Combustion noise - University of Cambridge

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Proceedings of the Combustion Institute 35 (2015) 65–100

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CombustionInstitute

Combustion noise

Ann P. Dowling ⇑, Yasser Mahmoudi

Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, United Kingdom

Available online 22 November 2014

Abstract

Combustion noise is becoming increasingly important as a major noise source in aeroengines andground based gas turbines. This is partially because advances in design have reduced the other noisesources, and partially because next generation combustion modes burn more unsteadily, resulting inincreased external noise from the combustion. This review reports recent progress made in understandingcombustion noise by theoretical, numerical and experimental investigations. We first discuss the fundamen-tals of the sound emission from a combustion region. Then the noise of open turbulent flames is summa-rized. We subsequently address the effects of confinement on combustion noise. In this case not only is thesound generated by the combustion influenced by its transmission through the boundaries of the combus-tion chamber, there is also the possibility of a significant additional source, the so-called ‘indirect’ combus-tion noise. This involves hot spots (entropy fluctuations) or vorticity perturbations produced by temporalvariations in combustion, which generate pressure waves (sound) as they accelerate through any restrictionat the exit of the combustor. We describe the general characteristics of direct and indirect noise. To gainfurther insight into the physical phenomena of direct and indirect sound, we investigate a simple configu-ration consisting of a cylindrical or annular combustor with a low Mach number flow in which a flamezone burns unsteadily. Using a low Mach number approximation, algebraic exact solutions are developedso that the parameters controlling the generation of acoustic, entropic and vortical waves can be investi-gated. The validity of the low Mach number approximation is then verified by solving the linearized Eulerequations numerically for a wide range of inlet Mach numbers, stagnation temperature ratios, frequencyand mode number of heat release fluctuations. The effects of these parameters on the magnitude of thewaves produced by the unsteady combustion are investigated. In particular the magnitude of the indirectand direct noise generated in a model combustor with a choked outlet is analyzed for a wide range of fre-quencies, inlet Mach numbers and stagnation temperature ratios. Finally, we summarize some of theunsolved questions that need to be the focus of future research.� 2014 The Authors. Published by Elsevier Inc. on behalf of The Combustion Institute. This is an openaccess article under the CC BY license (http://creativecommons.org/licenses/by/3.0/).

Keywords: Combustion noise; Acoustic perturbation; Entropy perturbation; Vorticity perturbation; Choked nozzle

http://dx.doi.org/10.1016/j.proci.2014.08.0161540-7489/� 2014 The Authors. Published by Elsevier Inc. onThis is an open access article under the CC BY license (http://

⇑ Corresponding author.E-mail address: [email protected] (A.P. Dowling).

1. Introduction

In the last four decades noise emission hasdeveloped into a topic of increasing concern tosociety. This mainly stems from the adverse phys-iological impacts on those exposed to noise over

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66 A.P. Dowling, Y. Mahmoudi / Proceedings of the Combustion Institute 35 (2015) 65–100

lengthy periods. As a result, in air, road and railtransport technologies, the control of noise emis-sions is central to social acceptance and economiccompetitiveness. Due to its intermittent nature,aircraft noise is deemed to be the most annoyingtransportation noise, with road noise being theleast likely to annoy. Noise has an immediateeffect upon observers at the time of emission,causing annoyance and physiological change,and it also impedes the efficiency of observers.Longer term effects of noise are physiologicalimpairment, e.g. hearing damage, speech andsleep interference. Although individual aircrafthave become less noisy over the last 30 years,the increase in air traffic means that many citizensare concerned by aircraft noise. According to theInternational Civil Aviation Organization [1], glo-bal air transportation is anticipated to doubleover the next couple of decades. It is thereforeexpected that the negative social and environmen-tal impacts of its noise emission will increase.

Aircraft noise is noise pollution produced byany aircraft or its components during variousphases of a flight: on the ground such as auxiliarypower units, while taxiing, on run-up, duringtake-off, underneath and lateral to departure andarrival paths, or during landing. The primarysource of noise in an aircraft can be contributedto the fan, compressor, combustor, turbine, andjet exhaust [2] (see Fig. 1). On approach the air-frame is also a significant source of noise.

During the last few decades, research effortshave enabled a significant reduction of jet, fanand external aerodynamic noise. The reductionof jet noise was mainly achieved by introducingultra-high-bypass ratio turbofan engines. Fannoise has been reduced through effective acousticliners and complex designs of fan blade geometry.These efforts on the reduction of jet and fan noisehave left combustion noise as an importantremaining contributor [2,3]. Figure 2 shows thesignificance of combustion noise relative to othernoise sources for a typical turbojet application.

Combustion noise appears to be the third dom-inant source in the whole turbojet engine noiseafter fan and jet noise, especially at approachand cut back conditions. Furthermore, recentstudies on low-NOx combustors such as lean pre-mixed prevaporized (LPP) combustion show con-siderable increase in noise emission [4]. This isbecause lean premixed and stratified combustionburns more unsteadily [5–7]. As discussed in thesereferences, lean premixed combustors can also besusceptible to an instability arising from the feed-back interaction between unsteady combustionand acoustic waves. Such an instability occurs ata discrete tone related to the acoustic resonancesof the combustor usually shifted slightly by theflame response. Even when the self-excited insta-bility has been eliminated by a careful combustordesign, which reduces the sensitivity of the rate of

combustion to acoustic waves, LPP systems gen-erate substantial broad-band noise, which can beheard outside the engine. Furthermore, noise fromauxiliary power units (APU) is an important con-tributor to the overall level of ramp noise (thenoise generated by an aircraft while it is on theground, typically parked at the ramp). A signifi-cant component of APU noise is combustionnoise [8,9]. It is therefore crucial to investigate thisbroad-band combustion noise and develop meth-ods to predict and reduce it, in order to enablethe introduction of low noise, green technologieson next generation low-NOx combustors and pre-mixed burners in aero-engines [7,10,11].

2. Motivation: Gas turbine combustion noisesources

The total noise radiated by a gas turbine com-bustion chamber system consists of direct andindirect combustion noise [11,12]. The direct noisesources are related to the unsteady processes ofvolumetric expansion and contraction in the reac-tive region [13,14]. This is generated by the fluctu-ation of heat release rate associated with thechemical reaction [11,15]. It would occur even ifthe combustion were in unbounded space,although in the gas turbine the sound radiated ismodified from the open flame result by its trans-mission from the combustion chamber to thefar-field through the turbine and jet. The indirectcombustion noise, identified by the early works ofMarble and Candel [12] and Morfey [16] in theseventies, is an additional noise generated whena fluid with a non-uniform entropy or vorticitydistribution is accelerated, as it is when convectedthrough the choked nozzle located at the outlet ofthe combustion chamber in a gas turbine [17,18].A summary of these different core noise sourcesis illustrated in Fig. 3.

The spatial and temporal variation of the rateof combustion produces hot and cold spots whichare convected by the mean flow [11]. The couplingbetween hot spots generated by combustion andparticle acceleration in the mean flow then givesrise to pressure perturbations. Since the entropyof the hot spots is different to that of the sur-rounding, indirect noise is also called entropynoise. The vital role of acceleration/decelerationcaused Ffowcs Williams and Howe [17] to referto it as acoustical bremsstrahlung. The combustionalso generates unsteady shear leading to vorticityperturbations [19], which also convect and gener-ate pressure perturbations as they acceleratethrough the turbine nozzle guide vanes. Hence,at the combustor exit the interaction of the vorti-cal and entropic perturbations with the meanstreamwise velocity gradient results in energytransfer to an acoustic mode [19,20]. Accelerationof entropy and vorticity waves in the choked

Fig. 2. Typical contribution of noise sources on a turbojet engine at approach. (Fig. 2 is from SAFRAN Snecma (http://www.safran-group.com/), published with permission).

Fig. 1. Summary of engine noise sources (Rolls-Royce Trent 1000, copyright Rolls-Royce, published with permission).

A.P. Dowling, Y. Mahmoudi / Proceedings of the Combustion Institute 35 (2015) 65–100 67

nozzle results in generation of pressure waves thatpropagate upstream (from where they can bereflected from the flame zone and/or the upstreamgeometry altering the downstream propagatingwaves) and downstream from the turbine stageas indirect combustion noise [12,17,18].

In this review we discuss combustion noisearising from fluctuations in the rate of heat releasein a turbulent flame. It is broad-band in natureand is distinct from the discrete tones observedduring combustion instability. Instability occurswhen feedback is established between unsteadycombustion and pressure perturbations in a com-bustion chamber. The mechanism is that the

unsteady combustion generates pressure waves,which reflect at the boundaries of the combustorand the velocity or equivalence perturbation asso-ciated with these waves causes more unsteadycombustion. If the phase relationship is suitable[21,22], self-excited oscillations grow. This leadsto discrete tones at resonant frequencies associ-ated with the acoustic characteristics of the com-bustor and can be accompanied by high noiselevels and severe pressure oscillations that cancause structural damage to engine components[5–7]. The vibrations induced by the oscillationscan cause fatigue cracking of combustor linersand reduce the lifetime of a combustor by a factor

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Fig. 3. Illustration of combustion noise sources in a gas turbine: generation of direct combustion noise and indirectcombustion noise in aero-engines.


of two or more [6,23]. During an instability theheat release and the pressure perturbationsthroughout the combustor are at a fixed frequencyand coherent.

In contrast, under normal stable operatingconditions the combustion noise sources are inco-herent throughout the combustor, and the com-bustion noise is a random output of turbulentcombustion and radiates across a broadband offrequencies. However, some of the numericaland experimental techniques used to understandcombustion instability are relevant to combustionnoise, particularly the modelling of the linearwaves (acoustic, entropic and vortical) generatedby unsteady combustion within the complicatedgeometry of a gas turbine.

An additional link between broadband com-bustion noise and combustion instability is thatthe generation of pressure oscillations by entropywaves at the choked combustor, which is the mainsource of indirect combustion noise, has beenstudied extensively because it may play a role inthe feedback mechanism for combustion instabil-ity [24–27]. Indeed pressure waves generated inthis way are considered to be the key feedback-mechanism for a very low frequency combustioninstability [24,26,28,29] called ‘rumble’. Rumbleusually only occurs at start-up of aero-engines atsub-idle and idle conditions, and exhibits suchlow frequencies (in the range of 50 Hz to150 Hz) that it does not couple to the combustoracoustic resonances [28]. The feedback mechanismis thought to involve the entropy waves generatedby the unsteady combustion convecting down-stream to the combustor exit, where accelerationthrough the nozzle leads to an upstream propagat-ing pressure wave which perturbs the combustionfurther. It is the long convection time from flamezone to combustor exit that leads to the low

frequencies of self-excitation. The dissipation ordispersion of these entropy waves can stabilizeor destabilize the modes of the system, dependingon the configuration of the combustor and theform of the coupling [26,30].

In contrast to combustion instability, in broad-band combustion noise the unsteadiness in therate of combustion is mostly caused by the localturbulent flow and mixing, and is only veryweakly influenced by reflected waves from theboundaries. Predicting the sources of both directcombustion noise and entropy noise requires anunderstanding of the characteristics of the under-lying heat release fluctuations [10,31]. Thus,studies of combustion noise are inevitably linkedto the understanding of combustion modes offlames. Previous investigations on combustionnoise indicated that combustion noise is broad-band having random phase and the peak intensityis typically in the low frequency range around200–1000 [32–34].

This paper contains a review of progress inthese various areas and a discussion of currentchallenges. It focuses on comparison betweendirect and indirect (entropy and vorticity) com-bustion noise. The classical theory of combustionnoise of open flames is presented in Section 3. Sec-tion 4 presents the governing equation of combus-tion noise derived from the exact equations ofconservation of mass, momentum and energy. Itenables the different terms describing noisesources (direct combustion, accelerating entropyand vorticity) to be identified within a unifiedframework. Combustion noise of confined flamesand comparison between direct and indirect(entropy and vorticity) is presented in Section 5.The challenges and unexplored issues inunderstanding the combustion noise are discussedin Section 6. To give more insight about the


generation mechanism of combustion noise, ana-lytical solutions are presented in Section 7 for amodel combustor based on a low Mach numberapproximation. The numerical procedure of line-arized Euler equations based on the low-orderthermo-acoustic network model (LOTAN) andthe specific boundary conditions are brieflydescribed in Section 8. Section 9 discusses theresults obtained based on the analytical andnumerical solutions. First the validity of the ana-lytical solution, based on the low mean flow Machnumber approximation, is tested against thenumerical solution of the linearized Euler equa-tions in a modal combustor. Then the effects ofdifferent parameters such as frequency, stagnationtemperature ratio across the flame, and inletMach number on the magnitude of the acoustic,entropic and vortical waves are studied. Resultsfor the choked outlet nozzle and the magnitudeof indirect and direct noise generated are pre-sented in Section 9.3 for a range of different fre-quencies, inlet Mach numbers and stagnationtemperature ratios. Conclusions of the paper,open questions and possible areas for futureresearch are presented in Section 10.

3. Classical theory of combustion noise

A classical experiment on noise emitted fromcombustion was conducted by Thomas andWilliams [14] who filled a soap bubble with ahomogeneous premixed fuel–air mixture andignited it by a spark at the centre of bubble (seeFig. 4a). The constant pressure ignition produceda transient burning of mixture which resulted in atransient increase of the volume of bubble gases.As a consequence, a spherically symmetric pres-sure wave was emitted. Since the bubble size is

Fig. 4. (a) Instantaneous snapshots of an expanding spherical flproduced by expansion obtained for C2H4-O2-N2 mixture wcomputed pressure (Thomas and Williams [14] by permission

small compared to the acoustic wavelength, thistype of emission can be considered as a monopolepoint source of sound. The pressure in the soundwave emitted by the flame was recorded as a func-tion of time, Fig. 4b, in which good quantitativeagreement was obtained between the measuredpressure field and that calculated by measuringand differentiating the bubble radius-time curves.They showed that the sound pressure dependedlinearly on flame radius and on the square ofburning velocity. In this early study it was foundthat at a distance, r from the ignition position,the pressure fluctuation, p0(r,t) was given by

p0ðr; tÞ ¼ q0

4pr€V ðt � r=c0Þ; ð1Þ

where V(t) is the volume of the bubble. q0 and c0

are the mean air density and speed of sound in far-field through which the sound is propagating. Theprime denotes a perturbation from the mean and adot a time derivative. Thus €V ðtÞ ¼ d2V =dt2 is thesecond rate of change of the volume of the bubble.Equation (1) represents the fundamental principleof combustion noise: the sound is generated byunsteadiness in the rate of expansion by combus-tion of gas. In an open flame, it is the rate ofchange of the expansion rate that generates sound,€V ðtÞ. Steady combustion, leading to a constantrate of expansion, is silent. Although, in this sim-ple experiment the flame was laminar, the resultshave been proven to be equally applicable to noiseemission from open turbulent flames. In the phe-nomenological theory of Bragg [13] on direct com-bustion noise, it was assumed that the turbulentflame was a collection of eddies, which have theirown heat release rate. Each eddy acts as a mono-pole source of sound which is statistically inde-pendent of the neighbouring eddies.

ame front after ignition in time and (b) acoustic pressureith burning velocity of 55 cm s�1, + measured and oof the Royal Society).


Many studies (e.g. [11,35–37]) have tried toaddress the problem of combustion noise inturbulent flames and assumed that the turbulentflame can be regarded acoustically as an assemblyof monopole sound sources with differentstrengths and phases that are distributed in thereaction zone of the flame. Price et al. [35] andHurle et al. [36] found that the far-field soundpressure radiated by a turbulent premixed flamedepends on the rate of volume increase of the fueland oxidant in the combustion region, in whichthe rate of volume increase is proportional tothe rate of consumption of the fuel and oxidantin the flame. Price et al. [35] also pointed out thatthe same result holds for turbulent diffusionflames, if it is assumed that the fuel and air burnin stoichiometric proportions. According to thismodel, it has been shown that the instantaneoussound pressure p0iðr; tÞ in the sound waves thatemanate from the ith source element in the turbu-lent flame is given by

p0iðr; tÞ ¼q0

4prðE � 1Þ€viðt � r=c0Þ; ð2Þ

where r is the distance of observer from thesource, _vi is the volume rate of consumption offuel and oxidant in the flame element and E isthe volumetric expansion ratio of burned tounburned gases. For combustion at constant pres-sure E is equal to qu/qb where qb and qu are themean density of the burnt and unburnt gases.They also commented that for a compact flame,the total sound pressure in the far field is givenby the simple summation over all elements ofthe component pressures, p0iðr; tÞ in Eq. (2). Hurleet al. [36] confirmed this relationship experimen-tally by using the light emission from short-livedCH or C2 free radicals to monitor the rate of com-bustion. Since their pioneering work, the relation-ship between acoustic pressure perturbation andrate of heat release in combustion has been veri-fied using more advanced signal processing byShivashankara et al. [38].

3.1. Far-field pressure distribution of an openturbulent flame

Consider an example of an open turbulent pre-mixed flame shown in Fig. 5. Dowling and FfowcsWilliams [15] and Crighton et al. [39] showed thatby using Lighthill’s analogy [40] the far-fieldsound pressure fluctuation resulting from thedirect noise is given by [15,39]

p0ð~x; tÞ ¼ ðc� 1Þ4pc2

0j~xj

Zv

€qð~y; t � j~xj=c0Þd3~y; ð3Þ

where v is the volume containing the combustionregion or the flame brush. In the derivation ofEq. (3) ~x is considered to be in the far-field and

the flame is compact. Then r�1 ¼ j~x�~yj�1 is

approximately j~xj�1 and the retarded timet � j~x�~yj=c0 simplifies to t � j~xj=c0 [39]. In thisexpression it has been assumed that c the ratioof specific heats is independent of temperatureand that the combustion takes place at ambientpressure, so that qc2 ¼ cp0 ¼ q0c2

0. Ambient con-ditions in the region surrounding the combustionregion are denoted by the suffix 0 and _qð~y; tÞ isthe instantaneous rate of heat release per unitvolume in the combustion region. Equation (3)is valid in the far-field, i.e. for j~xj >> j~yj andj~xj >> c0=x, where x is a typical radian frequencyof the sound produced. The compact assumptionrequires that j~yj << c0=x for positions ~y withinthe combusting zone, i.e. the dimension of flamelength is small compared with the acoustic wave-length/2p. The same expression as Eq. (3) forpressure perturbation has also been derived byStrahle [41] (Eq. (3) slightly differs from the origi-nal one derived by Strahle [41], who erroneouslyconsidered the monopole to be related to the rate

of change ofR

S qb~u � d~S rather q0

RS~u � d~S, where

S is any surface enclosing the combustion zoneand ~u is the fluid velocity. Strahle’s result shouldtherefore be multiplied by the factor q0/qb).

Equation (3) clearly states that the rate ofchange in the total heat release rate generatespressure fluctuations. This expression applies toturbulent premixed, non-premixed and partiallypremixed flames as noted by Hurle et al. [36].Although, the characteristics of heat release mech-anisms and the physics underlying the combustionprocess in different modes affect the characteristicsof pressure fluctuation, Eq. (3) clearly manifeststhat noise depends on the combustion modethrough the temporal change of heat release ratein the combustion region [36,41].

For turbulent combustion, the acoustic pres-sure will be a stochastic variable and we are moreinterested in its statistical properties than anyinstantaneous value. The mean square pressureperturbation p02ð~xÞ can be obtained by squaringEq. (3) and taking the time average. This leads to

p02ð~xÞ ¼ c� 1

4pc20j~xj

� �2Z€qð~y; tÞd3~y �

Z€qð~z; tÞd3~z;

where the overbar denotes a time mean. Substitut-ing for ~z ¼~y þ~D, where ~D is the separationvector, results in

p02ð~xÞ¼ c�1

4pc20j~xj

� �2Zv

Zvt

€qð~y; tÞ€qð~yþ~D; tÞd3~Dd3~y;

ð4Þ

where vt is the volume over which the integrand isnon-zero. The ratio of the volume integralR

€qð~y; tÞ€qð~y þ~D; tÞd3~D to the maximum value of

its integrand €q2ð~y; tÞ is defined to be correlationvolume, which we denote by V corð~yÞ. It is the

Fig. 5. Schematic of turbulent flame illustrating coordi-nates for the description of the acoustic field.


volume over which €q in the turbulent combustionis correlated. WritingZ

€qð~y; tÞ€qð~y þ~D; tÞd3~D ¼ €q2ð~y; tÞV corð~yÞ;

Equation (4) becomes

p02ð~xÞ ¼ c� 1

4pc20j~xj

� �2 Zv

€q2ð~y; tÞV corð~yÞd3~y: ð5Þ

Equation (5) applies to any flame mode. It isclear that to predict the mean square far-fieldpressure perturbation we need to know €q2ð~y; tÞand V corð~yÞ. A flame is characterized by its meanrate of heat release �_qð~yÞ and there are well devel-oped methods, computational and experimental,to determine this. From Eq. (5) we can decomposethe requirements for modelling combustion noiseof an open turbulent flame into the need to modelthe correlation volume V corð~yÞ and the way inwhich €q2ð~yÞ is related to �_qð~yÞ. The ratio€q2ð~yÞ=�_q2ð~yÞ can be rewritten as the product

€q2ð~yÞ_q02ð~yÞ

� _q02ð~yÞ�_q2ð~yÞ :

The first term is the ratio of the mean square valueof the time derivative of _q0ð~y; tÞ to the mean squarevalue of _q0ð~y; tÞ itself. Hence, it represents a fre-quency squared. The second term is the squareof the rms value of _qð~y; tÞ to its mean, which is anon-dimensional measure of the amplitude ofthe perturbation.

Many scaling laws and empirical relations havebeen proposed to understand the physics of

combustion noise. The acoustic pressure waveperturbation given by Eq. (5) can be applied toall modes of turbulent combustion, premixed[10,11,33,35,42–48], non-premixed [34,49–53] andpartially premixed [54,55]. Candel et al. [10] givea review of empirical and computational model-ling of the noise source for premixed flames. Morerecently for premixed flames Swaminathan et al.[31,56] and Liu et al. [57] have analysed DirectNumerical Solutions (DNS) and advanced laserdiagnostics of premixed flames to obtain two-point spatial correlation of the rate of change ofthe fluctuating heat release rate. This approachleads to the development of a generic form forthe cross-correlation. The results has then beenapplied to predict the far-field Sound PressureLevel (SPL) using Eq.(5) from open flamesreported in [46].

The sound field from non-premixed and par-tially premixed flames is less well characterizedthan that radiated by premixed flames. This isdue to the difficulty in modelling the rate of heatrelease and the complexity of measuring thisquantity experimentally. Furthermore, in experi-ments the level of mixing of fuel and oxidizerwould cause the generated sound to feature char-acteristics of premixed flame sound. Nonetheless,a number of investigations studied the combus-tion noise radiated by non-premixed [34,49–53]and partially premixed flames [54,55] to under-stand the region bridging these combustionmodes. It was pointed out that higher levels ofsound are generated by turbulent non-premixedflames than by premixed flames at a similar veloc-ity [58]. For premixed flames, the region that themaximum acoustic output occurs are located closeto the flame tip [37], while, for turbulent non-premixed flames predominant sound sources aredistributed in the rear region of the reaction zone[35,58]. Further, using Mach number (M) scaling[54] it was demonstrated that the rms sound pres-sure generated by methane partially premixedflames scales with M5 compared to M3 for turbu-lent non-premixed methane flames.

4. General thermoacoustic sources and combustionnoise

In Section 3 we picked out the dominant noisesource when unsteady combustion occurs in anunbounded flow (Eq. (3)). But further insightcan be obtained by working from the full equa-tions of motion. In this way we are able to setcombustion noise within a framework thatextends to include jet noise and to identify entropyand vorticity as sources of sound. The startingpoint is Lighthill’s equation [40,59]. Lighthillshowed that the exact mass and momentum con-servation equations can be written in the form[40,59]:


@2q0

@t2� c2

0

@2q0

@xi@xi¼ @2T ij

@xi@xjð6Þ

where q0 = q � q0 and T ij ¼ quiuj þ p � p0ð�c2

0ðq� q0ÞÞdij � sij is the Lighthill stress tensor.The Kronecker delta is denoted by dij. q is thedensity, p the pressure and~u the particle velocity.q0, p0 and c0 denote the mean density, pressureand speed of sound in the acoustic far-field. sij isthe viscous stress tensor. In a high Reynolds num-ber isothermal flow Tij reduces to quiuj showingthat the unsteady Reynolds stresses generatesound in the same way as would a distributionof quadrupoles [39]. This is the basis of Lighthill’stheory of jet noise. However, when there is com-bustion, there are significant regions of the flowwhere the density is significantly different fromambient and the term c2

0ðq� q0Þdij within Tij isimportant. Equation (6) can be rearranged tohighlight this by arranging the wave operator tobe on the pressure fluctuations and writing

1

c20

@2p0

@t2� @2p0

@xi@xi¼ @2

@xi@xjðquiuj� sijÞ�

@2qe

@t2; ð7Þ

where qe ¼ q� q0 � ðp � p0Þ=c20 is the so-called

‘excess density’ [39]. This quantity vanishes inthe far-field but is non-zero in regions where theentropy is significantly different from that in thesurroundings. The thermoacoustic sound sources,o2qe/ot2 and o2sij/oxioxj are nonzero in the regionswith irreversible processes. Viscosity is not a sig-nificant parameter in flow noise at least in regionsaway from the boundary layers, so that there isvery little loss of generality in assuming flowmotions to be inviscid. Thus, in Eq. (7) the viscousterm o2sij/oxioxj is small and can be ignored. Ther-modynamic relationships can be used to deter-mine the strength of the source o2qe/ot2. Thedetails are given by Dowling in Crighton et al.[39]. First, the partial time derivative is writtenin a form of material derivative as

@qe

@t¼ Dqe

Dt� qe

qDqDt� @ðuiqeÞ

@xi; ð8Þ

where the material derivative D=Dt ¼@=@t þ~u � r and the continuity equation has beenused to replace oui/oxi with � q�1Dq/Dt. Theenergy equation for a gas made up of N chemi-cally reacting species results in [31,39]

DqDt¼ 1

c20

DpDtþ a

cp

@qi

@xi�sij

@ui

@xjþXN

n¼1

@h@Y n

��q;p;Y m

qDY n

Dt

!;

ð9Þwhere Yn is the mass fraction of the nth species, hthe enthalpy and qi is the heat flux vector given byqi ¼ �k@T=@xi þ q

PNn¼1Y nhnV n;i with k the

thermal conductivity of the mixture and Vn,i thediffusion velocity of the nth species in direction i.The specific heat at constant pressure is cp, the

volumetric expansion coefficient is a and for anideal gas is equal to 1/T, and a=cp ¼ ðc� 1Þ=c2

0.Equation (9) demonstrates that the density of amaterial particle changes because of the pressurevariations in a compressible fluid and also becauseof the expansion caused by heating. If qe in Eq. (8)is written explicitly as q� q0 � ðp � p0Þ=c2

0, Eq.(9) can be used to replace Dq/Dt this leads to

@qe

@t¼ aq0

cpq@qi

@xi�sij

@ui

@xjþXN

n¼1

@h@Y n

��q;p;Y m

qDY n

Dt

!�@ðuiqeÞ

@xi

� 1

c20

1�q0c20

qc2

� �DpDt�ðp�p0Þ

qDqDt

� �:

Substituting into Eq. (7) results in

1

c20

@2p0

@t2� @2p0

@xi@xi

¼� @@t

aq0

cpq

XN

n¼1

@h@Y n

��q;p;Y m

qDY n

Dtþ@qi

@xi� sij

@ui

@xj

!" #

þ @2

@xi@xjðquiuj� sijÞþ

1

c20

@

@t1�q0c2

0

qc2

� �DpDt

��ðp�p0Þ

qDqDt

�þ @2

@xi@tðuiqeÞ: ð10Þ

Using the chain rule of differentiationXN

n¼1

@h@Y n

��q;p;Y m

qDY n

Dt

¼XN

n¼1

@h@Y n

��T ;p;Y m

xn �@h@q

��p;Y n

XN

n¼1

@q@Y n

��T ;p;Y m

xn

!

�XN

n¼1

@h@Y n

��q;p;Y m

r:Jn; ð11Þ

where xn is the production rate per unit volume ofspecies n by reaction, and Jn,i = qVn,iYn is the fluxof species n by diffusion in the direction i. The firstterm on the right hand side of Eq. (11) representsthe heat release rate per unit volume and the sec-ond term describes the volumetric expansion dueto non-isomolar combustion. The third term isdetermined by the effects due to species diffusion.

Truffaut et al. [60] examined the additionalsource of noise associated with non-isomolarcombustion described by the second term on theright-hand side of Eq. (11). When fuels are burntin air, the reactive species are strongly diluted ininert nitrogen and the contribution of molarexpansion is small compared to the direct heatrelease term (the first term of the right-hand sideof Eq. (11)). Thus, for air-breathing combustionsystems the molar production term can beneglected [61]. However in some industrial appli-cations, such as welding torches, fuels are burntin pure oxygen and then the change in the numberof moles is no longer negligible, partly because thechemical species are no longer diluted in nitrogen


but also because the high combustion temperature(>3000 K) leads to strongly dissociated combus-tion products [60].

Substituting Eq. (11) into Eq. (10) leads to aninhomogeneous wave equation which can be writ-ten as

1

c20

@2p0

@t2� @2p0

@xi@xi¼ W 1

:

þW 2 þ W 3

:

þW 4

:

; ð12aÞ

where

W 1 ¼ �aq0

cpq

XN

n¼1

@h@Y n

��T ;p;Y m

xn �@h@q

��p;Y n

�XN

n¼1

@q@Y n

��T ;p;Y m

xn

!�XN

n¼1

@h@Y n

��q;p;Y m

r:Jn

þ @qi

@xi� sij

@ui

@xj

�; ð12bÞ

W 2 ¼@2

@xi@xjðquiuj � sijÞ; ð12cÞ

W 3 ¼1

c20

1� q0c20

qc2

� �DpDt� ðp � p0Þ

qDqDt

� �; ð12dÞ

W 4 ¼@

@xiðuiqeÞ ð12eÞ

The details of the derivation of this equation aregiven explicitly by Dowling in Crighton et al.[39]. Bailly et al. [62] obtained a similar equationbased on an extension of Lilley’s equation [63]to reactive flows. Lilley’s equation includes somemean flow propagation effects with propagationthrough a parallel mean shear flow being includedin the operator on the left-hand side of theequation.

Assuming that the terms on the right hand sideof Eq. (12a) are known, the problem is equivalentto an inhomogeneous wave equation. These termsrepresent the various sources of sound generation.The first term W 1 is of monopole type anddescribes the sound generated by irreversible flowprocesses from the unsteady heat release rate andthose of non-isomolar combustion, species diffu-sion, heat diffusion and viscous dissipation[64,65]. When exothermic reactions take place,the first term in Eq. (12b) due to the rate of heatrelease is the most significant. The term r �~q inEq. (12b) is the heat-conduction noise source, dis-cussed by Sinai [66]. The second term W2 is thefamiliar quadrupole source of Lighthill’s jet-noisetheory associated with the velocity fluctuations. Itis well known that it leads to an acoustic intensitythat scales with Mach number to the eight power,M8 [39]. As noted by Powell [67], in a low Machnumber isothermal flow W2 is approximately

equal to r � ðq~x�~uÞ þ r2 12qj~uj2

� �. It is this

source that generates sound when vorticity isaccelerated and, when combustion occurs in a

confined geometry with a downstream constric-tion, leads to the indirect combustion noise sourcewe introduced in Section 2. We will discuss thisfurther in Section 5. The third term W3 also leadsto an M8 scaling [39] but with a different coeffi-cient from the classical Lighthill’s term. It is onlyappreciable if qc2 in the combustion region is notequal to q0c2

0, the value in the far-field. Sinceqc2 = cp and pressure is nearly constant through-out a low Mach number flow, this noise sourceshould be very small [10]. As pointed out by Sinaithis term is only effectively nonzero in a fluid withvariations in its specific heats and even then hefound that this mechanism of noise generationto be insignificant [66]. The last term W4 is dipolein nature and describes the effect of momentumchanges on density inhomogeneities; it is theentropy noise source discussed in Section 2.

In the combustion region xn the rate of pro-duction species n is nonzero and the first termon the right hand side of Eq. (12a) describes thestrong monopole sound source of combustion[39]. It has been shown by Flemming et al. [52]and Ihme et al. [53] that when unsteady combus-tion occurs in a low Mach number flow, this termis about two orders of magnitude larger than theother sources demonstrated in Eq. (12a). If theaverage molecular weight is constant, the secondterm in the first part of Eq. (12a) is neglected.When the diffusion of species in neglected thenfor the combustion of hydrocarbons in air

�XN

n¼1

@h@Y n

��q;p;Y m

xn ¼ _q; ð13Þ

where _q is the heat release rate per unit volume.For an ideal gas a/cp = (c � 1)/c2. If c is assumedto be independent of temperature and the com-bustion takes place at ambient pressure, as is thecase for example in an open turbulent flame thenwe have qc2 ¼ cp0 ¼ q0c2

0. Then applying Eq. (13)into Eq. (12a) results in an equation for combus-tion noise as

1

c20

@2p0

@t2� @

2p0

@x2i¼ ðc� 1Þ

c20

€qð~y; tÞ: ð14Þ

Using the free-space Green function [39] gives thefar-field pressure perturbation due to an openflame as

4pc20j~xjp0ð~x; tÞ¼ ðc�1Þ

Zv

€qð~y; t�j~xj=c0Þd3~y; ð15Þ

recovering the result of the simplified analysis inEq. (3). Equation (15) predicts that in the far-fieldfor a compact flame, i.e. when the wavelength ofthe emitted sound is large compared to the flamelength, the acoustic pressure perturbation is pro-portional to the time derivative of the total heatrelease evaluated at a retarded time. The variationsof c arising due to temperature inhomogeneities


can generate monopole sound when it is subjectedto pressure fluctuations, as noted by Dowling [68],Strahle [69] and Sinai [66], but this effect is negligi-ble compared to the strong combustion source inEq. (15). In free space, the contributions fromthe entropy and vorticity terms (in Eqs. (12e) and(12c)) are also smaller than the term retained inEq. (15) but they can have significant effects whenthere are strong variations in the mean flow. Addi-tional combustion noise is generated when theseentropy perturbations are accelerated by the flowthrough constrictions in an engine [11,18]. The rel-ative contribution to the overall radiated noisefrom indirect and direct combustion noise compo-nents is an ongoing issue [9,10] that we examinefurther in the following sections.

5. Combustion noise of confined flames

In practical applications it is more common forcombustion to be confined than to occur inunbounded space. We introduce the noise of con-fined flames by investigating the first the smallamplitude flow perturbations that can occur inthe flow through a duct/pipe of constant cross-sectional area with uniform mean flow properties.Dowling and Stow [5] showed that by rewritingthe flow properties as a sum of time-averaged(denoted by an overbar) and fluctuating (denotedby a prime) components, the linearized conserva-tion equations maybe rearranged to describe thedevelopment of acoustic, entropy and vorticityfluctuations. It was shown that with no heatsource, any linear perturbation can be thoughtof as the sum of three types of disturbances [5],(i) an acoustic disturbance that is isentropic andirrotational; (ii) an entropic disturbance that isincompressible and irrotational; and (iii) a vorticaldisturbance that is incompressible and isentropic.These three types of disturbances are independentand when linear can be considered separately andsuperimposed [5]. Acoustic disturbances involvevelocity, pressure, density, and temperature fluc-tuations. For the acoustic disturbance the entro-pic and vortical fluctuations are zero, i.e. S0 = 0

and ~n0 ¼ 0, hence

1

c20

D21

Dt2�r2

� �p0 ¼ 0; ð16Þ

where D1/Dt is the convected derivative@=@t þ �~u � r and �~u is the mean fluid velocityassumed to be uniform. Disturbances of this typeare acoustic waves and, relative to the fluid, theypropagate at the speed of sound.

The entropic disturbances do not involve fluidmotion or pressure disturbances. They consist oftemperature and density fluctuations. Incompress-ible and irrotational entropy fluctuations, i.e.p0 = 0 and ~u0 ¼ 0, are expressed as

D1S0

Dt¼ 0: ð17Þ

This disturbance is stationary relative to the fluid,that is, it is convected with the mean flow. Thisdisturbance can be thought of as an entropy wave,and is sometimes referred to as a ‘convected hotspot’.

Vortical waves consist purely of vortical fluidmotion without pressure, density, and tempera-ture perturbations. For the vortical disturbance,p0 = q0 = 0 and hence the development of theseincompressible and isentropic vorticity fluctua-tions are described by

D1~n0

Dt¼ 0: ð18Þ

The vorticity wave also convects with the meanflow.

The linearized governing equations for thesethree types of perturbations indicate that theyare uncoupled in an uniform mean flow, and thatthe acoustic waves propagate with the speed ofsound relative to the mean flow, whilst entropyand vorticity fluctuations are advected with themean flow. If the mean flow is zero, the entropicand vortical waves do not convect and only theacoustic waves propagate. Using the full non-linear Navier–Stokes equations, Chu and Kov-asznay [19] analyzed the interaction of these threewaves. They found that in a quiescent medium theinteraction would be a second-order effect [19].The entropy mode, which is silent in a uniformmean flow, may transfer energy to acoustic andvorticity modes and vice versa by a nonlinearinteraction. When spatial gradients in steady flowproperties exist, such as a combustion zone or theacceleration of the mean flow through a nozzle, allthree types of linear waves can become coupled[70]. Palies et al. [70] investigated a vorticity wavegenerated by such an acoustic wave incident on anaerofoil cascade. The interaction of perturbationswith blade rows has been extensively investigatedin the field of turbomachinery, see for example theclassical textbook of Horlock [71] and references([72–74]). Acoustic waves can also be generatedas entropy waves are convected by the mean flowthrough a nozzle [18,75]. Conversion of entropyfluctuations by a cascade has been worked outanalytically by Cumpsty and Marble [76] andCumpsty [77]. It is shown in these references thatthe interaction gives rise to acoustic pressurewaves.

Many real combustion chambers featurechoked exit nozzles. Numerous studies in the pasthave studied reflection and generation of sound bythe passage of acoustic and entropic perturbationsthrough a nozzle in the linear (e.g. [12,18,26,78–80]) and nonlinear regime [81,82] and for a broadrange of frequencies. At mid to high frequencies,the wavelength of the entropy perturbations, i.e.


2pj~uj=x, is small compared with the turbulentlength scale. Thus, these perturbations diffusedue to high turbulent mixing in the burned gasesbefore the nozzle guide vanes and hence have neg-ligible amplitude by the time they reach the nozzle[83,84]. Whilst, for low frequencies below a fewhundred Hz, entropy perturbations have longwavelengths compared with the turbulent lengthscale. Hence, although these perturbations arealso subjected to intense turbulent mixing, thereis no significant diffusion nor dissipation of suchentropy perturbations as they convect betweenthe flame region to turbine inlet [82]. Further,recent studies have shown both numerically andexperimentally the significance of entropy noiseas a feedback for combustion instabilities andthermo-acoustic oscillations [26,85–87]. Thus, weknow that at these lower frequencies entropy per-turbations persist to the nozzle guide vanes withsignificant amplitudes.

In the linear regime, the response of the com-bustor exit to incoming perturbations has beenstudied theoretically by Marble and Candel [12].They considered the choked nozzle to be one-dimensional and compact, an approximation thatworks well when all wavelengths are much longerthan the nozzle. They derived analytical expres-sions for the relationship between the amplitudeof the acoustic waves generated and the tempera-ture fluctuations of the incident entropy waves.The generation of acoustic waves is determinedby the condition that the non-dimensional massflow rate be constant at the choked end. The indi-rect combustion noise is therefore related to theunsteady mass flux that must occur when temper-ature inhomogeneities enter the outlet throat. Thecompact nozzle theory of Marble and Candel [12]was extended by Cumpsty and Marble [76,88] andCumpsty [77] to study entropy noise produced in agas turbine at the combustor outlet and turbineblade stages. A significant increase of entropynoise generation was found with the increase ofpressure drop over a turbine stage. Stow et al.[79] studied both analytically and numericallythe reflection and transmission of downstreampropagating acoustic, entropy and vorticity wavesincident on a choked exit nozzle. They reportedthat for a compact choked nozzle the boundarycondition of Marble and Candel [12] is applicableeven for circumferential waves.

Bohn [80] investigated the one-dimensionalresponse of a subsonic nozzle flow to pressureand entropy disturbances and pointed out theimportance of nozzle shape on the noise generatedat higher frequencies when the nozzle is no longercompact [80]. Stow et al. [79] and in a separatestudy Goh and Morgans [30] extended the com-pact solution of Marble and Candel [12] fromlow to mid frequencies through an asymptoticexpansion of the linearized Euler equations fornon-compact nozzles. They showed that the

reflection and transmission across a nozzle canbe extended to non-zero frequencies using aneffective nozzle length that depends on the meanflow through the nozzle and hence on the nozzleshape. Moase et al. [89], for choked nozzle andsupersonic diffusers, and Giauque et al. [90], forsubcritical nozzles, extended the approach ofMarble and Candel [12] to non-compact nozzlesby solving the linearized Euler equations. Theycomputed the linear transfer functions for theupstream and downstream acoustic wavesgenerated by an incident entropy disturbance.Duran and Moreau [91] studied analytically theacoustic and entropy transfer functions of quasi-one-dimensional nozzles for both subsonic andchoked flows with and without shock waves. Theirstudy extended both the compact nozzle solutionobtained by Marble and Candel [12] and the effec-tive nozzle length proposed by Stow et al. [79] andby Goh & Morgans [30] to arbitrary frequenciesthrough an asymptotic expansion of the linearizedEuler equations. Their study can be applied to anynozzle geometry and different flow conditions.They found that for both choked and subsonicflows, the modulus and the phase of the transferfunctions of linear waves across the nozzle havea strong dependence on the frequency.

Bloy [81] investigated the non-linear regimeusing the method of characteristics in one-dimensional flow and calculated the pressurefluctuations generated by the convection of a tem-perature disturbance (entropy wave) with low fre-quency and large amplitude passing through asubsonic nozzle. Huet and Giauque [82] alsoextended the compact nozzle model of Marbleand Candel [12] to the nonlinear domain in orderto study the generation of sound by the passage ofacoustic or entropy perturbations through a noz-zle in the low frequency limit. The accuracy oftheir proposed second order model for largeamplitude entropy perturbations was evaluatedby varying the inlet and outlet Mach numbers ofthe nozzle. They found that it is accurate for asupercritical nozzle for both types of wavesconsidered.

Leyko et al. [78] used a simple quasi-1-D com-bustor model for an aero-engine terminated by achoked nozzle with the assumption of isentropicmean flow and quantified both direct and indirectnoise. The analysis performed was in a cold flame,a simplification in which a flame is considered togenerate only fluctuations in heat release, withno mean rate of heat release. With this simplifica-tion, the mean flow conditions upstream anddownstream of the flame were assumed to be thesame. They showed that the linearized acoustictheory of Marble and Candel [12] agrees well atlow frequencies with the numerical resultsobtained by solving the Euler equations ofmotion. Further, they [78] showed that the ratioof indirect-to-direct noise depends on the Mach


number in the flame zone and the Mach numberat the nozzle outlet (see Fig. 6). This ratio is smallfor laboratory experiments but large in most realaeroengines, indicating that indirect noise is thenthe dominant source and can exceed the directnoise by one order of magnitude but is negligiblein most laboratory experiments [78].

In an extension to the work done by Leykoet al. [78], Duran and Moreau [91] studied theratio of indirect to direct noise in a cold flamefor different frequencies and Mach numbers. Theyproduced a generalization of the indirect to directnoise ratio for different frequencies and concludedthat for high frequencies, the indirect to directnoise ratio decreases as frequency increases, whichmeans indirect noise is only significant at low fre-quencies and could be neglected at high frequen-cies. Using coherence analysis, Muthukrishnanet al. [92] deduced from experiments on a test rigfor an aero-engine combustor that broadbandindirect entropy noise is the principal source ofthe engine core noise if the combustor is chokedat the exit. But up until now, there has been littlesystematic direct experimental proof of theimportance of entropy noise in the ambient noiseradiated from a complete aeroengine. This is pre-sumably because of the difficulty of separating thedifferent noise generating processes leading todirect and indirect combustion noise. To do this,an experiment, generating entropy waves by elec-trical heaters was performed by Bohn [80] andZukoski and Auerbach [93] in seventies. In theseexperiments, however, the amplitude of theinduced temperature fluctuation generated wastoo low (only about 1 K), there was little scopefor parametric variations, and the data acquisitionand processing system did not have a sufficientlyhigh-resolution to separate out the direct and indi-rect noise components. The problem was studiedagain recently when Deutsches Zentrum fur Luft-und Raumfahrt (DLR) built a small-scale generictest rig Entropy Wave Generator (EWG) [94–96]and Vorticity Wave Generator (VWG) [97] tostudy the indirect combustion noise. They showedthat small non-uniformities of temperature canindeed generate significant noise when acceleratedwithin a nozzle.

Figure 7a shows the experimental measure-ments of Bake et al. [95] for the amplitude ofthe generated sound pressure versus the amplitudeof the temperature fluctuation for two nozzleMach numbers. This figure confirms the linearizedapproach of Marble and Candel [12] in which theamplitude of the generated entropy noise isassumed to increase linearly with the amplitudeof the temperature pulse. They also investigatedthe variation of the amplitude of the generatedpressure wave for fixed inlet temperature variationas the mean inlet nozzle Mach number increased.The results are shown in Fig. 7b and demonstratethat the pressure amplitude increased nonlinearly

with an increase of nozzle Mach number up to acertain amount (Fig. 7b). Further increase inMach number resulted in decrease of the pressurewave amplitude [95].

The DLR experiment generated interest in thecommunity and initiated multiple theoretical andnumerical methods to explain the experimentalresults [78,98–101]. Leyko et al. [99] studiednumerically and analytically the DLR experi-ments in an axisymmetric and a fully three-dimensional configuration. They reported thatthe linear acoustic and the low frequency compactnozzle hypothesis of Marble and Candel [12] isapplicable for the experimental results of EWGof DLR [95]. They also demonstrated that thepressure signals measured in the DLR experimentresult from the conversion of entropy perturba-tions into acoustic waves due the strong meanvelocity gradient in the nozzle, as well as fromthe acoustic wave reflection with exhaust systemdownstream of the nozzle at the outlet [99]. TheEWG has also been studied numerically usingthree-dimensional compressible URANS byMuhlbauer et al. [100]. The pressure fluctuationsand spectra they calculated agreed well with theexperiments in the EWG. They pointed out thatthe minor discrepancies between their simulationand the experiments could be removed by usinga time-domain application of the measuredimpedance boundary condition describing acous-tics of the exhaust as a function of the frequency.In contrast to the supersonic nozzles and subsonicconvergent nozzles studied in [78], a subsonic con-figuration has been studied by Giauque et al. [90]and Duran et al. [102]. The analytical results ofDuran et al. [102] based on a compact nozzlehypothesis for unchoked subsonic nozzle (withthroat Mach number of 0.7) showed that thestrength of indirect noise is two orders of magni-tude smaller than the direct noise suggesting thatthe measured pressure waves at the outlet ofEWG were not caused by entropy waves butrather by the acoustic waves generated by theheating device. Their numerical results for anunchoked subsonic nozzle based on non-compactnozzle theory illustrated that the strength ofindirect noise increases with frequency and thecompact nozzle solution is only correct for verysmall Helmholtz number, f ‘=�c (zero frequencies)where ‘ is the nozzle length [102]. However, Leykoet al. [78] showed that the compact hypothesis isvalid for dimensionless frequencies up tof ‘=�c ¼ 0:2 for supersonic and subsonic conver-gent nozzles.

With the aim of calculating the combustion-generated noise at the outlet of the aero-engine,Leyko et al. [103] and Duran et al. [104] combinedLES with analytical models of Cumpsty andMarble [88] and applied them to an isolated bladerow to differentiate the contribution of direct andindirect noise sources. Mishra and Bodony [105]

Fig. 6. Evaluation of the ratio of indirect to direct noise generated by a model combustor with a compact isentropicdownstream nozzle. The noise is calculated in the downstream duct (based on Eqs. (34) and (35) in Leyko et al. [78]).

Fig. 7. (a) Acoustic pressure amplitude of generated entropy noise as a function of temperature perturbation for twodifferent nozzle Mach numbers M = 0.15 and 1; (b) acoustic pressure wave versus nozzle Mach number for two differentamplitudes of temperature perturbation [95].


also studied application of actuator disk theorynumerically by comparison of two-dimensionalsimulations with the analytical model of Cumpstyand Marble [88] in the presence of small ampli-tude entropy perturbations. They [103–105] deter-mined the limits of the compact nozzle model fora wide range of frequencies and evaluated theindirect combustion noise as entropy waves con-vect through stator and rotor blade rows.

6. Current challenges

It is noted that currently there is, in general, lit-tle agreement on the importance of indirect noiseto the problem of combustion noise. This perhapsstems from a number of issues on understandingand modelling of entropic and vortical waves incombusting flows. For instance, hitherto, theoutput of the combustion chamber in terms of


acoustic (direct combustion noise), entropic (con-vecting hot and cold spots) and vortical perturba-tions (convecting vortex structures) has not beenwell characterized. The relative magnitudes ofthe waves generated by unsteady combustion cer-tainly depend on the frequency and mean flowconditions. Furthermore, the decay of entropyand vorticity perturbations in a combustion cham-ber is not clear. In addition, the general problem ofindirect noise requires that the impedance condi-tion and transmission property at the combustorchoked outlet be known. Incident entropy andvorticity waves cause both reflection and transmis-sion of acoustic waves. But the reflected wavescome back into the combustor where they reflectfrom both the flame zone and the combustor inlet.On occasion, they can even perturb the rate ofcombustion, which can contribute to a thermoa-coustic instability. Hence, the transmission andreflection of indirect noise needs to be evaluated.The difficulties in solving these issues may bebroadly summarized specifically as follows:

� Limitations in the modelling of the exitnozzle dynamics as the convertor of entro-pic and vortical disturbances into acousticwaves;

� Little quantitative information on the dis-sipation of entropy and vortical waves;

� Lack of a comprehensive understanding ofthe generation of linear waves in reactiveflows.

6.1. Exit nozzle dynamics

The analytical derivations of Marble and Can-del [12] and the extensions described in Section 5have been the main tools to model the responseof choked exits. These models assume a mean flowuniform over the cross-sectional area and onlyaxial and circumferential perturbations. AlthoughLeyko et al. [78] have shown that they work wellon the one-dimensional mean flow of the experi-ment of Bake et al. [95] their experimental valida-tion on geometries representative of aeroenginecombustors is currently very limited.

Hield and Brear [106] extended the two micro-phone method of Seybert and Ross [107] by addinga thermocouple to the measurement system toquantify the incident entropy wave and so be ableto investigate of the nozzle response to an entropicdisturbance. They [106] applied this new method toboth an open and a choked exit of a premixed com-bustor with a bluff-body flame stabiliser undergo-ing thermo-acoustic limit cycles. For the openexit, the acoustic reflection showed good agree-ment with the classical theory. However, in the caseof the choked exit neither the reflected pressure orentropy response agreed with Marble and Candel’s[12] linear boundary condition. It was argued that

this may be expected, given the large amplitude ofthe forcing and the possible nonlinearity of thechoked exit due to periodic nozzle unchoking, assuggested by Moase et al. [89].

The Marble and Candel [12] theory neglectsany radial variation in the mean and unsteadyflow field. While three-dimensional disturbanceswere investigated by Crocco and Sirignano [108]further work would be needed if incident three-dimensional disturbances are important in aeroen-gine combustors. There is an urgent need forexperiments on aeroengine combustors, wherethe mean flow will vary around the annulus andacross the radius, to enable the boundary condi-tions to be validated.

6.2. Dispersion of entropy waves

Between the flame, where the entropy waves aregenerated, and a choked combustor exit, thesewaves are convected by a flow that is non-uniformover the combustor cross-section and has intenseturbulent fluctuations. As stated by Sattelmayer[84], the non-uniformity in the mean-flow profileleads to residence times that vary across thecross-section. Thus, entropy disturbances gener-ated at different radial positions arrive at the nozzlewith different time delays and hence phases. Thiswill tend to reduce their cumulative effect and hencereduce the reflected acoustic wave, an effect that ismore important at high frequencies. Sattelmayer’sstudy [84] considers some essential physics of thedispersion problem. Partial diffusion of these waveschanges their relative phase and influences thesound generation generated by their interactionwith the nozzle, which can modify the resultantentropy noise [84]. The turbulence in the flow canalso diffuse the entropy disturbances and thereforereduce their potential for sound generation. Theimportance of the dispersion of entropy waveshas been noted by a number of authors[5,84,106,109]. However, these effects have notbeen considered in simplified analytical and numer-ical models of entropy noise (e.g. [12,30,78,89,91]).Thus, attempts to analyse this effect are quite rare.

Very recently Morgans et al. [109] studiedentropy wave advection within a turbulentchannel-flow. The study was based on DirectNumerical Simulation of entropy wave convectionwithin a simplified fully-developed turbulent flowbetween two parallel plates. The effect of heataddition due to combustion was neglected, thecoefficient of transport phenomena was consideredto be constant, and the flow assumed to be incom-pressible. Thus, the acoustic fluctuations wereassumed to be negligible and the temperature andentropy fluctuations scale with one another. Theyintroduced a Gaussian perturbation in the temper-ature field at the inlet of the channel and studiedthe transport of the entropy fluctuations [109].They found the entropy wave dissipation to be


negligible between the flame position and the endof the combustor. They pointed out that loss inentropy wave strength during differential convec-tion by a mean flow which varies across a combus-tor cross-section will in practice be more importantin a real aeroengine combustor [109]. Nevertheless,because of the general complexity of transport anddiffusion in turbulent exhaust flows some impor-tant aspects of the problem are still unexploredand most of the current understanding of the dis-persion process is rather intuitive.

6.3. Dynamics of linear wave generation

Survival of entropy waves throughout thecombustor is influenced by both dispersion andby the initial strength of the disturbance. Thosedisturbances with larger initial amplitude havenaturally a better chance of reaching the exit noz-zle. The relative strengths of the pressure andentropic waves generated by unsteady combustiondepend on the frequency and on the mean flow. Aflame generates stronger entropy disturbances atlow frequencies [110]. The current modelling ofentropy generation invariably assumes that thedisturbances are one-dimensional [5]. Thus thedynamics of entropy generation become a func-tion of the global heat release. The spatial varia-tion of entropy generation within the flameregion is ignored. While this might be a reason-able approximation for laboratory experimentswhich have combustion within cylindrical tubes,aeroengine combustors are usually annular andcombustion noise is important at frequencieswhere the first circumferential acoustic modearound the annular propagates. The unsteadycombustion can therefore excite disturbanceswhich vary with azimuthal angle around the cir-cumference. In the next following sections we willinvestigate linear wave generation and reflectionin annular combustors.

1 While our analysis for n > 0 is for annular combus-tors, the plane wave case n = 0 is relevant for any cross-sectional shape.

7. Analytical study of combustion noise

In this section we will discuss the general solu-tion for linear unsteady flow in a cylindrical orannular duct of uniform cross-sectional area. Itcan be thought of as a sum of acoustic, entropicand vortical waves. We then determine the gener-ation of these waves by unsteady heat input andcompare their relative magnitudes.

7.1. Linear waves in an annular duct

We consider the form of perturbations thatoccur in the gap between two concentric cylinders.The straight duct is assumed to have cross-sectional area a, and the mean flow is axial andindependent of circumferential or radial position.The flow is assumed to be inviscid, with pressure

p, density q and velocity~u. We will assume a per-fect gas, p = RgasqT, and so, T may be written interms of p and q. A linear disturbance in a straightduct can be thought of a sum of acoustic, entropyand vorticity waves, with acoustic waves propa-gating both upstream and downstream, whileentropy and vorticity disturbances convect withthe mean flow. Using cylindrical polar coordi-nates, x,r,h, the velocity field is denoted by~u ¼ ðu; v;wÞ and the duct has inner and outermean radii ri and ro, respectively. In annular gasturbines, the radial gap ro � ri of the combustoris typically much shorter than the circumference.In such situations we may approximate the flowby considering the case when the annular gap isnarrow, that is ri � ro with any variation of pres-sure in the radial direction negligible with thenthe radial velocity, v, to be zero [79].

This flow is taken to be composed of a steadyaxial mean flow denoted by bars and a small per-turbations denoted by primes, e.g. p ¼ �pðxÞþp0ðx; h; tÞ. The disturbances are assumed to havefrequency x in which the temporal dependenceis of the form eixt. We restrict attention to positivex without loss of generality. Further, the angulardependence of the perturbations is taken to be ofthe form einh with circumferential wavenumber n,which we will take to be a non-negative integer.Hence, we consider the disturbances to be of theform p0 ¼ Re½pðxÞeixtþinh� etc [111]. Since,�v ¼ �w ¼ 0 for the uniform mean flow, the solutionof the convected wave equation in Eq. (16) is[79,112,113]

p0ðx; r; tÞ ¼ ðAþeikþx þ A�eik�xÞeixtþinh; ð19Þwhere the axial wavenumbers for the two acousticwaves are

�ck� ¼Mx� x2 � x2

cut�off

h i1=2

1�M2; ð20Þ

where �c is the mean speed of sound, xcut�off is the

cut-off frequency of the duct ðn�c=rÞ=ð1�M2Þ1=2

and M is the mean Mach number. Whenx > xcut�off, A+ represents a downstream propa-gating acoustic wave and A� represents anupstream propagating wave. For x < xcut�off,the term within the square root in Eq. (20) is neg-ative, its square root is a purely imaginary num-ber, and thus the waves have exponentialbehaviour. Then, k+ and k� are chosen such thatthe Im(k+) > 0 and Im (k�) < 0. Then A+ repre-sents a downstream decaying disturbance andA� represents an upstream decaying disturbance.The plane wave n = 0 propagates at all frequen-cies.1 The first circumferential mode propagates


at frequencies above ð�c=rÞ=ð1�M2Þ1=2. The exact

value of this frequency depends on the meanradius of the annular combustor and the meantemperature and Mach number within it. This istypically in the range of few hundred Hz and istherefore where combustion noise is mostimportant.

Entropy and vortical disturbances convectwith the mean flow as shown in Eqs. (17) and(18) and so for disturbances with frequency xand mode number n, their space–time dependenceis eixtþinhþik0x, where k0 ¼ �x=�u. The entropyS = cv loge(p/qc) and for linear perturbationsS0 ¼ cvp0=�p � cpq0=�q. Rearranging this leads to

q0 ¼ p0

�c2� S0�q

cp: ð21Þ

We choose to write the constant amplitude of theentropy wave as AEcp=ð�q�c2Þ, then after substitut-ing for p0 from Eq. (19)

q0 ¼ 1

�c2Aþeikþx þ A�eik�x � AEeik0x

eixtþinh: ð22Þ

The velocity perturbations ~u ¼ ðu; 0;wÞ can befound from the equation of continuity for linearperturbations

ixq0 þ �u@q0

@xþ �qr �~u0 ¼ 0; ð23Þ

or from the momentum equations for the pertur-bations which become

ix�qu0 þ �q�u@u0

@x¼ � @p0

@x; ð24aÞ

ix�qw0 þ �q�u@w0

@x¼ � in

rp0: ð24bÞ

The contribution to the velocity perturbationsfrom acoustic wave can be calculated easily fromEq. (24):

u0 ¼ �kþ�qaþ

Aþeikþx þ�k��qa�

A�eik�x

� �eixtþinh; ð25aÞ

w0 ¼ �nr�qaþ

Aþeikþx þ �nr�qa�

A�eik�x

� �eixtþinh; ð25bÞ

where a� ¼ xþ �uk�. To this we can add any vec-tor that has zero divergence and zero convective

Fig. 8. One-dimensional disturbances in a duct wit

derivative, since such a vector will give no contri-bution to either Eq. (23) or (24). We choose towrite this additional vector as

u0 ¼ n�q�c

AV eixtþinhþik0x; ð26aÞ

w0 ¼ � k0r�q�c

AV eixtþinhþik0x: ð26bÞ

The radial component of the velocity needs to bezero to satisfy rigid wall boundary conditions onthe wall of the thin annular combustor. The fullexpression for the velocity perturbations are givenfrom a superposition of Eqs. (25) and (26),

u0 ¼ �kþ�qaþ

Aþeikþxþ�k��qa�

A�eik�xþ n�q�c

AV eik0x

� �eixtþinh; ð27aÞ

w0 ¼ �nr�qaþ

Aþeikþxþ �nr�qa�

A�eik�x� k0rqc

AV eik0x

� �eixtþinh: ð27bÞ

The strength of vorticity wave can be obtained bysubstitution for u0 and w0 from Eq. (27) into thedefinition of the vorticity, ~n ¼ r�~u. Its radialcomponent is given by

f0 ¼ ð1=rÞ@u0=@h� @w0=@x; ð28aÞand after substitution for u0 and w0 is found to beequal to

f0 ¼ iq�crðn2 þ ðk0rÞ2ÞAV eixtþinhþik0x: ð28bÞ

In this section we have shown that the perturba-tions can be expressed as the sum of four waves:two acoustic waves, an entropy wave and a vortic-ity wave and have the form in Eqs. (19), (22) and(27). The radial velocity component v is zero inthis narrow annular gap approximation.

7.2. Analytical solution for the linear waves gener-ated by unsteady combustion

We investigate the perturbations generated inan annular duct by unsteady combustion. For def-initeness we will consider the annular geometryintroduced in Section 7.1 except that now thereis heat input at a flame zone concentrated onx = 0. The geometry is shown schematically inFig. 8. Upstream and downstream of the flamezone, linear waves have the form determined in

h a mean flow and mean rate of heat release.


Section 7.1. They become coupled across the zoneof heat input and their dependence can be foundfrom the equations of conservation of mass,momentum and energy [82,114,115].

An infinite duct is considered with the heatinput concentrated at the fixed plane x = 0. Theenergy of flow is increased across x = 0 by the rateof heat input, Q ¼ Qþ Q0. The mean heat releaserate, Q, is defined by specifying the value of themean stagnation temperature after combustion.The linear fluctuations in the rate of heat inputhave a forced frequency x and azimuthal modenumber n and are denoted by Q0ðtÞ ¼ bQeixtþinh.No disturbances arrive from upstream nordownstream.

The inlet mean flow is prescribed by setting thepressure, temperature and mass flux. Upstream ofthe flame region, only an upstream propagatingacoustic wave will be present and S0 = f0 = 0.Applying this to Eqs. (19), (22) and (27), the per-turbation in flow quantities can be written in theform of

p01 ¼ A�1eik�1xþixtþinh; ð29aÞ

q01 ¼1

�c21

A�1eik�1xþixtþinh; ð29bÞ

u01 ¼�k�1

�q1a�1

A�1eik�1xþixtþinh; ð29cÞ

w01 ¼�n

r�q1a�1

A�1eik�1xþixtþinh; ð29dÞ

for x < 0. A�1 denotes the strength of the acousticpressure wave.

Downstream of the flame region, there may beconvected entropy and vorticity waves in additionto the downstream propagating acoustic wave,and so

p02¼Aþ2eikþ2xþixtþinh; ð30aÞ

q02¼1

�c22

ðAþ2eikþ2xþixtþinh�AEeik02xþixtþinhÞ; ð30bÞ

u02¼�kþ2

�q2aþ2

Aþ2eikþ2xþixtþinhþ n�q2�c2

AV eik02xþixtþinh; ð30cÞ

w02¼�n

r�q2aþ2

Aþ2eikþ2xþixtþinhþ�k02r�q2�c2

AV eik02xþixtþinh;

ð30dÞfor x > 0. A+2 denotes the strength of the down-stream propagating acoustic wave, AV and AE

are related to the strength of the entropy and vor-ticity wave respectively, and k02 ¼ �x=�u2. Thefour constants A�1,A+2,AE and AV are relatedby conservation conditions across the flame zone[82,114,115].

If the heat input is at constant position x = 0,then conservation of mass across flame disconti-nuity is

�q1u01 þ q01�u1 ¼ �q2u02 þ q02�u2; ð31aÞwhere the suffices 1 and 2 denote the flow quanti-ties at x = 0� and x = 0+, respectively.

Conservation of axial momentum across x = 0reveals

p01þq01�u21þ2�q1�u1u01¼ p02þq02�u2

2þ2�q2�u2u02: ð31bÞ

Conservation of radial momentum across x = 0 is

�q1�u1w01 ¼ �q2�u2w02: ð31cÞ

The perturbation in the energy flux out of a con-trol volume around x = 0 exceeds the energy fluxinto it by Q0(t), i.e.

cpT 01 �q1u01 þ q01�u1

þ �q1�u1 cpT 01 þ �u1u01

þ Q0=a

¼ cpT 02 �q2u02 þ q02�u2

þ �q2�u2 cpT 02 þ �u2u02

;

ð31dÞwhere T 01 is the mean stagnation temperature, i.e.T þ �u2=ð2cpÞ, and a is the cross-sectional area ofthe duct. The temperature fluctuation can berelated to the pressure and density fluctuationthrough the perfect gas relationship, p = RgasqT,which for linear disturbances gives

T 0 ¼ T ðp0=�p � q0=�qÞ ð32ÞEquations 31a, 31b and 31d are similar for planewaves, n = 0, as presented by Dowling [116].Equations (31) and (32) can be solved to deter-mine the strength of the waves in terms of the rate

of heat input Q0ðtÞ ¼ bQeixtþinh but the analysis istedious. We will give numerical solutions for thatin Section 9. Here we concentrate on a low Machnumber approximation in which the equationsbecome more tractable.

In the limit �u1; �u2 ! 0, Eq. (31b) simplifies top01 ¼ p02, and substituting for p01 and p02 from Eqs.(29a) and (30a) leads directly to

A�1 ¼ Aþ2 ¼ A; say: ð33ÞEqual strength pressure waves are sent upstreamand downstream by the unsteady heat release.We have designated the strength of these acousticswaves by the complex amplitude A.

The strength of the vorticity wave then followsdirectly by substituting for w01 and w02 from Eqs.(29d) and (30d) into (31c) and using the mean flowconservation of mass flux condition, �q1�u1 ¼ �q2�u2.This gives

�nr�q1a�1

A�1 ¼�n

r�q2aþ2

Aþ2 þ�k02r�q2�c2

AV : ð34aÞ

In the low Mach number limit, a+ = a� = x andusing the definition of k02 ¼ �x=�u2 and Eq. (33),we obtain after rearrangement,

AV ¼n�c2�u2

ðrxÞ21� �q2

�q1

� �A: ð34bÞ

Since in this low Mach number limit, the combus-tion takes place at uniform pressure, according tothe perfect gas law we have �q2=�q1 ¼ T 1=T 2. Also

2 In practice, the reflected wave AR would propagateto the flame zone where it would be partially reflected,subsequently changing strength of the downstreampropagating waves. The resultant complex multiply-reflected wave pattern can lead to resonances and isdetermined numerically in Section 9.3. Here we justconsider the first reflection.


T 1 and T 2 are equal to T 01 and T 02 respectivelyand so Eq. (34b) can be written as

AV ¼n�c2�u2

ðrxÞ21� T�1

R

A; ð34cÞ

where T R ¼ T 02=T 01 is the stagnation temperatureratio across the flame.

In a similar way using the equation of conser-vation of mass across flame (Eq. (31a)) and substi-tuting for u01; u

02; q

01 and q02 from Eqs. (29) and (30)

leads to an expression for AE in terms of A. Somecare needs to be taken with the evaluation of thelow Mach number limit of �u2q02. A first sight itmight be thought to be zero but that is incorrect.AE only enters the jump conditions in Eq. (31) asthe product �u2AE. In the limit �u2 ! 0; jAEj tends toinfinity, in such a way as to keep the product �u2AE

finite. Hence the low Mach number limit ofEq. (31a) is

�q1�k�1

�q1x

� �A ¼ �q2

�kþ2

�q2xAþ n

�c2

AV

� �þ �u2

�1

�c22

� �AE:

ð35aÞ

After substituting for AV from Eq. (34b), we findthat it leads to a term of order Mach numbersmaller than the other terms which simplify togive

AE ¼�c2

2

�u2xð�kþ2 þ k�1ÞA: ð35bÞ

In the limit �u! 0, for x > xcut�off, the axialwavenumbers for the two acoustic waves are

k�1 ¼½x2 � ð�c1n=rÞ2�1=2

�c1

; ð36aÞ

kþ2 ¼ �½x2 � ð�c2n=rÞ2�1=2

�c2

; ð36bÞ

while for x < xcut�off, k�1 and/or k+2 are purelyimaginary, the magnitudes of the waves are xdependent and the axial wave numbers for thetwo acoustic waves are expressed as

k�1 ¼ �i½ð�c1n=rÞ2 � x2�1=2

�c1

; ð36cÞ

kþ2 ¼ þi½ð�c2n=rÞ2 � x2�1=2

�c2

; ð36dÞ

where in every case the positive square root istaken of the term in square brackets.

Equations 34c and 35b give the magnitudes ofAV and AE respectively in terms of A, the magni-tude of the pressure waves. Their importance ascontributors to indirect and direct sound isdetermined by their relative contributions to thedownstream boundary condition. As discussed inSection 6.1, the appropriate boundary condition

for a choked exit is as given by Marble andCandel [12],

2u0

�uþ q0

�q� p0

�p¼ 0: ð37aÞ

The terms are to be evaluated in the straight ductjust upstream of the nozzle constriction, where themean flow has been denoted by the suffix 2 andthe downstream propagating waves are describedin Eq. (30) and produce a reflected upstream prop-agating reflected acoustic wave whose strength wedenote by AR

2. After substitution for the waves, inthe low Mach number limit, Eq. (37a) gives

2

�u2

�kþ2

�q2aþ2

Aeikþ2L þ �k�2

�q2a�2

AReik�2L þ n�q2�c2

AV eik02L

� �þ 1

�q2

�1

�c22

� �AEeik02L ¼ 0; ð37bÞ

where the nozzle is considered to be at x = L. Theratio of contributions of the direct incident soundwave and the indirect (entropy) noise to thereflected wave AR is therefore determined by

Ratio of direct=indirect ðentropyÞ

¼ 2kþ2�c22

�u2aþ2

AAE

� �eikþ2L

�� : ð38aÞ

After substitution for A/AE from Eq. (35b) andusing the low Mach number form a+2 = x, weobtain

Ratio of direct=indirect ðentropyÞ

¼ 2kþ2

kþ2 � k�1

� �eikþ2L

�� : ð38bÞ

For plane waves, n ¼ 0; k�1=kþ2 ¼ ��c2=�c1 andthis ratio is independent of frequency and is equalto 2=ð1þ �c2=�c1Þ which is equivalent to

2=ð1þ T 1=2R Þ, where TR is the mean stagnation

temperature ratio across the flame. When thereis negligible temperature rise across the flame,the entropy wave contributes as much to thereflected wave as the incident acoustic wave. Asthe mean temperature increases, the indirect con-tribution from the entropy wave becomes larger.This ratio is plotted in Fig. 9a. For circumferentialwaves, the ratio in Eq. (38b) is a function offrequency and mode number as well as mean


temperature ratio. From inspection of k�1 andk+2, the ratio depends on frequency only in thecombination xr=ðn�c1Þ and on L in the non-dimensional term Ln/r. At frequencies well belowthe k+2 cut-off, the acoustic wave is exponentiallysmall by the time it reaches the nozzle. The indi-rect component of the reflected wave is thereforemuch more substantial than the direct, but thisin turn will decay exponentially as it propagatesaway from the nozzle. The contours of the ratioof the magnitude of direct to indirect (entropy)noise for circumferential modes with Ln/r = 15and 0.5 are plotted in Fig. 9b and c, respectively.For a large value of Ln/r (Fig. 9b), there are twodistinct regions for the values of the ratio are vis-ible. For frequencies below the cut-off frequency(based on �c2), the ratio is very small, effectivelyzero. While for x > xcut�off, the ratio of direct toindirect noise increases with frequency and

asymptotes to the plane wave value 2=ð1þ T 1=2R Þ

at high frequencies. The ratio has a maximummagnitude of 1 for TR = 1, indicating that indirectentropy noise is always higher than the directnoise. For a nozzle which is only a short non-dimensional length from the combustion region,Ln/r = 0.5, Fig. 9c shows three different regionsfor the ratio of the magnitude of direct to indirect(entropy) noise. For frequencies x > xcut�off

(based on �c2) the ratio in Fig. 9c behaves like thatin Fig. 9b and asymptotes to the plane wave valueat high frequencies. At the cut-off frequency basedon �c2; kþ2 ¼ 0 and it is evident from Eq. (38b) thatthe contribution from the direct sound wave

vanishes. The zero along xr=ðn�c1Þ ¼ T 1=2R is clearly

seen in Fig. 9c. However, for frequencies justbelow the cut-off frequency, even though k+2 isimaginary the decay due to the exponential eikþ2L

is modest, because the non-dimensional lengthLn/r is small. Hence the magnitude of the ratiois dominated by the value of the fraction2kþ2=kþ2 � k�2j j in Eq. (38b) and this has a maxi-

mum value of 2 when xr=ðn�c1Þ ¼ 1. This localmaximum for each TR is seen clearly in Fig. 9c.As the frequency decreases further, both k�1 andk+2 are purely imaginary and of opposite signand so 2kþ2=kþ2 � k�2j j gradually decreases.

The relative contribution of the vorticity wavecan be determined in a similar way and isdescribed by

Magnitude of ratio of direct=indirect ðvorticityÞ

¼ kþ2�c2

naþ2

AAV

� �eikþ2L

�� ;ð39aÞ

which, after substituting for A/AV from Eq. (34c)leads to

Ratio of direct=indirect ðvorticityÞ

¼ xrn�c2

� �2�c2kþ2

xM2

1� T�1R

�1eikþ2L

��: ð39bÞ

Because this involves division by the mean flowMach number M2, this ratio is large in the lowMach number limit and the vorticity wave is notan important contributor to indirect noise.

7.2.1. Solution for the acoustic waveThe solution for the acoustic wave strength A

in terms of the unsteady rate of heat input

Q0ðtÞ ¼ bQeixtþinh follows from substituting for AE

and AV from Eqs. (35b) and (34c) into the equa-tion of conservation of energy across the flamezone, Eq. (31d). It is convenient to begin an inves-tigation of low mean flow Mach numbers by not-ing that Eq. (31a) may be used to recast the energyEq. (31d) in the form:

�q2�u2 cpT 02þ�u2u02

¼Q0=a� cpðT 02�T 01Þ �q1u01þq01�u1

þ �q1�u1 cpT 01þ�u1u01

: ð40Þ

All the terms involving multiplication of the meanflow velocity can be neglected in the low Machnumber approximation except the term �q2�u2cpT 02.T 02 includes a contribution from the entropy wavedescribed by AE which tends to infinity as �u2 ! 0in such a way as to keep �u2T 02 finite. We evaluate�u2T 02 by first using the linearized form of the per-fect gas equation in Eq. (32), T 0 ¼ T ðp0=�p�q0=�qÞ, and then substituting for q02 from Eq.(30b). In the low Mach number limit this simpli-fies to

�q2�u2cpT 02 ¼�u2cpT 2

�c22

AEeik02xþixtþinh

¼ �u2

ðc� 1ÞAEeik02xþixtþinh: ð41aÞ

After substituting this form for �q2�u2cpT 02 intoEq. (40) and using the form for u01 in Eq. (29c)we obtain,

�u2

ðc� 1ÞAE ¼bQa� cpðT 02 � T 01Þ

�k�1

a�1

� �A: ð41bÞ

Substituting for AE from Eq. (35b) finally gives

AbQ�� ¼ 1

ac� 1

�c21

T�1R

x

�kþ2 þ T�1R k�1

�� ; ð42aÞ

so that

pbQ�� ¼ 1

ac� 1

�c21

T�1R

x


eikþ2x

�� ; ð42bÞ

(a) n = 0

(b) / 15Ln r = (c) / 0.5Ln r =

TR

Magnitude of ratio of direct to indirect (entropy)

Fig. 9. Ratio of the magnitude of direct acoustic to indirect (entropy) noise as a function of temperature ratio andfrequency obtained using Eq. (38b) for (a) plane wave with n = 0, and for circumferential waves with Ln/r equal to (b) 15and (c) 0.5.


where k+2 and k�1 are presented in Eqs. (36a)–(36d). For x < xcut�off, k+2 and/or k�1 are purelyimaginary and hence the acoustic waves are decayalong the duct, and hence the wave amplitudes arex dependent. For x > xcut�off, the magnitude oftransfer function of acoustic wave to the unsteadyrate of heat addition is independent of x and isobtained as

pbQ�� ¼ 1

ac� 1

�c21

T�1R

x


�� : ð42cÞ

For plane waves (n = 0), Eq. (42c) is simplified as

pbQ��n¼0

¼ c� 1

a�c1

1

1þ T 1=2R

; ð42dÞ

an expression that is independent of frequency.Equation (42d) shows that in the case of negligiblemean heat release when T 02 ¼ T 01 and TR = 1, themagnitude of the transfer function (the magnitudeof the ratio of the pressure perturbation to thefluctuation in heat release rate that generated it)tends to ðc� 1Þ=ð2a�c1Þ. Further, it shows that as

temperature ratio across the flame increasesthe magnitude of the transfer function wavedecreases.

7.2.2. Solution for the entropy waveThe amplitude AE was defined such that the

entropy fluctuation is given by

bS ¼ cp

�q2�c22

AEeik0x: ð43Þ

Substituting for AE and A presented in Eq. (35b)into Eq. (42d) results inbSbQ�� ¼ 1

�mT�1

R 1þ ð1� T�1R Þk�1


�� ; ð44aÞ

using cp=R ¼ c=ðc� 1Þ;�c21 ¼ cRT 01 and mass flow

rate m = q1u1a has been introduced. Since theentropic waves convect with the mean flow, Eq.(44a) holds for all frequencies x whether belowor above xcut�off.

The result in Eq. (44a) can be interpreted byexpanding cpT 02 and substituting for Q=�m ¼ cpðT 02


�T 01Þ and m0=a ¼ �q1u01 þ q01�u1 into Eq. (40) whichleads to

�q2�u2�c22a

ðc� 1ÞcpS0 ¼ Q0 � Q

m0

�m: ð44bÞ

In this form it is clear that the entropy wave isgenerated by fluctuations in Q=m, the rate of heatreleased/mass flow rate. Even if the fluctuation inthe rate of heat release is zero, i.e. Q0(t) = 0, fluc-tuations in inlet mass flow m0 will still generateentropy fluctuations if the mean rate of heatrelease Q is nonzero.

Equation (44a) shows that as T R ¼T 02=T 01 ! 1, the magnitude of the entropy wave

transfer function bS=bQ tends to �m�1. As we pre-dicted in Section 7.2, Eq. (44a) shows that in thelimit �u! 0; �m! 0 the strength of the entropy

wave jbS j tends to infinity. However, �mS remainsfinite. So when comparing results for varyingMach numbers in Section 9, it is convenient to

plot ðbS=cvÞ=ðbQ=QÞ, where cv is the specific heatat constant volume and Q is the mean heat releaserate as �mcpðT 02 � T 01Þ. It follows directly fromEq. (44a) that

bS=cvbQ=Q

�� ¼ cð1� T�1

R Þ 1þ ð1� T�1R Þk�1


�� : ð44cÞ

For plane waves (n = 0), Eq. (44c) simplifies tobS=cvbQ=Q

��n¼0

¼ c T 1=2R � T�1=2

R

� �: ð44dÞ

Equation (44d) clearly shows that, similar to theacoustic waves, the magnitude of the entropywave transfer function for plane waves is indepen-dent of the frequency.

It should be pointed out that while the resultsin Eqs. (44a) and (44c) always apply just down-stream on the flame zone, they can only be appliedat large x if the dissipation of entropy wave isnegligible. In a low Mach number flow the wave-length of the entropy wave, 2p�u2=x, is very shortindeed for high-frequency disturbances [5]. Thenturbulent mixing and diffusion tend to smoothout the entropy fluctuations as they convectdownstream. As a consequence, although a strongentropy fluctuation may be generated in the flamezone, the amplitude of a high frequency entropicdisturbance may be negligible by the time thewave reaches the exit of the combustor. Judgmentis needed, based on the ratio of mixing to convec-tion time, to decide whether the entropy wavespersist as far as any downstream contraction. Ifthey do not, q0 should be replaced by its acousticcontribution p0=�c2 in Eq. (30b) far downstream ofthe flame zone. We would expect the entropy fluc-tuations to only be important at combustor exitfor mid to low frequencies [5].

7.2.3. Solution for the vorticity waveIn this section, we investigate the flow physics

involved in the generation of the vorticity waveand determine the wave strength analytically.Using Eq. (28b) and knowing that in the lowMach number limit n2 is negligible compared to(k02r)2, the vorticity wave is obtained as

f0 ¼ i�q2�c2

k202rAV eixtþinhþik02x: ð45Þ

Substituting for AV from Eq. (34c) into Eq. (45)results in

f0 ¼ �inr�q1�u2

1� �q1

�q2

� �Aeixtþinhþik02x: ð46aÞ

Since �q2�u2 ¼ �q1�u1, after some rearrangementEq. (46a) leads to

f0 ¼ �1

r�u1

� 1

�q21

ð�q2 � �q1Þ � inAeixtþik02xþinh: ð46bÞ

Knowing that op0/oh = in Aeixt +inh and substitut-

ing for inAeixt+inh in Eq. (46b) gives

f0 ¼ �1

�u1�q21

� �� ð�q2 � �q1Þ �

1

r@p0

@h

� �on x

¼ 0: ð46cÞ

Equation (46c) demonstrates that f0 / ð�q2 � �q1Þ�ð1r

@p@hÞ: the vorticity generated at the flame on

x = 0 is due to the baroclinic ($p � $q) vorticityproduction mechanism [117], involving the gradi-ent of the mean density across the flame and thegradient of acoustic pressure fluctuation in thecircumferential direction.

The magnitude of the vorticity wave can beobtained by substituting for A from Eq. (42a)and using k02 ¼ �x=�u2; �u2 ¼ �m=ð�q2aÞ ¼ �mT 02=

ðT 01�q1aÞ;�c2 ¼ �c1ðT 02=T 01Þ1=2

. This gives

fbQ ¼ nðc� 1Þ�mr�c2

1

1� T�1R

T�1

R

� x


ieik02x: ð47aÞ

The magnitude of transfer function for the vortic-ity wave is expressed as

fbQ�� ¼ nðc� 1Þ

�mr�c21

ð1� T�1R ÞT�1

R

x


�� :ð47bÞ

Just as for the entropy waves, since the vorticitywaves convect with the mean flow, Eq. (47b) holdsfor all frequencies x whether they are below orabove xcut�off. Furthermore, this equation showsthat magnitude of the vorticity transfer functionhas the same variation with frequency abovexcut�off as that for the pressure wave presentedin Eq. (42b).


It is seen that in the case of no mean heatrelease, T 02 ¼ T 01; T R ¼ 1, the magnitude of thevorticity perturbation is zero. The explanation issimple. Then there is no change in the mean den-sity across the flame. Therefore, according to thebaroclinic vorticity production mechanism in Eq.(46c), the amount of vorticity produced is zero.It is also clear from Eq. (47b) that for the planewave (n = 0) there is no vorticity generation and

hence jf=bQj ¼ 0. In this case the circumferentialpressure gradient is zero.

Equation (47b) shows that at fixed frequencyand TR, the transfer function of the vorticity wave

jf=bQj is proportional to �m�1 and so tends to infin-

ity in the limit �u! 0. Of course in practice jbQjtends to zero if it is proportional to Q becausethe oncoming mean velocity tends to zero. Hencein the numerical results in Section 9 we chose torepresent the transfer function for vorticity wave

in terms of jf=ðbQ=QÞj, which is given by

fbQ=Q

�� ¼ n

r1� T�1

R

2 x


�� : ð48Þ

8. Numerical results

The main application of high-fidelity numeri-cal methods is to enhance the understanding ofthe fundamental physical processes. In realindustrial design environments there is an addi-tional need for less computational time intensiveprediction tools that can be used before the fullgeometry has been defined and that are sufficientlyquick to be useable for design optimisation. Thisis particularly important for noise generationand transmission because the regions throughwhich acoustic waves propagate are extensiverequiring large computational domains. There-fore, low-order models are required. These toolsprovide the fastest approach for the design andoptimisation of combustors and combustor-turbine coupling. A computer program LOTAN(Low-Order Thermo-Acoustic Network model)is one such low-order model. It calculates numer-ically the linear waves (acoustic, entropic andvortical) generated by unsteady heat input in acomplicated network of pipes provided suitableboundary conditions are applied. The programis intended for circular or thin annular combus-tors, typical of aeroengines, so axial and circum-ferential variations are included but radialdependence is ignored. This thermoacoustic net-work model was developed to predict combustioninstabilities. If a flame model is prescribed torelate unsteady combustion to flow perturbations,the code can be used to obtain the frequencies ofthe resonant modes, their stability and modeshapes. It models plane wave and circumferential

modes of combustion instabilities [5,7,79,118].Alternatively, LOTAN can be used to determinethe forced response to specified unsteady heatinput at frequency x and circumferential modenumber n [4]. We will use LOTAN to investigatethe linear waves generated by unsteady heatrelease in the geometry introduced for the analyt-ical study in Section 7. There the solutionsobtained were for low mean flow Mach numbers.Using LOTAN we can also investigate the varia-tion with Mach number. The results are described

in terms of transfer functions jp=bQj; jbS=cvj=jbQ=Qjand jfj=jbQ=Qj which describe the flow response tounsteady heat input of unit amplitude at fre-quency x and circumferential mode number n.

8.1. Description of the model

The model used in the LOTAN programassumes that the flow is composed of a steady axialmean flow and small amplitude perturbations. Themean flow is assumed to be one-dimensional andthe perturbations vary axially and circumferen-tially but are independent of radius. The model isformulated in terms of a network of modulesdescribing the features of the geometry. For ductmodules of constant cross-sectional area and noheat input, acoustic wave propagation and convec-tion of entropy and vorticity waves are used torelate the perturbations at one end of the duct tothose at the other. Quasi-steady conservation lawsfor mass, momentum and energy are used to relateflow perturbations across zones of heat input andacross duct junctions. The application of inletand outlet boundary conditions then enablesdetermination of the linear perturbations. A per-fect gas is assumed and the specific heats of cp,cv

and c are assumed to be temperature dependentas cp(T) = 973.6 + 0.1333 � T, where T is in (K)and cp is in J/kg K [111,118,119].

8.2. Formulation

Combining the expressions for the four wavespresented in Section 7 we have

ðp; q; u; wÞT ¼ F~W ðxÞ; ð49aÞwhere

~W ðxÞ ¼ ðAþeikþx; A�eik�x; AEeik0x; AV eik0xÞT; ð49bÞ

F ¼

1 1 0 01�c2

1�c2 � 1

�c2 0�kþ�qaþ

�k��qa�

0 n�q�c

�nr�qaþ

�nr�qa�

0 �k0r�q�c

0BBBB@1CCCCA: ð49cÞ

At the flame discontinuity the jump conditions aresatisfied by conserving the mass flux, m = qua,axial momentum flux, fx = pa + mu and radialmomentum flux, fh = rmw. Across the flame or

Table 1Summary of the geometry.

Axial position of the flame, x = 0.0Mean radius of duct, r = 0.3 mTotal cross-sectional area, a = 0.3 m2

Distance between the flame and the outlet, L = 0.5 m


zone of heat input the energy flux, e = mcp

T + m(u2 + w2)/2 = acpu/(c � 1) + m(u2 + w2)/2,increases due to the rate of addition of heat. Theperturbations of these fluxes are given by [111,119]

ðm; f x; f h; eÞT ¼ Gðp; q; u; wÞT; ð49dÞ

where

G ¼ a

0 �u �q 0

1 �u2 2�q�u 0

0 0 0 r�q�uc�u

c�112�u3 c�p

c�1þ 3

2�q�u2 0

0BBB@1CCCA: ð49eÞ

Using the matrices F and G we can easily convertbetween wave amplitudes, flow perturbations andflux perturbations. Given p; q; u and w at the start

of the duct we can find ~W ðx0Þ (where x0 is the startof the duct) by inverting Eq. (49a). The waveamplitudes at the end of the duct (x = x0+L) are

then ~W ðx0 þ LÞ ¼ PðLÞ~W ðx0Þ where P(L) is thepropagation matrix,

PðLÞ ¼

eikþL 0 0 0

0 eik�L 0 0

0 0 eik0L 0

0 0 0 eik0L

0BBB@1CCCA: ð49fÞ

We will also require the perturbations in entropyand vorticity that are determined using Eqs. (21)and (28b), respectively [111,119].

8.3. Boundary conditions

At the inlet the mean flow is prescribed by set-ting the pressure, temperature and mass flux. Theperturbations are given by the type of inlet. For asemi-infinite inlet pipe only an upstream propa-gating acoustic wave will be present and so wetake, A+ = AE = AV = 0 and A� will be calculatedas part of the solution. At the outlet (x = L) thereis a boundary condition on flow perturbations.For a semi-infinite outlet pipe, the boundary con-dition is that there is no upstream wave, A� = 0[111,119].

For the second part of the results presented inSection 9.3 for a nozzle, the outlet boundary isconsidered to be choked, which is a good approx-imation for the turbine inlet, and the nozzle isassumed to be sufficiently short to be treated ascompact. Then for n = 0, the boundary conditionfound by Marble and Candel [12] gives

2u�uþ q

�q� p

�p¼ 0: ð50Þ

Using the Euler equations for a narrow annulargap, Stow et al. [79] have shown that the bound-ary condition found by Marble and Candel [12]

also applies for circumferential modes providedthe nozzle is compact.

9. Results

In this section the results of the analyticalexpressions presented in Section 7 and the numer-ical results obtained using LOTAN described inSection 8 are compared. First the errors intro-duced by the approximation of neglecting themean flow Mach number presented in Section 7are investigated. Then the numerical results foramplitudes of acoustic, entropy and vorticitywaves are presented as functions of frequency,Mach number and temperature ratio across theflame. The geometry considered in the numericalsimulation is summarized in Table 1 and we con-sider the inlet stagnation pressure and stagnationtemperature to be 12 bar and 653 K respectively.

9.1. Validation of zero mean flow approximation

In this section, the analytical results of Section 7are compared with numerical results from LOTANobtained at the duct outlet (x = L) to investigatethe range of validity of the low-Mach numberapproximation. With the conditions summarizedin Table 1, LOTAN calculations are performedfor a wide range of forced frequencies, f = x/2p,from f = 200 � 1200 Hz, temperature ratio acrossthe flame T R ¼ T 02=T 01 from 1.1 to 5 and twodifferent inlet Mach numbers, M01 ¼ 0:06 andM01 ¼ 0:1. The specific heats are assumed to betemperature dependent. However, to aid compari-son with analytical results, the numerical computa-tion is also performed with uniform specific heatsthat is computed as cpðT averageÞ ¼ 973:6þ 0:1333�ðT 02 þ T 01Þ=2, where T is in (K) and cp is in J/kg K.

Figure 10 shows the wave amplitudes obtainedby the analytical solution and LOTAN withuniform cp, and LOTAN with temperaturedependent cp as a function of frequency forM01 ¼ 0:06; T 02=T 01 ¼ 3:0 and n = 1 with a non-reflective downstream boundary condition. Theresults are presented for a distance L downstreamof the combustion zone. The peak observed in theamplitude of the three waves corresponds to thecut-off frequency based on the sound speed down-stream of the flame, i.e. xcut�off/2p � 460 Hz. Theacoustic, entropic and vortical transfer functionsobtained analytically are in very good agreement

Fig. 10. Comparison between analytical solutions basedon low Mach number mean flow approximation (dashedline with symbol) obtained using Eqs. (42b), (44c) and(48) with LOTAN for uniform cp (dashed line) andLOTAN with temperature dependent cp (solid line), as afunction of frequency, x/2p, for T 02=T 01 ¼ 3:0;M01 ¼ 0:06 and for the first circumferential mode n = 1at the non-reflecting duct outlet x = L.


with the results of LOTAN with uniform cp, thedifference in general being less than the error fromneglecting the variation cp with temperature.Therefore, the low mean flow Mach numberapproximation captures reasonably well the main

characteristics of the wave transfer functions,including their general behavior as a function offrequency. In addition, the results presented inFig. 10 clearly show the effect of specific heat vari-ations on the transfer functions. For example athigh temperature ratios there is up to 15% differ-ence between the results for uniform cp comparedwith those for a temperature dependent cp.

The validity of low the Mach number approx-imation is tested as a function of temperatureratio for a fixed frequency of x/2p = 200 Hz andcircumferential mode number n = 1. This fre-quency is below the cut-off frequencies based onthe sound speeds downstream of the flame thatchange between 300 Hz-600 Hz for the range oftemperature ratios considered. Results for theamplitudes of the transfer functions for the differ-ent waves are shown in Fig. 11 for a mean flowMach number of M01 ¼ 0:06 and Fig. 12, forM01 ¼ 0:10. It is seen that for all three wavesand both Mach numbers, the results obtainedusing the low mean Mach number approximationare in excellent agreement with the results ofLOTAN with uniform specific heat for modesttemperature ratios. The maximum error is forthe vorticity wave at the higher Mach numberand highest temperature ratio where the error is

about 20%. However, since M02 ¼ M01T 1=2R , at

those conditions the downstream Mach numberis about 0.22, stretching the limit of the low Machnumber approximation where we neglect terms oforder M in comparison with unity. ComparingFigs. 11 and 12 shows that the transfer functionsare only weakly influenced by the mean inletMach number and that the normalization wechose for the wave amplitudes (i.e. dividing the

entropy and vorticity wave amplitudes by bQ=QÞcaptures much of the variation with Machnumber.

Generally speaking the results presented in thissection show that for a practical range of frequen-cies (few hundred Hz) and low temperature ratios(T 02=T 01 � 2 to 3) the results obtained based onthe low Mach number approximation are in excel-lent agreement with LOTAN. Therefore, the lowmean flow approximation captures reasonablywell the main characteristics of the acoustic,entropy and vorticity waves. Nonetheless, for highfrequencies and high temperature ratios theresults show that the low mean flow approxima-tion may lead to error in determining the strengthof the three waves and then a computation of thelinearized Euler equations (LEE) as provided, forexample, by LOTAN is required.

9.2. Characteristics of the waves

Downstream of the heat release zone, thedownstream-travelling/attenuating acoustic wave,the entropy and vorticity waves generated by the

Fig. 11. Comparison between analytical solutions basedon zero mean flow approximation (dashed line withsymbol) obtained using Eqs. (42b), (44c) and (48) withLOTAN for uniform cp (dashed line) and LOTANwith temperature dependent cp (solid line), as a functionof T 02=T 01, for f = 200 Hz, M01 ¼ 0:06 and for the firstcircumferential mode n = 1 at the non-reflecting ductoutlet x = L.

Fig. 12. Comparison between analytical solutions basedon zero mean flow approximation (dashed line withsymbol) obtained using Eqs. (42b), (44c) and (48) withLOTAN for uniform cp (dashed line) and LOTAN withtemperature dependent cp (solid line), as a function ofT 02=T 01, for f = 200Hz, M01 ¼ 0:1 and for the firstcircumferential mode n = 1 at the non-reflecting ductoutlet x = L.


unsteady combustion move towards the duct exit.Results are presented first for a non-reflecting

downstream boundary. We will investigate thecharacteristics of the three waves for circumferen-tial wave numbers, n = 0, 1, 2 by running LOTAN

Fig. 13. Magnitude of three waves as a function of inletMach number obtained for a frequency of 200 Hz andT 02=T 01 ¼ 3, for circumferential wave numbers n = 0(solid line with triangles), n = 1 (dashed line withsquares) and n = 2 (dotted line with circles) at the non-reflecting duct outlet x = L.


for temperature dependent specific heats andinvestigate the effects of frequency, temperatureratio and inlet Mach number. In the previous sec-tion, we demonstrated the usefulness of the lowMach number analytical solution and this willbe used to interpret the numerical results.

9.2.1. Effect of inlet Mach numberTo study further the effects of a non-negligible

mean flow Mach number, the numerical calcula-tions are performed for a wide range of inlet Machnumbers and three circumferential wave numbers,n = 0, 1, 2. The magnitude of the acoustic, entropyand vorticity waves at the duct outlet (x = L) areshown for a temperature ratio of T 02=T 01 ¼ 3and frequency of x/2p = 200 Hz in Fig. 13. It isseen that, with the normalization we have chosen,the magnitudes of the transfer functions for theacoustic, entropy and vorticity waves tend to finitenon-zero numbers showing that our choice of nor-malization (i.e. dividing the entropy and vorticitywave strength by bQ=Q) has been appropriate tocapture the variation with Mach number at lowMach numbers. Figure 13a shows that for planewaves, n = 0, as the Mach number increases themagnitude of the acoustic wave decreases. The dif-ference between the magnitudes of the acousticwaves obtained for the higher mean Mach num-bers compared with the low Mach number asymp-tote is about 25%. The same difference also occursat higher circumferential mode numbers and isconsistent with terms of order Mach number beingneglected in comparison with unity in the lowMach number asymptotic form. For higher cir-cumferential wavenumbers, the frequency of200 Hz is below the cut-off frequency and themagnitude of the acoustic wave at a distancedownstream of the zone of combustion is exponen-tially small. For instance when n = 2 the magni-tude of the acoustic wave at the duct exit isjpj=jbQj ¼ 7:0� 10�6 Pa J�1s in the low Machnumber limit.

Figure 13b and c show the effect of the inletMach numbers on the entropy and vorticity wavesrespectively. Figure 13b shows that similar to theacoustic wave, although for the plane waves themagnitude of the entropy waves decreases withan increase of mean Mach number, the differencebetween the maximum and minimum values ofjbS=cvj=jbQ=Qj is less than 10%. The variation ofinlet Mach number has more influence on themagnitude of the vorticity wave than on theacoustic and entropic waves. Nonetheless, it isseen that for M01 < 0:1 , the magnitude of the vor-ticity wave is almost independent of the value ofM01 .

9.2.2. Effect of frequencyIn this section the effect of the frequency of the

heat release rate fluctuation on the amplitudes of

the acoustic, entropy and vorticity waves isstudied computationally for an inlet Machnumber M01 ¼ 0:06 and T 02=T 01 ¼ 3. For frequen-cies below cut-off, xcut�off/2p (~460 Hz and

Fig. 14. Magnitude of three waves as a function offrequency obtained for M01 ¼ 0:06 and T 02=T 01 ¼ 3, forcircumferential wave numbers n = 0 (solid line withtriangles), n = 1 (dashed line with squares) and n = 2(dotted line with circles) at the non-reflecting duct outletx = L.


920 Hz for n = 1 and n = 2 respectively, based onthe sound speed downstream of the flame �c2Þ, theacoustic wave decays with downstream distanceand its magnitude is x dependent. The resultspresented here are at a non-reflecting duct exit adistance L = 0.5 m downstream of the combus-tion zone. Figure 14a shows that for the planewave n = 0, the amplitude of the acoustic waveis independent of frequency. This is consistentwith the analytical result in Eq. (42d) whichclearly demonstrated that for n = 0, the amplitudeof the acoustic wave transfer function onlydepends on temperature ratio and is independentof frequency. Figure 14a shows that for n > 0, atlow frequencies the pressure amplitude away fromthe combustion zone is exponentially smallbecause the waves are cut-off. The magnitude of

jp=bQj increases up to a critical frequency. Theexact value of the critical frequency can beobtained by investigating the zero of the deriva-tive of Eq. (42b) with respect to x. It is close tobut not at the cut-off frequency based on �c2. Con-sequently, it is n dependent, as Fig. 14a shows. Asthe frequency increases further, the magnitude of

jp=bQj decreases and asymptotes to the plane waveresult. This is because as x tends to infinity,k�1 � x=�c1 and �kþ2 � x=�c2 and Eq. (42b) sim-plifies to the plane wave result in Eq. (42d).

Figure 14b illustrates the variation of theentropy wave as a function of frequency for thethree circumferential wave numbers. Just as forthe acoustic wave, for n = 0 the amplitude of theentropy wave transfer function remains constantas the frequency changes. The same conclusioncan be made by referring to Eq. (44d) whichshows that for the plane waves jbS=cvj=jbQ=Qj isindependent of frequency depending only on cand the temperature ratio T 02=T 01. For n > 0, asthe frequency tends to zero the magnitude of theentropy wave tends to a non-zero value whichfrom Fig. 14b seems to be independent of nbecause the results for n = 1 and n = 2 are similar.This can be confirmed from the analytical form inEq. (44c). Substituting the low-frequency forms,k�1 = � in/r and k+2 = in/r into Eq. (44c) showsthat the value of jbS=cvj=jbQ=Qj is indeed indepen-dent of n. In addition it is seen that, similar tothe acoustic wave solution, as the frequencybecomes high, the results for n > 0 asymptote toa value independent of frequency and tend tothe solution for plane wave. For a fixed circumfer-ential wave number, there is a minimum and amaximum in the magnitude of entropy wavetransfer function. Inspection of Eq. (44c) showsthat for n > 0 the maximum value of the entropytransfer function occurs at a frequency close tothe cut-off frequency based on �c2 where k+2 isnearly zero, while the minimum occurs close tothe cut-off frequency based on �c1 where k�1 issmall in magnitude.

Fig. 15. Magnitude of three waves as a function ofT 02=T 01 obtained for M01 ¼ 0:06 and f = 200 Hz, forcircumferential wave numbers n = 0 (solid line withtriangles), n = 1 (dashed line with squares) and n = 2(dotted line with circles) at the non-reflecting duct outletx = L.


The variation of the magnitude of vorticitytransfer function with frequency is shown inFig. 14c. For plane waves there are no vorticityperturbations and hence the value of jfj=jbQ=Qj iszero for n = 0. Comparison of Fig. 14c and ashows that for n > 0 the vorticity transfer functionhas a similar dependence on frequency as the pres-sure perturbation. As noted in Section 7.2.3, thisis also clear from a comparison of Eqs. (42b)and (47b). The reason is that for a fixed tempera-ture ratio, the gradient of mean density is con-stant. Thus, according to Eq. (46c), op0/oh = inp0 determines the strength of vorticity wavegiving the magnitude of the vorticity wave thesame dependence on frequency as the acousticpressure at the combustion zone.

9.2.3. Effect of temperature ratioEffects of the temperature ratio across the com-

bustion zone on the acoustic, entropy and vortic-ity waves are shown in Fig. 15 for an inlet Machnumber of 0.06 and frequency of 200 Hz at a dis-tance x = L downstream of the combustion zone.Since this frequency is below the cut-off frequen-cies for circumferential modes n = 1 and 2, thepressure transfer function is exponentially small,e.g. for n = 2 and T 02=T 01 ¼ 1:5 we havejpj=jbQj ¼ 2:0� 10�5 Pa J�1s, and for T 02=T 01 ¼ 5we obtain jpj=jbQj ¼ 4:0� 10�6 Pa J�1s. Figure 15ashows that as T 02=T 01 ! 1, the magnitude ofjp=bQj tends to a non-zero number. This can bereadily obtained by substituting TR = 1 into Eq.(42b). As noted in Section 7.2.1, heat release fluc-tuations generate acoustic waves even when thereis no mean heat input. As the temperature ratioacross the combustion zones increases, the magni-tude of jp=bQj decreases. This is consistent with thelow Mach number analytical solution in Eq. (42d)where we found the magnitude to be proportionalto ð1þ T 1=2

R Þ�1

for n = 0. In addition, it is seenthat for a fixed temperature ratio, as n increasesthe magnitude of the acoustic wave decreasesand n = 0 yields the highest magnitude. This isbecause 200 Hz is below the cut-off frequencyfor n – 0 and the acoustic waves decay exponen-tially with distance along the duct with a rate thatincreases with n.

Figure 15b shows the variation of jbS=cvj=jbQ=Qjas a function of temperature ratio. It tends to zeroas the temperature ratio T 02=T 01 ! 1. It is seenthat for a fixed n as the temperature ratioincreases the magnitude of this entropy transferfunction increases but not as rapidly as linearly.For high temperature ratios, it is constant anddoes not change with T 02=T 01. This is all consis-tent with the low Mach number analytical formin Eq. (44c). The strength of the entropy waveitself described by jbS=bQj actually decreases asthe temperature ratio decreases. This can alsobeen seen from Eq. (44c). Interesting to note is

that as T 02=T 01 ! 1, the magnitude of the entropywave jbS=bQj tends to a finite non-zero number(although jbS=cvj=jbQ=Qj ! 0Þ. As noted in Sec-tion 7.2.2, entropy waves can be generated by heat

Fig. 16. One-dimensional disturbances in simple combustor with a choked exit nozzle and a mean flow and mean rate ofheat release.


release fluctuations even in the case of zero meanheat release rate. In contrast to Fig. 15a, in whichthe magnitude of the acoustic pressure wave ishighly n dependent, Fig. 15b shows that changesin n only have a weak effect on the magnitude ofentropy wave. This is particularly true for lowtemperature ratios, where as shown in Eq. (44c),it is independent of n.

Figure 15c for the vorticity wave confirmsagain that for plane waves the magnitude of thevorticity wave is zero. It is further seen that asT 02=T 01 ! 1; jf=ðbQ=QÞj tends to zero. As the tem-perature ratio increases, jfj=jbQ=Qj increases butnot as fast as linearly. For small frequencies andn – 0, k�1 = � in/r and k+2 = in/r. Substitutinginto Eq. (48) shows that as temperature ratioincreases, jfj=jbQ=Qj increases as xð1� T�1

R Þ, i.e.independent of n. This is consistent with theresults in Fig. 15c which show that the differencebetween the magnitude of vorticity wave for thetwo circumferential wave numbers n = 1 andn = 2 is small.

9.3. Results for choked outlet nozzle

In this section the interaction of acoustic,entropy and vorticity waves at an exit nozzle isstudied. The configuration considered in this sec-tion is shown in Fig. 16. The outlet is choked sothat the non-dimensional mass flow rate is con-stant there.

Upstream of the zone of heat input there is anupstream propagating acoustic wave ðp0U Þ. Down-stream of the combustion zone, acoustic ðp0DÞ,entropy (S0) and vorticity ðn

!0Þ waves produced

by the flame, travel downstream toward the nozzlecontraction, where they are partially reflected togive an upstream propagating acoustic waveðp0RÞ. This interacts with the flame to produce atransmitted wave altering ðp0U Þ upstream of theflame in x < 0. It also alters the downstream prop-agating acoustic, entropic and vortical waves.LOTAN calculates this complex wave pattern.The mean flow is uniform upstream and down-stream of the zone of heat release and so linearperturbations in those regions will be the superpo-

sition of disturbances of the form shown in Sec-tion 7.1. Equations of conservation of mass,momentum and energy are used across the flamezone to relate the waves on either side as discussedin Section 7.2 and the choked end boundary con-dition in Eq. (50) is applied in the straight-walledduct just upstream of the nozzle contraction.

As well as determining the linear waves gener-ated, we also want to separate out the individualcontributions from the direct noise (acousticwaves generated directly by unsteady combustion)and indirect noise (acoustic waves generated bythe interaction of entropy and vorticity waveswith the exit nozzle). To calculate the directsound, we still calculate the entropy and vorticitywaves generated at the flame but ignore their con-tribution when applying the downstream bound-ary condition. This means that the reflectedwave depends only on the incident sound waveand the acoustic pressure fluctuations are uncou-pled from entropy and vorticity disturbances.The direct sound field in x > 0 is then obtainedby adding the downstream and upstreampropagating acoustic waves. The indirect noisedue to the entropy and vorticity waves incidenton the choked nozzle can be determined by sub-tracting the direct noise from the full sound fieldcalculated with all the incident waves contributingto the acoustic wave reflected from the nozzle. Themagnitude of the direct and indirect noise transferfunctions have been investigated numericallyusing LOTAN for three circumferential modenumbers, a wide range of frequencies, stagnationtemperature ratios and inlet Mach numbers.

In order to study the effect of a choked exit onthe magnitude of the acoustic waves produced bythe flame, the transfer function jp=bQj is shown asa function of temperature ratio for the plane waven = 0 in Fig. 17. It is seen that compared to thenon-reflecting case, the choked exit approximatelydoubles the magnitude of the pressure perturba-tion at the exit. This is not surprising. In a lowMach number mean flow, the choked end bound-ary condition in Eq. (50) approximates to a hardend boundary condition u ¼ 0 for an incidentplane sound wave. This leads to a reflected waveof equal amplitude, and hence to a pressure

Fig. 17. Magnitude of the indirect entropy noise (solidline with triangle), direct noise (solid line) and acousticnoise for non-reflecting outlet (dashed line) as functionof temperature ratio obtained for plane wave (n = 0),M01 ¼ 0:06 and f = 550Hz just upstream of the nozzle.

Fig. 18. Magnitude of indirect vorticity noise (dottedline), indirect entropy noise (solid line with triangle),direct noise (solid line) and acoustic noise for non-reflecting outlet (dashed line) as function of temperatureratio obtained for circumferential mode (n = 1),M01 ¼ 0:06 and f = 550Hz just upstream of the nozzle.


doubling just upstream of the exit nozzle. In addi-tion, the choked end converts an incoming con-vected entropy disturbance into indirect noise.

It is seen that although the indirect noise gen-erated is similar in magnitude to the incidentacoustic wave in the case with a non-reflectingboundary, it is usually smaller than the exit pres-sure in the multiple reflected field that makes updirect noise with a choked exit. Nevertheless, itis large enough to contribute to the high amountof total noise. The indirect noise has more compli-cated dependency on the temperature ratio thanthe direct acoustic wave which decreases graduallywith the temperature ratio. In contrast, the indi-rect sound has multiple peaks and troughs. Thisis due to resonances caused by multiple reflectionsof waves between the choked nozzle and theupstream geometry, an effect that is clearer inthe variation with frequency plotted in Fig. 19.In the simple geometry we are considering witha non-reflecting inlet, the upstream reflectionoccurs at the flame zone and is due to the changein the mean flow conditions across the combus-tion zone. Hence as the temperature ratio tendsto 1, the reflection from the combustion zone getsweaker and the variation in the amplitude of indi-rect noise is reduced.

Figure 18 shows results for circumferentialmode number n = 1. Just as in Fig. 17, the trans-fer function jp=bQj at the duct exit is shown for afrequency of 550 Hz (which is above the cut-offfrequency) and varying temperature ratios. Againit is seen that the choked end increases the magni-tude of the direct noise, as it did for the planewave in Fig. 17. At low temperature ratios theamount of indirect noise due to the vorticity per-

turbation is found to be two orders of magnitudesmaller than the direct wave. Although the magni-tude of the direct noise decreases at higher tem-perature ratios, the indirect noise due to thevorticity wave is still one order of magnitudesmaller. Thus, it is concluded that the vorticitywave has negligible contribution to the noisegeneration at the low mean flow Mach numberstypical of gas turbine combustors. This is in agree-ment with the simple analytical solution presentedin Eq. (39b). Just as for the plane wave, the indi-rect noise due to the entropy has multiple peaksand troughs due to the resonances. At low temper-ature ratios the direct noise is nearly twice theindirect entropy noise, whereas at a particularhigher temperature ratio a resonance can makethe magnitude of the entropy noise comparableto that of the direct noise.

The transfer function jp=bQj just upstream ofthe nozzle at the duct exit is shown as a functionof frequency in Fig. 19 for a choked exit nozzle,three circumferential mode numbers and an inletMach number of 0.06 and stagnation temperatureratio of 3. Resonances are observed in both thedirect and indirect sound, but with different fre-quency intervals between the peaks and troughs.For the plane wave in Fig. 19a we expect reso-nances in the direct acoustic wave to be relatedto the sum of the time taken for acoustic wavesto travel a distance L from combustion zone to

nozzle ðL=ð�c2 þ u(

2ÞÞ and back ðL=ð�c2 � u(

2ÞÞ. This

adds up to 2L�c2= �c22 � u

(22

� �which is approxi-

mately 2L=�c2 in this low Mach number mean flow.So we expect the gap between resonant frequen-cies to be about �c2=ð2LÞ Hz (approximately900 Hz for the conditions in Fig. 19). For the

Fig. 19. Variation of indirect noise (dotted line), directnoise (solid line) and total noise (green line) as functionof frequency obtained for M01 ¼ 0:06 and T 02=T 01 ¼ 3just upstream of the nozzle for circumferential mode (a)n = 0, (b) n = 1 and (c) n = 2.


indirect entropy noise, the relevant time is basedon convection from the combustion zone to the

nozzle ðL=u(

2Þ and acoustic propagation back

upstream ðL=ð�c2 � u(

2ÞÞ. For a low Mach numbermean flow this cycle time is approximately

ðL=u(

2Þ leading to a very much smaller frequency

interval u(

2=L Hz (approximately 180 Hz for theconditions in Fig. 19). When acoustic and entro-pic resonances coincide, the indirect noise is seento be much higher than the direct noise. It contrib-utes significantly to the overall pressure perturba-tion throughout this frequency range.

For the first circumferential mode (n = 1),Fig. 19b shows that at frequencies below cut-off(460 Hz), the acoustic wave is exponentiallysmall and thus the magnitude of the indirect noiseis higher than direct noise but even this is smallcompared with the comparable transfer functionfor the plane wave for low frequencies (<200 Hz). For frequencies above cut-off, the acous-tic waves propagate and the magnitude of thedirect noise increases and becomes comparableto the indirect noise. Again similar to the planewave, resonances are observed in both the directand indirect noise with the frequency gap betweenresonances being much smaller for the indirectthan for the direct noise. When acoustic andentropic resonances coincide particularly highcontributions occur from the indirect noise. Thesecombined resonances occur at frequencies of 461,935, 1770, 2601 and 3450. Only the lowest is likelyto be of practical interest because the entropywaves will be attenuated by diffusion at the higherfrequencies.

A similar pattern is observed for the higher cir-cumferential mode (n = 2) shown in Fig. 19cwhere the cut-off frequency is now about920 Hz. At frequencies below cut-off, the magni-tude of the indirect noise is predicted to be higherthan the direct noise. It is seen that the maximumvalue of the direct noise occurs at the frequencyclose to the cut-off frequency of 920 Hz. This issimilar to the results presented in Fig. 14a forthe non-reflecting end condition.

The results presented in this section haveshown that the main indirect source is due toentropy: the vorticity wave makes only a smallcontribution to the overall sound pressure in thelow Mach number flows in a gas turbine. The indi-rect sound due to entropy fluctuations convectingthrough a choked nozzle can be comparable to thedirect sound due to the acoustic waves directlygenerated by the unsteady combustion. Bothdirect and indirect sound exhibit resonances, anda prediction of their relative importance requiresa detailed calculation of the linear waves in thecombustor and depends the geometry, boundaryconditions and on the mean flow. The indirectsound pressure can be particularly intense at fre-quencies that are harmonics of both the entropicand the acoustic resonances.


10. Conclusions, open questions and future research

This paper has discussed some of the progressmade in understanding combustion noise andexamined the challenges to be faced, focusing onthe contribution from the different combustionnoise sources in an aeroengine combustor.

Early studies investigated the noise from flamesin unbounded space. For such open flames, there isanalytical solution for the pressure fluctuationsgenerated by specified fluctuations in heat releaserate: the pressure perturbation depends on the rateof change of the rate of combustion at an earliertime which takes into account the time taken forsound to travel from the combustion source tothe listener and decays inversely with distance fromthe source. When the combustion zone is compact(i.e. its linear dimension is small in comparison withthe speed of sound divided by the frequency) andthe listener is far away compared with the wave-length, this time delay is effectively the same acrossthe combustion zone. The mean square pressuregenerated by open flames then depends on theproduct of mean square value of the rate of changeof heat release multiplied by the correlation volumeof this stochastic source, integrated over the com-bustion zone (see Eq. (5)). This illustrates how theinvestigation of combustion noise decouples intotwo distinct elements: one is the determination ofthe statistics of the unsteady heat release in a turbu-lent flame and the other is the generation of noise bythese heat release fluctuations. The latter is anacoustic problem, there is an analytical solutionin free space and even with the complicated geom-etry of a gas turbine semi-analytical techniquesbased on linear waves can be used. The evaluationof the unsteady heat release and how this quantityscales with underlying turbulence parameters is afar more difficult problem, right at the heart of tur-bulent combustion modelling, and has many chal-lenges and uncertainties. Therefore, most of theearly studies used basic physical scaling, or semi-empirical correlations of these unknown parame-ters, in order to derive combustion noise models[31,32,56].

For an aeroengine real combustor, combustiontakes place in confined geometry and sound gener-ation occurs not only directly due to fluctuationsin the rate of heat release in the combustion zonebut also indirectly as hot-spots convect throughthe nozzle. In addition, the directly generatedacoustic waves are multiply reflected by theboundaries of the combustor. High-fidelity meth-ods for compressible reacting flows, e.g. LargeEddy Simulation (LES) have been used exten-sively in the field of turbulent combustion to pre-dict the mean flow fields [34,120,121] in turbulentflames. They have been also utilized in the investi-gations of combustion instabilities and combus-tion noise in swirled combustors [122–124]. TheLES methodology can be applied in either fully

compressible or low Mach number tools for react-ing flows coupled to computational aeroacoustics(CAA) approaches for the propagation of acous-tic waves [52,125]. Direct Numerical Simulation(DNS) of simple flames has been used to investi-gate the correlation volume of fluctuations in therate of heat release, a vital parameter when pre-dicting the combustion noise [31].

Hybrid methods or two-step methods havebeen developed over the last decade. Theseapproaches take advantage of the length scale sep-aration between a subsonic flow field and theacoustic field which is generated by this flow[126–129]. In low Mach number flows, the fluiddynamic and acoustic length scales are separatedby more than an order of magnitude [125]. In afirst step, the turbulent flow is computed by ahighly resolved solution of the conservation equa-tions. This information is used to determine thesource terms of some set of acoustic equationswhich are then solved to give the sound field. Inthis method it is necessary to determine the dom-inant combustion sources, which can be efficientlycomputed using standard methods such as LES toanalyse flame physics [52,130,131], and then use aCAA technique to predict the acoustic field gener-ated by these sources.

The acoustics of a gas turbine combustorrequires consideration of the whole geometrythrough which acoustic waves propagate whichis from compressor exit to turbine entry. Theanalysis of linear waves is made easier when thecross-stream dimension of the combustor is smallcompared with the acoustic wavelength. Thenacoustic modes with variations across the cross-section are ‘cut-off’, decaying with axial positionrather than propagating, and variations of theacoustic waves across the cross-section can beneglected. This leads to plane waves in a cylindri-cal combustor and axial and circumferentialwaves in an annular combustor. The frequenciesof interest for combustion noise in aeroenginesare sufficiently low that this is often a goodapproximation. Then the linear wave equationscan be solved semi-analytically by a networkapproach. This enables physical insight intoimportant mechanisms. Even without full knowl-edge of turbulent combustion noise sources somegeneral conclusions can be made about the com-bustion noise in gas turbine combustors. This isbecause the direct combustion noise and theentropy fluctuations, which are the cause of theindirect noise, are both related to the unsteadyrate of combustion. By looking at the transferfunctions between acoustic and entropic wavesand the rate of heat release we can investigatethe relative importance of the direct and indirectsound at different mean flow conditions for differ-ent frequencies and mode numbers.

We looked first at the generation of acoustic,entropic and vorticity perturbations in a model


combustor. We found that the strength of the gen-erated entropy wave was such that its first reflec-tion at a choked nozzle always made a biggercontribution to the reflected acoustic wave thanthe incident sound wave (see Fig. 9). Multiplereflections can change this simple conclusion withthe acoustic and entropic disturbances having res-onances due to multiple reflections in the combus-tor geometry. Each can be enhanced to be themost important at its resonant frequency. In par-ticular, the indirect entropy sound pressure can beparticularly intense at frequencies that are har-monics of both the entropic and the acousticresonances.

For circumferential mode numbers n P 1, vor-ticity fluctuations are generated due to the baro-clinic vorticity production mechanism involvingthe product of the gradient of pressure wave inthe circumferential direction and the gradient ofmean density across the flame. However, we haveshown that this vorticity wave does not generatesignificant noise in the low mean flow Mach num-bers in aeroengine combustors.

Although the simple calculations describedhere and other recent efforts explain the basicmechanism of generation and propagation ofacoustic, entropy and vorticity waves, the predic-tion of the contribution of indirect to direct com-bustion noise that would occur in real aeroenginecombustors still poses important scientific chal-lenges. There are a number of issues and questionsthat need further investigation. For example, topredict absolute levels rather than comparisonbetween relative levels of direct and indirect noisebetter models of the statistics of the unsteadycombustion are needed. Other open questionsinclude: How do better 2-D acoustic computa-tions of choked exit nozzle, affect the computedamount of indirect noise generated? What is thedominant dispersion mechanism of entropy wavesin combusting flows and how does this affect theamount of indirect noise generated?

The exit nozzle is the bridge between entropydisturbances and the indirect noise they radiate[7]. The existing theories describing the nozzledynamics make a number of simplifying assump-tions, such as one-dimensionality and compact-ness (e.g. [12,30,78,89,98]). Recent experimentalstudies, however, have showed that some of theseassumptions, particularly on one-dimensionality,are questionable [106]. To develop a two-dimensional model for the response of chokednozzle to entropic disturbances, further numericalsimulations need to be conducted to determine theacoustic waves reflected and transmitted whenentropy fluctuations convect through an extensivenozzle.

Another important issue in the problem ofcombustion noise is the dispersion of entropywaves. Turbulent mixing and diffusion can con-tribute to the reduction in the amplitude of the

entropy disturbances as they convect from thecombustion zone to the nozzle [109]. In additionvariations of the mean flow velocity across thecombustor will tend to spread out the entropyfluctuations due to differences in their convectiontime. The decay of entropy disturbances and itsspatial distribution should be further explored.

In real aeroengine combustors, the impact andimportance of entropy noise is a controversial issueas pointed out by Strahle [41], mainly due to thelack of comprehensive experimental data. One ofthe main challenges in experimental investigationof entropy noise is the separation of direct andindirect combustion noise. For example, character-izing the entropy waves entering a turbine stage ora nozzle requires measurement of temperature fluc-tuations spatially and temporally in a high-speedreacting flow. Acoustic waves generated by thisinteraction are also difficult to measure and requiregreat care. Ideally, theory, experiments and numer-ical methods need to be used together to under-stand noise sources in real engines. To achievethis goal, two difficulties need to be overcome: (1)Acoustic boundary conditions of the combustionchamber must be known and preferably controlla-ble. (2) The entropy fluctuations need to be mea-sured. On the experimental side separation ofdifferent effects can be obtained by controllingand independently introducing acoustic, entropicor vortical waves. Experiment [94] and numericalinvestigations [78] have shown that indirect com-bustion noise can be a significant source of noisein an experimental combustor configuration. How-ever, the relative contribution of the two acousticsource mechanisms in a realistic aero-engine com-bustor has not been fully demonstrated. In fact,recent measurements suggest that this is dependenton geometry, operating conditions, and frequencyrange [8]. The extreme operating conditions addcomplexity and make this a challenging experimentunder real engine conditions.

In this paper we only discussed the indirectnoise generated by the interaction of the entropicdisturbances with the choked nozzle at exit to thecombustion chamber. However, in an aeroengine,any entropy fluctuations that persist into the tur-bine will similarly generate acoustic waves as theyinteract with each stage of the turbine. This mech-anism modifies the overall level of combustionnoise radiated in far field and must therefore bebetter understood. There are turbomachineryapproaches based on actuator-disk theory thatenable such interactions to be evaluated, but cur-rently most low-order models ignore the disper-sion of entropy waves. The entropy and acousticwave interaction needs to be investigated experi-mentally on a realistic turbine and the results ofCFD turbomachinery solvers and low-order ana-lytical codes compared to the experimental resultsto investigate the dispersion of the entropy wavesas they pass through the turbine blade rows.


Currently lack of knowledge about the entropywave dispersion adversely affects the ability ofsuch models to predict the radiated combustionnoise from aeroengines.

Acknowledgements

The authors acknowledge financial supportthrough European Commission in the SeventhFramework Programme FP7 under Grant Agree-ment No. ACP2-GA-2012–312444 in RECORD(Research on Core Noise Reduction) project.We would like to thank the SAFRAN Snecmacompany for providing us with Fig. 2 showingthe contribution of noise sources on a turbojetengine.

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Combustion noise - University of Cambridge