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NASA Technical Memorandum 83587

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

Ronald G. Huff Lewis Research Center Cleveland, Ohio

Prepared for the Winter Annual Meeting of the American Society of Mechanical Engineers New Orleans, Louisiana, December 9-13, 1984

NASA A THEORETICAL PREDICTION OF THE ACOUSTIC PRESSURE GENERATED BY TURBULENCE-FLAME FRONT INTERACTIONS Ronald G. Huff National Aeronautics and Space Administration Lewis Research Center Cleveland, Ohio 44135

ABSTRACT combustor-turbine configurations and the turbopump preburner-turbine configurations used to pump propel- The equations of momentum and continuity are lants in earth to orbit propulsion systems (5). Two combined and linearized yielding the one dimensional of the primary differences are the propellants and nonhomogeneous acoustic wave equation. Three terms method of mixing the propellants prior to combustion. in the nonhomogeneous equation act as acoustic sources In both systems, however, after the fuel is mixed and are taken to be forcing functions acting on the with the oxidizer, the combustion process proceeds, homogeneous wave equation. The three source terms the resulting gas expands through the turbine and are: fluctuating entropy, turbulence gradients, and then flows through the remaining ducting to either turbulence-flame interactions. Each source term is the core engine nozzle or to the main rocket engine discussed. The turbulence-flame interaction source dome where it passes over liquid oxygen (LOX) tubes is used as the basis for computing the source acoustic in semi-crossflow. For the aircraft engine the pres- pressure from the Fourier transformed wave equation. sure fluctuations are the source of unwanted sound. Pressure fluctuations created in turbopump gas gener- For the space propulsion system the pressure fluctu- ators and turbines may act as a forcing function for ations become an unsteady driving force causing turbine and propellent tube vibrations in earth to vibrations in the LOX tubes that could be destructive orbit space propulsion systems and could reduce their if not accounted for in their design. life expectancy, A preliminary assessment of the The combustor has been the subject of numerous acoustic pressure fluctuations in such systems is theoretical and experimental investigations. Ref- presented. erence (6) discusses a few of the reported studies on combustion noise. The objective of the present paper INTRODUCTION is to present a simplified theoretical model, and prediction of the acoustic pressure generated in the Acoustic pressure fluctuations created in flow- combustor. Using this model a better understanding ing systems may be the source of unwanted noise and of the sound sources in combustors can be obtained serve as the unsteady driving force acting on mechan- enabling one to modify and predict their behavior. ical systems associated with the flow. Aircraft type All symbols used in this paper are defined in turbine engine noise has been a problem that has re- the symbol list, Appendix A. ceived attention over the past several years. High bypass turbofan engines with lower jet velocities have resulted in large reductions of aircraft jet noise. With the reduction of jet noise, other noise sources have emerged presenting additional problems COMBUSTION NOISE SOURCE MODEL for researchers working in the noise reduction field. The combustor is one of the acoustic sources whose The model for combustion noise considers one- noise characteristics are shown to be discernible in dimensional duct flow (Fig. 1). Fuel is sprayed into the farfield under some circumstances U-2J. Aircraft the airstream where turbulence generators, termed engines operating at reduced power settTngs have been turbulators, promote mixing of the fuel with the air shown to emit low frequency acoustic waves from the (oxidizer). A flame front perpendicular to the duct combustor that are transmitted through the turbine, walls exists downstream of the turbulators at X = ducting and nozzle to the farfield (3-4_). Combustor 0. The combustion process begins at the flame front noise may be expected from any turbulent combustion and extends downstream to X = Lt, the point where process. burning ceases. The region between X = 0 and L^ Of particular timely interest is the similarity is called the combustion zone. The duct extends to between components used in aircraft turbine engine infinity in the X direction. THE GOVERNING DIFFERENTIAL EQUATIONS AND ASSUMPTIONS (8) kX = - InI 9 .b.) The second order partial differential equation governing the noise generated within an inviscid, adiabatic, perfect gas, in a one-dimensional duct with where 9n u are the Fourier transforms of negligible fuel mass may be derived using the the terms on tine right side of Eq. (7). These terms equations for conservation of mass are the acoustic sources, i.e. forcing functions, and momentum respectively given here as: driving the acoustic wave equation. The source terms are of great interest since knowledge of their char- 3pV acteristics will give insight on how the fluctuating at 3X (1) pressures found in the duct are generated. The pri- mary thrust of this work is to define these source terms in detail. Before this can be accomplished, 3pV however, the nonhomogeneous equation (Eq. (8)) must (2) at 3X 3X be solved.

ACOUSTIC SOURCE AND WAVE EQUATIONS SOLUTION OF NONHOMOGENOUS WAVE EQUATION The acoustic wave equation describing the sound By using the methods given in Ref. (8) the com- propagation in a duct is derived by first taking the plete solution of the homogeneous part (i.e., with partial derivative of Eq. (1) with respect to time; 9 = 0) of Eq. (8) is then taking the partial derivative of Eq. (2) with respect to X, adding equations; -ikx fu.,0 l C2 e (9) 2 2 and then adding (1/CQ ) (3 P/3t ) to both sides of the resulting equation giving: With the above complete solution known, the Green's function (9) may be used to write the equation for the fluctuating pressure in the duct propagating 1 a"P 3 P _ 3 -p 1 3"P B'pV (3) away from the combustion zone which constitutes the 3X solution of the nonhomogeneous wave equation (Eq. (8)). For the left and right running waves respec- tively the pressure is given by: The assumption is made that the instantaneous pres- sure P, density p and velocity V are the sum of ikx mean and fluctuating components, that is: (10) P = P(x) + p(x,t) (4) P = p(x) + «(x,t) (5) -ikx V = V(x) + v(x,t) (6) 1 o,R = ~2TiT Jf (ID Substituting Eqs. (4) to (6) into Eq. (3), ex- panding and equating like order terms yields the wave equation as: where

- (9 (12) 3X (7) The Fourier transformed source terms (i.e. Fourier transform of terms on the right side of Eq. (7)) are To this point in the derivation nothing has been defined as: assumed concerning CQ in the above equation. In this work, CQ, is assumed to be the sonic velocity. (13a) Equation (7) then becomes the acoustic wave equation. »»,! It .is assumed that outside the combustion zone the sonic velocity is constant, though different, in the upstream and downstream portions of the duct. It is further assumed that turbulence generated pressure or from Appendix B, Eq. (B14) in terms of fluctuating fluctuations anywhere in the duct external to the entropy combustion zone are less than the acoustic pressures generated by the combustion process and that no ex- ternally generated acoustic waves pass through the (13) combustion zone. The acoustic pressure is usually given in terms of finite-bandwidth, mean-square pressure fluctuations as a function of frequency. This is accomplished by taking the. Fourier transform of Eq. (7), which removes time as a variable and thus simplifies the method of (14) solution. The Fourier transform of Eq. (7) in terms of wave number k is: (15) -xx. (18)

The source terms given by Eqs. (13) to (15) rep- resent the acoustic sources due to: fluctuating heat ACOUSTIC SOURCE TERM EVALUATION release, turbulence, and energy addition (or with- drawal) respectively. Each has been developed in In this paper no attempt will be made to evaluate the fluctuating entropy or cold flow acoustic sources Appendix B and will be discussed in greater detail anc below. *l,u ' *2 a- The fluctuating entropy source 9^ (0, Inould have low pressure levels if PHYSICS OF NOISE GENERATION PROCESS the burne? operates stably. The cold flow noise term ^2,00 appears to act as an upstream fluctuating The fluctuating entropy source term Eq. (13) may density generator and as such is amplified when pas- be written from Appendix B, Eq. (B16) as: sing through the combustion zone. Thus the third source term 93^ is expected to be the dominant one. pHv fO,o To evaluate p^ (Eq. (10)), the source term (16) '3,00 mus* be given as a function of distance V x.' It is shown in Ref. (j>) and Appendix B that for Mach numbers much less than unity the velocity is Where the local fuel/air ratio, fg, is allowed directly proportional to the heat energy input to the to fluctuate either because of unsteady turbulent air (Appendix B, Eq. (822)). mixing or because of some externally generated dis- By assuming that the fuel droplet mass decreases turbance such as reflected acoustic waves passing exponentially with axial distance, the velocity dis- through the fuel/air mixing region upstream of the tribution due to the energy released to the air can be flame front causing increased mixing. Thus this determined as a function of x. The second derivative source may represent the energy addition due to with respect to x is then substituted into Eq. (15). acoustic feedback. As such one might expect this The Fourier-transformed source term from Appendix B, source to yield information on instability in com- (Eq. (21)) is bustors. In order to evaluate this source more in- formation is needed on the local fuel/air ratio and H -xx the effect that turbulence and acoustic waves have on "c V (19) the mixing of the fuel with the air (oxidizer). The second source term (Eq. (14)), describes the source of pressure fluctuations associated with the fluctuating velocity gradient. Thus the generation Inserting Eq. (19) into Eq. (10), assuming that of fluctuating velocities by the turbulators or tur- the fluctuating density is constant with distance bulent mixing of fuel and oxidizer in rocket engines through the combustion zone, and integrating yields is a source of noise in these flowing systems. This the acoustic pressure in polar form as (see derivation source term depends only on the mass flow per unit in Appendix B, of Ref.(^). cross sectional area and the second derivative of the fluctuating velocity with respect to distance. (The i[kx + (7I/2) tarrl second derivative with respect to distance is here- R Wo x e * (-l inafter called the Laplacian). Hence it is considered TO?~ as the cold flow noise source that exists even if [1 combustion is absent. This source term has been (20) developed in Appendix B for turbulence generated over a given length, L-r. The turbulence at the end of where 0 < X < L^. Equation (20), with dimensions the generator is assumed to have reached its maximum of pressure per unit angular frequency, is used for value and decay of the turbulence downstream of the calculating the overall sound pressure level and the generator is neglected. The second source term from narrowband acoustic spectrum. The amplitude for the Appendix B, Eq. (B20) is: right and left running waves are calculated using Eq. (20) with the appropriate sonic velocity inserted -X X, in the wave number, k. In deriving Eq. (20) the conversion of turbulent = -2 (17) energy to acoustic energy was assumed to be complete. This, in reality, cannot be accomplished as evidenced by the fact that turbulence does exist downstream of Thmee thirtmrad acoustic source term 93 ^tq. a combustor. To account for this, an acoustic pres- (15)) is due to the Laplacian of the product of the sure efficiency np is introduced and defined as turbulent density and the mean velocity. It has been the ratio of the experimentally measured acoustic assumed in this work that the turbulent density fluc- pressure generated by the combustor p,,, to the tuations in the combustion zone are constant. With acoustic pressure that would be generated if all of this assumption this source term becomes a function the turbulent energy had been converted to an acoustic of the Laplacian of the mean velocity, the result of pressure P^. From Ref. (6) np is given by the 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). is (21) expressed as: Thus the acoustic pressure efficiency can now be Where the local static temperature is used in determined for any specific combustor. The value of calculating the sonic velocity, C. rip will probably depend on the combustor type and The fluctuating pressure in the combustor of the configuration. Therefore it is necessary that np CF6-50 turbofan engine has been measured (10-11). be determined for a number of combustors so that the The measured overall fluctuating pressure (assumed to proper value can be selected when making predictions be acoustic) in the combustor at 3.8 percent of design of combustor pressure fluctuations. thrust has been used in conjunction with Eq. (26) to calculate the acoustic pressure efficiency, np = OVERALL SOUND PRESSURE LEVEL 0.030, for the CF6-50 combustor. From this acoustic pressure efficiency the over- An expression for the overall sound pressure all sound pressure level in the combustor was pre- level can be obtained by multiplying Eq. (20) by the dicted over the engine thrust range by using Eq. (25). acoustic pressure efficiency np, squaring and then The results are shown in Fig. 2. The theory, given integrating over the frequency range of interest. by the solid line, agrees well with the measured The resulting equation for P^ is fluctuating pressure, given by the symbols. In light of this discussion it can be concluded that the theory predicts the trends in the combustor (22) overall acoustic pressure with engine operating con- Jref> ditions and, for CF6-50 combustors and similar geome- tries, an acoustic pressure efficiency of 0.030 can be used in Eq. (25) to predict the combustor overall where the term y contains all of the frequency acoustic pressure. The acoustic pressure efficiency dependent term and is defined as has also been determined for a YF102 engine combustor. This combustor, unlike the CF6-50, is a reverse-flow combustor and is much smaller than the CF6-50. Its x acoustic pressure efficiency is 0.016 at 30 percent IT df of design speed. Using the acoustic pressure effi- [1 ciency of 0.016 yields only a 0.2-dB error in pre- dicting the OASPL at the 95-percent speed. Comparing the acoustic pressure efficiencies shows that the CF6-50 combustor converts the turbulence to acoustic xC 2 [l 2, , -l/2«f\l I L pressure more efficiently than the YF102 combustor. = T~ 7 f TrC tan IC —p— II I (23) 2, |_ * V )\ I f NARROW-BAND SOUND PRESSURE LEVEL SPECTRUM The theoretical sound pressure spectrum is given Equation (23) indicates that ^"2 is inversely by Eq. (20). Discarding the phase information con- proportional to the frequency. Therefore the magni- tained in the exponential term and applying Eq. (21) tude of will be determined by the lower limit on to account for the efficiency of conversion of turb- frequency, providing that the upper limit is much ulence to acoustic pressure, Eq. (20) is written as greater than the lower limit. In determining the OASPL the combustion noise frequency limits are of AL, P b 1 1 (27) the order of 50 and 2000 Hz. For a fuel mass decay S w c f ' 1/2 constant x, defined as 2ir/L(,, and reasonable sonic Vc 0) Wo (fL /C)2] velocities, Eq. (23) can be approximated by using the b lower frequency limit by The right side of Eq. (27) represents the spectral -1 shape of the acoustic pressure given by Eq. (20). sec (24) The sound pressure level spectrum is given by

Wo Af SPL = 201og10^P-^ « y 1/2 The overall sound pressure level (OASPL) equation ef b T.O 2 can now be written for the combustor from Eqs. (22) (fL/C) ] and (24) as (28) VcC 6(i ) _W V0nH v f0. OASPL = 20 log (25) This equation states that the acoustic pressure level 10 p AC T 2 L p T,0 is inversely proportional to the square of the fre- Pref b quency since the term under the radical is small for fLb < C. The acoustic pressure efficiency tin can The predicted spectral shape is compared with be determined from the measured overall sound pressure the measured narrow-band spectra for the CF6-50 en- level and Eq. (25). The equation for experimentally gine (10) in Fig. 3. The measured fluctuating pres- sure level in the combustor of the CF6-50 turbofan determining the value of np is engine operating at 3.8 percent of design thrust is shown in Fig. 3(a). Also shown in Fig. 3(a) is a 2 L Pref t, L p A CVT.T O OASPL/20 ,.,,, plot of Eq. (28) with the level matched to the n x 11f)0 25 measured spectrum at a frequency of 2000 Hz. The P = ~ s~worrr < ) acoustic pressure efficiency is determined from Eq. (28) and is given by C T P engine dome. Figure 5 compares the predicted pressure 2_^_^b p T,0 ref 10SPL(f)/20 f v H f C spectrum in the preburner to the measured pressure \ <» w O v O ^ downstream of the turbine. The turbine will attenuate some of the preburner generated acoustic signal but it does not decrease the theoretically predicted value enough to cause serious disagreement. (29) CONCLUSIONS At 2000 Hz the measured SPL is equal to 110 dB and By using the theory developed herein, expressions the narrow-band spectral acoustic pressure efficiency, for the acoustic pressure generated by the combustion computed by using Eq. (29), is process have been derived. It has been shown that the overall acoustic pressure in large turbofan engine n = 4.5x10-2 combustors can be predicted over the range of engine p operating speeds, providing that the acoustic pressure efficiency, defined herein, can be determined. The when 6^/p is assumed to be 0.3. following conclusions are based on the theory: The trends in the spectral shape of both theory 1. The major source of combustor pressure fluc- and experiment agree very well at the 3.8-percent- tuations is the interaction of the turbulence and the thrust point given in Fig. 3(a). The comparison at mean internal energy (i.e., heat) additions in the 99.8-percent thrust shown in Fig. 3(b), however, is combustor. not as good. The theory given for the 99.8-percent- 2. For a typical large turbofan engine the ratio thrust point uses the acoustic pressure efficiency of the measured overall acoustic pressure in the com- obtained from the 3.8-percent-thrust point, namely bustor to the theoretical acoustic pressure (obtained nn = 4.5xlO~2. Examining Fig. 3(b) shows that by assuming that all of the turbulent energy is con- trie general trend is correct but that large oscilla^ verted to acoustic pressure) is a constant with a tions in the spectrum are not predicted by Eq. (28). magnitude of the order of 0.030. For smaller reverse- Acoustic reflections from the turbine may account for flow combustors the acoustic pressure efficiency has these oscillations. been found to be of the order of 0.020. From this discussion it is concluded that the 3. The turbulence-flame-generated noise is spectrum predicted by Eq. (28) agrees substantially directly proportional to the square of the sonic with the experimental measurements made in an opera- velocity, the mass flow rate per unit cross-sectional ting engine. It can also be concluded that the area, the ratio of heat energy added per unit mass of simplified combustion noise theory given herein sub- air to the inlet enthalpy, the inlet velocity, the stantially predicts the trends in the acoustic pres- combustion efficiency, and the turbulence intensity. sures generated by the combustion process of turbine It is inversely proportional to the square of the engine combustors. frequency and the burning length of the combustor. 4. Because combustor size limits turbulence 1/3-OCTAVE-BAND SOUND PRESSURE SPECTRUM scales, larger scale low frequency turbulence is suppressed. The 1/3-octave-band spectrum of the combustor noise can be calculated for a simplified spectrum by APPENDIX A neglecting the term under the radical contained in Eq. (28) and integrating over the 1/3 octave Symbols bandwidth: 2 cross-sectional flow area, m aSPL = 10 log. 0.23 (30) sonic velocity, m/sec; constant of integration specific heat at constant pressure, J/kg K This equation has been plotted in Fig. 4 with its level matched, as for the narrow-band spectrum, at flow or sonic velocity used in deriving the 2000-Hz band for the CF6-50 operating at the 3.8- generalized acoustic wave equation, m/sec percent-thrust point. The 1/3-octave-band data given by the symbols match the theoretical trends for fre- constants used in complementary function quencies above 100 Hz. At frequencies less than 100 Hz the data are significantly over predicted by for solving differential equations the theory; a result attributable to the absence of frequency parameter, Eq. (23) low frequency turbulence. frequency, Hz; function EARTH TO ORBIT PROPULSION SYSTEM PRESSURE SPECTRUM lower limit on frequency, Hz Turbopump systems used in earth to orbit propul- upper limit on frequency, Hz sion systems are subjected to fluctuating pressures '0 ratio of mass of fuel to mass of air that could cause a decrease in their life expectancy Af frequency bandwidth, Hz due to high cycle loading of structures. Equation 2 (28) was used to predict the fluctuating pressures in g acceleration of gravity, m/sec the ducting downstream of the hydrogen pump preburner using, as a first approximation, the CF6-50 turbofan H-r total enthalpy, equals heat energy released acoustic pressure efficiency reported herein. Ref- by combustion, J/kg (cal/kg) erence (5) gives the geometry of the hydrogen pump preburner, turbine, and ducting leading to the main heating value of fuel, J/kg (cal/kg) J mechanical equivalent of heat, J/kg m to angular frequency, 2irf, rad/sec k acoustic wave number, u/Cg, 1/m. _ bar over a quantity indicates time averaged

Lb burning length (i.e., axial distance Subscripts: measured from flame front to point in c 1/3-octave center frequency combustor where chemical reaction ends), m L left running

LT length of turbulence generating region, m n acoustic source numbers M Mach number R right running SPL sound pressure level, dB (re 20 wPa) T total conditions OASPL overall sound pressure level, dB (re 20 yPa) U Fourier-transformed quantity; spectrum P instantaneous or mean static pressure, Pa 0 upstream axial location P time-fluctuating part of instantaneous 1 axial location downstream of combustion pressure, Pa zone; first source term Pref reference pressure used in calculating sound 2 second source term pressure level, 20 yPa 3 third source term p Fourier-transformed fluctuating pressure, w acoustic pressure generated in combustor, APPENDIX B Pa sec Acoustic Source Term Derivation p , acoustic pressure for left-running wave (i.e., x < 0), Pa sec The acoustic source terms as defined in Ref. (6) are: p fj acoustic pressure for right-running wave (i.e., x > L ), Pa sec b (Bl) Q heat energy, J/kg (cal/kg) •1'^f-V3t c ' R universal gas constant for air, m/K H o S' fluctuating component of entropy, J/kg K T mean temperature, K (B2) T' instantaneous or fluctuating temperature, K t time, sec U instantaneous or mean internal energy, J/kg (B3) V mean velocity, m/sec v instantaneous or fluctuating velocity, m/sec For ease in solving the acoustic wave equation for the particular integral and because information W mean mass flow rate, kg/sec concerning the acoustic pressure is usually presented x^ axial distance measured from flame front, as a function of frequency the Fourier transforms of positive in direction of flow, m the source terms are required, they are: Xj distance measured from turbulence generator ^ - S 1 (B4) inlet, m '1,0 2 u I aSPL sound pressure level difference « time-fluctuating component of density, 2 W kg/m3 P '2,0, = 2 (B5) n combustion efficiency n_ acoustic pressure efficiency e phase angle, deg (B6) A fuel mass decay constant, assigned a value

Of 2n/Lb 5 axial distance in combustor measured from The first source term is related to the fluctuating heat release in the combustion zone. From the first flame front location, m law of thermodynamics in differential form: p instantaneous or mean density of fluid, Pdp kg/m3 dQ = dU - (B7) 9 Fourier-transformed source terms (Eq. (8)) dividing by the temperature and specifying a calor- This acoustic source contains the turbulent ically and thermally perfect gas yields this expres- velocities generated by either turbulators in air sion in terms of entropy as breathing turbine engine combustors or by the mixing of the propel 1 ants in rocket engine combustion dS dT R dp (B8) chambers. To represent the turbulence source gen- CT=T-- eration the turbulent velocity will be assumed to be an exponential function of distance given by:

dS dT , dp (B9) -xXT - = - (T- — V , 1 - e (B18) oi,L 4

For the thermally perfect gas where Xj is measured from the upstream side of the turbulator and L-j- is defined as the axial dT _ dj^ dp_ (BIO) distance measured from the front of the turbulence T - P P generator to the point where the turbulent velocity v(i>(Mr) reaches its maximum. The value of x is taken as 2n so that at X = Lj, the turbulent Substituting Eq. (BIO) into (B9) yields velocity v reaches the maximum. Taking the Laplacian of vu dS pdP . - - (Bll)

For the mean sonic velocity given by - v (B19)

c0= (B12) Substituting Eq. (B19) into Eq. (B17) gives:

Equation (Bll) maybe written as: - XX,

(B20) (B13) - 2

The third source term 93 w Eq. (B6) has been investigated in Ref. (6).' An exponential decay The first source term Eq. (B4) may, by substi- of the fuel droplet with distance from the flame front tuting Eq. (B13), be written in terms of the was assumed due to mixing and burning of the fuel fluctuating entropy as: droplet. Steady flow was assumed and the source term written for low Mach number flow as: , p 2 u> (B14) W (0 ,, .-XX '3,

For dQ = d(Hyf0)

REFERENCES s1 H"fr (B15) *• Huff> R- 6-> Clark, B. J., and Oorsch, R. 6., "Interim Prediction Method for Low Frequency Core Engine Noise," NASA TM X-71627, 1974. 2. Mathews, D. C. and Peracchio, A. A., "Progress in and Eq. (B14) becomes Core Engine and Turbine Noise Technology," AIAA Paper No. 74-948, Aug. 1974. 3. Reshotko, M. and Karchmer, A., "Core Noise Measurements from a Small, General Aviation Turbofan Engine," NASA TM-81610, 1980. 4. Huff, R. G., Groesbeck, D. E., and Goodykoontz, J. H., "Low Frequency Noise The only term in Eq. (B16) that is fluctuating in a Quiet, Clean, General Aviation Turbofan with time is the fuel-air ratio. Hence this source Engine," NASA TM-83520, 9ct. 1983. term describes the contribution of the fluctuating 5. Liang, P. Y., "An Inviscid Three-Dimensional fuel-air ratio to the acoustic pressure. Analysis of the Space Shuttle Main Engine Hot-Gas The second source term, 92,w, ^l- ("5} may Manifold," AIAA Paper No. 83-1523, June 1983. be written for steady mean flow'as: 6. Huff, R. G., "A Simplified Combustion Noise Theory Yielding a Prediction of Fluctuating Pressure Level," NASA TP-2237, 1983. - 7. Bird, R. B., Stewart, W. E., and Lightfoot, E. = 2 (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 Noise Mathematics, 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. T93?:1980. (NASA CR-159749.) 10. Doyle, V. L., "Core Noise Investigation of the CF6-50 Turbofan Engine, Data Report," R79AEG247, General Electric Co., Aircraft Engine Group, Cincinnati, Ohio, Jan. 1980. (NASA CR-159598.) SPRAY NOZZLES AND TABULATORS I 1 — • SOURCE REGION (p 1 — 1 I ^ j FLAME FRONT kr 7 FLOW f /' lp+6)0 (p + 6)j (P+P) k " (P + p)j 0 ^ (T+T)° [v + x|i 0 r * ; ^ r - '

L / \ / \ x =0^ ^-x = Ljj -oo «- X x-» + °° •• COMBUSTION ZONE

Figure 1. - Combustion noise source model.

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

0 20 40 60 80 ENGINE THRUST, PERCENT OF DESIGN Figure 2. - Combustor pressure fluctuations as a function of CF6-50 turbofan engine thrust—experimental data and theory. (a) 3.8-Percent thrust.

800 1200 1600 2000 FREQUENCY, Hz (b) 99.8-Percent thrust. Figure 3. - Comparison of theory to measured sound pressure level spectrum—CF6-50 turbofan engine combustor. Acoustic pressure efficiency, np. 4 5x10"': dimensionless fluctuating density, 6JP. 0.3. /U 9 THEORY, 10 log J l/f'df> fL 10 Iog10 0.233/fc -10 CF6-50 COMBUSTOR SPL vl -CNJ (REF. 10); 3.8-PERCENT 00 _J }= y Q- -H -20 SPEED; 42° SENSOR; THEORY MATCHED TO _T' DATA AT 2000 Hz -b -30 CO

-40 400 800 1200 1600 2000 1/3-OCTAVE-BAND CENTER FREQUENCY, Hz Figure 4. - 1/3-Octave-band fluctuating pressure level spectrum shape for CF6-50 turbofan engine combustor. Thrust, 3.8 percent of design.

10.0 o LU az Of 1.0 oo THEORY - EQ. (28) LU ct: CONVERTED TO

111oo c oQ^.C o z 5 .01 - ENGINE 0107 TEST DATA-'

,001 400 800 1200 1600 2000 FREQUENCY, Hz Figure 5. - Comparison of the theoretically predicted earth to orbit propulsion systems hydrogen pump preburner fluc- tuating pressure spectrum to the measured spectrum down- stream of the turbine. 1. Report No. 2. Government Accession No. 3. Recipient's Catalog No. NASA TM-83587 4. Title and Subtitle 5. Report Date A Theoretical Prediction of the Acoustic Pressure Generated by Turbulence-Flame Front Interactions 6. Performing Organization Code 506-60-12

7. Authors) 8. Performing Organization Report No. E-1962 Ronald G. Huff 10. Work Unit No.

9. Performing Organization Name and Address National Aeronautics and Space Administration 11. Contract or Grant No. Lewis Research Center Cleveland, Ohio 44135 13. Type of Report and Period Covered 12. Sponsoring Agency Name and Address Technical Memorandum National Aeronautics and Space Administration Washington, D.C. 20546 14. Sponsoring Agency Code

15. Supplementary Notes Prepared 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 the one 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 used as the basis for computing the source acoustic pressure from the Fourier trans- formed wave equation. Pressure fluctuations created in turbopump gas generators and 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 life expectancy. A preliminary assessment of the acoustic pressure fluctuations in such systems is presented.

17. Key Words (Suggested by Authors)) 18. Distribution Statement Noise; Combusti on : noise; Combustion Unclassified - unlimited instability; Combustor noise; Flow STAR Category 71 noise; Forced vibration

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