Linear response of stretch-affected premixed flames to flow oscillations

Linear response of stretch-affected premixed flames to flow oscillations

Combustion and Flame 156 (2009) 889–895 Contents lists available at ScienceDirect Combustion and Flame www.elsevier.com/locate/combustflame Linear ...

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Combustion and Flame 156 (2009) 889–895

Contents lists available at ScienceDirect

Combustion and Flame www.elsevier.com/locate/combustflame

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

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.