Low-Order Modelling of Extinction of Hydrogen Non-Premixed Swirl Flames

Abstract

Predicting the blow-off (BO) is critical for characterising the operability limits of gas turbine engines. In this study, the applicability of a low-order extinction prediction modelling, which is based on a stochastic variant of the Imperfectly Stirred Reactor (ISR) approach, to predict the lean blow-off (LBO) curve and the extinction conditions in a hydrogen Rich-Quench-Lean (RQL)-like swirl combustor is investigated. The model predicts the blow-off scalar dissipation rate (SDR), which is then extrapolated using Reynolds-Averaged Navier–Stokes (RANS) cold-flow simulations and simple scaling laws, to determine the critical blow-off conditions. It has been found that the sISR modelling framework can predict the BO flow split ratio at different global equivalence ratios, showing a reasonable agreement with the experimental data. This further validates sISR as an efficient low-order modelling flame extinction tool, which can significantly contribute to the development of robust hydrogen RQL combustors by enabling the rapid exploration of combustor operability during the preliminary design phases.

1. Introduction

Hydrogen propulsion can be considered an efficient solution for the decarbonization of the aviation industry. Designing aircraft engines powered by hydrogen presents unique challenges due to hydrogen’s distinct combustion properties. One major issue is hydrogen’s high adiabatic flame temperature, which can lead to high nitrogen oxide (NO_x) emissions [1]. Since strict emissions regulations govern aircraft engine certification, controlling NO_x output becomes critical for the development of hydrogen-powered aircraft. To address that, advanced combustor configurations like the Rich-Quench-Lean (RQL) design have shown promise [2]. The RQL combustor minimizes NO_x emissions by, first, burning fuel in a rich primary zone, then rapidly cooling the hot gases with dilution air jets [2]. This approach helps control flame temperatures and reduces harmful emissions, making it a promising solution for hydrogen combustion in aviation. One of the key parameters that governs the combustion in an RQL combustor is the dilution ratio. The dilution ratio (R) is defined as the fraction of the dilution air flow rate ($\dot{m_d}$) to the total air flow rate, which includes both primary and dilution air ($\dot{m_p}+\dot{m_d}$) [3,4]. The precise control of dilution is critical for managing flame stability, temperature distribution, and emissions, particularly NO_x. Despite hydrogen’s high reactivity and wide flammability range, excessive dilution poses a critical stability risk. Injecting too much dilution air cools the reaction zone and increases turbulent mixing/scalar dissipation rates, which can locally quench the flame [3]. If the dilution ratio exceeds a critical threshold, localized extinctions can result in global flame blow-off (BO), even for highly reactive hydrogen flames. Therefore, accurately predicting the critical blow-off dilution ratio is essential for designing reliable hydrogen RQL combustors. Moreover, lean blow-off (LBO) remains a fundamental operability limit, irrespective of fuel type, and accurately predicting the LBO curve remains important during the preliminary combustor design phases.

Large Eddy Simulation (LES) has shown some success in the prediction of localized extinction in laboratory-scale flames, such as the Sandia jet, Sydney swirl flame, or Cambridge swirling flames. Zhang and Mastorakos [5] used LES with Conditional Moment Closure (LES-CMC) to predict the entire lean blow-off (LBO) curve of a swirling flame, achieving around 25% accuracy in blow-off limits and durations. However, such comprehensive studies are rare, mainly due to the expensive computational cost of LES. To make BO prediction more feasible during the design phase, there is a growing need for lower-cost modelling methods that can assess a wider range of geometries and conditions without relying entirely on high-fidelity CFD. Gkantonas and Mastorakos [6] developed a low-order modelling tool that can predict LBO behaviour. The developed model is based on a stochastic representation of the Imperfectly Stirred Reactor (ISR) model [7,8,9], termed the sISR method. The model provides a map that shows the extinction probability (i.e., the proportion of a flame’s stoichiometric iso-surface undergoing local extinctions) at different values of scalar dissipation rates (SDRs). It relies on a key assumption, supported by experimental and numerical studies [5,10,11,12,13,14], which suggests that global flame blow-off occurs when 30% of the stochiometric iso-surface experiences local extinction. Thus, from the extinction threshold, the model predicts the blow-off SDR, which is then extrapolated using reference CFD data and simple scaling laws, to determine critical blow-off conditions such as velocity or dilution ratio. Gkantonas and Mastorakos validated the sISR against atmospheric Cambridge swirl spray flames (ethanol/kerosene fuels), demonstrating its capability to predict blow-off velocities within 13% of the experimental values [6]. Although sISR has been previously validated for hydrocarbon fuels, the sISR model itself is fuel-agnostic and provides a general framework extinction prediction at a wide range of fuels and combustor geometries [6]. Several core insights—such as the stochasticity of extinction events and a critical extent of localized extinction before full blow-off—are conceptually valid regardless of the used fuel. However, hydrogen’s strong differential diffusion effects are known to influence flame extinction and can pose modeling challenges to the sISR approach. The inclusion of differential diffusion to the sISR approach is possible following previous studies using the Imperfectly Stirred Reactor Network (ISRN) framework [15], which show that differential diffusion effects can be approximately recovered through additional terms in the conditional transport equations and species-specific Lewis numbers (e.g., H2 and H). This study extends the sISR validation by investigating its applicability in predicting the LBO curve and blow-off dilution ratio in a swirl hydrogen Rich-Quench-Lean (RQL)-like combustor. The paper is structured as follows: Firstly, the investigated model and its governing equation are discussed. Then, the experimental step, numerical implementation, and methodology are introduced. The results are then discussed, and, finally, the conclusions are summarised.

2. Methods

2.1. Governing Equation

The Imperfectly Stirred Reactor (ISR) model is based on the spatially integrated form of the Conditional Moment Closure (CMC) equations [9,15,16]. The model is called ‘stochastic’ as it is based on a stochastic representation of the SDR. Unlike a perfectly stirred reactor, the ISR allows for mixture fraction inhomogeneities, which is essential when modelling local extinction and finite-rate chemical effects. The governing equations for the conditional enthalpy and mass fractions are as follows [15]:

$${\displaystyle \frac{\partial Q_h}{\partial t}}=N_{\eta }^{\ast \ast }{\displaystyle \frac{{\partial }^2{Q}_h}{\partial {\eta }^2}}$$

(1)

$${\displaystyle \frac{\partial Q_{\alpha }}{\partial t}}=N_{\eta }^{\ast \ast }{\displaystyle \frac{{\partial }^2{Q}_{\alpha }}{\partial {\eta }^2}}+\dot{{\omega }_{\alpha }}|\eta$$

(2)

where $\eta$ is the space sample in mixture fraction space, $Q_h$ indicates a conditional specific enthalpy (i.e., $Q_h=h|\eta$), and $Q_{\alpha }$ indicates a conditional mass fraction (i.e., $Q_{\alpha }=Y_{\alpha }|\eta$). $\dot{{\omega }_{\alpha }}|\eta$ stands for the conditional chemical source term for a given species $\alpha$ and $N_{\eta }^{\ast \ast }$ is the conditional profile of the SDR. In the present analysis, the effect of differential diffusion is neglected as the investigated combustor involves high Reynolds numbers. Therefore, species diffusion effects are less significant compared to turbulent mixing and the LBO curve and extinction dilution ratio predictions are not expected to significantly change when including the effect of preferential diffusion. To model this term, the Amplitude Mapping Closure (AMC) model [16,17] is used, which expresses it as follows:

$$N_{\eta }^{\ast \ast }=N_0^{\ast \ast }\left(t\right)G(\eta )$$

(3)

with $G(\eta )$ being a shaping factor defined as follows:

$$G\left(\eta \right)=\mathrm{e}\mathrm{x}\mathrm{p}(-2[{{\mathrm{erf}}^{-1}(2\eta -1)]}^2)$$

(4)

Here, $N_0^{\ast \ast }\left(t\right)$ is a representative SDR that varies in time according to a stochastic differential equation (SDE), capturing temporal fluctuations. Following similar approaches to Pitsch and Fedorov [18] and Gkantonas and Mastorakos [19], the stochastic behaviour of $N_0^{\ast \ast }$ is governed by the following [6]:

$$d{N}_0^{\ast \ast }=f\left(N_0^{\ast \ast }\right)dt+{\sigma }_N\varphi \left(N_0^{\ast \ast }\right)d{W}_t$$

(5)

where $d{W}_t$ is a Wiener process, and the functions $f$ and $\varphi$ define the deterministic and stochastic components, respectively. They are given by the following [6]:

$$f\left(N_0^{\ast \ast }\right)=-(\mathit{log}\left(N_0^{\ast \ast }\right)-\mathit{log}\left({\displaystyle \frac{〈N_0^{\ast \ast }〉}{\mathit{exp}\left(0.5{\sigma }_N^2\right)}}\right){\displaystyle \frac{N_0^{\ast \ast }}{{\tau }_N}}$$

(6)

$$\varphi \left(N_0^{\ast \ast }\right)=〈N_0^{\ast \ast }〉\sqrt{{\displaystyle \frac{2}{{\tau }_N}}}$$

(7)

In this formulation, ${\sigma }_N$ defines the lognormal distribution width and is computed from the fluctuation parameter $F={\displaystyle \frac{\langle {(N_0^{\ast \ast \prime })}^2\rangle }{〈N_0^{\ast \ast }〉}}$ as follows [6]:

$${\sigma }_N=\sqrt{log(F^2+1)}$$

(8)

The fluctuation parameter $F$ can be approximated using the turbulent Reynolds number $R{e}_l$ as follows [6]:

$$F={\left(15R{e}_l\right)}^{{\displaystyle \frac{1}{8}}}$$

(9)

Finally, the characteristic timescale ${\tau }_N$ introduces a memory effect, capturing the autocorrelation time of SDR fluctuations. Since such fluctuations are concentrated in the smallest turbulent scales, prior work [19,20] suggests that ${\tau }_N\approx 1.6{\tau }_k$, with ${\tau }_k$ being the Kolmogorov timescale. More details regarding the model are found in Ref. [6].

2.2. Extraction of sISR Parameters from CFD

The sISR prediction approach relies on three global parameters extracted from the CFD simulation: (1) mean scalar dissipation rate $〈N_0^{\ast \ast }〉$, (2) fluctuation scaling parameter F, and (3) characteristic timescale ${\tau }_N$. These parameters are computed using volume-weighted averages of standard CFD quantities (density, turbulence properties, etc.) within a control region surrounding the stoichiometric mixture fraction iso-surface. The mean scalar dissipation rate (SDR) is estimated as follows [2]:

$$〈N_0^{\ast \ast }〉={\displaystyle \frac{N^{\ast \ast }}{{\int }_0^1{P}_{\eta }^{\ast \ast }{G}_{\eta }d\eta }}$$

(10)

where $G_{\eta }$ is a prescribed profile in mixture fraction space, derived from the AMC model, and $N^{\ast \ast }$ and $P_{\eta }^{\ast \ast }$ are found based on a mass-based integration of all CFD cells within a volume surrounding the iso-surface of the stoichiometric mean mixture fraction. This is carried out as follows [2]:

$$P_{\eta }^{\ast \ast }={\displaystyle \frac{\int \overline{\rho }{P}_{\eta }d{V}^{\prime }}{\int \overline{\rho }d{V}^{\prime }}}$$

(11)

$$N^{\ast \ast }={\displaystyle \frac{\int \overline{\rho }Nd{V}^{\prime }}{\int \overline{\rho }d{V}^{\prime }}}$$

(12)

with N defined at the CFD level as follows [2]:

$$N={\displaystyle \frac{C_d}{2}}{\displaystyle \frac{\epsilon }{k}}{\xi }^{\prime 2};with{C}_d=2.0$$

(13)

where $\overline{\rho }$ is the mean density, $\epsilon$ is the turbulence dissipation rate, $k$ is the turbulent kinetic energy, and ${\xi }^{\prime 2}$ is the mixture fraction variance. The integration region is typically a band surrounding the stoichiometric iso-surface and, in this analysis, is taken as ${\xi }_{st}+/-$ 70%. The fluctuation parameter $F$ and the characteristic time scale ${\tau }_N$ are obtained based on mass-based integration over the same region as in Equations (11) and (12) and are approximated using the local Reynolds number and the turbulent timescale, respectively [2].

$$F={\displaystyle \frac{\int \overline{\rho }{\left(R{e}_l\right)}^{\frac{1}{8}}d{V}^{\prime }}{\int \overline{\rho }d{V}^{\prime }}}={\displaystyle \frac{\int \overline{\rho }{\left(15\frac{k^2}{\nu \epsilon }\right)}^{\frac{1}{8}}d{V}^{\prime }}{\int \overline{\rho }d{V}^{\prime }}}$$

(14)

$${\tau }_N={\displaystyle \frac{\int \overline{\rho }(1.6{\tau }_k)d{V}^{\prime }}{\int \overline{\rho }d{V}^{\prime }}}={\displaystyle \frac{\int \overline{\rho }{1.6\left(\frac{\nu }{\epsilon }\right)}^{\frac{1}{2}}d{V}^{\prime }}{\int \overline{\rho }d{V}^{\prime }}}$$

(15)

2.3. Investigated Combustor

The RQL swirl combustor developed at Cambridge University is here investigated [3,4]. A schematic representation of the burner is presented in Figure 1. The primary air enters the combustor through a swirler, which introduces the flow at an 80-degree angle relative to the main axis. Fuel injection occurs through a 4 mm diameter hole positioned at the centre of a bluff body. Additional dilution air is supplied via four side jets, each inclined at 45 degrees to the combustor wall and located 54 mm downstream from the inlet. Both the air and fuel are introduced at an initial temperature of 300 K, and the system operates under atmospheric pressure.

Two equivalence ratios are used to characterize the operating condition: the primary equivalence ratio (${\varphi }_p$), which is defined as the ratio of the fuel flow to the primary airflow; and the global equivalence ratio (${\varphi }_g$), which is defined as the ratio of the fuel flow to the total airflow. The aim of the present investigation is to predict the LBO curve using a limited number of CFD simulations far from BO. These CFD simulations are selected at conditions far from BO at a fixed dilution ratio and different global equivalence ratios, which span the LBO curve ${\varphi }_g$ range. The operating conditions for the simulated CFD cases are shown in

Table 1. Operating conditions for the reference CFD cases. R is the dilution ratio.

Case	${\mathit{\varphi }}_{\mathit{g}}$	$\mathit{R} = \dot{{\mathit{m}}_{\mathit{d}}}/(\dot{{\mathit{m}}_{\mathit{d}}}+\dot{{\mathit{m}}_{\mathit{p}}})$
A	0.05	0.1
B	0.07	0.1
C	0.09	0.1

.

2.4. Methodology and Numerical Implementation

The sISR model is solved using the CLIO code [16], which solves the governing equation of the CMC equations. The sISR model is initialized using a 0D-CMC solution, where the initial scalar dissipation rate $N_0$ is set to be equal to the time-averaged value $N_0=〈N_0^{\ast \ast }(t)〉$. For the numerical implementation, an operator splitting approach is applied, using the same solvers and discretization methods as those described in Ref. [16]. The predicted extinction probability depends on the choice of the chemical mechanism, as it directly affects the scalar dissipation rate at which extinction occurs [6]. Thus, the sISR predictions can be highly sensitive to the chosen chemical kinetics. A chemical mechanism, which consists of 10 species and 31 reactions, has been employed for hydrogen chemistry [21]. This mechanism has been validated against experimental data in terms of the ignition delay time and the laminar flame speed, revealing good agreement with the experimental data [21]. Moreover, the used mechanism shows better performance compared to other hydrogen chemical mechanism [21]. The mixture fraction space is discretized into 76 grid points refined close to the stochiometric mixture fraction for hydrogen (i.e., ${\xi }_{st}$ = 0.028). The stochastic differential equation (SDE) is solved using the Milstein method [19] and the total simulation time is taken as 0.15 s, which corresponds to approximately 800 ${\tau }_N$, to provide enough sampling time for the SDE and the sISR equations to develop. Given that $N_0^{\ast \ast }$ evolves stochastically, multiple sISR simulations are necessary in order to estimate the probability of extinction. Here, an upper limit on $N_0^{\ast \ast }$ is imposed by clipping its lognormal distribution. A statistical limit is used, which is defined relative to the distribution’s percentiles (e.g., the 99.999th percentile). This helps eliminate rare events and outliers. In this work, an extinction event is defined based on the temperature at the stoichiometric mixture fraction ($T_{stoi}=T|{\eta }_{st}$) falling below a predefined threshold—typically within the range of 1000 K to 1500 K. If $n_r$ denotes the total number of realisations and $n_{ext}$ is the number of those in which extinction occurs, then the extinction probability $p_{ext}$ is calculated (i.e., $p_{ext}=n_{ext}/n_r$). In the present work, 100 different realisations are used to estimate the extinction probability. In this work, the flame is considered extinguished if the extinction probability is higher than 30% following Ref. [6].

The cold reference CFD cases are solved using the in-house Rolls-Royce CFD code PRECISE-UNS [22,23]. The simulations are conducted using the RANS framework with the $k-ϵ$ as a turbulence model. The LES mesh consists of approximately 4M tetrahedral cells. An in-flow boundary condition is employed at the air and fuel inlet, whereas an outlet boundary condition is used at the combustor exit. And second-order accuracy scheme is used for the spatial discretization of all the variables and convergence is assessed by monitoring the relative change in key solution quantities over successive iterations. The procedure for predicting the extinction dilution ratio and lean blow-off (LBO) curve using the sISR framework involves the following steps:

(1)

Parameter Extraction from CFD: Key parameters—including SDR, fluctuation parameter $F$, and a characteristic timescale ${\tau }_N$—are extracted from a reference CFD simulation corresponding to a known operating condition.

(2)

The Construction of Extinction Probability Map: A series of sISR simulations is performed over a broad range of SDR values at the fixed values of $F$ and ${\tau }_N$ obtained in Step 1. These simulations yield extinction probabilities, from which an extinction probability map is constructed. The map is then refined by averaging the extinction probabilities over additional sets of $F$ and ${\tau }_N$ to ensure robustness. The extinction SDR is obtained at an extinction probability that corresponds to 30%.

(3)

The Identification of Extinction SDR: From the probability map, the critical SDR corresponding to an extinction probability of 30% is identified and defined as the extinction SDR.

(4)

Estimation of Extinction Dilution Ratio: Finally, the extinction dilution ratio $R_{ext}$ is estimated by scaling the reference dilution ratio $R_{ref}$ using the ratio of extinction SDR (step 3) to the reference SDR (step 1). A schematic of the sISR prediction methodology is shown in Figure 2 and more details can be found in Ref. [6].

3. Results

3.1. Cold-Flow Reference Simulations

Figure 3 and Figure 4 show the velocity magnitude, mixture fraction, and scalar dissipation rate (SDR) fields for three operating conditions with increasing global equivalence ratio—Case A (${\varphi }_g$ = 0.05), Case B (${\varphi }_g$ = 0.07), and Case C (${\varphi }_g$ = 0.09)—at a 45-degree oblique slice spanning the combustor corners. In all cases, the fuel mass flow rate remains constant, and ${\varphi }_g$ is varied by reducing the air mass flow. It is also worth noting that all cases have the same dilution ratio. Each column corresponds to a different case, while the rows display (from top to bottom) the velocity, mixture fraction, and SDR fields. At the lowest equivalence ratio (Case A), the high primary air flow results in a strong jet momentum, which induces a pronounced recirculation zone downstream of the jet. This intense recirculation limits the expansion of the fuel jet, confining the mixture and resulting in a higher local velocity and sharper scalar gradients. As ${\varphi }_g$ increases (Cases B and C), the reduced airflow leads to a weaker jet momentum and a less intense recirculation zone, allowing the fuel to spread more broadly into the combustion chamber. The SDR fields also reflect this behaviour: Case A shows the most intense dissipation rates near the injector, driven by a strong velocity and mixture gradients, while the SDR magnitude decreases with increasing ${\varphi }_g$. This can be confirmed by estimating $〈N_0^{\ast \ast }〉$ within a volume surrounding the stochiometric mixture fraction iso-surface according to Equation (9), which reveal $〈N_0^{\ast \ast }〉$ = 21.1, 11.3, and 7.8 ${\mathrm{s}}^{-1}$ for case A, B, and C, respectively. The SDR behaviour is also consistent with the blow-off dilution ratio at each ${\varphi }_g$, with case A being closer to the blow-off compared to cases B and C.

3.2. Extinction Probability

Figure 5 and Figure 6 show the change in temperature at the stoichiometric mixture fraction and $N_0$ with time for a fully burning realisation and a blow-off realisation. As shown in Figure 6, the stochastic representation of the SDR allows for temperature fluctuations for both realisations. However, in some instances, the SDR attains higher values for the blow-off case compared to the fully burning cases, resulting in a flame blow-off after about 40 ms. These are two realisations out of 100 realisations used to obtain the extinction probability at $〈N_0^{\ast \ast }〉=60 {\mathrm{s}}^{-1}$. In that case, the extinction probability is 30%, suggesting that 30 realisations out of 100 result in a blow-off behaviour. Following the sISR approach and as depicted in Figure 2, a “probability map” for extinction events can be constructed, by obtaining the extinction probability at different values of SDRs. Figure 7 illustrates the numerical estimate of the mean extinction probability, $p_{ext}$, as a function of the mean SDR, $〈N_0^{\ast \ast }〉$ for different cases spanning various combinations of parameters $F$ and ${\tau }_N$. The figure also shows the associated uncertainty in the estimated $p_{ext}$, represented by confidence intervals derived from binomial statistics across multiple parameter scenarios. It can be seen in Figure 7 that the relationship resembles a sigmoid, with $p_{ext}$ increasing with a higher $〈N_0^{\ast \ast }〉$, indicating a greater likelihood of blow-off (BO) as one moves to the right along the curve. By adopting a critical probability threshold of approximately 30% (indicated by the red star symbol), the right region of the plot can be identified as an unstable regime where BO is likely to occur, whereas the left region indicates a stable flame regime. It is worth noting that the influence of hydrogen’s high laminar flame speed and adiabatic flame temperature is implicitly accounted for in the model through the extinction scalar dissipation rate threshold $N_{0,ext}$, which directly affects the extinction probability curve. For instance, in the present analysis, the SDR at which blow-off occurs is approximately $60 {\mathrm{s}}^{-1}$ (i.e., $SD{R}_{BO}\approx 60 {\mathrm{s}}^{-1}$) for hydrogen flames. In contrast, for hydrocarbon fuels such as ethanol at atmospheric conditions, the corresponding threshold is significantly lower (e.g., $SD{R}_{BO}\approx 8 {\mathrm{s}}^{-1}$) [6].

3.3. LBO Curve and Blow-Off Dilution Ratio Predictions

As outlined by Klimenko and Bilger [9], the volume-weighted conditional scalar dissipation rate (SDR) within a control volume, denoted as $〈N_{\mathsf{\eta }}^{\ast \ast }〉$, can be derived by performing a double integration of the mixture fraction probability density function (PDF) transport equation. This results in the following expression [6]:

$${〈N_{\mathsf{\eta }}^{\ast \ast }〉}_{\mathrm{b}\mathrm{u}\mathrm{r}\mathrm{n}\mathrm{e}\mathrm{r}}={\displaystyle \frac{1}{{\tau }_{res}{P}_{\eta }^{\ast \ast }}}{\int }_0^{\eta }{\int }_0^{{\eta }^{\prime }}[{\left(P_{{\eta }^{″}}^{\ast }\right)}_{in}-{\left(P_{{\eta }^{″}}^{\ast }\right)}_{out}]d{\eta }^{″}d{\eta }^{\prime }$$

(16)

where ${\tau }_{res}$ is the residence time within the burner and $P_{\eta }^{\ast \ast }$ is the mixture fraction PDF at either the inlet or outlet. If we assume that changes in operating conditions do not significantly affect the shape or magnitude of ${〈N_{\mathsf{\eta }}^{\ast \ast }〉}_{\mathrm{b}\mathrm{u}\mathrm{r}\mathrm{n}\mathrm{e}\mathrm{r}}$ through variations in the PDF, then the conditional SDR becomes primarily governed by the residence time, which is inversely proportional to the total mass flow rate. Assuming a constant fuel mass flow, the total mass flow rate—and, hence, $〈N_0^{\ast \ast }〉$—is directly proportional to the bulk velocity $U_b$. In this work, the dilution ratio is changed by increasing the dilution jet velocity, while maintaining a constant total mass flow rate. Thus, in the present investigation, $〈N_0^{\ast \ast }〉$ is directly proportional to the dilution ratio. This allows the extraction of the extinction dilution ratio from the reference SDR and the dilution ratio as follows:

$$R_{ext}={\displaystyle \frac{SD{R}_{BO}\times R_{ref}}{SD{R}_{ref}}}$$

(17)

The prediction of the LBO curve and extinction dilution ratio for the hydrogen swirl RQL-like combustor is shown in Figure 8. It can be observed that the extinction low-order modelling approach, sISR, is capturing the trending behaviour of the LBO curve compared to the experimental data. As the global equivalence ratio increases, the extinction dilution ratio also increases, a behaviour that is predicted by sISR. This further confirms the applicability of sISR in predicting the LBO curve using a limited number of CFD simulations, which can be far from BO and can also be based on a cheap CFD (i.e., cold flow using RANS).

4. Conclusions

The applicability of the low-order modelling extinction prediction tool, the stochastic Imperfectly Stirred Reactor (sISR), to predict the lean blow-off (LBO) curve and the extinction condition of a flow split between the primary and secondary zone is investigated for a hydrogen-fueled RQL-like swirl combustor. The model predicts the flame global extinction using a limited set of Computational Fluid Dynamics (CFD) simulations as reference conditions, which is, here, obtained using cold-flow RANS. It provides a map that shows the extinction probability (i.e., the proportion of a flame’s stoichiometric iso-surface undergoing local extinctions) at different values of scalar dissipation rates (SDRs). It has been found that the sISR modelling framework is capable of predicting blow-off (BO) dilution ratios across various global equivalence ratios, with the predictions showing a good agreement with the experimental data. The results support the applicability of the sISR model as a low-order tool for predicting flame extinction in RQL-like hydrogen combustors. Its application can play a valuable role in the early-stage design of hydrogen-rich RQL combustors by facilitating rapid assessments of combustor stability and operability.