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Response of non-premixed flames to bulk flow perturbations
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Proceedings of the
Proceedings of the Combustion Institute 34 (2013) 963–971
Combustion Institute www.elsevier.com/locate/proci
Response of non-premixed flames to bulk flow perturbations Nicholas Magina ⇑, Dong-Hyuk Shin, Vishal Acharya, Timothy Lieuwen School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150, USA Available online 4 August 2012
Abstract This paper describes the dynamics of non-premixed flames responding to bulk velocity fluctuations, and compares the dynamics of the flame sheet position and spatially integrated heat release to that of a premixed flame. The space–time dynamics of the non-premixed flame sheet in the fast chemistry limit is described by the stoichiometric mixture fraction surface, extracted from the solution of the -equation. This procedure has some analogies to premixed flames, where the premixed flame sheet location is extracted from the G = 0 surface of the solution of the G-equation. A key difference between the premixed and non-premixed flame dynamics, however, is the fact that the non-premixed flame sheet dynamics are a function of the disturbance field everywhere, and not just at the reaction sheet, as in the premixed flame problem. A second key difference is that the non-premixed flame does not propagate and so flame wrinkles are convected downstream at the axial flow velocity, while wrinkles in premixed flames convect downstream at a vector sum of the flame speed and axial velocity. With the exception of the flame wrinkle propagation speed, however, we show that that the solutions for the space–time dynamics of the premixed and non-premixed reaction sheets in high velocity axial flows are quite similar. In contrast, there are important differences in their spatially integrated unsteady heat release dynamics. Premixed flame heat release fluctuations are dominated by area fluctuations, while non-premixed flames are dominated by mass burning rate fluctuations. At low Strouhal numbers, the resultant sensitivity of both flames to flow disturbances is the same, but the non-premixed flame response rolls off slower with frequency. Hence, this analysis suggests that non-premixed flames are more sensitive to flow perturbations than premixed flames at O(1) Strouhal numbers. Ó 2012 Published by Elsevier Inc. on behalf of The Combustion Institute. Keywords: Non-premixed flame; Linear flame response; Velocity coupled response; Combustion instabilities; Flame transfer function
1. Introduction This paper describes the dynamics of confined, non-premixed flames to harmonic flow perturba-
⇑ Corresponding author. Address: Aerospace Combustion Lab, 635 Strong St. NW, Atlanta, GA 30318, USA. Fax: +1 404 463 0888. E-mail address: [email protected] (N. Magina).
tions and compares their response characteristics to that of premixed flames. This work is motivated by the problem of combustion instabilities [1]. During an instability, unsteady heat release couples with one or more of its acoustic modes, leading to self-excited oscillations. The development of lean, premixed combustion systems has motivated significant interest in this problem over the last decade. Enormous efforts have gone into characterizing the linear and nonlinear response
1540-7489/$ - see front matter Ó 2012 Published by Elsevier Inc. on behalf of The Combustion Institute. http://dx.doi.org/10.1016/j.proci.2012.06.155
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Nomenclature diffusion coefficient flame length Markstein number and rescaled Markstein number, respectively Pe Peclet number Stf ; StLf ; StW Strouhal numbers, StLf = cos2 h; fLf =ux;0 ; fW II =ux;0 , respectively Un normal component of mean velocity W I ; W II radial distance from centerline to confining wall and fuel nozzle lip, respectively mixture fraction f forcing frequency sd ; sc flame displacement and consumption speed, respectively D Lf ^c rc ; r
of premixed flames to flow perturbations, through analytic modeling, detailed computations, and experiments. As a result of this work, the physics controlling the response of laminar flames to harmonic flow forcing appears to be understood and capabilities have been developed to predict the space–time dynamics of the flame position and heat release [2–4]. Furthermore, while some fundamental questions remain, exciting progress has been made in obtaining similar predictive capabilities in highly turbulent flows as well [5,6]. Substantially less work has gone into the nonpremixed flame problem. However, the problem of instabilities in partially premixed or non-premixed systems has grown in prominence lately, particularly with the development of low NOx designs for aircraft engines or other liquid-fueled systems. Work is similarly needed to develop analytic models to describe the dynamics of non-premixed flames. Recently, several important studies by the IIT Madras group led by Sujith et al. [7–9] and Chakravarthy et al. [9,10] have addressed this problem. These studies have shown how this problem can be analyzed within the -equation formulation for the mixture fraction. They developed solutions for the flame position and heat release for several problems, including the response of the flame to axial velocity and mixture fraction oscillations. This paper closely follows these studies, with particular emphasis on the axial flow oscillation problem. We consider a linearized version of the problem considered by Balasubramanian and Sujith [7], and derive explicit solutions for the flame sheet position in Section 2.3. This solution allows us to explicitly illustrate the effects of velocity fluctuations in inducing flame wrinkles and axial flow in convecting wrinkles downstream. This solution is then used to examine the spatially integrated heat release response characteristics, as well as
ux x; y e h n uOx x ð Þ0 ; ð Þ1
axial flow velocity axial and radial coordinate, respectively small perturbation parameter local angle of the flame surface with respect to axial coordinate, see Eq. (19) radial flame sheet position stoichiometric mass ratio of oxidizer to fuel angular frequency mean/steady state and fluctuating component, respectively
the relative contributions of flame surface area fluctuations and mass burning rate fluctuations to the instantaneous heat release in Section 2.4. These results also allow us to compare the response characteristics of premixed and nonpremixed flames. We show that there are important similarities and differences in their local and global response characteristics. 2. Main body 2.1. Physics of governing equations This section discusses some basic features of non-premixed flame dynamics in the fast chemistry, thin reaction sheet limit. In addition, we assume that all species have equal diffusivities so that the mixture fraction formulation can be used. The space–time dynamics of the mixture fraction are described by the equation [11]: @ þ~ ur @t
¼ r ðDr Þ
ð1Þ
In the fast chemistry limit, the reaction sheet collapses to a surface defined by the equation ð~ x; tÞ ¼ ¼ 1=ð1 þ uOx Þ, where uOx is the
Fig. 1. Illustration of the forced non-premixed flame model problem.
N. Magina et al. / Proceedings of the Combustion Institute 34 (2013) 963–971
stoichiometric mass ratio of oxidizer to fuel. We define the instantaneous position of this reaction sheet by y ¼ nðx; tÞ, as shown in Fig. 1. It is helpful to compare the dynamics of the mixture fraction equation for non-premixed flames with the G-equation used to analyze the dynamics of premixed flames [2,11,12], which is given below: @G þ~ u rG ¼ sd jrGj @t
ð2Þ
The two expressions have the same convection operator on the left-hand side which illustrates the importance of flow perturbations in the direction normal to the flame sheet in pushing the flame sheet around. However, the right-hand sides of these two expressions are different. The premixed flame expression has the normal flame propagation operator, sd jrGj, while the non-premixed flame expression has a diffusion operator, r ðDr Þ. This difference is significant and reflects, among other things, the fact that non-premixed flames do not propagate. Moreover, the premixed flame dynamics equation is nonlinear, while the non-premixed flame dynamics equation is linear (assuming that Un and D are not functions of ). Another significant, yet subtle difference is that the G-equation is physically meaningful and valid only at the flame itself where Gð~ x; tÞ ¼ 0 (i.e., although it can be solved away from the flame, the resulting G values have no physical significance [13]). In contrast, the -equation describes the physical values of the mixture fraction field everywhere. This observation has important consequences for both solution approaches of these problems, as well as the ð~ x; tÞ ¼ flame sheet dynamics that are discussed next. Substantial analytical progress has been made to analyze “single valued” premixed flames with the substitution Gðx; y; z; tÞ ¼ x nðy; z; tÞ. This leads to an explicit expression for the flame front position [14–18]. However, note that this substitution arbitrarily assigns values to the G field away from the flame itself, namely that G varies linearly with coordinate x away from the flame. Since the G field is completely arbitrary away from the flame, this is allowable. In the non-premixed flame problem, the goal is similarly to only consider the dynamics of the reaction sheet surface ð~ x; tÞ ¼ , and so the space–time dynamics of the rest of the field is of lesser interest. However, we cannot make an analogous substitution, such as ðx; y; z; tÞ ¼ y nðx; z; tÞ, as this assigns valfield away from ð~ x; tÞ ¼ . This ues to the cannot be done since the -equation describes the entire spatial distribution of the mixture fraction field. The implication of this fact is that the entire mixture fraction field must be solved and the ð~ x; tÞ ¼ surface extracted from the resulting solution field (which generally cannot be written as an explicit expression).
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This discussion also reflects important underlying physics of the two flames. Consider a premixed and non-premixed flame in a velocity field given by ~ uðx; y; z; tÞ, where the velocity field just upstream of the flame sheet is given by ~ uðx; nðx; z; tÞ; z; tÞ ¼ ~ uu . The premixed flame dynamics are only a function of ~ uu ; this implies that for a given ~ uu , its space–time dynamics are the same for a variety of different velocity fields ~ uðx; y; z; tÞ. In contrast, the space–time dynamics of the non-premixed reaction sheet are a function of the entire velocity field, ~ uðx; y; z; tÞ, not just its value at the reaction sheet. This result is a direct manifestation of the fact that the -equation is an elliptic partial differential equation. In contrast, the linearized, stretch-free G-equation is a hyperbolic partial differential equation. 2.2. Formulation This section formulates the problem considered in this paper, which is illustrated in Fig. 1. We consider a two-dimensional flame in a uniform axial flow field, ux;0 . At the inlet (x = 0), fuel and oxidizer flow into the domain as indicated in the figure, leading to the following inflow conditions: 1 for 0 6 jyj < W II ð3Þ ðx ¼ 0; yÞ ¼ 0 for W II 6 jyj < W I The solution can also be easily generalized to include more general inflow fuel/oxidizer compositions (e.g., such as if the fuel were diluted) by shifting and rescaling the value of . Enforcing this boundary condition enables an analytic solution of the problem. However, in reality there is axial diffusion of fuel into the oxidizer and vice versa, so that the solution must actually be solved over a larger domain that includes the fuel/oxidizer supply systems. As such, the boundary condition in Eq. (3) implicitly neglects axial diffusion at x ¼ 0, a point we will return to later. Assuming symmetry at y ¼ 0 and no diffusion through the walls at y ¼ W I , leads to the following two additional boundary conditions: @ @y
ðx; y ¼ 0Þ ¼ 0
@ @y
ðx; y ¼ W I Þ ¼ 0
ð4Þ
We will derive the solution in the limit of small perturbations, and so expand each variable as ð Þðx; y; tÞ ¼ ð Þ0 ðx; yÞ þ ð Þ1 ðx; y; tÞ. Since the governing equation, Eq. (1), is linear, this procedure is not necessary in order to obtain an analytic solution and, in fact, this assumption was not made in the work by Balasubramanian and Sujith [7]. However, this expansion is useful in analyzing controlling features of the flame dynamics at the forcing frequency and, very significantly, it enables an explicit analytic expression for the
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space–time dynamics of the flame position, n1 ðx; tÞ, which is otherwise not possible. The mixture fraction field in the absence of forcing is obtained from Eq. (1): ux;0
@ 0 @2 0 @2 0 ¼D 2 þD 2 @x @y @x
ð5Þ
Similarly the dynamical equation for
1
is given
" 1 X
2=ðnpÞ sinðAn Þeb 1¼ PeStW 2pi þ b þ A2n b2 =Pe2 1 y x cos An exp b W II PeW II x exp b exp½ixt PeW II hom where
by: @ 1 @ 1 @2 1 @2 1 @ 0 þ ux;0 D 2 D 2 ¼ ux;1 @t @x @y @x @x
ð6Þ
The solution to these equations can be derived in an analogous way as the Burke–Schumann solution using separation of variables. The full solution, including axial diffusion, for the steady state mixture fraction field is given by: 0
¼
1 W II X 2 sinðAn Þ þ W I n¼1 np qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi31 0 2 2 2 4 2 y x 4Pe Pe þ 4Pe An 5A cos An exp @ 2 W II PeW II
ð7Þ
where An ¼ npðW II =W I Þ are the eigenvalues and the Peclet number, Pe, is given by Pe ¼ ðux;0 W II Þ=D. The Peclet number physically corresponds to the relative time scales for convective and diffusive processes to transport mass over s
W 2 =D
u
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pe4 þ 4Pe2 A2n
2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pe2 Pe4 þ 4Pe2 A2n 8piPe3 StW
ð10Þ
ð11Þ
ð12Þ 2 Again we will focus the subsequent analysis in the absence of axial diffusion, i.e. in the Pe 1 limit, whose solution is: " # 1 X ieðAn Þ2 ð2=npÞ sinðAn Þ 1 ¼ 2pStW Pe n¼1 y x exp A2n cos An W II PeW II x exp½ixt ð13Þ 1 exp 2piStW W II bhom ¼
where the Strouhal number based on the halfwidth of the fuel nozzle is defined as StW ¼ ðfW II Þ=ux;0 . This expression can also be written in terms of 0 as:
W
diffusion ¼ W IIII=ux;0 ¼ x;0D II . As such, a distance W II ; sconvection the Pe 1 limit physically corresponds to the limit where convective processes are much faster than diffusive ones over the length scale WII. For the subsequent analysis, we will focus on the following simplified version of the solution that neglects axial diffusion, since we have already done so implicitly in formulating the boundary condition in Eq. (3). The steady state mixture fraction field solution is: 1 W II X 2 sinðAn Þ þ 0 ¼ np WI n¼1 y x exp A2n ð8Þ cos An W II PeW II
This equation can be derived by solving Eq. (5) and neglecting the axial diffusion term or, equivalently, taking the Pe 1 limit of Eq. (7). We next consider the solution for the fluctuating flame position responding to uniform bulk fluctuations in flow velocity: ux;1 ¼ eux;0 exp½ixt
b ¼
Pe2
#
ð9Þ
The full solution for the fluctuating mixture fraction field, 1, is:
1
¼
ieW II @ 0 x 1 exp 2piStW exp½ixt W II 2pStW @x ð14Þ
2.3. Space–time dynamics of reaction sheet While these expressions for 0 and 1 provide solutions for the mixture fraction field over the entire domain, we are particularly interested in the reaction sheet location. An implicit expression for the flame sheet position, n0 ðxÞ, can be determined from the coordinates where 0 = , yielding: ¼
1 W II X 2 sinðAn Þ þ np WI n¼1 n ðxÞ x exp A2n cos An 0 W II PeW II
ð15Þ
Similarly, the position of the fluctuating flame can be determined from the implicit expression: ¼
0
ðx; n0 ðxÞ þ n1 ðx; tÞÞ þ
þ n1 ðx; tÞ; tÞ
1
ðx; n0 ðxÞ ð16Þ
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Expanding Eq. (16) to first order leads to the following explicit expression for fluctuating flame position: n1;n ðx; tÞ ¼
¼
ðx; y ¼ n0 ðxÞ; tÞ
1
@ 0 ðx; y @n
¼ n0 ðxÞÞ
ðx; y ¼ n0 ðxÞ; tÞ jr 0jðx;y¼n0 ðxÞÞ
1
ð17Þ
where n1;n is measured normal to the mean flame surface in the direction of the oxidizer. This can be re-written as: " # ! @ 0=@x n1;n ðx; tÞ ie ¼ 2pStLf Lf jr 0j ðx;y¼n0 ðxÞÞ x exp½ixt 1 exp 2piStLf Lf ;0 ð18Þ where StLf ¼ ðf Lf ;0 Þ=ux;0 is the Strouhal number @ 0=@x based on flame length. The term can be jr 0j
written in terms of the local angle of the flame, using the relations: " 2 2 #1=2 @ 0 @ 0 @ 0=@x ð19Þ þ ¼ jr 0j ¼ sin h0 ðxÞ @x @y where h0 denotes the angle of the mean flame with respect to the axial coordinate. Using these results, the solution for n1;n ðx; tÞ can be written as: ieux;0 x sin h0 ðxÞ 1 exp i2pStLf n1;n ðx; tÞ ¼ Lf ;0 2pf exp½i2pft
ð20Þ
This expression is an important contribution of this paper and very significant in that it is an explicit equation for the space–time dynamics of the flame position. For reference, the corresponding fluctuations of an attached premixed flame with constant burning velocity subjected to bulk flow oscillations are given by [14]: ieux;0 x n1;n ðx; tÞ ¼ sin h 1 exp i2pStf 2pf Lf ;0 exp½i2pft
ð21Þ
where Stf is the flame Strouhal number for premixed flames, defined in the nomenclature section, and the angle h is a constant (the expression is more involved if h is varying, which would occur if the flow or flame speed varies spatially). Notice the similarities in the premixed and non-premixed solutions, with the exception of the spatial phase dependence, 1 ei2pfx=ux;0 term. This difference reflects the influence of premixed flame propagation on wrinkle convection speeds; i.e., the wrinkle
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convection speed in the axial direction, ux;0 = cos2 h, is the vector superposition of the axial flow velocity and the axial projection of a vector pointing normal to the flame with a magnitude equal to the burning velocity, sd . In contrast, the non-premixed flame does not propagate and wrinkles convect downstream at a speed of ux;0 . In both cases, local maxima and minima arise through this 1 ei2pfx=ux;0 term, due to interference between wrinkles generated at the x = 0 boundary and disturbances excited locally. This can be seen by re-writing it as: 1 ei2pfx=ux;0 ¼ 2 sinðpfx=ux;0 Þeiðpfx=ux;0 p=2Þ
ð22Þ
For both premixed and non-premixed flames, wrinkles are generated at the boundary because of flame attachment. For the premixed flame, this is invoked through the attachment boundary condition i.e., n1;n ðx ¼ 0; tÞ ¼ 0. In the non-premixed case, wrinkles are generated through the assumption of constant mixture fraction at the burner outlet, i.e., 1ðx ¼ 0; tÞ ¼ 0. Finally, we note that incorporating stretch effects into the premixed flame analysis modifies Eq. (21) by multiplying the complex exponential inside the braces by the factor expð4p2 rc St2f x=Lf Þ [19], where rc is the Markstein length normalized by the burner half-width, WII. For a thermo-diffusively stable flame, this stretch correction leads to an exponential decay in wrinkle magnitude because of the flame front curvature. We next present several illustrative solutions of the space–time dynamics for the flame position. Note that the solution is a function of the four dimensionless parameters StLf , W II =W I , Pe, and . The temporal evolution of the flame position is plotted in Fig. 2 at two Strouhal numbers. Note the bulk axial pulsing of the flame at lower Strouhal numbers, and the spatial wrinkling at higher values. An alternative way to visualize these results is through the magnitude and phase of n1;n , illustrated in Fig. 3. The nodes and local maxima and minima referred to above are clearly evident in the figure. The phase rolls off linearly with axial distance, again reflecting the convection process described by the 1 ei2pfx=ux;0 ¼ 2 sinðpfx=ux;0 Þ eiðpfx=ux;0 p=2Þ term, and jumps 180° across the nodes. 2.4. Heat release analysis Having considered the local space–time wrinkling of the flame, we next consider the spatially _ integrated heat release, QðtÞ, which is given by the following surface integral over the reaction sheet: Z _ m_ 00F hR dA QðtÞ ¼ ð23Þ A
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Fig. 2. Snapshots showing four instantaneous positions of a forced non-premixed flame at two different forcing = 0.3 and Pe ¼ 50 (a) e ¼ 0:2, StW ¼ 0:0018, StLf ¼ 0:3, (b) e ¼ 1:0, StW ¼ 0:012, frequencies using nominal values of StLf ¼ 2:0. The unforced flame is indicated by dashed lines.
The term m_ 00F is the reactant mass burning rate per unit area, and hR is the heat release per unit mass of reactant consumed. For premixed flames, the mass burning rate can be written in terms of the burning velocity as m_ 00F ¼ qu suc , where qu is the density and suc is the laminar consumption speed of the unburned reactant, yielding: Z _ qu suc hR dA ð24Þ Premixed flame : QðtÞ ¼
Decomposing , Lf, and n into their mean and fluctuating components results in: 2
ð1 þ uOx Þ _ qhR D QðtÞ ¼ uOx Z Lf ;0 þLf ;1 ðtÞ @ 0 ðx; n0 þ n1 ðx; tÞÞdx @y 0 Z Lf ;0 þLf ;1 ðtÞ @ 1 ðx; n0 þ n1 ðx; tÞÞdx þ ð28Þ @y 0
flame
For non-premixed flames, the reactant mass burning rate is: m_ 00F ¼ m_ 00Ox þ m_ 00Fuel ¼ qD ¼ ð1 þ uOx ÞqD
Then, linearizing this expression yields: Z Lf ;0 @ 0 ð1 þ uOx Þ2 _ qhR D dx QðtÞ ¼ uOx @y 0 ) Z Lf ;0 Z Lf ;0 @ 1 @2 0 dx þ n1 2 dx þ @y @y 0 0
@Y Ox @Y Fuel qD @n @n
@Y Fuel @n
ð25Þ
where n represents the direction normal to the flame surface into the oxidizer. By also relating the fuel mass fraction and the mixture fraction gradients, the heat release can be written as: Non-premixed flame : Z ð1 þ uOx Þ2 _ QðtÞ ¼ qhR uOx flame @ @ sin h D cos h dA ð26Þ D @x @y
Note that fluctuations in Lf do not contribute to this linearized expression since @ 0=@y is zero along the centerline and, therefore, at the flame tip (axial diffusion would provide a non-zero contribution). We will denote transfer functions by F, defined as follows: F¼
FN ;mbr ¼
0
@ 1ðx;n0 Þ dx @y
þ
Q_ 1 =Q_ 0 ux;1 =ux;0
ð30Þ
where premixed or non-premixed transfer functions will be denoted as FP and FN , respectively. It is useful to decompose the results in Eq. (29) into contributions from mass burning rate and flame area fluctuations; i.e., F ¼ Fmbr þ Fa . This decomposition requires replacing the dx by cos hdA in Eq. (29) (see Eq. (26)) and similarly expanding the solution. The
We will assume fixed composition fuel and oxidizer, so that the heat of reaction and mixture density are constant. The first term in Eq. (26) includes the effect of axial diffusion and is, consequently, neglected in the following analysis, yielding: Z Lf ðtÞ ð1 þ uOx Þ2 @ ðx; nðx; tÞÞ _ QðtÞ ¼ dx qhR D @y uOx 0 ð27Þ R Lf ;0
ð29Þ
R Lf ;0 0
@ 1ðx;n0 Þ @x
e
R Lf ;0 0
sinðh1 Þdx þ @ 0ðx;n0 Þ dx @y
R Lf ;0 0
n1 ðx; tÞ
@ 2 0ðx;n0 Þ dx @y 2
ð31Þ
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b
a
eu
x;0 Fig. 3. Axial dependence of (a) magnitude and (b) phase of flame response, where nref ¼ Pe2pf , and using nominal values = 0.3 and Pe ¼ 50 for StLf values of 0.3 and 2.0. Note the abscissa in (b), (x/ux,0/f), can equivalently be written as of ðx=Lf ÞStLf .
resulting mass burning rate contribution to the transfer function is: The mass burning rate term contributes to heat release oscillations due to the fluctuations in spatial gradients of the mixture fraction for non-premixed flames. For premixed flames, the mass burning rate fluctuations are linked to the stretch sensitivity of the burning velocity, which fluctuates because of the oscillatory curvature of the wrinkled front [19]. Similarly, the area contribution is given by: R Lf ;0 FN ;wa ¼
@ 0ðx;n0 Þ @y
0
e ¼
R Lf ;0 0
R Lf ;0
@ 0ðx;n0 Þ dx @y
@ 0ðx;n0 Þ @x
0
e
cosðh0 ÞdA1
R Lf ;0 0
sinðh1 Þdx
@ 0ðx;n0 Þ dx @y
ð32Þ
As this area term is weighted by the time averaged burning rate (which, unlike premixed flames, is spatially non-uniform for non-premixed flames considered here), it will be called the weighted area. The un-weighted area transfer function (important for constant burning velocity premixed flames) is given by: 1
FN &P ;a ¼ R Lf ;0 e 0 ½1 þ ðdn0 =dxÞ2 1=2 dx Z Lf ;0 @n1;y ðx; tÞ dn0 =dx dx 2 1=2 @x ½1 þ ðdn0 =dxÞ 0
ð33Þ
or equivalently: 1 FN &P ;a ¼ R Lf ;0 e 0 ½1= cos h0 ðxÞdx Z Lf ;0 sin h0 sin h1 dx 2h cos 0 0
ð34Þ
There are significant variations in time averaged heat release rate along the non-premixed flame (e.g., no heat release at the tip in the absence of axial diffusion). Thus, the weighting of flame area is a very significant effect influencing how area fluctuations lead to heat release. Moreover, the characteristics of the weighted and unweighted area transfer functions are quite different for non-premixed flames, while they are identical for premixed flames with spatially uniform burning velocities. For example, in the low Strouhal number limit, the non-premixed flame weighted and un-weighted area transfer functions differ in phase by 180° and have appreciably different magnitudes. The solutions for the premixed flame transfer functions are simpler, as the unforced flame is flat in a uniform velocity field. Following Wang et al. [19], and retaining only leading order terms in Markstein length, the transfer function is: n o 2 2 ^c Þð1 e2piStf 4p r^c Stf Þ ^c ð1 þ 2piStf r Fp ¼ r |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} n
¼FP ;mbr
o 2 2 þ 1=ð2piStf Þ 1 e2piStf 4p r^c Stf ð35Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼FP ;a
^ c is given by r ^c ¼ rc sin h tan h. where r Typical solutions for the overall unsteady heat release, as well as the contributions from flame area and mass burning rate are shown in Fig. 4 for the non-premixed and premixed flame. Both premixed and non-premixed transfer functions have magnitudes of identically unity at zero St, and then roll off with increasing St. Starting with Fig. 4(a), note how the non-premixed flame heat release fluctuations for StLf 1 are dominated by mass burning rate fluctuations over the entire Strouhal number range. For StLf 1, the mass
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Fig. 4. Strouhal number dependence of the magnitude of the heat release, area, and mass burning rate transfer functions ^c ¼ 0:05 and = 0.06, Pe ¼ 10, and (b) premixed flame with parameters r for a (a) non-premixed flame with parameters Lf =W II ¼ 932.
burning rate contributions to F are of O(1) and O(Stf) for non-premixed and premixed flames, respectively. In contrast, premixed flames at low and O(1) Strouhal numbers are dominated by area fluctuations, shown in Fig. 4(b). The mass burning rate fluctuations are a much smaller effect and only exert a comparable response as area fluctuations at high Strouhal numbers where Stf Oð^ r1 c Þ [19]. The mass burning rate fluctuations do also exert an indirect influence on the flame area perturbations when Stf Oð^ r1=2 Þ, by c smoothing out flame wrinkles, causing the “smoothing” of the area gain curve in the premixed case, relative to the much more oscillatory curve for the non-premixed flame.
Fig. 5. Strouhal number dependence of the magnitude and phase of the heat release transfer function for a nonpremixed and premixed flame with the same properties as Fig. 4.
Direct comparisons of the gain and phase response of the premixed and non-premixed flame results are shown in Fig. 5. Significantly, these results show that non-premixed flames are significantly more sensitive to flow perturbations than premixed flames when StLf > Oð1Þ, an important and somewhat unexpected conclusion. Although not proven here, it can be shown that the St 1 response of both flames scales as F Oð1=StÞ. Interestingly, this figure shows that the non-premixed flame has an intermediate region where its response rolls off more slowly; we were unable to derive this regions St sensitivity analytically, but computations suggest that FN Oð1=StL1=2 Þ in this f region. The 1/St scaling is less obvious for this premixed flame case as stretch effects do modify the results for the Strouhal number ranges shown in the plot. The corresponding phases of the premixed and non-premixed flame transfer functions are also included in Fig. 5. Both curves start at zero for low Strouhal numbers, indicating that low frequency flow modulation induces heat release fluctuations that are in phase. The curves roll off with different slopes toward negative values and asymptote to 90° (for a stretch-insensitive flame; as shown in the graph, stretch modifies this result), indicating the delay in heat release relative to the forcing, due to convection of disturbances along the front. Note also the nearly constant phase in the non-premixed flame in the intermediate Strouhal number range discussed above. The undulations in phase for the premixed flame correspond to ripples in the gain, and reflect the influence of interference processes in controlling the flame area. The differences in phase between the two flames again reflects the different processes
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controlling unsteady heat release. The corresponding phases of the area contributions alone are much closer between the two flames for a ^1=2 broader StLf range for r 1. c 3. Conclusions This paper has presented results for the flame sheet motion and unsteady heat release induced by bulk axial excitation of a non-premixed flame. A key contribution of this paper is Eq. (20), which is an explicit expression for the space–time dynamics of the flame sheet. This expression shows the role of axial convection in propagating flame wrinkles downstream, leading to nodes and anti-nodes in the flame response, similar to premixed flames. In addition, it was shown that their heat release dynamics are quite different, premixed flames being dominated by area fluctuations and non-premixed flames by mass burning rate fluctuations. Their gain sensitivities both tend towards unity at low St values, but the non-premixed flame response is larger than premixed flames for St O(1).
Acknowledgments This work has been partially supported by the US Department of Energy under contracts DEFG26-07NT43069 and DE-NT0005054, contract monitor, Mark Freeman.
References [1] T. Lieuwen, H. Torres, C. Johnson, B.T. Zinn, J. Eng. Gas Turb. Power 120 (1998) 294–302.
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[2] T. Lieuwen, J. Prop. Power 19 (5) (2003) 765–781. [3] F. Baillot, D. Durox, R. Prud’Homme, Combust. Flame 88 (2) (1992) 149–152, IN1, 153–168. [4] A. Bourehla, F. Baillot, Combust. Flame 114 (3–4) (1998) 303–318. [5] V. Acharya, B. Emerson, U. Mondragon, et al., ASME Turbo Expo, 2011. [6] Y. Huang, V. Yang, Prog. Energy Combust. Sci. 35 (2009) 293–364. [7] K. Balasubramanian, R. Sujith, Combust. Sci. Technol. 180 (1–3) (2008) 418–436. [8] K. Balasubramanian, R. Sujith, J. Fluid Mech. 594 (2008) 29–58. [9] M. Tyagi, S. Chakravarthy, R. Sujith, Combust. Theory Model. 11 (2) (2007) 205–226. [10] M. Tyagi, N. Jamadar, S. Chakravarthy, Combust. Flame 149 (3) (2007) 271–285. [11] F.A. Williams, Combustion Theory, second ed., Benjamin/Cummings Publishing Co., Menlo Park, CA, 1985. [12] G. Markstein, Nonsteady Flame Propagation, Published for and on behalf of Advisory Group for Aeronautical Research and Development, North Atlantic Treaty Organization by Pergamon Press, 1964. [13] M. Oberlack, H. Wenzel, N. Peters, Combust. Theory Model. 5 (3) (2001) 363–383. [14] T. Preetham, K. Sai, H. Santosh, T. Lieuwen, J. Prop. Power 26 (3) (2010) 524–532. [15] S. Hemchandra, Preetham, T. Lieuwen, J. Prop. Power 24 (6) (2008) 1390–1402. [16] T. Schuller, D. Durox, S. Candel, Combust. Flame 134 (1–2) (2003) 21–34. [17] L. Boyer, J. Quinard, Combust. Flame 82 (1) (1990) 51–65. [18] M. Fleifil, A.M. Annaswamy, Z.A. Ghoneim, A.F. Ghoneim, Combust. Flame 106 (1996) 487–510. [19] H.Y. Wang, C.K. Law, T. Lieuwen, Combust. Flame (2009) 889–895.
Proceedings of the
Proceedings of the Combustion Institute 34 (2013) 963–971
Combustion Institute www.elsevier.com/locate/proci
Response of non-premixed flames to bulk flow perturbations Nicholas Magina ⇑, Dong-Hyuk Shin, Vishal Acharya, Timothy Lieuwen School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150, USA Available online 4 August 2012
Abstract This paper describes the dynamics of non-premixed flames responding to bulk velocity fluctuations, and compares the dynamics of the flame sheet position and spatially integrated heat release to that of a premixed flame. The space–time dynamics of the non-premixed flame sheet in the fast chemistry limit is described by the stoichiometric mixture fraction surface, extracted from the solution of the -equation. This procedure has some analogies to premixed flames, where the premixed flame sheet location is extracted from the G = 0 surface of the solution of the G-equation. A key difference between the premixed and non-premixed flame dynamics, however, is the fact that the non-premixed flame sheet dynamics are a function of the disturbance field everywhere, and not just at the reaction sheet, as in the premixed flame problem. A second key difference is that the non-premixed flame does not propagate and so flame wrinkles are convected downstream at the axial flow velocity, while wrinkles in premixed flames convect downstream at a vector sum of the flame speed and axial velocity. With the exception of the flame wrinkle propagation speed, however, we show that that the solutions for the space–time dynamics of the premixed and non-premixed reaction sheets in high velocity axial flows are quite similar. In contrast, there are important differences in their spatially integrated unsteady heat release dynamics. Premixed flame heat release fluctuations are dominated by area fluctuations, while non-premixed flames are dominated by mass burning rate fluctuations. At low Strouhal numbers, the resultant sensitivity of both flames to flow disturbances is the same, but the non-premixed flame response rolls off slower with frequency. Hence, this analysis suggests that non-premixed flames are more sensitive to flow perturbations than premixed flames at O(1) Strouhal numbers. Ó 2012 Published by Elsevier Inc. on behalf of The Combustion Institute. Keywords: Non-premixed flame; Linear flame response; Velocity coupled response; Combustion instabilities; Flame transfer function
1. Introduction This paper describes the dynamics of confined, non-premixed flames to harmonic flow perturba-
⇑ Corresponding author. Address: Aerospace Combustion Lab, 635 Strong St. NW, Atlanta, GA 30318, USA. Fax: +1 404 463 0888. E-mail address: [email protected] (N. Magina).
tions and compares their response characteristics to that of premixed flames. This work is motivated by the problem of combustion instabilities [1]. During an instability, unsteady heat release couples with one or more of its acoustic modes, leading to self-excited oscillations. The development of lean, premixed combustion systems has motivated significant interest in this problem over the last decade. Enormous efforts have gone into characterizing the linear and nonlinear response
1540-7489/$ - see front matter Ó 2012 Published by Elsevier Inc. on behalf of The Combustion Institute. http://dx.doi.org/10.1016/j.proci.2012.06.155
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Nomenclature diffusion coefficient flame length Markstein number and rescaled Markstein number, respectively Pe Peclet number Stf ; StLf ; StW Strouhal numbers, StLf = cos2 h; fLf =ux;0 ; fW II =ux;0 , respectively Un normal component of mean velocity W I ; W II radial distance from centerline to confining wall and fuel nozzle lip, respectively mixture fraction f forcing frequency sd ; sc flame displacement and consumption speed, respectively D Lf ^c rc ; r
of premixed flames to flow perturbations, through analytic modeling, detailed computations, and experiments. As a result of this work, the physics controlling the response of laminar flames to harmonic flow forcing appears to be understood and capabilities have been developed to predict the space–time dynamics of the flame position and heat release [2–4]. Furthermore, while some fundamental questions remain, exciting progress has been made in obtaining similar predictive capabilities in highly turbulent flows as well [5,6]. Substantially less work has gone into the nonpremixed flame problem. However, the problem of instabilities in partially premixed or non-premixed systems has grown in prominence lately, particularly with the development of low NOx designs for aircraft engines or other liquid-fueled systems. Work is similarly needed to develop analytic models to describe the dynamics of non-premixed flames. Recently, several important studies by the IIT Madras group led by Sujith et al. [7–9] and Chakravarthy et al. [9,10] have addressed this problem. These studies have shown how this problem can be analyzed within the -equation formulation for the mixture fraction. They developed solutions for the flame position and heat release for several problems, including the response of the flame to axial velocity and mixture fraction oscillations. This paper closely follows these studies, with particular emphasis on the axial flow oscillation problem. We consider a linearized version of the problem considered by Balasubramanian and Sujith [7], and derive explicit solutions for the flame sheet position in Section 2.3. This solution allows us to explicitly illustrate the effects of velocity fluctuations in inducing flame wrinkles and axial flow in convecting wrinkles downstream. This solution is then used to examine the spatially integrated heat release response characteristics, as well as
ux x; y e h n uOx x ð Þ0 ; ð Þ1
axial flow velocity axial and radial coordinate, respectively small perturbation parameter local angle of the flame surface with respect to axial coordinate, see Eq. (19) radial flame sheet position stoichiometric mass ratio of oxidizer to fuel angular frequency mean/steady state and fluctuating component, respectively
the relative contributions of flame surface area fluctuations and mass burning rate fluctuations to the instantaneous heat release in Section 2.4. These results also allow us to compare the response characteristics of premixed and nonpremixed flames. We show that there are important similarities and differences in their local and global response characteristics. 2. Main body 2.1. Physics of governing equations This section discusses some basic features of non-premixed flame dynamics in the fast chemistry, thin reaction sheet limit. In addition, we assume that all species have equal diffusivities so that the mixture fraction formulation can be used. The space–time dynamics of the mixture fraction are described by the equation [11]: @ þ~ ur @t
¼ r ðDr Þ
ð1Þ
In the fast chemistry limit, the reaction sheet collapses to a surface defined by the equation ð~ x; tÞ ¼ ¼ 1=ð1 þ uOx Þ, where uOx is the
Fig. 1. Illustration of the forced non-premixed flame model problem.
N. Magina et al. / Proceedings of the Combustion Institute 34 (2013) 963–971
stoichiometric mass ratio of oxidizer to fuel. We define the instantaneous position of this reaction sheet by y ¼ nðx; tÞ, as shown in Fig. 1. It is helpful to compare the dynamics of the mixture fraction equation for non-premixed flames with the G-equation used to analyze the dynamics of premixed flames [2,11,12], which is given below: @G þ~ u rG ¼ sd jrGj @t
ð2Þ
The two expressions have the same convection operator on the left-hand side which illustrates the importance of flow perturbations in the direction normal to the flame sheet in pushing the flame sheet around. However, the right-hand sides of these two expressions are different. The premixed flame expression has the normal flame propagation operator, sd jrGj, while the non-premixed flame expression has a diffusion operator, r ðDr Þ. This difference is significant and reflects, among other things, the fact that non-premixed flames do not propagate. Moreover, the premixed flame dynamics equation is nonlinear, while the non-premixed flame dynamics equation is linear (assuming that Un and D are not functions of ). Another significant, yet subtle difference is that the G-equation is physically meaningful and valid only at the flame itself where Gð~ x; tÞ ¼ 0 (i.e., although it can be solved away from the flame, the resulting G values have no physical significance [13]). In contrast, the -equation describes the physical values of the mixture fraction field everywhere. This observation has important consequences for both solution approaches of these problems, as well as the ð~ x; tÞ ¼ flame sheet dynamics that are discussed next. Substantial analytical progress has been made to analyze “single valued” premixed flames with the substitution Gðx; y; z; tÞ ¼ x nðy; z; tÞ. This leads to an explicit expression for the flame front position [14–18]. However, note that this substitution arbitrarily assigns values to the G field away from the flame itself, namely that G varies linearly with coordinate x away from the flame. Since the G field is completely arbitrary away from the flame, this is allowable. In the non-premixed flame problem, the goal is similarly to only consider the dynamics of the reaction sheet surface ð~ x; tÞ ¼ , and so the space–time dynamics of the rest of the field is of lesser interest. However, we cannot make an analogous substitution, such as ðx; y; z; tÞ ¼ y nðx; z; tÞ, as this assigns valfield away from ð~ x; tÞ ¼ . This ues to the cannot be done since the -equation describes the entire spatial distribution of the mixture fraction field. The implication of this fact is that the entire mixture fraction field must be solved and the ð~ x; tÞ ¼ surface extracted from the resulting solution field (which generally cannot be written as an explicit expression).
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This discussion also reflects important underlying physics of the two flames. Consider a premixed and non-premixed flame in a velocity field given by ~ uðx; y; z; tÞ, where the velocity field just upstream of the flame sheet is given by ~ uðx; nðx; z; tÞ; z; tÞ ¼ ~ uu . The premixed flame dynamics are only a function of ~ uu ; this implies that for a given ~ uu , its space–time dynamics are the same for a variety of different velocity fields ~ uðx; y; z; tÞ. In contrast, the space–time dynamics of the non-premixed reaction sheet are a function of the entire velocity field, ~ uðx; y; z; tÞ, not just its value at the reaction sheet. This result is a direct manifestation of the fact that the -equation is an elliptic partial differential equation. In contrast, the linearized, stretch-free G-equation is a hyperbolic partial differential equation. 2.2. Formulation This section formulates the problem considered in this paper, which is illustrated in Fig. 1. We consider a two-dimensional flame in a uniform axial flow field, ux;0 . At the inlet (x = 0), fuel and oxidizer flow into the domain as indicated in the figure, leading to the following inflow conditions: 1 for 0 6 jyj < W II ð3Þ ðx ¼ 0; yÞ ¼ 0 for W II 6 jyj < W I The solution can also be easily generalized to include more general inflow fuel/oxidizer compositions (e.g., such as if the fuel were diluted) by shifting and rescaling the value of . Enforcing this boundary condition enables an analytic solution of the problem. However, in reality there is axial diffusion of fuel into the oxidizer and vice versa, so that the solution must actually be solved over a larger domain that includes the fuel/oxidizer supply systems. As such, the boundary condition in Eq. (3) implicitly neglects axial diffusion at x ¼ 0, a point we will return to later. Assuming symmetry at y ¼ 0 and no diffusion through the walls at y ¼ W I , leads to the following two additional boundary conditions: @ @y
ðx; y ¼ 0Þ ¼ 0
@ @y
ðx; y ¼ W I Þ ¼ 0
ð4Þ
We will derive the solution in the limit of small perturbations, and so expand each variable as ð Þðx; y; tÞ ¼ ð Þ0 ðx; yÞ þ ð Þ1 ðx; y; tÞ. Since the governing equation, Eq. (1), is linear, this procedure is not necessary in order to obtain an analytic solution and, in fact, this assumption was not made in the work by Balasubramanian and Sujith [7]. However, this expansion is useful in analyzing controlling features of the flame dynamics at the forcing frequency and, very significantly, it enables an explicit analytic expression for the
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space–time dynamics of the flame position, n1 ðx; tÞ, which is otherwise not possible. The mixture fraction field in the absence of forcing is obtained from Eq. (1): ux;0
@ 0 @2 0 @2 0 ¼D 2 þD 2 @x @y @x
ð5Þ
Similarly the dynamical equation for
1
is given
" 1 X
2=ðnpÞ sinðAn Þeb 1¼ PeStW 2pi þ b þ A2n b2 =Pe2 1 y x cos An exp b W II PeW II x exp b exp½ixt PeW II hom where
by: @ 1 @ 1 @2 1 @2 1 @ 0 þ ux;0 D 2 D 2 ¼ ux;1 @t @x @y @x @x
ð6Þ
The solution to these equations can be derived in an analogous way as the Burke–Schumann solution using separation of variables. The full solution, including axial diffusion, for the steady state mixture fraction field is given by: 0
¼
1 W II X 2 sinðAn Þ þ W I n¼1 np qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi31 0 2 2 2 4 2 y x 4Pe Pe þ 4Pe An 5A cos An exp @ 2 W II PeW II
ð7Þ
where An ¼ npðW II =W I Þ are the eigenvalues and the Peclet number, Pe, is given by Pe ¼ ðux;0 W II Þ=D. The Peclet number physically corresponds to the relative time scales for convective and diffusive processes to transport mass over s
W 2 =D
u
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pe4 þ 4Pe2 A2n
2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pe2 Pe4 þ 4Pe2 A2n 8piPe3 StW
ð10Þ
ð11Þ
ð12Þ 2 Again we will focus the subsequent analysis in the absence of axial diffusion, i.e. in the Pe 1 limit, whose solution is: " # 1 X ieðAn Þ2 ð2=npÞ sinðAn Þ 1 ¼ 2pStW Pe n¼1 y x exp A2n cos An W II PeW II x exp½ixt ð13Þ 1 exp 2piStW W II bhom ¼
where the Strouhal number based on the halfwidth of the fuel nozzle is defined as StW ¼ ðfW II Þ=ux;0 . This expression can also be written in terms of 0 as:
W
diffusion ¼ W IIII=ux;0 ¼ x;0D II . As such, a distance W II ; sconvection the Pe 1 limit physically corresponds to the limit where convective processes are much faster than diffusive ones over the length scale WII. For the subsequent analysis, we will focus on the following simplified version of the solution that neglects axial diffusion, since we have already done so implicitly in formulating the boundary condition in Eq. (3). The steady state mixture fraction field solution is: 1 W II X 2 sinðAn Þ þ 0 ¼ np WI n¼1 y x exp A2n ð8Þ cos An W II PeW II
This equation can be derived by solving Eq. (5) and neglecting the axial diffusion term or, equivalently, taking the Pe 1 limit of Eq. (7). We next consider the solution for the fluctuating flame position responding to uniform bulk fluctuations in flow velocity: ux;1 ¼ eux;0 exp½ixt
b ¼
Pe2
#
ð9Þ
The full solution for the fluctuating mixture fraction field, 1, is:
1
¼
ieW II @ 0 x 1 exp 2piStW exp½ixt W II 2pStW @x ð14Þ
2.3. Space–time dynamics of reaction sheet While these expressions for 0 and 1 provide solutions for the mixture fraction field over the entire domain, we are particularly interested in the reaction sheet location. An implicit expression for the flame sheet position, n0 ðxÞ, can be determined from the coordinates where 0 = , yielding: ¼
1 W II X 2 sinðAn Þ þ np WI n¼1 n ðxÞ x exp A2n cos An 0 W II PeW II
ð15Þ
Similarly, the position of the fluctuating flame can be determined from the implicit expression: ¼
0
ðx; n0 ðxÞ þ n1 ðx; tÞÞ þ
þ n1 ðx; tÞ; tÞ
1
ðx; n0 ðxÞ ð16Þ
N. Magina et al. / Proceedings of the Combustion Institute 34 (2013) 963–971
Expanding Eq. (16) to first order leads to the following explicit expression for fluctuating flame position: n1;n ðx; tÞ ¼
¼
ðx; y ¼ n0 ðxÞ; tÞ
1
@ 0 ðx; y @n
¼ n0 ðxÞÞ
ðx; y ¼ n0 ðxÞ; tÞ jr 0jðx;y¼n0 ðxÞÞ
1
ð17Þ
where n1;n is measured normal to the mean flame surface in the direction of the oxidizer. This can be re-written as: " # ! @ 0=@x n1;n ðx; tÞ ie ¼ 2pStLf Lf jr 0j ðx;y¼n0 ðxÞÞ x exp½ixt 1 exp 2piStLf Lf ;0 ð18Þ where StLf ¼ ðf Lf ;0 Þ=ux;0 is the Strouhal number @ 0=@x based on flame length. The term can be jr 0j
written in terms of the local angle of the flame, using the relations: " 2 2 #1=2 @ 0 @ 0 @ 0=@x ð19Þ þ ¼ jr 0j ¼ sin h0 ðxÞ @x @y where h0 denotes the angle of the mean flame with respect to the axial coordinate. Using these results, the solution for n1;n ðx; tÞ can be written as: ieux;0 x sin h0 ðxÞ 1 exp i2pStLf n1;n ðx; tÞ ¼ Lf ;0 2pf exp½i2pft
ð20Þ
This expression is an important contribution of this paper and very significant in that it is an explicit equation for the space–time dynamics of the flame position. For reference, the corresponding fluctuations of an attached premixed flame with constant burning velocity subjected to bulk flow oscillations are given by [14]: ieux;0 x n1;n ðx; tÞ ¼ sin h 1 exp i2pStf 2pf Lf ;0 exp½i2pft
ð21Þ
where Stf is the flame Strouhal number for premixed flames, defined in the nomenclature section, and the angle h is a constant (the expression is more involved if h is varying, which would occur if the flow or flame speed varies spatially). Notice the similarities in the premixed and non-premixed solutions, with the exception of the spatial phase dependence, 1 ei2pfx=ux;0 term. This difference reflects the influence of premixed flame propagation on wrinkle convection speeds; i.e., the wrinkle
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convection speed in the axial direction, ux;0 = cos2 h, is the vector superposition of the axial flow velocity and the axial projection of a vector pointing normal to the flame with a magnitude equal to the burning velocity, sd . In contrast, the non-premixed flame does not propagate and wrinkles convect downstream at a speed of ux;0 . In both cases, local maxima and minima arise through this 1 ei2pfx=ux;0 term, due to interference between wrinkles generated at the x = 0 boundary and disturbances excited locally. This can be seen by re-writing it as: 1 ei2pfx=ux;0 ¼ 2 sinðpfx=ux;0 Þeiðpfx=ux;0 p=2Þ
ð22Þ
For both premixed and non-premixed flames, wrinkles are generated at the boundary because of flame attachment. For the premixed flame, this is invoked through the attachment boundary condition i.e., n1;n ðx ¼ 0; tÞ ¼ 0. In the non-premixed case, wrinkles are generated through the assumption of constant mixture fraction at the burner outlet, i.e., 1ðx ¼ 0; tÞ ¼ 0. Finally, we note that incorporating stretch effects into the premixed flame analysis modifies Eq. (21) by multiplying the complex exponential inside the braces by the factor expð4p2 rc St2f x=Lf Þ [19], where rc is the Markstein length normalized by the burner half-width, WII. For a thermo-diffusively stable flame, this stretch correction leads to an exponential decay in wrinkle magnitude because of the flame front curvature. We next present several illustrative solutions of the space–time dynamics for the flame position. Note that the solution is a function of the four dimensionless parameters StLf , W II =W I , Pe, and . The temporal evolution of the flame position is plotted in Fig. 2 at two Strouhal numbers. Note the bulk axial pulsing of the flame at lower Strouhal numbers, and the spatial wrinkling at higher values. An alternative way to visualize these results is through the magnitude and phase of n1;n , illustrated in Fig. 3. The nodes and local maxima and minima referred to above are clearly evident in the figure. The phase rolls off linearly with axial distance, again reflecting the convection process described by the 1 ei2pfx=ux;0 ¼ 2 sinðpfx=ux;0 Þ eiðpfx=ux;0 p=2Þ term, and jumps 180° across the nodes. 2.4. Heat release analysis Having considered the local space–time wrinkling of the flame, we next consider the spatially _ integrated heat release, QðtÞ, which is given by the following surface integral over the reaction sheet: Z _ m_ 00F hR dA QðtÞ ¼ ð23Þ A
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Fig. 2. Snapshots showing four instantaneous positions of a forced non-premixed flame at two different forcing = 0.3 and Pe ¼ 50 (a) e ¼ 0:2, StW ¼ 0:0018, StLf ¼ 0:3, (b) e ¼ 1:0, StW ¼ 0:012, frequencies using nominal values of StLf ¼ 2:0. The unforced flame is indicated by dashed lines.
The term m_ 00F is the reactant mass burning rate per unit area, and hR is the heat release per unit mass of reactant consumed. For premixed flames, the mass burning rate can be written in terms of the burning velocity as m_ 00F ¼ qu suc , where qu is the density and suc is the laminar consumption speed of the unburned reactant, yielding: Z _ qu suc hR dA ð24Þ Premixed flame : QðtÞ ¼
Decomposing , Lf, and n into their mean and fluctuating components results in: 2
ð1 þ uOx Þ _ qhR D QðtÞ ¼ uOx Z Lf ;0 þLf ;1 ðtÞ @ 0 ðx; n0 þ n1 ðx; tÞÞdx @y 0 Z Lf ;0 þLf ;1 ðtÞ @ 1 ðx; n0 þ n1 ðx; tÞÞdx þ ð28Þ @y 0
flame
For non-premixed flames, the reactant mass burning rate is: m_ 00F ¼ m_ 00Ox þ m_ 00Fuel ¼ qD ¼ ð1 þ uOx ÞqD
Then, linearizing this expression yields: Z Lf ;0 @ 0 ð1 þ uOx Þ2 _ qhR D dx QðtÞ ¼ uOx @y 0 ) Z Lf ;0 Z Lf ;0 @ 1 @2 0 dx þ n1 2 dx þ @y @y 0 0
@Y Ox @Y Fuel qD @n @n
@Y Fuel @n
ð25Þ
where n represents the direction normal to the flame surface into the oxidizer. By also relating the fuel mass fraction and the mixture fraction gradients, the heat release can be written as: Non-premixed flame : Z ð1 þ uOx Þ2 _ QðtÞ ¼ qhR uOx flame @ @ sin h D cos h dA ð26Þ D @x @y
Note that fluctuations in Lf do not contribute to this linearized expression since @ 0=@y is zero along the centerline and, therefore, at the flame tip (axial diffusion would provide a non-zero contribution). We will denote transfer functions by F, defined as follows: F¼
FN ;mbr ¼
0
@ 1ðx;n0 Þ dx @y
þ
Q_ 1 =Q_ 0 ux;1 =ux;0
ð30Þ
where premixed or non-premixed transfer functions will be denoted as FP and FN , respectively. It is useful to decompose the results in Eq. (29) into contributions from mass burning rate and flame area fluctuations; i.e., F ¼ Fmbr þ Fa . This decomposition requires replacing the dx by cos hdA in Eq. (29) (see Eq. (26)) and similarly expanding the solution. The
We will assume fixed composition fuel and oxidizer, so that the heat of reaction and mixture density are constant. The first term in Eq. (26) includes the effect of axial diffusion and is, consequently, neglected in the following analysis, yielding: Z Lf ðtÞ ð1 þ uOx Þ2 @ ðx; nðx; tÞÞ _ QðtÞ ¼ dx qhR D @y uOx 0 ð27Þ R Lf ;0
ð29Þ
R Lf ;0 0
@ 1ðx;n0 Þ @x
e
R Lf ;0 0
sinðh1 Þdx þ @ 0ðx;n0 Þ dx @y
R Lf ;0 0
n1 ðx; tÞ
@ 2 0ðx;n0 Þ dx @y 2
ð31Þ
N. Magina et al. / Proceedings of the Combustion Institute 34 (2013) 963–971
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b
a
eu
x;0 Fig. 3. Axial dependence of (a) magnitude and (b) phase of flame response, where nref ¼ Pe2pf , and using nominal values = 0.3 and Pe ¼ 50 for StLf values of 0.3 and 2.0. Note the abscissa in (b), (x/ux,0/f), can equivalently be written as of ðx=Lf ÞStLf .
resulting mass burning rate contribution to the transfer function is: The mass burning rate term contributes to heat release oscillations due to the fluctuations in spatial gradients of the mixture fraction for non-premixed flames. For premixed flames, the mass burning rate fluctuations are linked to the stretch sensitivity of the burning velocity, which fluctuates because of the oscillatory curvature of the wrinkled front [19]. Similarly, the area contribution is given by: R Lf ;0 FN ;wa ¼
@ 0ðx;n0 Þ @y
0
e ¼
R Lf ;0 0
R Lf ;0
@ 0ðx;n0 Þ dx @y
@ 0ðx;n0 Þ @x
0
e
cosðh0 ÞdA1
R Lf ;0 0
sinðh1 Þdx
@ 0ðx;n0 Þ dx @y
ð32Þ
As this area term is weighted by the time averaged burning rate (which, unlike premixed flames, is spatially non-uniform for non-premixed flames considered here), it will be called the weighted area. The un-weighted area transfer function (important for constant burning velocity premixed flames) is given by: 1
FN &P ;a ¼ R Lf ;0 e 0 ½1 þ ðdn0 =dxÞ2 1=2 dx Z Lf ;0 @n1;y ðx; tÞ dn0 =dx dx 2 1=2 @x ½1 þ ðdn0 =dxÞ 0
ð33Þ
or equivalently: 1 FN &P ;a ¼ R Lf ;0 e 0 ½1= cos h0 ðxÞdx Z Lf ;0 sin h0 sin h1 dx 2h cos 0 0
ð34Þ
There are significant variations in time averaged heat release rate along the non-premixed flame (e.g., no heat release at the tip in the absence of axial diffusion). Thus, the weighting of flame area is a very significant effect influencing how area fluctuations lead to heat release. Moreover, the characteristics of the weighted and unweighted area transfer functions are quite different for non-premixed flames, while they are identical for premixed flames with spatially uniform burning velocities. For example, in the low Strouhal number limit, the non-premixed flame weighted and un-weighted area transfer functions differ in phase by 180° and have appreciably different magnitudes. The solutions for the premixed flame transfer functions are simpler, as the unforced flame is flat in a uniform velocity field. Following Wang et al. [19], and retaining only leading order terms in Markstein length, the transfer function is: n o 2 2 ^c Þð1 e2piStf 4p r^c Stf Þ ^c ð1 þ 2piStf r Fp ¼ r |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} n
¼FP ;mbr
o 2 2 þ 1=ð2piStf Þ 1 e2piStf 4p r^c Stf ð35Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼FP ;a
^ c is given by r ^c ¼ rc sin h tan h. where r Typical solutions for the overall unsteady heat release, as well as the contributions from flame area and mass burning rate are shown in Fig. 4 for the non-premixed and premixed flame. Both premixed and non-premixed transfer functions have magnitudes of identically unity at zero St, and then roll off with increasing St. Starting with Fig. 4(a), note how the non-premixed flame heat release fluctuations for StLf 1 are dominated by mass burning rate fluctuations over the entire Strouhal number range. For StLf 1, the mass
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Fig. 4. Strouhal number dependence of the magnitude of the heat release, area, and mass burning rate transfer functions ^c ¼ 0:05 and = 0.06, Pe ¼ 10, and (b) premixed flame with parameters r for a (a) non-premixed flame with parameters Lf =W II ¼ 932.
burning rate contributions to F are of O(1) and O(Stf) for non-premixed and premixed flames, respectively. In contrast, premixed flames at low and O(1) Strouhal numbers are dominated by area fluctuations, shown in Fig. 4(b). The mass burning rate fluctuations are a much smaller effect and only exert a comparable response as area fluctuations at high Strouhal numbers where Stf Oð^ r1 c Þ [19]. The mass burning rate fluctuations do also exert an indirect influence on the flame area perturbations when Stf Oð^ r1=2 Þ, by c smoothing out flame wrinkles, causing the “smoothing” of the area gain curve in the premixed case, relative to the much more oscillatory curve for the non-premixed flame.
Fig. 5. Strouhal number dependence of the magnitude and phase of the heat release transfer function for a nonpremixed and premixed flame with the same properties as Fig. 4.
Direct comparisons of the gain and phase response of the premixed and non-premixed flame results are shown in Fig. 5. Significantly, these results show that non-premixed flames are significantly more sensitive to flow perturbations than premixed flames when StLf > Oð1Þ, an important and somewhat unexpected conclusion. Although not proven here, it can be shown that the St 1 response of both flames scales as F Oð1=StÞ. Interestingly, this figure shows that the non-premixed flame has an intermediate region where its response rolls off more slowly; we were unable to derive this regions St sensitivity analytically, but computations suggest that FN Oð1=StL1=2 Þ in this f region. The 1/St scaling is less obvious for this premixed flame case as stretch effects do modify the results for the Strouhal number ranges shown in the plot. The corresponding phases of the premixed and non-premixed flame transfer functions are also included in Fig. 5. Both curves start at zero for low Strouhal numbers, indicating that low frequency flow modulation induces heat release fluctuations that are in phase. The curves roll off with different slopes toward negative values and asymptote to 90° (for a stretch-insensitive flame; as shown in the graph, stretch modifies this result), indicating the delay in heat release relative to the forcing, due to convection of disturbances along the front. Note also the nearly constant phase in the non-premixed flame in the intermediate Strouhal number range discussed above. The undulations in phase for the premixed flame correspond to ripples in the gain, and reflect the influence of interference processes in controlling the flame area. The differences in phase between the two flames again reflects the different processes
N. Magina et al. / Proceedings of the Combustion Institute 34 (2013) 963–971
controlling unsteady heat release. The corresponding phases of the area contributions alone are much closer between the two flames for a ^1=2 broader StLf range for r 1. c 3. Conclusions This paper has presented results for the flame sheet motion and unsteady heat release induced by bulk axial excitation of a non-premixed flame. A key contribution of this paper is Eq. (20), which is an explicit expression for the space–time dynamics of the flame sheet. This expression shows the role of axial convection in propagating flame wrinkles downstream, leading to nodes and anti-nodes in the flame response, similar to premixed flames. In addition, it was shown that their heat release dynamics are quite different, premixed flames being dominated by area fluctuations and non-premixed flames by mass burning rate fluctuations. Their gain sensitivities both tend towards unity at low St values, but the non-premixed flame response is larger than premixed flames for St O(1).
Acknowledgments This work has been partially supported by the US Department of Energy under contracts DEFG26-07NT43069 and DE-NT0005054, contract monitor, Mark Freeman.
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