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NASA Technical Memorandum 83587

A Theoretical Prediction of the AcousticPressure Generated by Turbulence-Flame Front Interactions

Ronald G. HuffLewis Research CenterCleveland, Ohio

Prepared for theWinter Annual Meeting of the American Society of Mechanical EngineersNew Orleans, Louisiana, December 9-13, 1984

NASA

https://ntrs.nasa.gov/search.jsp?R=19840018315 2020-02-06T09:17:26+00:00Z

A THEORETICAL PREDICTION OF THE ACOUSTIC PRESSURE GENERATED BY TURBULENCE-FLAME FRONT INTERACTIONS

Ronald G. HuffNational Aeronautics and Space Administration

Lewis Research CenterCleveland, Ohio 44135

ABSTRACT

The equations of momentum and continuity arecombined and linearized yielding the one dimensionalnonhomogeneous acoustic wave equation. Three termsin the nonhomogeneous equation act as acoustic sourcesand are taken to be forcing functions acting on thehomogeneous wave equation. The three source termsare: fluctuating entropy, turbulence gradients, andturbulence-flame interactions. Each source term isdiscussed. The turbulence-flame interaction sourceis used as the basis for computing the source acousticpressure from the Fourier transformed wave equation.Pressure fluctuations created in turbopump gas gener-ators and turbines may act as a forcing function forturbine and propellent tube vibrations in earth toorbit space propulsion systems and could reduce theirlife expectancy, A preliminary assessment of theacoustic pressure fluctuations in such systems ispresented.

INTRODUCTION

Acoustic pressure fluctuations created in flow-ing systems may be the source of unwanted noise andserve as the unsteady driving force acting on mechan-ical systems associated with the flow. Aircraft typeturbine engine noise has been a problem that has re-ceived attention over the past several years. Highbypass turbofan engines with lower jet velocitieshave resulted in large reductions of aircraft jetnoise. With the reduction of jet noise, other noisesources have emerged presenting additional problemsfor researchers working in the noise reduction field.The combustor is one of the acoustic sources whosenoise characteristics are shown to be discernible inthe farfield under some circumstances U-2J. Aircraftengines operating at reduced power settTngs have beenshown to emit low frequency acoustic waves from thecombustor that are transmitted through the turbine,ducting and nozzle to the farfield (3-4_). Combustornoise may be expected from any turbulent combustionprocess.

Of particular timely interest is the similaritybetween components used in aircraft turbine engine

combustor-turbine configurations and the turbopumppreburner-turbine configurations used to pump propel-lants in earth to orbit propulsion systems (5). Twoof the primary differences are the propellants andmethod of mixing the propellants prior to combustion.In both systems, however, after the fuel is mixedwith the oxidizer, the combustion process proceeds,the resulting gas expands through the turbine andthen flows through the remaining ducting to eitherthe core engine nozzle or to the main rocket enginedome where it passes over liquid oxygen (LOX) tubesin semi-crossflow. For the aircraft engine the pres-sure fluctuations are the source of unwanted sound.For the space propulsion system the pressure fluctu-ations become an unsteady driving force causingvibrations in the LOX tubes that could be destructiveif not accounted for in their design.

The combustor has been the subject of numeroustheoretical and experimental investigations. Ref-erence (6) discusses a few of the reported studies oncombustion noise. The objective of the present paperis to present a simplified theoretical model, andprediction of the acoustic pressure generated in thecombustor. Using this model a better understandingof the sound sources in combustors can be obtainedenabling one to modify and predict their behavior.

All symbols used in this paper are defined inthe symbol list, Appendix A.

COMBUSTION NOISE SOURCE MODEL

The model for combustion noise considers one-dimensional duct flow (Fig. 1). Fuel is sprayed intothe airstream where turbulence generators, termedturbulators, promote mixing of the fuel with the air(oxidizer). A flame front perpendicular to the ductwalls exists downstream of the turbulators at X =0. The combustion process begins at the flame frontand extends downstream to X = Lt, the point whereburning ceases. The region between X = 0 and L^is called the combustion zone. The duct extends toinfinity in the X direction.

THE GOVERNING DIFFERENTIAL EQUATIONS AND ASSUMPTIONS

The second order partial differential equationgoverning the noise generated within an inviscid,adiabatic, perfect gas, in a one-dimensional duct withnegligible fuel mass may be derived using theequations for conservation of massand momentum respectively given here as:

at

3 p Vat

3pV3X

3X 3X

(1)

(2)

ACOUSTIC SOURCE AND WAVE EQUATIONS

The acoustic wave equation describing the soundpropagation in a duct is derived by first taking thepartial derivative of Eq. (1) with respect to time;then taking the partial derivative of Eq. (2) withrespect to X, adding equations;

and then adding (1/CQ ) (32P/3t2) to both sidesof the resulting equation giving:

1 a"P 3 P _ 3 -p 1 3"P B'pV(3)

3X

The assumption is made that the instantaneous pres-sure P, density p and velocity V are the sum ofmean and fluctuating components, that is:

P = P(x) + p(x,t)

P = p(x) + «(x,t)

V = V(x) + v(x,t)

(4)

(5)

(6)

Substituting Eqs. (4) to (6) into Eq. (3), ex-panding and equating like order terms yields the waveequation as:

(7)

3X

To this point in the derivation nothing has beenassumed concerning CQ in the above equation. Inthis work, CQ, is assumed to be the sonic velocity.Equation (7) then becomes the acoustic wave equation.It .is assumed that outside the combustion zone thesonic velocity is constant, though different, in theupstream and downstream portions of the duct. It isfurther assumed that turbulence generated pressurefluctuations anywhere in the duct external to thecombustion zone are less than the acoustic pressuresgenerated by the combustion process and that no ex-ternally generated acoustic waves pass through thecombustion zone.

The acoustic pressure is usually given in termsof finite-bandwidth, mean-square pressure fluctuationsas a function of frequency. This is accomplished bytaking the. Fourier transform of Eq. (7), which removestime as a variable and thus simplifies the method ofsolution. The Fourier transform of Eq. (7) in termsof wave number k is:

kX = - n ..II 9 b)(8)

where 9n u are the Fourier transforms ofthe terms on tine right side of Eq. (7). These termsare the acoustic sources, i.e. forcing functions,driving the acoustic wave equation. The source termsare of great interest since knowledge of their char-acteristics will give insight on how the fluctuatingpressures found in the duct are generated. The pri-mary thrust of this work is to define these sourceterms in detail. Before this can be accomplished,however, the nonhomogeneous equation (Eq. (8)) mustbe solved.

SOLUTION OF NONHOMOGENOUS WAVE EQUATION

By using the methods given in Ref. (8) the com-plete solution of the homogeneous part (i.e., with9 = 0) of Eq. (8) is

fu.,0 l C2 e-ikx

(9)

With the above complete solution known, theGreen's function (9) may be used to write the equationfor the fluctuating pressure in the duct propagatingaway from the combustion zone which constitutes thesolution of the nonhomogeneous wave equation (Eq.(8)). For the left and right running waves respec-tively the pressure is given by:

ikx

-ikx

where

o,R = ~2TiT

- (9

f1J

(10)

(ID

(12)

The Fourier transformed source terms (i.e. Fouriertransform of terms on the right side of Eq. (7)) aredefined as:

»»,!(13a)

or from Appendix B, Eq. (B14) in terms of fluctuatingentropy

(13)

(14)

(15)

The source terms given by Eqs. (13) to (15) rep-resent the acoustic sources due to: fluctuating heatrelease, turbulence, and energy addition (or with-drawal) respectively. Each has been developed inAppendix B and will be discussed in greater detailbelow.

PHYSICS OF NOISE GENERATION PROCESS

The fluctuating entropy source term Eq. (13) maybe written from Appendix B, Eq. (B16) as:

-xx.

pHv fO,o

V(16)

Where the local fuel/air ratio, fg, is allowedto fluctuate either because of unsteady turbulentmixing or because of some externally generated dis-turbance such as reflected acoustic waves passingthrough the fuel/air mixing region upstream of theflame front causing increased mixing. Thus thissource may represent the energy addition due toacoustic feedback. As such one might expect thissource to yield information on instability in com-bustors. In order to evaluate this source more in-formation is needed on the local fuel/air ratio andthe effect that turbulence and acoustic waves have onthe mixing of the fuel with the air (oxidizer).

The second source term (Eq. (14)), describes thesource of pressure fluctuations associated with thefluctuating velocity gradient. Thus the generationof fluctuating velocities by the turbulators or tur-bulent mixing of fuel and oxidizer in rocket enginesis a source of noise in these flowing systems. Thissource term depends only on the mass flow per unitcross sectional area and the second derivative of thefluctuating velocity with respect to distance. (Thesecond derivative with respect to distance is here-inafter called the Laplacian). Hence it is consideredas the cold flow noise source that exists even ifcombustion is absent. This source term has beendeveloped in Appendix B for turbulence generated overa given length, L-r. The turbulence at the end ofthe generator is assumed to have reached its maximumvalue and decay of the turbulence downstream of thegenerator is neglected. The second source term fromAppendix B, Eq. (B20) is:

-X X,

= -2

The third acoustic source term

(17)

me tmra acoustic source term 93 ^tq.(15)) is due to the Laplacian of the product of theturbulent density and the mean velocity. It has beenassumed in this work that the turbulent density fluc-tuations in the combustion zone are constant. Withthis assumption this source term becomes a functionof the Laplacian of the mean velocity, the result ofthe heat energy added by the combustion process.This source is investigated in detail in Ref. (6).The third source term from Appendix, B, Eq. (B2T). isexpressed as:

(18)

ACOUSTIC SOURCE TERM EVALUATION

In this paper no attempt will be made to evaluatethe fluctuating entropy or cold flow acoustic sources*l,u anc' *2 a- The fluctuating entropysource 9^ (0, Inould have low pressure levels ifthe burne? operates stably. The cold flow noise term2̂,00 appears to act as an upstream fluctuatingdensity generator and as such is amplified when pas-sing through the combustion zone. Thus the thirdsource term 93^ is expected to be the dominantone.

To evaluate p^ (Eq. (10)), the source term'3,00 mus* be given as a function of distancex.' It is shown in Ref. (j>) and Appendix B that forMach numbers much less than unity the velocity isdirectly proportional to the heat energy input to theair (Appendix B, Eq. (822)).

By assuming that the fuel droplet mass decreasesexponentially with axial distance, the velocity dis-tribution due to the energy released to the air can bedetermined as a function of x. The second derivativewith respect to x is then substituted into Eq. (15).The Fourier-transformed source term from Appendix B,(Eq. (21)) is

"cHV -xx(19)

Inserting Eq. (19) into Eq. (10), assuming thatthe fluctuating density is constant with distancethrough the combustion zone, and integrating yieldsthe acoustic pressure in polar form as (see derivationin Appendix B, of Ref.(̂ ).

R Wo x ei[kx + (7I/2) * tarrl(-l

TO?~ [1(20)

where 0 < X < L^. Equation (20), with dimensionsof pressure per unit angular frequency, is used forcalculating the overall sound pressure level and thenarrowband acoustic spectrum. The amplitude for theright and left running waves are calculated using Eq.(20) with the appropriate sonic velocity insertedin the wave number, k.

In deriving Eq. (20) the conversion of turbulentenergy to acoustic energy was assumed to be complete.This, in reality, cannot be accomplished as evidencedby the fact that turbulence does exist downstream ofa combustor. To account for this, an acoustic pres-sure efficiency np is introduced and defined asthe ratio of the experimentally measured acousticpressure generated by the combustor p,,, to theacoustic pressure that would be generated if all ofthe turbulent energy had been converted to an acousticpressure P^. From Ref. (6) np is given by

(21)

Thus the acoustic pressure efficiency can now bedetermined for any specific combustor. The value ofrip will probably depend on the combustor type andconfiguration. Therefore it is necessary that npbe determined for a number of combustors so that theproper value can be selected when making predictionsof combustor pressure fluctuations.

OVERALL SOUND PRESSURE LEVEL

An expression for the overall sound pressurelevel can be obtained by multiplying Eq. (20) by theacoustic pressure efficiency np, squaring and thenintegrating over the frequency range of interest.The resulting equation for P^ is

Jref>(22)

where the term y contains all of the frequencydependent term and is defined as

xIT [1

df

xC 2 [l 2, , -l/2«f\l I L

= T~ 7 Tr tan I —p— II I2, |_f *C VC )\ I f

(23)

Where the local static temperature is used incalculating the sonic velocity, C.

The fluctuating pressure in the combustor of theCF6-50 turbofan engine has been measured (10-11).The measured overall fluctuating pressure (assumed tobe acoustic) in the combustor at 3.8 percent of designthrust has been used in conjunction with Eq. (26) tocalculate the acoustic pressure efficiency, np =0.030, for the CF6-50 combustor.

From this acoustic pressure efficiency the over-all sound pressure level in the combustor was pre-dicted over the engine thrust range by using Eq. (25).The results are shown in Fig. 2. The theory, givenby the solid line, agrees well with the measuredfluctuating pressure, given by the symbols.

In light of this discussion it can be concludedthat the theory predicts the trends in the combustoroverall acoustic pressure with engine operating con-ditions and, for CF6-50 combustors and similar geome-tries, an acoustic pressure efficiency of 0.030 canbe used in Eq. (25) to predict the combustor overallacoustic pressure. The acoustic pressure efficiencyhas also been determined for a YF102 engine combustor.This combustor, unlike the CF6-50, is a reverse-flowcombustor and is much smaller than the CF6-50. Itsacoustic pressure efficiency is 0.016 at 30 percentof design speed. Using the acoustic pressure effi-ciency of 0.016 yields only a 0.2-dB error in pre-dicting the OASPL at the 95-percent speed. Comparingthe acoustic pressure efficiencies shows that theCF6-50 combustor converts the turbulence to acousticpressure more efficiently than the YF102 combustor.

NARROW-BAND SOUND PRESSURE LEVEL SPECTRUM

Equation (23) indicates that ^"2 is inverselyproportional to the frequency. Therefore the magni-tude of will be determined by the lower limit onfrequency, providing that the upper limit is muchgreater than the lower limit. In determining theOASPL the combustion noise frequency limits are ofthe order of 50 and 2000 Hz. For a fuel mass decayconstant x, defined as 2ir/L(,, and reasonable sonicvelocities, Eq. (23) can be approximated by using thelower frequency limit by

sec-1 (24)

The theoretical sound pressure spectrum is givenby Eq. (20). Discarding the phase information con-tained in the exponential term and applying Eq. (21)to account for the efficiency of conversion of turb-ulence to acoustic pressure, Eq. (20) is written as

Vc

AL,P bS0)

w Woc1f '

1

(fLb/C)2]1/2 (27)

The right side of Eq. (27) represents the spectralshape of the acoustic pressure given by Eq. (20).

The sound pressure level spectrum is given by

The overall sound pressure level (OASPL) equationcan now be written for the combustor from Eqs. (22)and (24) as

OASPL = 20 log 10VcC

2PrefLb

6 W VnH f.(i) _ 0 v 0p ACpTT,0

(25)

The acoustic pressure efficiency tin canbe determined from the measured overall sound pressurelevel and Eq. (25). The equation for experimentallydetermining the value of np is

nP =2PrefLt, C TL p A VT.O 1f)OASPL/20 ,.,,,~ s~worrr x 10 <2 5)

SPL = 201og10^P-^ « Wo Af

ef b T.O y

(fL/C)2]1/2

(28)

This equation states that the acoustic pressure levelis inversely proportional to the square of the fre-quency since the term under the radical is small forfLb < C.

The predicted spectral shape is compared withthe measured narrow-band spectra for the CF6-50 en-gine (10) in Fig. 3. The measured fluctuating pres-sure level in the combustor of the CF6-50 turbofanengine operating at 3.8 percent of design thrust isshown in Fig. 3(a). Also shown in Fig. 3(a) is aplot of Eq. (28) with the level matched to themeasured spectrum at a frequency of 2000 Hz. Theacoustic pressure efficiency is determined from Eq.(28) and is given by

2_^_^b CpTT,0Pref 10SPL(f)/20 f\ <» w vOHvfOC ^

(29)

At 2000 Hz the measured SPL is equal to 110 dB andthe narrow-band spectral acoustic pressure efficiency,computed by using Eq. (29), is

-2np = 4.5x10

when 6^/p is assumed to be 0.3.The trends in the spectral shape of both theory

and experiment agree very well at the 3.8-percent-thrust point given in Fig. 3(a). The comparison at99.8-percent thrust shown in Fig. 3(b), however, isnot as good. The theory given for the 99.8-percent-thrust point uses the acoustic pressure efficiencyobtained from the 3.8-percent-thrust point, namelynn = 4.5xlO~2. Examining Fig. 3(b) shows thattrie general trend is correct but that large oscilla^tions in the spectrum are not predicted by Eq. (28).Acoustic reflections from the turbine may account forthese oscillations.

From this discussion it is concluded that thespectrum predicted by Eq. (28) agrees substantiallywith the experimental measurements made in an opera-ting engine. It can also be concluded that thesimplified combustion noise theory given herein sub-stantially predicts the trends in the acoustic pres-sures generated by the combustion process of turbineengine combustors.

1/3-OCTAVE-BAND SOUND PRESSURE SPECTRUM

The 1/3-octave-band spectrum of the combustornoise can be calculated for a simplified spectrum byneglecting the term under the radical contained inEq. (28) and integrating over the 1/3 octavebandwidth:

aSPL = 10 log. 0.23 (30)

This equation has been plotted in Fig. 4 with itslevel matched, as for the narrow-band spectrum, atthe 2000-Hz band for the CF6-50 operating at the 3.8-percent-thrust point. The 1/3-octave-band data givenby the symbols match the theoretical trends for fre-quencies above 100 Hz. At frequencies less than100 Hz the data are significantly over predicted bythe theory; a result attributable to the absence oflow frequency turbulence.

EARTH TO ORBIT PROPULSION SYSTEM PRESSURE SPECTRUM

Turbopump systems used in earth to orbit propul-sion systems are subjected to fluctuating pressuresthat could cause a decrease in their life expectancydue to high cycle loading of structures. Equation(28) was used to predict the fluctuating pressures inthe ducting downstream of the hydrogen pump preburnerusing, as a first approximation, the CF6-50 turbofanacoustic pressure efficiency reported herein. Ref-erence (5) gives the geometry of the hydrogen pumppreburner, turbine, and ducting leading to the main

engine dome. Figure 5 compares the predicted pressurespectrum in the preburner to the measured pressuredownstream of the turbine. The turbine will attenuatesome of the preburner generated acoustic signal butit does not decrease the theoretically predictedvalue enough to cause serious disagreement.

CONCLUSIONS

By using the theory developed herein, expressionsfor the acoustic pressure generated by the combustionprocess have been derived. It has been shown thatthe overall acoustic pressure in large turbofan enginecombustors can be predicted over the range of engineoperating speeds, providing that the acoustic pressureefficiency, defined herein, can be determined. Thefollowing conclusions are based on the theory:

1. The major source of combustor pressure fluc-tuations is the interaction of the turbulence and themean internal energy (i.e., heat) additions in thecombustor.

2. For a typical large turbofan engine the ratioof the measured overall acoustic pressure in the com-bustor to the theoretical acoustic pressure (obtainedby assuming that all of the turbulent energy is con-verted to acoustic pressure) is a constant with amagnitude of the order of 0.030. For smaller reverse-flow combustors the acoustic pressure efficiency hasbeen found to be of the order of 0.020.

3. The turbulence-flame-generated noise isdirectly proportional to the square of the sonicvelocity, the mass flow rate per unit cross-sectionalarea, the ratio of heat energy added per unit mass ofair to the inlet enthalpy, the inlet velocity, thecombustion efficiency, and the turbulence intensity.It is inversely proportional to the square of thefrequency and the burning length of the combustor.

4. Because combustor size limits turbulencescales, larger scale low frequency turbulence issuppressed.

APPENDIX A

'0Af

gH-r

Symbols2

cross-sectional flow area, m

sonic velocity, m/sec; constant of

integrationspecific heat at constant pressure, J/kg K

flow or sonic velocity used in deriving

generalized acoustic wave equation, m/sec

constants used in complementary function

for solving differential equations

frequency parameter, Eq. (23)

frequency, Hz; function

lower limit on frequency, Hz

upper limit on frequency, Hz

ratio of mass of fuel to mass of airfrequency bandwidth, Hz

2acceleration of gravity, m/sec

total enthalpy, equals heat energy released

by combustion, J/kg (cal/kg)

heating value of fuel, J/kg (cal/kg)

M

SPL

OASPLP

P

Pref

J mechanical equivalent of heat, J/kg m

k acoustic wave number, u/Cg, 1/m.

Lb burning length (i.e., axial distance

measured from flame front to point in

combustor where chemical reaction ends), m

LT length of turbulence generating region, m

Mach number

sound pressure level, dB (re 20 wPa)

overall sound pressure level, dB (re 20 yPa)

instantaneous or mean static pressure, Pa

time-fluctuating part of instantaneous

pressure, Pareference pressure used in calculating sound

pressure level, 20 yPap Fourier-transformed fluctuating pressure,w

acoustic pressure generated in combustor,Pa sec

p , acoustic pressure for left-running wave

(i.e., x < 0), Pa sec

p fj acoustic pressure for right-running wave

(i.e., x > Lb), Pa sec

Q heat energy, J/kg (cal/kg)

R universal gas constant for air, m/K

S' fluctuating component of entropy, J/kg K

T mean temperature, KT' instantaneous or fluctuating temperature, K

t time, sec

U instantaneous or mean internal energy, J/kg

V mean velocity, m/sec

v instantaneous or fluctuating velocity, m/sec

W mean mass flow rate, kg/secx^ axial distance measured from flame front,

positive in direction of flow, mXj distance measured from turbulence generator

inlet, m

aSPL sound pressure level difference

« time-fluctuating component of density,kg/m3

n combustion efficiency

n_ acoustic pressure efficiency

e phase angle, deg

A fuel mass decay constant, assigned a value

Of 2n/Lb5 axial distance in combustor measured from

flame front location, m

p instantaneous or mean density of fluid,kg/m3

9 Fourier-transformed source terms (Eq. (8))

to angular frequency, 2irf, rad/sec

_ bar over a quantity indicates time averaged

Subscripts:

c 1/3-octave center frequency

L left running

n acoustic source numbers

R right running

T total conditions

U Fourier-transformed quantity; spectrum

0 upstream axial location

1 axial location downstream of combustion

zone; first source term

2 second source term

3 third source term

APPENDIX B

Acoustic Source Term Derivation

The acoustic source terms as defined in Ref. (6)are:

•1'̂ f-V'3tHco (Bl)

(B2)

(B3)

For ease in solving the acoustic wave equationfor the particular integral and because informationconcerning the acoustic pressure is usually presentedas a function of frequency the Fourier transforms ofthe source terms are required, they are:

'1,0

'2,0,

^ - S 12 u I

= 22 PW

(B4)

(B5)

(B6)

The first source term is related to the fluctuatingheat release in the combustion zone. From the firstlaw of thermodynamics in differential form:

dQ = dU - Pdp (B7)

dividing by the temperature and specifying a calor-ically and thermally perfect gas yields this expres-sion in terms of entropy as

dS dT R dpC T = T - -

dS dT ,- = - (T-

dp—

For the thermally perfect gas

dT _ dj^ dp_T - P P

Substituting Eq. (BIO) into (B9) yields

dS pdP .- -

For the mean sonic velocity given by

c 0=

(B8)

(B9)

(BIO)

(Bll)

(B12)

This acoustic source contains the turbulentvelocities generated by either turbulators in airbreathing turbine engine combustors or by the mixingof the propel 1 ants in rocket engine combustionchambers. To represent the turbulence source gen-eration the turbulent velocity will be assumed to bean exponential function of distance given by:

V ,oi,L

1 - e-xXT4

(B18)

where Xj is measured from the upstream side ofthe turbulator and L-j- is defined as the axialdistance measured from the front of the turbulencegenerator to the point where the turbulent velocityv(i>(Mr) reaches its maximum. The value of x istaken as 2n so that at X = Lj, the turbulentvelocity v reaches the maximum. Taking theLaplacian of vu

- v (B19)

Substituting Eq. (B19) into Eq. (B17) gives:

Equation (Bll) maybe written as: - XX,

(B13)

The first source term Eq. (B4) may, by substi-tuting Eq. (B13), be written in terms of thefluctuating entropy as:

, p2 u> (B14)

For dQ = d(Hyf0)

s1 H"fr

and Eq. (B14) becomes

The only term in Eq. (B16) that is fluctuatingwith time is the fuel-air ratio. Hence this sourceterm describes the contribution of the fluctuatingfuel-air ratio to the acoustic pressure.

The second source term, 92,w, ^l- ("5} maybe written for steady mean flow'as:

= 2

- 2 (B20)

The third source term 93 w Eq. (B6) hasbeen investigated in Ref. (6).' An exponential decayof the fuel droplet with distance from the flame frontwas assumed due to mixing and burning of the fueldroplet. Steady flow was assumed and the source termwritten for low Mach number flow as:

W (0 ,,

'3,<o - ~ A~~ 0.-XX (B21)

REFERENCES

(B15) *• Huff> R- 6-> Clark, B. J., and Oorsch, R. 6.,"Interim Prediction Method for Low Frequency CoreEngine Noise," NASA TM X-71627, 1974.

2. Mathews, D. C. and Peracchio, A. A., "Progress inCore Engine and Turbine Noise Technology," AIAAPaper No. 74-948, Aug. 1974.

3. Reshotko, M. and Karchmer, A., "Core NoiseMeasurements from a Small, General AviationTurbofan Engine," NASA TM-81610, 1980.

4. Huff, R. G., Groesbeck, D. E., and Goodykoontz,J. H., "Low Frequency Noisein a Quiet, Clean, General Aviation TurbofanEngine," NASA TM-83520, 9ct. 1983.

5. Liang, P. Y., "An Inviscid Three-DimensionalAnalysis of the Space Shuttle Main Engine Hot-GasManifold," AIAA Paper No. 83-1523, June 1983.

6. Huff, R. G., "A Simplified Combustion NoiseTheory Yielding a Prediction of FluctuatingPressure Level," NASA TP-2237, 1983. -

7. Bird, R. B., Stewart, W. E., and Lightfoot, E.(B17) N., Transport Phenomena, Wiley, New York, 1960,

pp. 322-323.

8. Wylie, C. R., Jr., Advanced Engineering 11. Doyle, V. L., and Moore, M. T., "Core NoiseMathematics, 2nd ed., McGraw-Hill, New York, 1960. Investigation of the CF6-50 Turbofan Engine,

9. Hildebrand, F. B., Methods of Applied Final Report," R79AEG395, General Electric Co.,Mathematics, Prentice-Hall, New York, Aircraft Engine Group, Cincinnati, Ohio, Jan.T 9 3 ? : 1 9 8 0 . (NASA CR-159749.)

10. Doyle, V. L., "Core Noise Investigation of theCF6-50 Turbofan Engine, Data Report," R79AEG247,General Electric Co., Aircraft Engine Group,Cincinnati, Ohio, Jan. 1980. (NASA CR-159598.)

SPRAY NOZZLES AND TABULATORS

I

1 — •11

lp+6)0

(P+P)0

(T+T)°0

-oo «- X

SOURCE REGION (p

1 I^ j FLAME FRONT

kr 7FLOW f /'

k "^

r * ;

r - '

L

//

x =0^

•• COMBUSTION ZONE

—

(p + 6)j(P + p)j

[v + x|i^

\\^-x = Ljj

x-» + °°

Figure 1. - Combustion noise source model.

THEORY-EQ. (25); ACOUSTICPRESSURE EFFICIENCY H • 0.030

0 20 40 60 80ENGINE THRUST, PERCENT OF DESIGN

Figure 2. - Combustor pressure fluctuations as a functionof CF6-50 turbofan engine thrust—experimental dataand theory.

(a) 3.8-Percent thrust.

800 1200 1600FREQUENCY, Hz

(b) 99.8-Percent thrust.

Figure 3. - Comparison of theory to measured soundpressure level spectrum—CF6-50 turbofan enginecombustor. Acoustic pressure efficiency, np.4 5x10"': dimensionless fluctuating density,6JP. 0.3.

2000

vl -CNJ

00 _J }=y Q- -H

-10

-20

_T '

-b -30CO

-40

/U 9THEORY, 10 log J l / f 'd f>

fL10 Iog10 0.233/fc

CF6-50 COMBUSTOR SPL(REF. 10); 3.8-PERCENTSPEED; 42° SENSOR;THEORY MATCHED TODATA AT 2000 Hz

400 800 1200 16001/3-OCTAVE-BAND CENTER

FREQUENCY, Hz

2000

Figure 4. - 1/3-Octave-band fluctuating pressurelevel spectrum shape for CF6-50 turbofan enginecombustor. Thrust, 3.8 percent of design.

oLUaz

OfooLUct:

oo1 1 1 c

Q^C

o.oz

5

10.0

1.0

.01 -

,001

THEORY - EQ. (28)CONVERTED TO

ENGINE 0107TEST DATA-'

400 800 1200FREQUENCY, Hz

1600 2000

Figure 5. - Comparison of the theoretically predicted earth toorbit propulsion systems hydrogen pump preburner fluc-tuating pressure spectrum to the measured spectrum down-stream of the turbine.

1. Report No.

NASA TM-835872. Government Accession No. 3. Recipient's Catalog No.

4. Title and Subtitle 5. Report Date

A Theoretical Prediction of the Acoustic PressureGenerated by Turbulence-Flame Front Interactions 6. Performing Organization Code

506-60-12

7. Authors)

Ronald G. Huff

8. Performing Organization Report No.

E-1962

10. Work Unit No.

9. Performing Organization Name and Address

National Aeronautics and Space AdministrationLewis Research CenterCleveland, Ohio 44135

11. Contract or Grant No.

12. Sponsoring Agency Name and Address

National Aeronautics and Space AdministrationWashington, D.C. 20546

13. Type of Report and Period Covered

Technical Memorandum

14. Sponsoring Agency Code

15. Supplementary NotesPrepared for the Winter Annual Meeting of the American Society of Mechanical.Engineers, New Orleans, Louisiana, December 9-13, 1984.

16. Abstract

The equations of momentum and continuity are combined and linearized yielding theone dimensional nonhomogeneous acoustic wave equation. Three terms in the non-homogeneous equation act as acoustic sources and are taken to be forcing func-tions acting on the homogeneous wave equation. The three source terms are:fluctuating entropy, turbulence gradients, and turbulence-flame interactions.Each source term is discussed. The turbulence-flame interaction source 1s usedas the basis for computing the source acoustic pressure from the Fourier trans-formed wave equation. Pressure fluctuations created in turbopump gas generatorsand turbines may act as a forcing function for turbine and propellent tube vibra-tions in earth to orbit space propulsion systems and could reduce their lifeexpectancy. A preliminary assessment of the acoustic pressure fluctuations insuch systems is presented.

17. Key Words (Suggested by Authors))

Noise; Combusti on : noise; Combustioninstability; Combustor noise; Flownoise; Forced vibration

19. Security Classif. (of this report)Unclassified

18. Distribution Statement

Unclassified - unlimitedSTAR Category 71

20. Security Classif. (of this page)Unclassified

21. No. of pages 22. Price*

*For sale by the National Technical Information Service, Springfield, Virginia 22161

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Washington, D.C.20546

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