GT Proceedings of the ASME Turbo Expo 2013 GT 2013 June 3-7, 2013, San Antonio, USA

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1 Proceedings of the ASME Turbo Expo 213 GT 213 June 3-7, 213, San Antonio, USA GT PREDICTION OF THE NONLINEAR DYNAMICS OF A MULTIPLE FLAME COMBUSTOR BY COUPLING THE DESCRIBING FUNCTION METHODOLOGY WITH A HELMHOLTZ SOLVER. Alexis Cuquel [1,2], Camilo Silva [3], Franck Nicoud [4], Daniel Durox [1,2], Thierry Schuller [1,2] [1] CNRS, UPR 288 Laboratoire d Energétique Moléculaire et Macroscopique Combustion (EM2C), Grande Voie des Vignes, Châtenay-Malabry, France [2] Ecole Centrale Paris, Grande Voie des Vignes, Châtenay-Malabry, France [3] Technische Universität Müchen D Garching, Germany [4] Université Montpellier II and CNRS, UMR 5149 Institut de Mathématiques et Modélisation de Montpellier (I3M), 3495 Montpellier, France ABSTRACT This study focuses on the numerical determination of thermo-acoustic instabilities using a combination of the Flame Describing Function (FDF) methodology and a numerical code solving the Helmholtz equation. In this framework, the FDF is defined by a set of Flame Transfer Functions (FTF) that depend on both the frequency and amplitude of acoustic perturbations. The FDF methodology has been recently used in combination with acoustic network methods to examine the nonlinear stability of generic configurations with simplified geometries. Its extension to complex 3D geometries requires the use of numerical tools such as a Helmholtz solver. In the present work, that combination is validated on a multiple injection combustor. The implementation of the FDF methodology in the Helmholtz solver is detailed before examining numerical predictions obtained by the use of an experimentally determined FDF in the Helmholtz solver. The instability frequencies and growth rates are determined for each perturbation level and different nonlinear behaviors are exhibited depending on the combustor geometry. The case of linearly unstable modes reaching limit cycles is first examined. A more complex case involving mode switching is then examined when two unstable modes are present. In this situation, the most unstable mode in the linear regime triggers another Address all correspondence to this author: alexis.cuquel@ecp.fr unstable mode at a higher perturbation level. These numerical calculations are compared with experimental data and exhibit a good match in terms of amplitude and frequency reached by the limit cycle. NOMENCLATURE F (ω) Flame Transfer Function (FTF). F (ω, ṽ 1 ) Flame Describing Function (FDF). G(ω) FTF or FDF gain. ϕ(ω) FTF or FDF phase lag. ω Angular frequency. ω r Real part of the angular frequency. ω i Growth rate or imaginary part of the angular frequency. Q Global heat release rate (integrated over the flame volume). q Local volumetric heat release rate. v Velocity vector. u,v Radial and axial velocity components (scalars). p Pressure. c Speed of sound. γ Heat capacity ratio. ζ (ω) Specific impedance. x Vector of coordinates. K a Rayleigh conductivity. h Perforated plate thickness. 1 Copyright c 213 by ASME

2 σ a d Perforated plate porosity. Perforated plate hole radius. Inter-orifice spacing of the perforated plate. Superscripts relative to the downstream perforated plate surface. + relative to the upstream perforated plate surface. Subscripts Steady value. 1 1 st -order perturbation. relative to the reference point used for FTF determination. ref INTRODUCTION By limiting the operating range of modern gas turbines, thermo-acoustic instabilities have become an issue at the design stage of combustors. It has been long known that these selfsustained flow oscillations involve a strong coupling between the burner acoustics and the combustion dynamics, eventually leading to severe damages to the combustor itself [1 3]. A robust way to predict these instabilities is still needed. Early investigations relying on acoustic modal analysis have shown that the instability frequencies lie close to the acoustic eigenmodes of the combustor, but this type of analysis does not provide any information on the system stability. Later, in order to predict the onset of such instabilities, linear stability analyses have been developped and performed to determine the growth or damping rate of the identified modes [4 6]. This type of analysis takes into account the flame response to flow perturbations by including the effects of heat release rate perturbation into the acoustic description of the system dynamics. It is usually done by means of the Flame Transfer Function (FTF) [7], relating the dimensionless heat release rate perturbations Q 1 / Q to the dimensionless axial velocity modulations ṽ 1 /v at some location in the fresh reactant stream: F (ω) = G(ω)e iϕ(ω) = Q 1 / Q ṽ 1 /v (1) where ω refers to the angular frequency, G(ω) stands for the FTF gain, ϕ(ω) is the FTF phase lag and ṽ 1 denotes the velocity fluctuation at the probe location. In this approach, the gases are supposed to be perfectly premixed and equivalence ratio fluctuations are not considered here. The FTF can be determined experimentally [8 1], numerically [11 13] or even analytically in simplified geometries [14, 15]. It can be coupled to a low-order acoustic network model [4,16] to give a dispersion relation D(ω) = which solutions ω = ω r +iω i may be used to determine the system stability, depending on the sign of ω i. It also provides the corresponding instability frequency ω r as well as the mode structure. This type of analysis is however limited to the linear domain where perturbation amplitudes are small and thus fails in predicting nonlinear instability characteristics such as the frequency and amplitude of a limit cycle. While nonlinear acoustics can be of significant importance in the case of rocket engines [17], it was shown that the flame is often the main nonlinear component of the system [9, 18 21]. Recently, new methods have been developed to take into account this nonlinearity. The concept of Flame Describing Function (FDF) was introduced by Dowling [18] in a theoretical study where the gain was considered to saturate at some constant value above a certain perturbation level threshold. This method was generalized and first used by Noiray et al. [2] by considering a set of FTF that depend on both the frequency and perturbation level: F (ω, ṽ 1 ) = G(ω, ṽ 1 )e iϕ(ω, ṽ 1 ) = Q 1 / Q ṽ 1 /v (2) where ṽ 1 denote the velocity perturbation amplitude. Each FTF in this description is determined for a fixed input level. Combined to a low-order acoustic model of the system studied, the method leads to an extended dispersion relation D(ω, ṽ 1 ) = that also depends on the perturbation level. The solution of that equation ω( ṽ 1 ) = ω r ( ṽ 1 ) + iω i ( ṽ 1 ) is thus parametrized by the perturbation level. It is then possible to describe the evolution of the instability from its onset in the linear regime up to the limit cycle in the nonlinear regime. Morever, it enables to predict the limit cycle amplitude by examining the growth rate evolution and finding the perturbation level for which ω i ( ṽ 1 ) =. This method has been used to predict nonlinear thermo-acoustic behaviors in different configurations. The original setup comprises a multiple injection system where a collection of laminar conical flames are anchored on a perforated plate in the absence of a confinement tube [2]. Validations with a confinement tube [22, 23], and also in a confined swirling flame case [24] were recently conducted as well. This method was proven successfull in predicting the instability onset as well as the limit cycle frequency and amplitude. Moreover, several typical non-linear phenomena were retrieved such as hysteresis, mode switching and triggering. The extension of this methodology to practical combustors featuring 3D complex geometries requires the use of numerical tools. One possible way to achieve that is to solve the sound wave equation in the frequency domain, i.e. the Helmholtz equation [25]. These simulations were already used to determine the modal distribution [26] and more recently to characterize the linear stability of combustor with FTF [27 3]. These calculations lead to the same type of results than low-order acoustic meth- 2 Copyright c 213 by ASME

3 ods, but they enable to take into account the real geometry of the combustor as well as accurate speed of sound and temperature fields in the combustor, acoustic damping associated to multiperforated liners and complex boundary conditions at the system extremities. In the present work, the FDF methodology is used in combination with an acoustic solver to perform a nonlinear stability analysis on a generic configuration exhibiting thermo-acoustic instabilities. This type of simulation was already considered in a recent article [31] to investigate limit cycles of linearly unstable modes with simplified acoustic boundary conditions. It is used here to investigate linearly and nonlinearly unstable modes in a configuration featuring complex impedances and a complex response of the injection unit. The acoustic solver AVSP used for that purpose is presented in the first section with a focus on the treatment of the forced Helmholtz equation and the boundary conditions. The configuration used to validate the simulations is introduced in the section 2. The experimental setup is first briefly described. A more thorough description of the numerical setup is then done including numerical domain, mesh and boundary conditions. In section 3, a first validation without combustion is carried out. Finally, results from the nonlinear stability analysis are presented and are systematically compared to experimental data from [23]. The case of linearly unstable modes is examined first. A more complex case involving mode switching is then examined when two unstable modes are present. 1 THE AVSP ACOUSTIC SOLVER The AVSP code is a solver developped at CERFACS and University Montpellier II that uses a finite volume formulation on unstructured grids with tetrahedral elements [25,32]. It solves the wave equation with a volumetric source term in the frequency domain, i.e. the Helmholtz equation written here with a right hand side forcing term corresponding to a volumetric heat release rate source term q 1 (W.m 3 ): (c 2 p 1 ) + ω 2 p 1 = iω(γ 1) q 1 (3) where c stands for the speed of sound, γ represents the heat capacity ratio and ω is the angular frequency. Inclusion of the FDF in the AVSP formalism Effects of steady combustion are reflected by a change in the speed of sound appearing in the left hand side of Eq. (3). Unsteady combustion effects appear in the right hand side of the Eq. (3) through heat release rate perturbations q 1. This quantity is a key point to model in order to predict thermo-acoustic instabilities because it provides the energy necessary to trigger and sustain the instability. Equation (3) is modified to make the axial velocity perturbations appear: (c 2 p 1 ) + ω 2 p 1 = iω(γ 1) q 1 (4) [ ] q = iω(γ 1) 1 / q q ṽ 1 (x ref )/v (x ref ) v (x ref )ṽ1(x ref ) (5) where ṽ 1 (x ref ) is the axial velocity perturbation at a reference point location, v (x ref ) is the steady axial velocity at the same reference point considered for the FTF determination and q is the steady volumetric heat release rate. The term between brackets in Eq. (5) is thus defined as a local Flame Transfer Function: F loc (ω r, ṽ 1 (x ref )/v (x ref ) ) = q 1 / q ṽ 1 (x ref )/v (x ref ) The velocity perturbation at the reference point can be related to the pressure gradient at the same location using the linearized momentum balance: (6) ṽ 1 (x ref ) = 1 iωρ ref ref p 1 (x ref ) n ref (7) where ρ ref is the density at the reference point and n ref is the unit vector along the direction of the velocity component used for FTF determination, corresponding here to the axial direction. It leads to the following expression: (c 2 p 1 ) + ω 2 p 1 = (γ 1) q ρ ref v (x ref ) F loc ref p 1 (x ref ) n ref (8) The remaining issue is to link the local FTF F loc to the global FTF F defined in Eq. (1). An integration of the local heat release rate perturbations over the flame volume yields: Q 1 = q 1 dv (9) V F where V f is the flame volume. It can be shown that for a compact flame, i.e. a flame which dimensions are much smaller than the acoustic wavelength, integration of Eq. (6) leads to [31]: F loc (ω, ṽ 1 (x ref )/v (x ref ) ) = F (ω, ṽ 1 (x ref )/v (x ref ) ) (1) The local and global FTF are equal for a compact flame. 3 Copyright c 213 by ASME

4 Boundary condition treatment in AVSP The acoustic solver AVSP allows to consider non-perfectly reflective boundary conditions through the use of specific impedances that are frequency dependent: ζ (ω) = p 1 ρ c ṽ 1 n (11) where ρ is the fresh gas density, c is the speed of sound and n is the unit vector normal to the boundary surface. Another important characteristic of the AVSP solver is that it allows to take into account discontinuities in the sound field such as those introduced by modeling the acoustic response of compact perforated plates by a pressure jump between two boundary faces noted (+) and ( ) and by the continuity of the acoustic volume flowrate: ṽ + 1 (x+ ) n b = ṽ 1 (x ) n b (12) + [ p 1 (x + ) p 1 (x ) ] = iωρ d 2 ṽ 1 (x) n b K a (13) where n b is the unit vector normal to the perforated plate surface, d is the inter-orifice spacing of the perforations and K a is the Rayleigh conductivity of a perforation. The latter parameter has to be modeled to enforce the pressure jump through the perforated plate. In [27, 29, 33], the Rayleigh conductivity K a was linked to the perforated plate characteristics using the model derived by Howe [34]. This model was implemented in the ASVP code, validated against an analytical solution in the case of a simplified annular geometry and then used to compute the eigenmodes of a full helicopter chamber [33, 35]. In the present work, the presence of small conical flames anchored on the perforated plate prevent the flow from generating vortices [16]. It thus makes the Howe model not valid for the present configuration. We used instead an alternative model developped by Melling [36] taking acoustic dissipation within the perforation into account, but neglecting external dissipation due to vortex shedding at the perforation outlets. This model leads to the following expression for the pressure jump discontinuity: + [ p 1 (x + ) p 1 (x ) ] [ = iωρ h = h σ ( 1 + l ν a 1 + l ν (1 + i) a ] ṽ1 (x ) n b (1 + i) σ ) (14) p 1 (x ) n b (15) where h is the plate thickness, l ν = (2ν/ω) 1/2 is the viscous acoustic boundary layer thickness, ν is the kinematic viscosity, a is the perforated plate hole radius and σ is the perforated plate porosity. The pressure jump is imposed between the two surfaces of the perforated plate. FDF methodology for the AVSP acoustic solver The discretization of the Helmholtz equation Eq. (8) along with the boundary conditions over an unstructured grid with a finite volume method leads to an eigenvalue problem that reduces to the following form [25]: A P + ωb(ω)p + ω 2 C P = D(ω)P (16) where P is a vector containing the eigenvector values, A and C are matrices containing coefficients coming from the discretization of the homogeneous Helmholtz equation, B(ω) contains the information about the boundary conditions (see Eqs. (12) and (15) ) and D(ω) represents the unsteady flame source term including the FDF. When using frequency dependent boundary or jump conditions (such as an impedance condition or pressure jump through a perforated plate) or a frequency dependent heat release rate perturbation field (when using a FTF or a FDF for example), the matrices B(ω) and D(ω) are a function of the angular frequency. This results in a nonlinear eigenvalue problem that is resolved with a fixed-point iterative method [25]. Equation (16) is first reduced to a linear eigenvalue problem that is defined for the k th iteration, by: [A + Ω k B(Ω k )) D(Ω k )]P + ω 2 k C P = (17) where Ω k = f (ω k 1 ) is a function of the previous iteration result. This linear eigenvalue problem is solved by an Arnoldi iterative method. More details about that procedure can be found in [25, 32, 35]. The fixed-point method is pursued until the error defined by ε = ω k Ω k /ω k is lower than a specified value, typically of the order of 1%. Two fixed-point algorithms have been tested here. The simplest algorithm possible states that Ω k = ω k 1. It uses the result from the previous iteration to compute the acoustic variables at the boundaries and to estimate unsteady flame effects. This method is effective when it comes to find attractive fixed points but it is not able to retrieve repulsive fixed points. To tackle that problem, a second algorithm was developped [31] by introducing a relaxation coefficient β and stating that Ω k = (1 β)ω k 1 + βω k 1. When using the relax value β =, this algorithm reduces to the first fixed point method. In cases where the first algortihm does not converge, a value of β =.5 is usually used with the second algorithm. 4 Copyright c 213 by ASME

5 FIGURE 2. One quarter of the numerical domain is represented here along with the mesh used for the simulations. The location of the perforated plate is delimited by the empty space in between the cavities. The refined mesh right above the plate correspond to the flame anchoring region. FIGURE 1. Sketch of the experimental configuration used to validate the simulations. Reproduced from [37]. 2 CONFIGURATION The configuration studied in this article is sketched in Fig. 1. The dynamics of this combustor was extensively investigated in previous studies [22, 23]. It comprises two cavities which are related to each other by a perforated plate. Small conical flames are anchored on the perforated plate and constitute a compact flame zone with respect to the acoustic wavelength. The feeding manifold length L 1 can also be modified by moving the piston up or down. Numerical setup The numerical domain comprises two distinct domains that correspond to each cavity of the experimental configuration explored. The feeding manifold is taken as a cylinder of radius 36.1 mm and variable length L 1. The combustion chamber is a cylinder of radius 65 mm and of length L 2 = 91 mm. The perforated plate separating the two cavities has a radius R pp = 78 mm and thickness h = 3 mm. A quarter of the numerical domain along with the mesh is presented in Fig. 2. It is worth noting that the mesh has been refined above the perforated plate, where the flame region is located. Effects of steady combustion are taken into account through a jump in gas temperature. The plenum temperature is taken as 3 K and the confinement tube temperature is taken as 3 K for cold flow acoustic eigenmode calculations and as 9 K for thermo-acoustic instability simulations. Eventhough no explicit heat losses have been included in the model, a burnt gas temperature lower than the adiabatic flame temperature is considered here. This value results from estimates based on measurements along the flame tube external surface which were averaged. More details are provided in [37]. Heat release rate perturbations are taken into account as described in section 2. The volumetric source term appearing in the right hand side term in Eq. (3) is distributed over the region indicated in red in Fig. 3. This distribution corresponds to a cylindrical zone that is defined by a radius of 35 mm and a height of 2 mm, and it is located 1 mm above the perforated plate upper wall. These dimensions are of the same order of magnitude than those measured during operation of the burner. The unsteady heat release rate field is thus made compact with respect to the acoustic wavelengths considered in the present study ( f < 1 khz). 5 Copyright c 213 by ASME

6 with the classical expression used for an unflanged open pipe [38, 39]: Z(ω) = 1 4 ( ωr c ) 2 i.61 ωr c (18) FIGURE 3. Location of the source term appearing in the right hand side in (Eq. 3). Measurements A hot wire, a photomultiplier and microphones were used to measure the velocity, heat release rate and sound pressure level at different locations in the plenum. The data were used to retrieve the frequency, amplitude and structure of the unstable modes. FDF measurements were determined in a different experimental setup equipped with a loudspeaker to fix the perturbation level and frequency of the acoustic perturbations. The FDF used in the present simulations is plotted in Fig.4-left for several perturbation levels. The FDF gain exhibits a low-pass filter behavior and severely drops when the perturbation level is increased. The FDF phase increases regularly with frequency but also depends on the input level. The corresponding reflection coefficient is plotted in Fig. 4- right for two different values of the gas temperature in the confinement tube: T b = 3 and 9 K. Calculations conducted with the first temperature are used to determine the acoustic eigenmodes of the combustor without combustion. Simulations with combustion were conducted with the second temperature. The temperature of the burnt gases is fixed in these simulations with values found in the experiment [37]. The outlet reflection coefficient is a function of frequency and one can observe that the modulus drops at high frequency. The reflection coefficient at the burner outlet also depends on the burnt gas temperature chosen. These boundary conditions have already been used in [37] coupled to low-order acoustic network models. Results from these analytical models were compared to the results from models derived with simple boundary conditions and to a large experimental database gathered about the different self-sustained oscillations. These comparisons actually showed that taking into account such realistic boundary conditions enables to obtain more accurate predictions. Moreover, no depedency on the perturbation level has been considered for the boundary conditions. It was shown in the experiments that the reflection coefficient from the piston head does not depend on the sound level. The sound level reached at the combustor outlet is limited indicating that the linear mode chosen to describe its response is a reasonable approximation. Boundary conditions The eigenmode determination being very sensitive to boundary conditions, a special care has been put in using realistic inlet and outlet impedances. At the inlet, the piston head reflection coefficient has been measured in a separate experiment [37] and is plotted as a function of frequency in Fig. 4-center. One can see that the reflection coefficient modulus exhibits significant differences with a perfectly reflecting boundary condition R inlet = 1, but its phase remains close to zero. The reflection coefficient has been measured with gas flowing out of the piston as in the experiments with thermo-acoustic instabilities. Measurements were also performed in the abscence of flow to compare predictions of the acoustic eigenmodes in the absence of combustion. At the combustor exit, the radiation impedance is modeled 3 RESULTS The results obtained by a combination of the acoustic solver AVSP and the FDF methodology are presented in this section. A first case without combustion is presented along with an experimental validation. Then, linearly unstable modes are shown to reach a limit cycle which amplitude and frequency compare well with measurements. Results from the AVSP/FDF methodology are also compared to predictions from a low-order acoustic network model (LOM) [22, 23, 37]. This model features two main acoustic cavities separated by a perforated plate. An annular cavity is also considered to model the space between the perforated plate and the confinement tube and the boundary conditions previously presented are also used in this description. 6 Copyright c 213 by ASME

7 Gain R R Phase (rad) R (rad) π/2 π/2 π 5 1, R (rad) 3π/2 π π/ FIGURE 4. (left) Flame Describing Function gain (top) and phase lag (bottom) as a function of the frequency for increasing perturbation levels with a constant increment of.5. The perturbation levels ṽ 1 /v are indicated in the legend only for five curves, including the highest and lowest perturbation level values. (center) Piston-head reflection coefficient determined experimentally with flow (black line with ) and without flow (grey line with ). (right) Theoretical outlet reflection coefficient plotted for different gas temperatures T = 3 K (black line) and T = 9 K (grey line). Acoustic modes without combustion A first calculation is performed here to determine the acoustic eigenmodes from the experimental configuration investigated without combustion. The acoustic eigenfrequencies were also determined experimentally on the configuration sketched in Fig. 1. An external loudspeaker was used to perturb the flow out- Plenum length Exp. AVSP LOM L 1 =.12 m 462 Hz 472 Hz 482 Hz 882 Hz 881 Hz 99 Hz L 1 =.35 m 225 Hz 219 Hz 228 Hz 64 Hz 68 Hz 69 Hz 792 Hz 814 Hz 844 Hz 119 Hz 1194 Hz 1225 Hz TABLE 1. Acoustic eigenmodes of the combustor determined experimentally, numerically with the AVSP solver and with a low-order model (LOM). In the simulations, the following temperature were considered: T plenum = 3 K and T tube = 3 K. side of the confinement tube while two microphones were set to gather the pressure time signals in front of the loudspeaker and at the bottom of the plenum. A transfer function between these signals was computed enabling the determination of the resonnant frequencies of the sytem. The acoustic eigenfrequencies were determined numerically with a constant temperature of 3 K in the whole combustor. The numerical results from these simulations as well as the predictions from the LOM are compared to the experimental data in Tab. 1 for two different geometrical configurations corresponding to two plenum lengths L 1 =.12 and.35 m. The predicted and calculated eigenfrequencies of the combustor match well, within 3 % with those from measurements. Linearly unstable modes From that point, all the following results have been obtained by taking into account steady and unsteady combustion effects. A temperature of 3 K is considered in the plenum while the burnt gas temperature in the confinment tube is set to 9 K. A first geometrical configuration is considered here for a plenum length of L 1 =.35 m. Preliminary results obtained from the LOM indicate that the second acoustic mode of the plenum is unstable (i.e. featuring a positive growth rate ω i > ) and experimental data reveal a limit cycle which frequency corresponds to 7 Copyright c 213 by ASME

8 1 1 ω i (rad.s 1 ) 5 ω i (rad.s 1 ) v 1 / v v 1 / v FIGURE 5. Results from a non-linear stability analysis carried out on the second acoustic eigenmode for L 1 =.35 m. The instability frequency f and growth rate ω i are plotted against the perturbation level v 1 /v. Numerical results from AVSP ( ) and from LOM (-) are compared to experimental data ( ) from [23]. FIGURE 6. Results from a non-linear stability analysis carried out on the third acoustic eigenmode for L 1 =.54 m. The instability frequency f and growth rate ω i are plotted against the perturbation level v 1 /v. Numerical results from AVSP ( ) and from LOM (-) are compared to experimental data ( ) from [23]. that mode [23]. The results from the numerical simulations carried out with the AVSP solver are presented in Fig. 5 in terms of instability frequencies f and growth rates ω i as a function of the perturbation level v 1 /v. This mode is linearly unstable (ω i > at small perturbation levels) but the growth rate decreases as the perturbation level increases until reaching a zero value around v 1 /v =.45. This value determines the limit cycle amplitude and allows one to read the predicted limit cycle frequency that is roughly equal to 68 Hz. It is worth noticing that results from AVSP simulations (blue circles) are quite close to the LOM predictions (blue line), especially when considering the growth rate. A comparison between the different predictions and experimental data measured at the limit cycle (red square - extracted from Figs. 9 and 11 in [23]) show that the AVSP solver is able to retrieve correctly the limit cycle amplitude found experimentally at v 1 /v =.4. Mode switching A second validation is carried out with a different geometrical configuration for a longer plenum length L 1 =.54 m. Results from the simulations are plotted in Figs. 6 and 7 for the third and the second acoustic eigenmodes, respectively. By studying the growth rate evolution of the third mode plotted in Fig. 6, one can conclude that the mode is linearly unstable and that the insta- bility will grow untill the perturbation amplitude reaches a value of v 1 /v =.35. If the instability amplitude is further increased, the growth rate becomes negative and higher fluctuation levels cannot be sustained for this mode. Considering now the second mode in Fig. 7, simulations indicate that it is triggered only for perturbation levels larger than v 1 /v =.3 and is unstable. This mode is thus nonlinearly unstable in the intermediate range of perturbation levels. Its growth rate decreases for higher oscillation amplitudes and reaches zero around v 1 /v =.65, yielding the limit cycle amplitude predicted by these simulations. In Fig. 6, results from the simulations are compared to LOM predictions and experimental data. One can notice that predictions from AVSP or LOM are very close, with differences limited to 2 Hz. The experimental data presented in Fig. 6 were determined from the velocity signal measured by the hot wire after ignition (see Fig. 9 in [22]). In the experiments, the third mode is unstable and thus initially growing but also vanishing at a later time when it gives way to the second mode. This signal was thus filtered around the third mode frequency to isolate the third mode component and the third mode frequency was determined by calculating the power spectral density distribution. The amplitude represented in Fig. 6 by the dashed red line is the largest one reached by this filtered signal, i.e. the one for which ω i =. These experimental data match predictions very closely. Predictions for the second mode and experimental data are compared 8 Copyright c 213 by ASME

9 ω i (rad.s 1 ) 1 5 remains positive up to a larger amplitude v 1 /v =.65. This mode switching phenomenon can thus be retrieved also with the AVSP code even though slight differences were observed in the way the system reaches the limit cycle. CONCLUSION v 1 / v FIGURE 7. Results from a non-linear stability analysis carried out on the second acoustic eigenmode for L 1 =.54 m. The instability frequency f and growth rate ω i are plotted against the perturbation level v 1 /v. Numerical results from AVSP ( ) and from LOM (-) are compared to experimental data ( ) from [23]. in Fig. 7. A first difference can be observed between predictions from AVSP and LOM. At small amplitude, calculations from the LOM indicate a very small growth rate but the AVSP solver does not detect any mode. At intermediate perturbation levels comprised between.3 and.5, the growth rates calculated with the LOM and AVSP slightly differ, but the oscillation frequencies match well. At high perturbation level, predictions from AVSP and LOM collapse leading to the same limit cycle with a frequency of 46 Hz and an amplitude v 1 /v =.65. In [22], this case was thoroughly studied and experiments exhibited a situation where the third mode oscillation starts to grow and then naturally switches to the second mode oscillation for a certain oscillation threshold level. This phenomenon was well retrieved by the LOM predictions. By comparing the growth rate evolutions in Figs. 6 and 7, the AVSP results show that the third mode is linearly unstable and features growth rates larger than those associated to the second mode. The situation changes for a certain threshold level when the oscillation reaches an amplitude v 1 /v =.35. At that amplitude, the growth rate of the third mode drops to zero, what means that this mode has reached limit cycle. Simultaneously, the growth rate of the second mode features large positive value, meaning that the second mode amplitude keeps increasing. At larger amplitude, the growth rate of the third mode is negative, meaning that this mode is damped. Therefore, only the second mode remains because its growth rate The stability of a multipoint injection combustor with respect to thermo-acoustic instabilities has been assessed by using a combination of a numerical code that solves the inhomogeneous Helmholtz equation and the Flame Describing Function (FDF) methodology. The latter enables to take into account the nonlinear dependence of the flame frequency response to flow perturbations while the former allows to deal with real geometries and complex boundary conditions. The methodology is validated by comparing predictions with experiments and with predictions from a low-order model. It is shown that it is possible to reproduce linearly and nonlinearly unstable modes with the AVSP/FDF solver as well as the saturation of acoustic energy transfer that leads to limit cycles. Both the frequency and amplitude of these limit cycles were retrieved. These validations in a situation where the flame is compact in a generic combustor with a simplified geometry demonstrate the potential of this nonlinear AVSP/FDF methodology to adress more complex configurations where it is more difficult to develop a low-order model of the system dynamics. Finally, a highly nonlinear phenomenon was retrieved involving switching between two unstable modes. Future work will concern more complex configurations to test the impact of the combustor geometry and the spatial distribution of heat release rate perturbations in the combustion chamber in the case of non-compact flames. ACKNOWLEDGMENT The authors would like to express their gratitude to F. Boudy, PhD at EM2C, for sharing the experimental data and his help on low-order modeling as well as E. Motheau, PhD student at CER- FACS, for its support on the acoustic solver AVSP. The research leading to these results has received funding from the European Community s Seventh Framework Programme (FP7/27-213) under Grant Agreement #ACP8-GA This is part of the 4-year KIAI project started in May 29, which is a European initiative financed under the FP7 and addresses innovative solutions for the development of new combustors in aero-engines. It aims at providing low NOx methodologies to be applied to design these combustors. REFERENCES [1] Rayleigh, L., The explanation of certain acoustic phenomena. Nature, 18, pp Copyright c 213 by ASME

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