Thermoacoustic FEM modeling of partly autoignition and propagation-stabilized flames

Thermoacoustic FEM modeling of partly autoignition and propagation-stabilized flames Symposium on Thermoacoustics in Combustion: Industry meets Academia (SoTiC 2025) Sept. 8 - Sept. 11, 2025 Trondheim, Norway Paper No.: 52 ©The Author(s) 2025 Simon M. Heinzmann1,2 , Nino A. Leder1, Harish S. Gopalakrishnan1and Mirko R. Bothien1,3 Abstract This paper describes a methodology to thermoacoustically account for different flame types in a single Finite-Elementcomputation. To do so, the flame segmentation mechanism presented in previous studies is used to characterize the different flame zones. Differentiation is done between propagation-stabilized shear-layer flames and autoignition flames that respond to acoustic perturbations very differently. While autoignition flames mainly respond to acoustic pressure and temperature fluctuations, propagation-stabilized flames respond to acoustic velocity perturbations. Thus, flame transfer functions specific to each flame type are analytically implemented within the frequency domain Finite-Elementcomputation. Using the novel framework, it is shown that the global FTF obtained from Computational-Fluid-Dynamics (CFD) simulations of a backward facing step reheat combustor can be reproduced accurately in the low-frequency regime for an operating point where the flame is forced in a planar manner. The investigated operating point operates on hydrogen fully premixed at lean and autoignitive conditions. The autoignition framework is validated by comparison to one-dimensional direct-numerical-simulation (DNS). The time averaged CFD heat release rate result is validated with large eddy simulation (LES) data. Keywords thermoacoustic, autoignition,reheat-combustor, FEM Novelty and Significance Statement The novelty of this work lies in the incorporation of two different types of flame in a single thermoacoustic FEM calculation. To the best of our knowledge, this has not yet been shown in the literature before. The significance of this work lies within industrial applications. In industrial reheat combustion chambers, the autoignition flame is typically not purely autoignition stabilized, and flame anchoring regions are present which contribute to the stability of the flame. Nomenclature Abbreviations PV Progress Variable CFD Computational Fluid Dynamics CFL Courant-Friedrichs-Lewy FEM Finite-Element-Method FTF Flame Transfer Function HRR Heat Release Rate LES Large eddy simulation RANS Reynolds Averaged Navier-Stokes Greek ¯ ˙ωP V Averaged turbulent source term of PV ˙ωP V Source term of P V γRatio of specific heats ωAngular frequency in rad/s ρDensity σSpread of time delay σ˙q,ai HRR width of autoignition flame σ˙q,ps HRR width of propagation-stabilized flame τTime delay Yi,0Initial mixture composition Roman (·)′Perturbation in time domain (·)0Time-averaged quantity (·)init Initial (·)ai Autoignition (·)ps Propagation-stabilized ˙qInstantaneous total HRR ˙q0,a,ai Volumetric HRR integrated along the streamwise direction of the autoignition flame [W/m2] ˙q0,a,ps Volumetric HRR integrated along the streamwise direction of the propagation-stabilized flame [W/m2] ˆ (·)Fourier transform of a fluctuating quantity ˆ ˙qVolumetric HRR perturbation of entire combustion chamber ˆ ˙qa,ai Autoignition flame integrated HRR perturbation 1ZHAW Zurich University of Applied Sciences, Institute of Energy Systems and Fluid-Engineering, Winterthur, Switzerland 2ETH Zürich, Department of Mechanical and Process Engineering, Zürich 3Department of Energy and Process Engineering, NTNU, Norway Corresponding author: S. M. Heinzmann, ZHAW Zurich University of Applied Sciences, Institute of Energy Systems and Fluid-Engineering, Winterthur, Switzerland Email: [email protected] 2Symposium on Thermoacoustics in Combustion: Industry meets Academia (SoTiC 2025) Paper No.(52) ˆ ˙qa,ps Propagation-stabilized flame integrated HRR perturbation ˆ f1Downstream traveling acoustic wave ˆg1Upstream traveling acoustic wave ˆ h1Entropic wave ˆxig Autoignition flame movement perturbation c0Speed of sound F1, F2, F3Autoignition flame HRR FTFs fiMass fractions e.g. fuel, hot gas, carrier air FTFps Propagation-stabilized flame HRR FTF G1, G2, G3Autoignition flame movement FTFs pPressure P(·)Probability pop Operating pressure pref Pressure at reference location TTemperature tTime uVelocity x, y, z x-,yand z-coordinates xig,0Mean ignition length Introduction To fulfill the Paris agreement (1), a large-scale integration of renewable energy systems in the energy landscape is essential (2–4). To stabilize and balance the grid, gas turbines running on carbon-free fuels can be an ideal asset (3;5–9). However, these non-conventional fuels possess very different combustion properties when compared to natural gas. Thus, novel gas turbine combustor development is faced with new challenges to enable the fuel flexible firing capability of an engine. The thermoacoustic analysis to aid the mitigation of combustion instabilities remains a key component in the development phase (10). There are a variety of methods to perform thermoacoustic analysis of combustion chambers. Direct numerical simulation (DNS) (11;12) and large eddy simulation (LES) (13) are considered high-fidelity analysis tools, which are capable of analyzing certain thermoacoustic effects in the highest detail. However, they come with a high computational cost and are unfeasible for entire industrial combustor geometries. Alternatively, Helmholtz solvers in combination with flame transfer/describing functions extracted from computations or experiments have been shown to accurately capture the thermoacoustics of combustion chambers (14–20). Nicoud et al. (14) showed that using a FEM Helmholtz solver the stability of combustion chambers with propagation-stabilized flames could be captured accurately. To do so, the flame transfer function (FTF) was described by an n−τmodel (21). Laera et al. (15) also used a Helmholtz solver together with a flame describing function to correctly capture the stability of a laboratory scale annular combustion chamber comprised of multiple matrix burners. Silva et al. (19) combined a Helmholtz solver with a flame describing function for a swirl-stabilized combustion chamber to predict the acoustic pressure amplitudes of limit-cycles. Generally, the application of Helmholtz solvers to perform thermoacoustic analysis significantly reduces computational time compared to DNS and LES. Alternatively, networkmodels have shown to be able to reproduce the combustion chamber acoustics and flame interaction with the highest computational efficiency (22–26). In this paper, we present an FEM based approach to thermoacoustically model flames that are partly stabilized by autoignition. The overall flame, which consists of a propagationand an autoignition part, is analytically described in FEM. The segmentation mechanism from prior studies (17;18) is leveraged to define the different flame types. The segmentation mechanism (17;27) was initially designed to thermoacoustically model non-compact reheat flames under the influence of transverse combustor eigenmodes. In the case of transverse eigenmodes, the acoustic wavelength is of similar length as the flame, thus, resulting in an acoustically non-compact flame. To deal with this, the segmentation mechanism divides the flame into multiple subflames in transverse direction. Each of the very thin subflames is then considered acoustically compact without variation of the fluctuation quantities across its height. Thus, single values for the pressure, temperature and velocity fluctuations act on each subflame. For each of the subflames, a transfer function can be assigned. In the study presented here, the flame is acoustically compact and as such no segmentation is needed. However, the segmentation mechanism is leveraged to assign a specific FTF to each heat release rate (HRR) region depending on its flame stabilization type. To obtain the flame characterization parameters, RANS simulations are done using Fluent with an in-house reheat flame model, which is an adaptation and extension of the model derived by Kulkarni et al. (28;29). The thermoacoustic analysis is done for a planarly forced backward facing step (BFS) burning hydrogen at elevated pressure. The presented research is relevant, because in industrial reheat combustors the autoignition flame is never only stabilized by autoignition. Typically, the reheat flame consists of propagation-stabilized HRR regions as well as regions predominantly stabilized by autoignition. Propagationstabilized flames occur within the shear layers of recirculation zones. Their stabilization is governed by the balance of the flame consumption speed and the flow velocity. With respect to thermoacoustics, the acoustic waves affect the propagation-stabilized flame regions by velocity fluctuations, which modulate the equivalence ratio. In contrast, autoignition stabilized flames are governed by the balance of the autoignition time scale and the flow residence time scale. The fluctuating pressure and especially the isentropic temperature fluctuations induced by acoustic waves affect the local reaction rates and thus the ignition chemistry. This can strongly alter the ignition time and therefore the ignition length, which results in HRR fluctuations. Thereby, the history effect of the fluctuations during the ignition phase matters for reheat flames and is accounted for in current state-of-the-art flame response modeling tools, as presented in Refs. (17;27;30–33). Heinzmann et al. 3 Validation data The segmentation mechanism developed by Heinzmann et al. (17;18) was already validated in prior studies and initially designed to numerically deal with acoustically non-compact flames. This segmentation mechanism can also be efficiently used for the planar wave case to incorporate different flame types. The model is compared to 1D DNS (32) for a flame solely stabilized by autoignition, and to RANS computations for the partly autoignitionand propagation-stabilized flame in a BFS combustion chamber. 1D DNS simulation The CFD study used for validation in Ref. (34) is a compressible 1D DNS computation burning hydrogen fully premixed at lean conditions. The operating pressure is atmospheric with an inlet temperature of 1100 K and an inlet flow speed of 200 m/s. The duct of length 0.3 m with nonreflecting boundary conditions was forced with acoustic and entropic waves at the inlet. For more details on the numerical setup the reader is referred to Ref. (34). The duct is modeled in 2D in FEM with a length of 0.3 m and a height of 0.05 m. The boundary conditions were set to non-reflecting and the ˆ f1and ˆg1acoustic forcing state from the CFD initialized by forcing with ˆ f1,b,ˆ h1,b and ˆg2,b waves (Fig. 1), where the subscript bstands for boundary. Because the Helmholtz equation does not account for any mean flow effects, entropic waves incident at the inlet were forced directly within the mathematical flame formulation via a harmonic source term. Autoignition flame RANS computation The 2D RANS CFD computations are performed for a BFS Fig. 2with a total length of 0.3 m. The BFS is located at 0.15 m, with a symmetric area expansion from 0.01 m to 0.02 m. The investigated operating point burns hydrogen and air fully premixed at lean and autoignitive conditions (mass fractions: H2(0.0053), O2(0.1842), H2O(0.0521), N2(0.7584)). The inlet velocity measures 150 m/s, the mean operating pressure is 12.9 bar, and the inlet temperature is 1143 K. The simulation is done using ANSYS Fluent 2024 R1 (35) with an in-house combustion model which is based on the work of Kulkarni et al. (28;29). In the combustion model, the radical buildup during the induction phase is represented by a normalized progress variable (PV ). The PV and its source term is obtained by tracking representative intermediate and product species (here HO2and H2O) in 0D reactors. For normalization, the sum of the tracked species Figure 1. 1D DNS configuration schematic of the modeled duct in the FE-solver. The boundaries are forced with ˆ f1,b,ˆ h1,b and ˆg2,b waves. mass fractions is then divided by its sum at equilibrium. It can be shown, that for fully premixed conditions, the PV source term only depends on the temperature, pressure and the current ignition progress (36). ˙ωPV = ˙ωPV(T, p, PV )(1) Assuming, that the temperature does not change significantly during the radical build up, it can be divided into two parts: The initial temperature T0,init and the fluctuating temperature T′due to the acoustic forcing. Similarly, the pressure is divided into the operating pressure pop and the fluctuating part p′. ˙ωPV = ˙ωPV(T0,init, T′, pop, p′, PV )(2) By assuming isentropic correlation of pressure and temperature for acoustic waves, T′can be expressed using p′,pop and T0,init: T′=T0,init p′ pop + 1γ−1 γ +T0,init (3) As pop and T0,init are constant and known a priori, the PV source term is defined by: ˙ωPV = ˙ωPV(p′, PV )(4) For the turbulence chemistry interaction (TCI) a presumed beta PDF (37) approach is used. The mean turbulent PV source term is then obtained by folding the source term over a probability distribution P(PV )which is defined by the PV ’s mean and variance: ˙ωPV =Z1 0 ˙ωPV(p′, PV )P(PV )dPV (5) The reaction rate is then computed by multiplying the Eddy Dissipation Model’s (38) reaction rate with the value of the PV in order to delay it until the residence time of a fluid particle is equal to the ignition delay time. The advantage of this approach is that the correct ignition delay time can be achieved without transporting all the intermediate species and computing their reaction rates. Instead, the PV source terms are computed a priori using a detailed mechanism (SanDiego (39)) and stored in a lookup table. For the CFD simulations the realizable k−ϵturbulence model and second order temporal and spatial discretization is used with a timestep of 6e-6 s to obtain an acoustic CFL of 0.96. The structured uniform mesh consists of hexahedral elements with a size of 0.5 mm. A mesh independence study has been made comparing element sizes of 0.2 mm and 0.125 mm. The RANS mean field is validated by comparison to data from a compressible LES computation of the same geometry and at the same operating point. The OpenFoam code (40) is used to solve the Navier-Stokes equations. The partially-stirred reactor (PaSR) turbulent combustion model is employed. The LES simulation is setup identically to the shorter BFS shown in Gopalakrishnan et al. (33). 4Symposium on Thermoacoustics in Combustion: Industry meets Academia (SoTiC 2025) Paper No.(52) Figure 2. Schematic of the backward facing step (BFS) geometry. Methodology Flame response and segmentation The flame response to planar acoustic waves is characterized differently for the propagation-stabilized shear layer flames and the autoignition flame. For the shear layer flame, an n−τ−σmodel is implemented, similarly to (41), using Eq. 6. The FTF thereof (FTFps Eq. 7) relates the HRR perturbations to the velocity fluctuations at the dump plane, and is derived from the initial n−τmodel from Crocco (21). The magnitude nhas been chosen equal to 1 to satisfy the low frequency FTF limit for fully premixed propagation-stabilized flames (gain equal to 1) (42). The second exponential term exp(−σ2 psτ2 ps/2) is a measure for the delay spread across the flame and acts as a low-pass filter at higher frequencies. They are calculated by fitting Gaussian kernels to the timely averaged HRR field and evaluating the location and spread of ignition lengths. ˆ ˙qa,ps ˙q0,a,ps =exp(iωτps)exp(−σ2 psτ2 ps/2) ˆux(xref,y) ¯u0 (6) FTFps = ˆ ˙qa,ps ˙q0,a,ps ˆux(xref,y ) ¯u0 (7) For the autoignition flame, the framework from Gopalakrishnan et al. (32) is used to quantify the flame response. The total autoignition HRR response of the flame is assumed to be a linear superposition of the individual flame responses (FTFs) to the acoustic and entropic waves (31). Eq. 8 represents the HRR FTFs Fi(ω)and Eq. 9characterizes the flame movement using the Gi(ω)FTFs. ˆ ˙qa,ai/˙q0,a,ai is the normalized heat release rate perturbation, ˆxig/xig,0 the normalized flame movement, ˙q0,a,ai (W/m2) is the mean heat release rate per unit area and xig,0the mean ignition length (32). ˆ f1is the downstream traveling acoustic wave, ˆg1the upstream traveling acoustic wave and ˆ h1the with the flow transported entropic wave (32). ˆ ˙qa,ai ˙q0,a,ai =F1(ω)ˆ f1 p0 +F2(ω)ˆg1 p0 +F3(ω)ˆ h1 p0 (8) ˆxig xig,0 =G1(ω)ˆ f1 p0 +G2(ω)ˆg1 p0 +G3(ω)ˆ h1 p0 (9) FEM implementation To characterize the flame in the FEM simulation and solve the inhomogeneous Helmholtz equation (Helmholtz equation with source terms), the numerical segmentation mechanism derived by (17;18) is used. A main benefit of the segmentation is, that simulations can be done for non-compact flames, i.e. flames that are under the influence of transverse eigenmodes. Fig. 3schematically visualizes the segmentation mechanism. Differentiation is made between propagation-stabilized subflames (blue circles) and the autoignition subflames (magenta circles). Due to the fine segmentation, it can be assumed that for each subflame a single value for the pressure and velocity perturbation is present. Thus, the flame also reacts with a single value for the HRR and the flame movement. For each of the subflames, specific FTFs can be implemented to capture the different flame responses of propagation-stabilized and autoignition flames. Hence, the segmentation mechanism allows different flame types in the same FEM computation. For the propagation-stabilized flame parts (Eq. 10) as well as the autoignition flame regions (Eq. 11), Gaussian kernels are used to analytically position the flame in the computational domain. σ˙q,ps and σ˙q,ai are measures for the HRR width, xf,ps(y)and xig,0(y)the mean flame positions. ˙qps(x, y) = ˙q0,a,ps(y) σ˙q,ps(y)√2πexp −1 2x−xf,ps(y) σ˙q,ps(y)2! (10) ˙qai(x, y) = ˙q0,a,ai(y) σ˙q,ai(y)√2πexp −1 2x−xig,0(y) σ˙q,ai(y)2! (11) The parameters for the HRR width, the mean flame position and the integrated HRR can be calculated from either CFD mean field solutions or experimental chemiluminescence imaging if available. Using these quantities and the assumption of the Gaussian kernel in x-direction (multiple studies used Gaussian distributions (17;30;31;34;43)), the average flame can be rebuilt fully in FEM. Whilst the procedure here is done in 2D on the xy-plane, the previous study (27) derived the identical procedure in 3D. Hence, the new method shown here can also be used in 3D. At this point, the average flame is fully characterized in the FEM simulation and what remains to be characterized is the representation of the fluctuating HRR perturbations. This is achieved by obtaining the instantaneous HRR after multiplying the mean HRR by the FTF Eq. 8(for the propagation-stabilized flame Eq. 6respectively) and accounting for the flame motion in the Gaussian distribution using Eq. 9to obtain Eqs. 12. For the propagation-stabilized Heinzmann et al. 5 Figure 3. Numerical segmentation method for a perturbed partly autoignition and propagation-stabilized flame in 2D. The blue circles indicate the propagation-stabilized flame parts, and the magenta circles represent the autoignition flame. flame region, no flame motion is introduced due to the anchored nature of the flame. ˆ ˙qa,ps(y) = ˙q0,a,ps(y)FTFps ˆux(xref , y) ¯u0 ˆ ˙qa,ai(y) = ˙q0,a,ai(y) F1(ω)ˆ f1 p0 +F2(ω)ˆg1 p0 +F3(ω)ˆ h1 p0! (12) Using Gaussian kernel distributions for both flames the instantaneous volumetric HRR is obtained (Eq. 13). ˙q(x, y) = ˙q0,a,ps(y) σ˙q,ps(y)√2π... exp −1 2x−xf,ps(y) σ˙q(y)2! ˆ ˙qa,ps(y) ˙q0,a,ps(y)+ 1!+... ˙q0,a,ai(y) σ˙q,ai(y)√2πexp −1 2x−xig,0(y)−ˆxig(y) σ˙q,ai(y)2!... ˆ ˙qa,ai(y) ˙q0,a,ai(y)+ 1!(13) The fluctuating HRR ˆ ˙q(x, y), which is the source term for the Helmholtz equation, is consequently computed by subtracting the mean HRR from the instantaneous counterpart (Eq. 14). ˆ ˙q(x, y) = ˙q(x, y)−( ˙qps(x, y) + ˙qai(x, y)) (14) The equation for the inhomogeneous Helmholtz equation, which is solved in FEM (in the frequency domain), thus becomes Eq. 15 by adding the source term to the righthand side of the homogeneous Helmholtz equation. From the computations the eigenfrequencies and linear growth rates are extracted. In this study COMSOL 6.2 is used. ∇·1 ρ0∇ˆp+ω2 γp0 ˆp=−iω γ−1 γˆ ˙q(x, y)(15) Results Results obtained by implementing the proposed methodology are shown in this following section. 1D DNS forced Fig. 4shows the intermediate result obtained in comparison with the DNS simulation to validate the FEM implementation of a flame solely stabilized by autoignition. An excellent match is achieved in comparison with the DNS result in both phase and magnitude. The small deviation in phase could result from neglecting mean flow effects by employing the Helmholtz equation. 0 0.005 0.01 0.015 0.02 -0.04 -0.02 0 0.02 0.04 Figure 4. Comparison of the HRR fluctuations for the 1D DNS configuration forced at 100Hz. The solid line represents the DNS data and the dashed line the FEM result. 6Symposium on Thermoacoustics in Combustion: Industry meets Academia (SoTiC 2025) Paper No.(52) Figure 5. Mean field solutions of the RANS simulation: a) y-velocity, b) x-velocity, c) temperature and d) pressure. Figure 6. Comparison of the mean HRR between LES (top) RANS (middle) and FEM (bottom). The backward facing step is located at x= 0.15m. Mean fields of the BFS burning hydrogen Figure 5shows the time-averaged mean fields from the RANS simulation. Figure 5a) shows the y-velocity. The recirculation zone is visible by the blue and red areas. Figure 5b) displays the x-velocity. The recirculation zone can again be nicely observed by the negative velocity areas in blue. The temperature distribution is given in Fig. 5c) and matches well with the results from CANTERA simulations. Lastly, the pressure is shown in Fig. 5d), where a pressure drop downstream of the area jump is observed and a pressure increase downstream of the flame. The time-averaged flame shape is computed using the in-house autoignition reheat flame model described in the CFD section. To validate the mean field qualitatively, comparisons to center-plane data from the described LES simulation are made. The comparison between the results from the LES and the RANS simulations shows that the model can replicate the flame to a certain extent. The mean ignition-length and overall flame shape is captured well (length scales are small). The RANS simulation however predicts the propagation-stabilized shear layer flames to be further upstream compared to the validation data. Experience shows, that the occurrence of an earlier timed ignition is linked to the RANS solver used in combination with the reheat flame model. Further comparisons can be made by analyzing the volumetric HRR of Fig. 6for two different sections in Fig. 7. The three dash-dotted blue lines show the normalized volumetric HRR along a section (y= 0.001m) across the shear layer flame. The diamonds represent the LES data, the circles the RANS data and the stars the FEM data, respectively. As was already indicated before, the HRR peak of the shear layer flame occurs further upstream in the RANS simulation at 0.158m compared to 0.169m. The FEM value matches the one of the RANS closely with 0.157m. The peak value is over-predicted by the RANS compared to the LES simulation. The solid magenta lines provide the same comparison for a section at a y-coordinate of 0.001m, close to the symmetry line. In this region the flame is stabilized by autoignition. The peak of the RANS simulation is at an Heinzmann et al. 7 Figure 7. Comparison of the volumetric HRR of different sections through the shear layer flame (blue dash-dotted lines.) and the autoignition stabilized part (solid lines). The comparison is made between the LES (diamonds), RANS (circles) and the FEM (stars). Figure 8. FTF Comparisons between the CFD results (diamonds) and the FEM computation (circles). The blue dashed line represents the used n−τ−σmodel. x-coordinate of 0.188m. It is predicted further downstream compared to the LES at around ∼0.18m. The mean ignition length of the FEM matches the RANS closely with 0.186m. The HRR distribution is spread more symmetrically around the peak compared to the RANS, which is a prerequisite of the introduced mathematical model (Eq. 13). More generally, the flame appears more compact in the LES compared to the RANS computation. Due to the match of the overall flame shape, the RANS mean field can still be used for the subsequent study. Simulations with harmonic acoustic forcing at the inlet with a pressure amplitude of 0.5% are performed using ANSYS Fluent. The values of the n−τ−σmodel are computed to be n= 1 (low frequency limit for fully premixed propagation stabilized flames (42)), σ= 0.099ms and τ= 0.12ms using the method described in the methodology section. The forced FEM simulation is set up identically to the RANS computation by forcing harmonically at the inlet with a pressure amplitude of 0.5%. The outlet is set to 8Symposium on Thermoacoustics in Combustion: Industry meets Academia (SoTiC 2025) Paper No.(52) Figure 9. FTF Comparisons between the n−τ−σmodel (blue dashed) and the autoignition FTF (magenta dash-dotted line). The solid black line is the combination of both FTFs. The circles represent the FTF from the FEM, and the diamonds the FTF from the RANS simulations. fully anechoic to mitigate reflections and upstream traveling acoustic g-waves. Fig. 8shows the FTFs ( ˙q′/˙q0)/(u′/u0)of both the RANS simulations and the FEM simulations. The FEM computation (circles) is able to capture the phase very accurately. The gain of the FTF shows slight differences between the CFD simulation (diamonds) and the FEM computation. The overall match of the FTF is good, and the newly introduced segmentation model to combine differently behaving reheat flame regions for FEM computations works. Further, the comparison to the dashed blue line, which represents the n− τ−σmodel, indicates that the investigated flame response is dominated by the propagation-stabilized flames in the shear layer. Consequently, the proposed methodology can be used in a further step for stability calculations of longitudinal eigenmodes. For completeness, Fig. 9shows the used FTFs (with respect to the velocity fluctuations ( ˙q′/˙q0)/(u′/u0)) for a larger frequency range. At higher frequencies, the assumption of no mean flow velocity breaks down in the FEM. However, the FTFs can still be analyzed to understand the flame response at higher frequencies. Looking at the n−τ−σFTF (blue dashed line) of the shear layer flame zones, the low-pass filter behavior can clearly be observed. A negligible HRR response contribution from the shear layer flames from 4000Hz onwards is indicated. The FTF from the solely autoignition stabilized central HRR zone is computed using the Lagrangian framework (32) and is shown by the magenta dash dotted line. The FTF indicates that at higher frequencies the autoignition flame zone contributes to a large extent to the overall flame response. However, no low-pass filter is contained in the model and thus the gain is estimated to be slightly lower in reality. The Lagrangian framework (32) was validated with 1D DNS simulations up to 4000Hz, where deviations of +20% were observed in the gain between DNS and the model. The black solid line represents a superposition of both the shear layer flame FTF and the autoignition flame FTF with an overall autoignition weighting of 0.32 and a shear layer flame weighting of 0.68 respectively. This gives an indication of how the flame behaves globally at higher frequencies, given that the assumption of a superposition of the HRR response of a propagation-stabilized flame region and an autoignition flame region is valid. Conclusion With reheat combustors becoming of increasing interest in industrial gas turbines, autoignition flames have shifted into the scope of current research. Especially in industrial reheat combustion chambers, an autoignition flame will likely have certain flame regions that are propagation-stabilized due to either flow recirculation or vortex breakdown. So far, research on thermoacoustics, especially the flame dynamics, has been performed separately for propagationstabilized flames and autoignition-stabilized flames. In this paper, we propose an analytical model to predict the response of a composite flame partly stabilized by propagation and partly by autoignition. We leverage a previously developed segmentation method to incorporate both flame types in the same FEM computation. Using the segmentation mechanism, different FTFs are prescribed to the propagationand autoignition-stabilized flame regions. For the propagation-stabilized HRR regions, a classical n−τ−σmodel is used. For the autoignition-stabilized HRR regions, the FTF obtained from a Lagrangian particle tracking based approach is used. Results showed a very good Heinzmann et al. 9 fit in comparison to forced CFD data. Consequently, the framework can be used to compute longitudinal eigenmode stability predictions of partly autoignition and propagationstabilized flames and can aid the development of novel emissions-free industrial gas turbine combustors. Acknowledgements The research work presented in this manuscript was carried out within the FLEX4H2 project. The FLEX4H2 project is supported by the Clean Hydrogen Partnership and its members European Union, Hydrogen Europe and Hydrogen Europe Research (GA 101101427), and the Swiss Federal Department of Economic Affairs, Education and Research, State Secretariat for Education, Research and Innovation (SERI). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or any other granting authority. Neither of them is liable for any use that may be made of the information contained therein. References [1] United Nations. Paris agreement, 2015. [2] IEA. World energy outlook 2022, 2022. [3] Bundesamt für Energie. Energieperspektiven 2050+, 2021. [4] Expert Group Security of Supply ETH Zurich. Energy security in a net zero emissions future for switzerland, 2023. [5] Ciani A, Bothien M, Bunkute B et al. Superior fuel and operational flexibility of sequential combustion in Ansaldo Energia gas turbines. 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