FLAME-ACOUSTIC INTERACTIONS IN A COUNTERFLOW FIELD

A.C. Zambon† and H.K. Chelliah,§

Department of Mechanical and AerospaceEngineeringUniversity Virginia

Charlottesville, VA 22904

Abstract

The formulation of counterflow, quasi one-dimensional, unsteady, compressible conservationequations are presented, together with the decom-position of pressure into mean pressure along theaxis of symmetry, unsteady or acoustic (in axialdirection) and radial components. The discretizedequations are integrated numerically using a Mac-Cormack predictor-corrector scheme. The Navier-Stokes characteristic boundary conditions are usedto accurately represent the perfectly reflecting andpartially reflecting boundary conditions. For well-resolved simulations, the occurrence of self-excitedflame-acoustic instabilities is analyzed for a rangeof flow strain rates and two finite-rate kinetic mod-els. It is shown that for identical flow strain rates,the one-step global model promotes growth of theunsteady pressure, while the detailed kinetic modeldoes not. Detailed analyses of the characteristic timescales are presented in an attempt to better under-stand the exact coupling mechanism.

Nomenclature

a = strain ratec = acoustic velocity

ht = total enthalpy

† Graduate Student§ Associate Professor, Senior Member AIAA

J = radial pressure eigenvaluekj = specific reaction rate constant of reaction j

L() = amplitude of characteristic wavesl = nozzle separation distance

Ma = Mach numberp = pressure

p̄0 = mean pressurep′0 = unsteady or acoustic pressure

r = radial distanceT = temperaturet = time

U = gradient of u in r-direction at r = 0u = velocity in r-directionv = velocity in x-direction

Vk = diffusion velocity of kth speciesWk = molecular weight of kth speciesYk = mass fraction of kth speciesx = axial distance

ω̇k = molar production rate of kth speciesλ = thermal conductivity of gas mixtureµ = viscosity of gas mixtureρ = density of gas mixture

Subscripts−∞ = conditions in the oxidizer stream∞ = conditions in the fuel stream

Introduction

Flame dynamic modelling plays a vital role in the de-velopment of model-based active control strategiesto suppress combustion instabilities. While flamedynamic modelling of real combustors has been pur-sued by many investigators,1 computational require-ments impose severe limitations on such efforts for

1

42nd AIAA Aerospace Sciences Meeting and Exhibit5 - 8 January 2004, Reno, Nevada

AIAA 2004-459

Copyright © 2004 by Zambon and Chelliah. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

a fore-seeable future. Recognizing these limitation,flame modelling efforts have focused on simplifiedreacting flow geometries, eg. tube combustors2 anddump combustors.3 Both these configurations offersome features common to real combustors; how-ever, the combustion processes are still quite com-plex and significant uncertainties exist in the flame-acoustic modelling. An alternate flow configura-tion that offers considerable simplifications in ana-lyzing coupled acoustic wave propagation with flamechemistry, species and energy transport is exploredhere. The flow configuration chosen is the wellknown counterflow field, with two opposed streamsof non-premixed methane and air. The major dis-tinction of the present formulation from previous un-steady counterflow investigations is the inclusion offlow compressibility effects.4−6 Beside non-premixedmethane-air flames, other fuel and oxidizer combina-tions as well as premixed flames can be investigatedin this flow configuration.

Numerical results presented in this paper highlightthe importance of implementing accurate formula-tion at the low Mach number (Ma) limit, as well asthe boundary conditions. In particular, the influenceof fully-reflecting and partially-reflecting numericalboundary conditions on the growth rate of unsteadypressure in the counterflow field is presented. By im-plementing two different finite-rate chemical kineticmodels, the need to implement accurate kinetic mod-els in simulation of flame-acoustic coupling phenom-ena is also highlighted.

Formulation

An illustration of the counterflow field, togetherwith the expected dominant unsteady or acousticpressure mode shape is shown in Fig. 1. Themain advantage of the counterflow field is relatedto the planar flame structure and the associatedself-similar mathematical formulation. It is assumedthat the flame structure remains planar for a widerange of flow conditions, eg. flow strain rates,fuel-air mixtures, etc. For steady counterflowflames, the numerical integration of the discretizedequations is now well established.7 For previousunsteady counterflow flames,4 under incompressible

flow assumption, the flame solutions are obtainedwith external perturbation of the flow velocity atthe boundaries.

x

Fuel

Flame

Stagnation Plane

Mixing layer

Oxidizer

P’

Figure 1: Illustration of the counterflow field ofa non-premixed methane and air flame, with thedominant unsteady pressure mode shape.

While retaining the same self-similar approxima-tion for planar flames, full unsteady effects, includ-ing flow compressibility, are considered here. Suchan analysis requires the introduction of the axial mo-mentum equation and the associated unsteady pres-sure terms. The resulting unsteady formulation ofplanar counterflow flame can be written as

∂ρ

∂t+

∂ρv

∂x+ 2ρU = 0, (1)

∂ρYk

∂t+

∂ρvYk

∂x+

∂ρYkVk

∂x+ 2ρYkU = Wkω̇k, (2)

∂ρv

∂t+

∂ρvv

∂x+

∂p0

∂x+ 2ρvU =

∂

∂x

[43µ(

∂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

∂xv − 4

3µ(

∂v

∂x− U)U. (5)

Here, ht is the total enthalpy per unit volume, givenby

ht(x, t) = h(x, t) +12[v(x, t)]2

= h0f +

∫ T (x,t)

T0

cp(τ) dτ +12v2, (6)

while the equation of state, p0 = ρRT , yields den-sity (or pressure). The associated thermochemicaland transport properties are obtained from SANDIAChemkin and transport codes.8,9

In the self-similar formulation, the pressure termexpanded about the axis of symmetry can be writtenas

p(x, r, t) = p0(x, t) + p2(x, t)r2 + . . . . (7)

The second-order term p2 accounts for the pressuregradient in the radial direction, which directly in-fluences the radial velocity component U(x, t). Inthe steady-state case, for low Mach number flowsthe isobaric assumption ∂p0/∂x ' 0 is usuallyintroduced.10 In the present unsteady formulation,for low Mach numbers p0(x, t) can be written as thesum of the mean pressure and unsteady or acousticpressure, i.e. p0 = p̄0 + p′0(x, t), so that the totalpressure is given by

p(x, r, t) = p̄0 + p′0(x, t) +12J r2, (8)

where J ≡ 2p2 is the pressure eigenvalue(1/r)(∂p/∂r). In the limit Ma → 0, the leadingorder term of Eq. (3) yields p̄0 = constant while thefirst order term yields

∂ρv

∂t+

∂p′0∂x

= 0. (9)

As shown later, Eq. (9) provides an accurate repre-sentation of the axial momentum equation.

In steady-state flames, each value of J yields aunique value for the flow strain-rate defined hereas a = dv/dx outside the mixing layer on theair-side.10 In the present unsteady flame analysiswith J=constant, oscillations of strain-rate canoccur because of the unsteady pressure effects.

Boundary Conditions:For the two opposed nozzles, the pressure nodesare assumed to be located close to the nozzle exits(i.e. open tube boundary conditions). With con-stant temperature and species mass fractions of in-flow streams, the following physical boundary con-ditions are imposed at x = x−∞ = −l/2

p = p−∞, Yk = Yk,−∞, U = 0, T = T−∞, (10)

and at x = x∞ = +l/2

p = p∞, Yk = Yk,∞, U = 0, T = T∞. (11)

The axial velocities at the boundaries are pre-scribed through the implementation of Navier-StokesCharacteristic Boundary Conditions (NSCBC).11

The choice of NSCBC can yield reflective or non-reflective acoustic waves at the boundaries. In theNSCBC approach, there are generally two character-istic waves travelling into the domain through inflowboundary (their amplitudes being L2 and L5), andone characteristic wave travelling from the interiorof the domain to the inlet with amplitude L1, givenby

L1 = (v − c)[∂p0

∂x− ρc

∂v

∂x

], (12)

L2 = 0, (13)L5 = −L1. (14)

The pressure (or equivalently density) and axial ve-locity are updated by integrating the following equa-tions at the boundaries

∂ρ

∂t+

1c2

[L2 +

12(L5 + L1)

]= 0, (15)

∂ρv

∂t+ v

1c2

[L2 +

12(L5 + L1)

]+ ρ

12ρc

[L5 − L1] = 0.

(16)

It is possible to relax the assumption of pressurenodes located at the nozzle exits to allow partialreflections. In this case, the predicted amplitudeof incoming waves can be modified by varying theparameter σ in L5 = σ(p0 − p̄0), where p̄0 is theatmospheric pressure in the far-field.

Numerical Approach:The integration of the governing equations is per-formed using an explicit MacCormack scheme.12 Thesystem of partial differential equations is recast inthe conservative form in terms of the conservedvariables, convective flux, diffusive flux and sourceterm vectors. The method is based on a predictor-corrector approach, where the conserved variablesvector is updated by upwind differencing the convec-tive flux vector in the predictor step, and by down-wind differencing in the subsequent corrector step.At all iterations, the derivative of diffusive flux is al-ways discretized in the direction opposite to that ofthe flux vector derivative. In order to eliminate anydirectional bias, at the next iteration the differencingdirection of all spatial derivatives is reversed.

The initial conditions for the time-dependent gov-erning equations are given by the steady-state coun-terflow flame solution, evaluated at the same meanflow conditions. This initial solution is obtainedfrom a modified version of Smooke’s code9 and re-lies on the isobaric assumption. When employedas input in the unsteady code, the solution adjuststo the full compressible equations described above.The subtle differences between the steady-state for-mulation and the full unsteady formulation (mainlydue to treatment of pressure) leads to a broad fre-quency perturbation. After a brief transient period,depending on the selection of the boundary condi-tions, acoustic waves with the natural frequency ofthe spatial domain is amplified, while the remainingnoise is dissipated.

Because of the need to resolve the flame struc-ture and capture the flame dynamics, a high spatialresolution is required. For a selected grid size of∆x ∼ 10µm − 15µm, based on the speed of sound,the largest time step size is estimated around 10−8s.Even for a very short spatial domain (l = 1 cm), a

uniform grid would result in extremely long compu-tational times. In order to speed up the executionof the unsteady code, a stretched grid is employedwith a high resolution in the chemically reacting re-gion and at the boundaries, while in the rest of thespatial domain the grid is smoothly stretched.12

Results and Discussion

Non-Reacting Case:For a non-reacting counterflow field, the absence ofheat release implies that the source term drivingthe pressure instability is zero. This case servesas a good test for the self-similar formulation andthe numerical approach adopted. Figure 2 showsthe spatial variation of initial velocity components(axial v and radial U), temperature T and unsteadyor acoustic pressure (p′0(x, t) = p0(x, t) − p̄0, withp̄0 ≡ patm). When the full axial momentum equa-tion given by Eq. (3) is adopted, with perfectlyreflecting boundary conditions corresponding toσ = ∞, the predicted acoustic pressure is seen toincrease as shown in Fig. 3. The initial perturbationoscillations seen is due to the starting solution used.In the absence of any source term, the predictedgrowth in Fig. 3 is a consequence of the self-similarformulation for the axial momentum equationadopted.

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−500

0

500

1000

x [ cm ]

v [

cm/s

], U

[ s

−1 ],

p0−

p atm

[ d

ynes

/cm

2 ], T

[ K

]

v

U

p0−p

atm

T

Figure 2: Initial values of velocity components vand U , temperature T , and pressure p′0 used for thenon-reacting case.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10−3

−5000

−4000

−3000

−2000

−1000

0

1000

2000

3000

4000

5000

Time [ s ]

p 0′ (x,

t)

[ dyn

es /

cm2 ]

Figure 3: Unsteady pressure vs. time, for a non-reacting counterflow field with Eq. (3).

For low Mach number flows considered here,appropriate non-dimensionalization of axial momen-tum equation, Eq. (3), yields an alternate equationfor acoustic pressure given by Eq. (9). When Eq.(9) is implemented with perfectly reflecting bound-ary conditions, the predicted acoustic pressureindeed decays for the non-reacting case as shown inFig. 4.

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

−100

−50

0

50

100

Time [ s ]

p0′ (

x, t

)

[ d

yne

s /

cm2 ]

Figure 4: Unsteady pressure vs. time, for the non-reacting counterflow field with Eq. (9).

As stated earlier, the initial unsteady pressureoscillation seen in Fig. 4 is due to the startingsolution used. This initial pressure amplitudecan be modified with an imposed perturbation,typically over several cycles. Irrespective of theinitial pressure amplitude, the rate of decay of theunsteady pressure strictly depends on the boundaryconditions imposed. While results shown in Figs.3 and 4 are based on perfectly reflecting boundaryconditions (i.e. σ = ∞), for other partially reflectingboundary conditions, a rapid decay of the unsteadypressure is obtained as shown in Fig. 5. Resultsin Fig. 5 reiterate the importance of implementingaccurate boundary conditions that may promotethe growth of resonant acoustic energy within thecomputational domain.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10−3

−2000

−1500

−1000

−500

0

500

1000

1500

2000

Time [ s ]

p 0(x, t

) − p

e [ d

ynes

/ cm

2 ]a) σ = 107

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10−3

−1500

−1000

−500

0

500

1000

1500

Time [ s ]

p 0(x, t

) − p

e [ d

ynes

/ cm

2 ]

b) σ = 106

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10−3

−300

−200

−100

0

100

200

300

Time [ s ]

p 0(x, t

) − p

e [ d

ynes

/ cm

2 ]

c) σ = 105

Figure 5: Unsteady pressure vs. time, for the non-reacting case as in Fig. 4 with varying values of σ.

In the following reacting flow simulations, the ax-ial momentum equation given by Eq. (9) and Eqs.(1)-(2), (4)-(5) for mass, species, momentum in r-direction and energy are integrated subject to theperfectly reflecting boundary conditions, i.e. σ = ∞.

Reacting Case:Previous laboratory investigations with tube com-bustors have indicated that under certain condi-tions, flame and acoustics can interact leading tothe growth of the unsteady pressure. These flame-acoustic coupling phenomena are known to occuronly for certain tube lengths, corresponding to acharacteristic acoustic time scale associated with thecombustor. Furthermore, because of losses at theboundaries in experiments, a limit cycle behavior isoften observed.

Analogous to such tube combustor investigations,it is shown here that for specific flow conditions,flame-acoustic coupling can exist in the counterflowfield as well. Because of the small geometry of theflow field, the resonant acoustic frequency is muchhigher, as expected. However, the small geometryallows accurate implementation of the chemical ki-netic models to describe the overall reaction rateand the heat release. Two kinetic models were im-plemented, (i) one-step global model of the formCH4 + O2 → CO2 + 2H2O, with a Arrhenius reac-tion rate constant, and (ii) a detailed kinetic modelthat includes 17 species in 39 reactions.13 The colli-sion frequency and activation energy of the one-stepglobal model was selected such that the flame extinc-tion condition of a methane-air non-premixed flamematched that of the detailed model.

Figure 6 shows the time history of the acous-tic pressure p′0 near the stagnation plane of a non-premixed methane-air flame, predicted using theone-step model. Results are shown for two differ-ent flow strain rates, a low-strain rate of 50 s−1

and a high-strain rate of 416 s−1 (near extinction).At the low strain rate, the predictions indicate avery slow growth rate of acoustic pressure ampli-tude with the perfectly reflecting boundary condi-tions implemented, while for the high strain rate arapid growth of the acoustic pressure is observed.As the flow strain rate is increased, the characteris-tic flame time scales decrease and approach that ofthe acoustic time scale of the geometry, leading tothe rapid growth rate of pressure observed in Fig.6b.

Unlike in previous unsteady counterflow flames,7,8

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01−1000

−500

0

500

1000

Time [ s ]

p0′ (x

, t)

[

dyn

es /

cm

2 ] a) Strain Rate = 50 s−1

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01−1000

−500

0

500

1000

Time [ s ]p

0′ (x

, t)

[

dyn

es /

cm

2 ] b) Strain Rate = 416 s−1

Figure 6: Unsteady pressure vs. time for two differ-ent flow strain rates, using one-step global kineticmodel.

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−1000

−500

0

500

1000

x [ cm ]

p0′ (

x,

t)

[ d

yn

es /

cm

2 ]

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−200

−100

0

100

200

Axia

l V

elo

city

[ c

m /

s ]

p0′ (x, t)

Axial Velocity

Figure 7: Velocity and pressure oscillations savedat every 1000 time steps, for the high-strain resultsshown in Fig. 6.

the above pressure oscillations are self-excited, i.e.no external velocity perturbations are introduced.The large pressure oscillations of ±100 pa leads to

significant velocity oscillations at the boundaries, asseen in Fig. 7. These predictions are for the high-strain case shown in Fig. 6b, at about 10 msec., andsaved at every 1000 time steps. The pressure modeshapes shown indicate the dominant half-wave modeestablished within the counterflow field. A FFTanalysis based on the pressure history is shown inFig. 8, where the highest power corresponds to thedominant half-wave frequency of 19,500 s−1. Otherhigher order modes corresponding to full-wave,3/2-wave, etc. indicate much lower amplitude orpower as shown in Fig. 8.

0 20 40 60 80 100 120 140 160 180 200

10−4

10−3

10−2

10−1

100

101

102

103

Frequency [ kHz ]

Po

we

r S

pre

ctru

m

Figure 8: FFT analysis results of unsteady pressurerecorded at the mid-point for high-strain resultsshown in Fig. 6.

The characteristic time scales that yield flame-acoustic interactions shown in Fig. 6b are presentedin Fig. 9. These time scale comparisons across theflame structure imply that the flame-acoustic cou-pling is most likely due to finite-rate kinetics inter-acting with acoustic waves, rather than the transporteffects coupling with acoustics. A detailed sensitiv-ity analysis is needed to establish this conclusively.

Irrespective of the exact coupling mechanism,the predicted heat release and pressure oscillationsshown in Fig. 10 satisfy the Rayleigh criterion.While the Rayleigh criterion is a useful tool, itdoes not provide a priori information about the

occurrence of flame-acoustic coupling. Only detailedanalysis based on characteristic time scales as shownin Fig. 9 can provide a priori conditions and discernexact coupling phenomena.

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.210

−6

10−5

10−4

10−3

10−2

10−1

x [ cm ]

Ch

ara

cte

rist

ic T

ime

Sca

les

[

s ]

τacoustic

τchem, CH

4

τchem, O

2

τchem, H

2O

τchem, CO

2

τconv

τdiff, CH

4τdiff, O

2τdiff, N

2τdiff, H

2O

τdiff, CO

2

Figure 9: Characteristic convective, diffusive, chem-ical and acoustic tine scales across the flamestructure, for the high-strain case shown in Fig. 6b.

0.01 0.0101 0.0102−4000

−3000

−2000

−1000

0

1000

2000

3000

4000

Time [ s ]

p 0(x, t

) − p

e [ d

ynes

/cm

2 ], 1

03 v′ (

x, t)

[cm

/s],

1011

∫q′ (

x, t)

dx

[erg

/cm

2 /s] v′ (x, t)

∫q′ (x, t) dx

p0(x, t) − p

e

Figure 10: Unsteady pressure and heat release forconditions in Fig. 6b satisfying the Rayleighcriterion.

In contrast to the one-step reaction model usedabove, when a detailed chemical kinetic modelconsisting of 17 species in 39 elementary reactionsis employed, for the same flow conditions thepredicted flame-acoustic coupling is rather weak asshown in Fig. 11. While the chemical time scaleanalysis is complex because of the large numberof species and reactions involved and is currentlyunder investigation, the present results imply thatthe one-step global model may yield erroneousflame-acoustic interactions.

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

−50

0

50

Time [ s ]

p0′ (x

, t)

[ dynes / c

m2 ] a) Strain Rate = 50 s−1

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

−50

0

50

Time [ s ]

p0′ (x

, t)

[ dynes / c

m2 ]

b) Strain Rate = 416 s−1

Figure 11: Unsteady pressure vs. time for two dif-ferent flow strain rates, using a detailed kineticmodel.

Conclusions

The purpose of this work was to investigate the oc-currence of self-sustained flame-acoustic interactionsin the well-defined counterflow flame configuration.A self-similar mathematical formulation and a so-lution algorithm were developed to simulate acous-tic waves propagating back and forth between the

nozzles and their interaction with a laminar non-premixed flame.

For the opposed flow geometry, it was shown thatthe the dominant acoustic mode shape correspondsto a half-wave mode, consistent with the open-tubepressure boundary conditions. The growth rate ofthe unsteady acoustic pressure was shown to be afunction of flow strain-rate, as well as the finite-ratechemical model implemented. In particular, the one-step global kinetic model with an activation energyof 30 kcal/mole indicated a positive growth rate forall strain rates ranging from 50 to 450 s−1. Thedetailed 17 species, 39 elementary reaction modelwith identical boundary conditions, however, showedmuch smaller unsteady pressure growth rates. Infact, the detailed model indicated a decay of thepressure amplitude at the lower strain rate of 50 s−1.The significant differences in flame-acoustic couplingobtained with one-step global model and the detailedmodel highlight the risks of using highly simplifiedkinetic models in theoretical/computational investi-gations.

Acknowledgment

This work was supported by the University of Vir-ginia and Virginia Space Grant Consortium.

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