NASA/TM--1998-208488 AIAA-98-3764

A Simplified Model for the Investigation of

Acoustically Driven Combustion Instabilities

D. Dane Quinn

The University of Akron, Akron, Ohio

Daniel E. Paxson

Lewis Research Center, Cleveland, Ohio

Prepared for the

34th Joint Propulsion Conference

cosponsored by AIAA, ASME, SAE, and ASEE

Cleveland, Ohio, July 12-15, 1998

National Aeronautics and

Space Administration

Lewis Research Center

November 1998

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AIAA-98-3764

A SIMPLIFIED MODEL FOR THE INVESTIGATION OF ACOUSTICALLY DRIVEN

COMBUSTION INSTABILITIES

D. Dane Quinn *

Dept. of Mechanical Engineering

The University of Akron

Akron, OH 44325-3903

Daniel E. Paxson t

NASA Lewis Research Center

MS 77-1

Cleveland, OH 44135

ABSTRACT

A simplified one-dimensional model of reactive flow is

pre_ented which captures features of aeropropulsion systems, including acoustically driven combustion instabilities. Although the resulting partial differential equations are one-dimensional, they qualitatively describe

observed phenomena, including resonant frequencies and

the admission of both steady and unsteady behavior.

A number of simulations are shown which exhibit both

steady and unsteady behavior, including flame migration

and thermo-acoustic instabilities. Finally, we present examples of unsteady flow resulting from fuel modulation.

NOMENCLATURE

Sc = & viscosity

D mass diffusion coefficient'

Le = (_/cp) thermal diffusion coefficient

D mass diffusion coefficient '

= (Q/c_)vo heat-release, To

8 = _ activation energy, RTo

# = __a = _ acoustic time scale

co to characteristic time scale'

6 = v_..p_.._ = _ characteristic time scale

pozo_o td diffusive time scale '

A K= characteristic time scale

= _o--_e ---_tr reactive time scale '

7 = _ specific-heat ratio,

*Summer Faculty Fellow at NASA Lewis Research Center,

quinn@ uakron.edu

*Member AIAA, dpaxson@lerc.nasa.gov

INTRODUCTION

The current drive toward lean operating, low-emission,

low-loss combnstors in next generation aeropropulsion

systems makes them susceptible to various instabilities

and makes the understanding of those instabilities critical. Combustion instabilities, specifically acoustically

driven instabilities, often arise from the interaction between the energy released by the chemical reactions and

the underlying fluid flow. This coupling can dramatically

alter the dynamical behavior of the combustion process,

and result in unstable behavior. Typical combustion instabilities in propulsion systems can often be chaxacterized by oscillatory pressure fluctuations coupled with an

unsteady heat release. This instability can lead to flame

blowout, reduced performance, and decreased engine life.
Althoughthemagnitudeof the pressure oscillations

can be substantial, the total energy associated with the

unsteady behavior is typically only a small fraction of

the total energy generated by the combustion process.

As a result, the pressure fluctuations can often be identiffed with the acoustic modes of the enclosed volume.

The acoustic behavior of the propulsion system induces

an oscillatory release of heat, which in turn drives the

acoustic modes, or vice versa.

The fundamental equations of reactive gas dynamics can be described as a set of reaction-diifusion equations governing the energy release associated with the

combustion and transport of the chemical species, coupled to the Navier-Stokes equations for the underlying

fluid flow, subject to an equation of state. Because of the

complexity of both the fluid and transport equations, as

well as the coupling mechanisms, the ability to model

and accurately predict the dynamical behavior and performance of propulsion systems is often limited. As a

result, the simulation and characterization of unstable

combustion behavior is critical to the development of

future propulsion systems and is the subject of intense

interest [1].

A common approach to modeling thermo-acoustic

combustion instabilities is to simplify the governing

equations to those of one-dimensional flow [1, 2]. The

use of a constitutive model with a sensitive time-lag formulation is then employed to couple the unsteady heat

release with the underlying fluid flow [3]. As a result,

the time-lag, or n-v formulation, eliminates the transport equations which govern the chemical kinetics, and

reduces the complexity of the resulting model. Although

this reduction presents a simplification of the model, it

has met with great success in characterizing these instabilities. Despite this success, we study these instabilities

with the inclusion of simplified chemical kinetics. Our

goal is to obtain fundamental insight into the behavior

of these systems, in particular the coupling between the

unsteady heat release and the acoustic response.

This work presents a numerical investigation of

both the convection and transport equations governing

one-dimensional reactive flow. These equations represent a simplified model for the flow in aeropropulsion

combustors, and hence cannot capture such phenomena as flow oscillations driven by shear layer instabilities or the liquid-fuel vaporization process. However,

they nonetheless exhibit self-excited thermc_acoustic instabilities and can provide a tool for studying the qualitative behavior of such systems, including the sensitivity

of the system to parametric variations in the convection

and transport parameters. In particular, we study the

dynamics of this system with an Arrhenius model for the

chemical kinetics and a description of the flame holder.

The simulation will be described, including the governing

equations, justification for the simplifications, and integration techniques. The results will be presented and

contrasted with respect to the development of instabilities.

GOVERNING EQUATIONS

We consider a premixed gas flowing in a finite length

combustor with constant cross sectional area, which undergoes a single-step irreversible chemical reaction. In

this formulation, we assume that the mixture contains

a single deficient component, the amount of which controls the reaction. In addition, this reactive component

is assumed ._o exist in an inert bath, in which the total

concentration of the reactive component is small--as the

combustion process occurs, the overall gas parameters of

the mixture remain unchanged.

In a single spatial dimension, the basic nondimensional equations of motion governing reactive gas dynamics are:

V

with:

v, + _'x = s, (1)

P

pu

pE

PY

F

p_

puE + L_pu

puy

0

s= 0 + Le(6Tx)_+ ](__ Z)_ Sc6_

-m + (_y_)_

E represents the total energy and is defined as:

E= T +p2('Y_l)u_

-y

Time is nondimensionalized by the acoustic time scale,

so that p = 1.00, and z(.) = Oz/i)(.). These equations

are completed by the ideal gas law, p = pT.

We assume that there is no external heat transfer,

although these equations will be augmented below to

include the effect of a flame holder. These equations are

written in conservation form, with p(x, t) and u(x, t) the

total density and mass weighted velocity, respectively, of

the gas. The temperature is described by T(x, t), p(x, t)

is the press.ire, and y(x, t) represents the mass fraction

of the react mt in the combustible gas.

The scurce terms within S contain the thermodynamic effec s of the reactive flow, as well contributions

from dilrmsional terms. Q represents the rate of heat

release by _he chemical reaction, while zh is the mass

conversion -ate of the unreacted fuel. These equations

have been used by a number of authors to study flame

propagatior= through high activation energy asymptotics

[4, 5, 6, 7, _] and the behavior of premixed combustion

in wave rot._rs [9].

The r ondimensionalization has been performed

with respect to the unreacted gas and the dimensional

combnstor length I. At the inlet, x -- 0, we impose

y(0, t) = 1. The boundary conditions on the pressure

and tempex ature are dependent on the geometry of the

combustion chamber. In addition, physically we expect

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y(x,t) --+0 as x --¢I.It should be noted that the diffusive time scale isnot necessarilycontrolledby the fluid

properties but may well be controlledby the flowfield

(e.g.through turbulent diffusion).

Heat-release

We describe the instantaneous rate of heat release,Q,

associated with the combustion process as a one-step,

irreversiblereaction,where (_isproportional to the mass

conversion rate of the unreacted fuel. In addition, we

assume that Th is proportional to the mass of unreacted

fuel,with a temperature dependent reaction rate:

=

where f_is the nondimensional heat release parameter

and rh isthe mass conversion rate of the unreacted fuel.

p and y are the density of the gas and the mass fraction

of the premixed fuel,respectively,so that the quantity

py physicallyrepresentsthe mass per unit volume of fuel

available to the combustion process. A is a parameter

associated with the rate of mass conversion and k(T) is

the temperature-dependent component of the reaction

rate.

With this choice for k(T), the ratio between the maximum rate of reaction at T = Tl and the initialrate of

reaction at T = Ti isapproximately:

Flame Holder

In the absence of a flame holder,a stable flame requires

a delicatebalance between the upstream diffusionof the

flame and the downstream convection of the fluid. As

a result,for most allgas parameters, this balance will

not occur and the flame willeither propagate upstream

to the inlet (flashback),or propagate downstream and

out of the combustor, a condition known as blow-out.

The flame holder acts to limit the propagation of the

flame upstream. We note that flow effects,such as a

spatiallyvarying flow velocitydue to cross-sectionalarea

variations,can also serve to anchor the flame.

We assume that the effectsof the flame holder are

twofold: (i)the diffusionconstant 5 becomes spatially

dependent, resultingfrom increased turbulent diffusivity

downstream ofthe flame holder,and (ii)the flame holder

transfers heat to and from the fluid. We incorporate

both effectsin this model.

Mixture thermodynamics. The thermodynamic properties of the premixed gas are contained within the parameters f_and A. Ifwe subsequently consider varying

mixtures of oxidant and reactant, both parameters are

limited above by some maximum values, say flmax and

Ar_a.x.

The flame temperature resultingfrom the combustion process scaleswith f_,the nondimensional heat release,so that the temperature at the exit is approximately T1 = T(1, t) = To +_, where To = T(0,t).

As such, for a given reactant, the maximum v_ue of

roughly corresponds to stoichiometric conditions, so

as f_ approaches f_ma_, the equivalence ratio of the mixture approaches unity. Also, A, the DamkShler number,

scales the rate of the chemical reaction. These quantities

parallel the equivalence ratio and the heat content of the

premixed fuel.

Chemical kinetics. In this simulation we model the

chemical reaction by a single-step,first-orderArrhenius

model, so that the temperature dependent reaction rate

is:

k(T)Arrhenlus_" e 0(1-1/T),

with 0 the activation energy [4]. In addition, we also

incorporate an ignition temperature Ti, below which no

combustion occurs. As a result, k(T) takes the form:

f O, T < T_,

k(T)

e aO-1/T), T >_ Ti.

Diffusional effects. We assume that the flame holder

acts to increase the diffusiondue to induced turbulent

mixing. This implies that the scaleof the turbulent eddies is largerthan the width of the laminar flame zone.

As a result,we model the spatialdistributionof 5(x) as:

5(x)=_1 [1 (1-- __.o)(tanh(20(x-a))q-1)+ _o].

In the limits x -+ +co the diffusionalcoefficientapproaches 5o and 61 respectively.

Thermodynamic effects. The heat transfereffectsare assumed to be spatiallydependent as well,of the form:

f_he=_.J(T(=,t), X) =

loexp [(I - (50(x -a))')] x

(TI- T(x,t)) ,

where g(x) is the local convection transfer coefficient,

and T I is the nominal temperature for the flame holder

[10].Thus g(x) describes the spatialdistributionof the

convective effectsof the flame holder. We augment the

energy equation, by f(T, x). In addition to the increased

diffusionaleffectsdiscussed above, the flame holder acts

to limitthe propagation of thermal energy upstream of

the flame. This form for the heat transfer coefficient

provides an analyticapproximation which islocalizedin

the neighborhood of x = a.

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NUMERICAL SIMULATIONS

With prescribed initial conditions, the system governing

reactive gas dynamics in the form of Eq. (1) is simulated using MacCormack's method [11],which issecondorder accurate in both space and time. The flow is

solved in the interval0 _< x _< 1 for t _> 0, subject to

characteristic-basedboundary conditions [12],and the

_ional terms are incorporated as a source using a

second order spatialdifferencingscheme. To verifythe

qualitativeaccuracy of our integrationscheme we have

varied both the number of grid points,as well as the size

of the time step.

Unless noted, we assume _f= 1.40 and choose the

flow parameters:

Sc = 1.00, Le = 1.00, # = 1.00,

3 = 1.50, 0 = 2.00, T_ = 1.00.

The flame holder located to be at 25% of the combustor

length (a = 0.25),with:

60 = 0.025.

fo = 0.1, T I = 1.00, ¢_1----0.01, 6-_

In the following, we present several simulations to

illustrate the dynamical characteristics of the model, ineluding both steady and unsteady behavior. In addition,

we consider both choked and open inlet conditions, as

well as a number of different parametric conditions. In

the figures, darker shades indicate lower amplitudes.

Choked-open Boundary Conditions

The inletconditions are choked and isothermal,so that:

p(O, t)u(O, t) = 0.20, T(O, t) = 0.75,

while the exit is open:

p(1, t) = 1.00.

The remaining boundary conditions are determined from

the interior of the domain of integration x E [0, 1]. In

Figure 1, we show the steady response for A = 2.50. We

note that this response is stable.

Flame Mil_ration

By starting the flame away from its equilibrium position, we can also observe the transient behavior of the

flame. As seen in Figure 2, combustion initially occurs

at x = 0.40, well away from the flame holder located

at x = 0.20. The simulation parameters are chosen as

above, with A = 1.50. As time evolves, the flame moves

upstream, eventually stabilizing on the flame holder.

Open-open Boundary Conditions

We now apply open boundary conditions and assume the

flow is isothermal at the inlet, so that at x = 0:

p(0, t) = po = 1.05, T(0, t) = 0.90,

while the exit is open:

p(1, t) = 1.00,

which impli_ a 5.0% pressure drop across the combustor length. The remaining boundary conditions are determined from the interiorof the domain of integration

x E [0,1].

Unsteady Response

For A = 1.501 the asymptotic solution is unsteady, and

the fundamental mode is excited (see Figure 3). In Figure 3a we show the time trace of the pressure at x = 0.25.

As time increases the amplitude of the pressure oscillations initially grow before limiting around t = 35. For

comparison with Figure 5, in Figures 3c,d we show the

steady solution, which for this value of A is unstable.

In Figure 4 we show the time-varying components

of the flow, denoted by (-)', including the unsteady rate

of heat release Q'(x, t). We define the local Rayleigh

index _(x,t) = Q'(x,t). lJ(X,t),and further define the

Rayleigh index as the time average of this local index

over one oscillation:

G(x) =- _ 7_(x, t)dt,

where T is the period of oscillation.7_(x,t) represents

the work done to the fluidby the chemical heat releaseat

time t and spatiallocation x. As time evolves the local

Rayleigh index, and hence the work done,takes on both

positiveand negative values. However, G(x) represents

the average work done over one cycle. The well-known

Rayleigh's criterionstates that if G(x) > 0 energy is

added to the system and the amplitude of the pressure

oscillation is expected to increase, while for G(x) < 0

the flame-acoustic interaction removes energy from the

system. As expected, examination of Figure 4 indicates

that over most of the flow field the unsteady heat release

is in-phase with the oscillating pressure component--the

Rayleigh index is positive and we indeed find an unsteady response.

Parametric Variations

Darling, et al. experimentally studied the acoustic respouse in se_,eralcombustor configurations[13].In their

work, itwa:_ noted that " ... the resonances were very

sensitiveto :hanges in the mean flow conditions,and the

effectsof thesechanges were not always consistent."For

example, at low inlettemperatures the amplitude of a

resonant acoustic mode was increased as the equivalence

ratio was increased. However, for larger inlettemperaturesthe reonant peak was decreased as the equivalence

ratio was in :reased.

Although, as mentioned above, we do not attempt

to make dilect comparisons between our fuel parameters (_,A) .rod the equivalence ratioand heat capacity,

we do note that as we change _ and A this model possesses similarqualitativebehavior. For both A = 0.75

and A = 400 the asymptotic response, shown in Figure 5, is st_'ady,while, as seen in Figures 3 and 4, an

unsteady rtsponse exists for A = 1.50. This type of

behavior is _mmon--there isan intermediate range of

A for whic] the steady-state solution is unstable. For

4

American Institute of Aeronautics and, kstronautics
1

°o°... ..........................

0-

- , , . I • . , I , , , I , , . I , , ,

0 0.2 0.4 0.6 0.8

X

.o.. .......................

.-

s

0

• , , • I , , , I • , , I , • , I , • , -

0 0.2 0.4 0.6 0'.8

X

a) O_(=),_,,_(x)) ,,s. b) (Teq(x),ycq(x)) vs. x

FIGURE I: Choked-open boundary conditions--steady response (A = 2.50). The flow velocity and temperature are

shown dashed in a) and b) respectively, while the pressure and species fraction are solid curves.

6 2

08

1.5

0.5

0 0

0 0.2 04 06 0.8 1 0 0.2 0.4 06 0.8 1

X X

08

a) b)

FIGURE 2: Flame migration and stabilization with choked-open boundary conditions. Figure b) is an expanded view

for 0 _<t < 2 which emphasizes the migration of the flame, initiallylocated at x = 0.4, upstream to the flame holder

at x =0.2.

5

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1.5

0.5 , . , J , . . I . , . I , , , f , , ,

0 2O 4O 60 80

t

I00

1.5

0.5

I00 100.5 101 101.5 102

t

a) p(0.25, t) vs. t b) (p(O.25, t),T(O.25, t)) vs. t

1.2

1

0.8

0.6

0.4

0.2

0

_'''l'''l'''l'''l''

oo...o.. .....................

, , , i, ,, I ,,,i,,, I,,,

0 0.2 0.4 0.6 0.8

X

2.5

2

1.5

1

0.5

0

' ' I ' ' ' I ' ' ' I' ' ' I '' '

7°°°.°." *'°°" - °-° ............

/

i_ i A a t

0 0.2 0.4 0.6 0.8

X

c) (p,,(x),_,(x))vs. x d) (TeqCx),y.q(x)) vs. x

FIGURE 3: Open-open boundary conditions ((_1 -- 0.02). For A _-- 1.25 tae response is unsteady. Figure a) presents

the growth of the pressure response as time increases, eventually approaching a periodic oscillation. In b) the steadystate oscillation in pressure and temperature at x = 0.25 are shown as the solid and dotted line respectively, while

in c) and d) we show the time-independent profile which is unstable. T_e flow velocity and temperature are shown

dashed in c) and d) respectively, while the pressure and species fraction _re solid curves.

6

American Institute of Aeronautics and Astronautics
10

00l_

lO

6

4

0 0

0 0.2 0.4 0.6 0.8 I 0 0.2 04 06 0.8 I

X X

-00075

a) p'(_,t) b) T'(=,t)

10 I0

-01 -0.02.

o o

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

X X

c)u'(=,t) d)O'(x,t)

FIGURE 4: Open-open boundary conditions (61 _---0.02). Unsteady components of the response shown in Figure 3.

The time interval shown is while the amplitudes of the oscillations are small As seen previously in Figure 3, large

amplitude oscillations eventually develop.

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A su_ciently small or large, the steady-state is stable.

Reflecting back on our discussion between the relationship between /3 and _, and the equivalence ratio, this

implies that for near-stoichiometric mixtures, our model

leads to an unsteady asymptotic response. As A --_ 0

we numerically encounter flame blowout, while large A

admits the possibility of flashback.

Finally, we note that for 8 = 2.0 the ratio between

the final and initial burning rate is approximately 3.2.

If we decrease the activation energy to 8 -- 1.0, so that

ratio between the final and initial burning rate is approximately 1.8 the response is stable for all values of

A.

Fuel Flow Modulation

Looking forward to possible attempts to implement control strategies based on fuel modulation, we study the

effects of small amplitude oscillations in the fuel species

parameter y at the inlet:

y(0, t) = 1.0 + 0.01 sin(21rwt).

In Figure 6a, we show the amplitude of the pressure

oscillation at z = 0.25 as the modulation frequency w

varies from w = 0.25 to w = 1.00. When the fuel species

is modulated near the fundamental acoustic frequency

at w ,,, 0.6, modulation amplitudes of 1% lead to large

amplitude acoustic oscillations. Figure 6b presents the

growth of the pressure oscillation for w = 0.6. We note

that in these simulations the flame zone is located at approximately 20% of the combustor length. As a result,

diffusional effects reduce the amplitude of the fuel modulation at the flame zone when compared to the inlet

fluctuations. Thus we expect that if the fuel was modulated slightly upstream of the flame zone, rather than at

the inlet, the resulting acoustic response would increase

further.

SUMMARY

This work focuses on numerical simulations of reactive

flow which, while retaining the simplification of onedimensional flow, contains equations for species transport as well as fluid convection and diffusion. As evidenced by the representative figures shown, it is possible

to study thermo-acoustic instabilities, flame evolution,

and the behavior resulting from parametric variations.

In addition, the model for the chemical kinetics contains

two effects--Arrhenius temperature dependence and an

extinction temperature. We also investigate the dynamical behavior with the inclusion of both thermodynamic

and diffaasional effects from the flame holder.

Examination of the unsteady temperature distribution indicates two distinct regions along the width of the

combustor--i) the flame zone, in which large temperature variations occur due to the chemical reactions, and

ii) remainder of the combustor, in which the temperature

variations are, in fact, almost isentropic.

Although the model is too simplified to obtain

quantitative comparisons with the phenomena observed

in experimental tests, it provides an ideal framework to

study the qcalitative behavior of thermo-acoustic coupling and cyst be used as a basis for the investigation of

the control o.¢ these oscillations and the implementation

of various control strategies, including modulation of the

fuel flow.

[1]

REFERENCES

OYEDIRAN, A., DARLING, D., AND RADHAKRISHNAN, K. Review of combustion-acoustic instabiliti_. Technical Memorandum NASA TM107020, AIAA-95-2469, NASA, August 1995. 31st

AIAA/ASME/SAE/ASEE Joint Propulsion Conference.

[2] MCMANUS, K. R., POINSAT, T., AND CANDEL,

S.M. A review of active control of combustion

instabilities. Progress in Energy and Combustion

Science, 19:1-29, 1993.

[3] CROCCO, L. AND CHENG, S.-I. Theory of Combustion Ir_tability in Liquid Propellant Rocket Motors.

Butterworths Scientific Publications, London, 1956.

[4]

Is]

BUCKMASTER, J. D. AND LUDFORD, G. S. S. Theory of Laminar Flames. Cambridge University

Press, Cambridge, 1982.

CLAVIN, P., PELC_, P., AND HE, L. Onedimensional vibratory instability of planar flames

propagating in tubes. Journal of Fluid Mechanics,

216:299-322, 1990.

[6] LEDDER, G. AND KAPILA, A. K. The response of

premixed flames to pressure perturbations. Combustion Science and Technology, 76:21-44, 1991.

[7] MATALON, M. AND MATKOWSKY, B. J. Flames as

gasdynamic discontinuities. Journal of Fluid Mechanics, 124:239-259, 1982.

[8] MCINTOSH, A. C. The effect of upstream acoustic

forcing and feedback on the stability and resonance

behavior of anchored flames. Combustion Science

and Te_.hnology, 49:143-167, 1986.

[9] NALIM, M. R. AND PAXSON, D. E. A numerical

investigation of premixed combustion in wave rotors. _ SAfe Journal of Engineering for Gas Tarbines a_d Power, 119(3):668-675, 1997.

[10] ENTEZ.,M, B., VAN MOORHEM, W. K., AND MAJDALAb I, J. Modeling of a Rijke-tube pulse combustor asing computational fluiddynamics. AIAA

Paper AIAA 97-2718, AIAA, July 1997. 33rd

AIAA/ASME/SAE/ASEE Joint Propulsion Conference.

[11] ANDERSON, D. A., TANNEHILL, J. C., AND

PLETCHER, R. H. Computational Fluid Mechanics

and H_at 7_'ansfer. McGraw-Hill, New York, 1984.

8

American Institute of Aeronautics and Astronautics
' ' ' I ' ' ' I ' ' ' I ' ' ' I ' ' '

o _

,s

i

\ .

°. .......

, , , I , i , I , , , I , , , I , , ,

0.2 0.4 0.6 0.8

x

3 , , ,

2 '*

)

t

0

, = , I , , , I , , • I • • . I . . •

0 0.2 0.4 0.6 0.8

x

a) A = 0.75 b) A = 4.00

FIGURE 5: Open-open boundary conditions--for A = 0.75 and A = 4.00 the time-independent response is stable,

shown in a) and b) respectively. As A increases the length of the flaxne zone decreases. In each figure the states axe

denoted by: pressure __...._; temperature ..... ; velocity _ _ _; species ....

12

1.15

_. li

1,05

l

I

l

"llii" ,.,,,,,,,IHnlllll II-,,,,.............

I

i , I i i i I , i i I , , i

0.4 0.6 0,8

1

0 ,50 1O0 150 200 250

t

a) St(0,t) = 1.0+ 0.01 sin(2rrcat) b) p(0.25, t); a) = 0.60

FIGURE 5: Acoustic oscillations resulting from fuel flow modulation (0 = 1.50, x = 0.25). In a) the amplitude of

the response is plotted versus ca the forcing frequency, while in b) the pressure is shown for w = 0.6--the initial

conditions in b) coincide with the steady response for no fuel modulation.

9

American Instituteof Aeronautics and Astronautics
[12] HIRSH, C. Numerical Computation of Internal and

External Flows, volume 2. Wiley-Interscience, New

York, 1988.

[13] DARLING, D., RADHAKRISHNAN, K., AND OYEDIRAN, A. Combustion noise at elevated pressures

in a liquid-fueled premixed combustor. Technical

Memorandum NASA TM-107481, ASME-97-GT308, NASA, June 1997. Turbo-Expo 97.

10

American Instituteof Aeronautics and A_tronautics

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November 1998 Technical Memorandum

4. _IILE AND SUBTITLE 5. FUNDING NUMBERS

A Simplified Model for the Investigation of Acoustically

Driven Combustion Instabilities

6. AUTHOR(S)

D. Dane Quinn and Daniel E. Paxson

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135-3191

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

WU-523-26-13--00

8. PERFORMING ORGANIZATION

REPORT NUMBER

E-11267

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASATM--1998-208488

AIAA-98-3764

11. SUPPLEMENTARY NOTES

Prepared for the 34th Joint Propulsion Conference cosponsored by AIAA, ASME, SAE, and ASEE, Cleveland, Ohio,

July 12-15, 1998. D. Dane Quinn, The University of Akron, Departmenl of Mechanical Engineering, Akron, Ohio

44325-3903 and Summer Faculty Fellow at NASA Lewis Research Cen:er; Daniel E. Paxson, NASA Lewis Research

Center. Responsible person, Daniel E. Paxson, organization code 5530, (216) 433-8334.

12a. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified - Unlimited

Subject Category: 07 Distribution: Nons:andard

This publication is available from the NASA Center for AeroSpace Information, (301) 621-0390.

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

A simplified one-dimensional model of reactive flow is presented whict_ captures features of aeropropulsion systems,

including acoustically driven combustion instabilities. Although the resulting partial differential equations are one

dimensional, they qualitatively describe observed phenomena, including resonant frequencies and the admission of both

steady and unsteady behavior. A number of simulations are shown which exhibit both steady and unsteady behavior,

including flame migration and thermo acoustic instabilities. Finally, we present examples of unsteady flow resulting

from fuel modulation.

14. SUBJECT TERMS

Combustion; Computational

17. SECURITY CLASSIFICATION

OF REPORT

Unclassified

18. SECURITY CLASSIFICATION

OF THIS PAGE

Unclassified

NSN 7540-01-280-5500

19. SECURIT_ CLASSIFICATION

OF ABSTRACT

Unclassified

15. NUMBER OF PAGES

16

16. PRICE CODE

A03

20. LIMITATION OF ABSTRACT

Standard Form 298 (Rev. 2-89)

Prescribed by ANSI Std. Z39-18

298-102