Effect of air preheating temperature on the dynamic
behavior of a swirled spray flame

Marie Truffot$^{a,b,*}$, Antoine Renaud$^{a}$, Franck Richecoeur$^{a}$, Laurent Zimmer$^{a}$, Yoann Méry$^{b}$

Abstract

The effects of the air inlet temperature on the dynamic behaviour of a spray flame are reported for different air flow
rates. Dodecane droplets are generated through a pressurised nozzle and introduced in the combustion chamber
fed with air via two co-rotating swirlers. The flow rate of dodecane is kept constant (1.24 g s⁻¹) while the air
flow rate is changed from 28 g s⁻¹ to 35 g s⁻¹ and its temperature from 70 °C to 220 °C. Microphones are used to
record the different dynamics observed while a camera and a Phase Doppler Anemometry system allow to monitor
the flame and the fuel spray.

For low flow rates (between 28 g s⁻¹ and 32 g s⁻¹, leading to an overall equivalence ratio ranging between 0.6 and 0.7), the pressure fluctuations start by increasing with an increase of the temperature until, for a given temperature, the pressure fluctuations suddenly drop. This threshold temperature depends on the air flow rate. When displaying the pressure fluctuation level as a function of the bulk velocity (taking both air flow rate and temperature into account), the curves collapse into a similar shape. The instability frequency monotonically increases with the bulk velocity. For high flow rates (between 32 g s⁻¹ and 35 g s⁻¹, leading to overall equivalence ratio ranging between 0.5 and 0.6), a hysteresis phenomenon is found. Pressure fluctuations are relatively high for constantly decreasing temperature conditions and have almost similar levels as those obtained for lower flow rates. When constantly increasing the temperature, the levels are much lower, showing a bi-stable behaviour for the pressure fluctuations at a given operating point. A simple analysis based on the acoustic energy balance equation allows to qualitatively retrieve the behavior of the pressure fluctuations amplitude and frequency for the low flow rate cases.

Keywords: Spray combustion; Thermo-acoustics; Preheating

<footnote>$^{a}$EM2C Laboratory, CNRS-CentraleSupélec-Université Paris Saclay, 8-10 rue Joliot Curie, 91190 Gif-sur-Yvette, France</footnote>

<footnote>$^{b}$Safran Aircraft Engines, Site de Villaroche, Rond-Point René Ravaud, 77550 Moissy Cramayel, France</footnote>
# 1. Introduction

Engine manufacturers increasingly favour a lean premix combustion regime to tackle the environmental challenges and stricter regulation. The fuel's multi-injection in recent reacting systems leads to a more uniform distribution while limiting long mixing chambers. A small central pilot is used for sustainable flames to enable easier anchoring while preventing blow-off issues. However, this regime has some drawbacks, such as developing instabilities. It is then of strong interest to understand these systems' dynamics to characterise and eliminate the development of these combustion instabilities (CI, [1]). Thus, several recent studies have highlighted that the preheating airflow temperature plays a crucial role in triggering CI, even with gaseous fuels. The temperature tends to increase the velocity by thermal expansion and therefore reduces the delay between injection and flame ([2]). An increase in the airflow temperature has been shown to broaden the equivalence ratio range where CI occur [3]. In many cases, once CI is triggered, further increasing of the temperature did not lead to situations for which the oscillations decrease. The temperature mainly acts as a criterion for bifurcation that separates the two regions: stable or unstable [4].

While this is already difficult to predict in gaseous conditions, the situation becomes more complex with liquid fuel as vaporisation plays a key role in the way combustion occurs for spray flames. It is shown to impact flame stabilisation, pollutant emissions, and flame dynamics. The vaporisation of the fuel leads to additional time delays that may have positive or negative impacts on the development of CI [5]. A recent study, [6] shows the importance of the fuel composition in triggering instabilities while preheating the airflow. Three different single component hydrocarbons have been studied by slightly changing the air temperature from 383 K to 483 K. It is found that the swirl stabilised flame changes from stable to mildly unstable to unstable for a change within 40 K for n-decane. Using multi-component fuels will lead to several delay times, corresponding to the relative evaporation of the different components. Experiments with kerosene [7] lead to situations for which an increase in the inlet temperature is accompanied by a decrease in the pressure fluctuations. The switch between unstable and stable conditions happens for a temperature of 570 K. Therefore, for liquid fuel, it has been shown that even if CI is triggered, further increasing the temperature leads to situations for which the pressure oscillations decrease. As seen, a change in the temperature of the air may have positive or negative impacts.

Injecting the fuel at different locations to achieve lean premixed conditions may further complicate the dynamics as several time delays will exist, depending on the injection location. In this case, fuel staging may lead to several mean flame shapes depending on the fuel split strategy and the way to achieve it ([8],[9]).

Such bi-stable conditions may also occur in more

simple situations, which would allow a more extensive analysis of the conditions leading to such bi-stability. Furthermore, in many studies, the bifurcation point is usually chosen as being the stoichiometric ratio. The purpose of the current study is to investigate the temperature as being the trigger for a bifurcation in a simple hollow cone spray nozzle. The characteristics of the droplets will be kept constant while air flow rate and temperature will be systematically varied around a nominal operating point. In the following of the paper, the setup will be first described. Steady behaviours of the system for fixed air flow rate and temperature will be first described before analysing transient cases in which temperature is varied.

# 2. Materials and methods

## 2.1. Experimental set up

The experimental device (called BIMER, [9]) is equipped with a multipoint staged injector based on an industrial geometry where fuel and air can mix before the combustion chamber. The injector connects the plenum with a rectangular chamber ($500 \times 150 \times 150 \text{ mm}^3$) equipped with two quartz walls providing optical access for the observation of the combustion region. Three stainless steel water-cooled walls complete the chamber. The airflow is controlled with a Bronkhorst In-Flow mass flow controller (less than 2% relative uncertainty for the considered operating points) and preheated with a Sylvania SureHeat Max device.

The injector comprises two stages, each including its own air swirler and fuel injection. The upstream stage, called the pilot stage, contains an 18-vane swirler, with a geometrical swirl number of 0.6, that is designed to allow around 13% of the global air flowrate to flow through this stage. The fuel injection is made with a pressurized nozzle (MMP 6390-0-0-03) associated with a controlled fuel flow (Bronkhorst Mini Cori-Flow, 0.2% relative uncertainty). The central injection nozzle creates a hollow cone of droplets with a 30° half angle. The downstream stage, named the take-off stage, contains a 20-vane swirler, with a geometrical swirl number of 1, through which the remaining 87% of the air flows. In this study, no fuel is injected through this stage. The diameter of the outlet of the complete injection system is 50 mm. A 3D-sketch of the supply line and the combustor is shown in figure 1.

The liquid fuel (pure dodecane in the present case) is only injected through the pilot stage with a fuel flow rate of $1.24 \text{ g s}^{-1}$, which allows the conservation of the initial properties of the droplets through the different operation points. The thermal power generated by the combustion is then kept constant at around 55 kW. The equivalence ratio is varied from 0.5 to 0.7 by changing the air flowrate from $35 \text{ g s}^{-1}$ to $28 \text{ g s}^{-1}$ with $1 \text{ g s}^{-1}$ steps. For each operating point, the airflow temperature varies from around $70 \text{ }^{\circ}\text{C}$ to $220 \text{ }^{\circ}\text{C}$.
<figure><img src="image_1.png" /></figure>

<figure><img src="image_2.png" /><figcaption>Fig. 1: a. 3D sketch of the BIMER test bench. b. Detail of the swirler and fuel injection system.</figcaption></figure>

## 2.2. Diagnostics

The test bench is equipped with several diagnostics to study the flame dynamics. A K-type thermocouple monitors the airflow temperature inside the plenum. Pressure fluctuations are measured in the plenum and along the chamber axis with four microphones (B&K 4938, water-cooled and located in a semi-infinite tube configuration, M1-4 in Fig. 1). For the current analysis, only signals from M2 will be used. This microphone is placed inside the combustion chamber and close to the inlet, where the flame stabilises. A photomultiplier (Hamamatsu H10722-110) is placed at 90° from the combustor axis to collect the light emitted from the beginning of the chamber, where the flame is located. It is equipped with a bandpass filter centered around 310 nm to focus on OH* emission. Both temperature, OH* emission and pressure fluctuations signals are recorded at a sampling rate of 20 kHz for the steady plenum temperature states (section 3) and 10 kHz during the transition cases (section 4).

The fuel spray is characterized by Phase Doppler Anemometry (PDA) system made by the TSI corporation. The PDA measurements are carried out 25 mm from the bottom of the chamber and spaced vertically by 2.5 mm. The acquisition is set to stop after the measurement of 5000 drops, knowing that the detection frequency ranges from 100 drops per second in the sparsely populated areas to 8000 drops per second in the denser regions of the spray. The acquisition duration therefore changes from 625 ms to 50 s depending on the probed point but, since both values are much higher than the characteristic flow time-scales

found in the experiment, convergence of the average is validated.

A high-speed camera (Photron Fastcam SA-5) is used to record images of the flame at a rate of 7 kHz. The camera has a resolution of 1024×640 pixels, and images are acquired at 90° from the chamber axis with a 60 mm AF-DC Nikon lens and an aperture of 2.8 for mean shape measurements. This setting allows imaging the full height of the chamber at the exit of the injector. No interferential filter is used in order to avoid a detrimental decrease of the light signal. No inverse Abel transform has been applied to those images and results will be provided only on a relative intensity scale.

## 3. Flame response at various airflow temperature

In this section, the flame behaviour for steady conditions is investigated. Three values of air mass flow rate (30 g s⁻¹, 32 g s⁻¹ and 35 g s⁻¹, referred as OP30s, OP32s and OP35s respectively) are chosen and, for each of these flow rates, several different electric heater power are set. This gives different values of preheating temperature, defined as the temperature measured by the thermocouple placed in the plenum. Each experiment is carried out in a steady state when the preheating temperature variation does not exceed 1 °C within the recording duration (1 min). Overall, more than 100 experiments are conducted with this protocol.

Signals from the microphones and the thermocouple are sampled at a frequency of 20 kHz for one minute. The mean values of the plenum temperature and the fluctuations (RMS) of M2 are computed and plotted in Fig. 2, where each dot represents one experiment.

<figure><img src="image_3.png" /><figcaption>Fig. 2: Effects of the preheated air temperature on the pressure fluctuations for three different air flowrates.</figcaption></figure>

A large range of preheating temperature is investigated, from 70 °C to 220 °C. For each air flowrate, the overall amplitude of pressure fluctuations in the chamber increase with temperature and reach a peak before decreasing suddenly back to lower levels. For 35 g s⁻¹, the fluctuations are always weaker than for the other investigated flow rates. On the contrary,
30 g s⁻¹ and 32 g s⁻¹ have similar fluctuations levels with a peak of overall pressure fluctuations around 650 Pa. Their shapes evolve similarly with temperature as if the change in the air flowrate involves only a horizontal shift to the curve. The position of this peak shifts to lower preheating temperature with an increase in the air flowrate. The levels reached by all conditions for higher temperatures are of the order of 100 Pa. Thus there is a minimum preheating temperature for all operating conditions leading to low fluctuations levels. This lower-limit temperature also seems to depend on the air flowrate.

The increase in the preheating temperature may impact both the droplet diameter and the air velocity at the inlet of the burner. Thus, the modifications of some properties of the spray could be the reason for the sudden drop in pressure fluctuations. To investigate the changes in the properties induced by a change in the temperature, the droplet velocity profiles measured by PDA at different preheating temperatures are reported in Fig. 3. The position of the chosen datasets in the temperature vs pressure fluctuations plane is also highlighted in Fig. 2.

<figure><img src="image_4.png" /><figcaption>Fig. 3: Profile of the axial and radial velocity for 30 g s⁻¹ measured at 25 mm from the dump plane for different air preheating temperatures.</figcaption></figure>

Whatever the preheating temperature, the velocity profiles shown in Fig. 3 present a characteristic shape for swirling flows: they show that the spray is diverging and the velocity droplet profiles are governed by the presence of the internal recirculation zone and the external recirculation zone. These two zones confine the drops so as to create a conical spray where the axial speeds are high. The major difference between

the different conditions is the value of the maximum droplet velocity, which increases with air temperature. Indeed, at a constant mass flow rate, the increase in air temperature leads to an increase in air-flow velocity. Taking this effect into account allows the collapse of the velocity curves. Thus, the differences seen on Fig. 3 are due to the thermal expansion of the air entering the combustion chamber.

The PDA system also gives information on the diameter of the measured droplets. The average diameter ($D_{10}$) as well as the Sauter Mean Diameter SMD ($D_{32}$) profiles are plotted in Fig. 4. The profiles show a very small decrease in $D_{10}$ and $D_{32}$, of the order of 2 µm, when the preheating temperature increases from 140 °C to 205 °C. To check the relevance of such differences, the repeatability associated with the PDA diameter measurement is calculated using three separate measurements taken at 180 °C. It is found that a typical fluctuation around 2.8 µm is found within an experiment (keeping all parameters identical). Thus, given the order of magnitude found for the different preheating temperatures compared to the measurement variability, the reduction in diameter with the air temperature increase may be considered negligible. Thus, it seems that the diameter of the droplets ($D_{10}$ and $D_{32}$) remains generally constant when the preheating temperature varies. Furthermore, by taking into account the measurement fluctuations, it appears that the distribution of the diameter of the droplets inside the spray can be considered quite homogeneous.

<figure><img src="image_5.png" /><figcaption>Fig. 4: Sauter Mean Diameter for 30 g s⁻¹ measured at 25 mm from the dump plane for different air preheating temperatures.</figcaption></figure>

To check if the different pressure fluctuations levels can be explained by different flame anchoring mechanisms, data from the high-speed camera is then analyzed. On average, the different conditions investigated (air flow rates and temperatures) lead to similar mean spontaneous emission. Figure 5 presents flames recorded for maximum and minimum pressure fluctuations obtained with a flow rate of 30 g s⁻¹.
These conditions correspond to the circles in Fig. 2, with fluctuations of 620 Pa for 178 °C and 100 Pa for 206 °C. For both conditions, the figure shows only half of the combustor with a signal scaled from 0 to 1, 1 being the maximum within the two images.

<figure><img src="image_6.png" /><figcaption>Fig. 5: Mean image for 30 g s⁻¹ at 178 °C (top) and 206 °C (bottom) preheated air temperature</figcaption></figure>

For both cases, the flames show a diverging shape and light emission can be observed already at the outlet of the injector. These two observations indicate a flame stabilization close to the nozzle in a V shape, typical from pilot flames [10]. Furthermore, the maximum intensity is detected at positions close to the injector outlet. The similarities between the two images indicate that the flame keeps the same overall structure, regardless of pressure fluctuations (around 620 Pa for 178 °C compared to 100 Pa for 206 °C). Therefore the sharp decrease of pressure fluctuations at high temperature in *OP30s* and *OP32s* is not due to a change of overall flame structure as could have been observed in other works.

The previous results have shown that no strong differences exist on average between the different preheating temperatures, from the spray and flame points of view. To investigate dynamical phenomena, spectra from the microphone and photomultiplier signals for 178 °C and 206 °C at 30 g s⁻¹ of air are presented on Fig. 6.

At 178 °C, the spectrum of the pressure fluctuations signal (in red Fig. 6) shows a strong peak at 322 Hz, reaching 150 dB. This peak is linked to the strong pressure fluctuations observed for this operating point. The spectrum of OH* emission also shows a strong peak at this frequency, indicating an interaction between pressure and heat release rate. This is the trace of a strong thermo-acoustic instability and its frequency is close to the quarter-wave acoustic mode of the chamber, as observed in previous experiments on the same setup [9] and often found in combustors

<figure><img src="image_7.png" /><figcaption>Fig. 6: Power Spectral Density of the microphone (Micro, red) and OH* photomultiplier (PM, black) signals for 30 g s⁻¹ of air at 178 °C (top) and 205 °C (bottom) preheating.</figcaption></figure>

open to the atmosphere at the outlet. For 206 °C, a dominant peak around 350 Hz can still be observed in the microphone spectrum but its amplitude is just above 120 dB, compared to 150 dB earlier. So, while there is still a thermo-acoustic instability, its amplitude is really small which is linked to the low value of pressure fluctuations recorded for this operating point. A secondary peak can be found in both the microphone and photomultiplier spectra at 1579 Hz. Previous studies have linked this frequency to the presence of the Precessing Vortex Core (PVC); which disrupts the fuel droplets as they are injected which, in turn, mark the flame through fluctuations in heat release rate, as demonstrated in [11]. This phenomenon can also be detected at 178 °C, with a slightly smaller frequency than the 206 °C case due to lower thermal expansion.

## 4. Effect of continuous airflow temperature variation

### 4.1. General behavior for different air flow rates

The previous results were obtained with successive and independent steady air temperature experiments. In this section, we investigate the flame dynamics under transient conditions with eight different air mass flow rate values, ranging from 28 g s⁻¹ to 35 g s⁻¹ with a 1 g s⁻¹ step. We generate several decreases and increases of the plenum temperature for each flow rate by reducing or increasing the electric heater power during recording of the test. The preheating temperature, then, varies over a range of around 100 °C within the recorded duration of 2 min. The sampling frequency of microphone and thermocouple signals is equal to 10 kHz allowing for a longer recording duration.

Figure 7 presents the pressure fluctuations versus the plenum temperatures. For the temperature ramp cases, the temperature and microphone signals are divided in 2 s-long windows. During this short duration, temperature changes are below one degree. The average temperature and the RMS of pressure fluctuations
are computed for each of these windows. The results labelled $OPxx^+$ correspond to measurements performed with an increasing air temperature, whereas those labelled $OPxx^-$ correspond to decreasing temperatures. In addition, we report the values obtained during steady-state experiments (red circles), as discussed in the previous section.

For the OP30 cases, the transient pressure fluctuations match the steady-state measurements, and the results are similar for increasing and decreasing conditions. For the OP32 cases, the transient pressure fluctuations show two close but distinct shapes where the steady states correspond to the lower state, obtained with increasing temperatures. It is worth noting that the fluctuations for the lowest and the highest temperatures are identical for increasing/decreasing temperatures. The presence of these two states is even more noticeable at OP35 where the peak fluctuation of the lower states is half the values of the higher state. As for the previous flow rates, the steady-state corresponds to the lower state obtained with increasing temperatures. It is worth pointing out that the steady states have been obtained without any preferential direction for the temperature variation required to reach the recorded operating points presented in the previous section.

<figure><img src="image_8.png" /><figcaption>Fig. 7: Comparison of the measured pressure fluctuations between steady-states and dynamic variations of the temperature as function of three different air flowrates.</figcaption></figure>

Systematic measurements with the selected eight air mass flow rates show that a hysteresis phenomenon between increasing an decreasing temperatures appears for the highest flow rates between $32 \text{ g s}^{-1}$ and $35 \text{ g s}^{-1}$. This phenomenon is still un-

der investigation and, in the rest of the paper, focus is put on the flow rates between $28 \text{ g s}^{-1}$ and $32 \text{ g s}^{-1}$ where the system's response is unique and independent on the sign of temperature variation.

## 4.2. Scaling for lower flow rates conditions

Changing the air flow rate between $28 \text{ g s}^{-1}$ to $32 \text{ g s}^{-1}$ leads to a variation of the equivalence ratio between 0.7 and 0.6. For this group, there is no difference between increasing and decreasing temperature ramps and the curves all look quite similar with an increasing then decreasing behavior of the pressure fluctuations with respect to temperature. However, one can see in Fig. 2 and 7 that the position of the curve's maximum is dependent on the chosen air flow rate and that temperature may therefore not be the right parameter to investigate the underlying physical phenomenon. A first hypothesis is that the plenum temperature mainly induces a modification of the flow velocity that then drives the flame dynamics. In order to test this, a mean bulk velocity is computed for all cases in order to take into account the increase in the volumetric flow rate and the effects of thermal expansion induced by higher plenum temperatures. This bulk velocity is based on the overall cross-section of the injector (with a diameter of 50 mm).

<figure><img src="image_9.png" /><figcaption>Fig. 8: Pressure fluctuations for the lower flowrates cases with respect to the mean bulk velocity for the inlet conditions.</figcaption></figure>

Figure 8 shows the evolution of pressure fluctuations with the bulk velocity. It can be seen that almost all the curves collapse to a single one, the only exception being the OP28 cases, for which the decreasing part occurs slightly before the other cases. Remarkably, the shapes are all similar, with a substantial decrease observed after the peak for high bulk velocities where a bulk velocity increase smaller than 6% (from a bulk velocity of $19.5 \text{ m s}^{-1}$ to $20.5 \text{ m s}^{-1}$) induces pressure fluctuations to drop by a decade. On the other side (lower bulk velocities), the slope is less sharp, and pressure fluctuations levels do not reach the lowest values achieved for high velocities. The self-similarity with a bulk velocity scaling shows the importance of fuel transportation to the flame front to explain the dynamics of those flames. As the mean flame shape is not changed, it is expected that the evaporation front will be relatively similar for all those conditions. Therefore, a similar bulk velocity
implies a similar convective time of the droplets to the flame front, which then appears to be the governing parameter here.

To further investigate the dynamics of this set of flames, the frequency of the instability (taken as the maximum peak in the power spectral density of each 2 s-long window of microphone signal) is computed and plotted versus the bulk velocity on Fig. 9. All curves show that the frequency of the instability increases with the bulk velocity. This occurs in two steps: first, the increase is limited until around 20 m s⁻¹ where the slope becomes steeper. This breaking point matches the region of sharp decrease of the instability amplitude found in Fig. 8. The curves appear shifted vertically with the case at 28 g s⁻¹ showing the highest frequencies and the case at 32 g s⁻¹, the lowest frequencies for a given bulk velocity. This is probably an effect of the higher equivalence ratio for the lower air flow rates. Indeed, in this lean region, a higher equivalence ratio means a higher flame temperature and therefore a higher combustor temperature. Given that the acoustic resonant frequency scales with the speed of sound which depends on the square root of the temperature, obtaining higher frequencies for richer (and therefore hotter) cases is not unexpected.

<figure><img src="image_10.png" /><figcaption>Fig. 9: Evolution of the instability frequency with respect to the bulk velocity for different air flow rates.</figcaption></figure>

Section 4.3 aims at explaining two features found in Fig. 8 and 9: why the instability amplitude first grows then decreases with the bulk velocity and why the frequency monotonically increases with the same parameter.

### 4.3. Modelling and discussion

The analysis is based on an acoustic energy balance approach. Details of the mathematical derivations can be found in Schuller et al. [12] and only important results are recalled here when necessary. For an instability to grow, the source term in the acoustic energy balance equation (based on the product of pressure and heat release rate fluctuations) must be higher than the damping term (mostly constituted by the acoustic losses at the boundaries of the system). This translates into the following equation:

$$K \cos (\phi_{\zeta} - \phi_{\text{FTF}}) > D. \quad (1)$$

The left-hand side represents the Rayleigh source-term. K is a quantity regrouping, among others, the gain of the flame transfer function $G_{FTF}$ and the modulus of the acoustic impedance $\zeta$ linking acoustic velocity and pressure fluctuations at the flame. $\phi_{\zeta}$ and $\phi_{FTF}$ represent respectively the phase of the impedance at the flame and the phase of the flame transfer function. The cosine therefore represents the phase matching between pressure and heat-release rate fluctuations. On the right-hand side, D represents the damping term of the acoustic energy balance equation. D evolves with frequency and has high values away from the acoustic resonances of the system and lower values when one gets closer to a resonant frequency. Incidentally, setting $D = 0$ allows to retrieve the so-called Rayleigh-criterion which provides a necessary condition for the appearance of thermo-acoustic instabilities. It is important to note that all the quantities in this equation are dependent on frequency.

It is possible to simply rewrite Eq. 1 as:

$$\cos (\phi_{\zeta} - \phi_{\text{FTF}}) > \frac{D}{K}. \quad (2)$$

This allows to set a criterion on $\phi_{FTF}$ for an instability to grow. When D is larger than K, the quotient on the right-hand side is above 1 and no solution exist to fulfill the inequality: the system is stable. However, as one gets closer to a resonant frequency, $\frac{D}{K}$ decreases and it is then possible to find values of $\phi_{FTF}$ that fulfill Eq. 2. Finally, moving further away from this resonant frequency leads to $\frac{D}{K}$ increasing again to values above 1 with no possible solutions for the inequality. One can then define an "instability island" which comprises the values of $\phi_{FTF}$ for which the driving term is higher than the damping term. A graphical representation of it is visible on Fig. 10.

<figure><img src="image_11.png" /><figcaption>Fig. 10: Graphical representation of Eq. 2 and Eq. 3 showing the condition for thermo-acoustic instability growth.</figcaption></figure>

To apply this model to the observed phenomena, it is now necessary to determine how $\phi_{FTF}$ evolves with the preheating temperature. A simple time-lag model can be used for the transfer function which
leads to:

$$ \phi_{\text{FTF}} = \omega\tau = 2\pi f\tau, \quad (3) $$

where f is the frequency and τ is the time lag between a velocity perturbation at the injection and the heat release rate response from the flame. This time-lag is mostly driven by the transportation of fuel to the flame zone, which can be expressed as a characteristic length of the flame divided by a convective velocity for the perturbations. As seen on Fig. 5, the flame shape is mostly insensitive to the preheating temperature so the characteristic length can be considered constant. The convective velocity, on the other hand, is expected to scale linearly with the preheating temperature as is the case for the bulk velocity. Therefore, an increase of the preheating temperature leads to an increase of the velocity which in turn implies a decrease in the time delay. This is represented graphically by the three solid lines in Fig. 10.

Analyzing Fig. 10, one can now understand the behaviour of the instability amplitude and frequency as a function of the bulk velocity on Fig. 8 and 9. As the temperature increases, the slope of the FTF phase curve decreases until it enters the instability band on the low-frequency side. When the FTF phase curve crosses the instability island, multiple solutions are possible in theory. However, recalling the meaning of Eq. 1, it is likely that the system will move as close as possible to the center of the instability island to maximize the growth rate in order to reach a higher limit cycle. This does not simply appear with this linear analysis as it requires modelling the evolution of the damping rate with amplitude and using a flame describing function framework for the flame response. As the temperature keeps on increasing, the phase curve moves away from the center of the instability island until it exits it on the high-frequency side. Therefore, from this analysis, one expects to see the amplitude increase then decrease with the bulk velocity and the frequency monotonically increase when the bulk velocity is increased, which is what is found experimentally. This remains however only a qualitative analysis for the moment and many phenomena, such as the change in slope in Fig. 9, remain to be explained. Measuring the flame describing functions and the impedance of the injection system in this setup is expected to lead to a more quantitative analysis.

## 5. Conclusions

For the present spray flame, increasing the preheating temperature leads to increasing and then decreasing pressure fluctuations. It is shown that the spray and the mean flame shape are only weakly affected by the change in preheating temperature although the dynamic behavior is clearly different. Performing continuous sweeps in temperature leads to two different responses depending on the air flow rate. For high air flow rates, it can be seen that the behavior is different depending on the direction of the temperature

change. This hysteresis phenomenon remains to be understood.

For lower air flow rates, the curves of the pressure fluctuations all collapse when plotted as a function of the bulk velocity, indicating that it is the governing parameter rather than the preheating temperature itself. The instability frequency is seen to increase with the bulk velocity, with a noticeable change in slope which occurs when the pressure fluctuations strongly decrease. Qualitatively, these observations can be retrieved by considering a simple model based on the energy balance equation and a time-lag approach for the flame transfer function.

For further investigations, several paths can be followed. Measuring flame describing functions for different preheating temperatures may help in understanding the phenomena at stake. This would however not be sufficient to explain the observed hysteresis phenomenon for the higher air flow rates and, for this purpose, adding more diagnostics would allow to investigate the changes in flame behavior during the experiment.

## Declaration of competing interest

The authors have no competing interest to declare.

## Acknowledgments

This study has received support from SAFRAN AIRCRAFT ENGINES and ANRT (CIFRE n°2020/0067).

## References

[1] T. Poinsot, Prediction and control of combustion instabilities in real engines, Proceedings of the Combustion Institute 36 (1) (2017) 1–28.

[2] J. Bae, S. Jeong, Y. Yoon, Effect of delay time on the combustion instability in a single-element combustor, Acta Astronautica 178 (2021) 783–792.

[3] S. A. Pawar, M. Raghunathan, K. V. Reeja, P. R. Midhun, R. I. Sujith, Effect of preheating of the reactants on the transition to thermoacoustic instability in a bluff-body stabilized dump combustor, Proceedings of the Combustion Institute 38 (4) (2021) 6193–6201.

[4] Y. Huang, V. Yang, Bifurcation of flame structure in a lean-premixed swirl-stabilized combustor: transition from stable to unstable flame, Combustion and Flame 136 (3) (2004) 383–389.

[5] S. Tachibana, K. Saito, T. Yamamoto, M. Makida, T. Kitano, R. Kurose, Experimental and numerical investigation of thermo-acoustic instability in a liquid-fuel aero-engine combustor at elevated pressure: Vability of large-eddy simulation of spray combustion, Combustion and Flame 162 (6) (2015) 2621–2637.

[6] F. Chen, C. Ruan, T. Yu, W. Cai, Y. Mao, X. Lu, Effects of fuel variation and inlet air temperature on combustion stability in a gas turbine model combustor, Aerospace Science and Technology 92 (2019) 126–138.

[7] X. X. Han, D. Laera, A. S. Morgans, Y. Lin, Z. Zhang, X. Hui, C. Sung, Inlet temperature driven supercritical bifurcation of combustion instabilities in a lean premixed prevaporized combustor, Experimental Thermal and Fluid Science 109, 109857 (2019).
[8] S. Hermeth, G. Staffelbach, L. Y. Gicquel, V. Anisimov, C. Cirigliano, T. Poinsot, Bistable swirled flames and influence on flame transfer functions, Combustion and Flame 161 (1) (2014) 184–196.

[9] T. Providakis, L. Zimmer, P. Scouflaire, S. Ducruix, Characterization of the Acoustic Interactions in a Two-Stage Multi-Injection Combustor Fed With Liquid Fuel, Journal of Engineering for Gas Turbines and Power 134 (11), 111503 (2012).

[10] L. C. Mesquita, A. Vié, L. Zimmer, S. Ducruix, Numerical analysis of flame shape bifurcation in a two-stage swirled liquid burner using large eddy simulation, Proceedings of the Combustion Institute 38 (4) (2021) 5971–5978.

[11] A. Renaud, S. Ducruix, L. Zimmer, Experimental Study of the Precessing Vortex Core Impact on the Liquid Fuel Spray in a Gas Turbine Model Combustor, Journal of Engineering for Gas Turbines and Power 141 (11), 111022 (2019).

[12] T. Schuller, T. Poinsot, S. Candel, Dynamics and control of premixed combustion systems based on flame transfer and describing functions, Journal of Fluid Mechanics 894 (2020).