Generalization of Turbulent Swirl Flame Transfer

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Generalization of Turbulent Swirl Flame Transfer Functions in Gas Turbine Combustors a

K. T. Kim & D. A. Santavicca

a

a

Center for Advanced Power Generation, Department of Mechanical and Nuclear Engineering, The Pennsylvania State University, University Park, Pennsylvania, USA Accepted author version posted online: 09 Apr 2013.

To cite this article: K. T. Kim & D. A. Santavicca (2013): Generalization of Turbulent Swirl Flame Transfer Functions in Gas Turbine Combustors, Combustion Science and Technology, DOI:10.1080/00102202.2012.752734 To link to this article: http://dx.doi.org/10.1080/00102202.2012.752734

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ACCEPTED MANUSCRIPT Generalization of Turbulent Swirl Flame Transfer Functions in Gas Turbine Combustors K. T. Kim1,*, D. A. Santavicca1 1

Center for Advanced Power Generation, Department of Mechanical and Nuclear Engineering, The Pennsylvania State University, University Park, Pennsylvania, USA

Downloaded by [Otterbein University] at 05:56 10 April 2013

Corresponding author: K.T. Kim, PhD GE Global Research Center, One Research Circle, Niskayuna, NY, 12309, Tel: 518-387-4049, E-mail: [email protected]

Abstract Experimental investigations of the forced response of swirl-stabilized turbulent flames to upstream flow disturbances were performed in an industrial scale gas turbine combustor operating with natural gas fuel and CO2/air. We measured flame transfer functions (FTFs) for a wide range of forcing frequency over a broad range of operating conditions with 50−120 kW thermal power. A sensitivity analysis was then performed in order to identify the key dimensionless parameters controlling the forced swirl flame dynamics. Two different dimensionless parameters, St1 = ( f δ sw ) U and St2 = ( fL f ) U , are used as dimensionless frequencies, while the dependence of the FTF gain on the turbulent flame speed is taken into account using a dimensionless flame length, ξ = L f Dc . The implementation of the non-dimensionalization strategy using several time and length

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ACCEPTED MANUSCRIPT scales reveals that all FTFs are well characterized by either St1 or St2 , but the best result is obtained from a combination of St1 and St2 , which accounts for the interference mechanisms of vortical and acoustic disturbances in the system. As a consequence, the occurrence of local minima and local maxima, a clear manifestation of destructive and


constructive interferences of acoustically forced swirl-stabilized flames, is well captured by the dimensionless numbers. This methodology is then applied to extensive FTF data measured from a different gas turbine combustor. A comparison of the FTFs for the two different gas turbine combustion systems provides insight into generalized transfer function behaviors in the dimensionless domain. This study focuses on velocity-coupled combustion instability and these generalizations may not extend to situations where other coupling mechanisms dominate.

KEYWORDS: Acoustic forcing; Gas turbine combustion; Flame transfer function; Lean-premixed; Swirl-stabilized

1. INTRODUCTION Flame transfer functions are an essential element of the prediction of acoustically coupled combustion instabilities. In combination with a linear acoustic theory, the flame transfer


ACCEPTED MANUSCRIPT function provides a powerful predictive capability for linear/nonlinear combustion instability characteristics, including the onset, temporal evolution, and limit cycle oscillations in a combustion system. Experiment, theory, and numerical combustion simulations have been extensively performed to obtain flame transfer functions as a


function of forcing frequency and amplitude in laminar and turbulent flows (Balachandran et al., 2005; Palies et al., 2010; Bellows et al., 2006; Preetham et al., 2008; Birbaud et al., 2008; Noiray et al., 2008; Chong et al., 2011; Schimek et al., 2011). In these investigations, the response of a flame to a single or multiple perturbations has been explored in relation to acoustic/convective wave interference mechanisms (Kim et al., 2010a; Tyagi et al., 2007).

One of the open topics in the study of flame dynamics subjected to acoustic excitation is the development of a methodology for non-dimensionalization of flame transfer functions. Schuller et al. (2003) proposed a unified model for the prediction of laminar premixed flame transfer functions for conical and V-flame dynamics based on a linearization of the G-equation. It was found that the laminar flame dynamics is governed by two dimensionless parameters, a reduced frequency,

ω* = ω R ( S L [1 − ( S L / U )2 ]1/ 2 ) ,

and the ratio of

the flame burning velocity to the mean flow velocity, S L / U . Lieuwen (2005) performed


ACCEPTED MANUSCRIPT an analytic investigation of the nonlinear dynamics of premixed flames responding to harmonic velocity disturbances, concluding that the linear transfer function depends on the Strouhal number, St = ω L f U , and the ratio of the flame length to width, β = L f R . Recently, Duchaine et al. (2011) presented direct numerical simulation results on the


sensitivity of laminar flame transfer functions to the variation of flame speed, the expansion angle of burnt gas, inlet air temperature, and inlet duct temperature. They showed that the flame speed and the inlet duct temperature are the key parameters controlling the transfer function phase.

The presence of turbulent flows and swirling flow structures complicates the unsteady processes of combustion/acoustic interactions. It has been shown that a Strouhal-number based non-dimensionalization method is useful for swirl FTFs of turbulent reacting flows (Buchner et al., 1993; Bunce et al., 2011; Palies et al., 2010; Orawannukul et al., 2011).Although relevant FTF data are abundant in the literature, however, it is difficult to extrapolate the existing data to other regimes, even in the same combustor configuration, primarily because the FTF is highly sensitive to each input parameter. A slight modification of an inlet parameter can result in a substantial change in the FTFs, and amplify uncertainties related to linear/nonlinear stability predictions. Therefore, the need


ACCEPTED MANUSCRIPT for strategies to describe generalized FTF behaviors in practical applications has remained a challenging problem.

Here, we report a non-dimensionalization method for turbulent swirl FTFs, which were


determined over a number of inlet conditions in order to accurately evaluate the sensitivity of the FTFs in response to the variation of each inlet parameter, including mean velocity, inlet mixture temperature, equivalence ratio, CO2 concentration, and axial swirler position. The objectives of this paper are to quantitatively describe how the swirl flame response varies with respect to the change of each inlet parameter, to identify critical time and length scales in the context of the sensitivity analysis, and to propose a method for generalizing all measured FTFs using nondimensional numbers which include information on the key physical processes controlling the unsteady dynamics.

2. EXPERIMENTAL METHODS 2.1. Lean Premixed Gas Turbinecombustor An axisymmetric, industrial-scale, lean-premixed gas turbine combustor was used in this investigation. This rig was previously used for velocity-forced FTF measurements at elevated pressure (Bunce et al., 2011) and fuel-forced FTF measurements (Orawannukul


ACCEPTED MANUSCRIPT et al., 2011). As schematically illustrated in Figure 1, this combustor consists of an air inlet section, a siren, a swirl injector, an optically-accessible quartz combustor section, a steel combustor section, and an exhaust section. A siren-type modulation device with a rotating plate and a stator was used to generate acoustic velocity fluctuations at the inlet


of the combustion chamber for a range of forcing frequency. The amplitude of acoustic forcing can be independently varied by controlling the amount of fuel/air mixture that bypasses the siren. The center body has a diameter of 31.7 mm and the outer tube has an inner diameter of 54.1 mm. The center body is centered in the mixing tube, and positioned such that its downstream end is 20 mm recessed from the combustor dump plane. The combustor consists of a quartz cylinder enclosed in a stainless steel box with quartz windows on two sides to allow optical access to the flame. The quartz cylinder is connected to the exhaust which consists of a 911 mm long 102 mm diameter stainless steel pipe with a valve at the end to allow pressurization of the combustor. Detailed dimensions are included in Figure 1.

2.2. Instrumentation And Operating Conditions High frequency-response, water-cooled, piezoelectric pressure transducers (PCB model 112A04) were used to measure pressure oscillations. The two-microphone method was


ACCEPTED MANUSCRIPT employed to measure acoustic velocity fluctuation at the inlet of the combustion chamber. A photomultiplier tube (PMT, Hamamatsu model H7732-10) coupled with a band-pass interference filter (432 ± 5 nm) was used to measure the global CH* chemiluminescence emission intensity from the whole flame. An ICCD camera (Princeton Instruments model


PI-MAX) with a CH* band pass filter centered at 430 nm (10 nm FWHM) was used to record the unforced flame images. A three-point Abel deconvolution method was utilized to reconstruct the two-dimensional flame structure from the line-of-sight integrated intensity information. Experimental data were recorded with a National Instruments data acquisition system controlled by the Lab view software program at a sampling frequency of 8192 Hz. A total of 16,384 data points were taken during each test, resulting in a frequency resolution of 0.50 Hz and a time resolution of 0.122 msec.

The flame transfer function measurements were performed at combustor pressure of 1 atm and inlet mixture temperatures of 100−250 °C, over a range of equivalence ratios from 0.60 to 0.75, a range of inlet mean velocities from 25.0 to 40.0 m/s, and axial swirler positions of 0.077 and 0.096 m. For the present experiments, the fuel used was natural gas. The fuel nozzle was located far upstream of the swirl injector to eliminate any non-homogeneous distributions of equivalence ratio. In addition, CO2 concentrations


ACCEPTED MANUSCRIPT from 0% to 50% were considered in an attempt to investigate the influence of exhaust gas recirculation (EGR) on combustion dynamics. Forcing frequencies were varied between 100 and 500 Hz in 25 Hz steps, and the forcing amplitude was kept constant at 5% to ensure the linear response (Kim et al., 2010b).Nonlinear heat release evolution at high


forcing amplitude is not considered in the present paper.

3. RESULTS AND DISCUSSION 3.1. Swirl Flame Transfer Functions The frequency dependence of the global heat release response was investigated in the lean-premixed gas turbine combustor. Figure 2 shows a typical example of the FTF gain and phase plotted against forcing frequency. The flame transfer function is defined as the ratio of the heat release oscillation ( q ' ) to the inlet velocity oscillation ( u ' ), where both are normalized by their corresponding time-averaged values ( q , u ): FTF ( f ) =

q '( f ) q u '( f ) u

(1)

Auto- and cross power spectral density calculations of the measured velocity and heat release signals were conducted to obtain the FTFs. In Figure 2, multiple data points at a constant forcing frequency denote repeated measurements at the same inlet condition.


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Slightly scattered data distributions are seen in the FTF gain and phase plots, except for the phase plot at forcing frequencies of f = 225 Hz and 475 Hz. It is clearly seen that the transfer function gain is characterized by a periodic occurrence of local minima and local


maxima, a fundamental characteristic of swirl flame transfer functions. It is known that this behavior is caused by destructive and constructive interference mechanisms of upstream perturbations (see Preetham et al. (2008) and Jones et al. (2011)).Furthermore, Figure 2a shows that the first FTF maximum value is larger than the second transfer function maximum, and both gain values exceed unity due to constructive interference. The first local minimum occurs at approximately 225 Hz and the second at about 475 Hz.

The transfer function phase evolves linearly with frequency for 100 ≤ f ≤ 175 Hz and 300 ≤ f ≤ 450 Hz, with the slope of the phase plots being nearly the same for the two regimes.

A non-monotonic evolution of the transfer function phase is, however, observed at the local minima. It is interesting to note that at the local minima, the FTF phase has multiple solutions. Specifically, the FTF phase at the first local minimum, fG min1 , varies between 14.3° and 26.4° when the negative phase values are unwrapped. This behavior is more noticeable at the second local minimum, fG min2 , where the FTF phase varies between 85.2°


ACCEPTED MANUSCRIPT and 126.1°. It is important to note that the multiplicity of the FTF phase values at the local minima is not related to uncertainties of the measurements, but is associated with a fundamental characteristic of acoustically perturbed swirl flame dynamics. Although the amplitude of acoustic excitation, A = 0.05,is sufficient to generate flow disturbances, the


resultant amplitude of the heat release oscillations is nearly zero in the vicinity of the local minimum points, due to the destructive interference of the two competing phenomena. As a result, the FTF phase is not clearly defined at the local minimum points.

The presence of alternating behavior and non-monotonic phase evolution in the frequency domain is the most prominent feature of swirl flame transfer functions. Similar observations have been made in investigations of high-pressure flame transfer functions(Bunce et al., 2011; Schuermans et al., 2009), multi-nozzle gas turbine flame transfer functions (Szedlmayer et al., 2011), and low-velocity swirled flame transfer functions (Palies et al., 2010). Moreover, this dynamic property isnot influenced by swirler geometry, i.e., axial or radial (see Palies et al., 2011a; Hirsch et al., 2005). It is known that this phenomenon is caused by the interference of vortical and acoustic disturbances, which have different propagation speeds and dissipation mechanisms.


ACCEPTED MANUSCRIPT Among several shear-layer induced vortical disturbances, a vorticity wave generated at the swirl vanes is found to be the most important source of convective perturbations (Komarek and Polifke, 2010; Palies et al., 2011b). Recently, this mechanism was demonstrated by Kim and Santavicca (2012),in a comprehensive investigation of self-


excited combustion instabilities in a model gas turbine combustor. In this study, the wave interference mechanisms are explored as a function of axial swirler location, mean nozzle velocity, and oscillation frequency, in the presence of temporal equivalence ratio oscillations.

The FTF presented in Figure 2 was measured at a fixed inlet operating condition. The modification of any input parameter has a significant impact on the FTF gain and/or phase, due to changes in local/global reaction zone structures, acoustic/convective disturbance propagation speeds, dissipation rates, and acoustic/thermal boundary conditions. For example, it is conjectured that inlet parameters related to reactant compositions, such as equivalence ratio and CO2 concentrations, may affect local and global combustion processes, whereas axial swirler position is thought to exert a considerable influence on disturbance convection processes. In the following section, a sensitivity analysis of the flame transfer functions is performed with respect to the


ACCEPTED MANUSCRIPT variation of each input parameter, to identify the dominant time and length scales. Based on the sensitivity analysis, we propose a method for the description of generalized swirl flame transfer functions for two different lean-premixed gas turbine combustors.


3.2. Sensitivity Analysis Numerous FTF data are plotted in Figure 3. The FTFs were measured at 90 different test conditions over a broad range of inlet operating conditions. Inlet mixture temperature ( Ti ), mean inlet velocity ( U ), equivalence ratio ( φ ), CO2 concentration ( X CO 2 ), and axial swirler position ( δ sw ) were varied. For a given inlet condition, the same measurement was repeated 2~4 times. It can be seen from Figure 3 that it is difficult to define a coherent behavior in the frequency domain, due to the different sensitivity of the FTFs to each input parameter. The FTF is by definition dimensionless, but the x-coordinate has a dimension, i.e., time. An important time scale is deduced through a systematic investigation of the linear transfer functions obtained from a number of test conditions. The time scale is then used to non-dimensionalize the forcing frequency.

First, the dependence of FTF gain minimum frequency on mean inlet velocity is presented in Figure 4a. This result highlights the importance of a convection mechanism


ACCEPTED MANUSCRIPT on the acoustically forced swirl flame dynamics. It is clearly seen that the FTF gain minimum frequencies, both fG min1 and fG min 2 , increase linearly with inlet velocity. The rate of increase is nearly constant at ∂fG min1 ∂U = 7.0 Hz/m/s for the gas turbine combustor investigated here. Assuming that the slope remains unchanged up to a high inlet velocity,


the first local minimum may occur at approximately 500 Hz for U = 70 m/s. This indicates that under such a condition the transfer function gain would exhibit a typical low-pass filter behavior for forcing frequencies below ~ 500 Hz. This hypothesis can be demonstrated using FTF data provided by Kim et al. (2010c), in which they found that a swirl-stabilized flame in the convectively compact regime does not exhibit local minima. In addition, it is noted from Figure 4a that the vertical distributions at a fixed bulk velocity are caused by the influence of other parameters, Ti , φ , and X CO 2 . These effects are excluded in the following analysis.

Consider next the acoustic effect. The influence of inlet temperature on the local minimum frequency is illustrated in Figure4b, in which the inlet mixture temperature is converted into the sound propagation speed in the nozzle. This measurement shows that the gain minimum frequency tends to increase gradually with the speed of sound, but the change is not significant relative to the effect of bulk velocity, as illustrated in Figure 4a.


ACCEPTED MANUSCRIPT Specifically, the slope of the first local minimum frequency with respect to the speed of sound is ∂fG min1 ∂ci = 0.64 Hz/m/s, which is one order of magnitude smaller than ∂fG min1 ∂U . As the inlet fuel/air mixture temperature increases from 100 to 250 °C, the

first FTF gain minimum frequency increases by 20%, which is consistent with the rate of


increase in the speed of sound for the variation of the inlet temperature considered here. It can be therefore concluded that the increase in the FTF gain minimum frequency is due primarily to the modification of the acoustic wave propagation mechanisms. It should also be noted, with respect to Figures 4a and 4b, that the second FTF gain minimum frequency is not necessarily an integer multiple of the first FTF gain minimum frequency. This implies that the occurrence of the local minima is not a purely acoustic phenomenon but a mixed convective-acoustic phenomenon.

To better understand the processes of convection of flow perturbations, the swirler was mounted at different axial positions and the ensuing effects on the forced flame response were investigated. Figure4c shows the transfer function gain minimum frequency for two different swirler positions, δ sw = 0.077 and 0.096 m. Here δ sw is the distance between the exit of the swirler and the flame attachment point. Note that different swirler locations may affect unforced flame structures through the modification of the swirl strength at the


ACCEPTED MANUSCRIPT combustor dump plane, which in turn affects the turbulent flame speed and flame angle. However, it was shown by Kim and Santavicca (2012) that the unforced flame structure remains almost unchanged for the conditions considered here. Figure 4c shows clearly that the swirler position itself exerts a significant influence on the linear response. The


FTF gain minimum frequency decreases with increasing δ sw for a fixed mean inlet velocity, because the convection time for vortical disturbances to travel to the combustion chamber increases with δ sw . This observation is physically important because it reveals that a vorticity wave generated at the swirl vane plays a crucial role in developing unsteady flow disturbance fields in a swirl-stabilized combustor. This measurement supports the previous findings of Komarek and Polifke (2010). Furthermore, Palies et al. (2010) showed that when an acoustic wave impinges on the swirler, azimuthal velocity perturbations are generated and the flow disturbances are convected with the mean flow. This interference of vortical and acoustic velocity disturbances in the presence of equivalence ratio oscillations was explored by Kim and Santavicca (2012).

The sensitivity of the FTF gain minimum frequency to equivalence ratio is presented in Figure 4d, which shows that the local minimum values are not sensitive to the variation of equivalence ratio. This implies that the variation of reactant compositions has a


ACCEPTED MANUSCRIPT negligible influence on the frequency dependence of the interference of acoustic/vortical disturbances. As will be demonstrated in Figure 5, however, equivalence ratio and CO2 concentration exert a significant influence on the transfer function gain.


Figures 5a-5c show the effects of inlet mixture temperature, equivalence ratio, and CO2 mole fraction on transfer function gains, in particular the second maximum values. Note that measurements of the first peak ( Gmax1 ) are not always possible due to forcing frequency limitations related to the two-microphone method (Waser and Crocker, 1984). It is expected that the trends observed in the second peak ( Gmax 2 ) to hold for the first peak ( Gmax1 ) as well, and these trends to be representative of the change in flame response amplitude overall. This is confirmed in the subsection 3.3 in the interpretation of generalized transfer function data. It is seen from Figure 5a that the FTF gain at the second peak decreases with increasing inlet temperature. Likewise, Gmax 2 tends to decrease with equivalence ratio, but Gmax 2 increases with CO2 concentration, as illustrated in Figures 5b and 5c, respectively. These phenomenological observations indicate that Gmax 2 correlates strongly with the flame burning velocity: the higher the flame

propagation speed, the lower the FTF gain. This could be attributed to coupled effects of flame length and kinematic restoration. For a short flame, the flame wrinkling amplitude


ACCEPTED MANUSCRIPT may not reach a peak because convective wavelength is larger than flame length. Also, flame wrinkles caused by convective disturbances would be damped faster for higher flame speed mixtures due to kinematic restoration effects, i.e., the flame propagation normal to itself leading to flame wrinkle destruction (Shin and Lieuwen, 2012).This


further suggests that the flame propagation speed is an essential parameter for the description of the flame transfer functions.

Precise determination of the flame propagation speed in practical applications is a challenging problem. In the present investigation, we instead used flame geometric parameters obtained from unforced CH* chemiluminescence images in an attempt to take the flame burning velocity effects into account. Normalized flame length ( ξ ) is defined in Eq. 2: ξ=

Lf Dc

(2)

Here, the diameter of the cylindrical combustion chamber, Dc , is used as a reference length scale. Figure 5d plots Gmax 2 as a function of the normalized flame length, ξ , for all test conditions considered. It clearly shows that Gmax 2 increases with ξ . This parameter is used for generalization of flame transfer functions in the next section.


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3.3. Generalization Of Flame Transfer Functions As discussed above, the sensitivity analysis is critical to provide insights about the velocity-forced flame dynamics. The sensitivity analysis of the flame transfer function


was performed by changing one parameter at a time. It would be interesting if we could consider higher-order effects on the FTFs, which may provide more insights into the flame response mechanisms. Here, two different definitions of Strouhal numbers are used to interpret the generalized FTF behaviors: St1 =

St2 =

f δ sw τ v  τ v τ c  1 =  − = ∆θ vθ −u U T  T T  2π

fL f U

(3)

(4)

where L f = LCH *max (measured from unforced flame images)

Here τ v and τ c are the propagation time of vortical and acoustic disturbances from the swirler to the combustor dump plane, U is the mean nozzle velocity, and ∆θ v −u is the θ

phase difference between azimuthal and acoustic velocity perturbations at the combustor inlet. The first definition, St1 , is the ratio of vortical disturbance convection time to the period of acoustic oscillation, which is equivalent to the phase difference between vortical and acoustic disturbances at the combustor inlet, because the contribution of


ACCEPTED MANUSCRIPT τ v T is one order of magnitude greater than the contribution of τ c T (see Eq. 3). This is

consistent with a comparison of the slopes ∂fG min1 ∂U and ∂fG min1 ∂ci , as presented in Figures 4a and 4b. In the second definition of a Strouhal number, St2 , a characteristic flame length is utilized (see Eq. (4)). The definition of this parameter is similar to that of


the Strouhal number, St = (ω0 L f ) u0 , introduced by Fleifil et al. (1996) and Wang et al. (2009).

The measured flame transfer functions presented in Figure 3 are plotted as a function of St1 and St2 in Figure 6. Note that the definition of St1 does not include parameters related

to combustion-acoustic interactions, but includes only inlet parameters, δ sw and U , together with f . Nonetheless, Figures 6a and 6b show clearly that all measured FTFs over a wide range of inlet conditions are well represented by a single nondimensional number, St1 . Note that the FTF gain is normalized by ξ to account for the dependence of the FTF gain on the turbulent flame speed. The transfer function gain decreases almost linearly with the nondimensional number, St1 , until the number reaches around0.61, at which point non-monotonic phase change occurs, as shown in Figure 6b. The phase evolves almost linearly with St1 , when the number is less than 0.61.


ACCEPTED MANUSCRIPT It is important to observe that all measured FTF gain exhibits the first local minimum when St1 = 0.61. Interestingly, this value is consistent with the measurement of Palies et al. (2011a), in that a Strouhal number, St = fL ucv , defined based on the FTF gain minimum frequency, the distance between the swirler outlet and the combustor dump


plane, and the convective velocity of azimuthal vortical disturbances, is approximately 0.67 for a laboratory-scale swirl combustor. Furthermore, a swirl flame transfer function investigated by Hirsch et al. (2005) yields a Strouhal number of 0.66, when the same definition of the Strouhal number is used. The definition of the Strouhal number considered by Palies et al. (2011a) is equivalent to the definition of St1 considered in the present study. The discrepancy may be attributed to the assumption that the convective velocity is equal to the bulk velocity.

The quantitative agreement of the Strouhal number, St1 , at the first FTF minimum point implies that the characteristic length/velocity scales used can possibly be applied to other, geometrically similar, gas turbine combustion systems. This result emphasizes the importance of the interference of azimuthal and acoustic velocity disturbances to the forced flame dynamics in relation to swirl number fluctuations. Provided that the contribution of the acoustic wave propagation time is included in the definition of St1 (see


ACCEPTED MANUSCRIPT Eq. 3), the phase difference between azimuthal and acoustic velocity disturbances at the first FTF minimum point is approximately 180°. Due to the out-of-phase interference of the convective/acoustic perturbations, the magnitude of swirl number fluctuations is the highest at that point. Local heat release perturbations induced by flame surface


kinematics are counteracted by the swirl number fluctuations. When the rate of rotation is limited, on the other hand, the flame response is amplified, leading to a FTF gain greater than unity.

Convection in the combustion chamber can be described by St2 , defined in Eq. 4. The corresponding FTF data are presented in Figures 6c and 6d. It is evident that the evolution of transfer functions is well characterized by St2 . The presence of local minima and local maxima is well captured in the nondimensional domain. It is noteworthy that in the low frequency region, the FTF phase is well represented by St2 , with a slope of about 7.78. This result implies that irrespective of inlet conditions, the initial slope of FTF phase, which controls the linear stability characteristics of the combustor, can be predicted when the Strouhal number is available (Dowling, 1997; Duchaine et al., 2011).


ACCEPTED MANUSCRIPT Consider next the physical significance of the convective time, L f U , used in the definition of St2 . Figure 7 shows the relationship between the flame’s characteristic response time delay (mean flame response time deduced from the FTF phase) and the convective time used in the definition of St2 . The ratio of flame length to bulk velocity,


L f / U , represents a convective time of a disturbance from the flame base to the maximum

heat release position. As demonstrated in Figure 7, these two time scales are almost linearly related. It should be observed, however, that the slope of the data is approximately 0.89, and therefore the use of L f / U as an indicator of flame’s characteristic response time results in a slight underestimation.

It is now appropriate to consider the combination of the two convection processes simultaneously, i.e., a vorticity wave convection mechanism in the mixing section and the flame’s response to the inlet perturbations. Figure 8presents the normalized FTFs as a function of the combination of St1 and St2 .In the magnitude and the phase plots, the nondimensionalization using St1 + St2 yields better results than the non-dimensionalization using a single dimensionless parameter. This means that the underlying physics of forced swirl flame dynamics is well captured by the dimensionless parameter. Specifically, Figure 8b shows that the normalized FTF phase increases linearly with St1 + St2 in region


ACCEPTED MANUSCRIPT A. In regions B and C, the dimensionless number St1 + St2 varies from 0.8 to 1.7. The FTF phase is characterized by a single value at St1 + St2 = 0.8 and 1.7, while for 0.8< St1 + St2