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Linear response of stretch-affected premixed ﬂames to ﬂow oscillations H.Y. Wang a , C.K. Law a,∗ , T. Lieuwen b a b

Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA

a r t i c l e

i n f o

Article history: Received 3 July 2008 Received in revised form 26 January 2009 Accepted 26 January 2009 Available online 21 February 2009 Keywords: Combustion instabilities Acoustic-ﬂame interaction Flame stretch Transfer function G-equation

a b s t r a c t The linear response of 2D wedge-shaped premixed ﬂames to harmonic velocity disturbances was studied, allowing for the inﬂuence of ﬂame stretch manifested as variations in the local ﬂame speed along the wrinkled ﬂame front. Results obtained from analyzing the G-equation show that the ﬂame response is mainly characterized by a Markstein number σˆ C , which measures the curvature effect of the wrinkles, and a Strouhal number, St f , deﬁned as the angular frequency of the disturbance normalized by the time taken for the disturbance to propagate the ﬂame length. Flame stretch is found to become important −1/2 ). Speciﬁcally, for disturbance when the disturbance frequency satisﬁes σˆ C St2f ∼ O (1), i.e. St f ∼ O (σˆ C frequencies below this order, stretch effects are small and the ﬂame responds as an unstretched one. When the disturbance frequencies are of this order, the transfer function, deﬁned as the ratio of the normalized ﬂuctuation of the heat release rate to that of the velocity, is contributed mostly from ﬂuctuations of the ﬂame surface area, which is now affected by stretch. Finally, as the disturbance frequency increases to St f ∼ O (σˆ C−1 ), i.e. σˆ C St f ∼ O (1), the direct contribution from the stretch-affected ﬂame speed ﬂuctuation to the transfer function becomes comparable to that of the ﬂame surface area. The present study phenomenologically explains the experimentally observed ﬁltering effect in which the ﬂame wrinkles developed at the ﬂame base decay along the ﬂame surface for large frequency disturbances as well as for thermal-diffusively stable and weakly unstable mixtures. © 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

1. Introduction This paper theoretically investigates the effects of ﬂame stretch on the linear response of premixed ﬂames to harmonic velocity disturbances. The study was motivated by the general interest in self-induced combustion-driven oscillations in combustion systems [1–3], and by recent investigations on the response of heat release to ﬂow modulations [4–8]. These recent analyses assume that the ﬂame speed is constant and hence is independent of stretch. This then implies that the heat release responds to disturbances only through modulations of the ﬂame surface area. However, recognizing that the local ﬂame propagation speed and hence the local burning rate are actually affected by the disturbances, it is reasonable to expect that ﬂuctuations of the heat release rate are correspondingly affected. In particular, since the wavelength of the ﬂame wrinkling induced by the acoustic forcing scales inversely with the disturbance frequency [4,6], variations in stretch-induced ﬂame speeds are expected to become signiﬁcant at high frequencies. For example, Baillot and co-workers [9–11] studied the response of Bunsen ﬂames of methane–air mixtures to velocity disturbances of varying amplitudes and frequencies. It was found that,

*

Corresponding author. E-mail address: [email protected] (C.K. Law).

at low disturbance frequencies and amplitudes, the ﬂame front wrinkles with constant amplitude from the ﬂame base to its tip. At higher frequencies but similar low amplitudes, a phenomenon referred to as “ﬁltering” was observed, wherein ﬂame wrinkling was evident only at the ﬂame base and decayed with streamwise location downstream. Preetham et al. [12] subsequently studied the response of lean propane ﬂames and demonstrated photographically the existence of the “ﬁltering” phenomenon, as shown in Fig. 1. It is seen that at a low disturbance frequency ( f = 100 Hz) the ﬂame wrinkles persist along the ﬂame front with nearly constant amplitude. However, when the frequency is doubled ( f = 190 Hz), the ﬂame wrinkles decay rather rapidly along the ﬂame front and become only evident at the ﬂame base. Lieuwen [13] suggested that this behavior could result from the growing signiﬁcance of ﬂame speed variation along the ﬂame due to the small radii and hence strong curvature of the ﬂame wrinkles at high disturbance frequencies. The primary objective of the present investigation is to study the role of ﬂame stretch through the curvature of the ﬂame wrinkles on the premixed ﬂame response to acoustic oscillations. This is motivated by the recognition that studies since the 1980s have conclusively identiﬁed the essential and signiﬁcant inﬂuence of stretch on the response of both premixed and diffusion ﬂames [14]. The inﬂuence is further augmented in the presence of nonequid-

0010-2180/$ – see front matter © 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.combustﬂame.2009.01.012

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Nomenclature A G GA GS Lf Le n Q su sou St St f u uo u v W

ﬂame surface area transfer function transfer function resulted from ﬂuctuations of the ﬂame surface area transfer function resulted from the ﬂame speed ﬂuctuation steady-state ﬂame length Lewis number normal vector on the ﬂame front pointing toward the unburned mixture rate of heat release laminar ﬂame speed with stretch effect laminar ﬂame speed without stretch Strouhal number, ωo L f /u o reduced Strouhal number, deﬁned in Eq. (19) streamwise component of ﬂow velocity mean streamwise ﬂow velocity amplitude of velocity disturbance transverse component of ﬂow velocity ﬂame half width

x y Ze

streamwise coordinate transverse coordinate Zel’dovich number

Greek symbols ﬂame aspect ratio, L f / W thermal thickness of ﬂame nondimensional amplitude of velocity disturbance ﬂame stretch rate Markstein length related to the curvature sensitivity of the ﬂame speed σC /β(1 + β)1/2 , deﬁned in Eq. (20) disturbance angular frequency instantaneous ﬂame position steady-state ﬂame position disturbed ﬂame position

β δ

ε κ σC σˆ C ωo ζ ζo ζ1

Overbars and accents −

steady-state values disturbed values

Fig. 2. Schematic of two-dimensional wedge shaped ﬂame geometry.

Under a harmonic disturbance velocity ﬁeld, u (x, t ), the ﬂame surface oscillates around its steady-state position. This leads to ﬂuctuations of the heat release rate that may couple to the disturbance ﬁeld, resulting in oscillations with either growing or decaying amplitude. Thus the basic problem of interest is to determine the response of the ﬂame position, ζ (x, t ), and the heat release rate of the ﬂame, to a given u (x, t ). The response of the ﬂame is evaluated by the transfer function, deﬁned as Fig. 1. Visualization of a 100 Hz (a) and 190 Hz (b) acoustically excited lean propane ﬂame (equivalence ratio 0.7). Images show ﬂame wrinkling with constant and damped amplitude, respectively.

iffusion because of the associated modiﬁcation of the ﬂame temperature. Since the present phenomena involve ﬂame wrinkling at various scales, it behooves us to assess how and to what extent they are affected by stretch and nonequidiffusion. We shall show in due course that such an inﬂuence is indeed signiﬁcant and as such needs to be accounted for in analyses of combustion instability. 2. Formulation Fig. 2 illustrates the geometry considered in the analysis, namely a two-dimensional wedge ﬂame stabilized by a bluff body. The streamwise and transverse dimensions of the ﬂame are given by the ﬂame length, L f , and its half width, W , without imposed disturbance. The instantaneous ﬂame-sheet location at the transverse location, y, is given by x = ζ ( y , t ) and is assumed to be a single-valued function of y.

G=

) ( Q / Q (u /u o )

(1)

where u o is the mean ﬂow velocity, the overbar and prime respectively denote the steady-state and disturbance values, Q is the global heat release rate of the ﬂame given by

Q (t ) =

ρu su h R d A

(2)

where ρu is the density of the unburned mixture, su the local ﬂame speed, h R the heat release per unit mass of the reactant, and the integral is over the entire ﬂame surface area, A. Equation (2) shows that there exist three fundamentally different sources of generating heat release disturbances in a premixed ﬂame, namely disturbances in the mass burning rate of the ﬂame, ρu su , the heat of reaction, h R , and the ﬂame surface area, A. Here, we shall assume constant h R and ρu ; while allowing su to vary due to effects of ﬂame stretch. As such, our subsequent analysis focuses on the quantity Q

Q

=

su d A s¯ u d A

+

A

A

(3)

in the linear limit. Thus, the transfer function consists of contributions from the disturbance to both the ﬂame speed and ﬂame

H.Y. Wang et al. / Combustion and Flame 156 (2009) 889–895

surface area, which are expected to be also coupled since variations in the ﬂame speed would cause corresponding variations in the shape of the wrinkles and hence the ﬂame surface area. This is to be contrasted to previous studies [4–8] in which the transfer function is only affected by ﬂuctuations of the ﬂame surface area because the ﬂame speed is assumed to be constant. The ﬂame speed and surface area are now dependent on the ﬂame shape, which can be solved from the G-equation considered next. 2.1. G-equation The analytical approach used here closely follows that of Baillot et al. [15] and Fleifel et al. [4]. The ﬂame dynamics are modeled with the front tracking equation [14,16]:

2 ∂ζ ∂ζ ∂ζ =u−v − su 1 + ∂t ∂y ∂y

(4)

where u and v denote the streamwise and transverse components of the ﬂow velocity, respectively. The ﬂame speed can be expressed as [17] su sou

= 1 − δ∇ · n +

Ze

2

1 Le

κ −1 δ o

su

(5)

where sou is the constant, planar laminar ﬂame speed, n the local normal on the ﬂame front pointing toward the unburned mixture, δ the thermal thickness of the ﬂame, Ze the Zel’dovich number, and κ the ﬂame stretch rate given by

κ = −n · ∇ × (v × n) + (V · n)(∇ · n)

(6)

where v = (u , v ) is the ﬂow velocity at the ﬂame front on the unburned side and V = dx/dt the local velocity of the ﬂame front. We shall limit our study to the case of weak stretch, namely small δ/ L f , and assume Ze(Le−1 − 1) ∼ O (1). It is seen from the second and third terms of the RHS of Eq. (5) that the modiﬁcation of the ﬂame speed by stretch is given by the sum of the pure curvature effect and the nonequidiffusion-related stretch effect. Following previous studies [4,15], we assume that the ﬂame remains anchored at the base, i.e.

ζ ( y = 1, t ) = 0.

(7)

For wedge ﬂames, the second boundary condition comes from the requirement that all information should ﬂow out of the ﬂame. This is a rigorous way of capturing the fact that the ﬂame tail is free to move around [12], i.e.

∂ ζ ( y = 0, t ) = 0. ∂ y2 2

(8)

The ﬂow is assumed to be purely streamwise, i.e. v = (u , 0). Then, the mean streamwise velocity u o is related to the laminar ﬂame speed, sou , by

uo sou

=

1+

Lf

2

W

−i ωo t

u (ζ, t ) = u o + u e

It is noted that Schuller et al. [18] employed a spatially nonuniform disturbance ﬁeld by incorporating the convective phase variation u (ζ, t ) = u o + u e i (kζ −ωo t ) ,

(9)

where u and ωo respectively denote the amplitude and angular frequency of the velocity disturbance.

v =0

(10)

where k is the convective wave number. The shape of the disturbed ﬂame front was found to result from the conjugating action of the wrinkles convected along the ﬂame induced by the bulk ﬂow oscillation, e −i ωo t , at the ﬂame base and those locally induced by the ﬂow nonuniformity, e ikζ . This disturbance, however, renders a nonzero divergence velocity ﬁeld, and hence does not satisfy the continuity equation. While including a transverse disturbance component in Eq. (10) in order to have a divergence-free velocity ﬁeld is straightforward, it leads to tedious algebra and does not contribute additional essential insight into the physics of the problem, at least at the level considered herein. Furthermore, it is emphasized that our goal here is not to simulate the exact disturbance ﬁeld of a particular experimental setup, but rather to elucidate the key physical processes and nondimensional parameters that control the damping of ﬂame wrinkling. The disturbance velocity ﬁeld, given by Eq. (9), yields the needed ﬂame wrinkling behavior for the present study. Nondimensionalizing the variables t, y, δ, u and ζ by L f /u o , W , W , u o and L f , respectively, the nondimensional front tracking equation is given by

∂ζ =u− ∂t

2 ∂ζ 2 su 1 + β ∂ y sou 1 + β2

(11)

and the nondimensional velocity ﬁeld can be written as u (ζ, t ) = 1 + ε e −iSt·t

(12)

where St =

ωo L f uo

is the Strouhal number and

ε = u /u o .

2.2. Solutions of ﬂame disturbance and transfer function In this section, we derive the expressions for the location of the disturbed ﬂame and the transfer function when the ﬂame speed is affected by stretch. In response to the velocity disturbance, the ﬂame position can be expanded as

ζ ( y , t ) = ζo ( y ) + ζ ( y , t ),

ζ ( y , t ) = ε ζ1 ( y )e −iSt·t + O ε 2

(13)

where

ζo ( y ) = 1 − y

(14)

is the steady-state ﬂame location. Substituting Eqs. (13) and (14) into Eq. (5), the ﬂame speed relation can be expressed as su

where the ratio of the ﬂame length to its half width, β = L f / W , plays an important role in the ﬂame dynamics. Furthermore, the ﬂame is assumed to be subject to a spatially uniform harmonic velocity disturbance of small amplitude

891

sou

=1−

σC βζ y y

(15)

(1 + β 2 )3/2

where the subscript “ y” denotes the spatial derivative with respect to y, and

σC = 1 −

Ze 2

1 Le

δ −1 W

(16)

is the Markstein number related to the curvature sensitivity of the ﬂame speed. It is seen from Eq. (16) that, as noted earlier, the

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ﬂame speed is modiﬁed by the curvature through a pure curvature effect, which is independent of Le, and the curvature component of the nonequidiffusion-related stretch. Consequently, the nonequidiffusional effect tends to strengthen the pure curvature effect when Le > 1, and weakens it when Le < 1. We further note that alternate expressions for the stretch-affected ﬂame speed exist, such as that of Matalon and Matkowsky [19]. However, once expanded in terms of the ﬂame position and ﬂow speed, they can be expressed in the same form as Eq. (15). The differences are lumped into the detailed expression for the Markstein number, σC . In this paper we shall study the effects of ﬂame stretch on the ﬂame response by employing different values of σC , which provide a direct interpretation of its inﬂuences on the ﬂame response. Thus, the present analysis is not restricted by the speciﬁc expression for the stretch-affected ﬂame speed, and as such is general in nature. We shall, however, restrict our investigation to positive values of σC , since we are interested in curvature-induced damping. Thus the mixtures of interest here are either diffusionally stable or mildly unstable, which conform to the experimental situations of Ref. [12]. It should also be pointed out that the Markstein number σC would become a function of frequency when the time scale of the ﬂow oscillation becomes comparable with that of diffusion through the ﬂame [20]. However, the current study reveals that the phenomenon of interest in this paper, namely damping of the ﬂame wrinkling, occurs at the time scale much larger than that of diffusion. Thus, the frequency-independent Markstein number, Eq. (16), and subsequent analysis in the following are still valid. Substituting Eqs. (13)–(15) into Eq. (11) and collecting O (ε ) terms, the evolution equation for the disturbed ﬂame location ζ1 can be derived as

β 2 ∂ζ1 ∂ 2 ζ1 + + iStζ1 + 1 = 0. 2 ∂y 1 + β2 ∂ y

σC β (1 + β 2 )3/2

(17)

The solution of Eq. (17), subject to the boundary conditions in Eqs. (7) and (8), is L1 y

ζ1 ( y ) = Ae

+ Be

where 1

L 1, 2 = A=−

2σˆ C

β2

(18)

1 − 4i σˆ C St f ,

L 22 e L 1 − L 21 e L 2 1 + β2

+C

C L 22

St f = St

σˆ C =

−1 ±

L2 y

B=

,

C L 21 L 22 e L 1 − L 21 e L 2

,

C =−

1 iSt

σC β(1 + β 2 )1/2

,

(20)

.

In the above St f , referred to as the reduced Strouhal number, combines effects of the aspect ratio of the ﬂame and the Strouhal number, and can be rewritten as ωo ( L f / cos θ)/(u o cos θ), where θ is the angle between the ﬂame surface, without disturbance, and the ﬂow direction. Thus it represents the angular frequency of the disturbance normalized by the time taken for the ﬂame disturbance to propagate the ﬂame length. Recognizing that in the limit of weak stretch, i.e. σˆ C → 0, we have L 2 → −∞, e L 1 e L 2 and e L 1 y e L 2 y except for the region very near the ﬂame tail ( y → 0), Eq. (18) can be simpliﬁed to

ζ1 = −

1 iSt

1−e

L 1 ( y −1)

.

e L 1 y e L 2 y is not closely satisﬁed. Furthermore, they only show very small difference at the ﬂame tail even for σˆ C = 0.5. Thus, hereafter we shall present the analysis based on the simpler solution, Eq. (21). Next, we consider the total heat release of the ﬂame. Since, by considering ﬂame stretch, the heat release responds to the disturbance through both the ﬂame surface area and ﬂame speed, ﬂuctuations of the heat release can be expressed as Q = Q A + Q S in the linear limit, where Q S =

(21)

Fig. 3 shows, for different values of σˆ C , the transverse distribution of the amplitude of the ﬂame oscillation, |ζ1 ( y )|, from the exact and approximate solutions, Eqs. (18) and (21), respectively. It is seen that the solutions agree well for σˆ C up to 0.2, which is a rather large value, even near the ﬂame tail region where

su d A,

Q A =

s¯ u d A

(22)

are their respective consequences, and A = (1 + β 2 )1/2 dy , d

(23)

∂ζ β2 d A = − dy , (1 + β 2 )1/2 ∂ y

(24)

su = −

(19)

,

Fig. 3. Transverse distribution of the ﬂame oscillation amplitude, |ζ1 ( y )|, from Eqs. (18) and (21), respectively, for different values of σˆ C with St = 10 and β = 2. Note that y = 1 and y = 0 correspond to the ﬂame base and tail, respectively.

σC βζ y y (1 + β 2 )3/2

.

(25)

Substituting Eqs. (8), (14) and (23)–(25) into Eq. (22), and then into Eq. (1), yields GS = − GA =

σˆ C β 2 1 + β2

∂ζ1 ( y = 1) ∂ζ1 ( y = 0) , − ∂y ∂y

β2 ζ1 ( y = 0) 1 + β2

(26) (27)

)/(u /u o ) and G A = ( Q / Q )/(u /u o ) are the where G S = ( Q S / Q A transfer functions contributed from ﬂuctuations of the ﬂame speed and ﬂame surface area, respectively. The overall transfer function is then given by G = G S + G A , which depends on two key parameters, St f and σˆ C . 3. Results and discussion 3.1. Baseline ﬂame response Since the inﬂuence of stretch on the ﬂame response to disturbances should be assessed based on comparisons between results with and without stretch, we shall ﬁrst present the baseline ﬂame response characteristics for the unstretched ﬂame. For this case,

H.Y. Wang et al. / Combustion and Flame 156 (2009) 889–895

(a)

Fig. 5. Transverse distribution of the ﬂame oscillation amplitude, |ζ1 ( y )|, for different values of σˆ C with St f = 47.5 and β = 2. Note that y = 1 and y = 0 correspond to the ﬂame base and tail, respectively.

(b) Fig. 4. Dependence of (a) the gain and (b) the phase of the transfer function on St f .

the transfer function is only contributed from ﬂuctuations of the ﬂame surface area. With σˆ C = 0, Eq. (21) becomes

ζ1 = −

1 iSt

1 − e iSt f (1− y ) .

(28)

Substituting Eq. (28) into (27) yields the transfer function G =−

1

1 − e iSt f

iSt f

(29)

with the gain given by

|G | =

2 sin(St f /2). St f

893

(30)

Fig. 4(a) shows the dependence of the gain of the transfer function, |G |, on St f , given by Eq. (30). It is seen that the gains are always less than unity and exhibit a series of peaks and nodes. In particular, the nodes in the gain occur at frequencies satisfying St f = 2nπ (n = 0, 1, 2, . . .). Fig. 4(b) shows that the phase of the transfer function increases with increasing St f , and has a jump of −π at St f = 2nπ as a result of the nodes in the gains at these values of St f . It is noted that the existence of nodes in the gain of the transfer function at St f = 2nπ does not mean that the ﬂame does not respond to disturbances at these frequencies. To demonstrate this point, the transverse distribution of the amplitude of the ﬂame oscillation, |ζ1 |, is shown in Fig. 5 for St f = 47.5 and β = 2. It is

seen that for this frequency there exist nodal points (|ζ1 | = 0) on the ﬂame surface in addition to the one at the ﬂame base ( y = 1). Thus the ﬂame segments within these nodal points are constrained by them and as such oscillate in the manner of a vibrating string. Since there is no nodal point at the ﬂame tail for this frequency, the ﬂame segment between the ﬂame tail and the nearest nodal point exhibits both bulk oscillatory movement and local wrinkling. Thus the ﬂuctuation of the ﬂame surface area is a consequence of the superposition of these two forms of ﬂame movement. Specifically, for frequencies corresponding to St f = 2nπ , a nodal point is located at the ﬂame tail so that the entire ﬂame surface is constrained by the nodal points, and the ﬂuctuation of the ﬂame surface area is only due to ﬂame wrinkling. In this case, it can be shown that the ﬂuctuation amplitude of the ﬂame surface area is O (ε 2 ), which is neglected by the linearization process. This is the reason that the transfer function shown in Fig. 4 has nodes for St f = 2nπ even though the velocity disturbance wrinkles the ﬂame. 3.2. Flame stretch effects We now consider the inﬂuence of stretch on the gain and phase of the transfer function. Fig. 5 shows the transverse distribution of the amplitude of ﬂame oscillation for different values of σˆ C , with the parameters (St f = 47.5 and β = 2) chosen to be consistent with the experiments of Bourehla and Baillot [11]. It is seen that in the presence of stretch, the amplitude of the ﬂame front wrinkling decays continuously from the ﬂame base ( y = 1) to the tail ( y = 0), in contrast to the constant amplitude for the unstretched ﬂame (σˆ C = 0). Thus, the experimentally observed damping in the ﬂame front oscillation away from the ﬂame base is reproduced. To further explore the damping mechanism of ﬂame wrinkling by stretch, we expand Eq. (21) for small σˆ C . In this limit,

L 1 ∼ −iSt f 1 − 2σˆ C2 St2f + σˆ C St2f . Then Eq. (21) becomes

ζ1 = −

1 iSt

1−e

σˆ C St2f ( y −1) −iSt f (1−2σˆ C2 St2f )( y −1)

e

.

(31)

It is seen that for suﬃciently small St f , Eq. (31) degenerates to that of the unstretched ﬂame

ζ1, N S = −

1 iSt

1 − e −iSt f ( y −1)

(32)

as is reasonable to expect. It is further seen from the comparison between Eqs. (31) and (32) that stretch damps the ﬂame wrinkling

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H.Y. Wang et al. / Combustion and Flame 156 (2009) 889–895

σˆ St2 ( y −1)

through the term e C f , and this damping effect increases exponentially toward the ﬂame tail, i.e. y → 0. This demonstrates that the extent of damping in the ﬂame wrinkling by stretch is controlled by the nondimensional parameter σˆ C St2f , and becomes O (1) as σˆ C St2f ∼ O (1), i.e. as the disturbance frequency satisﬁes −1/2

). This property is consistent with the plots in Fig. 5. St f ∼ O (σˆ C For example, even for the very small stretch, σˆ C = 0.0005, the damping is still quite evident especially near the ﬂame tail ( y = 0) because σˆ C St2f ≈ 1.13 ∼ O (1). For the case of σˆ C = 0.005 for which

σˆ C St2f ≈ 11.3, the damping is so strong that ﬂame wrinkling is only evident near the ﬂame base, consistent with the experimental observations of Bourehla and Baillot [11] and Preetham et al. [12]. Furthermore, the nondimensional parameter σˆ C St2f indicates that the damping effect increases quadratically with the disturbance frequency and hence is very sensitive to it. This is the reason that doubling the disturbance frequency is able to completely damp the ﬂame wrinkling except in the ﬂame base region, as shown by the observations of Preetham et al. [12] in Fig. 1. It is also seen from Eq. (31) and Fig. 5 that damping results in a more uniform ﬂame oscillation amplitude, indicating an increase of the relative contribution of the bulk oscillatory movement of the ﬂame to the ﬂuctuation of the ﬂame surface area. Equation (31) further shows that ﬂame stretch also modulates the wavelength of the wrinkling through the term 1 − 2σˆ C2 St2f in

Fig. 6. Variations of the gains of the overall transfer function G and the transfer functions resulted from the ﬂuctuations of ﬂame surface area and ﬂame speed, G A and G S , with St f for σˆ C = 0.05. The gain of the overall transfer function for unstretched ﬂame (σˆ C = 0) is also plotted for comparison.

−iSt (1−2σˆ 2 St2 )( y −1)

f C f , and this modulation effect is the exponential e O (1) for St f ∼ O (σˆ C−1 ). However, at such a large St f , wrinkling is damped such that its wavelength does not have much signiﬁcance. Thus, this effect can be neglected so that Eq. (31) can be further simpliﬁed to

ζ1 = −

1 iSt

1−e

σˆ C St2f ( y −1) −iSt f ( y −1) e

and the expansion of L 1 only needs to keep the ﬁrst two terms L 1 ∼ −iSt f + σˆ C St2f .

(33)

It is noted that by increasing St f to O (σˆ C−1 ), the expansion for L 1 , Eq. (33), becomes less accurate. However, the trend revealed for the ﬂame response at this order of frequency is still preserved. We next study effects of ﬂame stretch on the transfer function. Since the heat release rate mainly depends on the ﬂame surface area, which in turn depends on the ﬂame wrinkling, it is expected that ﬂame stretch starts to have an O (1) effect on the heat release and thereby on the transfer function for frequency St f from −1/2

O (σˆ C ). Substituting Eq. (21) into Eqs. (26) and (27), respectively, yields GS = −

L 1 σˆ C

1 − e−L1 , iSt f

(34)

GA = −

1

1 − e−L1 . iSt f

(35)

Fig. 6 shows variations of the gains of G, G A and G S , with the reduced Strouhal number, St f , for σˆ C = 0.05. The gain of the overall transfer function for the unstretched case, Eq. (30), is also plotted for the purpose of comparison. It is seen that in the presence of ﬂame stretch, the transfer function shows quite different behavior from the unstretched case. Speciﬁcally, the nodes at St f = 2nπ in the gain of the transfer function for the unstretched case (σˆ C = 0) are eliminated in the presence of stretch, as already shown in Fig. 5, leading to higher values of |G | for the stretched ﬂame around these frequencies. Relaxation of the ﬂame surface from the nodal points then enhances ﬂuctuation of the ﬂame surface area, through the bulk oscillatory movement, to a larger extent than the damping effect through reduced wrinkling, which is O (ε 2 ) for

Fig. 7. Variations of the phase of the overall transfer function G and the transfer functions resulted from the ﬂuctuations of ﬂame surface area and ﬂame speed, G A and G S , with St f for σˆ C = 0.05.

St f = 2nπ . Moreover, for the stretched case the overall transfer function G is very close to G A at small St f (<5), implying that contribution from the ﬂuctuation of the ﬂame surface area dominates that of the ﬂame speed. However, with increasing St f the relative contribution of G S increases and ﬁnally becomes comparable to G A at St f ∼ 30. It is noted that the gain of the overall transfer function, |G |, is not simply the sum of |G A | and |G S | because G A and G S are not necessarily in phase, as will be shown in Fig. 7. The dependence of the transfer functions, G A , G S and G, on the ﬂame stretch σˆ C and disturbance frequency St f can be further illustrated by substituting the expansion for L 1 , Eq. (33), into Eqs. (34) and (35), resulting in GS ≈ − GA ≈ −

1

iSt f

−σˆ St2 −i σˆ C St f + σˆ C2 St2f 1 − e C f e iSt f ,

1

−σˆ St2 1 − e C f e iSt f . iSt f

(36) (37)

Comparing Eq. (37) with the transfer function for the unstretched ﬂame, Eq. (29), shows that ﬂame stretch starts to have an O (1) effect on the transfer function as the disturbance frequency St f

H.Y. Wang et al. / Combustion and Flame 156 (2009) 889–895

satisﬁes σˆ C St2f ∼ O (1), as noted earlier. At this frequency, G S /G A ∼ 1/ 2 O ( ˆC )

σ and hence the contribution from the ﬂame speed ﬂuctuation is secondary relative to that of the ﬂame surface area. Thus, the overall transfer function G is mostly derived from ﬂuctuations of the ﬂame surface area, which however is still affected by ﬂame stretch through modulation of the shape of the wrinkles. Therefore we have

1/ 2 1

−σˆ St2 G ≈ GA ≈ − 1 − e C f e iSt f ∼ O σˆ C . (38) iSt f −1/2

) to O (σˆ C−1 ), the contribution With St f increasing from O (σˆ C from the ﬂame speed ﬂuctuation, G S , becomes comparable to G A . Furthermore, since ﬂame wrinkling is totally suppressed at this order of St f , ﬂuctuations of the ﬂame surface area are mainly due to the bulk movement of the ﬂame. Then, the transfer functions become 1

1 GS ≈ − GA ≈ − (39) −i σˆ C St f + σˆ C2 St2f , iSt f iSt f and the overall transfer function is given by G ≈−

1

1 + σˆ C2 St2f − i σˆ C St f ∼ O (σˆ C ).

iSt f

(40)

The above characteristics are consistent with Fig. 6, which shows that G and G A have almost identical values for St f < 5 (σˆ C St2f ∼ O (1)), while G A and G S contribute comparably to the overall transfer function G for St f > 20 (σˆ C St f ∼ O (1)). Fig. 7 shows variations of the phases of G, G A and G S with the reduced Strouhal number, St f , for σˆ C = 0.05. It is seen that, compared to the unstretched case, the −π jump in the phase resulting from the nodes in the gain of the transfer function is smoothed out, due to the elimination of these nodes in the presence of stretch. Furthermore, it is seen that at small St f , the phase of G follows closely that of G A , whereas with increasing St f it approaches the phase of G S due to the increased relative contribution of G S . This is the same trend as what was discussed for the gain of the transfer function in Fig. 6. 4. Conclusions In this study we have investigated the linear response of a 2D wedge-shaped premixed ﬂame to harmonic velocity disturbances, allowing for the dependence of the ﬂame speed on stretch. Different from previous studies, the transfer function now consists of contributions from ﬂuctuations of both the ﬂame surface area and ﬂame speed. Two nondimensional parameters, σˆ C St2f and

σˆ C St f , were identiﬁed to characterize their relative contributions

and thereby the inﬂuence of ﬂame stretch on the ﬂame response. Speciﬁcally, as the disturbance frequency satisﬁes σˆ C St2f ∼ O (1), −1/2

), ﬂame stretch starts to have O (1) effects on i.e. St f ∼ O (σˆ C the transfer function through damping of the disturbance-induced ﬂame wrinkling. At this order of the frequency, the contribution from the ﬂame speed ﬂuctuation is negligibly small. Thus ﬂame stretch affects the transfer function only through its modulation of the ﬂame shape and thereby its surface area, with this effect increasing with the square of the disturbance frequency. At larger frequencies such that σˆ C St f ∼ O (1), i.e. St f ∼ O (σˆ C−1 ), contributions from ﬂuctuations of the ﬂame surface area and ﬂame speed become comparable. It is noted that previously ﬂame stretch was thought to be not important in the response of ﬂames to disturbances. The suggested reason [4] is that while the ﬂame curvature and hence

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stretch effects could become large for large disturbance frequencies at which the wavelength of the disturbance-induced ﬂame wrinkling is small, the sensitivity of the ﬂame response diminishes signiﬁcantly at large frequencies. The present study has however demonstrated that ﬂame stretch is still important for “moderate” disturbance frequencies even for small σˆ C . This is because even when σˆ C is small, St f ∼ O (σˆ C−1 ) could assume values that are not very large but nevertheless would induce O (1) effects on the ﬂame response. While the present study has yielded useful insights into the effects of stretch on the ﬂame response upon being harmonically disturbed, especially on the role of self-induced curvature damping leading to the experimentally observed phenomenon of “ﬁltering,” and the critical Strouhal numbers at which stretch effects become important, there are additional issues that need to be investigated. In particular, the study has focused on damping situations because of our interest in understanding the damping phenomenon, and because they are suﬃcient to ensure stability in operations. It would however also be of interest to study situations in which the disturbance is either ampliﬁed or sustained, especially for small Le mixtures for which σC could become negative. Operationally, it has been suggested that resonant combustion could facilitate the heat transfer characteristics of burners. We also note that, by studying 2D instead of axisymmetric ﬂames, the effects of the azimuthal curvature of the bulk ﬂame on the development of wrinkles are suppressed. Studies on ﬂamefront cellular instability [14,21] have shown that these wrinkles tend to be moderated by positive stretch and aggravated by negative stretch, which are respectively manifested by the wedge and conical geometries. The richness of the potential ﬂame responses merits further investigation. Acknowledgment This work was supported by the Air Force Oﬃce of Scientiﬁc Research under the technical monitoring of Dr. Julian M. Tishkoff. References [1] J.C. Broda, S. Seo, R.J. Santoro, G. Shirhattikar, V. Yang, Proc. Combust. Inst. 27 (1998) 1849–1856. [2] C.O. Paschereit, E. Gutmark, W. Weisenstein, Proc. Combust. Inst. 27 (1998) 1817–1824. [3] A.P. Dowling, S.R. Stow, J. Propul. Power 19 (2003) 751–764. [4] M. Fleifel, A.M. Annaswamy, Z.A. Ghoniem, A.F. Ghoniem, Combust. Flame 106 (1996) 487–510. [5] A.P. Dowling, J. Fluid Mech. 346 (1997) 271–290. [6] A.P. Dowling, J. Fluid Mech. 394 (1999) 51–72. [7] S.H. Preetham, T. Lieuwen, AIAA Paper #2004-4035, 2004. [8] T. Lieuwen, Proc. Combust. Inst. 30 (2005) 1725–1732. [9] F. Baillot, A. Bourehla, D. Durox, Combust. Sci. Technol. 112 (1996) 327–350. [10] D. Durox, F. Baillot, G. Searby, L. Boyer, J. Fluid Mech. 350 (1997) 295–310. [11] A. Bourehla, F. Baillot, Combust. Flame 114 (1998) 303–318. [12] S.H. Preetham, T.S. Kumar, T. Lieuwen, AIAA Paper #2006-0960, 2006. [13] T. Lieuwen, in: T. Lieuwen, V. Yang (Eds.), Combustion Instabilities in Gas Turbine Engines, AIAA, Reston, VA, 2005, Chapter 12, p. 345. [14] C.K. Law, Combustion Physics, Cambridge Univ. Press, New York, 2006, pp. 416– 424. [15] F. Baillot, D. Durox, R. Prud’homme, Combust. Flame 88 (1992) 149–168. [16] A.R. Kerstein, W.T. Ashurst, F.A. Williams, Phys. Rev. A 27 (1988) 2728–2731. [17] S.H. Chung, C.K. Law, Combust. Flame 72 (1988) 325–336. [18] T. Schuller, S. Ducruix, D. Durox, S. Candel, Proc. Combust. Inst. 29 (2002) 107– 113. [19] M. Matalon, B. Matkowsky, J. Fluid Mech. 124 (1982) 239–260. [20] G. Joulin, Combust. Sci. Technol. 97 (1994) 219–229. [21] G.I. Sivashinsky, C.K. Law, G. Joulin, Combust. Sci. Technol. 28 (1982) 155–159.