Combustion Noise

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Combustion Noise
Dowling, A. P., & Mahmoudi, Y. (2015). Combustion Noise. Proceedings of the Combustion Institute, 35(1), 65-
100. https://doi.org/10.1016/j.proci.2014.08.016
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ofthe
Combustion
Institute
ProceedingsoftheCombustionInstitute35(2015)65–100
www.elsevier.com/locate/proci
Combustion noise
⇑
Ann P. Dowling , Yasser Mahmoudi
DepartmentofEngineering,UniversityofCambridge,CambridgeCB21PZ,UnitedKingdom
Availableonline22November2014
Abstract
Combustion noise is becoming increasingly important as a major noise source in aeroengines and
ground based gas turbines. This is partially because advances in design have reduced the other noise
sources, and partially because next generation combustion modes burn more unsteadily, resulting in
increasedexternalnoisefromthecombustion.Thisreviewreportsrecentprogressmadeinunderstanding
combustionnoisebytheoretical,numericalandexperimentalinvestigations.Wefirstdiscussthefundamen-
talsofthesoundemissionfromacombustionregion.Thenthenoiseofopenturbulentflamesissumma-
rized.Wesubsequentlyaddresstheeffectsofconfinementoncombustionnoise.Inthiscasenotonlyisthe
soundgeneratedbythecombustioninfluencedbyitstransmissionthroughtheboundariesofthecombus-
tionchamber,thereisalsothepossibilityofasignificantadditionalsource,theso-called‘indirect’combus-
tionnoise.Thisinvolveshotspots(entropyfluctuations)orvorticityperturbationsproducedbytemporal
variationsincombustion,whichgeneratepressurewaves(sound)astheyacceleratethroughanyrestriction
attheexitofthe combustor.Wedescribethegeneralcharacteristicsofdirectandindirectnoise.Togain
furtherinsightintothephysicalphenomenaofdirectandindirectsound,weinvestigateasimpleconfigu-
ration consisting of a cylindrical or annular combustor with a low Mach number flow in which a flame
zoneburnsunsteadily.UsingalowMachnumberapproximation,algebraicexactsolutionsaredeveloped
sothat the parameters controlling the generation ofacoustic, entropic andvortical wavescan beinvesti-
gated.ThevalidityofthelowMachnumberapproximationisthenverifiedbysolvingthelinearizedEuler
equationsnumerically for a wide rangeof inlet Mach numbers,stagnation temperatureratios, frequency
and mode number of heat release fluctuations. The effects of these parameters on the magnitude of the
wavesproducedbytheunsteadycombustionareinvestigated.Inparticularthemagnitudeoftheindirect
anddirectnoisegeneratedinamodelcombustorwithachokedoutletisanalyzedforawiderangeoffre-
quencies, inlet Mach numbers and stagnation temperature ratios. Finally, we summarize some of the
unsolvedquestions thatneed to bethe focusof future research.
(cid:2) 2014 The Authors. Published by Elsevier Inc. on behalf of The Combustion Institute. This isanopen
accessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/3.0/).
Keywords:Combustionnoise;Acousticperturbation;Entropyperturbation;Vorticityperturbation;Chokednozzle
1. Introduction
In the last four decades noise emission has
developed into a topic of increasing concern to
society.Thismainlystemsfromtheadversephys-
⇑ Correspondingauthor.
iological impacts on those exposed to noise over
E-mailaddress:[email protected](A.P.Dowling).
http://dx.doi.org/10.1016/j.proci.2014.08.016
1540-7489/(cid:2)2014TheAuthors.PublishedbyElsevierInc.onbehalfofTheCombustionInstitute.
ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/3.0/).
66 A.P.Dowling,Y.Mahmoudi/ProceedingsoftheCombustionInstitute35(2015)65–100
lengthy periods. As a result, in air, road and rail combustion to acoustic waves, LPP systems gen-
transport technologies, the control of noise emis- erate substantial broad-band noise, which can be
sionsiscentraltosocialacceptanceandeconomic heardoutsidetheengine.Furthermore,noisefrom
competitiveness. Due to its intermittent nature, auxiliarypowerunits(APU)isanimportantcon-
aircraft noise is deemed to be the most annoying tributor to the overall level of ramp noise (the
transportation noise, with road noise being the noise generated by an aircraft while it is on the
least likely to annoy. Noise has an immediate ground, typically parked at the ramp). A signifi-
effect upon observers at the time of emission, cant component of APU noise is combustion
causing annoyance and physiological change, noise[8,9].Itisthereforecrucialtoinvestigatethis
and it also impedes the efficiency of observers. broad-band combustion noise and develop meth-
Longer term effects of noise are physiological ods to predict and reduce it, in order to enable
impairment, e.g. hearing damage, speech and the introduction of low noise, green technologies
sleep interference. Although individual aircraft onnextgenerationlow-NOxcombustorsandpre-
have become less noisy over the last 30years, mixedburners in aero-engines[7,10,11].
theincreaseinairtrafficmeansthatmanycitizens
are concerned byaircraftnoise. According to the
InternationalCivilAviationOrganization[1],glo- 2. Motivation:Gas turbinecombustion noise
bal air transportation is anticipated to double sources
over the next couple of decades. It is therefore
expectedthatthenegativesocialandenvironmen- Thetotalnoiseradiatedbyagasturbinecom-
tal impactsof itsnoise emission will increase. bustion chamber system consists of direct and
Aircraft noise is noise pollution produced by indirectcombustionnoise[11,12].Thedirectnoise
any aircraft or its components during various sources are related to the unsteady processes of
phasesofaflight:onthegroundsuchasauxiliary volumetricexpansionandcontractioninthereac-
power units, while taxiing, on run-up, during tiveregion[13,14].Thisisgeneratedbythefluctu-
take-off, underneath and lateral to departure and ation of heat release rate associated with the
arrival paths, or during landing. The primary chemical reaction [11,15]. It would occur even if
source of noise in an aircraft can be contributed the combustion were in unbounded space,
to the fan, compressor, combustor, turbine, and although in the gas turbine the sound radiated is
jet exhaust [2] (see Fig. 1). On approach the air- modified from the open flame result by its trans-
frame isalso a significantsource of noise. mission from the combustion chamber to the
During the last few decades, research efforts far-field throughthe turbineandjet.The indirect
have enabled a significant reduction of jet, fan combustionnoise,identifiedbytheearlyworksof
and external aerodynamic noise. The reduction Marble and Candel [12] and Morfey [16] in the
of jet noise was mainly achieved by introducing seventies, is an additional noise generated when
ultra-high-bypass ratio turbofan engines. Fan a fluid with a non-uniform entropy or vorticity
noise has been reduced through effective acoustic distributionisaccelerated,asitiswhenconvected
linersandcomplexdesignsoffanbladegeometry. throughthechokednozzlelocatedattheoutletof
Theseeffortsonthereductionofjetandfannoise the combustion chamber in a gas turbine [17,18].
have left combustion noise as an important A summary of these different core noise sources
remaining contributor [2,3]. Figure 2 shows the isillustrated in Fig.3.
significance of combustion noise relative to other Thespatial andtemporal variation of the rate
noisesources for atypical turbojet application. ofcombustionproduceshotandcoldspotswhich
Combustionnoiseappearstobethethirddom- areconvectedbythemeanflow[11].Thecoupling
inant source in the whole turbojet engine noise between hot spots generated by combustion and
after fan and jet noise, especially at approach particle acceleration in the mean flow then gives
and cut back conditions. Furthermore, recent rise to pressure perturbations. Since the entropy
studiesonlow-NOxcombustorssuchasleanpre- of the hot spots is different to that of the sur-
mixedprevaporized(LPP)combustionshowcon- rounding, indirect noise is also called entropy
siderable increase in noise emission [4]. This is noise. The vital role of acceleration/deceleration
because lean premixed and stratified combustion caused Ffowcs Williams and Howe [17] to refer
burnsmoreunsteadily[5–7].Asdiscussedinthese toitasacousticalbremsstrahlung.Thecombustion
references, leanpremixedcombustorscanalsobe also generates unsteady shear leading to vorticity
susceptibletoaninstabilityarisingfromthefeed- perturbations[19],which also convectand gener-
back interaction between unsteady combustion ate pressure perturbations as they accelerate
and acoustic waves. Such an instability occurs at through the turbine nozzle guide vanes. Hence,
a discrete tone related to the acoustic resonances atthecombustorexittheinteractionofthevorti-
of the combustor usually shifted slightly by the cal and entropic perturbations with the mean
flame response. Even when the self-excited insta- streamwise velocity gradient results in energy
bilityhasbeeneliminatedbyacarefulcombustor transfertoanacousticmode[19,20].Acceleration
design,whichreducesthesensitivityoftherateof of entropy and vorticity waves in the choked
A.P.Dowling,Y.Mahmoudi/ProceedingsoftheCombustionInstitute35(2015)65–100 67
Fig.1. Summaryofenginenoisesources(Rolls-RoyceTrent1000,copyrightRolls-Royce,publishedwithpermission).
Fig.2. Typicalcontributionofnoisesourcesonaturbojetengineatapproach.(Fig.2isfromSAFRANSnecma(http://
www.safran-group.com/),publishedwithpermission).
nozzleresultsingenerationofpressurewavesthat unsteady combustion generates pressure waves,
propagate upstream (from where they can be which reflect at the boundaries of the combustor
reflectedfromtheflamezoneand/ortheupstream andthevelocityorequivalenceperturbationasso-
geometry altering the downstream propagating ciated with these waves causes more unsteady
waves) and downstream from the turbine stage combustion. If the phase relationship is suitable
asindirect combustion noise[12,17,18]. [21,22], self-excited oscillations grow. This leads
In this review we discuss combustion noise to discrete tones at resonant frequencies associ-
arisingfromfluctuationsintherateofheatrelease ated with the acoustic characteristics of the com-
in a turbulent flame. It is broad-band in nature bustor and can be accompanied by high noise
and is distinct from the discrete tones observed levels and severe pressure oscillations that can
during combustion instability. Instability occurs cause structural damage to engine components
when feedback is established between unsteady [5–7]. The vibrations induced by the oscillations
combustionandpressureperturbationsinacom- can cause fatigue cracking of combustor liners
bustion chamber. The mechanism is that the andreducethelifetimeofacombustorbyafactor
68 A.P.Dowling,Y.Mahmoudi/ProceedingsoftheCombustionInstitute35(2015)65–100
Fig. 3. Illustration of combustion noise sources in a gas turbine:generationof direct combustion noise and indirect
combustionnoiseinaero-engines.
of two or more [6,23]. During an instability the frequencies of self-excitation. The dissipation or
heat release and the pressure perturbations dispersion of these entropy waves can stabilize
throughoutthecombustorareatafixedfrequency ordestabilizethemodesofthesystem,depending
and coherent. on the configuration of the combustor and the
In contrast, under normal stable operating form ofthe coupling [26,30].
conditionsthecombustionnoisesourcesareinco- Incontrasttocombustioninstability,inbroad-
herent throughout the combustor, and the com- band combustion noise the unsteadiness in the
bustion noise is a random output of turbulent rate of combustion is mostly caused by the local
combustion and radiates across a broadband of turbulent flow and mixing, and is only very
frequencies. However, some of the numerical weakly influenced by reflected waves from the
and experimental techniques used to understand boundaries. Predicting the sources of both direct
combustioninstabilityarerelevanttocombustion combustion noise and entropy noise requires an
noise, particularly the modelling of the linear understandingofthecharacteristicsoftheunder-
waves (acoustic, entropic and vortical) generated lying heat release fluctuations [10,31]. Thus,
by unsteady combustion within the complicated studies of combustion noise are inevitably linked
geometry ofa gas turbine. to the understanding of combustion modes of
An additional link between broadband com- flames. Previous investigations on combustion
bustion noise and combustion instability is that noise indicated that combustion noise is broad-
thegenerationofpressureoscillationsbyentropy bandhavingrandomphaseandthepeakintensity
wavesatthechokedcombustor,whichisthemain is typically in the low frequency range around
source of indirect combustion noise, has been 200–1000[32–34].
studied extensively because it may play a role in This paper contains a review of progress in
the feedback mechanism for combustion instabil- these various areas and a discussion of current
ity [24–27]. Indeed pressure waves generated in challenges. It focuses on comparison between
this way are considered to be the key feedback- direct and indirect (entropy and vorticity) com-
mechanism for a very low frequency combustion bustionnoise.Theclassicaltheoryofcombustion
instability [24,26,28,29] called ‘rumble’. Rumble noiseofopenflamesispresentedinSection3.Sec-
usually only occurs at start-up of aero-engines at tion4presentsthegoverningequationofcombus-
sub-idle and idle conditions, and exhibits such tion noise derived from the exact equations of
low frequencies (in the range of 50Hz to conservation of mass, momentum and energy. It
150Hz) that it does not couple to the combustor enables the different terms describing noise
acousticresonances[28].Thefeedbackmechanism sources (direct combustion, accelerating entropy
isthoughttoinvolvetheentropywavesgenerated and vorticity) to be identified within a unified
by the unsteady combustion convecting down- framework. Combustionnoise of confinedflames
stream to the combustor exit, where acceleration and comparison between direct and indirect
throughthenozzleleadstoanupstreampropagat- (entropy and vorticity) is presented in Section 5.
ingpressurewavewhichperturbsthecombustion The challenges and unexplored issues in
further.Itisthelongconvectiontimefromflame understandingthecombustionnoisearediscussed
zone to combustor exit that leads to the low in Section 6. To give more insight about the
A.P.Dowling,Y.Mahmoudi/ProceedingsoftheCombustionInstitute35(2015)65–100 69
generation mechanism of combustion noise, ana- small compared to the acoustic wavelength, this
lytical solutions are presented in Section 7 for a typeofemissioncanbeconsideredasamonopole
model combustor based on a low Mach number pointsource ofsound. Thepressure inthe sound
approximation. The numerical procedure of line- waveemittedbytheflamewasrecordedasafunc-
arized Euler equations based on the low-order tion of time, Fig. 4b, in which good quantitative
thermo-acoustic network model (LOTAN) and agreement was obtained between the measured
the specific boundary conditions are briefly pressure field and that calculated by measuring
described in Section 8. Section 9 discusses the and differentiating the bubble radius-time curves.
results obtained based on the analytical and They showed that the sound pressure depended
numericalsolutions. Firstthe validityof theana- linearly on flame radius and on the square of
lyticalsolution,basedonthelowmeanflowMach burning velocity. In this early study it was found
number approximation, is tested against the that at a distance, r from the ignition position,
numerical solution of the linearized Euler equa- the pressure fluctuation,p0(r,t)was givenby
tions in a modal combustor. Then the effects of
q
differentparameterssuchasfrequency,stagnation p0ðr;tÞ¼ 0 V€ðt(cid:2)r=c Þ; ð1Þ
temperature ratio across the flame, and inlet 4pr 0
Mach number on the magnitude of the acoustic,
whereV(t)isthevolumeofthebubble.q andc
entropic and vortical waves are studied. Results 0 0
arethemeanairdensityandspeedofsoundinfar-
for the choked outlet nozzle and the magnitude
fieldthroughwhichthesoundispropagating.The
of indirect and direct noise generated are pre-
primedenotesaperturbationfromthemeananda
sented in Section 9.3 for a range of different fre-
quencies, inlet Mach numbers and stagnation dot a time derivative. Thus V€ðtÞ¼d2V=dt2 is the
temperature ratios. Conclusions of the paper, secondrateofchangeofthevolumeofthebubble.
open questions and possible areas for future Equation(1)representsthefundamentalprinciple
researchare presented inSection 10. of combustion noise: the sound is generated by
unsteadiness inthe rateofexpansionbycombus-
tion of gas. In an open flame, it is the rate of
3.Classical theory ofcombustion noise changeoftheexpansionratethatgeneratessound,
V€ðtÞ. Steady combustion, leading to a constant
A classical experiment on noise emitted from rateofexpansion,issilent.Although,inthissim-
combustion was conducted by Thomas and ple experiment the flame was laminar, the results
Williams [14] who filled a soap bubble with a havebeenproventobeequallyapplicabletonoise
homogeneous premixed fuel–air mixture and emission from open turbulent flames. In the phe-
ignited it by a spark at the centre of bubble (see nomenologicaltheoryofBragg[13]ondirectcom-
Fig.4a).Theconstantpressureignitionproduced bustion noise, it was assumed that the turbulent
atransientburningofmixturewhichresultedina flamewasacollectionofeddies,whichhavetheir
transient increase of the volume of bubble gases. ownheatreleaserate.Eacheddyactsasamono-
As a consequence, a spherically symmetric pres- pole source of sound which is statistically inde-
sure wave was emitted. Since the bubble size is pendentof the neighbouringeddies.
Fig.4. (a)Instantaneoussnapshotsofanexpandingsphericalflamefrontafterignitionintimeand(b)acousticpressure
produced by expansion obtained for C H-O-N mixture with burning velocity of 55cm s(cid:2)1, + measured and o
2 4 2 2
computedpressure(ThomasandWilliams[14]bypermissionoftheRoyalSociety).
70 A.P.Dowling,Y.Mahmoudi/ProceedingsoftheCombustionInstitute35(2015)65–100
Many studies (e.g. [11,35–37]) have tried to approximately j~xj(cid:2)1 and the retarded time
address the problem of combustion noise in t(cid:2)j~x(cid:2)~yj=c simplifies to t(cid:2)j~xj=c [39]. In this
0 0
turbulent flames and assumed that the turbulent expression it has been assumed that c the ratio
flamecanberegardedacousticallyasanassembly of specific heats is independent of temperature
of monopole sound sources with different and that the combustion takes place at ambient
strengths and phases that are distributed in the pressure, so that qc2¼cp ¼q c2. Ambient con-
reaction zone of the flame. Price et al. [35] and ditions in the regionsurro0undin0g0the combustion
Hurle et al. [36] found that the far-field sound region are denoted by the suffix 0 and q_ð~y;tÞ is
pressure radiated by a turbulent premixed flame the instantaneous rate of heat release per unit
dependsontherateofvolumeincreaseofthefuel volume in the combustion region. Equation (3)
and oxidant in the combustion region, in which is valid in the far-field, i.e. for j~xj>>j~yj and
the rate of volume increase is proportional to j~xj>>c =x,wherexisatypicalradianfrequency
0
the rate of consumption of the fuel and oxidant of the sound produced. The compact assumption
intheflame.Priceetal.[35]alsopointedoutthat requires that j~yj<<c =x for positions ~y within
0
the same result holds for turbulent diffusion the combusting zone, i.e. the dimension of flame
flames, if it is assumed that the fuel and air burn length is small compared with the acoustic wave-
in stoichiometric proportions. According to this length/2p. The same expression as Eq. (3) for
model, it has been shown that the instantaneous pressure perturbation has also been derived by
sound pressure p0iðr;tÞ in the sound waves that Strahle[41](Eq.(3)slightlydiffersfromtheorigi-
emanatefromtheithsourceelementintheturbu- nal one derived by Strahle [41], who erroneously
lent flameis givenby consideredthemonopoletoberelatedtotherate
p0iðr;tÞ¼4qp0rðE(cid:2)1Þ€viðt(cid:2)r=c0Þ; ð2Þ SoficshaannygesoufrfRaScqeb~eun(cid:3)cldo~Ssinragthtehreqc0oRmS~ubu(cid:3)sdti~So,nwzhoenree
where r is the distance of observer from the and~u is the fluid velocity. Strahle’s result should
source, v_i is the volume rate of consumption of therefore bemultiplied bythe factor q0/qb).
fuel and oxidant in the flame element and E is Equation (3) clearly states that the rate of
the volumetric expansion ratio of burned to change in the total heat release rate generates
unburnedgases.Forcombustionatconstantpres- pressure fluctuations. This expression applies to
sure E is equal to q /q where q and q are the turbulent premixed, non-premixed and partially
u b b u
mean density of the burnt and unburnt gases. premixed flames as noted by Hurle et al. [36].
They also commented that for a compact flame, Although,thecharacteristicsofheatreleasemech-
the total sound pressure in the far field is given anismsandthephysicsunderlyingthecombustion
by the simple summation over all elements of processindifferentmodesaffectthecharacteristics
thecomponentpressures,p0ðr;tÞinEq.(2).Hurle of pressure fluctuation, Eq. (3) clearly manifests
i that noise depends on the combustion mode
et al. [36] confirmed this relationship experimen-
through the temporal change of heat release rate
tally by using the light emission from short-lived
in the combustion region[36,41].
CHorC freeradicalstomonitortherateofcom-
2
For turbulent combustion, the acoustic pres-
bustion.Sincetheirpioneeringwork,therelation-
surewillbeastochasticvariableandwearemore
ship between acoustic pressure perturbation and
interested in its statistical properties than any
rate of heat release in combustion has been veri-
instantaneous value. The mean square pressure
fied using more advanced signal processing by
Shivashankaraet al.[38]. perturbation p02ð~xÞ can be obtained by squaring
Eq.(3)andtakingthetimeaverage.Thisleadsto
3.1. Far-field pressure distribution of an open (cid:2) c(cid:2)1 (cid:3)2Z Z
turbulent flame p02ð~xÞ¼ €qð~y;tÞd3~y(cid:4) €qð~z;tÞd3~z;
4pc2j~xj
0
Consideranexampleofanopenturbulentpre-
wheretheoverbardenotesatimemean.Substitut-
mixedflameshowninFig.5.DowlingandFfowcs ing for ~z¼~yþ~D, where ~D is the separation
Williams[15]andCrightonetal.[39]showedthat
vector, resultsin
by using Lighthill’s analogy [40] the far-field
sound pressure fluctuation resulting from the (cid:2) c(cid:2)1 (cid:3)2Z Z
direct noiseis givenby[15,39] p02ð~xÞ¼ €qð~y;tÞ€qð~yþ~D;tÞd3~Dd3~y;
4pc2j~xj
0 v vt
ðc(cid:2)1ÞZ ð4Þ
p0ð~x;tÞ¼ €qð~y;t(cid:2)j~xj=c Þd3~y; ð3Þ
4pc2j~xj 0
0 v wherev isthevolumeoverwhichtheintegrandis
t
where v is the volume containing the combustion non-zero. The ratio of the volume integral
region or the flame brush. In the derivation of R€qð~y;tÞ€qð~yþ~D;tÞd3~D to the maximum value of
Eq. (3)~x is considered to be in the far-field and
its integrand €q2ð~y;tÞ is defined to be correlation
the flame is compact. Then r(cid:2)1¼j~x(cid:2)~yj(cid:2)1 is volume, which we denote by V ð~yÞ. It is the
cor
A.P.Dowling,Y.Mahmoudi/ProceedingsoftheCombustionInstitute35(2015)65–100 71
combustion noise. The acoustic pressure wave
perturbation given by Eq. (5) can be applied to
all modes of turbulent combustion, premixed
[10,11,33,35,42–48], non-premixed [34,49–53] and
partially premixed [54,55]. Candel et al. [10] give
a review of empirical and computational model-
lingofthenoisesourceforpremixedflames.More
recently for premixed flames Swaminathan et al.
[31,56] and Liu et al. [57] have analysed Direct
Numerical Solutions (DNS) and advanced laser
diagnostics of premixed flames to obtain two-
point spatial correlation of the rate of change of
the fluctuating heat release rate. This approach
leads to the development of a generic form for
the cross-correlation. The results has then been
applied to predict the far-field Sound Pressure
Level (SPL) using Eq.(5) from open flames
reported in [46].
The sound field from non-premixed and par-
tially premixed flames is less well characterized
than that radiated by premixed flames. This is
due to the difficulty in modelling the rate of heat
release and the complexity of measuring this
Fig.5. Schematicofturbulentflameillustratingcoordi-
quantity experimentally. Furthermore, in experi-
natesforthedescriptionoftheacousticfield.
ments the level of mixing of fuel and oxidizer
wouldcausethegeneratedsoundtofeaturechar-
acteristics of premixed flame sound. Nonetheless,
volumeoverwhich€qintheturbulentcombustion a number of investigations studied the combus-
iscorrelated. Writing tion noise radiated by non-premixed [34,49–53]
Z and partially premixed flames [54,55] to under-
€qð~y;tÞ€qð~yþ~D;tÞd3~D¼€q2ð~y;tÞVcorð~yÞ; stand the region bridging these combustion
modes. It was pointed out that higher levels of
Equation (4)becomes sound are generated by turbulent non-premixed
(cid:2) c(cid:2)1 (cid:3)2Z flamesthanbypremixedflamesatasimilarveloc-
p02ð~xÞ¼ €q2ð~y;tÞV ð~yÞd3~y: ð5Þ ity [58]. For premixed flames, the region that the
4pc20j~xj v cor maximumacousticoutputoccursarelocatedclose
to the flame tip [37], while, for turbulent non-
Equation (5) applies to any flame mode. It is
premixed flames predominant sound sources are
clear that to predict the mean square far-field
distributedintherearregionofthereactionzone
pressure perturbation we need to know €q2ð~y;tÞ
[35,58]. Further, using Mach number (M) scaling
and V ð~yÞ. A flame is characterized by its mean
rate ofcohreat release (cid:2)q_ð~yÞ and there are well devel- [54]itwasdemonstratedthatthermssoundpres-
sure generated by methane partially premixed
oped methods, computational and experimental,
flamesscaleswithM5comparedtoM3forturbu-
todeterminethis.FromEq.(5)wecandecompose
lent non-premixedmethane flames.
the requirements for modelling combustion noise
ofanopenturbulentflameintotheneedtomodel
the correlation volume V ð~yÞ and the way in
which €q2ð~yÞ is related cotro (cid:2)q_ð~yÞ. The ratio 4. General thermoacousticsources andcombustion
€q2ð~yÞ=(cid:2)q_2ð~yÞ canberewritten as the product noise
€q2ð~yÞ q_02ð~yÞ InSection3wepickedoutthedominantnoise
(cid:4) :
q_02ð~yÞ (cid:2)q_2ð~yÞ source when unsteady combustion occurs in an
unbounded flow (Eq. (3)). But further insight
Thefirsttermistheratioofthemeansquarevalue can be obtained by working from the full equa-
ofthetimederivativeofq_0ð~y;tÞtothemeansquare tions of motion. In this way we are able to set
value of q_0ð~y;tÞ itself. Hence, it represents a fre- combustion noise within a framework that
quency squared. The second term is the square extendstoincludejetnoiseandtoidentifyentropy
of the rms value of q_ð~y;tÞ to its mean, which is a and vorticity as sources of sound. The starting
non-dimensional measure of the amplitude of point is Lighthill’s equation [40,59]. Lighthill
theperturbation. showed that the exact mass and momentum con-
Manyscalinglawsandempiricalrelationshave servation equations can be written in the form
been proposed to understand the physics of [40,59]:
72 A.P.Dowling,Y.Mahmoudi/ProceedingsoftheCombustionInstitute35(2015)65–100
@2q0 @2q0 @2T volumetric expansion coefficient is a and for an
@t2 (cid:2)c20@xi@xi¼@xi@xijj ð6Þ ideal gas is equal to 1/T, and a=cp ¼ðc(cid:2)1Þ=c20.
Equation (9) demonstrates that the density of a
where q0=q(cid:2)q0 and Tij ¼quiujþðp(cid:2)p0 material particle changes because of the pressure
(cid:2)c20ðq(cid:2)q0ÞÞdij(cid:2)sij is the Lighthill stress tensor. variationsinacompressiblefluidandalsobecause
The Kronecker delta is denoted by dij. q is the oftheexpansioncausedbyheating.IfqeinEq.(8)
density,pthepressureand~utheparticlevelocity. is written explicitly as q(cid:2)q (cid:2)ðp(cid:2)p Þ=c2, Eq.
0 0 0
q0, p0 and c0 denote the mean density, pressure (9)canbe usedto replaceDq/Dtthis leadsto
and speed of sound in the acoustic far-field. s is
tbheervisisoctohuesrmstarelsflsotwensToirj.rIenduacheisghtoRqeuyinuojlsdhsonwuiijmng- @@qte¼acpqq0 @@qxii(cid:2)sij@@xujiþXnN¼1@@Yhn(cid:4)(cid:4)(cid:4)(cid:4)q;p;YmqDDYtn!(cid:2)@ð@uxiqieÞ
that the unsteady Reynolds stresses generate 1(cid:5)(cid:2) q c2(cid:3)Dp ðp(cid:2)p ÞDq(cid:6)
sound in the same way as would a distribution (cid:2) 1(cid:2) 0 0 (cid:2) 0 :
c2 qc2 Dt q Dt
ofquadrupoles[39].ThisisthebasisofLighthill’s 0
theory of jet noise. However, when there is com-
Substituting intoEq. (7)resultsin
bustion, there are significant regions of the flow
where the density is significantly different from 1 @2p0 @2p0
ihbammiegpohbonliiergtnthahtnteatt.pnhrdiEsesqtbshuuyearteaitoreflrrnuamnc(tg6uci)n20aðtgqcioat(cid:2)nhnseqba0wnÞedadivjrewewaroriiprttihaennirgnagteoTdrijttioos c20¼@(cid:2)t2@@(cid:2)t"@acxqiq@0 xiXN @@Yh(cid:4)(cid:4)(cid:4)(cid:4) qDDYtnþ@@qxi(cid:2)sij@@xui!#
p n¼1 n q;p;Ym i j
1 @2p0 @2p0 @2 @2q @2 1 @(cid:5)(cid:2) q c2(cid:3)Dp
c20 @t2 (cid:2)@xi@xi¼@xi@xjðquiuj(cid:2)sijÞ(cid:2) @t2e; ð7Þ þ@xi@xjðquiuj(cid:2)sijÞþc20@t 1(cid:2)q0c20 Dt
where q ¼q(cid:2)q (cid:2)ðp(cid:2)p Þ=c2 is the so-called ðp(cid:2)p ÞDq(cid:6) @2
e 0 0 0 (cid:2) 0 þ ðuq Þ: ð10Þ
‘excess density’ [39]. This quantity vanishes in q Dt @x@t i e
i
the far-field but is non-zero in regions where the
entropy is significantly different from that in the Using thechain rule of differentiation
wsou2iqrthreo/oiurtnr2edavinnedrgssio.b2Tlseihj/peorxtohioceexrsjmsaeorsea.cnVooiusnscztoeicsriotsyoinuisnthdneosrtoeuagricosiengss-, XnN¼1@@Yhn(cid:4)(cid:4)(cid:4)(cid:4)q;p;YmqDDYtn
nificantparameterinflownoiseatleastinregions XN @h(cid:4)(cid:4) @h(cid:4)(cid:4) XN @q(cid:4)(cid:4) !
away from the boundary layers, so that there is ¼ @Y (cid:4)(cid:4) xn(cid:2)@q(cid:4)(cid:4) @Y (cid:4)(cid:4) xn
very little loss of generality in assuming flow n¼1 n T;p;Ym p;Yn n¼1 n T;p;Ym
tmeromtioon2ssitj/ooxbieoxinjvisisscmida.lTlahnuds,ciannEbqe.i(g7n)othreedv.iTschoeurs- (cid:2)XN @@Yh(cid:4)(cid:4)(cid:4)(cid:4) r:Jn; ð11Þ
modynamic relationships can be used to deter- n¼1 n q;p;Ym
mine the strength of the source o2q/ot2. The
e wherex istheproductionrateperunitvolumeof
details are given by Dowling in Crighton et al. n
speciesnbyreaction,andJ =qV Y istheflux
[39]. First, the partial time derivative is written n,i n,i n
ofspeciesnbydiffusioninthedirectioni.Thefirst
in a form ofmaterial derivative as
termontherighthandsideofEq.(11)represents
@q Dq q Dq @ðuq Þ theheatreleaserateperunitvolumeandthesec-
@te ¼ Dte(cid:2) qe Dt (cid:2) @xi e ; ð8Þ ond term describes the volumetric expansion due
i
to non-isomolar combustion. The third term is
where the material derivative D=Dt¼ determinedbytheeffectsduetospeciesdiffusion.
@=@tþ~u(cid:3)randthecontinuityequationhasbeen Truffaut et al. [60] examined the additional
used to replace oui/oxi with (cid:2)q(cid:2)1Dq/Dt. The source of noise associated with non-isomolar
energy equation for a gas made up of N chemi- combustion described by the second term on the
cally reacting speciesresults in [31,39] right-handsideofEq.(11).Whenfuelsareburnt
DDqt¼c120DDptþcap @@qxii(cid:2)sij@@xujiþXnN¼1@@Yhn(cid:4)(cid:4)(cid:4)(cid:4)q;p;YmqDDYtn!; iiennxeparatirn,snitoihtnreorgiseeanscmtiavanelldscpotehmceiepsacraoerndetrstitboruottnhiogenlydidorifelcutmtehdoelaainrt
release term (the first term of the right-hand side
ð9Þ
of Eq. (11)). Thus, for air-breathing combustion
whereY isthemassfractionofthenthspecies,h systems the molar production term can be
n
theenthalpyandq istheheatfluxvectorgivenby neglected [61]. However in some industrial appli-
i
q ¼(cid:2)k@T=@x þqPN Y h V with k the cations, such as welding torches, fuels are burnt
i i n¼1 n n n;i
thermal conductivity of the mixture and V the inpureoxygenandthenthechangeinthenumber
n,i
diffusion velocity of the nth speciesin direction i. ofmolesisnolongernegligible,partlybecausethe
The specific heat at constant pressure is c , the chemicalspeciesarenolongerdilutedinnitrogen
p
A.P.Dowling,Y.Mahmoudi/ProceedingsoftheCombustionInstitute35(2015)65–100 73
butalsobecausethehighcombustiontemperature confined geometry with a downstream constric-
(>3000K) leads to strongly dissociated combus- tion,leadstotheindirectcombustionnoisesource
tionproducts [60]. we introduced in Section 2. We will discuss this
Substituting Eq. (11) into Eq. (10) leads to an furtherinSection5.ThethirdtermW alsoleads
3
inhomogeneouswaveequationwhichcanbewrit- to an M8 scaling [39] but with a different coeffi-
tenas cientfromtheclassicalLighthill’sterm.Itisonly
appreciableifqc2inthecombustionregionisnot
c120 @@2tp20(cid:2)@@x2i@px0i ¼W:1þW2þW:3þW:4; ð12aÞ eqqcu2a=lctpoaqn0dc20p,retshseurveailsueneianrlythceonfsatra-nfiteltdh.roSuinghce-
out a low Mach number flow, this noise source
where
shouldbeverysmall[10].AspointedoutbySinai
W1¼(cid:2)acqq0 XN @@Yh(cid:4)(cid:4)(cid:4)(cid:4) xn(cid:2)@@qh(cid:4)(cid:4)(cid:4)(cid:4) tvhairsiatetiromnsisinoniltysesffpeecctiifivcelyhenaotnszaenrodinevaenfluthidenwihthe
p n¼1 n T;p;Ym p;Yn found that this mechanism of noise generation
XN @q(cid:4)(cid:4) ! XN @h(cid:4)(cid:4) tobeinsignificant[66].ThelasttermW4isdipole
(cid:4) @Y (cid:4)(cid:4) xn (cid:2) @Y (cid:4)(cid:4) r:Jn in nature and describes the effect of momentum
n¼1 n T;p;Ym n¼1 n q;p;Ym changes on density inhomogeneities; it is the
@q @u(cid:3) entropy noisesource discussedin Section 2.
þ@xi(cid:2)sij@xi ; ð12bÞ In the combustion region xn the rate of pro-
i j
duction species n is nonzero and the first term
@2 on the right hand side of Eq. (12a) describes the
W ¼ ðquu (cid:2)s Þ; ð12cÞ
2 @x@x i j ij strong monopole sound source of combustion
i j
1(cid:2)(cid:2) q c2(cid:3)Dp ðp(cid:2)p Þ Dq(cid:3) [39]. It has been shown by Flemming et al. [52]
W ¼ 1(cid:2) 0 0 (cid:2) 0 ; ð12dÞ and Ihme et al. [53] that when unsteady combus-
3 c2 qc2 Dt q Dt
0 tionoccursinalowMachnumberflow,thisterm
@ is about two orders of magnitude larger than the
W ¼ ðuq Þ ð12eÞ
4 @x i e other sources demonstrated in Eq. (12a). If the
i
average molecular weight is constant, the second
The details of the derivation of this equation are
term in the first part of Eq. (12a) is neglected.
given explicitly by Dowling in Crighton et al.
When the diffusion of species in neglected then
[39]. Bailly et al. [62] obtained a similar equation
for the combustion of hydrocarbonsin air
based on an extension of Lilley’s equation [63]
to reactive flows. Lilley’s equation includes some XN @h(cid:4)(cid:4)
mean flow propagation effects with propagation (cid:2) @Y (cid:4)(cid:4) xn¼q_; ð13Þ
throughaparallelmeanshearflowbeingincluded n¼1 n q;p;Ym
in the operator on the left-hand side of the where q_ is the heat release rate per unit volume.
equation. Foranidealgasa/c = (c(cid:2)1)/c2.Ifcisassumed
p
Assumingthatthetermsontherighthandside to be independent of temperature and the com-
ofEq.(12a)areknown,theproblemisequivalent bustion takes place at ambient pressure, as is the
toaninhomogeneouswaveequation.Theseterms casefor examplein anopenturbulent flamethen
representthevarioussourcesofsoundgeneration. wehaveqc2¼cp ¼q c2.ThenapplyingEq.(13)
The first term W1 is of monopole type and into Eq.(12a) res0ults i0n0an equation for combus-
describesthesoundgeneratedbyirreversibleflow tion noiseas
processesfromtheunsteadyheatreleaserateand
those of non-isomolar combustion, species diffu- 1 @2p0 @2p0 ðc(cid:2)1Þ
(cid:2) ¼ €qð~y;tÞ: ð14Þ
sion, heat diffusion and viscous dissipation c2 @t2 @x2 c2
0 i 0
[64,65]. When exothermic reactions take place,
Usingthefree-spaceGreenfunction[39]givesthe
the first term in Eq. (12b) due to the rate of heat
far-field pressure perturbation due to an open
release is the most significant. The term r(cid:3)~q in
flame as
Eq.(12b)istheheat-conductionnoisesource,dis-
cussed by Sinai [66]. The second term W is the Z
2 4pc2j~xjp0ð~x;tÞ¼ðc(cid:2)1Þ €qð~y;t(cid:2)j~xj=c Þd3~y; ð15Þ
familiarquadrupolesourceofLighthill’sjet-noise 0 0
theoryassociatedwiththevelocityfluctuations.It v
iswellknownthatitleadstoanacousticintensity recovering the result of the simplified analysis in
thatscaleswithMachnumbertotheeightpower, Eq.(3).Equation(15)predictsthatinthefar-field
M8 [39]. As noted by Powell [67], in a low Mach for a compact flame, i.e. when the wavelength of
number isothermal flow W is approximately the emitted sound is large compared to the flame
2
(cid:7) (cid:8) length, the acoustic pressure perturbation is pro-
equal to r(cid:3)ðqx~(cid:4)~uÞþr2 1qj~uj2 . It is this
2 portional to the time derivative of the total heat
source that generates sound when vorticity is releaseevaluatedataretardedtime.Thevariations
accelerated and, when combustion occurs in a of c arising due to temperature inhomogeneities