Nonlinear analytical flame models with amplitude-dependent time-lag distributions

Transcription

1 Special Isse TANGO Nonlinear analytical flame models with amplitde-dependent time-lag distribtions International Jornal of Spray Combstion Dynamics () 3! The Athor(s) 7 Reprints permissions: sagepb.co.k/jornalspermissions.nav DOI:.77/ jornals.sagepb.com/home/scd Sreenath M Gopinathan, Dmytro Irashev, Alessra Bigongiari Maria Heckl Abstract In the present work, we formlate a new method to represent a given Flame Describing Fnction by analytical expressions. The nderlying idea is motivated by the observation that different types of pertrbations in a brner travel with different speeds that the arrival of a pertrbation at the flame is spread ot over time. We develop an analytical model for the Flame Describing Fnction, which consists of a sperposition of several Gassians, each characterised by three amplitdedependent qantities: central time-lag, peak vale stard deviation. These qantities are treated as fitting parameters, they are dedced from the original Flame Describing Fnction by sing error minimisation nonlinear optimisation techniqes. The amplitde-dependence of the fitting parameters is also represented analytically (by linear or qadratic fnctions). We test or method by sing it to make stability predictions for a brner with well-docmented stability behavior (Noiray s matrix brner). This is done in the time-domain with a tailored Green s fnction approach. Keywords Flame transfer fnction, amplitde-dependence, mltiple time-lags, Green s fnction, stability analysis Date received: 3 May 7; accepted: 3 Jly 7. Introdction Power generation systems based on combstion of fels to extract energy operate with lean premixed flames in order to redce the polltion of the environment by exhast gases. However, sch systems are ssceptible to thermoacostic instabilities, which are characterised by high-amplitde acostic oscillations cased by the feedback between oscillations in pressre heat release rate. These oscillations lead to excessive vibration of strctral components, in extreme cases, major hardware damage. The relationship between the heat release rate the acostic field is a crcial element in modelling thermoacostic instabilities. This relationship may be given in the time-domain or freqency-domain. For example, the commonly sed n-law reads ðtþ ¼ n ðt Þ : ðþ in the time-domain, ^ð!þ ^ð!þ ¼ nei! : ðþ in the freqency-domain, where n is the copling coefficient is the time-lag. We se the following notation: e i!t for the time dependence, ðtþ for the flctation of the heat release rate in the timedomain, ^ð!þ for its freqency-domain eqivalent (Forier transform), for the mean rate of heat release; the same notation is sed for the acostic velocity, (at a chosen reference position pstream of the School of Chemical Physical Sciences, Keele University, Staffordshire, UK DICCA, Università degli Stdi di Genova, Genoa, Italy Corresponding athor: Maria Heckl, School of Chemical Physical Sciences, Keele University, Staffordshire ST55BG, UK. m.a.heckl@keele.ac.k Creative Commons CC BY-NC: This article is distribted nder the terms of the Creative Commons Attribtion-NonCommercial 4. License ( creativecommons.org/licenses/by-nc/4./) which permits non-commercial se, reprodction distribtion of the work withot frther permission provided the original work is attribted as specified on the SAGE Open Access pages (

2 International Jornal of Spray Combstion Dynamics () flame). In general, the time-domain expression is given in terms of a fnctional F, ðtþ ¼ F ðtþ : ð3þ the freqency-domain expression by the flame transfer fnction (FTF) denoted by Tð!Þ, Tð!Þ ¼ ^ð!þ= ^ð!þ= : ð4þ The FTF may depend on the amplitde of the velocity, in which case it is referred to as flame describing fnction (FDF). The FTF or FDF of a given brner can be measred, the reslt is a seqence of complex nmbers at discrete freqencies. For analytical modelling prposes, it is necessary to convert sch data into a continos fnction of freqency. This is typically done by some ad hoc crve-fitting procedre. Several examples can be fond in the literatre: Schermans 3 Schermans et al. 4 measred the FTF of a trblent partially premixed brner approximated it by a sperposition of two Gassian crves. Komarek Polifke 5 measred the FTF of a perfectly premixed swirl brner (the BRS brner ) approximated it with three Gassians. Each Gassian was centred arond a specific delay time, which corresponded to the travel of a specific pertrbation qantity, sch as swirl nmber. A similar approach was sed by Bade et al.: 6 they approximated the FTF measred for an annlar combstor by a sperposition of a low-pass filtered discrete time-lag term two distribted time-lag terms. Sbramanian et al. 7 approximated the FTF of the BRS brner by sing a sperposition of rational fnctions with fitting parameters. Noiray 8 Noiray et al. 9 considered also the amplitde-dependence; they measred the FDF of a matrix brner approximated it with a straightforward interpolation. C osic et al. measred the FDF of a partially premixed swirl brner approximated it in a similar way. The FTF or FDF of a given brner can also be determined by nmerical simlations. Some flame models are based on the idea that the time it takes a fel particle to travel from the nozzle to the point of combstion differs slightly from particle to particle, this can be described in terms of a histogram or distribtion. Polifke et al. simlated a brner with an elliptical premix nozzle by steady-state CFD sed Lagrangian particle tracking to determine a histogram of arrival times. This trned ot to be similar to a Gassian crve was approximated accordingly. Flohr et al.,3 performed a similar simlation for a dmp combstor; they fond a distribtion of timelags, which was reminiscent of a sperposition of two Gassian distribtions, centred arond two peak vales. Similar simlations were performed by Schermans 3 Schermans et al. 4 for their premixed swirl brner; again, two Gassian distribtions were obtained. These were fond to be associated with the travel times of two different physical phenomena: flctations in trblence intensity flctations in fel concentration. More recently, flame models have been obtained with accrate, bt comptationally deming, nsteady CFD simlations. These generate the time histories for heat release rate velocity; by application of the Wiener Hopf inversion, the FTF is then determined. An example is the work by Tay-Wo- Chong et al. 4 who calclated the FTF of the BRS brner in this way (sing LES). This work was extended into the nonlinear domain by Irashev et al., 5 who performed simlations with for different forcing amplitdes. Stability predictions for a particlar combstion system can be made by combining a model for the nsteady acostic field in the combstor with a model for its FTF. If additional amplitde information is available in the form of an FDF, predictions can also be obtained for limit cycle amplitdes. Linear stability predictions were made, for example, by Heckl 6 Tay-Wo-Chong et al.; 4 Tay-Wo-Chong et al. 4 fond that small errors in the FTF can make sch predictions qite inaccrate. Nonlinear stability predictions were obtained by some of the athors qoted above: Noiray; Noiray 8 ; Noiray et al. 9 treated the length L of the plenm chamber as a continosly variable control parameter prodced a stability map in the L A plane (A is the amplitde of the velocity pertrbation). These maps gave not only the linear stability behavior, bt also revealed limit cycle amplitdes hysteresis behavior. Palies et al. 7 stdied a premixed swirl combstor with the length of the combstion chamber as control parameter obtained similar stability maps. C osic et al. predicted limit cycle amplitdes for their brner, these were in good agreement with experimental reslts. Li Morgans 8 made nonlinear stability predictions for a horizontal Rijke tbe. They modelled the heat sorce analytically by an extended n-law with the following featres: single time-lag; no time-lag distribtion; satration amplitde imposed on the heat release rate; ct-off freqency imposed on the FTF to captre its low-pass filter behavior. They sed a wave based approach to model the acostic field. The crrent paper bilds on the work by Heckl, 6,9 who developed a simple time-lag law for Noiray s matrix flame. 8,9 This law inclded a time-delayed velocity term, ðt Þ, an instantaneos velocity term, (t), each with its own copling coefficient, bt zero

3 Gopinathan et al. 3 stard deviation. The amplitde-dependence was in the time-lag in the copling coefficients, it was represented by simple fnctions. This model captred some relevant featres of the FTF (excess gain, nearconstant phase), bt not the low-pass behavior. Nevertheless, the model gave good stability predictions (sing a tailored Green s fnction approach), which were in line with Noiray s stability map. Bigongiari Heckl also sed the same heat release rate law for a Rijke tbe obtained its stability behavior sing tailored Green s fnction. The aim of or paper is to introdce a generic FDF in analytical form, representing both the freqency-dependence the amplitde-dependence analytically. The nderlying idea is that the nit implse response (UIR), which is effectively a timelag histogram, is a sperposition of distribtions, each characterised by a mean time-lag, stard deviation peak vale. This idea is motivated by the following observations: (i) Different types of pertrbations (e.g. pertrbation of swirl nmber, pertrbation of fel concentration, vortices, pertrbation of trblence intensity) travel with different speeds. (ii) The arrival of a pertrbation at the flame is spread ot over time. We assme the individal distribtions to be Gassian, with central time-lags,,... stard deviations r, r,...; they are also described by generalised copling coefficients n, n,... These qantities will be treated as fitting parameters, which are amplitdedependent chosen to fit a given FDF; their amplitde-dependence will be described by simple fnctions (linear or qadratic). Or representation of the FDF is very general can be adapted to model any of the flames mentioned in the literatre srvey above. We expect that it can be applied to many more flames, given that the rationale behind it is motivated by the transport phenomena typically observed in combstion systems. Since its basic idea is rooted in the time-domain, or method also provides the time-domain representation of the heat release law in analytical form. This is very sefl for performing time-domain simlations with a minimm of nmerical effort. The mathematical formlation of or method is explained in section, followed by a case stdy in section 3. We se Noiray s matrix brner 8,9 as the case stdy example. The matrix brner its measred FDF are described in section 3.. The UIR of the matrix flame is given in section 3., is converted into an analytical FDF in section 3.3. The tailored Green s fnction for the matrix brner can be Acostic excitation calclated analytically, the reslts are given in section 3.4. It forms the basis of the stability analysis, which is given in section 3.5. Stability predictions are made discssed in section 3.6. The paper concldes with a smmary sggestions for frther work in section 4.. Gassian time-lag distribtions We consider the brner set-p shown in Figre. A flame of finite extent is located downstream of a nozzle; the exit plane of the nozzle is taken as reference position. Flow pertrbations leaving the exit plane reach the flame front at different time instances therefore there is a distribtion of time-lags casing delayed heat release rate flctations. This sggests the following generalised law for the heat release rate in the time domain Z ðtþ ¼ hðþ ðt Þ d: h is a generalised copling coefficient depends on the time-lag. Moreover, it is identical with the UIR of the flame, as has been shown in Gopinathan et al. The dynamic behavior of many flames is characterised by two or more prominent time-lags, by a distribtion of the heat release rate arond these timelags. Let s assme a generic heat release rate law with k prominent time-lags,,..., k, with a Gassian distribtion D centred arond each of them, ðtþ ¼ n ðt Þ Dð Þd ðt Þ þ n Dð Þd þ ðt Þ þ n k Dð k Þd; Rod Exit plane τ Flame Figre. Schematic of a flame showing distribtion of time-lags. τ ð5þ ð6þ

4 4 International Jornal of Spray Combstion Dynamics () where D is given by Dð j Þ¼ p ffiffiffiffiffi e ð ð jþ =j Þ, j ¼,,..., k ð7þ j L matrix flame perforated plate tbe Eqation (6) contains 3k parameters,,,..., k, n, n,..., n k, r, r,..., r k, which are treated as fitting parameters assmed to be amplitde-dependent. We presme that the distribtions are close to zero for negative ; then we can extend the integration range in eqation (6) from ð, Þ to ð, þþ apply the Forier transform. This leads to the FDF (for details, see Appendix ). T k ð!, AÞ ¼ ^ð!, AÞ= ^ð!, AÞ= ¼ n ðaþe! ðaþ = e i! ðaþ þ n ðaþe! ðaþ = e i!ðaþ þ þ n k ðaþe! k ðaþ = e i!kðaþ : The nknown fitting parameters in this representation are determined by minimising the difference between the original FDF, Tð!, AÞ, the expression for T k ð!, AÞ given in eqation (8). We do this by sing the optimisation rotine lsqnonlin in MATLAB Õ, with the error defined by the vector q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ¼ <½T ð!, AÞ T k ð!, AÞŠ, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi =½T ð!, AÞ T k ð!, AÞŠ : The vector e k contains the real part imaginary part of the difference; it has N components, where N is the nmber of data points along the x-axis. We also impose the additional constraint that jt k ð!, AÞj ¼ at x ¼, which gives X k j¼ n j ¼ : ð8þ ð9þ ðþ The error minimisation procedre is repeated for each amplitde A for which data are available. The analytical representation of any known FDF can be obtained sing this procedre if the complex vales of the FDF are known for a sfficient nmber of freqency points at specific amplitdes. 3. Case stdy: Noiray s matrix brner 3. Description of the matrix brner The matrix brner sed in Noiray s work (Figre ) consists of a circlar tbe with a piston (variable position) at one end a perforated plate at the other end. The perforated plate acts as the flame holder for a D array (matrix) of small flames. The tbe is essentially a qarter-wave resonator, with one rigid end one nearly open end. The acostic field in the tbe extends slightly beyond the downstream end reaches the flames, making thermoacostic interaction possible. Noiray et al. 9 measred the FDF by exciting a harmonic sond field in the tbe with a lodspeaker. The excitation freqencies were in the range [, Hz], 5 excitation amplitdes were sed: A= ¼ :3,.3,.4,.48,.54. The velocity flctations were measred by Laser Doppler Velocimetry at the base of the flame (i.e. the reference position for their FDF was the same as the flame position). The ensing flctations of the rate of heat released by the matrix flame (global heat release) were measred by chemilminescence, sing a photomltiplier tbe eqipped with OH* filter. The measred reslts are shown by the markers in Figre 3 for the gain (jftfj, left) the phase (ffftf, right). The gain (jftfj) shows a low pass filter behavior reaches vales in excess of at low freqencies. The maximm vale decreases with increasing excitation amplitde, the position of the maximm shifts to lower freqencies. The phase (ffftf) crve is approximately linear at low freqencies, with a slope that increases with increasing excitation amplitde. 3. UIR of the matrix flame rigid piston (position variable) Figre. Schematic of Noiray s matrix brner. In a first step towards obtaining an analytical expression for the measred reslts in Figre 3, we determine

5 Gopinathan et al. 5 (a) FTF.5 A/ ū =.3 Exp A/ ū =.3 Exp A/ ū =.4 Exp A/ ū =.48 Exp A/ ū =.54 Exp (b) FTF 5 A/ ū =.3 MTL A/ ū =.3 MTL A/ ū =.4 MTL A/ ū =.48 MTL A/ ū =.54 MTL Freqency (Hz) Freqency (Hz) Figre 3. Gain (left) phase (right) vs freqency for the FDF of Noiray s matrix flame. Markers: experimental vales from Noiray et al. 9 Dashed crves: analytical approximation based on eqation () sing the expressions given in Table. FDF: Flame Describing Fnction. hk (s) 8 3 Figre 4. Implse response of the matrix flame. the corresponding UIR. The UIR is the response of the flame (in terms of the heat release rate) triggered by a normalised velocity implse. Its general definition is given by eqation (5). For realistic flames (i.e. flames that satisfy casality), the UIR is zero for negative - vales (casality) again for -vales beyond some maximm (finite dration of response). We take this maximm vale to be eqal to the slope of the phase crve at low freqencies denote it by ff. Then eqation (5) becomes (after a sbstittion for the integration variable with t ðtþ ¼ Z tff hðþ ðt Þ A/ ū =.3 A/ ū =.3 A/ ū =.4 A/ ū =.48 A/ ū =.54 d: ðþ We calclated the UIR of the matrix brner flame by the inverse z-transform of the FTF as described in Polifke. To calclate the inverse z-transform, the time interval ½, ff Š is discretised into 5 eqally spaced points (this nmber of points was chosen becase it gives a smooth UIR vs time-lag crve). The reslts are shown in Figre 4 for the five amplitde vales A= ¼ :3,.3,.4,.48,.54. We observe that there are two prominent time-lags in the UIR for all vales of A=. For example, at A= ¼ :3, there is one near ¼ :s, where the UIR has a maximm, one near ¼ :s, where the UIR has a (negative) minimm. There are some frther maxima minima at larger time-lags, bt they are minor. We therefore conclde that the FDF can be approximated with a distribtion arond two central time-lags.asa= increases, both increase, the peak vale of the maximm decreases, the minimm vale arond becomes less negative. 3.3 Analytical description of the matrix flame We now proceed to represent the FDF shown in Figre 3 by the analytical expression in eqation (8) with only two terms inclded in the expression for the FDF, T ð!, AÞ ¼n ðaþe! ðaþ = e i! ðaþ þ n ðaþe! ðaþ = e i! ðaþ : ðþ The nknown fitting parameters n, n,,, r r are determined as described in section. This is done individally for each of the five available amplitde vales A= ¼ :3,.3,.4,.48,.54. The reslts are shown by the markers in Figre 5. All fitting parameters vary with amplitde. n r seem to decrease with increasing amplitde, while n r seem to increase; all have considerable scatter. The reslts for clearly increase with amplitde, this increase appears to be faster than linear. We model this amplitde-dependence analytically by linear fnctions for n, n, r r, by qadratic fnctions for. These fnctions are shown by the dashed crves in Figre 5 are listed in Table. The coefficients in these fnctions were obtained by linear least sqares estimation. Altogether, the combination of eqation () the fnctions in Table provides a flly analytical representation for the FDF in Figre 3. This also provides an extrapolation to freqency amplitde vales for which experimental data are not available. Plots of the

6 6 International Jornal of Spray Combstion Dynamics () (a) (b) (c) n k n n k (s) 5 5 k (s) A/ ū.5.5 A/ ū.5.5 A/ ū Figre 5. Amplitde-dependence of the fitting parameters n, n (part (a)), s, s (part (b)) r,r (part (c)). Markers: reslts for individal amplitdes. Dashed crves: approximation ( extrapolation) by the analytical fnctions in Table. Table. Analytical approximation of the amplitde-dependence of the fitting parameters. Parameter Dependence on A= n n s s r r :85 ða=þþ:99 :85 ða=þ :99 :8 3 ða=þ 4:74 5 ða=þ þ8:7 4 4:35 3 ða=þ þ 6:79 5 ða=þ þ: 3 5:86 4 ða=þþ:36 4 4:63 4 ða=þþ9:3 4 analytical expressions (eqation () Table ) have been added to Figre 3 in the form of dashed crves. A comparison with the experimental reslts (markers in Figre 3) shows that the analytical representation is very accrate. Maniplating the expression for the T ð!, AÞ in eqation (), we can write the gain phase as (derivations not given in this paper), T ð!, AÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ n e! þ n e! þ n n e! ð þ Þ cos!ð Þ; ð3þ argðt ð!, AÞÞ ¼ n sin! þ e!ð Þ n sin! n cos! þ e! ð ; ð4þ Þ n cos! Valable interpretations can be obtained from these eqations for the gain phase of T ð!, AÞ. Eqation (3) shows that the gain depends on the time-lag difference, bt not on the time-lags individally. Of the three terms nder the sqare-root, only the third term is oscillatory; cos!ð Þ is periodic along the x-axis has a maximm at the freqency vale! max ¼ =ð Þ, hence the gain attains a Figre 6. The modelled configration, showing the acostic waves reflection transmission coefficients. maximm there. Figre 3 shows that x max moves to lower x-vales as A= increases, we can therefore conclde that mst increase with increasing A=. This is also the case for the analytical representation, as can be seen in Figre 5(b). Eqation (4) shows that the phase of the FTF depends on the difference, bt not on r or r individally. If sin! cos! in eqation (4) are approximated to Oð!Þ, the phase becomes!ðn þ n Þ=ðn þ n Þ, i.e. the phase crve starts with a slope which is given by a weighted time-lag. 3.4 Analytical model for the brner configration in terms of its tailored Green s fnction In order to model set-p in Figre analytically, we make the following assmptions:. The sond field is prely D, not only inside the tbe, bt also beyond its downstream end. The wave transmitted beyond this end is of corse 3D, bt we ignore this instead assme that the tbe has a semi-infinite contination, which keeps the transmitted wave D. There are forward backward travelling acostic waves inside the tbe (shown in Figre 6 by a þ a ) jst a forward travelling wave (shown by c þ ) in the semi-infinite contination section.. We model the downstream end as a combined interface (see Figre 6) consisting of a perforated plate at

7 Gopinathan et al. 7 x ¼ L, an open end at x ¼ L þ ; each interface has a given reflection transmission coefficient; the distance between them is very small,!. 3. The pstream end is modelled as rigid wall with a reflection coefficient of R ¼. 4. The mean temperatre speed of sond (denoted by c) are niform throghot the semi-infinite tbe. The pressre reflection transmission coefficient for individal interfaces (perforated plate open end) the combined interface of the configration in Figre 6 have been derived in Heckl. 6 R AC T AC are the reflection transmission coefficients at the combined interface for waves travelling in the forward direction R CA T CA are the reflection transmission coefficients for waves travelling in the backward direction. The expressions for these reflection transmission coefficients are given in Appendix. Interested readers are advised to refer to Heckl 6 for more details. The Green s fnction Gðx, x, t t Þ is the response observed at position x time t to a point sorce at position x firing an implse at time t. Its governing eqation ¼ ðx x Þðt t Þ: ð5þ The tailored Green s fnction 6,3 is the soltion of eqation (5), which satisfies the same conditions at all bondaries interfaces as the acostic field (here expressed in terms of the velocity potential). Natrally, this is a sperposition of modes, Gðx, x, ðt t ÞÞ ¼ Hðt t Þ< X n¼ g n ðx, x Þ e i! nðt t Þ : ð6þ Hðt t Þ denotes the Heaviside fnction; it garantees casality. The qantities g n (Green s fnction amplitde of mode n) x n (modal freqencies if thermoacostic copling is absent) have been calclated analytically for the combstion system shown in Figre 6 (see Heckl 6 for details on the derivation); the reslts are: with g n ðx, x Þ¼i ^gðx, x,! n Þ! n Fð! n Þ ^gðx, x,!þ ¼ Dðx,!ÞCðx,!Þ for L 5 x 5 x Cðx,!ÞDðx,!Þ for x 5 x 5 ð7þ ð8þ Cðx,!Þ ¼ ic e ikðx LÞ ; ð9þ Dðx,!Þ ¼Fð!Þ R CA e ikðx LÞ þ e ikðx LÞ þ R T AC eikx ; ðþ Fð!Þ ¼e i!lc R R AC e i!lc : ðþ Fð!Þ is the fnction appearing in the characteristic eqation Fð! n Þ¼, which determines the modal freqencies x n of the Green s fnction. 3.5 Stability analysis In or case, the Green s fnction is a velocity potential. The governing eqation for the velocity potential in the presence of nsteady heating is given c ¼ Bqðx, ðþ where B ¼ ð Þ=c (abbreviation), is the specific heat ratio. The term q(x, t) is the flctating part of the heat release rate per nit mass of air, for a compact flame (located at x q ), we can pt qðx, tþ ¼qðtÞðx x q Þ: ð3þ Following the procedre in Heckl 6, we can convert eqation () into an integral eqation for the velocity ðtþ at the heat sorce, Z t ðtþ ¼B t x, t t x ¼ x q x ¼ x q qðt Þ dt : ð4þ We note that one has to distingish between the velocity at the heat sorce position (x q ) the velocity at the FDF reference position (x ref ). Here, however, we have the special case where x q x ref (Noiray s velocity measrements were taken at the base of the matrix flame), so we can denote both velocities with the same symbol ðtþ. There is a direct physical interpretation of this eqation: the acostic velocity at observer time t observer location x q is given by the sm of velocity responses to a seqence of implses prodced at previos times t at the same location. If the modes are assmed to be non-interacting, eqation (4) can be converted into an ODE for the velocity of an individal mode; the mathematical maniplations can be fond in Heckl, 6 the reslt is ;=ð! n Þ _ þ j! n j ¼ B=ð! n G n ÞqðtÞþB<ðG nþ _qðtþ; ð5þ

8 8 International Jornal of Spray Combstion Dynamics () where G n is given by G n nðx, x Þ x ¼ x ¼ q L R T AC ei! nx q =c : x ¼ x q ð6þ B<ðG n Þ n Dð Þ sinðþd þ n Dð Þ sinðþd : ð3þ Clearly, eqation (5) is the eqation for a damped harmonic oscillator, forced by the flctating heat release rate q(t) its time derivative _qðtþ. The problem is closed if we have an analytical expression for q(t) in terms of ðtþ. This is provided by eqation (6), which is an expression for the global heat release rate ðtþ can be converted into local form, qðtþ ¼ n ðt ÞDð Þd þn ðt ÞDð Þd ; ð7þ is a constant factor, given by ¼ S ; ð8þ S is the cross-sectional area of the tbe. In order to simplify the sbseqent calclations we assme that ðtþ is harmonic with an nknown freqency, given by ðtþ ¼A cos t: ð9þ The time-lag terms ðt Þ in eqation (7) can be rewritten sbseqent sbstittion into eqation (5) gives simply (for details, see Appendix 3) with, þ a _ þ a ¼ ; ð3þ a ¼ =ð! n Þ B=ð! n G n Þ n Dð Þ sinðþ d þ n Dð Þ sinðþ d B<ðG n Þ n Dð ÞcosðÞd þ n Dð ÞcosðÞd ; ð3þ a ¼! n þ B=ð!n G n Þ n Dð Þ cosðþd þ n Dð Þ cosðþd Eqation (3) is the eqation for a damped harmonic oscillator. a is the damping coefficient hence an indicator of the stability behavior: mode n is stable if a nstable otherwise. a is the sqare of the oscillator s eigenfreqency. 3.6 Stability predictions We make stability predictions for the matrix brner with the properties listed in Table. Two parameters were varied to constrct stability maps: the tbe length L in the range from : m to :8 m, the amplitde A= in the range. We extrapolated the fitting parameters to nphysically high vales of A= to see whether we can detect a tendency for higher amplitdes (rather than to get reliable stability predictions for high amplitdes). The stability map for mode n ¼ is shown in Figre 7(a). This was determined in the following way: for each point ðl, A= Þ in the map, we calclated the coefficient a from eqation (3) (with x n from eqation (), G n from eqation (6), both for mode n ¼ ), noted the sign of a. Points with a (nstable) were marked in grey; points with a 5 (stable) were marked in white. The nstable region has the shape of a b emerging from the bottom left corner agrees qalitatively with Noiray s reslts 8,9 in the region where the experimental measrements are available. For comparison, the border of the instability region fond by Noiray et al. 9 is also shown in Figre 7(a) by the dashed black crve. The qantitative agreement is less Table. Geometry other parameters of the matrix brner. Parameter Vale Tbe radis, a.35 m Length of the tbe, L..8 m (variable) Thickness of the perforated plate, h.3 m Nmber of perforations per nit area, N :9 5 /m Radis of perforations, r p. m factor relating local global 3 5 m /s heat release rate, Distance of flame from. m perforated plate, x q L Specific heat ratio,.4 Speed of sond, c 345 m/s

9 Gopinathan et al. 9 (a) (b) (c) A/ ū.5.5.5t =.5T = A/ ū A/ ū L (m) L (m) L (m) Figre 7. Stability map of a matrix brner sing a two time-lag heat release rate law Green s fnction approach. (a) Stability map (nstable regions are grey; the dashed crve shows the reslts from Noiray et al. 9 ), (b) Contors of ð:5t Þ (c) contors of ð:5t Þ. satisfactory. Or predictions overestimate the size of the instability region. The stability map given in Noiray 8 Noiray et al. 9 was also obtained by extrapolating the FDF, however, this was done by setting the FDF gain to zero otside the measred freqency range. It is likely that the different extrapolation methods are responsible for the qantitative discrepancy between or stability map that in Noiray 8 Noiray et al. 9 We now investigate the inflence of the time-lags their amplitde-dependence on the stability behavior. In a qarter-wave resonator with a single time-lag in the heat release law, the stability behavior switches from stable to nstable (or vice versa) as crosses the threshold :5T, where T ¼ 4L=c is the period of mode (the mode nder consideration). With an amplitdedependent, the switch happens when the amplitde reaches the vale where ¼ :5T. If there are two prominent time-lags,, in the heat release law, we expect that they both inflence the stability throgh their amplitde-dependence. In order to illstrate this, we have prodced Figres 7(b) (c). Figre 7(b) shows contors for the difference ð:5t Þ Figre 7(c) for the difference ð:5t Þ. The zerocontors, i.e. the contors :5T ¼ :5T ¼, have been sperimposed on the stability map in Figre 7(a). We observe that they are very similar in shape to the bondaries of the nstable region. 4. Conclsions otlook In this paper, we presented a systematic method to approximate a known FDF (obtained elsewhere by measrement or nmerical simlations) by analytical expressions. After calclation of the corresponding UIR, it is possible to identify the time-lags that are prominent in the flame dynamics. We presme that these correspond to the travelling times of different pertrbations (e.g. flctations in trblence intensity or flctations in fel concentration), that therefore there are only a few prominent time-lags. The existence of jst a few sch prominent time-lags, each srronded by a decaying distribtion, allows s to represent the UIR as a sperposition of Gassian time-lag distribtions. This representation is Forier-transformed into the freqency domain an analytical expression for the FTF or FDF is obtained; the freqency-dependence of the FTF/FDF is also a sperposition of Gassians. We formlate a heat release rate law in the timedomain freqency-domain with several fitting parameters: the prominent time-lags (,,...), the stard deviations (r,,...) characterising the spread arond each time-lag, generalised copling coefficients (n, n,...). Ths or model has 3k parameters for k prominent time-lags. Each of the fitting parameters is determined by minimising the discrepancy with the original FTF data. Or model is well-sited to captre the main featres of a typical FTF: a vale of at zero freqency, an excess gain at low freqencies, a decay to zero at high freqencies (low-pass filter), a phase crve with nearly constant slope at low freqencies. We also formlate the amplitde-dependence of the fitting parameters,,..., r,,..., n, n,... analytically. Altogether, a fll analytical description of a measred FDF is obtained. These analytical expressions the tailored Green s fnction can be sed to predict the stability behavior of the combstion system nder consideration. We applied this method to a specific laboratory brner (Noiray s matrix brner) determined the analytical representation of its FDF. This was fond to be very accrate (the percentage error in the gain is well below % for most freqencies amplitdes). We sbseqently sed this analytical FDF to make stability predictions (based on a Green s fnction approach) presented them in the form of a stability

10 International Jornal of Spray Combstion Dynamics () map. We obtained good qalitative agreement with Noiray s measred stability map. Also, we observed that the stability map is qite sensitive to the fitting parameters the analytical description of their amplitde-dependence. Additional physical insight can be sed to obtain good accracy. The mltiple time-lag model is an improvement of the extended n model, which is somewhat hypothetical in that it featres a single time-lag term ðt Þ an instantaneos time-lag term ðtþ; it has been sed by Heckl 6,9 Bigongiari Heckl. The mltiple time-lag model has the following advantages: It determines the time-lags from the UIR of the flame is ths not only physically more sond, bt also closer to the actal data. It captres all key featres of a typical measred FDF: a gain of nity at zero freqency, excess gain at low freqencies, low-pass filter behavior. It gives good stability predictions, which are qalitatively similar to those obtained by Noiray. Or paper gives a good analytical approximation for the nonlinear heat release rate (FDF). It also provides a fast prediction tool for the stability of a combstion system sing this heat release rate law tailored Green s fnction. Or mltiple time-lag model can be applied for any flame with known FDF. Or stability analysis, which is based on the Green s fnction, can be applied to any combstor configration, for which the tailored Green s fnction is known analytically. In this paper, we have calclated the tailored Green s fnction for a rather simple configration: a qarter-wave resonator with a temperatre jmp. This calclation can be extended to more complex configrations, provided that these can be modelled as a combination of D elements. Featres that can be inclded in or model are for example: dmp plane between tbe sections, orifice between tbe sections, freqency-dependent reflection coefficients at the inlet otlet. However, not inclded in this paper are losses in the combstion system ths we over-predict the nstable region. This will be addressed in a ftre paper. Acknowledgements The presented work is part of the Marie Crie Initial Training Network Thermoacostic Aeroacostic Nonlinearities in Green combstors with Orifice strctres (TANGO). Declaration of Conflicting Interests The athor(s) declared no potential conflicts of interest with respect to the research, athorship, /or pblication of this article. Fnding The athor(s) disclosed receipt of the following financial spport for the research, athorship, /or pblication of this article: We grateflly acknowledge the financial spport from the Eropean Commission nder call FP7-PEOPLE-ITN-. References. Liewen TC Yang V. Combstion instabilities in gas trbine engines: operational experience, fndamental mechanisms, modeling. In: Progress in astronatics aeronatics. Vol., Virginia, USA: American Institte of Aeronatics Astronatics, Inc., 5.. Crocco L. Aspects of combstion stability in liqid propellant rocket motors. Part I: Fndamentals. Low freqency instability with monopropellants. J Am Rocket Soc 95; : Schermans B. Modeling control of thermoacostic instabilities. PhD Thesis, École Polytechniqe Fe de rale De Lasanne, Schermans B, Bellcci V, Gethe F, et al. A detailed analysis of thermoacostic interaction mechanism in a trblent premixed flame. In: Proceedings of ASME Trbo Expo, Power for L, Sea Air, Vienna, Astria, 4 7 Jne Komarek T Polifke W. Impact of swirl flctations on the flame response of a perfectly premixed swirl brner. J Eng Gas Trbines Power ; 3: Bade S, Wagner M, Hirsch C, et al. Design for thermo-acostic stability: modeling of brner flame dynamics. J Eng Gas Trbines Power 3; 35: Sbramanian P, Blmenthal RS, Polifke W, et al. Distribted time lag response fnctions for the modeling of combstion dynamics. Combst Theory Model 5; 9: Noiray N. Linear nonlinear analysis of combstion instabilities, application to mltipoint injection systems control strategies. PhD Thesis, École Centrale Paris, Noiray N, Drox D, Schller T, et al. A nified framework for nonlinear combstion instability analysis based on the flame describing fnction. J Flid Mech 8; 65: C osić B, Moeck JP Paschereit CO. Nonlinear instability analysis for partially premixed swirl flames. Combst Sci Technol 4; 86: Polifke W, Kopitz J Serbanovic A. Impact of the fel time lag distribtion in elliptical premix nozzles on combstion stability. In: 7th AIAA/CEAS aeroacostics conference exhibit, Maastricht, The Netherls, 8 3 May.. Flohr P, Paschereit CO, van Roon B, et al. Using CFD for time-delay modeling of premix flames. In: Proceedings of ASME Trbo Expo, New Orleans, Loisiana, USA, 4 7 Jne. 3. Flohr P, Paschereit CO Bellcci V. Steady CFD analysis for gas trbine brner transfer fnctions. In: 4st

11 Gopinathan et al. AIAA aerospace sciences meeting exhibit, Reno, Nevada, USA, 6 9 Janary Tay-Wo-Chong L, Bomberg S, Ulhaq A, et al. Comparative validation stdy on identification of premixed flame transfer fnction. J Eng Gas Trbines Power ; 34: Irashev D, Campa G Anisimov V. Response of swirl stabilized perfectly premixed flame to high-amplitde velocity excitations. In: The 3rd International Congress on Sond Vibration, Athens, Greece, Jly Heckl MA. Analytical model of nonlinear thermoacostic effects in a matrix brner. J Sond Vib 3; 33: Palies P, Drox D, Schller T, et al. Nonlinear combstion instability analysis based on the flame describing fnction applied to trblent premixed swirling flames. Combst Flame ; 58: Li J Morgans AS. Time domain simlations of nonlinear thermoacostic behavior in a simple combstor sing a wave-based approach. J Sond Vib 5; 346: Heckl M. A new perspective on the flame describing fnction of a matrix flame. Int J Spray Combst Dyn 5; 7: 9.. Bigongiari A Heckl MA. A Green s fnction approach to the rapid prediction of thermoacostic instabilities in combstors. J Flid Mech 6; 798: Gopinathan SM, Bigongiari A Heckl MA. Time-domain representation of a flame transfer fnction with generalised n law featring a time-lag distribtion. In: Proceedings of the 3rd international congress on sond vibration, Athens, Greece, 4 Jly 6.. Polifke W. Black-box system identification for redced order model constrction. Ann Nclear Energy 4; 67: Heckl MA Howe MS. Stability analysis of the rijke tbe with a green s fnction approach. J Sond Vib 7; 35: Gradshteyn IS Ryzhik IM. Table of integrals, series prodcts, 4th ed. London: Academic Press, Howe MS. Acostics of flid strctre interaction. Cambridge: Cambridge University Press, Levine H Schwinger J. On the radiation of sond from an nflanged circlar pipe. Phys Rev 948; 73: Appendix. Derivation of the FTF for the heat release rate law with mltiple time-lags Gassian distribtion of time-lags We assme that the distribtions in eqation (6) are sch that Dð j Þ, for 5 (j ¼,,..., k); then we can extend the integration range from ð, Þ to ð, þþ apply the Forier Transform to eqation (6), is Term zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ Z F ðtþ ðt Þ ¼FBn Dð Term zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ ðt Þ þ n Dð Þd þ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ ðt Þ þ n k Dð k ÞdC A : Term k ð33þ The Forier transform of the LHS of eqation (33) F ðtþ ¼ ^ð!þ : ð34þ By applying the convoltion theorem, Term j in eqation (33) can be written as Also, F ðtþ ðt Þ F n j Dð j Þd ð35þ ¼ n j F ðtþ F Dð j Þ : ¼ ^ð!þ F Dð j Þ ¼FðDðÞÞe i! j ; ð36þ the Forier transform of the Gassian fnction DðÞ is 4 Term j now becomes F n j ðt Þ FðDðÞÞ ¼ e! : ð37þ Dð j Þd! ^ð!þ j ¼ n j ei! j e : ð38þ Combination of eqations (34), (38) (33) gives ^ð!þ ¼ n ^ð!þ þ n k ^ð!þ ei! e! ^ð!þ þ n ei! e! þ ei! k e! k ð39þ

12 International Jornal of Spray Combstion Dynamics () T k ð!þ ¼ ^ð!þ ^ð!þ ¼ n e i! e! þ n e i! e! þ þ n k e i! k e! k : ð4þ jt BC j ¼, its phase is chosen sch that the velocity has continos phase at the tbe end. This gives T BC ¼ jt BC je i, with jt BC j ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jr BC j ¼ Argð R BC Þ: ð45þ Appendix. Reflection transmission coefficients at the interfaces The pressre reflection transmission coefficient for the combined interface have been derived in Heckl.; 6 the reslts are: R AC ¼ R AB R AB R BA R BC þ T AB T BA T BC R BA R BC T AC ¼ T ABT BC : R BA R BC ð4þ R AB, T AB are the coefficients of the perforated plate; these are given by (see Howe, 5 p. 36) Appendix 3. Harmonic oscillator eqation for a single mode The local heat release rate flctation q ðx, tþ is related to its global conterpart ðtþby ðtþ ¼ R q ðx, tþ dv, where the integration is taken over the volme of the flame. This redces to qðtþ ¼ ðtþ=ð SÞ for a compact flame described by eqation (3). Hence, with eqations (6) (8), we can write q(t) as Z qðtþ ¼ ðtþ n S ¼ ðt ÞDð Þd þ n ðt ÞDð Þd ; ð46þ R AB ¼!! þ in ck T in ck AB ¼! þ in ck ; ð4þ where N is the nmber of holes per nit area, K is the Rayleigh condctivity. For a plate of thickness h with circlar holes of radis r p (see Howe 5,p. 356 ), its derivative as _qðtþ ¼ n _ ðt ÞDð Þd þ n _ ðt ÞDð Þd : ð47þ K ¼ r p r p = þ h : ð43þ R BC is the reflection coefficient of an nflanged open tbe end; for a tbe with radis a, it is given by Levine Schwinger 6 R BC ¼ ½ð=4Þð!a=c Þ ið!a=cþ:633š þ½ð=4þð!a=c Þ ið!a=cþ:633š : ð44þ The wave radiated from the open end is modelled by a complex transmission coefficient, T BC, which is constrcted in the following way: its magnitde is chosen sch that acostic energy is conserved, i.e. jr BC j þ In order to simplify eqation (46), we assme that ðtþ is harmonic with an nknown freqency, given by ðtþ ¼A cos t; _ ðtþ ¼ A sin t: ð48þ ð49þ The time-lagged terms ðt Þ _ ðt Þ can be written with the trigonometric addition formlae as ðt Þ ¼ ðtþ cos _ ðtþ sin ð5þ

13 Gopinathan et al. 3 : ðt Þ ¼ ðtþ sin þ _ ðtþ cos : ð5þ The local heat release rate flctation q(t) its derivative then become qðtþ¼ n Dð Þð ðtþsin þ _ ðtþcosþd þ n Dð Þð ðtþsin þ _ ðtþcosþd, ð5þ _qðtþ¼ n Dð Þ ðtþcos _ ðtþ sin d þ n Dð Þ ðtþcos _ ðtþ sin d : ð53þ The freqency can be approximated by Reð! n Þ. Sbstitting eqations (5) (53) into eqation (5), sorting the many terms, we get eqation (3).