kinetic schemes. The reduced chemistry provided in this article is then well-suited to describe the kerosene oxidation in the case of supersonic combustion. The wave drag specific to compressible flows is a required ingredient to represent the spray injection correctly, along with the knowledge of the droplet size distribution. Indeed, the larger the diameter of the drop, the greater the modification of the velocity field will be, with an impact on the behavior of the flame.

Finally, using a simple version of the PaSR model, but easy to integrate

in numerical solvers and inexpensive in CPU time, did not bring major changes to the flow structure, certainly because of a mesh already well adapted to the proposed jet flame configuration. Such modeling strategy must therefore be validated on other applications such as scramjets. This point will be addressed in future work, as well as the integration of a specific method for the numerical treatment of kerosene stiffness chemistry.

## CRediT authorship contribution statement

**Florian Kissel:** Writing – original draft, Software, Investigation.
**Guillaume Ribert:** Writing – review & editing, Supervision, Funding acquisition.
**Pascale Domingo:** Writing – review & editing, Supervision.

## Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Florian Kissel reports financial support was provided by Agence de l'innovation de Défense. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

The present work has been supported by the Région Normandie and the Defence Innovation Agency. Computational resources were provided by CRIANN and GENCI under the grant number A0162B00152.

## Appendix A. Numerical considerations, WB<sub>χ</sub> (spray)

<figure><img src="image_27.png" /><figcaption>Fig. A.25. Grid size (Δ) field along with the stoichiometric mixture fraction iso-line for the kerosene spray case.</figcaption></figure>
<figure><img src="image_28.png" /><figcaption>Fig. A.26. Field of $v_i/v$, i.e. the ratio between the subgrid eddy viscosity and the laminar viscosity, along with the stoichiometric mixture fraction iso-line for the kerosene spray case.</figcaption></figure>

## Appendix B. Performance of the reduced mechanism FK_18S_29R

<figure><img src="image_29.png" /><figcaption>Fig. B.27. Ignition delay of kerosene in vitiated air.</figcaption></figure>
<figure><img src="image_30.png" /><figcaption>Fig. B.28. Counterflow flames with kerosene (left) against vitiated air (right). K ≡ strain rate.</figcaption></figure>

## Data availability

Data will be made available on request.

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