# Evaluation of flamelet-based models for liquid ammonia combustion in a
temporally evolving mixing layer

Zhenhua An, Jiangkuan Xing∗, Abhishek Lakshman Pillai, Ryoichi Kurose
Department of Mechanical Engineering and Science, Kyoto University, Kyoto,
daigaku-Katsura, Nishikyo-ku, Kyoto
615–8540, Japan

###### Abstract

Liquid ammonia combustion can be enhanced by co-firing with small molecular
fuels such as methane, and liquid ammonia will undergo flash evaporation due
to its relatively low saturation pressure. These characteristics, involving
the presence of multiple fuel streams, a rapid phase change process, and
strong heat loss, pose challenges for flamelet modeling of liquid ammonia
combustion. To address these issues, this study aims to evaluate the
effectiveness of flamelet-based models for liquid ammonia combustion in a
turbulent mixing layer. Specifically, the extended flamelet/progress variable
(E-FPV), extended flamelet-generated manifolds (E-FGM), and extended hybrid
(E-Hybrid) models are developed and assessed. Firstly, a three-dimensional
Point-Particle Direct Numerical Simulation (PP-DNS) with detailed chemistry is
performed, where the turbulent flow is fully resolved, and the ammonia
droplets are described by the Lagrangian method, to investigate the combustion
characteristics of a liquid ammonia/methane co-fired flame and to provide
state-of-the-art validation data for flamelet modeling. The PP-DNS results
reveal distinct stages in the liquid ammonia/methane co-fired flame, namely,
the mixing, methane-dominated, and fully reacting stages. The phase change
process introduces significant heat loss due to the high latent heat of liquid
ammonia. Subsequently, flamelet-based models are developed to account for the
complex fuel streams, rapid phase change process, and strong local heat loss.
The performance of these models is evaluated through a priori analysis by
comparing the predictions with the PP-DNS results. The a priori results show
that the E-FGM model outperforms the E-FPV and E-Hybrid models. This superior
performance can be attributed to the rapid flash evaporation and sufficient
mixing of the superheated ammonia, resulting in the dominance of the premixed
combustion mode in liquid ammonia combustion.

Keywords: Liquid ammonia combustion; Point-Particle DNS; FPV; FGM; Hybrid
model; Turbulent mixing layer

*Corresponding author.
E-mail address<EMAIL_ADDRESS>(JK. Xing).

##  1 Introduction

As a hydrogen carrier and a renewable carbon-free fuel, ammonia plays a
critical role in the global energy supply chain, facilitating the achievement
of decarbonization goals. Over the past few decades, significant progress has
been made in ammonia combustion research [1, 2]. In recent years, there has
been a notable increase in research focused specifically on the combustion of
liquid ammonia, emerging as a new focus. Compared to gaseous ammonia, the
liquid ammonia combustion offers several advantages. It simplifies the fuel
supply and pre-evaporations systems due to the liquid storage form of ammonia.
Notably, liquid ammonia has an exceptional octane number exceeding 110, making
it suitable for internal combustion engines [3]. In addition, the use of
liquid ammonia can effectively reduce energy consumption during the
vaporization process and minimize start-up time for gas turbines [4, 5].

To date, some research have been conducted to investigate the spray and
combustion characteristics of liquid ammonia. Okafor et al. [4, 5, 6]
conducted experiments and simulations to examine the direct combustion of
liquid ammonia spray in a swirl combustor, focusing on stability and emission
characteristics. Mounaïm-Rousselle et al. [7, 8, 9, 10] systematically
conducted the research on liquid ammonia spray characteristics and evaporation
process through experiments and simulations. Li et al. [11, 3, 12, 13, 14]
systematically investigated the liquid ammonia combustion under internal
combustion engine conditions, covering aspects ranging from spray injection
characteristics to the combustion of diesel-ignited ammonia dual fuel. In
addition, Colson et al. [15] studied the effect of injection temperature and
nozzle geometry on ammonia flash spray. Scharl et al. [16] investigated the
ammonia spray combustion and mixture formation under internal engine-like
conditions. The above summarizes the majority of research conducted on the
spray and combustion of liquid ammonia, relying primarily on experimental
investigations. However, numerical simulations of liquid ammonia remain
limited, especially in the area of combustion simulation. Regarding spray
simulation, An et al. [17] developed a combined phase change model for liquid
ammonia. Meanwhile, Zhang et al. [18], Shin et al. [19], Zembi et al. [10],
and Huang et al. [20] also evaluated the phase change models and investigated
the special characteristics of ammonia flash spray. In terms of combustion
simulation, An et al. [21] conducted a numerical study of spherical flame
propagation in dispersed liquid ammonia droplets using the quasi-Direct
Numerical Simulation (DNS) method. Somarathne et al. [6] numerically studied
the liquid ammonia/air spray combustion in a gas turbine-like combustor using
Large Eddy Simulation/partially stirred reactor (LES/PaSR) approach.

The flamelet model has attracted a lot of attention in combustion simulations
due to its ability to reduce computational cost while considering detailed
chemistry [22]. However, to the best of the authors’ knowledge, there is
currently no relevant research on the simulation of liquid ammonia flames
using flamelet models. The unique flash evaporation of liquid ammonia leads to
rapid phase change and significant local heat loss. In the case of liquid
ammonia flames, stabilization often requires the assistance of pilot flames
fueled by small-molecule fuels, resulting in the presence of multiple fuel
streams. The applicability of flamelet-based combustion models to liquid
ammonia combustion and how to account for the presence of multiple fuel
streams and strong heat loss in liquid ammonia flames remain open questions.

To this end, the present study aims to assess the performance of flamelet-
based models for liquid ammonia combustion modeling. First, a three-
dimensional Point-Particle Direct Numerical Simulation (PP-DNS) is performed
to investigate the combustion characteristics of a liquid ammonia/methane co-
fired flame in a temporally evolving mixing layer. This PP-DNS simulation also
serves as a reference for evaluating the subsequent flamelet models. Second,
three flamelet models, including the extended flamelet progress variable model
(E-FPV), the extended flamelet generated manifold model (E-FGM), and the
extended hybrid model (E-Hybrid) that combines the FPV and FGM through flame
index, are developed and examined through the a priori analysis with the PP-
DNS solutions as benchmarks.

##  2 Mathematical approaches

###  2.1 PP-DNS coupled with detailed chemistry

In the PP-DNS, the gas phase governing equations, including the continuous,
momentum, enthalpy, and species equations, are solved without any averaging
and filtering. The detailed chemistry (59 species and 356 elementary reactions
[23]) and detailed molecular mixing are considered. For the dispersed phase,
the conservation equations, including the position, velocity, and temperature
are solved under the Lagrangian framework. The particular phase change process
of ammonia droplets is described using a combined model developed and
validated in our previous study [17]. The two-way coupling method is used to
describe the interactions between the gas phase and the droplets. The details
of all conservation equations are not presented here for brevity and the
interested reader can refer to our previous studies [21, 24].

###  2.2 Extended flamelet-based models

Flamelet-based models assume that the thermo-chemical state of the flame can
be tabulated and expressed as a function of several control variables. In
general, diffusion and premixed flames can be described by FPV [25] and FGM
[26] models, respectively. In the present work, three models are developed,
named the E-FPV, E-FGM, and E-Hybrid models. Compared with the traditional FPV
and FGM models, we introduced two additional parameters, $X$ (methane ratio in
the whole fuel stream) and $h$ (total enthalpy), to consider the complex fuel
streams and heat loss. Therefore, 4 control variables, i.e., progress variable
($c$), mixture fraction ($Z$), methane ratio ($X$), and total enthalpy ($h$),
are used to describe the liquid ammonia/methane co-fired flame. Specifically,
the thermochemical properties $\varphi$ can be tabulated as:

$\varphi=\varphi\left(c,Z,X,h_{norm}\right),$ (1)

where $c$ is defined as the sum of the mass fractions of H2O, CO2, and H2
obtained directly from the DNS results in the a priori study. $Z$ can be
calculated as $Z=Z_{\mathrm{CH_{4}}}+Z_{\mathrm{NH_{3}}}$.
$Z_{\mathrm{CH_{4}}}$, $Z_{\mathrm{NH_{3}}}$, and $Z$ are the methane,
ammonia, and total mixture fractions, respectively. The transportation
equations of $Z_{\mathrm{CH_{4}}}$ and $Z_{\mathrm{NH_{3}}}$ are also solved
in the DNS.
$X=Z_{\mathrm{CH_{4}}}/\left(Z_{\mathrm{CH_{4}}}+Z_{\mathrm{NH_{3}}}\right)$.
$h$ is normalized during the table lookup process.

In the flamelet-based models, the flow field is described by the conservation
equations of mass and momentum. Additional transport equations of progress
variable, mixture fractions, and total enthalpy are solved as:

$\frac{\partial\left(\rho{c}\right)}{\partial
t}+\nabla\cdot\left(\rho{\bm{u}}{c}\right)=\nabla\cdot\left(\rho
D_{c}\nabla{c}\right)+\dot{\omega}_{c}+\dot{S}_{c},$ (2)
$\frac{\partial\left(\rho{Z_{i}}\right)}{\partial
t}+\nabla\cdot\left(\rho{\bm{u}}{Z_{i}}\right)=\nabla\cdot\left(\rho
D_{i}\nabla{Z_{i}}\right)+\dot{S}_{i},$ (3) $\frac{\partial\left(\rho
h\right)}{\partial
t}+\nabla\cdot\left(\rho{\bm{u}}h\right)=\nabla\cdot\left(\rho D_{h}\nabla
h\right)+{\dot{S}_{h}},$ (4)

where $\rho$ and $\bm{u}$ are the gaseous phase density and velocity,
respectively. $D_{k}$ represents the diffusion coefficient of $k$.
$\dot{\omega}_{c}$ is the reaction rate derived from the flamelet table.
$Z_{i}$ represents the $Z_{\mathrm{CH_{4}}}$ or $Z_{\mathrm{NH_{3}}}$, and
$\dot{S}_{i}$ equals to 0 when $i=_{\mathrm{CH_{4}}}$.

To construct the flamelet tables, one-dimensional counter-flow diffusion
flames and freely-propagating flames are calculated under different $X$ values
and initial temperatures. All the flamelet tables are obtained using the
FlameMaster package [27] with the same reaction mechanism as in the DNS (59
species and 356 elementary reactions [23]). $X$ ranges from 0 to 1 with an
interval of 0.1. When $X$ value equals to 0 or 1, the local fuel would be pure
ammonia or methane, respectively. For E-FPV tables, one-dimensional counter-
flow diffusion flames are calculated at the different stoichiometric scalar
dissipation rate from 0.001 s-1 to a maximum value where the flame is
completely quenched. The fuel temperature $T_{f}$ is set to be equal to the
oxidizer temperature $T_{o}$ to represent the interphase heat transfer and
ranges from 250 to 2100 K. Note that the 250 K is used here to account for the
large heat loss caused by flash evaporation. For E-FGM tables, one-dimensional
freely-propagating flames are calculated at different equivalence ratios,
which range from 0.4 to 2.0 with an interval of 0.1. The flamelets under the
different equivalence ratios are then converted to $Z$ space. Different
initial temperatures from 250 to 700 K are considered to represent the heat
loss effect. In addition, pure oxidizer and fuel solutions are also created to
ensure the completeness of the E-FGM tables in the mixture fraction space. In
the present work, the E-FPV and E-FGM flamelet tables are generated with a
resolution of $150\times 51\times 11\times 51$ for $c\times Z\times X\times
h_{norm}$, respectively. For the E-Hybrid model, the flame index which is
defined as [28] to access the local combustion mode,
$FI=1/2\left(1+\left(\nabla Y_{F}\cdot\nabla
Y_{O_{2}}\right)\left(\left|\nabla Y_{F}\cdot\nabla
Y_{O_{2}}\right|+\epsilon\right)\right)$, where $Y_{F}$ and $Y_{O_{2}}$ are
the mass fractions of fuel and oxygen, respectively. $\epsilon$ is an
infinitesimal positive value to avoid dividing by zero. The final prediction
of the E-Hybrid model is obtained by
$\varphi_{E-Hybrid}=FI\varphi_{E-FGM}+\left(1-FI\right)\varphi_{E-FPV}$.

Figure 1 shows the flamelet solutions of the one-dimensional counter-flow
diffusion flames (shown as (a) and (b)) and freely-propagating flames (shown
as (c) and (d)) in the mixture fraction space and the progress variable space,
respectively, at different temperatures and $X$ values. While for the freely-
propagating flames, each flamelet has only one mixture fraction value, thus
they are shown in condition of equivalence ratio $\phi$ = 1 here. It is found
that as the fuel/oxidizer temperature increases, the general temperature
increases, indicating that heat loss is important in the flame behavior. As
the ammonia fraction in the fuel stream increases, the flame temperature
gradually decreases due to the relatively lower heat value of ammonia. For the
FGM flamelet data, as the methane fraction in the fuel stream increases, the
maximum progress variable and temperature increase.

Fig. 1: 1-D flamelet profiles of temperature against the $Z$ and $c$. (a) FPV
flamelet data at $X_{\mathrm{CH4}}$ = 0.0, $T_{0}$ = 250 - 2100 K. (b) FPV
flamelet data at $T_{0}$ = 300 K, $X_{\mathrm{CH4}}$ = 0.0 - 1.0. (c) FGM
flamelet data at $X_{\mathrm{CH4}}$ = 0.0 and $\phi$ = 1, $T_{0}$ = 250 - 700
K. (d) FGM flamelet data at $T_{0}$ = 300 K and $\phi$ = 1, $X_{\mathrm{CH4}}$
= 0.0 - 1.0.

##  3 Numerical setup

The computational domain of the turbulent mixing layer spans 24 mm in $x$ and
$y$ directions, and 12 mm in $z$ direction as shown in Fig. 2. A uniform grid
resolution of 50 $\mathrm{\mu m}$ is used, resulting in grid points of
480$\times$480$\times$240\. The mesh resolution is sufficient to resolve the
flame thickness and the Kolmogorov length scale. Liquid ammonia
($L_{\mathrm{NH_{3}}}$) droplets with a diameter of 25 $\mathrm{\mu m}$ (Note
that this is the value after the atomization process) and a premixed mixture
of methane/air are injected into the upper half of the computational domain at
a velocity of 20 m/s and a carrier gas preheating temperature of 500 K. The
heat fraction of ammonia in the total fuel stream is 0.7, the global
equivalence ratio is 1.0. All the droplets are randomly distributed in the
upper domain. The Stokes number of the droplets is less than 0.6. Then they
have good tracking property and the initial droplet velocity is set equal to
that of the carrier gas. The initial droplet temperature is 279 K, and the
initial ambient pressure is 0.25 MPa. The ammonia droplets undergo the flash
evaporation at this condition. The phase change process can be well captured
using the combined model [17]. This setup is referred to the experimental
configurations [4, 5]. For the lower part of the domain, the properties of co-
flow are determined by calculated results of $\mathrm{NH_{3}/CH_{4}}$/air
flames under the same condition. The velocity of the hot co-flow mixture is 1
m/s. The periodic boundary conditions are applied to left/right and front/back
sides. The zero-gradient boundary condition is used for the top and bottom
sides. To support development of turbulence in the mixing layer, the turbulent
flow is generated by the Passot-Pouquet isotropic kinetic spectrum [29] with a
fluctuation of 4% of the jet velocity and an integral length scale of
$L_{y}/6$. The turbulent flow is then superimposed on the main stream.

Fig. 2: Computational domain and boundary conditions for temporally evolving
liquid ammonia flame. The ammonia droplets are colored by diameter. The vortex
structure is obtained from the Q criterion (Q = $1\times 10^{7}$ $1/s^{2}$)
colored by the local gas temperature. The flow goes from left to right.

The PP-DNS and the a priori analysis are carried out on the supercomputer
Fugaku at the RIKEN Center for Computational Science with in-house solvers,
and the computational cost of the PP-DNS is approximately 800,000 core hours
by parallel computation with 7200 cores. The Euler framework is employed to
solve the governing equations for the gas phase, utilizing the SIMPLE
algorithm for calculations. Spatial and transient terms are integrated using a
second-order central differencing scheme, while temporal terms are integrated
using the Euler-implicit scheme. To maintain numerical stability, the Courant
number is constrained to be below 0.1, leading to a time step of approximately
0.4 $\mathrm{\mu s}$.

##  4 Results and discussion

###  4.1 Analysis of liquid ammonia/methane co-fired flame

Figure 3 shows the temporal evolution results of PP-DNS for the liquid
ammonia/methane co-fired flame from initial condition to burnt state. The
regions represent temperature fields and distributions of ammonia droplets.
Initially, the fresh mixture and the burned co-flow are located in the upper
and lower parts of the computational domain, respectively. Before the reaction
can proceed, the ammonia droplets begin to evaporate. As mixing layer
turbulence develops, the evaporated liquid ammonia and carrier gas mix with
the hot co-flow. Due to the flash evaporation of the ammonia droplets, the
temperature drops briefly before ignition. After 5 ms, the temperature
increases significantly and the diameter of the ammonia droplets continues to
decrease. As mixing and turbulence further develop, the flame is ignited and
propagates upward and downward. Due to flash evaporation, the liquid ammonia
evaporates completely before reaching the high temperature region of the
flame, making it difficult to see the interaction between the droplets and the
flame.

Fig. 3: Temporal evolution of gas phase temperature fields and distributions
of ammonia droplets for liquid ammonia flame from 1 - 9 ms.

The fuel stream contains ammonia droplets and methane. The phase change
process of liquid ammonia will absorb the heat from the surrounding gas,
resulting in strong local heat loss. The ignition delay time of methane is
much less than that of ammonia. Methane is consumed first, and then ammonia is
piloted by ignition of methane and consumed. Therefore, the combustion process
of liquid ammonia/methane co-fired flame can be divided into three stages, as
shown in Fig. 4. In the first stage (S1), the ammonia droplets undergo rapid
flash evaporation and the volume-averaged temperature decreases continuously,
indicating an inert evaporation and mixing stage. In this stage, the fraction
of ammonia in the total fuel stream increases significantly. In the second
stage (S2), liquid ammonia continues to evaporate. The heat release from the
reaction is greater than the heat loss caused by the phase change. The
temperature then begins to rise. The main contribution of HRR comes from the
reaction of methane. This stage is called the methane-dominated stage. In the
third stage (S3), the evaporation rate of liquid ammonia begins to be lower
than the combustion consumption rate of ammonia, and its mass fraction
decreases. In this stage, the ammonia continues to be consumed, and the
temperature continues to rise. This stage is called the fully reacting stage.
The a priori analysis is performed in the Section 4.2 for these three stages.

Fig. 4: Temporal evolution of the volume-averaged mass fractions of NH3 (black
solid line) and CH4 (black dash line), mean temperature of the carrier gas
(red line), evaporation rate of ammonia droplets (blue line), and fraction of
$Z_{\mathrm{NH_{3}}}$ in $Z_{\mathrm{NH_{3}}}+Z_{\mathrm{CH_{4}}}$ (green
line). S1, S2, and S3 represent mixing, methane-dominated, and fully reacting
stages, respectively.

Because of the large latent heat of ammonia droplets, the rapid phase change
process causes large heat loss. Figure 5 shows the relationship between total
enthalpy and mixture fraction at different times. The scatters are colored by
normalized $c$. The dashed lines represent the flamelet results with
$E_{\mathrm{NH_{3}}}$ = 0.7 and an initial temperature of 500 K. Before 5 ms,
the scatters are clearly divided into two clusters, with the cluster with a
larger value of $c$ near the dashed line and the cluster with a smaller value
of $c$ far below the dashed line, representing the hot co-flow and the fresh
mixture stream, respectively. With the development of mixing, the hot co-flow
is mixed with unburned gas, the cluster with a smaller value of $c$ decreases,
and $c$ increases. It can be seen that at all times, the vast majority of the
scatters are below the dashed line, indicating a significant heat loss.
Therefore, when simulating a liquid ammonia flame using flamelet-based models,
the consideration of enthalpy becomes essential to accurately capture and
describe the heat loss phenomenon.

Fig. 5: Scatter plots of total enthalpy and mixture fraction from the whole
computational domain at different times. The scatters are colored by
normalized $c$.

###  4.2 A priori analysis of the extended flamelet-based models

According to the analysis of PP-DNS, there are three stages for liquid
ammonia/methane co-fired flame. The a priori analysis is carried out to
evaluate the performance of the extended FPV, FGM, and hybrid models for these
three stages compared with the results of PP-DNS as a benchmark. The four
control variables ($c,Z,X,h_{norm}$) are used as the input parameters for the
flamelet look-up tables. $c$ is computed from the transported mass fraction.
The two mass fractions $Z_{\mathrm{CH_{4}}}$ and $Z_{\mathrm{NH_{3}}}$ are
transported. Then, $Z$ and $X$ are calculated by
$Z=Z_{\mathrm{CH_{4}}}+Z_{\mathrm{NH_{3}}}$ and
$X=Z_{\mathrm{CH_{4}}}/\left(Z_{\mathrm{CH_{4}}}+Z_{\mathrm{NH_{3}}}\right)$,
respectively. $h_{norm}$ is calculated form the transport equation of enthalpy
and normalized to 0-1. This analysis provides a direct assessment of the
ability of the flamelet-based model to accurately reproduce the reference
solution.

Figure 6 shows the temporal evolution of temperature fields at selected times
for DNS and flamelet-based models, in a slice at $z=L_{z}/2$. The DNS results
can be regarded as the state-of-the-art validation, as shown in Fig. 6(a). At
$t$ = 3 ms, the temperature has not yet risen. At $t$ = 5 ms, the fresh
mixture is ignited, and the temperature of the ignition kernel increases. The
temperature of fuel stream decreases slightly due to flash evaporation of the
ammonia droplets. The temperature of the hot co-flow does not change much.
Figure 6(b) shows the predictions of the E-FPV model. E-FPV overestimates the
temperature of the co-flow and does not predict the temperature non-uniformity
of the mixing layer. In particular, the temperature in the reaction zone is
not well predicted. Figure 6(c) shows the prediction of the E-FGM model. It
can be found that E-FGM prediction has a good agreement with DNS solution. The
flame structure, the temperature stratification, and high temperature zone
near the reaction zone are all well predicted. This is mainly due to the rapid
flash evaporation effect of liquid ammonia, which causes the droplets to
evaporate completely before coming into interact with the flame and has enough
time to mix with air, resulting in the reaction mainly taking place under the
premixed mode. Therefore, the E-FGM model can predict the results well and its
performance far exceeds that of E-FPV. As for the E-Hybrid model, since the
predictions of the E-FGM model are already accurate enough under the current
conditions, the overall prediction accuracy will be lowered after adding the
E-FPV model. There is a certain gap between the prediction of the E-Hybrid
model and the benchmark, but the difference is smaller than what the E-FPV
model predicts, as shown in Fig. 6(d).

Fig. 6: Gas phase temperature and ammonia droplets colored by their diameter
in a slice at $z=L_{z}/2$ at the selected times for the results of (a) DNS,
(b) E-FPV, (c) E-FGM, and (d) E-Hybrid models.

Figure 7 shows the comparisons between the DNS results (black dots) and the a
priori profiles of the E-FPV (blue line), E-FGM (red line), and E-Hybrid
(green line) models at $t$ = 1 ms, which represents the inert mixing stage,
along different spatial locations. The gray and light blue regions represent
the non-premixed and premixed combustion mode, respectively. In this stage,
the turbulence in the mixing layer has not yet developed, as seen in Fig. 3.
Numerous ammonia droplets evaporate and absorb heat from the carrier gas. The
ignition kernel is located around $y$ = 12 mm, and will evolve up and down
over time. For the temperature field, the predictions of the three models are
relatively accurate. It can be found that E-FPV slightly overestimates the
value. E-FGM slightly underestimates it on the co-flow side. For
$Y_{\mathrm{O_{2}}}$ and $Y_{\mathrm{H_{2}O}}$, it can be seen that these
three flamelet-based models all have good agreements with DNS. However, the
E-FPV model significantly overestimates the profile of CO and NO, especially
in the co-flow. This deviation could be attributed to the fact that co-flow is
a combustible mixture formed after the premixed gas burns out, and the FPV
model based on the diffusion flame surface is not accurate in predicting minor
components such as CO and NO. In contrast, the E-FGM model predicts the value
of NO very well, which is in good agreement with the results of DNS for all
the three spatial locations. The results of the E-Hybrid model are between
those of the E-FPV and E-FGM models, and also overestimate CO and NO. In
addition, it can be seen that the proportion of the premixed region (light
blue) is very small in stage S1.

Fig. 7: A priori analysis along different spatial locations (Left: $x/L_{x}$ =
1/4, middle: $x/L_{x}$ = 1/2, right: $x/L_{x}$ = 3/4) between the results of
DNS (black dots) and E-FPV (blue line), E-FGM (red line), E-Hybrid models
(green line) at $t$ = 1 ms (S1). The gray and light blue regions represent the
non-premixed and premixed combustion modes, respectively.

Figure 8 shows the comparisons between the DNS results and the a priori
profiles of the E-FPV, E-FGM, and E-Hybrid models at $t$ = 5 ms, which
represents the methane-dominated stage. In this stage, the ignition kernel
propagates towards lower and upper directions, and the temperature increases
significantly. Due to the rapid evaporation and mixing process, most of the
region features premixed combustion mode. For the temperature field, the E-FPV
model underestimates the result. The superiority of the E-FGM and E-Hybrid
models retains. For the prediction of the major components, such as O2 and
H2O, all three models have a good agreement with DNS. Note that E-FGM predicts
intermediate components well, although CO has some overestimation, it is
closer to the DNS results compared to other models.

Fig. 8: A priori analysis along different spatial locations (Left: $x/L_{x}$ =
1/4, middle: $x/L_{x}$ = 1/2, right: $x/L_{x}$ = 3/4) between the results of
DNS (black dots) and E-FPV (blue line), E-FGM (red line), E-Hybrid models
(green line) at $t$ = 5 ms (S2). The gray and light blue regions represent the
non-premixed and premixed combustion modes, respectively.

Figure 9shows the comparisons between the DNS results and the a priori
profiles of the E-FPV, E-FGM, and E-Hybrid models at $t$ = 9 ms, which
represents the fully reacting stage. In this stage, the turbulence is fully
developed, and most of the droplets have completely evaporated. The fresh
mixture continues to be consumed, and the temperature continues to rise. The
mixing proceeds thoroughly, the premixed mode dominates the whole region.
Therefore, the results of the E-FGM and E-hybrid models are nearly the same
(see red and green lines). The superiority of the E-FGM model retains,
especially for the prediction of NO mass fraction. For the gas temperature,
E-FGM and E-Hybrid models can also give better predictions.

Fig. 9: A priori analysis along different spatial locations (Left: $x/L_{x}$ =
1/4, middle: $x/L_{x}$ = 1/2, right: $x/L_{x}$ = 3/4) between the results of
DNS (black dots) and E-FPV (blue line), E-FGM (red line), E-Hybrid models
(green line) at $t$ = 9 ms (S3). The gray and light blue regions represent the
non-premixed and premixed combustion modes, respectively.

Overall, the extended flamelet-based models can give a reasonable prediction
for liquid ammonia/methane co-fired flame considering the complex fuel streams
and large heat loss. The E-FPV and E-Hybrid models give relatively large
differences in temperature and minor products, which is due to the combustion
mode being mainly premixed mode. The agreement between the DNS and a priori
E-FGM results is quite good for all the three stages. The above results
indicate the superiority of the E-FGM model in predicting the liquid
ammonia/methane co-fired flame.

##  5 Conclusions

In the present study, the extended flamelet-based models for the liquid
ammonia combustion were developed and evaluated in a temporally evolving
mixing layer. Specifically, a PP-DNS was firstly performed, and three reacting
stages were identified: mixing stage, methane-dominated stage, and fully
reacting stage. It was observed that the effect of heat loss is significant in
liquid ammonia combustion. Then, E-FPV, E-FGM, and E-Hybrid models were
developed and evaluated through the a priori study. By comparing the
predictions of the E-FPV, E-FGM, and E-Hybrid models with the DNS solutions as
benchmarks, it can be seen that for the simulation of liquid ammonia
combustion, the E-FGM model has a better performance than the E-FPV and
E-Hybrid models. This can be attributed to the rapid flash evaporation and
sufficient mixing of the liquid ammonia. In the future study, the E-FGM model
can be used for the numerical simulation of liquid ammonia combustion and
liquid ammonia/methane co-fired flame.

Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

Acknowledgments

This work was partially supported by MEXT as “Program for Promoting Researches
on the Supercomputer Fugaku” (Development of the Smart design system on the
supercomputer “Fugaku” in the era of Society 5.0) (JPMXP1020210316). This
research used the computational resources of supercomputer Fugaku provided by
the RIKEN Center for Computational Science (Project ID: hp230321, hp220141).
Z.H.A is grateful to the China Scholarship Council (Project No. 202206280012).
J.K.X and R.K especially thank the support from Japan Society for the
Promotion of Science (Grant No: 21P20351).

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