Leading edge dynamics of lean premixed flames stabilized on a bluff body

Combustion and Flame 191 (2018) 39–52

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Leading edge dynamics of lean premixed flames stabilized on a bluff body Dan Michaels1,∗, Ahmed F. Ghoniem Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA

a r t i c l e

i n f o

Article history: Received 22 May 2017 Revised 6 July 2017 Accepted 14 December 2017

Keywords: Unsteady laminar flame Flame leading edge Laminar premixed flame Bluff body flames

a b s t r a c t This paper examines the dynamics of the flame leading edge in a laminar premixed CH4 /air flame stabilized on a bluff body in a channel. Harmonic fluctuations and step velocity change are used to simulate the flame response to acoustic oscillations, which are of primary importance in the study of thermoacoustic instabilities. We use a fully resolved unsteady two-dimensional code with detailed chemistry and species transport, with coupled heat transfer to the bluff body. Calculations were conducted with different equivalence ratios, body materials, and steady state inlet velocity with step or harmonic perturbations. Results reveal that the flame leading edge dynamics displays a peak response around St = 0.5 suggesting that the leading edge motion is mainly due to the advection of appropriate ignition conditions as a result of the excitement of the wake recirculating flow. There is considerable augmentation of the flame wrinkles generated by the flame leading edge motion as result of the flow–flame interaction. Additionally, we show that a flame that anchors on average further upstream leads to stronger damping of the shear layer vortices and thus weaker vortex-flame interaction and heat release fluctuations. Hence, we identify two different mechanisms by which the flame leading edge location and oscillation amplitude impact heat release fluctuations. The study suggests a stronger dependence of the overall flame wrinkling and heat release fluctuations on the flame leading edge dynamics than recognized previously and the potential role it plays in combustion dynamics. © 2017 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

1. Introduction In industrial combustors and propulsion systems the reactants velocity is orders of magnitude higher than the laminar burning velocity, thus a recirculation zone generated at the wake of a bluff body is a common way to provide a low velocity region where the flame can stabilize. Bluff body stabilized flames are susceptible to combustion instabilities due to various coupling processes between combustion, flow structure and acoustic [1]. Such instabilities generate oscillations in pressure, heat release and heat flux to the walls, which can result in structural damage, flashback or blowoff. Combustion instability originating from vortex–flame interactions is frequently observed in premixed combustion systems [2]. The underlying mechanism [3–5] is thought to be the shedding of vortices due to velocity perturbation of the shear layer, and the periodic heat release oscillations associated with the vortex–flame

∗

Corresponding author. E-mail address: [email protected] (D. Michaels). 1 Current address: Faculty of Aerospace Engineering, Technion – Israel Institute of Technology, Haifa, Israel.

interaction, which couples with the pressure oscillations if they are in phase, according to the Rayleigh criterion. The relationship between the flame and the acoustic field in this regime can be described by a transfer function between the heat release and velocity (or pressure) fluctuations. A transfer function of a laminar premixed ducted flame was developed by Fleifil et al. [6] and later extended to conical flames by Schuller et al. [7]. The impact of the flame anchoring dynamics on conical flames has drawn significant attention. Schreel et al. [8] measured experimentally the acoustic transfer function of a burner stabilized premixed flat flame. The experiments showed a resonance with a peak gain that depends on the flame holder material, with lower amplification when using a lower thermal conductivity flame holder that resulted in a higher surface temperature. Rook et al. [9,10] developed analytical and numerical models that showed that the resonance response originates from heat transfer coupling between the flame consumption speed and heat loss mechanism. Atlay et al. [11] extended the planar flame models of perforated plate stabilized flames to include flame area oscillations, and demonstrated good agreement with experimental measurements. Kedia et al. [12,13] included the heat transfer in the flame holder in his analytical model and in the fluid–solid coupled numerical simulations,

https://doi.org/10.1016/j.combustflame.2017.12.020 0010-2180/© 2017 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

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D. Michaels, A.F. Ghoniem / Combustion and Flame 191 (2018) 39–52

Fig. 1. A sequence of flame chemiluminescence images (Reproduced from Ref. [20]) with interval of 2.5 ms between images, from an acoustically coupled backward facing step combustor at equivalence ratio of 0.85, inlet temperature of 300 K, Reynolds number of 6500 and ceramic step. The images show the periodic motion of the flame base/leading edge and the rollup of the flame surface downstream associated with vortex shedding.

and showed its importance for predicting the surface temperature and flame transfer function. Cuquel et al. [14] introduced the flame leading edge (or base) dynamics from Rook et al. [10] into a kinematic flame model of a conical flame and validated the model experimentally. Their model demonstrated the important role of the flame leading edge dynamics in determining the non-linear flame response to velocity fluctuations. Experimental investigation in a slot burner by Mejia et al. [15] demonstrated the impact of the flame holder temperature on the flame leading edge motion and the flame transfer function, which lead to transition from unstable to stable operation as the temperature of the flame holder increased. They show that at higher body temperature the flame leading edge motion is smaller, which leads to lower wrinkling of the flame according to kinematic considerations described by Cuquel et al. [14]. Application of a similar kinetic model to bluff body stabilized flames was accomplished by Dowling [16] and Lieuwen [17], assuming that under velocity fluctuations smaller than the mean velocity (no negative inlet velocity) the flame remains attached to the downstream corner of the bluff body. Experimental measurements by Shanbhogue et al. [18] on an acoustically forced bluff body stabilized flame were compared to a kinematic model, concluding that the key processes controlling the flame dynamics include the excitation of the flame front by the vortical structures formed by the oscillating flow, the anchoring of the flame on the body, and kinematic restoration of the flame surface. The first two processes are dominant near the body and the third is observed downstream where the vortices decay. The non-linear response leading to saturation of the response at higher forcing amplitudes was related to flame kinematics that smooth out the flame wrinkles. The assumption of flame attachment provided good correspondence between the model and experiment regarding the flame sheet displacement normal to the undisturbed flame close to the bluff body for forcing amplitudes of 1% of the mean inflow velocity. Preetham et al. [19] extended the kinematic model of laminar premixed flames forced by harmonic velocity perturbations, and included flow disturbances with arbitrary convective velocity. They showed that the flame wrinkles generated by the flame anchoring and velocity nonuniformities can either constructively or destructively superpose, depending on the flame shape mean flow velocity phase speed of the shear layer instability wave and frequency. In this work, we look at bluff body flames and relax the assumption of flame attachment to the bluff body, thus by allowing perturbations that originate from the attachment point to contribute to flame wrinkling. Moreover, we consider the flame and flow interaction though flame stretching, gas expansion, and baroclinic torque. Therefore, we expect to see the result of the superposition of flame surface oscillations originating from the shear layer instability and from the flame anchoring. The significant periodic displacement of the flame leading edge as result of acoustics is evident in Fig. 1, which is taken from the experimental investigation of Hong et al. [20] on lean premixed flames in a backward facing step combustor. In Fig. 1 we present a

sequence of flame images during relatively weak oscillations. The significant impact of vortices on the flame during acoustic oscillations leads to increase in the heat release and sustains combustion dynamics. The measurements showed that a ceramic step broadened the stable operating conditions in comparison to a steel step (shown in Hong et al. [20]). Kedia and Ghoniem [21] investigated the response to harmonic forcing of laminar premixed flames stabilized on a heat conducting bluff body. They showed difference in the heat release and heat flux to the bluff body for different flame holder materials, but the frequency dependence and associated physical mechanism were not studied in detail because they limited the analysis to a single forcing frequency. A possible mechanism for the impact of flame holder heat transfer properties on the flow structure was revealed in detailed numerical simulations of Michaels and Ghoniem [22] on steady laminar bluff body stabilized flame. It was found that for a flame holder with lower thermal conductivity the flame stabilizes further upstream and closer to the shear layer, resulting in stronger decay of vorticity downstream and thus weaker vortex–flame interaction. Berger et al. [23] analyzed the same configuration but with isothermal boundary conditions at the bluff body surface, and showed that as the flame holder temperature increased the flame moved further upstream and the recirculation zone shortened. In this work, we look at unsteady bluff body stabilized flames and investigate the impact of the flame leading edge dynamics on heat release oscillations. We examine if the impact of the flame leading edge location is mainly through modification of the flow field, as suggested by Michaels and Ghoniem [22], or there are also significant heat release oscillations due wrinkles produced by the flame leading edge motion, as suggested by Cuquel et al. [14] for conical flames. In the present paper, we conduct harmonic forcing simulations over a range of forcing frequencies and conditions, and also compute the response to an inlet velocity step in order to investigate: (a) The flame leading edge dynamics and (b) The Impact of the leading edge dynamics on heat release oscillations. We first look at the leading-edge kinematics, which displays a peak response at a certain Strouhal (St) number, defined by the frequency times the bluff body height and divided by the inflow velocity. Next, we analyze the conjugate heat transfer with the bluff body and the flame structure, and elucidate the impact of heat losses and flame stretch on the flame leading edge dynamics. Subsequently, we look at how the leading-edge dynamics influences the flow field. The results suggest two important mechanisms by which the flame leading edge dynamics impact heat release oscillations. The first mechanism is the advection of flame wrinkles generated by the motion of the leading edge, which result in peak response at St = 0.5 for the present study. The second mechanism is through the impact of the flame leading edge location on the flow field and strength of the vortices impinging the flame, which has a predominant impact on the magnitude of heat release oscillations. Finally, we provide insight on the role of the leading-edge dynamics on the

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phase and amplitude of heat release oscillations during acoustic forcing. 2. Model The model is based on a 2-D direct numerical simulation with detailed chemistry and species transport, and with no artificial flame anchoring boundary conditions. Capturing the multiple time scales, length scales and flame–wall thermal interaction was done using a low Mach number operator-split projection algorithm, coupled with a block-structured adaptive mesh refinement (see Safta et al. [24]) and an immersed boundary method for the solid body (see Kedia et al. [25]). 2.1. Governing equations and numerical methodology At the low-Mach limit, the continuity, momentum and scalar equations are written in the compact form as:

∇

1 Dρ ·v=− ρ Dt

(1)

1 ∂v = − ∇ p + CU + DU ∂t ρ

(2)

∂T = CT + DT + ST ∂t

(3)

∂ Yk = CYk + DYk + SYk , k = 1, 2, . . . , Ns ∂t

(4)

where v is the velocity vector, ρ the density, T the temperature, Yk the mass fraction of species k, p is the hydrodynamic pressure, Ns D is the number of chemical species, and the operator Dt represents the material derivative. The convection, diffusion and source terms in (2)–(4) are given by:

1

CU = −(v · ∇ )v,

DU =

C T = − ( v · ∇ )T ,

1 DT = ∇ · (λ∇ T ) − ρ cp

CYk = −(v · ∇ )Yk ,

∇ ·τ

DYk = −

Ns 1  hk ω˙ k , ρ cp

ST = −

ρ

(5)



1

k=1

 c p,kYk Vk

· ∇ T (6)

k=1

∇ (ρYk Vk )

(7)

ω˙ k ρ

(8)

ρ

SYk =

Ns 

where τ is the stress tensor given by:



 ∂ ui ∂ u j 2 τi j = μ + − δ ∇ ·v ∂ x j ∂ xi 3 i j

(9)

where μ is the dynamic viscosity, and λ is the mixture thermal conductivity. The diffusion velocity of species k is given by:

Vk = −

Dk,m Yk



∇ Yk +

Yk ∇ W¯ ¯ W



(10)

¯ where Dk, m is the mixture-averaged diffusivity of species k and W is the mixture-averaged molecular weight. Additionally, cp and cp, k are the specific heat at constant pressure for the mixture and for species k respectively, and hk and ω˙ k are the specific enthalpy and molar production rates. The NASA polynomials [26] are used to compute the thermodynamic properties. The mixture-averaged transport proper“ties are evaluated using a dipole-reduced formalism [27]. The Soret and Dufour effects, as well as radiation are ignored. The system of governing Eqs. (1)–(4) are closed with the

Fig. 2. Schematic drawing of the numerical domain. The black square is the bluff body.

equation of state for ideal gas and assuming that the thermodynamic pressure P0 is spatially uniform in the low Mach limit:

P0 =

ρ RT ¯ W

= const

(11)

where R is the universal gas constant. By restricting our focus to flows in open domains, P0 is constant. The coupled heat exchange between the immersed solid body and the surrounding fluid is incorporated by simultaneously integrating the governing equations of the reacting flow and the transient heat conduction equation inside the solid:

∂T 1 = ∇ · (λs ∇ T ) ∂t ρs c s

(12)

where ρ s , cs and λs are the density, heat capacity and thermal conductivity of the solid body. The numerical integration of the system of equations is performed in three stages: (1) A projection algorithm for the momentum equation (2) A symmetric Strang splitting scheme is implemented for the chemical source term and the convection and diffusion terms. (3) The projection algorithm is repeated for the momentum equation using the updated scalar fields. Dirichlet boundary conditions for the velocity and scalars are used for the inlet and channel wall boundary, while at the outlet a “convective” boundary condition is applied, and a Neumann (zero gradient) boundary condition is used for the pressure. A binary marker function is used as an indicator of the solid cells at all levels of mesh refinement. Layers of fictitious cells, called buffer zones, are created within the numerical domain near the solid–fluid boundary, and their values are filled so that the boundary conditions get imposed automatically when the symmetric stencils are used within the domain for derivatives and interpolations. The no penetration of the species is obtained with a single sided buffer zone and the temperature and heat flux matching conditions are imposed by using a dual buffer zone. 2.2. Numerical setup A schematic drawing with the dimensions of the numerical simulation is given in Fig. 2. The width of the square flame holder is d = 5 mm, and the channel height is 25 mm. At the inlet of the channel a parabolic profile of fully developed laminar flow is assumed. The average inlet velocity is 1.6 m/s and the inlet temperature is 300 K, thus the Reynolds number is 500 based on the flame holder width and the average inlet velocity. Calculations were initialized from steady conditions, and subjected to harmonic inlet velocity fluctuations at frequencies of 50–400 Hz and amplitudes of the perturbation to initial (or mean) inlet velocity (u’/U0 ) of = 0.1 or 0.2. The corresponding Strouhal number, defined as St = f U0 /d is in the range of 0.156–1.25. The results discussed in the present paper are at condition at which the flame

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and flow response became periodic, after about 10 forcing cycles. Additionally, the steady flow was perturbed by a step change in the inlet velocity, with perturbation amplitude relative to the initial velocity (u’/U0 ) of 0.01, 0.05 or 0.1. The simulation was terminated after 50 ms, when the flow reached a new steady. The multi-step C1 skeletal mechanism of Smooke and Giovangigli [28], which includes 16 species and 46 reversible reactions, was used for methane-air combustion. The ability of the mechanism to accurately predict the extinctions strain rate was shown to be critical for the accurate prediction of the flow and for flame stabilization [29]. The skeletal mechanism of Smooke and Giovangigli [28] was indeed shown to predict well both the burning velocity and the extinction strain rate [29]. In this study, we focus at premixed methane-air flames at equivalence ratio (φ ) of 0.6–0.8 at 1 atm and 300 K. In order to evaluate the impact of the bluff body thermal properties on the flame dynamics both steel and ceramics bodies materials were used. For steel, the thermal conductivity, density and heat capacity were λs = 12 W/mK, ρ s = 80 0 0 kg/m3 , and cs = 503 J/kgK. For ceramics, the thermal properties were: λs = 1.5 W/mK, ρ s = 673 kg/m3 , cs = 840 J/kgK. The channel outer walls were assumed to be isothermal, at the same temperature as the unburned mixture (300 K). The simulations were obtained with a constant time step of 2 μs. The size of the Cartesian structured grid coarsest level was 196 μm, with one additional levels of grid refinement. This is equivalent to a flame resolution of 98 μm. A grid independence analysis in [30] showed that a fine grid resolution of 100 μm is sufficient for successfully capturing the lean premixed CH4 –air flames. 3. Results and discussion In Fig. 3 we show representative results of the flame response to harmonic velocity forcing. The flame surface is defined by the contour of 10% of the maximum heat release rate for unit volume. The right column in Fig. 3 shows a close up of the flame leading edge with overlaid temperature and velocity vectors. The advection of the flame leading edge is clearly seen by the flame motion in the direction of the local velocity vector. The flame leading edge was found to be located in a region of low or negative axial velocity, hence within the wake. As shown on the right-hand side of the figure, the forcing generates vortices that displace the flame leading edge. The growth of these vortices downstream wrinkle the flame front and cause unsteady heat release fluctuations. Whereas vortex shedding downstream a free shear layer or one attached to a sudden expansion has been shown before [31,32], the highresolution results shown here emphasize the motion of the flame leading edge as well. For the present investigation, the interest is in the flame leading edge dynamics and its impact on flame wrinkling and heat release oscillations near the bluff body (up to the end of the recirculation zone) in response to the shear layer flapping and vortex shedding from the flame holder. In the steady flow case, the length of the recirculation zone is 5.00d; at forcing St numbers of 0.156–0.625 the time averaged recirculation zone length is close to the same value (between 4.97d and 5.02d). On the other hand, at St = 1.25, it reduces to 4.48d. Shanbhogue et al. [18] showed that the flame dynamics up to the wake end (average recirculation zone length) is governed by the flame interaction with the vortices which are shed from the body and the anchoring of the flame. The flame dynamics further downstream is governed by kinematic restoration that tends to smoothen the flame wrinkling, and the baroclinic torque most significantly impacts the flow field [33]. It is notable that at φ = 0.5 the simulations with u’/U0 = 0.1 resulted in permanent flame extinction after about 10 forcing cycles during which we observe local extinction and re-ignition as well as asymmetric flame response. Therefore, it would not be accurate

to assume that a symmetric solution should always be expected. Moreover, for non-reacting flow the structure is inherently nonsymmetric, with the formation of the von-Karman vortex street at St = 0.23 for the present geometry [30]. Significant displacement of the flame leading edge is displayed in Fig. 3. The impact of the equivalence ratio, inlet flow velocity, forcing frequency and flame holder material on the flame leading edge motion and flame dynamics is studied in Sections 3.1 using a step change in inlet velocity, and in Section 3.2 using harmonic forcing. In order to better understand the impact of the leading-edge motion on the heat transfer to the walls of the bluff body, and the impact of the bluff body thermal properties on the leading-edge dynamics, the conjugate heat transfer is investigated in Section 3.3. The detailed species distribution at the flame leading edge and further downstream are compared in Section 3.4 in order to identify variations in the flame structure during the flame motion. From the right column in Fig. 3 it is clear that the flame dynamics is impacted by vortex shedding, and in Section 3.5 the flame stretch is examined, as well as the impact of the flame location on the flow field downstream the bluff body. Finally the impact of the leading edge dynamic on heat release oscillations is analyzed in Section 3.6. 3.1. Flame leading edge dynamics – step velocity perturbation The transfer function between the inlet velocity and flame leading edge location was computed in order to identify frequencies of interest. The Bode plot in Fig. 4 was obtained from a Fourier transform of the response to a step velocity change, which was transformed to give the impulse response (the impulse response is the derivative of the step response). Fast Fourier transform was used to compute the discrete response. Because the flame response was calculated for a period of 50 ms, the lowest resolved frequency, and frequency increment is 20 Hz. Step responses with amplitude u’/U0 of 0.05 and 0.1 gave very similar Bode plots, confirming that the leading-edge response is in the linear regime. Figure 4 presents results of the transfer function between the inlet velocity and the translation of the flame leading edge in the direction normal to the stationary flame front, denoted by nLE . The angle between the flame front normal at the leading edge at stationary conditions and the x-axis was 132°. The translation normal to the un-perturbed flame front is of primary interest because it induces flame wrinkling. Cases with different inlet velocity, equivalence ratio and flame holder material display maximum gain at Strouhal (St) number of 0.45–0.5 and a second smaller peak at St of 1.19–1.25. The time lag is plotted, instead of the phase, to emphasize the advective nature of the response. The time lag, as defined in Fig. 4, is −1.2 ± 0.1 at the St numbers of peak response. Thus, at both of those peaks, the flame leading edge displacement and velocity are out of phase (approximately 1.1π and 3π , calculated from the results shown in Fig. 4). For St > 1.5 the gain becomes very low, and the time lag increases because of saturation of the phase at π . The weak gain at higher frequency is typical of dissipative phenomenon. Scaling of the response according to a St number suggests that the leadingedge motion is mainly due to advection as result of destabilization of the wake. The St number at peak response is approximately twice that of the non-reacting von-Karman vortex street St number (0.23), which may indicate the excitement of the second harmonic, that is a symmetric mode [34]. The frequency at the peak response at equivalence ratio (φ ) of 0.6 is practically identical to that φ = 0.7, despite the increase in the characteristic flame time δ /SL by a factor of 2.5. Previous work on the flame response to acoustic perturbations in flat flames [10] and conical flames [13,14], showed a peak response at a frequency that depends on the flame time scale δ /SL . The peak re-

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Fig. 3. Temperature contours at φ = 0.7 during 5 consecutive instances of the forcing cycle of u’/U0 = 0.1 and St number of 0.625. The right column shows close up of the flame leading edge and the velocity vectors. Overlaid are the heat release rate contour of 10% of the maximum value (thick black line), flame holder surface (white square) and a mark of the end of the recirculation zone (white + sign).

sponse was explained by coupling between the reactants enthalpy (and velocity) at the flame base location and the burning velocity at that location as a result of conjugate heat transfer with the bluff body. Unlike conical flames, in bluff body stabilized flames under the conditions investigated in the work, the flame leading edge resides inside a recirculation zone. This difference leads to more complex convection and diffusion of enthalpy fluctuations from the reactants to the flame leading edge, which reduces the coupling due to conjugate heat transfer with the body and leads

to the strong role of transport within the recirculation zone. The very different heat recirculation dynamics with the steel and ceramic bluff bodies, presented in a following section of this paper, provide further support for the hypothesis that the frequency at the peak response does not originate from conjugate heat transfer with the flame holder. The unique structure of the flame leading edge in bluff body stabilized flames can provide further explanation for the weak impact of conjugate heat transfer on the frequency of peak response.

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D. Michaels, A.F. Ghoniem / Combustion and Flame 191 (2018) 39–52

Fig. 4. Gain and time lag (phase/2π St) of the transfer function (nLE /d)/(u’/U0 ) between the inlet velocity and the leading-edge motion of the flame in the direction normal to the unperturbed flame front under steady conditions, denoted by nLE . The unperturbed flame normal direction is at an angle of 132° from the positive x axis.

Previous analysis of steady flames [22] showed a flame leading edge structure which is similar to that experimentally measured by Barlow et al. [35], and revealed that the flame leading edge is highly strained, with impact of heat exchange with the bluff body, recirculation of combustion products and back support by heat and radicals from combustion products. Thus, the different convection and diffusion processes can filter fluctuations in the reactants enthalpy at the thermal boundary layer of the bluff body before they reach the flame leading edge, preventing the same feedback mechanism through conjugate heat transfer with the flame holder as observed in conical flames. Despite the insensitivity of the St number at the peak response to the body thermal conductivity or equivalence ratio, the gain and time lag of the response are sensitive. By looking at the different responses to steel and ceramic flame holders, we see that the conjugate heat transfer has impact on the gain (maximum gain higher by 8%) and on the time lag as defined in Fig. 4 (up to 0.16 for St < 1.5). This can be attributed to the larger nominal distance of the flame leading edge from the steel bluff body (shown in Fig. 5) due to the higher heat flux to the body, and thus higher local velocity fluctuations. 3.2. Flame leading edge dynamics – harmonic velocity perturbations Based on the analysis of the linear response, Strouhal numbers of 0.43 and 1.25 were selected for analyzing the response to harmonic forcing, which is close to where peak response was observed (above 95% of the peak response). Additionally, St numbers of 0.156, 0.312 and 0.625 were selected (frequencies of 50,100 and 200 Hz), as well as St = 0.23, which corresponds to the von Karman vortex shedding frequency of the non-reacting flow. Only the results corresponding to the flame leading edge location during forcing for both ceramic and steel flame holders is seen in Fig. 5. Only the phase-plane plot of the leading edge of the upper flame is presented due to symmetry of the solutions. The leading-edge motion is clockwise. The impact of the forcing frequency on the leading-edge location is shown in Fig. 5a for ceramics and Fig. 5c for steel. When increasing the St number from 0.156 to 0.625 Hz there is an increase in the magnitude of the flame leading edge displacement in the axial direction. There is a maximum of the ellipse major axis at St = 0.43, as predicted by the transfer function (Fig. 4). Normal location and velocity oscillation are out of phase at St = 0.43 (see the marked circles in Fig. 5a for the instant of maximum inlet velocity), consistent with a phase of 1.1π at St = 0.43 according to

Fig. 4. When forcing at St = 1.25 the leading-edge motion amplitude becomes smaller and its mean location moves closer to the body. The time averaged recirculation zone length for the ceramic body also shows a drop from 4.97d to 5.03d at and below St = 0.625 to 4.48d at St = 1.25. Thus, both the flame leading edge moves, on average, further upstream and the mean recirculation zone length gets significantly shorter at St = 1.25. Moreover, the largest perturbation (although marginal) of the recirculation zone length occurs at the same St number as maximum leading edge motion. Hence, there is a strong correlation between the flame leading edge dynamics and the response of the wake flow to the imposed perturbations. These findings strengthen the hypothesis that the flame leading edge motion is mainly due to advection as result of the response of the wake flow. At St = 0.312 the calculations were also conducted at higher forcing amplitude (u’/U0 = 0.2) as seen in Fig. 5b. The magnitude of the leading-edge oscillations at u’/U0 = 0.2 is twice than that with u’/U0 = 0.1, showing the motion of the leading edge is proportional to the magnitude of the forcing under those relatively low forcing amplitudes. Thus, the response of the leading edge is linear to the oscillation amplitude. Comparison between the steel and ceramic bodies is shown in Figs. 5.a–c. With ceramics, the magnitude of oscillations is smaller by about 10% (as shown by the ellipse major axis), similar to the conclusions from the response to a step change in velocity. Considering the oscillation in the x or y direction independently, the oscillations are about 25% larger for steel. Additionally, with ceramics, the mean flame leading edge location remains significantly closer to the bluff body. Hence, the conjugate heat transfer impacts both the leading edge mean location and oscillation amplitude during harmonic forcing. The flame leading edge motion at different equivalence ratios is presented in Fig. 5d. At higher equivalence ratios, the difference in the flame leading edge location between steel and ceramics increases. For steel it can be seen that as φ increases the flame leading edge moves mainly closer to the shear layer (the positive ydirection). For ceramics, the displacement of the flame is mainly upstream, towards the body. Nevertheless, the magnitude of oscillations (the ellipse major axis) does appear to be only weakly dependent on the equivalence ratio, similarly the small difference in gain between φ = 0.6 and 0.7 in Fig. 4. The main conclusion from the above analysis is that the flame leading edge dynamics scale according to a Strouhal number, exhibiting a peak response at St≈0.5. The flame characteristics (equivalence ratio) and bluff body material (thermal conductiv-

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Fig. 5. Phase diagram of the flame leading edge location for the case of: (a) Ceramic bluff body at different forcing frequencies and u’/U0 = 0.1. The moment of maximal Uin is marked with a circle. (b) Different bluff body materials (cer. – ceramics) and forcing amplitudes. (c) Steel flame holder at different forcing frequencies (d) and different equivalence ratios. Only the location of the flame on the top side on the bluff body flame holder is shown due to symmetry. The time advances in clockwise direction.

ity) impact the response in terms of gain and mean location. A lower equivalence ratio or a higher thermal conductivity of the bluff body leads to larger oscillations and to a mean location further away from the bluff body. For inlet velocity perturbation of up to u’/U0 = 0.2 the flame leading edge dynamics change linearly with the perturbation amplitude. In the following sections, we will focus our attention on the results with the different body thermal properties as a way to identify the impact of leading edge dynamics on the conjugate heat transfer, the flame structure, the flow field and the heat release. 3.3. Conjugate heat transfer with the flame holder At steady state, there is a balance between the heat flux into the body from the high temperature combustion products in the wake region and the heat flux out of the body towards the reactants. The bluff body temperature at steady state, under nominal conditions (φ = 0.7 and U0 = 1.6 m/s) is 858–898 K for ceramics and 691-697 K for steel. The Fourier transform was computed from the response to a step change in the inlet velocity. The heat flux to the downstream side of the bluff body displayed a peak response at St = 0.45 ± 0.01, as shown in Fig. 6, similar to the flame leading edge motion frequencies at the peak response presented in Fig. 4. The gain in the heat flux increases with the thermal conductivity of the material, but the qualitative trends and phase response are very similar for both materials. The results cannot be considered a transfer function because the peak gain at u’/U0 of 0.01, 0.05, and 0.1 was nearly identical (see Fig. 6). This means that even for very small velocity fluctuation the response saturates. Because of the long thermal time scale associated with the bluff body, shown in the following analysis, the dynamics close to St = 0 can not be captured and thus the phase might be shifted by 2π . Having a peak response in the heat flux at the same frequency as that of the

leading-edge motion can have two possible explanations: (i) the peak response in flame dynamics originates from the heat transfer coupling, or (ii) the peak response in the heat flux originates from the leading edge dynamics. Because of the practically identical peak response frequency for steel and ceramics despite the difference of nearly an order of magnitude in heat conduction and thermal effusivity, ( λρ c), we conclude that the frequency at the peak response in the heat flux originates from the leading edge dynamics and is independent of the flame holder material. At higher frequency (St > 1.5) there is still large gain that becomes independent of frequency and with zero phase, indicating that any flow unsteadiness results in significant increase (up to 6fold) in heat flux to the flame holder. The frequency-independent gain is characteristic to a Fourier transform of a Dirac delta function in the time domain. This response can be explained by the instantaneous adjustment of the flow field to the step increase in the inlet velocity, which raises instantly the temperature gradient adjacent to the body. The peak gain at St = 0.5 suggests that this is a result of the flame leading edge dynamics. The significant increase in the heat flux to a bluff body flame holder at unsteady conditions was observed by Xavier et al. [36] for a rotating cylindrical bluff body made of steel. The interaction between the flame and a moving wall resulted in heat fluxes of up to 7 times higher than those obtained using a stationary wall. Figure 7 shows the fluctuation in the heat transfer to the different sides of the flame holder as a result of forcing at St = 0.43. Positive values indicate heat flux into the bluff body. The fluxes at steady inflow are also presented, showing heat transfer into the bluff body from the combustion products (the downstream side) and out towards the reactants (upstream side). The integral of the heat flux on the bottom and top sides is relatively small at steady inflow. For the ceramic bluff body, during forcing the heat flux at the upstream and downstream sides are similar to that of steady

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Fig. 6. Frequency response of the heat flux to the downstream side of the body to inlet velocity oscillations. The results are the Fourier transform of the derivative of response to inlet velocity using step input of size u’/U0 .

Fig. 7. Heat flux to the bluff body vs. inlet velocity at φ = 0.7 and forcing at St = 0.43 (138 Hz) with u’/U0 of 0.1. Positive sign is heat flux into the bluff body. Time advances is in the arrow direction.

conditions, whereas from the top and bottom sides there is much higher heat flux into the bluff body than under steady condition. This leads to net positive heat flux into the flame holder. For the case of steel, we see that the heat flux into the downstream side and out at the top and bottom sides occur at similar phase, displaying heat recirculation. However, there is significant hysteresis, resulting in higher heat flux into the downstream side and out of the top and bottom sides during flow deceleration. For the steel bluff body we see heat recirculation through the steel that tends to even out the temperature of the flame holder. For all the investigated forcing conditions, with the ceramic bluff body the net heat flux over each cycle is into whereas with the steel the net heat flux is out of the bluff body. The Biot number (hd/λs ) is about 0.4 for the ceramic body and 0.1 for the steel body, leading to higher temperature gradients in the ceramics relatively to that in the steel, as seen in Fig. 8. The heat convection coefficient, h, can be approximated from a time averaged heat flux of 500 W/m for ceramics and −10 0 0 W/m for steel, the surface temperature and the adiabatic flame temperature. In order to get a rough approximation of the characteristic time constant of the thermal process in the bluff body, we assume a uniform temperature in the bluff body (Biot number < 1) and use the energy conservation equation to obtain the following expression:

tc =

ρs c s d h

(13)

This results in a thermal time constant of 115 s for the steel and 25 s for the ceramic bodies. Because the time constant for both cases is much larger than the simulation time, only small temperature changes are captured. For example, during the last com-

puted forcing cycle at St = 0.43 there was a decrease of 0.005 K at the center of the steel bluff body, whereas for ceramics an increase of 0.02 K was noticed. As the bluff body temperature decreases/increases the heat losses from the flame will tend to decrease/increase and the flame leading edge location will move upstream/downstream. Therefore, the difference in location between the low and high conductivity materials (ceramic and steel) would increase with time. The seemingly harmonic response we observe after 10 forcing cycles would gradually shift after thousands of cycles. In order to get further insight into the impact of the conjugate heat transfer on the flame, it is useful to look at the normalized enthalpy defined as:

Hn =

H − H p,cold Had − H p,cold

(14)

where H is the total (sensible and chemical) enthalpy, Had is the reactants enthalpy and Hp,cold is the enthalpy of the equilibrium combustion products at 300 K. Hn is 1 under adiabatic conditions and 0 if the combustion products are quenched to 300 K (Hn > 1 means super adiabatic conditions), thus it could be considered a good measure of the heat losses. In Fig 8 we present Hn and the temperature at 2 instances of a forcing cycle at u’/U0 = 0.1 and St = 0.43 at φ = 0.7 for both ceramic and steel flame holders on the top and bottom halves of the figure, respectively. For the different forcing conditions shown in Fig. 5, the normalized enthalpy and temperature at the leading edge differ by less than 0.005 between the two materials. During flow forcing, the normalized values fluctuate by up to 0.02 from the steady-state value, reflecting the im-

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Fig. 8. Contours of normalized total enthalpy (H–Hp,cold )/(Had -Hp,cold ) and temperature (T-Tin )/(Tad -Tin ) showing the conjugate heat transfer with the body at φ = 0.7 during two instances of the forcing cycle of u’/U0 = 0.1 and Strouhal number of 0.43 (minimum and maximum inlet velocity) for both a ceramics and steel flame holders (top and bottom half of figure). Overlaid are the heat release rate contour of 10% of the maximum value (black line) and the body (square).

pact of flow unsteadiness on the conditions at the flame leading edge. As shown in Fig. 8, the lower heat losses associated with the ceramic bluff body lead to the further upstream location of the flame leading edge (shown in more detail in Fig. 5). There is increase in the enthalpy of the reactants at the upstream side of the bluff body due to the heat flux from the flame holder. The recirculating combustion products adjacent to the flame holder downstream side are subjected to the highest heat losses. Moreover, since the flame leading edge is inside the recirculation zone, there is mixing of the reactants with the lower enthalpy combustion products recirculating near the body, thus the increase in enthalpy at the upstream face of the bluff body seems to hardly influence the flame. Figure 8 shows transport of high temperature combustion towards and away from the flame holder during the harmonic forcing, which is the cause for the large heat flux fluctuations seen in Fig. 7 for the steel bluff body. Our analysis showed only small differences in the temperature at the flame leading edge between steel and ceramics despite the relatively large differences in the flame leading edge location and conjugate heat transfer with the body. Therefore, we conclude that the leading edge location follows necessary ignition conditions throughout the forcing cycle that depend on the local temperature and reactants mixture. Analysis of the local gas composition at the leading edge will be provided in the next section. The conjugate heat transfer with the flame holder essentially controls the location of the appropriate “ignition temperature”. Similar conclusions for steady flow by Kedia and Ghoniem [30], and the analysis above extends the conclusion for unsteady flames. Recent work by Miguel-Brebion et al. [37] on a circular bluff body show that neglecting the radiative heat losses can result in excessively high bluff body temperatures. Simulation results with different emissivity have shown that as long as the resulting bluff body temperature is lower than a certain threshold, the flame stabilized downstream the flame holder, and moves upstream above

a critical value. A similar result of the impact of the flame holder temperature on the flame stabilization location was shown by Berger et al. [23] by artificially varying the bluff body temperature for the same geometry used in this study. Moreover, it was shown [37] that as long as the flame is stabilized downstream of the bluff body, the impact of its emissivity on the flame holder temperature is small. In the present study, the flame stabilizes downstream the flame holder within the recirculation zone, so a decrease in the flame holder temperature as a result of radiative heat losses are expected to cause slightly further downstream flame stabilization location. Indeed, under steady conditions, a difference in about 200 K between the two flame holders causes the flame to stabilize 0.1d further downstream for the lower surface temperature. Moreover, during the unsteady simulations the flame holder temperature is practically constant, and the much larger heat flux fluctuations seen for steel result from the large difference in the thermal conductivity between the two materials. Hence, we do not expect radiative heat losses to alter observed heat transfer dynamics. 3.4. Flame structure The flame structures at the leading edge and at a downstream location are compared in Fig 9. The profile of different scalars is presented along a line normal to the flame downstream (x = 7d) and normal the leading edge. The normal to the flame is defined by the temperature gradient and the zero coordinate is defined at a normalized temperature of 0.72 (steady state temperature at the leading edge). From Fig. 9 one can see that the flame at the leading edge is very similar for steel and ceramics. Fluctuations in the species profiles are in correspondence with the variations of the temperature and enthalpy at the leading edge during flow forcing. One can also see that the difference between the scalar profiles at the leading edge and far from the flame holder are much higher than the fluctuations in the profiles during forcing. In summary, the results show that the flame structure at its leading edge, de-

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Fig. 9. Species mass fraction, temperature and heat release (HR) distribution along lines normal to the flame leading edge (top) and along a line normal to the flame at a downstream location (bottom), for ceramic (left) and steel (right) bluff bodies. The results are at steady conditions (thick line) and at nine evenly spaced instances during an harmonic forcing cycle at u’/U0 = 0.1 and Strouhal number of 0.43 (thin lines).

fined by 10% of the maximum heat release rate for unit volume, is distinctly different from the flame structure far from the bluff body, generally independent of conjugate heat transfer with the flame holder, and undergoes slight variations during flow forcing. The results provide strong indication that the flame leading edge dynamics are predominantly governed by local transport. This is in agreement with our hypothesis that the leading-edge motion is mainly due to advection as a result of excitation of the flow inside the recirculation zone, resulting in a scaling of the leading edge frequency of peak response according to a Strouhal number (Fig 4). Heat and mass transfer in this region will be discussed next. The flame far from the body is similar to that of a laminar flamelet. In contrast, the flame at the leading edge is much thicker. The distinctive structure at the leading edge is similar for both steady and unsteady flow, and its unique characteristics were presented and discussed by Michaels and Ghoniem [22]. It was shown that the structure of the flame leading edge is a result of high stretch rates, heat losses, recirculating combustion products and “flame back-support” (see Marzouk et al. [38]). In addition, the observed similarity to the flame leading edge structure measured by Barlow et al. [35] in turbulent flames, shows that the leading edge structure is independent from the exact geometry forming the recirculation zone. The relatively large fluctuations in both the temperature and OH radical on the burned side of the flame leading edge are related to the establishment of a flame in regions of combustion products, in accordance with the “back support” concept

[38]. The fluctuations in the scalar profiles in the preheat zone can be attributed to vortex flame interaction during the forcing cycle. By looking at the scalars profiles in the preheat zone we see that the fluctuations at the leading edge are qualitatively similar for steel and ceramics. These fluctuations in the preheat zone can be attributed to the high stretch rate fluctuations near the flame leading edge. Far from the leading edge the fluctuations are larger for steel. This is an indication that the vortex–flame interaction far from the body is stronger for the steel body, as will be shown in the following section.

3.5. Flow–flame interaction In Fig. 10 the maximum vorticity at each cross section and the flame stretch downstream the bluff body are shown for both the steel and ceramics bodies for forcing at St = 0.43. Results at forcing Strouhal numbers of 0.156–1.25 show qualitatively similar trends of lower vorticity and stretch downstream for ceramics. Figure 10 shows a vortex advecting downstream (vorticity peak) inducing flame stretching. The difference in vorticity and stretch between the two bluff body materials is small near the body and increases considerably as the vortex and flame are transported downstream. In the following, we analyze the impact of the bluff body material, and thus the leading edge dynamics, on the flow field. Subsequently, we discuss how this impact on the flow field influences stretching and wrinkling of the flame.

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Fig. 10. Maximum vorticity at each channel cross-section (left) and flame stretch (right) downstream the body at five equally spaced time intervals during a forcing cycle at Strouhal number of 0.43 and u’/U0 = 0.1. Results for both steel body (dashed lines) and ceramic body (the continues lines) are presented.

From Fig. 10a one can see that at the downstream side of the body (x/d = 3) the maximum vorticity values for both materials are similar. The slightly lower values for ceramics can be attributed to the higher temperature and thus higher viscosity near the sides of the body (see Fig. 8). It is clear that the differences in vorticity become larger with distance, with the decay being faster for the ceramics body. The recirculation zone ends at approximately x/d = 5, and it is seen that at the moment the vortex reaches this location (number 4 in Fig. 10) the vorticity is higher by 35% for the steel in comparison to the ceramics. For comparison, under steady flow conditions [22] the flame the vorticity at x/d = 5 was higher by 26% for the steel body. Understanding the source of the larger impact of the flame holder material on the vorticity decay under unsteady flow in comparison to steady flow requires examination of the various terms in the vorticity equation:

Dω = −ω (∇ · u ) − Dt

1  ∇ p × ∇ρ + ∇ × · τ ∇ ρ ρ2

(15)

In previous work the gas expansion term was shown to have the largest contribution to vorticity decay near the flame holder (see Mehta and Soteriou [39]). Analysis of the steady flow [22] revealed that under steady flow conditions the impact of the flame holder material on the flow field is through the gas expansion term (first terms in Eq. 15). The further upstream location of the flame leading edge for ceramics results in faster decay of the vorticity due to gas expansion closer to the shear layer (i.e. larger value of ω∇ · u). In harmonically forced flow, the vortex–flame interaction is more complex than under steady flow. The flame motion impacts the vorticity field through fluctuations in the gas expansion term because the relative location of the flame and the shear layer is not constant. As seen in Fig. 5, during forcing at Strouhal number of 0.43 the difference in leading edge location between the steel and ceramic bodies is larger than under steady conditions. This can lead to the larger differences in the flow field between steel and ceramics for the forced case as discussed above. Another mechanism by which the flame leading edge dynamics can impact the flow is by enhancing flame wrinkling that alters the baroclinic torque term in the vorticity equation. More specifically, flame wrinkling generated by the leading edge motion change the relative direction of density and pressure gradients along the flame and thus imposes temporal and spatial variations in the baroclinic torque. The results show that overall there is larger impact of the flame holder thermal properties on the flow field during harmonic forcing in comparison to steady flow, and this impact is attributed to the leading-edge dynamics.

From Fig. 10b we can see that at the flame leading edge the maximum stretch rate the flame can sustain with the ceramic body is 40% higher than with steel. This higher resistance to stretch for ceramics can be explained by the lower heat flux to the bluff body. At a distance of ∼1d from the bluff body the trend changes and the stretch rate becomes higher for steel due to larger velocity gradients as reflected by the larger vorticity downstream the steel body. At x = 5d (approximately at the reattachment point) the stretch rate for the steel body is 33% higher. The contribution of curvature to flame stretch was found to be less than 10% in the vicinity of the body (x/d < 5). The flame stretch does not decay in a similar manner as the vorticity because of the increasingly dominant contribution of curvature further downstream (x/d > 5). It is also interesting to see positive and negative stretch as would be expected in the field of a large vortex being convected downstream. At the leading edge, the flame sustains very high stretch rates. Hong et al. [40] experimentally measured similar orders of magnitude stretch rates in a backward facing step combustor. The high stretch rates near the bluffbody also results in lower flame displacement speed (positive Markstein number for lean methane flames) and hence less smoothing by kinematic restoration of flame wrinkles generated by the leading edge motion. The large differences in flame stretch rates between steel and ceramic bodies lead to noticeable differences in flame elongation and heat release near the body, as shown in the next section. The analysis in Sections 3.3 and 3.4 showed that the flame leading edge motion is caused by advection of “ignition” conditions (temperature and species) within the wake. Looking at the bottom two images of Fig. 3, one can see that the vortices that cause the advection of the flame leading edge are transported downstream alongside wrinkles generated by the flame leading edge motion. Hence, small flame wrinkles generated by the flame leading edge dynamics are impacted significantly by vortex–flame interaction. Preetham et al. [19] showed analytically that for long bluff body stabilized flames, such as those considered in this study, convective flow non-uniformities amplify the flame disturbances generated at the flame anchoring over a wide range of frequencies. Even though these results were obtained without considering the flame leading edge dynamics or flame stretching, this augmentation is expected and observed also in the present work. We show that this augmentation mechanism is also impacted by the leading edge dynamics by influencing the flow field, as seen in Fig. 10a. In summary, The results herein demonstrate that the leading edge dynamics not only generates flame wrinkling from kinematic considerations [14], but also impacts the flow field in the vicinity of the bluff body and hence the flame stretching by vortex–flame

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Fig. 11. Frequency response of the heat release to inlet velocity oscillations. The results are the Fourier transform of the derivative of response to inlet velocity step input of size u’/U0 = 0.01. The time delay is normalized by the characteristic flow time d/U0 .

interaction and augmentation of the flame wrinkles generated by the leading edge dynamics. The mechanism suggests a link between flame anchoring and flame–vortex interactions, which are the two processes that control the flame dynamics near a bluff body during harmonic velocity forcing according to previous studies [18,19]. 3.6. Heat release oscillations We focus our attention on the heat release between the flame holder and the end of the recirculation zone (x ≤ 5d) because the flame anchoring and vortices shed from the body dominate this region [18]. The frequency response of the heat release to inlet velocity oscillations is presented in Fig. 11. The results were obtained from the response to a step of 1% in the inlet velocity. Increasing the flow velocity, decreasing the equivalence ratio or lowering the thermal conductivity of the flame holder material result in higher peak gain in heat release, similar to the impact on the leadingedge motion (Fig. 4). The response of heat release of such a small step increase in velocity (u’/U0 = 0.01) was used because the heat release response to a step of u’/U0 = 0.1 or u’/U0 = 0.05 was found to be outside the linear regime. The results in Fig. 11 show a maximum gain at St≈0.55, indicating that there is a peak in the flame– vortex interaction at that St number. The higher peak for steel vs. ceramics can be related to the further downstream location of the leading edge that results in stronger vorticity damping as discussed above. However, because at u’/U0 = 0.01 the leading-edge motion amplitude is at the same order of magnitude as the grid resolution (98 μm), the leading edge motion can be considered negligible. Hence, in order to derive conclusions regarding the impact of the leading edge motion on the heat release larger forcing amplitudes must be considered. Thus, we will focus our analysis on harmonic forcing of u’/U0 = 0.1, conditions at which the leading edge motion is significant (see Fig. 5) and the leading edge motion is still linearly proportional to the perturbation. The flame response to harmonic forcing is shown in Fig 12 along with a comparison between steel and ceramic flame holders at similar flow condition (U0 = 1.6 m/s, u’/U0 = 0.1) and various frequencies. The flame area follows the heat release quite closely, thus it is not presented for the sake of brevity. The heat release is normalized by the time-averaged value in order to obtain the relative amplitude of the oscillations. At St = 0.31 (Fig. 12a) the heat release oscillation amplitude are 36% larger with steel in comparison to ceramics, whereas the leading edge oscillation amplitude (shown in Fig. 5) is very similar for both materials, with only a

shift in the mean location. From these results we conclude that regardless of the leading-edge oscillation amplitude, the further upstream mean location of the flame leading edge in the case of a steel bluff body leads to lower vorticity damping and larger flame wrinkling at all frequencies. At St = 0.43 (Fig. 12b) the heat release oscillations are 50% larger with steel, which can be attributed to both the further-fromthe-bluffbody location of the leading edge and the higher oscillations amplitude with the steel body (about 10% difference). Also, at St = 0.43 the heat release oscillations reach its highest value in all the cases shown in Fig. 12 for both steel and ceramics, corresponding to the maximum in flame leading edge motion normal to itself (see Fig. 5). When forcing at St = 0.63 (Fig. 12c) the heat release oscillation amplitude is 10% lower than at St = 0.43, but the difference between steel and ceramics is slightly larger, with 54% increase with steel in comparison to ceramics. At St = 1.25 (Fig. 12d) the amplitude of oscillations is similar for both materials. It is interesting to see that in Fig. 11, where the forcing amplitude is only u’/U0 = 0.01 and leading-edge motion is hardly noticeable, the difference in the peak gain between steel and ceramics is 31%. Hence, for forcing amplitude that is high enough to cause significant leading edge motion, the impact of the flame holder material becomes larger. We conclude that close to the St number of the peak response in the leading edge normal motion, the amplitude of the heat release oscillations reaches a maximum and the impact of the flame holder material on the vortex–flame interaction becomes larger. Both mechanisms, which originate from the mean and amplitude of leading edge oscillations, have very significant impact on the heat release oscillations. Looking at the time lag in Fig. 12, it can be seen that at St = 0.31 (Fig. 12a) there is minimal difference between steel and ceramics, whereas for St ≥ 0.43 (Fig. 12b–d) there is a larger time lag for steel. This longer time lag for steel in comparison to ceramics above St≈0.3 is consistent with the time lag of the leading edge motion shown in Fig 4. Previous studies [14] have shown that the amplitude of heat release oscillations resulting from the leading-edge motion should be proportional to the displacement amplitude normal to the nominal flame direction. It was also found [19] that flame area fluctuations originating from the flame anchoring are augmented as a result of flow non-uniformities. Here, we show that the flame leading edge dynamics not only generate heat release oscillation by wrinkling the flame, but also impact the flow field and thus have large impact on the amplification of those wrinkles.

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Fig. 12. Heat release near the bluff body (x < 5d) in response to harmonic velocity forcing. Forcing frequencies of 100, 138, 200 and 400 Hz correspond to St numbers of 0.312, 0.431, 0.625 and 1.25.

4. Conclusions By applying a fully resolved and coupled numerical approach for premixed flame stabilized on a heat conducting flame holder we were able to shed light on the flame leading edge dynamics and its impact on the flame heat release fluctuations. We have simulated the response to acoustic forcing in a coupled system by harmonic forcing of the inlet velocity at different frequencies and looking as well at the response to a step change in inlet velocity. The main contribution of the work is the identification of two mechanisms by which the leading-edge location impacts the heat release oscillation. First, as the leading edge is further away from the body, the vorticity decay downstream of the bluff body becomes slower, which leads to stronger flame wrinkling and heat release oscillations. This first mechanism impacts predominately the amplitude of heat release oscillations. Second, flame leading edge motion normal to the nominal flame surface results in wrinkling of the flame. The leading-edge dynamics exhibits a peak response at St ≈0.5 corresponding to excitement of the flow in the recirculation zone. Relatively small flame wrinkles generated by the flame leading edge motion are significantly amplified by the vortices, leading to large heat release fluctuations when the flame is excited at the leading edge frequency of peak response. Additionally, the time lag between the velocity and heat release oscillations is dictated by the flame leading edge dynamics. The conjugate heat transfer with the body impacts both mechanisms, because a further downstream location of the leading edge as result of higher heat flux to the flame holder leads also to larger fluctuations of the leading edge.

The dynamic response of flames to velocity fluctuations, that may result from combustion instabilities, are significantly impacted by the flame leading edge dynamics, and therefore should be accounted for in modeling. Since the flame structure at the leading edge is impacted by heat losses, preferential diffusion, recirculation combustion products and high strain [22], all of those physical processes should be considered in modeling the dynamics of bluff body stabilized flames. In this paper, we reveal that the leading edge dynamics have a central role in the amplitude and time lag of the flame response so artificial anchoring of the flame can lead to considerable errors. A previous study [19] showed that the flame kinematics resulting from the vertical structures can cause amplification of flame wrinkles generated as results of the flame attachment to the bluffbody. Here we show how the flame anchoring actually exhibits dynamics that impact the heat release oscillations by wrinkling the flame and also significantly impacting the flow field. Additionally, the impact of the conjugate heat transfer on the flame leading edge dynamics presented in this study explains the experimental measurements of lean premixed turbulent flame in a backward facing step combustor [20] which showed that a ceramic step increased significantly the stable operating conditions in comparison to a steel step.

Acknowledgments This work was partly supported by a MIT-Technion fellowship and partly by KAUST.

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