ONE-DIMENSIONAL MODEL DESCRIBING EIGENMODE FREQUENCY SHIFT DURING TRANSVERSE EXCITATION

A shift in transverse eigenmode frequency was observed in an experimental combustion chamber when exposed to large amplitude acoustic oscillations during oxygenhydrogen combustion tests. A shift in eigenmode frequency under acoustic conditions representative of combustion conditions is of critical importance when tuning acoustic absorbers or investigating injection coupled combustion instabilities. The experimentally observed frequency shift was observed both in the frequency domain and as an asymmetric amplitude response to a linear frequency ramp of an external excitation system in the time domain. The frequency shift was found to be dependent on amplitude and operating condition. A hypothesis is presented for the frequency shift based on change in speed-of-sound distributions due to §ame contraction when exposed to high amplitude pressure oscillations. A one-dimensional (1D) model was created to test the hypothesis. Model parameters were based on relationships observed in experimental data. The model was found to accurately recreate the frequency shifting asymmetric response observed in test data as well as its amplitude dependence. Further development is required to investigate the in§uence of operating conditions and chamber design on the quantitative modeling of the frequency shift.


INTRODUCTION
High-frequency (HF) combustion instabilities are an ongoing challenge to rocket development programs. Signi¦cant progress in understanding has been made since combustion instabilities were ¦rst identi¦ed in the 1940s. However, due to the complicated processes present during rocket combustion, further research of the physical process that leads to combustion instabilities as well as development of tools to predict and limit the risk of combustion instabilities are still required [1].
High-frequency combustion instability occurs when combustion processes couple with the local acoustic ¦eld and is characterized by high-amplitude pressure oscillations that can lead to the rapid degradation of the rocket combustion chamber [2]. The frequency of the pressure oscillations is de¦ned by the speed-of-sound and the combustion chamber geometry and is of critical importance when analyzing coupling as both injector resonance coupling and combustion chamber processes have been shown to be sensitive to oscillation frequency [3].
Investigation and identi¦cation of acoustic eigenmode frequencies can be made both experimentally and numerically [4,5]. Analytical or numerical calculations which assume a constant speed-of-sound are limited in accuracy. Analysis presented in [6] identi¦ed frequency shifts in the ¦rst transverse eigenmode (1T mode) based on axial position in the combustion chamber. Using the PIANO-SAT acoustic simulation software, the di¨erence in frequency was identi¦ed as originating from a nonuniform speed of sound distribution in the combustion chamber. This highlighted the importance of using a representative speed-ofsound distribution for accurate prediction of eigenmode frequencies under combustion conditions.
In a coaxial injection system, the length of the central liquid oxygen (LOx) jet has been shown to be highly dependent on acoustic amplitude [7]. When exposed to high acoustic amplitudes, the §ame contracts toward the injection plane leading to increased heat loads to the wall and the injection plate. Injection conditions, in particular, injection velocity ratio (VR), have also been shown to in §uence both the susceptibility of the combustion chamber to combustion instabilities [8] and the contraction length of the LOx jet [9,10]. The LOx jet contracts due to improved atomization and mixing which also reduces the length of the §ame in the combustion chamber [7]. The in §uence of §ame contraction on the combustion chamber speed-of-sound distribution has not yet been addressed in detail. Improving the understanding of how §ameacoustic interactions in §uences the speed-of-sound distribution is important for accurate tuning of acoustic resonance cavities and investigation of coupling of injector and combustion chamber resonance frequencies.
The combustor ¢H£ (BKH) is a rectangular combustion chamber used for the investigation of acoustic §ame interaction (Fig. 1). Combustion chamber acoustics can be externally forced producing high-amplitude pressure oscillations. The in §uence of these pressure oscillations on jet breakup and combustion zone length can be visualized through optical access windows located either side of the primary injection zone.
During investigation of acoustic dissipation, a frequency shift was observed during 1T-mode excitation, which manifested as an asymmetric amplitude response to a linear ramp of excitation frequency [11]. Examples of such asymmetric responses will be presented later in this paper (for example, in Fig. 4). In this work, the physical process leading to the frequency shift is investigated and Figure 1 Frequency of 1T eigenmode with on and o¨resonance excitation a hypothesis is formed. Based on the hypothesis, a 1D model was developed to demonstrate the phenomena and to investigate the mechanism.

COMBUSTOR H
BKH is a rectangular combustion chamber designed to investigate §ameacoustic interactions. The combustion chamber (see Fig. 1) has a length of 305 mm, width of 50 mm, and a height of 200 mm. The rectangular cross section limits the fundamental modes of interest to two-dimensional form. The dimensions of length and height were chosen based on typical length and diameter of upper stage §ight engines. This de¦nes acoustic eigenmodes with representative frequencies, but the relatively high contraction ratio results in bulk §ow Mach numbers an order of magnitude lower than those typical for real engines. BKH has ¦ve primary shear-coaxial injection elements arranged in a matrix pattern and is operated with the propellant combination oxygenhydrogen. Hydrogen is injected at either ambient (293 K) or cryogenic (60 K) temperature. The oxygen is injected at a cryogenic temperature of 120 K. Operating conditions are varied by varying the propellant mass §ow rates and by using two di¨erent primary injector con¦gurations (Fig. 2).
In addition to the primary injectors, secondary injection zones and window ¦lm cooling, both with ambient hydrogen, are employed in the operation of the combustion chamber. The secondary injection zones are used to limit recirculation above and below the primary injectors and to protect the combustion BKH is equipped with an excitation system. The excitation system excites combustion chamber acoustics by opening and closing a secondary nozzle perpendicular to the bulk §ow direction. The secondary nozzle is open and closed by a rotating toothed wheel. The rate at which the wheel rotates controls the frequency of the excitation.
Acoustic pressure §uctuations are measured by the use of §ush mounted Kistler type 6043A60 dynamic pressure sensors. The sensors have an accuracy of ±1%, a dynamic range of ±20 bar and are sampled with a rate of 100 kHz. A 30-kilohertz antialiasing ¦lter is applied to this signal.

Eigenmode Frequency Shift
A shift in eigenmodes frequencies was observed in a spectral analysis of the 1T and ¦rst combined longitudinal and transverse (1L1T) eigenmodes during excitation. Figure 3 shows two power spectral density functions (PSDs), one during onresonance excitation of the 1L1T mode and one during o¨-resonance excitation. A shift was observed in the ¦rst longitudinal (1L) mode, 1T, and 1L1T modes. The proximity of the excitation frequency to the 1L1T mode makes it di©cult The same phenomenon was observed in [12] when examining spectrograms of BKH excitation. The frequency shift of the 1L1T mode during 1T mode excitation was observed to be approximately 140 Hz. This is close to what is observable during 1L1T mode excitation in Fig. 3. The di¨erence in frequency shifts between those observed in [12] and those presented here can be attributed to the spatial distribution of each eigenmode. Due to di¨erent spatial distributions, each eigenmode is in §uenced di¨erently by changes in the speed-of-sound distribution.
In [12], the frequency increase of 140 Hz was estimated to correspond to an increase in bulk temperature of around 5.5%, or 62 K, which matched well with thermocouple measurements in the chamber wall. The increase in temperature was accompanied by an approximately 2% increase in c * combustion e©ciency, both of which are postulated to result from improved atomization and mixing due to the transverse velocity component of the acoustic ¦eld acting on the primary injection zone.

Asymmetric Amplitude Response
To excite an eigenmode, BKH is exposed to a linear frequency ramp of an external acoustic driving force. During testing, an asymmetric response of the 1T mode to the linear frequency ramp was observed. Figure 4 shows a standard test sequence with an upwards and a downwards ramp through the 1T mode. During both the up and down ramp, the amplitude responses are asymmetric with a skew to higher frequencies (see Fig. 4).
If the asymmetry is due to changes in sound speed distribution, as suggested by the previous ¦ndings, the asymmetry is in §uenced by the energy density and distribution of the §ame. Flame distribution and energy density in the nearinjection region are in §uenced by the ratio of oxidiser-to-fuel (ROF), the injection conditions, and acoustic amplitude. Subsequently, the in §uence of operational and acoustic conditions on asymmetry was investigated. Figure 5 shows the in §uence of acoustic amplitude on the symmetry of amplitude response to a linear frequency ramp.
The following equation that describes a Fano pro¦le was used to measure the asymmetry of the pro¦le in this study: where • is the full width at half maximum; ω is the given frequency; ω 0 is the central frequency of the eigenmode; and the Fano symmetry coe©cient q is the measure of the pro¦le symmetry [13]. The dependence of q on acoustic amplitude is given in Fig. 5. The Fano coe©cient is inversely proportional and responds nonlinearly to the asymmetry. For high values of q, in this case over 100, the ¦t approaches a standard Lorentzian pro¦le with no asymmetry. Values between 10 and 100 show low levels of asymmetry and as the value of q approaches 1, the response becomes highly asymmetric. The response to pressure amplitude was found to be exponential and, so, Fig. 5 presents the relationship on double logarithmic axes.
The symmetry was strongly a¨ected by pressure amplitude, with higher amplitudes decreasing the symmetry of the pro¦les. The response was linear on the double logarithmic axis. The uncertainty increased for higher q values due to the decreasing sensitivity of q for highly symmetric pro¦les.
Injection conditions, such as injection VR, in §uence the extent of the §ame and interactions between the §ame and pressure oscillations [8]. The in §uence of injection conditions on the amplitude response symmetry was investigated to examine the role of combustion in the frequency shift.
The symmetry was found to respond most consistently to changes in VR. Figure 6a shows the relationship between VR and symmetry, with decreasing VR leading to increased asymmetry. In oxygenhydrogen combustion, the VR is known to in §uence the susceptibility of the LOx core to interaction with the acoustic ¦eld [9], and the pressure amplitude has been shown to strongly in §uence LOx core length and heat release distribution [12]. It should be noted that the result for VR in Fig. 6a cannot be isolated from the in §uences of ROF and hydrogen temperature, both of which also modify the VR in BKH, apart from the two primary injector con¦gurations (see Fig. 2).
Considering the in §uence of hydrogen injection temperature in Fig. 6b, a higher asymmetry was generally observed in liquid hydrogen (LH 2 ) tests. However, there was a very large spread in asymmetry measured for gaseous hydrogen (GH 2 ) tests. The spread could be attributed to changes in primary hydrogen pressure drop, and an improved relationship with respect to pressure drop was observed (Fig. 6c). The asymmetry appeared to be less when the primary hydrogen pressure drop was above 10 bar and increased as pressure drop decreased toward zero.

Heat Release Distribution
In BKH, the response of the §ame to both periodic and continuous excitation can be examined using OH* and visible imaging. The contraction of the LOx Figure 7 Instantaneous OH* images for low acoustic pressure amplitude (o¨resonance excitation) (a); and high 1T-mode amplitude (on-resonance excitation) (b) core and the §ame toward the faceplate during combustion instability has been well documented in a number of combustion chambers [7,14,15] and can also been observed in the OH* images in Fig. 7. The extent of this contraction has been shown to be dependent on operating conditions and on acoustic pressure amplitudes [7].

Hypothesis of Physical Mechanism
Based on the physical observations made in subsections 3.13.3, a hypothesis of the physical origin of the frequency shift was formed: when a linear frequency sweep is applied, the acoustic pressure amplitude increases as the eigenmode frequency is approached. The increasing amplitude causes a contraction of the §ame toward the face plate, increasing the localised speed-of-sound. For the 1T mode, the change in speed of sound in the central region of the combustor results in an increase in eigenmode frequency. By this mechanism, the shift in frequency is proportional to the pressure amplitude.
An increase in eigenmode frequency means that the linear frequency sweep takes longer to reach the peak amplitude. In the approach to the eigenmode frequency, the excitation frequency is ¢chasing£ the eigenmode frequency. After the peak amplitude is reached, the pressure amplitude decreases and the §ame returns to its undisturbed state which decreases the eigenmode frequency and subsequently causes a more rapid decrease in pressure amplitude as the distance between the forcing frequency and the eigenmode frequency increases more rapidly than the ramp rate.
This mechanism generates the unidirectional asymmetry observed with a skew only toward higher forcing frequencies. The strength of the skew is both dependent on amplitude and on a variable representing a relationship between contraction of the §ame and a change in the speed of sound.

MODEL DESCRIPTION
A simple harmonic oscillator model capable of producing an eigenmode frequency shift based on acoustic pressure amplitude was developed based on the hypothesis presented in the previous section. This section describes the design of the model which is based on experimental observation.

Model Overview
The model implements a simple damped-driven harmonic oscillator with an input driving function and a feedback loop, based on pressure amplitude, to control (1) the driving function, which provides a linearly ramped sinusoidal chirp function to drive the harmonic oscillator; (2) the 1D damped driven harmonic oscillator with a variable eigenmode frequency input, which calculates the pressure response; (3) the §ame contraction submodel, which implements the relationship between pressure amplitude and contraction length in Fig. 8; and (4) a frequency factor submodel, which converts the contraction length into a change in the speed of sound.
The unexcited eigenmode frequency is supplied as a constant to the model. The driving function was chosen to ramp through an eigenmode frequency representative of the 1T mode in BKH. The ramp rate was the same as a typical BKH test (83.3 Hz/s). A sinusoid driving function was employed due to the simplicity of implementation. The damped-driven harmonic oscillator had a damping rate of 410 s −1 , which is representative of 1T mode damping rates observed in BKH [11]. The model is 1D and was implemented in Simulink. The model was run with a variable time step 4th-order RungeKutta integrator. Output sampling rate and calculation of passed variables were made at a constant time step of 10 µs.

Flame Contraction Model
The contraction of the §ame toward the faceplate occurs when the combustion zone is exposed to high-amplitude pressure and velocity oscillations. A relationship between normalized acoustic amplitude (P ′ /P cc ) and oxygen core contraction length has been presented previously for BKH (Fig. 9) [15]. An approximately linear response with constant gradient was observed until pressure amplitude reaches around 5.5% of chamber pressure. After this, another approximately linear response with a much lower gradient was observed. The contraction of the §ame is assumed to be represented by the contraction of the LOx core toward the faceplate. The relationships in Fig. 8 provide a physical basis for relating pressure amplitude to changes in speed-of-sound distribution through a change in heat-release distribution.
The §ame contraction submodel links pressure amplitude with §ame contraction length (Fig. 10). The input to the model is instantaneous pressure amplitude Figure 9 Retraction of LOx core length due to increasing pressure amplitudes (mod-i¦ed from [7]): 1 ¡ 60 bar; test A; 2 ¡ 60 bar, text B; 3 ¡ 60 bar, test C; 4 ¡ 40 bar, test A; and 5 ¡ 40 bar, test B. Filled signs refer to GH2 and blank signs refer to LH2. Line re §ects the trend for all 60-bar GH2 tests Figure 10 Overview of §ame contraction model sampled at 10 µs. The model uses a switch to recreate the relationship observed in Fig. 9. Two domains were selected, an upper and a lower domain each applying a linear relationship with di¨erent coe©cients of gradient and constants. The two domains were separated at the P ′ /P cc value of 5.5%. The model recreates this behavior using the input of the pressure amplitude to calculate the §ame contraction for both domains before selection of the domain based on input pressure. The output of contraction length is used by the next model to calculate the frequency shift.

Frequency Shift Model
In Fig. 8, the contraction length is calculated in units of core length as a ratio to oxygen jet diameter. The heat release density and, therefore, the speed-of-sound and frequency shift are inversely proportional to the contraction of the oxygen core; so, the reciprocal of this value was taken as input.
The ¦nal step is to multiply the signal by a constant termed the ¢frequency factor.£ The frequency factor can be viewed as a compound constant consisting of three related physical phenomena. First, it represents the increase in energy density near the injection plane due to the contraction of the §ame toward the faceplate. Second, the localized change in speed of sound due to the increased energy density. Third, the change in eigenmode frequency due to the localized change in speed-of-sound. This makeup of the frequency factor is illustrated in Fig. 11. In this simpli¦ed form, the parameter is di¨erent for each operating condition as the behavior of the §ame is di¨erent for each operating condition.

Figure 11 Overview of frequency shift model
The in §uence of the frequency factor variable, along with the pressure amplitude, on signal asymmetry is investigated in the results section.

RESULTS
The in §uence of model parameters on the asymmetry and frequency shift was investigated and compared to test data. Two key parameters were investigated, the excitation amplitude and the frequency factor. The following subsection shows the in §uence of each of these parameters on the asymmetry of the signal response to excitation.

Recreation of Asymmetrical Response
The acoustic amplitude was varied whilst holding the frequency factor constant at 250. The response to a range of pressure amplitudes representative of those observed in BKH is presented in Fig. 12. When the pressure amplitude increases the asymmetry of the pro¦le also increases. This demonstrates that the coupling between acoustic amplitude and asymmetric response in test data (see Fig. 5) is represented by the model (see Fig. 12). Additionally, a shift in eigenmode frequency is observed. The frequency shift was compared to test data for similar amplitudes and was found to be consistent. A more detailed comparison with experimental data, based on operating conditions, was not possible due to the simpli¦cation of the model parameters. Figure 13 compares the amplitude with the asymmetry as measured by the ¦t of a Fano pro¦le to the numerical model data. The asymmetry is represented by the Fano coe©cient which is inversely proportional to the asymmetry and is nonlinear. The increasing asymmetry with pressure amplitude in Fig. 13 is reproduced by the model for two frequency factors, 200 and 340.
Two additional points need to be made. Two domains of operation are observed for the two frequency factors examined, an upper and a lower, each of which correspond to the two domains of LOx core contraction implemented in the model. In the lower amplitude region, a strong negative gradient is observed as the symmetry decreases with increasing amplitude. In the higher amplitude region, no gradient in the amplitude-symmetry relationship is observed. The lower gradient observed in the LOx core length P ′ /P cc relationship corresponds to the lower gradient region of the model response. The behavior of the sym-metry coe©cient, q, with respect to amplitude for a frequency factor of 340 is inconsistent with that of a frequency factor of 200. The increased frequency factor leads to larger asymmetries which are not accurately accounted for by the asymmetric Fano ¦tting pro¦le when the amplitude is also high. This highlights a limitation in the application of Fano ¦tting pro¦les to asymmetric pressure response produced by a frequency shift.

In §uence of Frequency Factor
The in §uence of the frequency factor was investigated for constant amplitude excitation. In Fig. 14, the amplitude was set to 3.3 bar. Increasing frequency factor enhances the asymmetry and the frequency shift due to the contraction of the §ame. Frequency factors of up to 600 were investigated. However, frequency factors above 400 became increasingly nonrepresentative of the curves observed during testing.

Comparison with Test Data
The model was compared to test data to observe the response to both up and down ramping. Figure 15 shows the response of the model to up and down ramping overlaid on the up and down ramp presented in Fig. 4. Four curves are presented in each ¦gure. Both the frequency ramps from test data and from the simulation are compared in black. In grey, the amplitude response of the model is given and compared to test data given by the frequency-amplitude scatter. For In general, the asymmetry observed in the simulation data agrees well with that of the test data. The skew is always shifted to higher frequencies as is observed in experimental data. The response of the model can be tuned using the amplitude and frequency factor variables to match the test data. In Fig. 15, the up and down ramps were tuned to improve the comparison with experimental data. The turning point matches well between the model and the test data. However, after passing the resonance, the pro¦le does not follow the test data Figure 16 Comparison of ¦t quality between 1D model (a) (1 ¡ experimental amplitude response; and 2 ¡ computational amplitude response), and standard pro¦les (b) (1 ¡ dynamic pressure; 2 ¡ Lorentzian pro¦le; 3 ¡ asymmetric Lorentzian pro¦le; and 4 ¡ Fano pro¦le) amplitude response optimally. This has been attributed to the nonlinearity in the test data ramp when compared to the perfect linear ramp of the model (see Fig. 15). Figure 16 shows a comparison with a highly asymmetric pro¦le in the frequency domain which removes the in §uence of the deviation from a linear ramp observed in test data. The model showed good agreement with the asymmetry observed in test data.
The in §uence of white noise was tested by introducing an additional driving term into the model. The model was rerun to see if the noise had an impact on the turning point of the asymmetry or the ability of the model in general to match the experimental pro¦le. White noise with amplitude of 10% of the driving function amplitude was found to have no in §uence in the ability of the model to recreate the phenomena observed in experimental data.
The response of the frequency shift model was compared to three response pro¦les, the Fano pro¦le, which has been described in Eq. (1), the Lorenzian pro-¦le and the asymmetric Lorentzian pro¦le which are described by the following equations: (2) where α is the half width at half maximum and B, for the asymmetric Lorentzian pro¦le, is the asymmetry. Both the Fano and asymmetric Lorentzian pro¦les can take into account asymmetry with the coe©cient q for the Fano pro¦le and B for the asymmetric Lorentzian pro¦le. For the asymmetric Lorentzian pro¦le, the value of B varies between −1 and 1 with B = 0 reducing to the Lorentzian pro¦le without symmetry. The asymmetric Lorentzain pro¦le is typically used in investigation of helioseismology and a more detailed description of its applications in that ¦eld is given in [16]. The Lorentzian pro¦le without asymmetry cannot take this into account and, as a consequence, can poorly ¦t asymmetric pro¦les [11]. Figure 16 shows the comparison between the response of the frequency shift model and the three pro¦les described by Eqs. (1)(3). The quality of the ¦t using the frequency shift model is improved with respect to the asymmetric pro¦les used previously for measurement of acoustic dissipation. When comparing the di¨erent models directly, the frequency shift model ¦ts the test data the most closely. The asymmetric pro¦les o¨er an improved ¦t over the symmetric Lorentzian pro¦le, especially for amplitudes above 1/3 of peak amplitude, but do not capture the physical behavior of the asymmetry.

DISCUSSION
Understanding and prediction of eigenmode frequency shift under representative conditions are important for limiting the risk of combustion instabilities. The frequency shift observed in test data is well explained by a change in speed-of-sound distribution due to contraction of the §ame. Contraction of the §ame due to improved breakup and mixing processes through exposure to transverse acoustic oscillations has previously been observed in BKH [15]. A contracting §ame mechanism is consistent with the dependency of asymmetry on injection conditions, in particular, to VR which has been shown to in §uence the susceptibility of LOx core breakup to combustion instabilities [9,10].
The model presented here shows that the frequency shift model can account for the observations made in test data. However, due to the simpli¦ed nature of the model, it is limited when applied to di¨erent systems and taking into account complex processes such as the in §uence of injection conditions.
The two active blocks that relate acoustic amplitude with frequency shift, the §ame contraction model and the frequency shift model, are signi¦cant simpli¦cations on the processes they represent. For example, the contraction rate of the jet with respect to the pressure amplitude is dependent on operating conditions and geometry. In BKH, this is represented by the in §uence of VR on asymmetry. The model presented here is tuned for use in BKH by utilizing existing results of LOx core breakup. No data exist where the relationship of amplitude and LOx core contraction can be examined under representative conditions for cases outside of BKH. A more generalized relationship between acoustic amplitude, LOx core contraction, and injection conditions (such as VR) would help in further development of this model.
The second parameter used to relate pressure amplitude to eigenmode frequency shift is the frequency factor. The frequency factor relates the contraction of the jet to the change in sound speed and frequency. The parameter is a sim-pli¦cation of a complex set of relationships that are dependent on operating condition and propellant type. For example, if the same contraction occurred in a methane §ame would the subsequent speed-of-sound increase be the same? This is unlikely as the propellant combinations have di¨erent combustion temperatures and combustion products. For the model to be generalized and maintain accuracy, the incorporation of additional terms into the frequency factor block would be required.
The distribution of the §ame is restricted to the central axis of the combustor in BKH and is a specialized case. Above and below the §ame zone, secondary hydrogen dominates and in these zones, the speed-of-sound is not in §uenced by contraction of the §ame. In real engines, combustion is distributed across the entire cross section. This could lead to a larger frequency shift if the entire §ame zone was to retract. The height of the BKH combustion chamber is comparable with upper stage engines. However, the §ame occupies approximately one quarter of this region, restricting the response to only this region.
The model accurately reproduced the in §uence of amplitude on asymmetry over a limited amplitude domain. The inconsistency observed between numerical and experimental results was attributed to the ability of the asymmetric ¦tting functions to capture highly asymmetric pro¦les of this type. This is highlighted in Fig. 16. The Fano and asymmetric Lorentzian pro¦les provide a better ¦t than the symmetric Lorentzian pro¦le but are signi¦cantly worse than the frequency shift model at following the experimentally obtained response. This suggests that the Fano and asymmetric Lorentzian pro¦les should only be used as tools to improve the accuracy of the measurement of response peak width, and that they are not physically representative of the amplitude, §ame contraction, and speed-of-sound interactions.
With further re¦nement, the model could be used to tune acoustic resonators to resonance modes under unstable conditions. Additionally, the model could be used to investigate the natural limit cycle of injector-resonance coupling where a shift in chamber mode frequency shifts the chamber in and out of resonance with the injector. To accurately measure this in one dimension, the model would have to have a frequency dependent driving amplitude which will be implemented in future developments of the model.

CONCLUDING REMARKS
In experiments with combustor ¢H,£ an increase in combustion chamber eigenmode frequencies was observed during excitation of eigenmodes with a transverse velocity component. The frequency shift was observed as an asymmetric amplitude response to a linear excitation ramp. Acoustic amplitude and operating conditions in §uenced the strength of the asymmetry. This led to the hypothesis that a contraction of the §ame and the subsequent increase in temperature in the near faceplate region was the physical cause of the observed frequency shift in transverse modes.
To test the hypothesis, a 1D damped-driven harmonic oscillator model was developed. The model included a frequency shift term with the amount of shift being related to the instantaneous acoustic amplitude. When exposed to a linear frequency sweep, the acoustic model was found to accurately follow the asymmetries observed in test data. Two parameters, contraction length dependent on acoustic amplitude and a frequency factor, were incorporated into the model to in §uence the asymmetry. A parametric study of each of the parameters showed that the dependence of asymmetry on amplitude observed in test data was also reproduced by the acoustic model. The frequency factor can be tuned to account for the in §uence of operating condition and combustion chamber con¦guration. However, this tuning is currently dependent on the use of experimental data and generalized applicability of the model would ¦rst require the development of appropriate submodels to account for other operating conditions and con¦gurations.
This model is the ¦rst step in understanding the importance of acoustic sound speed distribution on combustion instability. The model is of particular relevance for tuning of acoustic resonators which have high sensitivity to acoustic frequency. Additionally, injection coupling is highly sensitive to the eigenmode frequencies of the combustion chamber and the injector. A slight shift in eigenmode frequency may mean the di¨erence between stable or unstable operation. Further development of the model is required to generalize the results and allow its use in the study of the frequency sensitivity of injection-coupled instabilities.