Identification of flame dynamics in a technically premixed combustor with compliant fuel injection

Abstract

This work investigates the response of a turbulent, technically premixed methane air flame to flow perturbations, with a focus on how fuel injector impedance affects system dynamics. The flame transfer function (FTF) is determined via system identification (SI) applied to time-series data from forced, high-fidelity large-eddy simulations (LES) and is validated against experimental mean flame shape and position. A Multiple Input, Single Output SI approach is used to separately identify the flame’s response to fluctuations in premixture velocity and equivalence ratio. The simulated premixture velocity FTF agrees well with the premixed experimental FTF, confirming that the simulations have a reasonable capture of flame dynamics. However, the airline fluctuation results show poorer agreement with the technically premixed measurements, indicating that injector impedance plays a significant role in those experiments. A low-order acoustic network model (LOM) is employed to evaluate the system transfer matrix, incorporating injector impedance along with the extracted transfer functions. This allows the injector impedance and the effects of fuel-line aperture and fuel-flow perturbations to be inferred. The results show that fuel injector impedance is significant even in the cold flow measurements, and that its combined effect with the flame’s response to fuel flow perturbations substantially influences the dynamics of the reacting case. These findings highlight the importance of ensuring injector acoustic stiffness across the full operating range in technically premixed systems, or alternatively, of characterizing the response to fuel flow perturbations and accounting for their influence on the measured flame dynamics.

Keywords

Thermoacoustics flame transfer function mixing transfer function system identification large eddy simulation turbulent combustion compliant fuel injection

Introduction

The development of low-emission combustion systems for propulsion and energy is key to addressing the global climate goals. Lean premixed combustion systems offer reduced emissions, making them attractive for manufacturers to meet stringent regulations. However, these systems are highly susceptible to thermoacoustic instability.¹ Analyzed initially by Rayleigh,² a constructive interaction between unsteady heat release and acoustic perturbations can cause thermoacoustic instabilities. If left uncontrolled, this can lead to a poor performance and even catastrophic failure in combustion systems. As such, acoustic characterization and stability analysis are imperative for developing combustors.

A low-order network model (LOM) can be used to assess system stability, enabling the study of multiple configurations and operating conditions at a low-computational expense.^3–5 Employment of these models requires knowledge of the dynamics of the flame, which is typically represented in the frequency domain by the flame transfer function (FTF) or its time-domain equivalent, the flame impulse response (FIR). Generally, the FTF relates fluctuations in global heat release to perturbations of a relevant flow variable at a position upstream of the flame. For a fully premixed (FP) combustor, where fuel and air are mixed at a distance far upstream of the burner outlet, the flame response has been studied extensively by experimental^6,7 and numerical investigations.^8–10 This type of flame is a single-input and single-output system, in which velocity perturbations at a reference position generate global heat release fluctuations

\frac{{\dot{Q}}^{'}}{\bar{\dot{Q}}} = F (ω) \frac{u_{r}^{'}}{\bar{u_{r}}}

(1)

In typical industrial applications, achieving perfect mixing is often unfeasible, resulting in a technically premixed (TP) system where acoustic perturbations in the mixing duct cause fluctuations in the equivalence ratio, which are then transported to the flame position by the mean flow. Perturbations in the equivalence ratio affect the dynamics of the flame through multiple mechanisms,^11–13 making them a significant driver of the system’s dynamics. Considering the interplay between velocity perturbations and mixing, TP flames offer a considerably more complex and richer set of dynamics than their perfectly mixed counterpart. As discussed by Huber and Polifke,^14,15 the total response of a technically premixed flame in the linear regime is constituted as a superposition of the contributions from each mechanism, leading to a multiple-input single-output (MISO) FTF

\frac{{\dot{Q}}^{'}}{\bar{\dot{Q}}} = F_{u} (ω) \frac{u_{r}^{'}}{\bar{u_{r}}} + F_{ϕ} (ω) \frac{ϕ_{r}^{'}}{\bar{ϕ_{r}}}

(2)

A typical assumption in the study of TP combustors is that of “stiff,” or non-compliant fuel injection,^16–20 where fuel mass flow rate can be considered unaffected by pressure perturbations. In such cases, only air mass flow rate fluctuations drive perturbations in equivalence ratio, which are then altered through diffusive and dispersive mechanisms; all of these processes can be quantified by a mixing transfer function. A compliant injector, on the other hand, responds to pressure perturbations, which modulate the fuel mass flow rate, leading to an additional frequency-dependent interaction between air- and fuel-line acoustics.^4,21 In the case of a compliant injection scheme, characterization of the fuel-line acoustic boundary condition is necessary to accurately model a given system in a reduced-order model.

The present work aims to identify the MISO FTFs $F_{u}$ and $F_{ϕ}$ of a TP, turbulent swirl flame. The novelty lies in the subsequent investigation of the further interaction provided by the acoustics of the fuel line, as the response to air mass flow rate fluctuations is insufficient to capture the overall flame response. The paper is organized as follows: First, we present the setup of an incompressible large eddy simulation (LES) of the reacting flow, comparing the mean flame against the experiment. Following this, we present the results from excited simulations, where a broadband signal forces the air and fuel inlets to excite the flame, yielding an output signal for the heat release rate. We then identify the characteristic filter using system identification (SI) methods.²² Finally, we use an LOM of the burner, including its premixed and technically premixed configurations, and plug the identified responses to investigate how injector impedance and fuel fluctuations influence the burner’s acoustic characteristics.

Experimental configuration

The case studied in this work is the GE15 burner, an atmospheric, dry, low NO $_{x}$ methane combustor capable of operating in both FP and TP modes.^17,23,24 Figure 1 shows a schematic of the computational domain. In the TP configuration, fuel is injected into the mixing section as a jet-in-cross flow (JICF) through 24 injectors. The flow field of this burner is typical of swirl-stabilized flames, featuring inner and outer recirculation zones, with a V-shaped flame anchored at the tip of the bluff body. One of the main features of this burner is that the bluff body is recessed compared to the back plate of the combustion chamber.

Figure 1.

Schematic of computational domain of the technically premixed GE15 burner.

Experimental results are available for various power ratings (streamwise velocity in the burner section) and global equivalence ratios. The present work investigates the operating conditions at an equivalence ratio of 0.625, summarized in Table 1. Inlet mass flows are calculated to match the burner tube velocity, assuming that both streams have the same temperature. For mean flame validation, the time-averaged line-of-sight (LOS) integrated $OH *$ chemiluminescence will be used.^25,26

Table 1.

Summary of operating conditions and boundary information.

Boundary	Type	Details
Air inlet	Mass flow inlet	$T = 573 K$ , $\dot{m} = [28, 32.6, 37.3] \times 10^{- 3} kg / s$
Fuel inlet	Mass flow inlet	$T = 573 K$ , $\dot{m} = [10.2, 11.9, 13.6] \times 10^{- 3} kg / s$
Chamber walls	No-slip isothermal wall	$T = 700 K$
Bluff body tip	No-slip isothermal wall	$T = 700 K$
Other walls (swirler,injectors, duct)	No-slip adiabatic wall	—
Outlet	Pressure outlet	—

Measurements of the acoustic transfer matrices, and ultimately the FTF, were done using the multi-microphone method (MMM). The model employed for extracting the FTF was the burner–flame transfer matrix (BFTM) approach.^3,27 This method is commonly used for the experimental measurement of FTFs and is particularly necessary for TP combustors.^17,28 Initially, the MMM is used to extract the burner transfer matrix (BTM), which relates the acoustic states up and downstream of the burner under cold conditions

{[\begin{matrix} \frac{p}{ρ c} \\ u \end{matrix}]}_{d} = B T M {[\begin{matrix} \frac{p}{ρ c} \\ u \end{matrix}]}_{u}

(3)

Then, the procedure is repeated for the reacting, or hot conditions, extracting the BFTM

{[\begin{matrix} \frac{p}{ρ c} \\ u \end{matrix}]}_{d} = B F T M {[\begin{matrix} \frac{p}{ρ c} \\ u \end{matrix}]}_{u}

(4)

The model assumes that the effects of the burner and the flame are sequential and that the burner transfer matrix is unaffected by the presence of the flame. Under these assumptions, the BFTM is a sequential multiplication of the BTM and the flame transfer matrix (FTM). Then, the hot and cold transfer matrices are used to calculate the FTM

F T M = B F T M (B T M)^{- 1}

(5)

Finally, the FTF is directly related to one coefficient of the FTM

F = \frac{{F T M}_{22} - 1}{T_{d} / T_{u} - 1}

(6)

where

T_{d}

and

T_{u}

correspond to the temperature downstream and upstream of the flame, respectively. The method never explicitly measures the resulting quantity

F

shown in equation (6), but instead it reconstructs it by using the hot and cold transfer matrices for the respective configuration, be it FP or TP.²⁹ In the case of GE15, the results for the BTM in both FP and TP modes exhibit a non-trivial change in behavior across these configurations, suggesting that the interaction between the two feeding lines contributes to the acoustic behavior of the system. Furthermore, during cold measurements of the TP case, fuel is not injected, resulting in drastic changes in behavior when operating in the same configuration under both cold and hot conditions, which contradicts the assumption that the BTM remains unaffected in the reacting case. The breaking of this assumption introduces a significant uncertainty in the experimental results for the TP FTF, as discussed in detail by Eder et al.³⁰

Methodology

System identification

We employed the well-established combination of LES and SI,^10,18,20,31 specifically MISO identification. The methodology consists of the imposition of a broadband noise signal at the inlet boundary of a statistically steady-state reactive LES, extracting time series of velocity, equivalence ratio, and heat release rate. The Wiener–Hopf (WH) equation³² relates the autocorrelation of the inputs with the cross-correlation to the output, and it is inverted to find the FIR

\begin{aligned} \frac{{\dot{Q}}_{i}^{'}}{\bar{\dot{Q}}} & = \frac{1}{\bar{u_{r}}} \sum_{k = 0}^{l - 1} h_{k}^{u} u_{r, i - k}^{'} \\ + \frac{1}{\bar{ϕ_{r}}} \sum_{k = 0}^{l - 1} h_{k}^{ϕ} ϕ_{r, i - k}^{'}, i = l, \dots, N \end{aligned}

(7)

The FIR for velocity $h^{u}$ and equivalence ratio $h^{ϕ}$ are composed of $l$ coefficients each. Once we extract them from the time series, they are related to the FTF via the $z$ -transform

F_{u} (ω) = \sum_{k = 0}^{l - 1} h_{k}^{u} e^{- i ω k Δ t}

(8)

F_{ϕ} (ω) = \sum_{k = 0}^{l - 1} h_{k}^{ϕ} e^{- i ω k Δ t}

(9)

The use of MISO identification requires that the input signals exhibit low cross-correlation. For this reason, we excite both inlets simultaneously, imposing decorrelated Daubechies wavelet signals³³ at the corresponding boundaries. Each signal contains a maximum of 10% of the mean value, with a cutoff frequency of 950 Hz.

Numerical setup

Ansys Fluent v2024R2 was CFD software of choice to run the simulations. The LES employed the WALE³⁴ subgrid-scale model with no wall function. The low-Mach approximation, which treats density as a function of temperature and assumes a constant static pressure, eliminates acoustic waves and increases the degree of control over the effective excitation signal, while still capturing the relevant time lags present in flame dynamics, provided the flame is acoustically compact as demonstrated by Eder et al.¹⁰ The flame in this study is of similar length compared to the BRS setup from Komarek and Polifke,⁶ while having a high degree of preheating making the speed of sound higher and thus making the system more compact. Given the length scales present in this burner, we conserve a Helmholtz number $He < 0.1$ until about $400 Hz$ , where some non-compact effects might influence results.

The BFER chemical mechanism,³⁵ a two-step global mechanism commonly used in methane flame simulations, is employed in this study. We implemented a user-defined function to calculate the cell-local equivalence ratio and adjust the pre-exponential factors in the Arrhenius equations. To model the interaction between turbulence and chemistry, we employed the dynamically thickened flame model^36,37 with the Charlette efficiency function,³⁸ using eight grid cells to resolve the flame. The model utilizes a laminar flame speed and thickness that are precomputed in one-dimensional (1D) simulations over equivalence ratios ranging from 0.3 to 1.8, and then fitted with scaled hyperbolic tangent functions to estimate values based on the local composition. Other transport properties use the assumption of a constant Lewis number $L e = 1$ , a constant Prandtl number $P r = 0.7$ , and a power law for viscosity as described by Franzelli et al.³⁵

An Ansys Fluent mesher generated the computational domain using the poly-hex core algorithm. The grid is composed of a total of $25.9 \times 10^{6}$ cells. To resolve the flame as accurately as possible, the maximum cell size in the region downstream of the burner exit is restricted to 0.5 mm. This grid size leads to a maximum thickening factor in the domain of 12, a value that, while relatively high, is similar to values found in other studies based on flame dynamics³⁹ with good results. In the mixing section, which includes the swirler and injector, the maximum cell size is slightly lower, at 0.4 mm. The coarsest level of refinement for the mesh uses a maximum cell size of 2 mm, and no cell is allowed to have a length lower than 0.25 mm, to maintain a problem size that is manageable given the available computational resources. Across all refinement levels, the fluent mesher attempts to stick to the maximum cell size, halving or doubling in size across transition zones, and resolving geometrical features. The average wall $y^{+}$ across the domain is 1.4, with all values being below 5. The highest values occur at the injector-hole walls, with a value of 3.2. However, the flow profile in the injector holes differs from the experimental one, as we simplified the geometry by removing the upstream duct and imposing a constant velocity at the fuel inlet. In light of this, this high value of $y^{+}$ is deemed acceptable.

As we stated previously, we investigate the burner at an equivalence ratio of 0.625 while varying axial velocities in the burner tube section. Both inlet streams have an equal temperature of 573 K. Previous work on the perfectly premixed flame iteratively modified the wall temperature of the combustion chamber and the bluff-body to accurately capture the mean flame shape. This approach concluded with a value of 700 K. Given that the flame topology is similar between the two configurations, this wall temperature is a sensible choice for the technically premixed case as well. All other walls use adiabatic boundaries.

A timestep of 1 $μ$ s was employed, with a fully implicit second-order time discretization, resulting in a maximum Courant-Friedrichs-Lewy (CFL) number of 3–5 in the fuel jet and injector hole regions, which have Mach numbers of approximately 0.2. The high value of CFL in this region is acceptable because resolving turbulence inside the injector holes is not of particular importance, rather in the shear layer of the JICF, which governs mixing, in which CFL numbers remain below unity as in all other sections. Mach numbers of 0.2 lie below the usual limit of 0.3 for low-Mach approximations, with a maximum density deviation of 2% which justifies the incompressible flow assumption.

The input signal for velocity is a density- and area-weighted average of axial velocity. In contrast, the equivalence ratio is calculated as a mass-flow-rate-weighted average. Equations (10) and (11) show the corresponding expressions. In both cases, the reference position “ $r$ ” is located at the tip of the bluff body, which coincides with the base of the flame

u_{r} = \frac{\int ρ u d A_{r}}{\int ρ d A_{r}}

(10)

ϕ_{r} = \frac{\int ρ u ϕ d A_{r}}{\int ρ u d A_{r}}

(11)

Finally, the output signal corresponds to the global sum of cell-local heat release.

CFD results

Mean flame validation

After the simulations have run for a sufficiently long time, the experimental measurement of the LOS-integrated $OH *$ chemiluminescence is compared against the LOS-integrated mean heat of reaction from simulations. Despite the established knowledge of a non-linear relationship between these two quantities due to variations in the equivalence ratio, previous work on TP flames^16,18–20 has demonstrated that these quantities provide qualitative information on the flame location and flame topology, making them a valuable means of validating numerical results.

Figure 2 shows the results of mean flame shape and position, as well as the axial distribution of heat release intensity. On both accounts, a good agreement can be observed between the numerical results and the experiments. Accurate capture of a V-shaped flame, with minimal heat release in the outer shear layer is evident, demonstrating a good agreement in position. The center of gravity of the flame differs by approximately 5 mm, which is close to 5% of the total flame length. Note that experimental figures show a clear asymmetry across the $z$ -axis.

Figure 2.

LOS integrated mean heat release from LES (top row), LOS integrated $OH *$ from experiment (middle row). The bottom row shows the normalized axial distributions of heat release (red line) and $OH *$ (black line), and the center of gravity with vertical dashed lines. LOS: line-of-sight; LES: large eddy simulation.

Perfectly premixed FTF

The forced simulations ran for a total of 330 ms of physical time under each operating condition. After removing 30 ms of initial transients, we resample the time series to a frequency fine enough to resolve the highest frequency of interest (800 $Hz$ ) but coarse enough to avoid an overly complex model, concluding in a resampling frequency of 10 $kHz$ . We then perform Wiener–Hopf inversion (WHI) on the resampled time series, building a model that is in accordance with equation (7), with the FIR essentially being a filter that associates the inputs to the output. The filter length is then chosen by starting from an initial guess of 15 $ms$ , and progressively increasing until no change in the FIR is observed, concluding in a length of 25 $ms$ . The classical WHI method was used because more complicated model structures like Box–Jenkins require the tuning of other parameters like the noise rational polynomial, without yielding any significant benefits in finite LES time series, as demonstrated by Jaensch et al.⁴⁰

Figure 3 compares LES/SI results for the premixture velocity FTF, $F_{u}$ —compare with the first term in equation (2)—with the experimentally measured FTF $F$ , as defined by equation (1). Because the flame topology is the same in the FP and TP configurations, and the respective contributions of the input signals $u_{r}^{'}$ and $ϕ_{r}^{'}$ to the total response are identified and separated in the MISO SI, agreement between the two dynamic responses may be expected in this particular configuration. This comparison provides validation of the dynamics present in the numerical simulations, without requiring consideration of the additional interplay between air and fuel feeding lines, since the latter were not present in the FP configuration of the burner.

Figure 3.

FTFs $F_{u}$ (black line) and $F_{ϕ}$ (red line) from LES and MISO SI, with $σ = 1.96$ confidence intervals. FTF $F$ ( $\circ$ ) from experiment, fully premixed condition. Mean burner tube velocities: $30 m / s$ (left), $35 m / s$ (middle), and $40 m / s$ (right). FTF: transfer function flame; LES: large eddy simulation; MISO: Multiple-Input Single-Output; SI: system identification.

Two important features of $F_{u}$ that must be accurately represented are its low-pass filter behavior and its low-frequency limit. The former requires that the gain at high frequencies tends toward zero. The latter is equivalent to a change in mean conditions. As discussed by Lawn and Polifke,¹¹ the zero-frequency limit for $F_{u}$ should be unity with zero phase, as an increase in mean mass flow should proportionally increase mean heat release. As for $F_{ϕ}$ , given that the flame is lean, a similar argument can be made for its low-frequency limit, because a mean increase in fuel will proportionally increase the heat release.

The FTFs in Figure 3 clearly show low-pass filter behavior with a decaying gain at higher frequencies. Similarly, all numerical results at low frequency tend toward unity gain with zero phase, within the confidence intervals. Capturing both of these features provides a baseline consistency check that the numerical results accurately reflect the physical constraints observed in experiments.

Another physical trend to look out for in comparing dynamics is the scaling of results with mean conditions. Since the mean flame center of gravity stays relatively constant across all conditions studied, increasing velocity will reduce the convective time lag, scaling the FTF in the frequency domain. For higher mean velocities, the slope of the phase curve should be flatter, a physical trend that is correctly captured when comparing numerical and experimental results. The phase curves should also collapse when using the Strouhal number to scale frequency, a feature that our results have, but is not shown here for brevity.

Comparing the results over all frequencies, we observe good agreement in phase slopes across all operating conditions, thereby increasing confidence in the consistency of the numerical results. Regarding gain, we observe that local minima are not well captured for the cases with 30 and 35 $m / s$ . Conversely, for the case with 40 $m / s$ , a high degree of agreement exists in phase and gain, with only a slight shift in the local minimum from 45 to 400 $Hz$ . For frequencies greater than 500 $Hz$ , both continue to agree. A final note on the experimental results for the case with 30 $m / s$ is that they do not tend toward unity gain at low frequencies, with the 20 $Hz$ gain being close to 3.2. This phenomenon is caused by the inability of loudspeakers to reliably force the system at such low frequencies.

Experimental measurements did not explicitly account for the influence of fuel-line acoustics on the transfer matrix, instead assuming a stiff fuel line and forcing only the air stream. As a result, a direct comparison requires extracting a BFTM from the low-order model. Additionally, constructing the TP FTF requires identifying a mixing transfer function^19,41 that connects equivalence ratio perturbations at the reference position with velocity perturbations upstream. This transfer function will be addressed in the following section.

Mixing transfer function

Equivalence ratio is a convective wave, similar to entropy, in that both are convected by bulk flow toward the flame. These convective waves have the property of being transported and spatially dispersed,⁴² leading to the commonly used model of 1D convection-diffusion.^12,43,44 Under this model, the equivalence ratio at the reference position is a function of the global, or nominal equivalence ratio $ϕ_{g}$

ϕ_{g} = s (\frac{{\dot{m}}_{f}}{{\dot{m}}_{a}})

(12)

where

s

is the stoichiometric factor, and

{\dot{m}}_{f}

{\dot{m}}_{a}

denote mass flow of fuel and air, respectively. It can also be referred to as the equivalence ratio at the injector. Linearization of this quantity gives the following correspondence with fuel and air fluctuations

\frac{ϕ_{g}^{'}}{\bar{ϕ_{g}}} = \frac{u_{f}^{'}}{\bar{u_{f}}} - \frac{u_{a}^{'}}{\bar{u_{a}}}

(13)

Finally, a mixing transfer function for

ϕ_{r}

can be expressed, which relates global equivalence ratio to the equivalence ratio at the reference position

\frac{ϕ_{r}^{'}}{\bar{ϕ_{r}}} = H_{g} (ω) \frac{ϕ_{g}^{'}}{\bar{ϕ_{g}}}

(14)

We expect that $H_{g}$ has a gain of unity at zero frequency, since a quasi-steady state increase in the global equivalence ratio must produce a proportional increase at the reference location. This behavior is expected for all premixed (or technically premixed) flames in which a coherent equivalence ratio is observable directly upstream of the flame, regardless of whether the mean or global composition is lean or rich. This model is based solely on the dispersion and dissipation of $ϕ$ through mean flow gradients and diffusion, respectively. Consequently, gain is expected to decay with increasing frequency, as the associated spatial gradients become amplified.

As mentioned by Garcia et al.²⁰ and experimentally observed by Bluemner et al.,²¹ the model from equation (14) fails to capture an additional hydrodynamic contribution. For this reason, a slightly more complex model for mixing was proposed,²⁰ and has been successfully applied to TP combustors.¹⁹ Mixture velocity, and its transfer function are added as contributions to equation (14)

\frac{ϕ_{r}^{'}}{\bar{ϕ_{r}}} = H_{u} (ω) \frac{u_{r}^{'}}{\bar{u_{r}}} + H_{g} (ω) \frac{ϕ_{g}^{'}}{\bar{ϕ_{g}}}

(15)

The perturbations arising from $H_{u}$ can be understood as local inhomogeneities caused by the simultaneous excitation of both air and fuel inlets, in the absence of a change in their mass flow ratios. This means that forcing both inlets proportionally and in-phase will still yield a change in $ϕ_{r}^{'}$ , even though $ϕ_{g}^{'}$ is zero as per equation (12). While an exact explanation for this particular phenomenon is still unavailable, swirl number fluctuations and vortex shedding in the mixing section are possible avenues through which the mixing process can be modulated, allowing the equivalence ratio to fluctuate in the absence of perturbations to the global or nominal composition.

It is also necessary to consider the contribution from each inlet to the total mixture velocity, scaling velocity fluctuations from $u_{f}^{'}$ and $u_{a}^{'}$ with their corresponding mean mass fraction of total inlet mass flow $\bar{Y_{f}}$ and $\bar{Y_{a}}$ , respectively

\frac{u_{r}^{'}}{\bar{u_{r}}} = \bar{Y_{f}} \frac{u_{f}^{'}}{\bar{u_{f}}} + \bar{Y_{a}} \frac{u_{a}^{'}}{\bar{u_{a}}}

(16)

which can then be plugged together with equation (13) into (15), to calculate the contribution from each inlet to the reference equivalence ratio, yielding

\frac{ϕ_{r}^{'}}{\bar{ϕ_{r}}} = \underset{H_{a}}{\underset{⏟}{(\bar{Y_{a}} H_{u} - H_{g})}} \frac{u_{a}^{'}}{\bar{u_{a}}} + \underset{H_{f}}{\underset{⏟}{(\bar{Y_{f}} H_{u} + H_{g})}} \frac{u_{f}^{'}}{\bar{u_{f}}}

(17)

The mixing transfer functions are shown in Figure 4, as well as $H_{a}$ , as described in equation (17). As can be seen, the low-frequency limit of $H_{g}$ approaches unity for all cases, with a decaying gain as frequency increases, in accordance with expectations.

Figure 4.

Mixing transfer functions $H_{u}$ (red line), $H_{g}$ (blue line) and, $H_{a}$ (black line) from LES and MISO SI, with $σ = 1.96$ confidence intervals. Mean burner tube velocity: $30 m / s$ (left), $35 m / s$ (middle), and $40 m / s$ (right). LES: large eddy simulation; MISO: Multiple-Input Single-Output; SI: system identification.

For the case of $H_{u}$ , zero gain is observed for low frequencies. This low-frequency behavior tracks with expectations, as an increase in mean mass flow that is proportional for both feeding lines should not cause any perturbations of the equivalence ratio, after an initial transient phase. Then, gain increases with frequency until a maximum is observed for both cases with 30 and 35 $m / s$ in mean velocity, but not yet reached for 40 $m / s$ . Of particular note, the maximum gain as well as the frequency at which it is observed are scaled with higher mean velocity.

For all cases at $f > 100 Hz$ yields $| H_{a} | > 1$ , hence a fluctuation in air velocity results in an even greater fluctuation of $ϕ_{r}$ . This, at first glance, seems paradoxical, given that the fuel mass flow remains constant. However, this explanation only considers a fluctuation in the global equivalence ratio $ϕ_{g}$ , which is convected and diffused to the reference position. If, in addition to this, we consider the effect from $H_{u}$ , the superposition of these two mechanisms can yield a gain in excess of unity in the equivalence ratio. A deeper investigation remains to be done on the nature of $H_{u}$ , to strengthen its physical interpretation, as well as understand its behavior at high frequencies.

Technically premixed FTF

Having identified $H_{a}$ and $H_{f}$ , they can be plugged into equation (2) to identify the distinct contribution that each inlet makes to the total flame response

\frac{{\dot{Q}}^{'}}{\bar{\dot{Q}}} = F_{a} (ω) \frac{u_{a}^{'}}{\bar{u_{a}}} + F_{f} (ω) \frac{u_{f}^{'}}{\bar{u_{f}}}

(18)

The formulation shown in equation (18) is more convenient for use in an LOM than that of equation (2), since all hydrodynamic effects that are not captured by the LOM are implicitly embedded in the FTF. If experiments had a stiff injector, the TP FTF should be compared directly to $F_{a}$ (as $u_{f}^{'} = 0$ ). To test the hypothesis that the experiments saw non-stiff injectors, this comparison is shown in Figure 5. The numerical results exhibit the correct zero-frequency limit, with the contributions from velocity and equivalence ratio canceling at this point, leading to zero gain and an arbitrary phase. At frequencies below 200 $Hz$ , a good agreement in phase and a similar trend in gain across all cases instills confidence in the physical accuracy of simulations. The approximate shape of the gain curve is predicted correctly, despite having more pronounced local modulations.

Figure 5.

FTFs $F_{a}$ (black line) and $F_{f}$ (red line) from LES and MISO SI, with $σ = 1.96$ confidence intervals and harmonic forcing (∙). FTF $F$ $(\circ)$ from experiment, technically premixed condition. Mean burner tube velocities: $30 m / s$ (left), $35 m / s$ (middle), and $40 m / s$ (right). FTF: transfer function flame; LES: large eddy simulation; MISO: Multiple-Input Single-Output; SI: system identification.

The gain curves show slight shifts in local maxima and minima. For the case with 30 $m / s$ , peak gain is observed at 170 $Hz$ , compared to around 220 $Hz$ for experiments, and the second local minimum is shifted to 525 $Hz$ , compared to 580 $Hz$ . The pattern of predicting the maximum gain at lower frequencies can also be observed in the cases with 35 and 40 $m / s$ . The local minima are predicted to be present at higher frequencies. Finally, the slope of the phase is significantly steeper for all numerical cases, meaning that key time delays are over-predicted compared to experimental results.

As for fuel velocity response $F_{f}$ , a direct comparison to experiments is impossible since this significant quantity is not usually available from experimental measurement. A zero-frequency limit of unity and a low-pass filter behavior are observed in the response to fuel-line fluctuations, with a significant gain at frequencies between 0 and 400 $Hz$ . This behavior agrees with expectations, since the main quantities that control this transfer function are $H_{g}$ and $F_{ϕ}$ , both of which have the same features.

To further validate the consistency of numerical results, harmonic forcing is performed for the reference operating condition 35 $m / s$ at frequencies of 250, 500, and 750 $Hz$ , both for fuel and air. In all cases, harmonic forcing agrees well with broadband simulations, providing confidence in the low influence of noise or non-linearities on the results.

With all this said, the clear pattern that emerges is that the numerical simulations exhibit a qualitative agreement and accurately capture physical trends, but fail to fully predict the behavior observed experimentally. This disagreement can be attributed to several uncertainties associated with the experiments. The temperature profile for the combustion chamber wall, as well as the exact temperature at which reactants reach the test section, was not explicitly measured. Additionally, fuel was not preheated to 573 K in experiments, and while this may not affect mixture temperature significantly, since fuel is only about 3% of total mass, it may affect the properties of the JICF, such as its momentum flux ratio, which in turn could affect mixing properties and the decay of $F_{ϕ}$ .

Nevertheless, as discussed earlier, a fair comparison between the experimental results and the numerical results should involve evaluating the transfer matrix in a network model, employing the extracted transfer functions, and physically superimposing their contributions.

Acoustics

Acoustic network model

To investigate the interaction between fuel and air-line acoustics, a low-order network model of the GE15 was built using the open-source code taX.⁴⁵ Experimental results for the transfer matrices are only available for the case with 35 $m / s$ mean velocity in the burner tube, and as such, this is the only case of concern for this section. For the fully premixed configuration, we construct the acoustic network with a contraction, a duct, and an expansion, with additional duct length added at each termination to account for phase shifts that may result from wave propagation or flow inertia. The technically premixed configuration remains the same, with the singular distinction of adding a T-junction element at an intermediate position in the burner duct, as shown in Figure 6. Initially, in the interest of investigating the isolated effect of the T-junction, we construct the two models for cold conditions, thereby avoiding the need to account for the flame.

The area jumps present in the network model use the $L$ – $ζ$ model^3,46

{[\begin{matrix} \frac{p}{ρ c} \\ u \end{matrix}]}_{d} = [\begin{matrix} 1 & - M ζ - i k L \\ 0 & \frac{A_{u}}{A_{d}} \end{matrix}] {[\begin{matrix} \frac{p}{ρ c} \\ u \end{matrix}]}_{u}

(19)

where

ζ

is the pressure loss coefficient,

L

is an effective length, and

A_{u}

and

A_{d}

denote upstream and downstream areas. However, the

L

value in the area change element itself is set to 0, effectively being replaced by additional lengths up and downstream of the network shown in Figure 7, due to a more straightforward interpretation of their meaning. The values for

ζ

and the lengths up and downstream then need to be inferred from the experimentally measured cold transfer matrices.

Figure 6.

Low-order acoustic network model of the technically premixed burner: contraction–duct–T-junction–duct–expansion.

We then repeat the procedure to infer the coefficients of the technically premixed version of the experiments. Since we are replacing a section of the duct with a T-junction, which is a three-dimensional element, this introduces an additional unknown: the exact position of this component. The T-junction element is also a three-port block, based on the linearized conservation of mass and the Bernoulli equation across two joining streams, but its effect can be modeled as a two-port with the frequency-dependent impedance of the fuel line as

{[\begin{matrix} \frac{p}{ρ c} \\ u \end{matrix}]}_{d} = [\begin{matrix} 1 & 0 \\ Z^{- 1} & 1 \end{matrix}] {[\begin{matrix} \frac{p}{ρ c} \\ u \end{matrix}]}_{u}

(20)

As shown in equation (20), a non-compliant fuel line, characterized by an infinite impedance, would not affect the resulting transfer matrix. Since the impedance at the injector has an associated reflection coefficient, we will refer to this quantity interchangeably.

Figure 7.

Low-order acoustic network model of the fully premixed burner: contraction–duct–expansion.

Figures 8 and 9 show the results for the transfer matrices of the cold, premixed, and technically premixed configurations, respectively. All coefficients of the fully premixed BTM show good agreement between the LOM and experiments across all frequencies, except for $T_{22}$ . This coefficient, which relates acoustic velocity perturbations upstream to those downstream of the burner, exhibits a poor match after 400 Hz, where measurements appear to become slightly less reliable.

Figure 8.

Perfectly premixed burner transfer matrix from taX (——) and experiments $(\circ)$ .

Figure 9.

Technically premixed burner transfer matrix from taX (——) and experiments $(\circ)$ .

Conversely, the agreement in the technically premixed model is slightly worse across all elements. That being said, $T_{12}$ , which relates upstream velocity perturbations to pressure perturbations downstream, shows a similar shape between experiments and the LOM, akin to $T_{22}$ . We note that for the technically premixed case, the measurements exhibit erratic behavior beyond 400 Hz; therefore, we refer only to qualitative trends. As can be observed, the behavior in experiments differs wildly across both configurations, even though the only difference is the opening of the injector holes, given that the cold measurements did not include any fuel flow. The parameters inferred in the optimization procedure also support this observed disparity, with the pressure loss coefficients being negligible in the case of the technically premixed model. Also, no additional length is necessary to account for phase differences upstream and downstream of the network, as indicated by the almost constant value of the $T_{11}$ coefficient, which relates pressure fluctuations from upstream to downstream of the burner. This coefficient has a constant value in the technically premixed case, while in the premixed configuration, it has a negative component that grows with frequency until it dominates, analogous to the $L$ value in equation (19). The only coefficient that remains similar in both experiments is $T_{21}$ , which relates pressure perturbations upstream to velocity downstream the burner. This final coefficient remains zero across both configurations for most of the frequencies of interest, and therefore also has an arbitrary phase value.

Figure 10 shows the reflection coefficient of the fuel line for the non-reacting case. In this configuration, the fuel line was modeled as a simple duct with an arbitrary reflection coefficient at the boundary, and we infer the value of the length and the reflection coefficient by fitting the transfer matrix of the model with the transfer matrix of experiments. The inferred value of the upstream reflection of the fuel line has a value of $R_{f} \approx - 0.6$ , and a duct length of $L_{d} \approx 0.35 m$ . These parameters suggest that the upstream boundary is relatively soft, with a significant phase shift at higher frequencies, and significant acoustic losses due to a non-ideal reflection.

Figure 10.

Technically premixed burner-flame transfer matrix from taX (—) and experiments (˚).

Finally, we repeat the optimization procedure for the hot, technically premixed case, as shown in Figure 11 for the resulting transfer matrix. For this case, we plug the FTFs inferred in the previous section and compute their superposition by extracting the overall transfer matrix. The reflection coefficient for this case is also shown in Figure 10.

Figure 11.

Inferred fuel-line reflection coefficient for cold (black line) and hot (red line) technically premixed burner.

The fuel-line impedance again changes significantly with respect to the non-reacting case. The significant change in behavior arises from the highly complicated acoustic and hydrodynamic behavior of the JICF. The inferred impedance implicitly captures this effect, but it can also be done by modeling the complete fuel line, adding pressure losses and further lengths that mimic this behavior. A more preferable method to model the behavior of this injector with fuel jets is to use Howe’s model of the Rayleigh conductivity⁴⁷ for an orifice with bias flow. However, this lies beyond the scope of this paper.

The fuel line in the reacting case seems to have a relatively anechoic behavior at very low frequencies, as indicated by a reflection coefficient close to 0, allowing for some additional contributions from the $F_{F}$ . As the frequency increases, the equivalent reflection coefficient approaches a stable magnitude of $R_{f} \approx 1$ , indicating that the fuel line becomes less compliant at higher frequencies. It is worth noting that the agreement between the model with inferred parameters and the experiments is not very accurate in this case either.

Nevertheless, a qualitative agreement between the BFTMs can be observed through the superposition of the FTFs identified in the previous section, as illustrated by the coefficient $T_{22}$ in gain and phase up until 400 Hz. This analysis reveals that the assumption of a non-compliant injector leads to inconsistencies in the acoustic characterization of a burner, and that precise modeling of the fuel line is necessary in cases where compliance is presumed to have played a significant role in the system’s dynamics.

Conclusions and outlook

In this work, we studied the dynamics of a technically premixed flame to understand the effect of fuel injector impedance on the acoustic characteristics of the system. By thoroughly analyzing the total contribution of each inlet as well as the mechanisms through which they affect the flame, a deeper knowledge of industrially relevant combustion systems is achieved. Specifically, the principal novelty of this paper lies in the investigation of fuel-line velocity fluctuations and an assessment of their contribution to flame dynamics.

The GE15 combustor was thoroughly examined at multiple operating points. A high-fidelity LES was set up and validated, leading to a good prediction in the mean flame shape and position. A time series of the forced heat release rate, as well as reference velocity and equivalence ratio, was generated, demonstrating good agreement with the experimentally measured fully premixed FTF, further validating the numerical results.

Another significant novelty found in this work was the seemingly paradoxical behavior of fuel transport, where air fluctuations can cause an equivalence ratio perturbation with a gain exceeding unity. This was conceptualized as the superposition of two distinct mechanisms that affect the equivalence ratio at the reference position. Future work should be conducted to investigate the mixing section directly and understand this phenomenon, thereby linking it to a hydrodynamic interaction between a swirler and an injector.

Results for the technically premixed FTF demonstrated theoretical consistency and qualitative agreement within a limited frequency range with the experimental results. Numerical results show excess gain in the range from 160 to 700 Hz, while a steeper slope is observed in the phase in the same range. Discrepancies between numerical simulations and experiments are attributed to the modeling of acoustic behavior in the injector that is not captured by incompressible simulations. However, significant uncertainties exist in boundary conditions, such as wall temperatures in the combustion chamber and the inlet temperature of reactants, which could alter the time lag from the base to the center of gravity of the flame, resulting in a different phase slope.

Finally, LOMs for the acoustics of the cold fully premixed, cold technically premixed, and hot technically premixed burners were constructed to study the acoustic transfer behavior of the configurations. Where relevant, the effective reflection coefficient of the fuel feed line was estimated, and its influence on the transfer matrix was examined. Both experiments and LOM predictions revealed significant differences between the cold fully premixed and technically premixed configurations, indicating that the inclusion of the injector had a non-trivial acoustic effect, which would not be present in the case of a non-compliant injector. Furthermore, the fuel line’s behavior changes markedly when mean flow through the injector becomes significant, as evidenced by the contrasting impedance between the cold no-flow case and the reacting case with flow. Moving forward, studies of technically premixed flames should ensure injector stiffness across all operating conditions. If this cannot be guaranteed, a detailed acoustic characterization of the fuel line as well as the flame response to fuel flow rate perturbations is required to describe completely and accurately combustor configurations of applied interest.

Footnotes

Acknowledgments

We wish to thank Layal Hakim, Sven Bethke, Samir Risa, Stefano Orsino, and Naseem Ansari for the discussion of results and advice on the setup of simulations. The authors gratefully acknowledge Jan Kaufmann and Manuel Vogel for helpful discussions and for providing experimental data. The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for providing computing time on the GCS SuperMUC-NG at Leibniz Supercomputing Centre (www.lrz.de).

ORCID iDs

Nicolas M Garcia

Marcel Désor

Joachim Ottinger

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research has been funded by GE Vernova, PO Numbers 740249957 and 740251162. Funding has also been provided by the German Federal Ministry for Economics and Climate Action under the Federal Aeronautical Research Program (LuFo VI, Call 3) under grant 20M2264C within the OptTuGen project.

Declaration of conflicting interests

The authors declare no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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