University of Virginia, Charlottesville, VA 22904-4746, USA
Under certain operating conditions, many combustion systems exhibit large ampli-tude pressure oscillations coupled with unsteadiness in the combustion processes (thermo-acoustic instabilities). Model-based active control approaches are being pursued to sup-press the potentially destructive instabilities, where a fundamental understanding of theflame-acoustics coupling mechanisms is critical. The formulation of a detailed numericalmodel for the investigation of the spontaneous resonant interaction of longitudinal acousticwaves with planar counterflow flames is presented, with the inclusion of compressibility andfinite-rate chemistry effects. For well-resolved simulations, the occurrence of self-sustainedthermo-acoustic instabilities in methane-air premixed counterflow flames is analyzed for arange of flow strain rates, employing both detailed and global one-step chemical kineticmodels. It is shown that the response of the flame to acoustic perturbations is dependenton the flow strain rate as well as on the reaction mechanism employed. In particular, unliketheir non-premixed counterpart, premixed counterflow flames modeled with detailed chem-ical kinetics show the potential for flame-acoustics coupling. The sensitivity of chemicalkinetic parameters on the acoustic response is investigated for several one-step models.
I. Introduction
The interaction of acoustic waves with methane-air non-premixed counterflow flames has been recentlyaddressed, with particular emphasis on the derivation of a fully unsteady formulation of the reacting flowfield, and on chemical kinetic models.1 Based on the same numerical model developed, the investigation isextended here to counterflow methane-air premixed flames.
The importance of flame-acoustics interaction is related to the establishment of large amplitude pressureoscillations in combustion systems operating under certain conditions. Such unsteady phenomenon is gen-erally referred to as thermo-acoustic instabilities. Specifically, the flame heat release rate is very sensitiveto disturbances in the flow field and, through a feedback mechanism, coupling of the oscillating heat releaserate with the natural acoustic modes of the combustor may reinforce acoustic pressure oscillations and leadto acoustic resonance.2
A wide range of combustion applications is affected by thermo-acoustic instabilities. Recent studies havefocused on combustion instabilities in gas turbine engines because of their importance to aircraft industryand land-based power generation. In particular, with recent environmental regulations, gas turbine enginesare required to operate in lean-premixed mode in order to reduce NOx emissions. This involves operationnear the lean flammability limit, with increased likelihood of heat release fluctuations and hence occurrenceof thermo-acoustic instabilities.3
Recently, model-based active control approaches are being pursued to suppress the potentially destructiveinstabilities.4,5 With this technique, the damping and suppression of pressure oscillations is accomplished bydynamic hardware components. However, detailed information about the flame-acoustics coupling is requiredin order to develop active control schemes for these devices. In particular, a fundamental understanding ofthe flame dynamics contributing to the unsteady heat release rate is essential.
The flame dynamics of practical combustion systems is affected by a variety of complex physical phenom-ena that may occur simultaneously on multiple time scales as well as length scales, e.g. flow compressibility
∗Graduate Student, Department of Mechanical and Aerospace Engineering, PO Box 400746, AIAA Student Member.†Associate Professor, Department of Mechanical and Aerospace Engineering, PO Box 400746, AIAA Associate Fellow.
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43rd AIAA Aerospace Sciences Meeting and Exhibit10 - 13 January 2005, Reno, Nevada
Figure 1. Illustration of twin premixed counterflow flames established in the counterflow configuration withidentical fuel/oxidizer inflow conditions. The predicted dominant acoustic pressure mode is also representedon the right.
effects,2,6–8 thermal-diffusive instabilities and cellular flames,9–11 flame surface area fluctuations due to hy-drodynamic effects,12,13 equivalence ratio fluctuations in premixed systems (near lean flammability limit),3
heat losses to the walls or flame holders,14,15 finite-rate chemistry effects,16 etc.As a result of the complexities of the phenomena involved, their interactions and their inherent non-
linearities, accurate modeling of thermo-acoustic instabilities is a very difficult task. It is well known thatnumerical simulation of actual full-scale combustion systems, with accurate resolution of the flame struc-ture, can be computationally very expensive. Therefore, most numerical investigations on thermo-acousticphenomena have focused on several scaled-down flame configurations (Rjike tubes and dump combustor).However, efforts to develop model-based active control schemes using such simplified flow geometries havemet limited success. The lack of progress is probably due to the complex interaction of several phenomenathat occur simultaneously in the reacting flow field.
The selected counterflow flame configuration is a scaled-down chemically reacting flow field that is com-putable with accurate resolution of the complex multi-scale combustion phenomena involved, includingdetailed transport properties and finite-rate chemistry. In the premixed regime, two planar axisymmetricflames are established in the mixing layer of two opposing jets, as shown in figure 1 together with thedominant acoustic pressure mode shape.
II. Formulation
The main advantage of the counterflow field is related to the planar flame structure and the associatedself-similar mathematical formulation, since it is assumed that the flame structure remains planar for a widerange of flow conditions, eg. flow strain rates, fuel-air mixtures, etc. For steady-state counterflow flames,the formulation and the solution method of the governing equations are well established.17
A. Governing Equations
While retaining the same self-similar approximation used for planar flames, fully unsteady effects, includingflow compressibility, are considered here. Such an analysis requires the introduction of the axial momen-tum equation and the associated unsteady pressure terms. The resulting unsteady formulation of planarcounterflow flame can be written as:
∂ρ
∂t+
∂ρv
∂x+ 2ρU = 0, (1)
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∂ρYk
∂t+
∂ρvYk
∂x+
∂ρYkVk
∂x+ 2ρYkU = Wkωk, (2)
∂ρv
∂t+
∂ρvv
∂x+
∂p0
∂x+ 2ρvU =
∂
∂x
[
4
3µ(
∂v
∂x− U)
]
+ 2µ∂U
∂x, (3)
∂ρU
∂t+
∂ρvU
∂x+ 3ρU2 = −2p2 +
∂
∂x
[
µ∂U
∂x
]
, (4)
∂ρht
∂t−∂p0
∂t+ ∂ρvht
∂x+ 2ρhtU = ∂
∂x
[
λ∂T∂x
]
−∑
k∂ρYkVkhk
∂x
+ ∂∂x
[
43µ( ∂v
∂x− U)v
]
+ 2µ∂U∂x
v − 43µ( ∂v
∂x− U)U. (5)
Here, ρ is the density, Yk the mass fraction of species k, v the axial velocity component, U the radialcomponent of the gradient of the radial velocity, T the temperature, and ht the total enthalpy per unitvolume, given by:
ht(x, t) = h(x, t) +1
2[v(x, t)]2. (6)
The thermal conductivity is designated with λ and the viscosity with µ. The diffusion velocity, the molarproduction rate and the molecular weight of the kth species are Vk, ωk and Wk, respectively. The thermo-dynamic effects of the radial pressure gradient due to p2 are negligible. The leading order term for pressurep0 is related to the other thermodynamic variables by the equation of state for a mixture of perfect gases:
p0(x, t) = ρRT. (7)
B. Pressure Description
Unlike other variables, in the self-similar formulation, the description of pressure requires a careful analysis.The second-order term p2 accounts for the pressure gradient in the radial direction, which directly affects theflow in the radial direction. In the steady-steady formulation, the isobaric assumption ∂p0/∂x ≃ 0 is usuallyintroduced for low Mach number flows,17,18 resulting in a uniform pressure field in the axial direction:
p(x, r, t) = p0 +1
2J r2, (8)
where both p0 and J are constants, the latter being the eigenvalue of a two-point boundary value problem:
J = (1/r)(∂p/∂r). (9)
In the present unsteady formulation, at the low Mach number limit, p0(x, t) can be written as the sum ofthe mean uniform pressure and the acoustic pressure fluctuation, i.e. p0 = p0 + p′0(x, t), so that the finalexpansion for pressure is given by:
p(x, r, t) = p0 + p′0(x, t) + p2(t)r2, (10)
where p2 ≡ 12J .
In steady-state counterflow flames, each value of J yields a unique value for the flow strain-rate, definedhere as a = |dv/dx| outside the mixing layer. Previous unsteady analysis of counterflow flames have retainedthe isobaric assumption by allowing a uniform pressure field to vary in time, i.e. p′0 = p′0(t). The disadvantageof these formulations is that it requires an externally imposed modulation of the uniform pressure field.
On the contrary, the present fully unsteady formulation relaxes the isobaric assumption and provides adescription of the acoustics of the flow field, i.e. p′0 = p′0(x, t). With J assumed to be constant, oscillationsof strain-rate can occur because of the unsteady pressure effects.
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C. Boundary Conditions
For the two opposed jets, the pressure nodes are assumed to be located close to the exits of the nozzles(i.e. open tube boundary conditions), as shown in figure 1. With constant temperature and species massfractions of the inflow streams, the following physical boundary conditions are imposed at x = x−∞ = −l/2
p = p−∞, Yk = Yk,−∞, U = 0, T = T−∞, (11)
and at x = x∞ = +l/2p = p∞, Yk = Yk,∞, U = 0, T = T∞, (12)
where l is the separation distance.The axial velocities at the boundaries are prescribed through the implementation of Navier-Stokes Charac-
teristic Boundary Conditions (NSCBC).19 The choice of NSCBC can yield reflective or non-reflective acousticwaves at the boundaries. In the NSCBC approach, there are generally two characteristic waves travellinginto the domain through the inflow boundary (their amplitudes being L2 and L5), and one characteristicwave travelling from the interior of the domain to the inlet with amplitude L1:
L1 = (v − c)
[
∂p0
∂x− ρc
∂v
∂x
]
, (13)
L2 = 0, (14)
L5 = −L1. (15)
The pressure (or equivalently density) and axial velocity are updated by integrating the following equationsat the boundaries:
∂ρ
∂t+
1
c2
[
L2 +1
2(L5 + L1)
]
= 0, (16)
∂ρv
∂t+ v
1
c2
[
L2 +1
2(L5 + L1)
]
+1
2c[L5 − L1] = 0. (17)
It is possible to relax the assumption of pressure nodes located at the nozzle exits to allow partial reflections.In this case, the predicted amplitude of incoming waves can be modified by varying the parameter σ inL5 = σ(p0 − p), where p is the atmospheric pressure in the far-field.
D. Numerical Method
The integration of the governing equations is performed using an explicit MacCormack numerical scheme,20
widely used in the literature for compressible flows. The system of partial differential equations is recastin the conservative vectorial form in terms of the conserved variables, convective flux, diffusive flux andsource term vectors. The associated thermo-chemical and transport properties are obtained from SANDIAChemkin and transport codes.21,22
The method is based on a predictor-corrector approach, where the conserved variables vector is updatedby upwind differencing the convective flux vector in the predictor step, and by downwind differencing inthe subsequent corrector step. At all iterations, the derivative of diffusive flux is always discretized in thedirection opposite to that of the flux vector derivative. In order to eliminate any directional bias, at the nextiteration the differencing direction of all spatial derivatives is reversed.
The initial conditions for the time-dependent governing equations are given by the counterflow steady-state solution, evaluated at the same mean flow conditions. This initial solution is obtained from a modifiedversion of Smooke’s code17 and relies on the isobaric assumption. When employed as input in the unsteadycode, the solution adjusts to the full compressible equations described in section A, after a brief transient.This transient results in the generation of acoustic waves over a broad frequency spectrum. Depending onthe selection of the boundary conditions, acoustic waves with the natural frequency of the spatial domaincan be reflected at the boundaries, while the remaining noise is dissipated.
The major limitation of the MacCormack scheme adopted is the explicit numerical integration, whichimposes a numerical time step size based on the CFL condition. Because of the need to resolve the flamestructure and capture the flame dynamics, a high spatial resolution is required in the chemically reactingregion. Depending on the strain-rate, the selected ∆x is in the range 10µm − 15µm. Based on the speed of
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−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50
500
Axial Coordinate, cm
Axi
al V
eloc
ity,
cm /
s, T
empe
ratu
re,
10 K
, H
eat R
elea
se R
ate,
10 9
erg
/ cm
3 /
s
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50
0.2
Mas
s F
ract
ion
of S
peci
es
Axial Velocity
Temperature
Heat Release Rate
Stagnation Plane
YCH
4
YO
2
YH
2O
YCO
2
Figure 2. For stoichiometric methane-air premixed counterflow flames with a separation distance of 1 cm andstrain rate, a, of 220 s−1, exploiting the symmetry of the twin flames with respect to x = 0, the left half of thefigure shows the spatial profiles of axial velocity v, temperature T and heat release rate q, whereas the righthalf shows the mass fraction profiles of the major chemical species.
sound, the largest time step size is estimated around 10−8 s. Even for a very short spatial domain (l = 1 cm),a uniform grid would result in extremely long computational times. In order to speed up the execution ofthe unsteady code, a stretched grid is employed with a high resolution in the chemically reacting region andat the boundaries, while in the rest of the spatial domain the grid is smoothly stretched using a stretchingfunction proposed by Roberts.20
III. Results and Discussion
The acoustic interaction of methane-air non-premixed flames in the counterflow configuration has beenstudied by the authors with both a detailed and one-step chemical mechanisms, and the importance ofchemical kinetic models in thermo-acoustic investigations was highlighted.1 A similar analysis is now ex-tended to methane-air premixed counterflow flames. In particular, here identical conditions for composition(Yk,−∞ = Yk,∞), temperature (T−∞ = T∞ = 300 K) and axial velocity (v−∞ = v∞) are employed at theexits of the opposed jets, resulting in the establishment of two twin premixed flames, one on either side ofthe stagnation plane, located at x = 0. All thermo-dynamic and fluid-dynamic variables are symmetric withrespect to the stagnation plane, with the only exception of the axial velocity component, which is symmetricwith respect to the point x = 0 and v = 0. The investigation is carried out for a separation distance betweenthe nozzles of 1 cm and a stoichiometric mixture of methane and air (YCH4
= 0.055, YO2= 0.220 and
YN2= 0.725). Considering the symmetry of the twin flames with respect to x = 0, the flow field and the
structure of the flames are shown in figure 2 for a strain rate of 220 s−1.The unsteady computations discussed next are performed employing perfectly reflecting boundary con-
ditions, corresponding to σ = ∞. The major distinction between the present and the previous modelingof unsteady counterflow flames is that the unsteady phenomena here are self-sustained, i.e. no externalvelocity perturbations are imposed. The growth of acoustic pressure fluctuations is a direct consequence ofthe resonant interaction between acoustic pressure fluctuations and flame dynamics. The resulting pressuremode shapes and the corresponding velocity oscillations are shown in figure 3. The mode shapes are basedon solutions saved every 100 time steps, near t = 0.01 s. The dominant half-wave mode shape of the acousticpressure is consistent with the open-tube boundary condition of the opposed nozzles. On the other hand,the fluctuating component of the axial velocity, v′ = v − v (the bar denotes the steady-state value), featuresanti-nodes at the exits of the nozzles, and a node overlapping the location of the stagnation plane, due to
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Figure 3. Fluctuating components of pressure, p′0, and axial velocity, v′, showing a self-excited first mode for
twin stoichiometric methane-air premixed counterflow flames with a separation distance of 1 cm and strainrate, a, of 1180 s−1 at 10 ms.
the symmetry of the flow field.
A. Chemical Kinetic Models
Similarly to the non-premixed flame investigated earlier,1 the thermo-acoustic investigation on premixedcounterflow flames is carried out analyzing the effects of chemical kinetic models, both detailed and one-step. Because of the simplicity of the latter model, particular emphasis is given to the effects of certainchemical kinetic parameters, such as activation energy and reaction order.
The detailed chemical kinetic model employed for methane-air mixtures consists of 17 chemical speciesin 39 elementary reactions.23 Compared to the global one-step model, the detailed model provides a moreprecise prediction of flame speeds, flame acceleration and extinction events, all of which are critical in theanalysis of the flame dynamics.
A global one-step irreversible reaction involving a stoichiometric mixture of methane and air can berepresented as:
CH4 + 2O2 → CO2 + 2H2O, (18)
where the reaction rate is expressed in Arrhenius form. The molar rate of progress of such reaction can thenbe expressed as:
ω = A[CH4]α[O2]
βexp
(
−Ea
RT
)
, (19)
where A is the pre-exponential factor, [ ] identifies the concentration of a chemical species, α and β deter-mine the reaction order for CH4 and O2 respectively, and Ea is the activation energy.
From the literature,18,24 values of activation energy for global models vary from 15 kcal/mole up to48.4 kcal/mole, depending on the specific problem dealt with. In the present analysis, first a one-stepmechanism with Ea = 30 kcal/mole and α = β = 1.0 is considered. Next, starting from this model, fouradditional mechanism are introduced by changing one parameter at the time, specifically Ea = 15 kcal/moleand Ea = 45 kcal/mole, and β = 0.5 and β = 2.0. Sensitivity analysis has shown that, with respect toreaction order, CH4 is less sensitive than O2. Therefore, only β is varied to study the effects of the reactionorder. Table 1 summarizes the values selected for the chemical reaction parameters in all global one-stepmechanisms.
Because of the intrinsic simplicity of the global one-step models, the proposed sets of chemical reactionparameters are not able to simultaneously satisfy the requirements on laminar burning speed, S0
L, extinction
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Table 1. Summary of chem-ical kinetic parameters usedin global one-step models.
A Ea* α β
1.42E12 15 1.0 1.0
1.61E11 30 1.0 0.5
2.48E14 30 1.0 1.0
4.48E20 30 1.0 2.0
1.97E16 45 1.0 1.0
* Units are kcal/mole.
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50
5
10
15
20
25
30
35
40
45
Equivalence Ratio
Lam
inar
Bur
ning
Spe
ed,
cm /
s
Ea = 15000 cal / mole, α = 1.0, β = 1.0
Ea = 30000 cal / mole, α = 1.0, β = 1.0
Ea = 45000 cal / mole, α = 1.0, β = 1.0
Detailed mechanism
(a)
0 500 1000 1500 2000 2500 30001400
1500
1600
1700
1800
1900
2000
2100
2200
2300
2400
Strain Rate, s −1
Max
imum
Tem
pera
ture
, K
a=220 s−1
a=1180 s−1
Ea = 15000 cal / mole, α = 1.0, β = 1.0
Ea = 30000 cal / mole, α = 1.0, β = 1.0
Ea = 45000 cal / mole, α = 1.0, β = 1.0
Detailed mechanism
(b)
Figure 4. Comparison between the detailed model and three one-step models with varying activation energyof a) predicted laminar burning speed as a function of equivalence ratio for a methane-air mixture, andb) predicted peak temperature as a function of strain rate for methane-air premixed counterflow flames atstoichiometric conditions.
strain rate, aext, and temperature of the detailed mechanism. Consequently, a trade-off between predictionson laminar burning speed and extinction is achieved by selecting a slightly lower value for S0
L of 31 cm/s atstoichiometric conditions and adjusting the pre-exponential factor accordingly.
A comparison of the predicted laminar burning speeds as a function of equivalence ratio, and peaktemperatures as a function of strain rate is shown in figure 4 for the detailed model and three one-stepmodels with varying activation energy. For a premixed counterflow flame, the position of the flame in thephysical domain is strongly dependent on S0
L. At φ = 1.0, a laminar burning speed of 31 cm/s for all one-step models results in a flame position very close to the one predicted by the detailed model, in particularfor strain rates above 400 s−1. In addition, depending on the choice of one-step model, extinction canoccur at much larger values of strain rate compared to aext = 2100 s−1 predicted by the detailed model.Consequently, in the unsteady calculations of thermo-acoustic interaction, a flow strain rates below 2100 s−1
are selected to perform comparisons between detailed and one-step models.
B. Flame-Acoustics Coupling
1. Effects of Chemical Kinetics: One-step and Detailed Models
As discussed in Ref. 1, for a non-premixed counterflow flame, computations employing a global one-stepmodel show an acoustic response of the flame considerably different from that obtained with detailed chem-istry. Figure 5 compares the evolution in time of the amplitude of the acoustic pressure fluctuations, |p′0|, for
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Figure 5. Comparison of the evolution in time of the amplitude of pressure fluctuations, |p′0|, at x = 0 for a
non-premixed counterflow flame at strain rates of a) 50 s−1 and b) 416 s−1, employing a one-step model withEa = 30 kcal/mole and α = β = 1.0, and the detailed model.
the two models employed, both at a) a low strain rate of 50 s−1 and b) close to extinction conditions. Whenthe detailed model is employed, decay of the pressure fluctuations or almost no change in their amplitudeare observed. On the other hand, the one-step model (Ea = 30 kcal/mole and α = β = 1.0) always showsa positive growth of p′0, resulting in large amplitude pressure fluctuations.
For premixed counterflow flames, figure 6 show the evolution in time of the acoustic pressure fluctuationsat the mid-point of the domain for strain rates of 220 s−1 and 1180 s−1 respectively, employing a) a one-step model with Ea = 30 kcal/mole and α = β = 1.0, and b) the detailed model. Unlike non-premixedcounterflow flames, the acoustic response of premixed counterflow flames always shows a positive growth,regardless of the strain rate or mechanism employed.
By extending the thermo-acoustic analysis from non-premixed to premixed counterflow flames, it isexpected to find a strong flame-acoustics coupling with the one-step model, which is the case. In fact, thepredicted amplitudes are now much larger, as shown in figures 6 a) and c). For instance, in the first case theinitial amplitude dramatically increases from 25 dynes/cm2 to a final value of about 3000 dynes/cm2 after10 ms. In the second case, with a strain rate of 1180 s−1, the final amplitude is much larger.
The new interesting result shown in figures 6 b) and d) is the amplification of pressure fluctuationsobserved in premixed flames with the detailed model. Although the predicted growth is smaller whencompared to the corresponding one-step results, this finding points out that, in premixed counterflow flames,significant coupling between acoustic waves and flame dynamics can be established and drive the instabilities.
Considering that computations are carried out employing perfectly reflecting boundary conditions andthat the quasi one-dimensional formulation of unsteady counterflow flames does not account for losses in theradial direction, the aforementioned result does not imply that all real premixed counterflow flames showresonant phenomena. The conclusion that can drawn is that, under certain conditions, actual premixedflames might show coupling mechanisms that can trigger and drive self-sustained thermo-acoustic instabili-ties, whereas computations with non-premixed flames do not indicate that, even under the most favorableconditions.
This conclusion suggests that, from a thermo-acoustic point of view, non-premixed and premixed coun-terflow flames show fundamental differences in the potential coupling mechanisms. Numerical modeling ofunsteady premixed counterflow flames can identify the conditions based on strain rate, equivalence ratioand separation distance, under which thermo-acoustic instabilities are most likely to be observed in actualsystems, provided that acoustic losses can be minimized.
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Figure 6. Comparison of the evolution in time of pressure fluctuations, p′0, at x = 0 for premixed coun-
terflow flames at strain rates of 220 s−1 (top) and 1180 s−1 (bottom), employing a one-step model withEa = 30 kcal/mole and α = β = 1.0 (right), and the detailed model (left).
2. Effects of Strain Rate
Similarly to the predictions of non-premixed flames with one-step models, the growth rates of premixedflames increase with increasing strain rates. Comparison of figures 6 a) and b) with figures 6 c) and d) showa difference of almost one order of magnitude in the amplitude after 10 ms for the detailed model, whereasthe one-step model is subject to a much larger difference.
3. Effects of Chemical Kinetic Parameters
Both for non-premixed and premixed counterflow flames, the adoption of one-step models always results inpositive growth rates of p′0. Figures 6 a) and c) showed this result for an activation energy of 30 kcal/moleand reaction order α = β = 1.0. Similar unsteady computations have been carried out for the other fourone-step models summarized in table 1.
Figure 7 compares the evolution in time of the amplitude of pressure fluctuations, |p′0|, at x = 0 fora premixed flame at a strain rate of 220 s−1, employing a one-step mechanism and a)varying activationenergy with α = β = 1.0, and b) varying the reaction order with Ea = 30 kcal/mole. In all cases, the initialamplitude of the pressure fluctuations is approximately the same. However, the predicted final amplitudesafter 10 ms are considerably different. The slope of each curve can be considered a measure of the growth
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Figure 7. Comparison of the evolution in time of the amplitude of pressure fluctuations, |p′0|, at x = 0 for
premixed counterflow flames at a strain rate of 220 s−1, employing one-step mechanisms with a) α = β = 1.0and varying activation energy, and b) Ea = 30 kcal/mole and varying the reaction order.
rate of the instability and the present results show that this growth rate is strongly dependent on the choiceof activation energy and reaction order.
Considering the model with Ea = 30 kcal/mole and reaction order α = β = 1.0 the reference one-stepmodel, the growth rate is decreased with smaller values of activation energies or with smaller values of thereaction order. On the other hand, an activation energy of 45 kcal/mole or a value of β of 2.0 result ina much stronger thermo-acoustic coupling. The final amplitudes of |p′0| differ over a range of several orderof magnitudes. For the same inflow conditions, only the one-step model with Ea = 15 kcal/mole andα = β = 1.0 is able to predict pressure fluctuations of the same order of magnitude of the ones obtainedwith the detailed model.
The present analysis suggests that chemical kinetic parameters are important in predicting thermo-acoustic phenomena. In particular, it points out that, in a detailed model, elementary reactions with largeactivation energies can be sensitive to acoustically induced pressure or velocity fluctuations. In addition,the results suggest that the overall activation energy and reaction order of the detailed model seem to beactually lower than the values selected in the reference one-step model.
IV. Conclusion
A detailed numerical model was developed to investigate the occurrence of self-sustained resonant inter-actions of longitudinal acoustic waves with planar counterflow flames. The counterflow flame configurationwas selected, being a scaled-down geometry that is computable with accurate resolution of the complexmulti-scale combustion phenomena involved, including detailed transport properties, finite-rate chemistryand compressibility effects. The fully unsteady, compressible, quasi one-dimensional formulation of thechemically reacting flow field was presented. The governing equations were integrated numerically basedon a MacCormack predictor-corrector scheme with the implementation of characteristics-based boundaryconditions imposed at the nozzle exits. The investigation was conducted with both a detailed model and aset of global one-step chemical kinetic models, where activation energies ranging from 15 to 45 kcal/mole,and reaction orders ranging from 1
2to 2 were selected.
Under certain conditions, coupling mechanisms between acoustics and flame dynamics can lead to theself-sustained amplification of the pressure fluctuations, with the establishment of resonant pressure modes inthe computational domain. In the counterflow configuration, the dominant acoustic mode shape correspondsto a half-wave mode, consistent with the open-tube pressure boundary conditions.
Regardless of the chemical kinetic model employed, the acoustic response of premixed counterflow flamesshowed a positive growth rate of the pressure fluctuations for all cases. In particular, unlike their non-
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premixed counterpart, premixed counterflow flames modeled with detailed chemical kinetics showed thepotential for flame-acoustics coupling. The growth rate was observed to be much larger when a one-stepmodel was employed instead of the detailed model, and as the strain rate was increased. For the one-stepmodel, the sensitivity of chemical kinetic parameters on the acoustic response was investigated by varyingactivation energy and reaction order with a constraint on the laminar burning speed, set to 31 cm/s. Thegrowth rate was shown to be strongly dependent on these parameters. A detailed analysis of the flamestructure and of the time scales needs to be performed to better understand these interesting results.
Acknowledgments
The authors would like to thank the University of Virginia and the Virginia Space Grant Consortium fortheir support.
References
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3Lieuwen, T. and Zinn, B., “The Role of Equivalence Ratio Oscillations in Driving Combustion Instabilities in Low NOx
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