Modeling the response of premixed flame transfer functions - Key elements and experimental proofs

5th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition 9-12 January 212, Nashville, Tennessee AIAA 212-985 Modeling the response of premixed flame transfer functions - Key elements and experimental proofs T. Schuller, A. Cuquel, P. Palies, J.P. Moeck, D. Durox, S. Candel. Laboratoire EM2C, CNRS and Ecole Centrale Paris, Châtenay Malabry, France A wide variety of analytical models has been proposed to describe the transfer function of premixed flames submitted to flow disturbances. These models generally rely on a kinematic description of the flame reaction surface which is perturbed around its steady state. The response then depends on the type of flow perturbation considered, the mean flow characteristics and flame properties. Systematic comparisons between model predictions and experimental results are however less numerous. This study revisits some of these theoretical descriptions and their underlying hypotheses to delineate their domain of application. The main features of the response of premixed flames to flow disturbances are highlighted. Different mechanisms are identified and their analytical representation is discussed. It is shown that experimental observations provide useful guidelines in the flame transfer function modeling. The cases investigated include laminar and turbulent configurations in conical or inverted V-flame configurations. Effects of swirl, flame root dynamics and confinement are emphasized. I. Introduction The objective of this study is to revisit current analytical approaches to the modeling of the response of premixed flames to incoming flow disturbances with emphasis on analytical descriptions of the flame transfer function. One question which immediately arises is that of the value of such models in a context where numerical simulation is progressing at a fast pace. With the rapid increase of computational resources, it is now possible to simulate the unsteady reacting flow in practical combustor configurations and examine the flame interaction with acoustic waves. This can then be used to calculate self-sustained combustion oscillations during unstable regimes (see for example 1 5 ) or to determine the response of the flame to externally imposed flow modulations (see for example 6 8 ). Perhaps the largest of such a calculation is that of a self-sustained instability in a full annular helicopter combustion chamber featuring 15 burners reported recently. 9,1 This type of simulations remains exceptional. They are time consuming and require large scale computing facilities that are not always available. They are generally used to reproduce some features observed experimentally, but are not sufficiently mature to envisage the prediction of unstable regimes at the design stage. It is also difficult to transfer results observed for the flame dynamics in these simulations to other situations corresponding to different operating conditions, thermal power or geometrical arrangements. It is however important to note that some recent work has demonstrated that it is possible to determine the flame transfer function over the frequency range of interest at relatively low computational cost using specific forcing signals and digital processing of the output data. 11 Theoretical analysis of flame response to flow disturbances, while limited to generic configurations with simplified geometry, offers a general framework to examine key elements controlling the flame transfer function and define scaling rules that may be used to ease the design of burners by taking into account unsteady effects. In low NOx fuel lean combustors, turbulent combustion mainly takes place in the premixed mode where the flame is wrinkled by turbulence and eventually by large scale flow disturbances under self-sustained combustion instabilities. It is then natural to treat the flame as a thin sheet and analyze how flow perturbations modify this interface. This problem is analyzed in section 2 for laminar flames, and more specifically when they are inclined with respect to the main flow direction. The transfer function of conical flames has been the subject of several theoretical investigations and is probably the best known to date. This FTF is Also with Institut Universitaire de France 1 of 17 Copyright 212 by Thierry Schuller. Published by the, Inc., with permission.

burned gases G > n = G G ξ(x) α flame front G(x;t) = w v S d n unburned gases G < Y ξ (X) y X η v x Figure 1. Left figure : Schematic representation of a flame sheet by an interface G =. S d is the flame front normal displacement speed with respect to the unburned gases, n = G/ G is the local unit normal, and v is the unburned gas velocity at the front location G = with respect to the laboratory frame. Right figure : inclined flame with an angle α with respect to the mean flow direction. The quantities ξ() and ξ(x) denote perturbations of the flame position at the flame root (X = ) and the current location X in the frame (X,Y) attached to the steady flame. examined in section 4 to reveal the main parameters controlling the flame response. The case of laminar inverted conical flames is studied in section 5 where interaction of large vortices with the flame adds a new difficulty to the description of the flame response to incident flow perturbations. This is used in section 6 to examine effects of the swirler on the flame response. II. Flame wrinkling When the internal reaction zone is not modified by turbulence, premixed wrinkled flames can be treated as a thin sheet modeled by an interface G(x,t) = separating unburned from burned gases. 12 This sheet is a solution of the level set propagation equation : G t +v G = S d(φ,x,t) G (1) where v is the local velocity at the flame front and S d is the normal flame displacement velocity with respect to the fresh gases. Using these notations the unit normal vector defined by n = G/ G is oriented from the burned gases (G > ) towards the fresh reactants (G < ) (Fig. 1). In the present analysis stretch effects are ignored and it is assumed that the flame displacement velocity S d is constant. One also considers that the flame velocity responds in a quasi-steady manner to fluctuations of the equivalence ratio φ. The former assumption is not valid for large perturbations levels where non-linear stretch effects reduce the front wrinkling. 13 The latter assumption is not adequate for high frequency perturbations 14 but since we are only dealing with perturbations of small amplitude at relatively low frequencies, we can consider that the flame displacement velocity is equal to the laminar burning velocity S L and that it is only a function of the local equivalence ratio φ : S d (φ,x,t) = S L (φ) (2) The dynamics of the flame sheet is investigated for small perturbations of the velocity field v and small fluctuations of the local equivalence ratio φ : v = v +v 1 and φ = φ +φ 1 (3) Disturbances φ 1 and v 1 are small compared to the mean values φ and v. The laminar burning velocity responds in a quasi-steady fashion and may be linearized around the steady state operating point : ( S L = S L 1+a φ ) 1 where a = φ ( ) SL φ S L φ φ=φ (4) 2 of 17

In this expression S L = S L (φ ) is the steady state laminar burning velocity. The coefficient a is the differential of laminar burning velocity S L (φ) with equivalence ratio around the operating point φ. For a methane/air mixture at φ =.8, S L =.26 m.s 1, one can estimate a = 2.3 using the evolution of the burning velocity with φ. 15 This approximation is only valid at low frequencies. Hysteresis effects observed at higher frequencies are obviously not described. 14 Introducing Eqs. (3) and (4) into the transport Eq. (1) and writing the G-field as a sum of mean and perturbed fields G(x,t) = G (x)+g 1 (x,t), one obtains the following transport equations for these fields : G 1 t v G = S L G (5) [ ] G + v S L G 1 = v 1 G +as L G φ 1 (6) G φ Using Eq. (5) and the definition of the unit normal vector n = G / G, the last expression can be rewritten : ( G 1 +v t G 1 = v 1 a φ ) 1 v n G (7) t φ where v t = v (v n )n is the mean flow velocity parallel to the mean flame front. This transport equation for the perturbed field G 1 shows how flow uniformities (φ 1 and v 1 ) wrinkle the flame. Small mixture composition fluctuations and velocity perturbations induce small disturbances of the flame position in the normal direction, which are then convected along the flame front by the projection of the mean local flow velocity v t in a direction parallel to the mean flame front. This generalizes a result from Boyer and Quinard 16 derived for uniform velocity modulations to any flow non-uniformities affecting the flame wrinkling. It is interesting to note that velocity and mixture composition perturbations wrinkle the flame in a similar way but with opposite signs. In the case of a lean premixed flame, a positive velocity perturbation leads to a positive flame displacement, while an increase in the equivalence ratio leads to a further increase in the laminar burning velocity and thus to a negative flame displacement. This argument was already used by Cho and Lieuwen 17 in their analysis of the response of conical flames to low frequency mixture composition oscillations. The main difference with velocity disturbances is that mixture composition perturbations also strongly modify the local reaction rate. 14 Local changes of consumption rate (flame displacement speed) and volumetric heat released per unit mass of mixture burned have to be taken into account in the description of the flame response. This problem is difficult to model and heuristic fit to experimental or numerical data are considered to obtain analytical expressions of the flame transfer function following a method proposed by Hubbard and Dowling. 13,17,18 Recent investigations of this problem have thus been envisaged numerically by solving the full Navier-Stokes equations to examine the response of an inverted conical flame 19 or conical flames submitted to mixture composition oscillations. 2 In the case of fully premixed systems kept at a fixed equivalence ratio and submitted to flowrate modulations, only velocity perturbations have to be considered. They may take different forms associated to acoustic waves, convected flow disturbances along the flame front or large scale coherent structures. 21 The local reaction rate may here eventually slightly depend on aerodynamic strain and flame curvature. These effects can be included in the level set description using corrections proposed by Markstein, 22 Clavin 23 and Matalon and Matkowski 24 for the flame displacement velocity. Results obtained for conical flames show however that predictions of the flame transfer function are barely modified compared to those obtained without these corrections, except in some specific cases. 25 One of the main effects is an attenuation of the resulting perturbations along the flame front. 26 This phenomenon was extensively studied and will not be considered in the present study. In many combustors the flame is generally inclined with respect to the main flow direction. The flame then often takes three typical shapes. The first corresponds to a conical shape which is a generic configuration widely studied in laboratories which typifies many burners used in boilers or radiant heaters where small conical flames are anchored over a collection of perforations. The second geometry is an inverted conical flame, also designated as V-flame or wedge flame, where the flame root may be stabilized aerodynamically as in many swirling flows or by a central bluff body as in some heating systems, afterburners and swirling burners. The last configuration also observed in turbulent swirling burners is associated to a M-shape like 3 of 17

flame stabilized between the inner and outer shear layers formed by the reactants jet with the inner and outer recirculation zones. In this case the flame root in the center may also be stabilized in some burners by a central bluff-body. In all these configurations, the flame is inclined and it is important to understand how flame wrinkles propagate along the flame front. To analyze effects of flame inclination on the unsteady response to flow disturbances, it is interesting to consider the two dimensional generic problem of an inclined flame with an angle α with respect to a uniform axial flow field characterized by v = (,v ) which is submitted to small velocity disturbances (Fig. 1). In this case the G-field can be considered as a single function of the axial coordinate y. Let G = y η (x) = designate the steady flame position in the laboratory reference frame (x, y). Using Eq. (5), the mean flame position η (x) is given by: [ 1/2 η (x) = x (v /S L ) 1] 2 = x/tanα (8) Small perturbations of the incident velocity field write v 1 = (u 1,v 1 ), where u 1 and v 1 are much smaller than v. Let X and Y designate the new co-ordinates with respect to the steady inclined flame (Fig. 1, right). In this new frame, the perturbed flame position is given by G = Y ξ(x,t) =, the velocity v has (U,V) components and Eq. (7) reduces to: ξ t +U ξ X = V 1(X,t) (9) where U = v cosα and V 1 = v 1 sinα u 1 cosα. The solution of this first order kinematic equation is straightforward and yields the normal flame displacement ξ(x, t) with respect to the steady position of the flame front as a function of the axial distance along the flame and time : ξ(x,t) = 1 U X ( V 1 X,t X ) )dx X +ξ (t XU U (1) The first contribution indicates that wrinkles are produced by the cumulated effects of the perturbed velocity field integrated along the flame front 16 and the second contribution corresponds to perturbations imposed at the flame boundary ξ (t) = ξ(x =,t). 27 For both types of perturbations, flame wrinkles ξ(x,t) are convected along the flame front with a celerity U = v cosα. This feature was recognized in the early work of Petersen and Emmons 28 for a flame sheet attached to a vibrating rod in a uniform flow. The convective nature of flame wrinkles for conical flames submitted to harmonic flow disturbances has been demonstrated in later experiments carried out by Baillot et al. 29 In a more recent investigation Kornilov et al. 3 showed that perturbations of the flame anchoring point or acoustic modulations yield the same types of wrinkles along the front of conical flames. These two mechanisms fully determine the response of premixed conical flames which are examined in the next section. III. Conical flames Many expressions have been derived for the transfer function of premixed conical flames submitted to harmonic flow modulations. 31 38 Detailed comparisons between predictions and measurements were however only carried out in a limited number of investigations. 34,38,39 The problem is generally examined for a single flame while practical configurations often feature a collection of small conical flames. The flame transfer function in these systems may significantly differ from the response of single flames. 4 In these cases, the response is often determined experimentally 41,42 or numerically, 43,44 except in the theoretical analysis reported by Altay et al. 45 It is possible to integrate the first contribution in Eq. (1) for prescribed structures of the mean flow and perturbed velocity fields. Harmonic flow perturbations are generally considered for this purpose and flow variables are decomposed in mean and fluctuating components a(t) = ā+ãexp( iωt), where ω denotes the angular frequency. When the dynamics of the anchoring point needs to be considered, the additional link between the perturbed velocity field and the resulting motion at the flame base must be modeled as well 45 to calculate the second contribution in Eq. (1). The transfer function of conical flames linking heat release 4 of 17

rate and harmonic velocity disturbances then takes the general form: F(ω) = Q 1 / Q ṽ 1 /v = F v1 +F ξ ξ()/r ṽ 1 /v (11) where F v1 and F ξ stand for filters associated to flame wrinkling due to velocity disturbances and flame root perturbations respectively, R being the burner radius and ṽ 1 /v the velocity perturbation level at the flame base. The last term vanishes when disturbances of the flame root can be neglected with respect to velocity perturbations. Estimates of this last contribution can be modeled by examining the unsteady heat transfer between the flame root and the burner. 45 47 ThefirstcontributionF v1 dependsontheshapeoftheconicalflame, themeanandperturbedflowfields. A single conical flame stabilized over a cylindrical burner with a flat uniform velocity profile at the burner outlet is considered here to examine F v1. It is then possible to derive analytical expressions for the flame transfer function for different types of velocity perturbations. The simplest one, corresponds to a uniform velocity modulation (model 1 in Tab. 1) simulating the response of the flame to velocity disturbances associated to acoustic waves whose wavelength is generally large compared to the flame length. The flame transfer function (FTF) F = Gexp(iϕ) is in this case only a function of the reduced frequency ω = ωr/(s L cosα). 34 The gain G is a low pass filter with a cut-off frequency ω 2π, where the phase lag saturates at ϕ = π/2. The second model (model 2 in Tab. 1) considers a velocity disturbance convected by the mean flow as suggested by experimental observations. 29 Further analysis has shown that this convected velocity wave originates from the feedback of flame wrinkling on the fresh reactant stream. 48 The FTF appears in this case as a function of the reduced frequency ω and the flame angle α separately. 49 For a given reduced frequency ω, elongated flames are more sensitive than small flames. The gain G is barely modified compared predictions obtained with the uniform modulation, while the phase is now regularly increasing with the forcing frequency ϕ = ωτ, where the time lag τ is a function of the flame angle α. The main weakness of model 2 is that mass balance cannot be satisfied for a purely axial convected velocity disturbance. It is then possible to use model 3 considering an incompressible convected velocity disturbance (model 3 in Tab. 1). In the region near the flame front, the flow perturbation induced by flame wrinkles is indeed incompressible. It has been shown that this model enables to reproduce the different shapes taken by the flame even at high frequencies. 5 Numerical integration of the G-equation with this type of model has also led to the best agreement with measurements for the flame transfer function. 39 An analytical expression for the FTF was recently proposed by Cuquel et al. 51 which is reproduced in Tab. 1. Table 1. Expressions of the conical FTF for different incoming velocity perturbations v 1 = (u 1,v 1 ). In these expression, k = ω/u indicates the wavenumber based on the mean flow velocity u, ω = ωr/(s L cosα) denotes the reduced angular frequency associated to flame wrinkles convected at the speed U along the flame front of length L: ω = ωl/u and k = cos 2 αω is the reduced frequency associated to velocity disturbances convected at the speed v in the axial direction over the flame height H: k = ωh/v. Model 1 Model 2 Model 3 ũ 1 = ũ 1 = ũ 1 = 1 2 ik(r x)v 1exp(iky) ṽ 1 = v 1 ṽ 1 = v 1 exp(iky) ṽ 1 = v 1 exp(iky) F = 2 ω 2 [1 exp(iω )+iω ] F = 2 1 ω F = 2 k2 i(k [2 ω ) (1 i 1 2 k 1 [ ] ( ) 2 1 exp(iω )+ ω k (exp(ik ) 1) exp(ik ) 1 ik + exp(iω ) 1 ( iω + exp(ik ) exp(ik ) 1 ik )] ) k k ω Predictions of the FTF obtained with these models are presented in Fig. 2 for two flames featuring different flame angles α = π/6 and π/3. The first condition explored corresponds to an elongated flame (α = π/6), while the second condition (α = π/3) is close to flashback and may be difficult to reach in practice. One can first note that the gain G features a stronger dependance with the flame angle for model 3 than for model 2. This is particularly visible for reduced frequencies larger than the cut-off frequency ω 2π, when the gain is lower than G.5. One finds here G =.4 and G =.15 for the elongated flame and the short flame respectively examined at the same reduced frequency ω = 4π. The evolution of the phase lag ϕ predicted with model 3 is also strongly influenced by the flame angle α. At very low 5 of 17

1 5π 1 5π Gain.8.6.4 4π 3π 2π Phase (rad) Gain.8.6.4 4π 3π 2π Phase (rad).2 π.2 π π 2π 3π 4π 5π 6π ω * π 2π 3π 4π 5π 6π ω * π 2π 3π 4π 5π 6π ω * π 2π 3π 4π 5π 6π ω * Figure 2. Gain G and phase lag ϕ of a conical FTF featuring a flame angle α with respect to the axial flow direction: α = π/6 (left figure) and α = π/3 (right figure). Predictions with model 3 (black line) are compared those obtained with model 1 (red line) and model 2 (green line). frequencies, the phase lag always matches that predicted by model 1. As the frequency increases the phase lag deviates from model 1 and approaches that predicted by model 2 in the intermediate range of frequencies. The asymptotic evolution at high frequency does however again strongly depend on the flame angle. For elongated flames, the phase lag regularly increases with reduced frequency and is close to the one predicted with model 2. For flames featuring a large flame angle, the phase lag saturates at high frequencies to a value close to that predicted with model 1. These conclusions for the phase lag are in agreement with experimental data presented for different setups operated under different flow conditions, 3,34,42 indicating that model 3 is probably more suitable to describe the phase lag of the conical flame transfer function. These predictions are now compared to measurements in Fig. 3 for methane/air conical flames stabilized on different burners operated at different flow rates and equivalence ratios. In these experiments, the velocity profile at the burner outlet is uniform and the height of the steady flame has been modified, either through a change in the bulk velocity v or burning velocity S L. The perturbation level at the burner outlet was also kept constant and is equal to v1 rms /v =.5. In the left figure, data are plotted for a burner nozzle radius R = 11 mm. The bulk velocity is here the varying parameter, while the equivalence ratio φ = 1.1 was kept constant. In the right figure, the burner nozzle has a radius R = 15 mm. Experiments with the large burner were conducted for a fixed bulk velocity v = 1.3 ms 1 and the equivalence ratio was changed to modify the flame speed. Measurements match well the phase lag predicted by model 3 in the left figure when the flowrate is varied for a fixed equivalence ratio, except at very high frequencies where the phase saturates. Even though the gain is over-predicted by model 3 over the entire range of frequencies, local extrema of the gain evolution are located at the correct reduced frequencies and the global trend of the undulation observed is well reproduced. Agreement with measurements slightly deteriorates in the right figure with the large burner when the equivalence ratio is varied for a fixed flow rate, although the general trend predicted by model 3 is still well reproduced for the phase lag. It is however important to note that the slope of the phase lag is very sensitive to the choice of the flame burning velocity. 44 This parameter was here fixed using values indicated by Vagelopoulos et al. 15 Numerical simulations of this problem have shown that it is possible to find a better collapse by scaling the results with the appropriate flame displacement speed reproducing the steady flame height. 52 The main conclusion is that the locally incompressible nature of the velocity perturbation convected along the flame is an essential feature that must be reproduced to determine the correct phase lag of conical flame transfer functions. An additional attenuation mechanism that damps the gain, but which does not modify the phase lag, is probably missing in model 3. It may originate from the flow perturbation which reduces with the distance to the burner 48,5,52 and may also be influenced by stretch effects. 26 It is now worth examining configurations with multiple conical flames. It is known that for a packed collection of conical flames the gain of the transfer function may largely exceed unity, a feature which is obviously not described by the previous models. This behavior has been observed for the transfer functions of multiple conical flames stabilized on regular arrays of circular perforations 4,53 and above rectangular slits as well. 42 It has been demonstrated, when elements from neighboring flames are in contact to each other close to the burner, that the unsteady heat transfer to the burner leads to a strong oscillation of the flame anchoring point resulting in additional flame wrinkles and heat release rate disturbances. 45 This 6 of 17

Gain 1.8.6.4 v = 1.32 m.s 1 v = 2.5 m.s 1 Models Gain 1.8.6.4 Model Φ =.8 Φ = 1.1.2.2 Phase (rad) 4π 2π Phase (rad) 4π 2π 2π 4π 6π 8π ω 2π 4π 6π 8π ω Figure 3. Gain and phase of the FTF for burners featuring beveled edges and different outlet radius. Left figure : R = 11 mm, φ = 1.1. Right figure: R = 15 mm, v = 1.3 m.s 1. Experimental data are also compared to predictions with different models. 51 mechanism activated for flame elements stabilized close to solid surfaces can be modeled by an enthalpy wave between a flat flame and a solid surface as shown by Rook et al. 46 It mainly involves two characteristic lengths associated to the preheat zone thickness and flame quenching distance with the solid wall. Under acoustic forcing this can be used to model the contribution ( ξ()/r)(ṽ 1 /v ) appearing in Eq. (11). 45,46,54 For methane/air flames a resonance behavior with a bell shape curve around the peak frequency at about 1-2Hz is found. It has also recently been shown 55 that this mechanism can be used to model saturation observed for the phase lag at lower frequencies as the perturbation level increases, providing some insight to model the flame describing function of conical flames. 4 The previous references fully document this mechanism and corresponding modeling. A second aspect related to the presence of multiple flames was however not considered so far in the determination of their frequency response to flow disturbances. In the presence of neighboring flames, the hot plume around a specific flame may not fully expand depending on the interspace between the burners. 56 For under-expanded hot plumes, the resulting pressure gradient in the hot gases acts on the fresh reactants leadingtoanaccelerationofthecentralflownearthe burneraxis. 57 The meanflowthuscannotbe considered uniform in these cases even for a flat uniform velocity profile at the injection plane. This phenomenon may explain the differences observed between simulations and measurements in the recent work of Duchaine et al., 44 where it was necessary to use a numerical domain with a slightly expanding angle to reproduce measurements of the velocity field in the fresh reactants and of the flame transfer function. This mechanism also operates for flames confined by hard walls. An experimental investigation was recently conducted by Cuquel et al. 51 where the flame transfer function was measured for laminar conical flames stabilized on a burner with 2R = 22 mm diameter with and without a quartz tube of internal diameter 2R t = 35 mm to confine the reaction zone. This situation is illustrated in Fig. 4 for a methane/air mass flowrate ṁ = 792 mg.s 1 at equivalence ratio φ =.8. In this case, the FTF becomes a function of η for confinement ratios η = R/R t.44. The frequency response for the gain is progressively shifted to higher frequencies when η increases and the phase lag φ reduces with increasing η for a fixed reduced frequency ω. It has been shown in this section that the transfer function of laminar premixed conical flames is a function of the reduced frequency ω and also of the flame angle α. This last feature is important to describe the correct behavior of the phase lag evolution with frequency over the useful frequency range of interest where the gain takes non-negligible values. Under acoustic forcing, the perturbed velocity field in the vicinity 7 of 17

Gain 1.2 1.8.6.4.2 4π η =.81 η =.6 η =.44 η = Phase (rad) 3π 2π π 2π 4π 6π ω Figure 4. Left: steady conical flame confined by different quartz tubes η = R/R t, where R is the burner radius and R t the quartz tube internal radius. The mean flow velocity is v = 792 mg.s 1 and equivalence ratio φ =.8. Right: Gain G and a phase ϕ of the transfer function measured as a function of the reduced frequency ω for different confinement ratios η. of the flame front has the structure of an incompressible convected velocity disturbance. The dynamics of the flame anchoring point ξ() must be considered when wrinkles at the flame roots cannot be neglected with respect to the relative velocity disturbance level at the same location. Unsteady heat transfer between flame elements stabilized close to solid boundaries should in this case be taken into account. When the hot plume of the flame cannot fully expand as in confined configurations or for multiple flame burners, the mean flow is modified and the transfer function is also altered. IV. V-flame The case of laminar V-flames constitutes a second interesting generic configuration because many swirling flames take a conical inverted shape. It is also a configuration of technological interest used in some domestic and industrial boilers. An early kinematic description of the flame dynamics stabilized behind bluff bodies and confined by a channel was considered by Marble and Candel. 58 This description was later used by Dowling 59 to obtain an expression for the flame transfer function of ducted V-flames submitted to low frequency acoustic disturbances. The frequency response of V-flames submitted to perturbations convected by the mean flow was then examined by Schuller et al. 49 This type of perturbation was shown to reproduce the several essential features of the flame transfer function of V-flames which is characterized by a large frequency range with a gain in excess of unity, a reduction in the gain very sensitive to the input level and a phase weakly modified by the disturbances amplitude. 4,6 62 This mechanism does not however constitute the unique contribution to heat release rate disturbances. V-flames are often stabilized near the dump plane of the combustor featuring an abrupt change in the cross section area. In this region the inclined flame front strongly interacts with the shear layer formed between the high-speed jet originating from the injector outlet and the low speed flow close to the chamber wall. This is illustrated in Fig. 5 for an unconfined methane/air inverted conical flame stabilized on a central rod of diameter 2a = 6 mm in a burner featuring a nozzle outlet diameter 2b = 22 mm. In the absence of flow modulation, the flame front is steady near the flame root and over most of the flame length (Left 8 of 17

Figure 5. Left figure : The left part with respect to the burner axis shows streamlines (represented in red) from the steady flow impinging the flame front in absence of flow modulation. The front is represented here in the symmetry plane of the burner after Abel deconvolution of chemiluminescence records. In the right part of the figure a superposition on a single frame of all successive positions taken by the flame front is represented when the flow is modulated at f = 15 Hz with a velocity perturbation v 1rms =.15 ms 1. Right figure : Iso-levels of vorticity contours (represented in white) extracted from PIV data superposed to an image of the trace of the flame front in the symmetry plane of the burner after Abel transformation of a chemiluminescence snapshot. Diagnostics were synchronized with respect to the acoustic modulation. Flow operating conditions: φ =.8, v = 1.87 ms 1. Central rod radius: a = 3 mm. Burner radius: b = 11 mm. 63 figure, left part with respect to the burner axis). The streamlines of the steady flow represented in red in this figure are slightly bent outward due to the adverse pressure gradient exerted by the sudden expansion of the surface area and by the flame on the fresh stream of reactants. Near the flame tip the flame front is blurred due to unsteady and incoherent interactions between the flame and the shear layer originating from the burner lip. In the presence of an acoustic modulation, the flame responds directly to these perturbations by wrinkles propagating along the flame front with the same mechanism already described for conical flames. The flame executes a cyclic oscillation near the flame root and flame wrinkles grow in amplitude as they progress along the front. This is highlighted in the right part of the left figure, where the superposition of the flame contours extracted during a modulation cycle clearly reveals the envelope of the flame motion along the flame front. This figure also indicates that the oscillation of the flame anchoring point participates to flame wrinkling even though its contribution to heat release rate disturbances remains weak due to the symmetry of the V-flame. The main observation is that the motion executed by the flame is dramatically accentuated near the flame tip when the flame front begins to interact with large vortices. This corresponds to a location where the image superposition is suddenly blurred over a large region near the flame tip. This additional mechanism must be taken into account. The snapshot of the vorticity field deduced from PIV measurements superimposed to the corresponding flame front at the same instant in the modulation cycle is plotted in the right figure. This superposition demonstrates that acoustic modulation synchronizes the formation of large coherent structures in the shear layer at the burner outlet which are swept by the flow and then strongly interact with the flame front at its periphery by enrolling the flame. These large coherent structures generate additional wrinkles which are also convected by the mean flow at an approximate velocity u c = v max /2, where v max is the peak axial velocity at the burner outlet along the radial direction, 61 but they do not propagate along the flame front. They originate from the burner outlet where flow separation takes place and impinge the flame front where the flame interacts with the shear layer. Preetham and Lieuwen 64,65 proposed to use the same kinematic description Eq. (1) with a prescribed velocity perturbation, which is convected at a reduced propagation speed along the flame front to take into account the contribution of vortices to flame wrinkles. The difficulty is that vortices are convected along the shear layer and begin to interact with the flame front only at a certain distance from the flame anchoring point which depends on the flame front position with respect to the shear layer position. It is then probably better to keep the correct 9 of 17

y α v 1 α y β α y 1 v 1 α β a b x a b b x x 1 Figure 6. Schematic of the steady V-flame front position in absence (left) and with (right) mean flow deflection at the burner outlet. a: radius of the central rod, b: burner nozzle outlet radius, α: flame angle with respect to the mean flow direction, and β: flow deflection angle with respect to the vertical axis. 63 convection velocity of ditursbances along the flame front and try to improve the modeling of the interaction between vortices and flame. It is possible here to capture the first contribution corresponding to perturbations convected along the flame front and examine differences between predictions and measurements of the FTF. This is done by using a modified version of the expression derived by Schuller et al. 49 for an axisymmetric V-flame anchored on a central rod of radius a in a uniform flow originating from a burner of radius b and inclined with an angle α with respect to the mean flow direction along the vertical axis (Fig. 6, left). This expression is given by : F CVR (ω,α,a,b) = 2 [ 1 b a ω 2 1 cos 2 exp(iω ) 1 exp(iω cos 2 ] α) 1 α b+a cos 2 α + 2i 1 b [ exp(iω ω 1 cos 2 cos 2 α) exp(iω ) ] (12) α b+a It was already indicated in Fig. 5 that the streamlines are bent outwards due to the abrupt area change at the burner outlet. This deflection in turn modifies the flame response. The streamlines are here deflected by a mean angle β = 15 o with respect to the burner axis. The flame angle with respect to the main flow direction is now given by α β as indicated in the schematic in Fig. 6 (right figure). The solution of this problem is given by Schuller, 63 but an approximate expression can be obtained by noting that the deflected flame position is now given by v sin(α β)/cosβ = S L and the width b a must be replaced by b a = (b a)cosβ. The flame transfer function of this V-flame featuring a flow which is deflected by an angle β with respect to the burner axis can be estimated as a first approximation by replacing the reduced frequency ω = ω(b a)/(s L cosα) in Eq. (12) by ω = ω(b a)cosβ/(s L cos(α β)). Predictions for the transfer functions of methane/air V-flames are compared to measurements obtained for lean conditions at an equivalence ratio φ =.8, different bulk flow velocities and a fixed perturbation velocity v 1rms =.16 m.s 1 in Fig. 7. The deflection angle β measured in the experiments has been taken into account to estimate the reduced frequency ω in this representation. Theoretical predictions with model 2 (solid line) and α = 3 o and β = 15 o yield the global trend of the gain of the flame transfer function at low and high frequencies, but do not capture the flame response at intermediate frequencies. It is shown that the the different results can be collapsed relatively well for the gain except for the values reached by local extrema in the flame response. They feature increasing values for elongated flames. The succession of a local minimum followed by a large maximum in the flame response indicates interferences between wrinkles convected from the flame root to the flame tip and the impact of vortices at the flame tip. 63 The measured phases of the different flame responses match well when they are plotted as a function of the reduced frequency ω. It is however shown that model 1 (dashed lines) or 2 (solid line) do not capture the phase of the flame transfer function, because this parameter is mainly controlled by the time lag for vortices originating at the burner lip to reach the flame front. 61 This interaction with vortical structures remains yet difficult to include in the kinematic description. The present models may however be used with caution to analyze more complex configurations when the flame takes an inverted conical shape. 1 of 17

gain 3. 2.5 2. 1.5 1. v=1.41 m/s v=1.64 m/s v=1.87 m/s v=2.1 m/s unif α=3 o, β=15 o phase (rad) 3 25 2 15 1 v=1.41 m/s v=1.64 m/s v=1.87 m/s v=2.1 m/s unif α=3 o, β=15 o.5 5. 1-1 1 1 1 1 2 ω * 1-1 1 1 1 1 2 ω * Figure 7. Flame transfer function of V-flames plotted as function of ω = ω(b a)cosβ/(s L cos(α β)) for different bulk flow velocities v. The equivalence ratio is φ =.8 and the perturbation velocity v 1rms =.16 m.s 1 at the fame base. Measurements are compared to predictions with a flame angle α = 3 o and a flow deflection angle β = 15 o. 63 V. Turbulent swirling flames The response to velocity disturbances of swirling turbulent flames is analyzed in this section. This case is considered in a recent article by Palies et al. 66 and only the main steps in the derivation are outlined in what follows. The instantaneous flame front position is described by Eq.(1), but turbulence induces random wrinkles of the flame front that must be distinguished from coherent perturbations associated to the modulation imposed to the flow. The turbulent velocity field can be decomposed in three components: v(x,t) = v (x)+v 1 (x,t)+v (x,t) (13) where v denotes the mean turbulent velocity field, v 1 the phase averaged velocity fluctuation around this mean field due to the modulation and v random velocity fluctuations. Turbulent effects are considered by taking an ensemble average of the G-equation to derive a Reynolds average equation: G t +v G = S T G (14) The turbulent flame speed S T being defined by : S T G = S d G v G (15) A phase average operation is then used to separate the perturbed G 1 field from the mean field G yielding: G 1 t v G = S T G (16) +(v +S T n ) G 1 = v 1 G +S T1 G (17) In this expresssion S T is the turbulent flame speed and S T1 indicate coherent perturbations in the flame speed associated to the flow modulation. Combining these last expressions, one is left with: ( G 1 +v t t G 1 = v 1 S ) T1 v G (18) S T where v t = v+s T n = v (v n )n. This result is the same as Eq. (7) describing the instantaneous flame sheet dynamics submitted to velocity and mixture composition oscillations, except that effects of mixture composition oscillations are replaced by fluctuations in the turbulent flame speed due to the coherent 11 of 17

Figure 8. Left : Lean swirling methane/air V-flame stabilized on a central rod in the combustion chamber. Right: Results from Large Eddy Simulations. Evolution of the heat release rate in the symmetry plane of the burner for harmonic modulations of the flow at f = 6 Hz and f = 9 Hz. Three instants in the modulation cycle are represented. Adapted from. 67 modulation. The response of turbulent swirling flames constitutes an important problem with many practical applications. 68,69 Equation (18) indicates that the transfer function results from the combined effects of the velocity modulation imposed at the burner outlet and turbulent burning velocity disturbances induced by the modulation. These two components may take different forms, depending on the type of perturbations considered. Assuming that flow perturbations are transported by the mean flow, it is then possible to make use of expressions derived in the preceding sections. This is illustrated here for an inverted swirling conical flame stabilized on a rod of diameter 2a in the center of a burner of diameter 2b (Fig. 8). Integration of Eq. (18) assuming that perturbations are convected by the mean flow leads to the following expression for the flame transfer function: 66 [ Q 1 ṽ 1 = F CV R (ω,α,a,b) S ] T1 Q v S T To close the problem, one needs to consider a model for disturbances of the turbulent burning velocity induced by the modulation. One feature often observed in swirling flows submitted to acoustic forcing is the oscillation of swirl number. 7 This has been observed in different gas turbine model combustors fed by gaseous 7 or liquid fuels. 71 It has also been demonstrated that the swirl vane location in the injection unit has a strong influence on swirling flame dynamics. 7,72 The response of the swirler to incident acoustic waves must then be taken into account to describe the perturbed flow impinging the flame. 73,74 Interaction of acoustic waves with the swirler leads to a mode conversion at the swirler trailing edge due to flow separation with a longitudinal acoustic wave and an azimuthal vortical mode in the flow downstream the swirler. 75 These two modes have different phase velocities and they induce fluctuations of the swirl number in the chamber. Using the actuation disk theory, it is possible to estimate the amplitude of these two modes and estimate the corresponding swirl number fluctuation in the chamber. 74 It is well known that the swirling strength modifies the flame burning rate. 76 When the swirl number increases the flame angle with respect to the flow direction increases. A fluctuation of the swirl number produces then an oscillation of the mean flame angle with respect to the mean flow direction. 67,74 It is then natural to link the coherent part of the burning velocity fluctuation S T1 to the azimuthal v 1θ and longitudinal v 1 velocity perturbation components at the burner outlet to reproduce this coherent flame root (19) 12 of 17

Figure 9. Swirling flame transfer function. U b = Figure 1. Swirling flame transfer function. U b = 2.67 m.s 1, φ =.7. 66 4.13 m.s 1, φ =.7. 66 angle oscillation: S T1 S T = χṽ1θ v θ +ζṽ1 v (2) The azimuthal and axial velocity components are out of phase at the burner outlet and are linked by: 77 ṽ 1θ v θ = ṽ1 v exp(iϕ) (21) In this expression the phase lag is given by ϕ = ωl/u b, where ω is the angularpulsation, U b the bulk velocity in the injection tube and L the distance between the swirler outlet and the burner nozzle outlet. These two relations can be used to obtain an expression for the flame transfer function of swirling flames submitted to velocity modulations: Q 1 Q = F s (ω,α)ṽ1 v where F s (ω,α) = F CV R (ω,α,a,b)[1 (ζ +χexp(iϕ))] (22) This formalism can be used to compare predictions with experimental data measured on the turbulent swirling burner presented in Fig. 8 for two bulk flow velocities. These flames are obtained for a mixture of methane and air at an equivalence ratio φ =.7 and a swirl number S =.55. Results are presented in Figs. 9 and 1 where the parameters χ =.4 and ζ =.4 were determined experimentally. 66 The phase shift between the azimuthal and axial velocity components at the burner outlet was also determined by Laser Doppler Velocimetry and it was found : Flame A : ϕ A = ωτ A +ϕ A, where τ A = 12. ms and ϕ A = 1. rad Flame B : ϕ B = ωτ B +ϕ B, where τ B = 8.5 ms and ϕ B = 1.5 rad These figures indicate that the general features of the transfer function are suitably retrieved. It is found in particular that the frequencies corresponding to minimum and maximum gains are reproduced, but that there are also some differences indicating that the processes are more complex than what has been modeled. It is finally interesting to examine two specific forcing conditions at f = 6 Hz and f = 9 Hz corresponding respectively to the minimum and maximum values for the gain of the flame transfer function of flame A (Fig. 9). Fluctuations of the swirling number and heat release rate are plotted in Fig. 11 as a function of the phase in the forcing cycle. This figure clearly shows that conditions leading to large swirl oscillations at a forcing frequency f = 6 Hz correspond to reduced heat release rate disturbances at the same frequency and 13 of 17

f= 6 Hz f= 9 Hz Figure 11. Relative fluctuations of the swirl number and corresponding heat release rate oscillations measured at f = 6 Hz and f = 9 Hz. U b = 2.67 m.s 1, φ =.7. Adapted from. 74 to a minimum of the flame transfer function. For a modulation at f = 9 Hz the situation is reversed. The swirl number oscillation is weaker in this case and large heat release rate oscillations take place leading to a maximum in the flame response. The mechanism associated to this behavior was elucidated by Palies et al. et al. 67 using Large Eddy Simulations (Fig. 8). Large vortices synchronized by the acoustic modulation are shed from the burner rim and impinge the flame front at the periphery contributing to large flame surface area destruction. The strength of this interaction depends on the relative distance between the flame front and burner rim. When swirl number fluctuations take place, the flame angle responds by an oscillation and the vortices shed from the burner rim cannot fully develop before interacting with the flame leading to a weaker flame destruction and a reduced heat release rate perturbation. VI. Conclusion The modeling of transfer function of flames submitted to external flow disturbances is reviewed in this article. A unified theoretical approach to this problem is provided in the framework of the perturbed level set G-equation. Several models derived for premixed flames featuring different geometries and flow interactions are confronted with measurements. It is shown that it is possible in many cases to reproduce the trends in flame response when the leading mechanisms are identified and suitably represented. In general, it is important to properly describe and account for the steady state behavior and geometrical shape of the flame front. In the case of conical flames, it is found that a proper representation of the perturbation field and a description of the dynamics of the anchor point are required to obtain the trends observed for the gain and phase. In the inverted conical flame case (the V-flame), it is important to represent interactions between the flame tip and vortices shed by the injector lips. In the swirling flame case, it is necessary to consider the mode conversion process which takes place at the swirler and generates a convective vorticity wave when this unit is impinged by longitudinal acoustic perturbations. The vorticity wave is accompanied by azimuthal velocity perturbations which induce fluctuations in swirl number and influence the unsteady flame dynamics. The descriptions obtained from these models are limited to the small disturbance range and do not account for nonlinear features which become important at high levels of perturbations. 14 of 17

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