On the generation of direct combustion noise in turbulent non …web.stanford.edu/group/ihmegroup/cgi-bin/MatthiasIhme/wp... · 2019. 1. 10. · aeroacoustics volume 11 · number

aeroacoustics volume 11 · number 1 · 2012– pages 25 – 78 25

On the generation of direct combustionnoise in turbulent non-premixed flames

Matthias Ihme* and Heinz Pitsch*Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI 48109

Institute for Combustion Technology, RWTH Aachen, Templergraben 64, 52056 Aachen, Germany

Received January 9, 2011; Revised October 6, 2011; Accepted October 8, 2011

ABSTRACTGeneration of combustion noise in an unconfined turbulent non-premixed flame is investigated.For this, a model is developed, combining Lighthill’s acoustic analogy with a flamelet-based combustion model to consistently express all thermochemical quantities by a set of reduced scalars. The model is applied in a large-eddy simulation, and the acoustic pressure in thefar field is obtained from an integral solution. Three relevant acoustic source terms with differentsource characteristics and Mach number scaling are identified. The spatial distribution andspectral characteristics of the acoustic sources are analyzed, and it is shown that the acousticsource due to chemical reaction is the main noise contributor, and is located in the upper part ofthe flame. Contributions from the acoustic sources due to Reynolds stresses and fluctuating massflux are found to be virtually insignificant at low frequencies. Discrepancies in the prediction ofhigh-frequency sound pressure level in the jet forward direction were analyzed and are attributedto high-frequency acoustic refraction effects due to variations in sound speed. The directivityexhibits a weak directionality in the 30° forward direction, and some phase cancellation betweenindividual acoustic sources is evident.

1. INTRODUCTIONAccording to the International Civil Aviation Organization [1], global air transportationis predicted to increase with an anticipated doubling in air traffic over the next twodecades. Associated with the increasing air traffic is the adverse environmental impactthrough noise and pollutant emissions. The primary noise source from aircraftoperating at subsonic flight conditions is the propulsion system, which can further beseparated in noise contributions from fan, compressor, combustor, turbine, and jetexhaust. Over recent years remarkable technological progress has been made leading tosignificant reductions in engine noise. These improvements were primarilyaccomplished through the reduction of jet noise by increasing the bypass ratio.

*Corresponding author. Email address: [email protected]

However, because of this significant reduction in jet noise and anticipatedadvancements in the noise suppression of turbomachinery, combustion noise isbecoming a leading contributor to the overall aircraft noise emission, and is nowconsidered as a lower aircraft noise limit [2, 3]. Furthermore, advanced combustionconcepts, such as rich-quench-lean combustors [4], staged fuel injection systems [5],and lean premixed prevaporized combustion [6] can lead to considerable increase innoise emissions. The main reasons for this are that lean premixed and stratifiedcombustion modes are receptive to combustion-driven oscillations, and spatiallyinhomogeneous heat release and turbulence fluctuations can amplify the noisegeneration in staged combustion systems [7].

The noise radiated from a combustor can be characterized as either direct or indirect[8]. Direct noise refers to the generation of noise due to the expansion of a spatiallycoherent gas volume through unsteady heat release. The hot gas volume performsmechanical work on its colder surrounding and generates acoustic pressureperturbations. In addition, the heat release by combustion affects the flow fieldstructure, so that the noise contribution from the turbulent velocity field is alsomodified. Indirect combustion noise, on the other side, is generated by the convectionof entropy non-uniformities through strong pressure gradients. In gas turbines, thenecessary pressure gradients are generated by contractions and turbine stages followingthe combustion chamber. The generation of indirect combustion noise from entropyinhomogeneities in nozzles has been described by Ffowcs Williams & Howe [9] andMarble & Candel [10]. Although Bake et al. [11] demonstrated that indirect combustionnoise is a significant source of noise in an experimental combustor configuration, therelative contribution of both acoustic source mechanisms in aircraft engines andauxiliary power units is currently not fully established, and recent measurementssuggest that this is dependent on geometry, operating conditions, and frequency range[12]. For instance, coherent noise measurements in combustors by Muthukrishnan et al.[13] and Miles [14] suggest that indirect combustion noise is particularly relevant forfrequencies below 200 Hz. For frequencies between 200 and 400 Hz, the pressuremeasurements show a higher coherence between combustion pressure and direct noise.This was attributed to the decorrelation of the pressure perturbations due to theirinteraction with the turbine.

Measurements in open and confined flames have shown that combustion noise isbroadband having random phase, and the peak intensity is typically in the lowfrequency range around 200–1000 Hz [15, 16]. In this respect, the broadband acousticperturbations interact only weakly with the heat release rate, so that this process is oftenconsidered to be passive [17]. However, theoretical analyses showed that noise inenclosed combustion systems, for instance in gas turbines and advanced low-emissionaircraft engines [6], plays a critical role as precursor for combustion instabilities [18,19], and even such weak interactions can trigger thermoacoustic instabilities, in whichthe flame becomes an active part of the acoustic system [20, 21, 22].

The subject of the present investigation is the generation of direct combustion noisefrom a non-premixed turbulent jet flame. For this a hybrid model is employedcombining large-eddy simulation (LES) for the reactive flow-field predictions with

26 On the generation of direct combustion noise inturbulent non-premixed flames

Lighthill’s acoustic analogy to determine the acoustic pressure field. By utilizing aflamelet-based combustion model, the source due to the entropy variation in Lighthill’sequation is reformulated to highlight the direct relation to the chemical production rate.In this respect, the acoustic analogy is fully consistent with the combustion model, andall relevant acoustic sources are extracted from a low-Mach number variable densityLES. After a brief review of related work on direct combustion noise in the next section,the combustion model and acoustic formulation are presented in Sec. 3. The hybridmodel is applied to a turbulent diffusion flame, and results are compared againstexperimental data. A basic requirement for the prediction of the acoustic pressure fieldis the accurate characterization of the underlying flow field. Therefore, statistical resultsfor flow field and scalar quantities together with two-point correlations are analyzed inSec. 5. Acoustic results are discussed in Sec. 6. First, numerical aspects of thecomputation and source term evaluation are presented, which is then followed by the discussion of the far field spectra, directivity, and the source term localization. Thepaper finishes with conclusions.

2. REVIEW OF RELATED WORK ON COMBUSTION-GENERATEDNOISEIn the following, an overview of experimental, theoretical, and computational work onthe generation of combustion noise is given. Note that the primary focus is on thecharacterization and prediction of direct combustion noise, and the interested reader isreferred to the review article by Candel et al. [23] containing a more detailed discussionon this subject, including indirect combustion noise and its relation to thermoacousticinstabilities.

The role of combustion noise was early recognized, and the noise emission fromturbulent premixed and non-premixed open jet flames was extensively studied. Knott[24] investigated the noise generation in coflowing and impinging jet flames usinghydrogen and ethylene as fuel. Effects of burner diameter and equivalence ratiovariations on the radiated power were studied, and it was shown that jet flames generatea broader frequency spectrum compared to non-reacting jets, having a dominant peakat lower frequencies.

The noise generation from multiple impinging jet diffusion flames operating withmethane and air was studied by Giammar & Putnam [25]. They identified firing rate asthe parameter that correlates with the noise emission, and it was argued that the centralfrequency of the acoustic spectrum is related to the fuel composition and reactionchemistry. In contrast to this finding, Bertrand & Michelfelder [26] concluded fromtheir experimental studies on swirling flames that the peak frequency depends on theswirl number but not on the firing rate. The directivity measurements in theirexperiments showed an omnidirectional sound emission for frequencies below 500 Hz,and a shift to a dipole behavior for increasing frequencies. In a theoretical study, Strahle[27] could show that this weak directionality can be attributed to refraction andconvection effects. Interestingly, Bertrand & Michelfelder [26] found that the preheatingof the oxidizer increases the sound power output and peak frequency, and related thisfrequency shift to changes in turbulence characteristics due to preheating.

aeroacoustics volume 11 · number 1 · 2012 27

Price et al. [28] characterized the noise emission from turbulent premixed, diffusion,and spray flames by measuring the far field sound pressure signal and the visible lightemission from CH and C2 radicals. These free radicals are formed in the high-temperature reaction zone, and are therefore suitable markers for the heat-release rate.Hydrogen, methane, and ethylene were considered as fuels. They successfullycorrelated the rate of change of the light emission to the pressure signal, and it wasshown that the C2 emission intensity is dependent on the volume flow rate butindependent of the turbulence.

The noise emission from non-premixed and premixed flames was also studied byKumar [29]. He observed significant differences in structure and noise characteristicsbetween premixed and diffusion flames. Contrary to Price et al. [28], he argued thatmodifications in the turbulence characteristics can have an important influence on thenoise emission. He suggested that variations in flame surface area in premixed flamesdue to the wrinkling of the flame front modify the heat release rate, resulting in theobserved monopole behavior. On the other side, the high-frequency components thatwere observed in the pressure spectra of the diffusion flame were attributed by Kumarto the abrupt volume expansion of the highly distorted and intermittent reaction zone.However, by utilizing the Shvab-Zeldovich formulation and expressing the density asfunction of mixture fraction, it can be shown that the dilatation is proportional to thediffusion of the mixture fraction. Since this term is typically small and slowly varying,his observation suggests that other physical mechanisms might be responsible for thehigh-frequency noise emission, such as local flame extinction or flame lift-off. In fact,photographs of the luminous diffusion flame [29] show indeed some evidence of a liftedflame which is detached from the nozzle.

Recently, Singh et al. [30] conducted experimental investigations on a partiallypremixed turbulent flame configuration at different operating conditions. They foundthat the spectral characteristics for these nitrogen-diluted methane/air flames show aqualitatively similar behavior for different fuel mass flow rates. Interestingly, they alsoobserved that the jet flame with the larger exit Reynolds number emits significantlymore noise at higher frequencies. Their measurements suggest that combustion noisesources are distributed over an extended spatial region, and the sound pressure level forthe diffusion flames exhibits a plateau at frequencies in the range 400–2000 Hz.

Investigations of the acoustic source term distribution in premixed and diffusionflames have been conducted by different groups. Price et al. [28] used the light emissionmeasurements from C2 and CH radicals to qualitatively identify the dominant acousticsource region in a diffusion flame. By masking parts of the flame and correlating the netchange in light emission with the pressure signal they found that noise in the frequencyrange between 200 and 1000 kHz was primarily emitted from the lower part of theflame in the nozzle-near region.

Shafer et al. [31] applied a similar optical measurement technique to a premixedpropane/air flame. By measuring the light emission from two regions in the flame withfocused optics they were able to quantify correlation length and magnitude of relevantconvective and chemical source terms that are directly related to the heat release. Theirresults show that both longitudinal and transverse correlations from the unsteady CH


radical emissions increase with increasing downstream direction, suggesting that theacoustic sources are located in the upper part of the flame.

Effects of turbulence intensity and variations in equivalence ratio φ in a premixedpropane/air flame were studied experimentally by Kotake & Takamoto [32]. While theyfound that acoustic power levels of flames with φ > 1.6 are rather insensitive to changesin the inflow turbulence, they showed that increasing turbulence levels in fuel-leanerflames can lead to significant increase in acoustic radiation, which they attributed toamplifications of high-frequency pressure perturbations. From measurements of thecorrelation between the acoustic pressure fluctuation and the time derivative of theradical emission intensity emitted from a focused volume in the flame, Kotake &Takamoto [32] reported remarkable differences between lean and fuel-rich flames. Inparticular, they found that the noise sources in lean premixed flames are mostlydistributed in the upper part of the flame above the maximum heat release, and its sourcestrength increases uniformly with increasing inflow turbulence. On the other side, thedominant sources in fuel-rich flames are located in the region below the maximumflame temperature, and increasing turbulence levels lead only to a shift in the noisesources towards the fuel nozzle without significant increase in the source magnitude.

Recently, Rajaram & Lieuwen [33] performed comprehensive spectral measurementsin a series of premixed flames for a range of nozzle diameters, equivalence ratios, andburner exit velocities. They found that the acoustic power is a function of equivalenceratio and exit velocity, and that the peak frequency decreases with increasing nozzlediameter. Interestingly, their experimental results also suggest that the acoustic spectraare independent of flame length and speed.

These experimental efforts on combustion noise characterization have beencomplemented by theoretical studies. Most notable is the work by Bragg [34] whoconsidered a turbulent flame as a distribution of uncorrelated monopole sources, and theacoustic power emitted by the flame is then related to the sum of the individual volumechanges due to the unsteady heat release.

Utilizing Lighthill’s acoustic analogy [35], Strahle [36] derived an expressionrelating the far field density perturbation to the second time derivative of the volumeintegral over the density variation in the combustion region. From this theory hedeveloped scaling relations for premixed and non-premixed flames, and was able toidentify the general monopole behavior and low frequency combustion noisecharacteristics. In subsequent work, he extended this theory and showed that the farfield pressure can be directly related to the volume integral of the Eulerian timederivative of the heat release rate evaluated at retarded time [2].

Chiu & Summerfield [37] derived a convective wave equation describing thegeneration and transmission of pressure perturbations in reacting flows. UnlikeLighthill’s analogy, which was derived from mass and momentum conservation, theyalso employ an energy equation to derive this wave equations. In this respect, theirmodel can be considered as extension of Phillips’ equation [38] to non-isothermalflows. In this model, effects due to variations in sound speed, thermo-viscous effects,and source term convection are considered. Chiu & Summerfield [37] applied the modelto premixed flames, and derived correlation functions for the flame structure in the


wrinkled and distributed reaction zone regimes. A similar inhomogeneous waveequation was also derived by Kotake [39]. In this work the source term contribution dueto the heat release rate was reformulated and expressed in terms of correlations betweenspecies and temperature fluctuations. This source term expression provided insight intothe low-frequency contribution of combustion noise, which Kotake attributed to thetemporal correlations of the thermochemical fluctuations.

Supported by theoretical models and experimental investigations, analytical andsemi-empirical scaling relations have been derived to characterize power spectra,directivity, and peak frequencies with respect to geometric parameters, operatingconditions, and fuel composition for various jet flame configurations.

Klein & Kok [17] developed an integral formulation for the prediction of noisegenerated from non-premixed flames. Their model was derived for the limit of infinitelyfast chemistry for which the acoustic source term can directly be related to the mixturefraction. The sound spectrum was then expressed in terms of the turbulence spectrumof the mixture fraction, and parameters describing its spectral shape were obtained froma numerical simulation. Comparisons of the modeling results with experimental data fora confined combustor configuration were in favorable agreement for mixing-controlledcombustion.

Miyauchi et al. [40] conducted detailed numerical simulations of the soundgeneration in a two-dimensional chemical reacting mixing layer. Results obtained fromacoustic analogies due to Lighthill and Powell [41] were compared with detailedsimulation results. From this analysis they concluded that Lighthill’s analogy providesaccurate predictions for the acoustic pressure field; however, they also showed that theseresults were sensitive to the extend of the spatial domain over which the acoustic sourceswere evaluated.

Direct numerical simulations of an axisymmetric isothermal and reacting jet havebeen conducted by Zhao & Frankel [42]. A one-step irreversible chemical reaction wasconsidered and the acoustic source terms were identified using Lighthill’s analogy.Their simulations showed that the reacting jet emits higher noise levels than a non-reacting jet at similar operating conditions, which the authors attributed to a frequencyshift of the unstable modes to lower values and the dominant contribution of the heatrelease rate to the overall noise emission. From a Fourier analysis they located thedominant source region in the flame. These results suggest that the acoustic source dueto heat release rate in the high and mid frequency range is located in the upstream partof the flame and shifts downstream for higher frequencies. Despite these interestingfindings the authors also recognized the limitations of their study, and suggested that theconsideration of three-dimensional effects and detailed reaction chemistry could lead todifferent sound characteristics.

Bui et al. [43] developed a hybrid approach in which the acoustic perturbationequations (APE) were extended to predict the noise generation in turbulent reactingflows. In this approach, LES was used to obtain the mean flow quantities and acousticsources. These flow field quantities were then transferred to the acoustic solver fromwhich the resulting pressure field was computed using the APE system. From the APEsystem for reacting flows, a sequence of equivalent acoustic source term formulations


was derived. These different formulations were analyzed by Bui et al. [44] with respectto sensitivities in the source term interpolation, source term complexity, and accuracy ofthe predictions for the spectral characteristics and directivity of the acoustic signal.

A hybrid approach for the prediction of direct combustion noise was also employedby Flemming et al. [45], in which LES and Lighthill’s analogy was combined to predictthe acoustic pressure field emitted from a nitrogen-diluted hydrogen diffusion flame. Inthis work the authors reformulated Lighthill’s second-order wave equation into a set offirst-order hyperbolic partial differential equations, and the acoustic source term,namely the second temporal derivative of the density, was obtained from a low-Machnumber LES computation. Computational results for the acoustic intensity wererescaled to the far field to facilitate comparisons with experimental data, showing goodagreement between simulations and measurements.

The present paper extends the previous work by Ihme et al. [46] in which importantsources of spurious noise were analyzed in the context of LES combustion noisesimulations. It was shown that these spurious noise sources contaminate the acousticspectra, and a low-pass filter was proposed to eliminate spurious noise. The presentwork addresses the acoustic source mechanisms and quantifies the acoustic source termdistribution, directivity, and far field characteristics for a nitrogen-diluted methane-hydrogen/air flame.

3. MATHEMATICAL MODELIn the present work Lighthill’s acoustic analogy [35] is employed for the prediction ofthe radiated sound field. In this analogy the noise-generating sources in a limited flowdomain, denoted by ΩF , are represented by an acoustically equivalent source termdistribution, embedded in a homogeneous stagnant fluid, ΩA (see Fig. 1), havingconstant reference properties aref , ρref and pref . Here, a, ρ, and p denote the speed ofsound, density, and pressure, respectively. The subscript “ref” refers to a constant


Source region ΩF

Acoustic region ΩA

(t, x)

(τ, y)

R

Figure 1: Model definition for aerodynamically generated sound; (τ, y) denotes apoint in the source region ΩF and (t, x) represents the point of perceptionin the acoustic domain ΩA.

reference quantity or a geometric property based on the jet fuel pipe. The exit Machnumber, M = Uref /aref , based on the jet exit bulk velocity and the speed of sound in airat room temperature, is sufficiently small so that variations in the density due topressure changes are negligible in the simulation of the turbulent reacting flow field.Therefore, the turbulent reactive flow field and the acoustic source term distribution arecomputed using a low Mach number variable-density LES code. For the prediction ofthe flow field and the species distribution in ΩF , a flamelet/progress variable (FPV)model [47, 48] is used.

3.1. Governing equationsThe instantaneous equations describing the conservation of mass and momentum,evaluated in ΩF, can be written in dimensionless form as

(1a)

(1b)

in which u is the velocity vector, σ= represents the viscous stress tensor, and Dτ =∂τ + u ⋅∇y is the substantial derivative. In the present investigation, externalvolume forces due to buoyancy are neglected because of the small Richardsonnumber of Ri ≈ O (10−5) in the considered problem. The viscous stress tensor,appearing in Eq. (1b), has the form:

(2)

Note that Eqs. (1) and (2) are written in non-dimensional form. The dimensionlessvariables used in these equations and in the remainder of the paper are defined as:

(3)

D Dτ τ σσ

= ∇ = ∇ = =∗ ∗∗

∗D

UD

U

Dref

refref

ref

, , ,y yuu rref

ref ref ref

ref ref

U

pU

ρ ν

ρρ

ρρ

ρ

,

,= =∗ ∗

rref ref p ref ref

ref

h hc2

, , ,,

νν

ν

αα

α

= =

=

∗ ∗

∗

∆Θ

,, , ,Z ZZ

C CC

W

ref refC

C re= = =∗ ∗ ∗

��

ωω ff

ref ref ref

ref

refC A

E

ρexp ,

R∆Θ

, ,χα

χZref

refZ

ref

refD ff D

U= = =∗

∗ ∗2

ΘΘ

∆Θ rref,

σ µ= − ∇ ⋅( )

= ∇2

1

3

1

2S I Sy yu with uu uy+ ∇( )

T.

ρ τD u y yRe,= −∇ + ∇ ⋅p 1 σ

Dτρ ρ ,= − ∇ ⋅y u


where ν is the kinematic viscosity, α is the equal molecular diffusivity for all species(assuming unity Lewis number), h is the enthalpy, cp is the specific heat capacity atconstant pressure, Z is the mixture fraction, and C is the progress variable. The chemicalreaction rate of C is denoted by ω⋅C , Θ is the temperature, W is the molecular weight,A is the frequency factor, E is the activation energy, R is the universal gas constant, andχZ is the scalar dissipation rate of the mixture fraction. The frequency is denoted by f.The quantities νref , αref , and cp,ref refer to properties of the fuel, and the asterisk denotesa dimensional quantity. Other reference quantities are based on the undisturbedreference field. Using this non-dimensionalization, the following similarity parameterscan be formed:

Values for these parameters for the configuration which is of interest in the present workwill be given in Sec. 4.1.

3.2. Combustion modelThe laminar flamelet concept was developed to describe non-premixed combustion. Inthis model, a turbulent diffusion flame is considered as an ensemble of laminarflamelets [49]. At sufficiently large Damköhler number or sufficiently high activationenergy, chemical reactions and heat transfer occur in thin layers. If the characteristiclength scale of this layer is smaller than that of the surrounding turbulence, the turbulentstructures cannot penetrate into the reaction zone and are unable to destroy the flamestructure [50]. The effect of turbulence in this so-called flamelet regime results in adeformation and stretching of the flame sheet. With this notion, a flamelet can be

Reynolds number :

Schmidt nu

Re = ,U Dref ref

refν

mmber :

Damköhler number :

Sc

Da

=

=

ν

αref

ref

reD

,

ff

ref

ref ref

ref

ref

refU

A

W

Eρexp −

R∆Θ

=

,

,

Eckert number : EcU

cref

p ref ref

2

∆Θ,,

,Richardson number :

Mach num

Ri =gD

Uref

ref2

bber :

Strouhal number :

M

St

=

=

U

f

ref

refα,

.


considered as a thin reaction zone surrounded by a molecular transport layer, which, inturn, is embedded within a turbulent flow [51]. The thermochemical state of the flamecan then be obtained from the solution of the steady laminar flamelet equations,

(4)

and is a function of the mixture fraction and scalar dissipation rate

(5)

In Eq. (4), ω⋅ is the vector of the source terms for species mass fractions andtemperature, which are collectively denoted by the vector φ. The solution of Eq. (4) atstoichiometric condition, Z = Zst, can be represented by the so-called “S-shaped curve,”and is shown in Fig. 2. The lower and upper branches of this curve describe the stableburning and non-reacting states, while the middle branch is unstable. Note that thelower branch essentially corresponds to the pure mixing of fuel and oxidizer at ambienttemperature, and the turning point between the upper branch and the middle branch ofthe S-shaped curve corresponds to χZ,st = χZ,q⋅ Since multiple solutions exist for acertain range of χZ,st, a parameterization using only Z and χZ,st cannot represent theentire solution space.

In order to overcome the ambiguity of the steady flamelet model a reaction progressparameter Λ is introduced in the FPV model [47, 48]. This parameter is based on areactive scalar, the progress variable C, and is defined to be independent of the mixture

χ αZ Z= ∇22

y .

−χZ

Z2∂2 φ ω= ,�


χZ,st

0.01 0.1 1 10 100 10000

0.25

0.5

0.75

1

1.25

0

0.25

0.5

0.75

1

1.25

0 0.25 0.5 0.75 1

Θst Θ

Z

Upper branch

Middle branch

Lower branch

χZ,qχZ,st = 0.20χZ,st = 150

Figure 2: S-shaped curve as solution of the steady flamelet equations showing thetemperature as function of the scalar dissipation rate at stoichiometricmixture fraction Zst = 0.167 for a N2-diluted CH4-H2/air diffusion flame.The right figure shows two flamelets for χZ,st = 0.2 and 150, respectively.

fraction. The reaction progress parameter has to uniquely represent each flamelet alongthe S-shape curve, and a possible definition that is used in the present work is

(6)

corresponding to the value of the progress variable of each flamelet evaluated atstoichiometric condition. Using this parameter, all thermochemical quantities, denotedby ψ = (ρ, ν, α, ω⋅C , Υ

T, Θ)T, can then be written as

(7)

The progress variable is a linear combination of some product mass fractions, and canbe obtained from Eq. (7) as

(8)

Under the assumption that FC is bijective, the reaction progress parameter in Eq. (7)can be eliminated by inverting the flamelet table, Eq. (8), and all thermochemicalquantities can then be expressed in terms of only Z and C :

(9)

with

In addition to the solution of the Navier-Stokes equations, the FPV model requiresthen the solution of the following transport equations describing the conservation of Zand C:

(10a)

(10b)

The equations of motion (1) and the combustion model (10) are then coupled throughthe density, obtained from the flamelet library

(11)ρ ρ= ( )G Z C, ,

ρ ρα ρτD C C C= ∇ ⋅ ∇( ) +1

ReScDa .y y �ω

ρ ρατD Z Z= ∇ ⋅ ∇( )1

Re,

Sc y y

G F F 1ψ ψZ C Z Z CC, , , .( ) = ( )( )–ψ ψ= ( )G Z C,

C ZC= ( )F , .Λ

ψ ψ= ( )F Z, .Λ

Λ =C Z ts ,


and viscosity, diffusivity, and chemical reaction rate of the progress variable are alsoobtained from the FPV relation.

3.3. Acoustic model 3.3.1. Wave equationIn an open turbulent reactive jet flame, the sound is generated both by the turbulent flowfield, primarily through unsteady Reynolds stresses, and by the combustion process inΩF . The sound propagates as acoustic waves to the point of perception (t, x) located inthe acoustic domain ΩA (see Fig. 1). Following Lighthill’s derivation [35], aninhomogeneous wave equation for the pressure disturbance can be derived from theexact equations of motion by performing the operation ∂τ(1a) −∇y · (1b) and adding theterm Μ2∂τ

2p to both sides. This leads to

(12)

The last term on the right hand side of Eq. (12) can be written in terms of the “excessdensity” [52],

(13)

Large spatial and temporal variations of entropy occur due to rapid heat release inchemically reacting flows, and hence this term becomes an important source incombustion-generated sound. The time derivative of ρe can be reformulated as [52]:

(14)

Using Eq. (11), the substantial derivative of the density can be expressed as

(15)

and with Eqs. (10) the wave equation can be reformulated as

(16)

M2

R M

∂ ∂2τ τγ ρ ρp p

T

′ − ∆ ′ = ≡ ∇ ⋅ ∇ ( )− ∇ ⋅ −( )( )y y y yF

uu u.� � �

1��

�

� ��

−

Da

R

∂ ∂τ ρρ

wCC

Q

−− ∇ ⋅ ∇( ) + ∇ ⋅ ∇( ){ }

1 12ReSc

∂ ∂ ∂τ ρρ ρα ρ ραΖ y y y yZ c C

QD

� ��

� �� Re+ −( )− ∇ ⋅ ∇ ⋅M2

P

∂2τ σp prefQ T

1y y

VV

� ,

D D Dτ τ τρ ρ ρ= +∂ ∂Ζ Z CC ,

∂ ∂τ τ τρ ρρ ρe p p= + ∇ ⋅ −( )( )− −( )

11D y refu M

2 .

ρ ρe refp p= −( )− −( )1 2M .

M21

∂ ∂2 2τ τρ σ ρp p− ∆ = ∇ ⋅ ∇ ⋅ −

−y y y uu Re−−( )M2p .


with p′= p − pref. Here, the terms T=R and T=V represent quadrupole sources due tounsteady Reynolds and viscous stresses. The term FM is a fluctuating mass flux term ofdipole nature, and QR, QD, and QP are monopole sources due to unsteady reaction rate,fluctuations due to diffusive transport, and pressure perturbations, respectively.

By employing a free-space Green’s function, a formal solution of Eq. (16) can bewritten as

(17)

3.3.2. Far-field approximation and acoustic powerThe magnitude of the individual source terms in Eq. (17) contributing to the far-field

pressure at the location x can be estimated as function of the non-dimensionalparameters using the far-field approximation |x| � |y| and the reciprocity relation. ThenEq. (17) can be written as

in which the square bracket [φ]R denotes φ(y, t – M|x – y|). Equation (18) shows thatthe monopole describing the unsteady reaction rate QR is independent of the Machnumber. The dipole term scales with the Mach number, and the terms containing T=R,T=V, and QP scale with M

2. For low Mach number combustion, the quadrupole termsdue to fluctuating viscous and Reynolds stresses are expected to be negligible, ascompared with the monopole source due to unsteady heat release. It is interesting tonote that the term T=R , which is small for the present case, is the dominant source ofsound in turbulent isothermal jets.

The total acoustic power Pac emitted from a turbulent reacting jet can be estimatedas

(19)

and the rate at which chemical and kinetic energy are supplied to the jet can becomputed from

(20)� �E m h uinp

= + +

Ec

ρ1

22

.

Pac p∝ ′M2,

(18)

p t Qt t′ ( ) = − ∂ + ⋅ ∂ +Ω∫∫∫,x xxx

F xx

x

1

42

2π F R RDa M MR M

2 ::

Re:

∂

+ ∂ −

t

t

T

Q

2

2

2

R

2P

2

MM

R

R

xx

x∂∂

− ∂

t t DT Q2 1

V Scd

R RRey,

p tt

′ ( ) =− −( )

−Ω∫∫∫,

,x

x y y

x yy1

4π

γ

F

Md .


In the case of turbulent flames, the thermal energy is usually dominant over thekinetic energy supply. With the above scaling relations and Eq. (18), the followingobservations can be made. The ratio between the acoustic power output generated bythe unsteady heat release and E

.in is

(21)

which scales linearly with the Mach number and quadratically with the Damköhlernumber. The contribution of F

Mto the acoustic power, normalized by E

.in, is

(22)

which scales with the third power of the Mach number. Furthermore, Lighthill’s well-known M5-scaling relation for the acoustic power radiated due to the unsteady Reynoldsstress can be written as

(23)

In the limit of large Reynolds number flows, the source term contributions due to theviscous stresses and species diffusion can be neglected.

By employing a low Mach number code for the computation of the acoustic sourceterms, the pressure appearing in Eq. (1b) contains only the hydrodynamic componentand acoustic pressure perturbations are not accounted for. Therefore, the term QP in Eq.(16) cannot be evaluated from the LES code that was employed in the present work.Assuming that this term is small, QP is neglected and the wave equation with thesimplified acoustic source term is then written as

(24a)

(24b)

In the following this equation is used to compute the acoustic pressure field emittedfrom a turbulent diffusion flame. Since the model is applied in the context of an LES,the Favre-filtered form of Eq. (24) is derived in the next section.

3.4. LES formulationThe acoustic model presented in the previous section was derived from theinstantaneous form of the governing equations. The objective of the present work,however, is to obtain the solution of the unsteady, three-dimensional, turbulent reactive

γ τ τa R M RDa .= ∇ ⋅ ∇ ⋅ − ∂ ∇ ⋅ − ∂y y FT Q

M a2 2∂ ′ − ∆ ′ =τ γp py ,

P T

Eac

in

R M( )

∝�5.

P

Eac

in

FM M( )

∝�3,

P Q

Eac

in

R MDa( )

∝�2,


flow field from a large-eddy simulation (LES). The LES technique is based on aseparation between large and small scales in a turbulent flow. While the large, energy-containing, and geometry-dependent scales of motion are numerically resolved, theeffect of the numerically unresolved scales onto the large scales requires modeling [53].The decomposition into the different scales is achieved by applying a filter operator tothe governing equations (1) and (10). For reacting flows, a density-weighted filter in theFavre-sense is employed, and a Favre-filtered quantity of a scalar ψ is defined as

(25)

where ∆ is the filter width, and G is the filter-kernel, satisfying the normalizationcondition

(26)

The residual field is then defined as ψ′′(τ,y) = ψ(τ,y) − ψ∼(τ,y), and Favre-filteredquantities are related to Reynolds-filtered quantities by ρ−ψ∼ = ρψ− .

In the FPV combustion model, all thermochemical quantities are obtained from thestate relation, Eq. (9), which is parameterized in terms of Z and C. For application inLES a state equation is required which relates Favre-filtered thermochemical quantitiesto the filtered quantities of mixture fraction and progress variable. Due to the strongnon-linearity of the state equation, moment methods, which are frequently employedas closure model in isothermal turbulent flows, are inadequate in turbulent reactingflows. Therefore, Favre-filtered quantities are modeled in a statistical sense byemploying a presumed probability density function (PDF) closure model, which can bewritten as

(27)

In principle, a transport equation for the joint scalar FDF P~(Z, Λ) can be solved [54, 55,56]. However, because of accuracy requirements for the modeling of the molecularmixing terms and the computational complexity of the numerical algorithm, we refrainfrom solving such a transport equation, and employ instead a presumed probabilitydensity function (PDF) approach to model P~(Z,Λ).

By making use of the fact that Z and Λ are defined to be statistically independent,the joint PDF can be expressed in terms of its marginal PDFs as

(28)

which results in a considerable simplification of the closure model for the presumedPDF. In this context, it is interesting to point out that Eq. (28) is a direct consequenceof the steady flamelet assumption and the definition of Λ, and its validity to flameletregimes has recently been analyzed in an a priori study [48].

� �P Z P Z P, ,Λ Λ( ) = ( ) ( )

� � �ψ ψ ψ= = ( ) ( )∫∫F F Z P Z Z, , .Λ Λ Λd d

G ξ ξ, ; .y ∆( ) =∫ d 1

�ψ τρ

ρ τ ψ τ, , , , ;y y( ) = ( ) ( ) ∆( )∫1 ξ ξ ξ ξ,G d


The marginal PDF of Z is modeled by a beta distribution, so that its shape is entirelydetermined by the first two moments of the mixture fraction, Z~ and Ζ′′2. A Diracdistribution is used to model P(Λ). By employing this presumed PDF in Eq. (27), ~ψ canbe evaluated and parameterized in terms of Z~, Ζ′′2, and ~Λ. By expressing ~Λ in terms ofthe Favre-filtered progress variable, this relation can be written as

(29)

which corresponds to the Favre-averaged form of the thermochemical library of Eq. (9).In order to evaluate this relation, two additional transport equations for Z~ and ~C are solved, and Ζ′′2 is obtained from a dynamic model [57].

By applying the LES filter operator to Eqs. (24), the Favre-filtered wave equationcan be obtained. Note, however, that this operation introduces two unclosed sourceterms, which require modeling. The first source term arises from the Reynolds stresstensor, which can be written as –ρ uu = –ρu~u~− =σ

– res. The superscript “res” refers here toa residual scale quantity. In the following application, the residual stress tensor isomitted so that

–T=R = –ρu~u~. This simplification can be justified by the fact that =σ– res

represents the velocity correlation of the unresolved scales corresponding to highfrequencies and wave numbers. Experimental measurements showed (see Sec. 2) thatcombustion noise is mainly relevant at low frequencies to which the contribution of theunresolved acoustic source term =σ

– res is negligible. Another argument arises from thesource term efficiency of T=R to the far-field pressure, which scales with M5. It wasshown in the previous section, that this term is only a minor contributor to the acousticpower in low Mach number flows, so that this simplification has only a secondary effecton the total sound pressure level.

The filtered source term due to the unsteady reaction rate is closed using the

presumed PDF model, and –QR = is precomputed and stored in the

thermochemical library, Eq. (29). With this, the filtered form of the acoustic model forapplication to LES can be written as

(30a)

(30b)

and p′ = –p – pref . In the following the acoustic pressure field is computed from Eq. (30)and the acoustic sources are obtained from the combustion LES.

3.5. Numerical implementationThe sound field at the observer location (t,x) is computed from the integral formulationwith the simplified source term shown in Eq. (30b). Rather than determining the time-dependent pressure fluctuation at this location, it is usually of more interest to analyzethe power spectral density S(ω) and sound pressure level (SPL) as function of thefrequency ω. Introducing the forward and inverse Fourier transformations

M

Da

2a

a R M R

∂ ′ − ∆ ′ =

= ∇ ⋅ ∇ ⋅ − ∂ ∇ ⋅ − ∂τ

τ τ

γ

γ

2p p

T Qy

y y F

,

,

�ω ρC C∂ ln( )

ψ ψ

= ′′

G Z Z C, , ,

2


(31a)

(31b)

the Fourier-transformed integral equation for the far-field pressure can be written as

(32)

with

(33)

The spatial derivatives in Eq. (33) can be transferred to the pre-exponential term usingintegration by parts, giving

(34)

with

(35a)

(35b)

(35c)

where Cartesian index notation was used and Ri = xi – yi so that R = x – y.

3.6. Model limitationsLighthill derived his acoustic analogy by representing the noise emission from aturbulent flow as an equivalent source term distribution which is embedded in anotherwise homogeneous flow. The separation between source and propagation regionsintroduces the limitation that this theory cannot directly account for acoustic refractioneffects due to mean flow interactions. Furthermore, the heat release in a turbulentreacting flow leads to changes in the local sound speed through the relation

µ ω ω= −iR

i RDa Mexp{ },

λ ω ω ωiii

R

Ri R i R= +( ) −{ }

31 3 M Mexp ,

κω

ωδ

iji j ijR R

R Ri

R R= + −

−

3 2

2 2

3

3 3 MM 11+( )

−{ }i R i Rω ωM Mexp ,

′( ) = + ⋅ −( )∫∫∫ˆ , : ˆ ˆ ˆ ,p T Qω π κ µx F y1

4 ΩF R M Rdλ

ˆ ˆ ˆ ˆ .γ ω ωa R M R= ∇ ⋅ ∇ ⋅ − ∇ −y y yT i i DaQF

′( ) =− −{ }

−( )

Ω∫∫∫ˆ ,exp

ˆ , ,p w xx y

x yy y1

4π

ωγ ω

F

i Mda

ψπ

ψ ω ω ω( ) ˆ exp ,t i t d= ∫ ( ) { }−∞∞1

2

ˆ( ) ( )exp{ } ,ψ ω ψ ω= ∫ −−∞∞ t i t dt


(36)

where γ denotes the specific heat ratio. Acoustic waves that propagate to the far fieldinteract with the sound speed gradients. This interaction can result in acoustic refractionand modification in the directivity pattern. Strahle [27] investigated the sound refractioneffects by velocity and temperature gradients in a turbulent jet flame. His analysisshowed that the high frequency sound emission is primarily affected by refractionleading to reduction in the power output. Effects of acoustic refraction for the presentjet flame configuration will be discussed in Sec. 6.2.

For the evaluation of the acoustic pressure, a free-space Green’s function was used,which implicitly assumes that all acoustic sources radiate undisturbed to the far field,and effects due to diffraction at the nozzle or other obstacles are negligible. Diffractioneffects are typically only relevant for high frequencies, and its significance is dependenton the directivity angle. As such, they have usually insignificant effect on the radiationin forward direction, but become increasingly important with increasing angle to the jetaxis. Nevertheless, in the experiment that will be considered in the following, sucheffects were minimized by covering the orifice and tubing with sound-absorbingmaterial [30].

Despite these limitations, Lighthill’s acoustic analogy is a general formulation, andwas successfully applied to a range of isothermal and reacting flow configurations. Theobjective of this work is to investigate the applicability of this analogy in the context ofa hybrid LES/CAA approach for the prediction of noise generation from a turbulentdiffusion flame.

4. APPLICATION TO DLR FLAME A4.1. Experimental configurationThe N2-diluted CH4-H2/air flame, considered here, was experimentally studied byMeier and coworkers [58, 59, 60]. The burner configuration for the non-premixedflame consists of a central fuel nozzle of diameter Dref which is surrounded by a co-flow nozzle of square shape. The fuel bulk velocity is Uref. Co-flowing air is suppliedat an axial velocity of 7.11 × 10–3Uref . All parameters used in the calculation aregiven in Tab. 1. The jet fluid consists of a mixture of 22.1 % methane, 33.2%hydrogen, and 44.7% nitrogen by volume with a stoichiometric mixture fraction ofZst = 0.167. It was reported that the DLR flame A burns very stably and that lift-offis not observed [58].

The Reynolds number, based on the nozzle diameter, pipe bulk exit velocity, andkinematic viscosity of the fuel mixture, is Re = 14,740. The Schmidt number is Sc= 0.486, the Mach number is M = 0.123, and the Eckert number is Ec = 5.4 ×10−4. The Damköhler number is the ratio of the characteristic flow time scale to thechemical time scale. To compute this ratio, an appropriate chemical time scale mustbe determined. Here, this time scale comes from the definition of the progressvariable

aW

= γR

Θ,


C = YCO2 + YCO + YH2o + YH2. (37)

The water mass fraction has the largest contribution to the progress variable and asensitivity analysis at a condition near flame quenching was performed, which showedthat water is primarily formed by the fast shuffle reaction

OH + H2 H2O + H. (38)

Therefore, the chemical time scale appearing in the Damköhler number is determinedusing the frequency factor and activation energy of Eq. (38). Performing this calculationgives Da = 0.644.

4.2. Computational setupThe Favre-filtered conservation equations for mass, momentum, mixture fraction, andprogress variable are solved in a cylindrical coordinate system in ΩF with y = (r, ϕ,y)

T.The governing equations are discretized on a staggered grid, and spatial derivaties arediscretized by a central difference scheme. A fractional-step method is employed toadvance the continuity and momentum equations, and the scalar transport equations aresolved semi-implicitly. More details on the numerical scheme can be found in Ref. [61].The residual stresses and scalar fluxes that appear in the Favre-filtered transportequations are modeled by a dynamic procedure [47, 62].

The geometry was non-dimensionalized by the jet nozzle diameter Dref, and thecomputational domain is 40 Dref × 2π × 120Dref in radial, circumferential, and axialdirections, respectively. The radial direction is discretized by 160 unevenly spaced gridpoints concentrated in the fuel nozzle. For the discretization of the jet radius 30 grid

�


Table 1: Reference parameters for the reactive jet simulation.

Parameter Value Units Dref 8 × 10

−3 mUref 42.2 m/saref 344.33 m/sρref (air) 1.169 kg/m

3

νref (fuel) 2.291 × 10−5 m2/sαref (fuel) 4.710 × 10−5 m2/scp,ref (fuel) 1,815.73 J/(kg K)Θref 293 K∆Θref 1,817 KEref 1.435 × 10

4 J/molAref 216 m

3/(mol s)Wref (air) 2.877 × 10

−2 kg/mol

points are used. The circumferential direction is equally spaced and uses 64 points. Thegrid in axial direction uses 320 points and is, beginning at the nozzle exit, stretched instreamwise direction. The corresponding grid stretching diagrams for axial and radialdirections are shown in Fig. 3. The total number of grid points used for the simulationis approximately 3.28 million. The minimum and maximum filter widths in the domainare ∆min = 3.04 × 10

–2 Dref (shear layer in the nozzle-near region) and ∆max = 1.94Dref(outermost grid cell at the outflow plane).

The turbulent inlet velocity profile is generated by separately performing a periodicpipe flow simulation by enforcing a constant mass flux [63]. Convective outflowconditions are used at the outlet and slip-free boundary conditions are employed at theradial boundaries.

The numerical simulation is run over ten flow-through-times to obtain a statisticallystationary flow field. Statistics are collected thereafter over ten flow-through-times,corresponding to 227.5 ms physical time.

The computed mean and resolved variance are obtained by averaging in temporal andazimuthal directions, υiz.

(39a)

(39b)

where 〈ψ∼〉 denotes the mean value of ψ∼ and 〈~ψ′′2

5. FLOW FIELD RESULTS5.1. Flow field analysisIn this section, statistical flow field results along the centerline and in radial directionare compared with available experimental data. Computational results are shown aslines, and experimental data and their estimated uncertainties are denoted by symbols.

Centerline profiles for axial velocity, mixture fraction, and temperature are shown inFig. 4. Mean and resolved rms values for the axial velocity are compared withexperimental data in the top row. Overall, the numerical results are in good agreementwith experimental data for the mean velocity, and only small differences betweenpredictions and measurements for the mean decay rate can be observed for 10 ≤ y ≤ 20.

Mixture fraction results are illustrated in the middle row of Fig. 4. The dashed lineindicates the value of stoichiometric mixture fraction, and the stoichiometric flame


00.250.5

0.751

1.251.5

0

0.05

0.1

0.15

0.2

0

0.25

0.5

0.75

1

0

0.05

0.1

0.15

0.2

0 20 40y y

60 80 0 20 40 60 800

0.25

0.5

0.75

1

1.25

0

0.05

0.1

0.15

0.2

ExperimentSimulation

〈u~〉

〈Z~ 〉

〈Θ~ 〉

Local mach number〈u

''2〉1

/2, M

(y)

<∼

〈Θ''2

〉1/2

<∼

〈Z''2

〉1/2

<∼

Figure 4: Comparison of measured (symbols) and calculated (solid lines) mean andrms statistics of axial velocity and mixture fraction along the centerlinefor DLR flame A. Experimental data are plotted with estimateduncertainties. The dashed line in the plot of the mean mixture fractioncorresponds to Zst = 0.167. The dashed line in the upper right figureshows the local Mach number, M(y) = 〈~u〉/〈~a〉.

length corresponds to the centerline location at which 〈Z~〉 = Zst. The measuredstoichiometric flame length is 64.25 diameters and the predicted value is 68.3, which isslightly over-predicted by six percent. The centerline decay rate for 〈Z~〉 is in excellentagreement with experimental results for the fuel-rich part of the flame.

The axial evolution of the mean temperature and resolved temperature fluctuationsalong the centerline are compared with experimental data in the bottom row of Fig. 4.Good agreement between simulation and experiment is obtained for the meantemperature up to the stoichiometric flame length. Thereafter, the temperature is over-predicted by approximately 150–200 K. This is mainly attributed to the over-predictionof the mixture fraction in the fuel-lean part of the flame and the neglect of heat lossesby thermal radiation. It has been observed in partially premixed methane/air flames atcomparable Reynolds numbers that the consideration of radiation in the optically thinlimit reduces the flame temperature by 100–150 K in the region above thestoichiometric flame length [64, 65]. An extension of the FPV model that accounts forsuch heat loss effects has recently been developed in the context of the prediction ofnitric oxide formation [66]. Since, however, the major species distribution is essentiallynot affected by such weak interactions between radiation and chemistry, and theacoustic source term distribution is mainly located inside the flame, this model was notemployed here.

Radial profiles for velocity, mixture fraction, and temperature at different axiallocations in the jet flame are shown in Figs. 5 and 6. Radial profiles of the velocity areshown in Fig. 5(a). Results for both mean and resolved fluctuation of the velocity are inoverall good agreement with experimental data. The resolved shear stress 〈u~″

(40)

and êy corresponds to the unit vector in axial direction, êr is the unit vector in radialdirection, and êst refers to the tangential vector along the stoichiometric surface. This

Riφ ξφ τ φ τ

φ τ φ( , )

( , ) ( , )

( , ) (y

y

y=

′′ ′′

′′ ′′

< <

< <

� �

� �

ξ

2 2 ξξξ

, )

ˆ ,

ˆ ,

ˆ ,τξ

ξ

with =

y e

y e

y e

+

+

+

y

r

ξ

st


0

0.25

0.5

0.75

1

1.25

0

0.1

0.2

0.3

0

0.25

0.5

0.75

1

1.25

0

0.1

0.2

0.3

0

0.25

0.5

0.75

1

1.25

0

0.1

0.2

0.3

0

0.25

0.5

0.75

1

1.25

0.30.20.10 0.30.20.10−0.1−0.2−0.30

0.1

0.2

0.3

y = 5

y = 10

y = 20

y = 40

y = 5

y = 10

y = 20

y = 40

r/yr/y r/y

(a) Axial velocity (b) Shear stress

0

0.005

0.01

0.015

0

0.005

0.01

0.015

0

0.005

0.01

0.015

0

0.005

0.01

0.015


〈u~〉

〈u~〉

〈u~'' v~

'' 〉<

<〈u~

'' v~'' 〉

<<

〈u~'' v~

'' 〉<

<〈u~

'' v~'' 〉

<<

√〈u~

'' 2〉

<√

〈u~'' 2

〉<

√〈u~

'' 2〉

<√

〈u~'' 2

〉<

〈u~〉

〈u~〉

Figure 5: Comparison of measured (symbols) and calculated (solid lines) mean andresolved rms statistics of (a) axial velocity and (b) shear stress at differentaxial locations for DLR flame A. Experimental data are plotted withestimated uncertainties.

directional dependence is explicitly indicated in the superscript i of the correlationcoefficient, with i = {y, r, st}. The integral length scale can then be computed as

(41)

which accounts for flow inhomogeneities in non-azimuthal directions by weighting Riφfrom both upstream and downstream directions. Two-point correlation coefficients inaxial direction (dashed lines) and the direction along the stoichiometric surface (solid

L Ri iφ φ ξ ξ( ) ( ) ,–y y,=

∞

∞

∫1

2d


y = 5

y = 10

y = 20

y = 40

y = 5

y = 10

y = 20

y = 40

0

0.2

0.4

0.6

0.8

1

0

0.1

0.2

0.3

0.4

0.5

0

0.2

0.4

0.6

0.8

1

0

0.1

0.2

0.3

0.4

0.5

〈Θ∼〉

〈Θ∼〉

0

0.2

0.4

0.6

0.8

1

0

0.1

0.2

0.3

0.4

0.5

0

0.2

0.4

0.6

0.8

1

0.30.20.1−0.1−0.2−0.3 0 0.30.20.1−0.1−0.2−0.3 0

0.1

0.2

0.3

0.4

0.5

r/yr/y

(a) Mixture fraction

0

0.25

0.5

0.75

1

1.25

0

0.1

0.2

0.3

0.4

〈Θ∼〉

0

0.25

0.5

0.75

1

1.25

0

0.1

0.2

0.3

0.4

〈Θ∼〉

0

0.25

0.5

0.75

1

1.25

0

0.1

0.2

0.3

0.4

0

0.25

0.5

0.75

1

1.25

0

0.1

0.2

0.3

0.4

r/yr/y

(b) Temperature


〈Z~〉

〈Z~〉

〈Z~〉

〈Z~'' 2

〉<

〈Z~'' 2

〉<

〈Z~'' 2

〉<

〈Z~'' 2

〉<

〈Θ~'' 2

〉<

〈Θ~'' 2

〉<

〈Θ~'' 2

〉<

〈Θ~'' 2

〉<

〈Z~〉

Figure 6: Comparison of measured (symbols) and calculated (solid lines) mean andresolved rms statistics of (a) mixture fraction and (b) temperature atdifferent axial locations for DLR flame A. Experimental data are plottedwith estimated uncertainties.

lines) for the fluctuations of u~, ~v, ~Θ, and Y~OH are shown in Fig. 7 for y = 5, 10, 15,20, and 25. The computed correlation coefficient for the axial velocity fluctuations isshown in Fig. 7(a). The negative correlation coefficient for y = 5, computed along thestoichiometric surface, suggests some degree of periodicity in the shear layer, which isnot so pronounced for the correlation in axial direction. While the spatial evolution ofboth correlation coefficients of u~″< are very similar, this is not the case for the radialvelocity component, shown in Fig. 7(b). In fact, the predictions for Ryv, evaluated in thestreamwise direction, show only a weak spatial dependence and exhibit a small negativecorrelation for y = 25, which is not evident for Rstv along the stoichiometric surface. Thecorrelation coefficients for the temperature fluctuation are illustrated in Fig. 7(c), andthe shapes of both correlation functions are very similar and symmetric around ξ = 0.

Two-point correlations of hydroxyl (OH) radical are shown in Fig. 7(d). Hydroxyl isan important intermediate species that participates in the hydrogen abstraction ofhydrocarbon fuels and in the carbon monoxide oxidation [67]. Correlation coefficientsfor this species are computed and are shown in Fig. 7(d). It is interesting to point outthat RyOH (along the axial direction) is mainly skewed towards the downstreamdirection; however, RstOH (along the stoichiometric surface) is approximately symmetric.

Integral length scales computed from the correlation coefficients are shown in Fig. 8and the corresponding similarity coefficients are summarized in Tab. 2. Figure 8 shows


01

0 10 20 30 0 10 20 30y + ξ y + ξ

Ry u

(y,ξ

),R s ut

(y,ξ

)R

y Θ (y

,ξ),

Rs Θt

(y,ξ

)

Ry O

H (y,

ξ), R

st OH

(y,

ξ)

(a) (b)

(c) (d)

y = 25y = 20y = 15y = 10y = 5 R

y v (y

,ξ),R

st (y,

ξ)u

Figure 7: Two-point correlation coefficients in axial direction (dashed line) andalong the stoichiometric surface (solid lines) for (a) ~u′′

that the integral length scales along the stoichiometric surface exhibit self-similarity andthe growth rate for all quantities is comparable (see Tab. 2). Interestingly, such apronounced similarity in the growth rate of the integral length scales along the axialdirection cannot be observed, and the computational results predict nearly constantvalues for both Lyv and L

yΘ.

Figure 9 compares computed radial correlation coefficients and integral length scalesfor the OH radical with experimental data from Zhang [68]. The interested reader isreferred to Ref. [69], in which measurement technique and experimental procedure aredescribed in detail. The radial position for which the correlation is computed in the simulation corresponds to the measurement location of maximum OH. Forreference, the dotted line in Fig. 9(a) shows the computed location where 〈Z~〉 = Zst.


LyΘ,L

st ΘLy u

,Lst u

LyO

H,Ls

t OH

Lyυ,

Lst υ

00.250.5

0.751

1.251.5

10 20 30 40 10 20 30 400

0.250.5

0.751

1.251.5

yy

(a) (b)

(c) (d)

Along axial distanceAlong stoich. surface

Figure 8: Spatial evolution of the integral length scales in axial direction (dottedlines with circles) and along the stoichiometric surface (solid lines withsquare symbols). The solid and dotted lines correspond to the regressionlines with corresponding regression coefficients given in Tab. 2.

Table 2: Comparison of integral length scale coefficients for correlations in axialdirection and the direction along the stoichiometric surface.

Liu/y Liv/y L

iΘ/y L

iOH/y

axial direction (i = y) 0.021 0.004 0.001 0.014along stoichiometric surface (i = st) 0.028 0.026 0.022 0.023

Figure 9(a) shows that the computed radial correlation coefficients are in goodagreement with experimental data for y ≤ 20, and RrOH is slightly over-predicted withfurther downstream locations. A possible reason for the apparent difference can beattributed to the relatively narrow region of the OH layer and the discrepancy betweenthe predicted and experimentally determined OH peak location.

The difference between computed and measured correlation coefficients is alsoreflected in the evolution of the integral length scale, which is shown in Fig. 9(b).Although the simulation over-predicts the values for LOH by approximately 20 %, theaxial evolution is accurately reproduced by the simulation. A comparison of the integrallength scales shows that LOH in the direction along Zst (compare Fig. 8(d)) isapproximately 2.5 times larger than LOH in radial direction. This is attributed to thepreferred orientation of the reaction zone and the difference in the turbulence structureof jet flame.

5.3. Combustion modelTurbulent non-premixed combustion is strongly affected by the interaction of turbulentmixing at small scales, by molecular diffusion between fuel and oxidizer, and bychemical reactions. For an assessment of only the combustion model, it is essential toperform a flow-field independent comparison of the simulation results for chemicalspecies and temperature with experimental data. In the following, this is done byanalyzing mixture fraction-conditioned data. The mean value of a scalar ψ~, conditionedon mixture fraction Z~ is determined by

(42)� � � � � �ψ ψ ψ ψ| | dZ = ∫ P( ) ,Z


0 1 2 3 40

10

20

30

40

50

60

60504030201000

0.2

0.4

0.6

Rr O

H (y

,ξ),y

r + ξ y

(a) (b)

〈Z~

〉 > Zst 〈Z~

〉 < Zst

Lr OH


Figure 9: Comparison of computational results and experimental data for hydroxylradical correlation, evaluated at (y,r) = (10, 1.31), (20, 2.21), (30, 2.61),(40, 3.26), and (50, 3); (a) radial two-point correlation coefficient and (b)integral length scale. The dotted line in (a) shows the location where (~Z) = Zst.

where P(ψ~Z~) = P(ψ~,Z~)/P(Z~) is the conditional PDF. Note that P is evaluated fromthe resolved instantaneous filtered flow field results, and, contrary to the FDFintroduced in Eq. (27), does not contain information about unresolved quantities.

Conditional data for the Favre-filtered temperature and mass fractions of H2O, CO2,CO, and OH are presented in Fig. 10. The predictions of the conditional meantemperature, shown in the first row, are in good agreement with measurements throughoutthe flame. Conditional data for the mass fraction of water, shown in the second row, arewell predicted for y ≥ 10, and the under-prediction at the first measurement station isattributed to the under-prediction of the hydrogen-fuel consumption on the fuel-rich sideof the flame (not shown). Conditional statistics for the two main carbon-containingproduct species CO2 and CO are illustrated in the third and fourth row, respectively. In thisflame, the fuel is the primary source of CO and CO2, so that the discrepancies betweensimulation and experiment can be related to the fuel consumption. Indeed, an analysis ofthe CH4 results confirms that the simulation over-predicts the fuel consumption on thefuel-rich side of the flame at the first two measurement locations, resulting in the apparentover-predictions of CO and CO2. With increasing distance to the nozzle the CH4consumption is better predicted, which also results in a better agreement of the CO2 andCO profiles between experiment and simulation.

Conditional data for the OH radical are illustrated in the last row of Fig. 10. It canbe seen that OH is formed in a narrow region of the flame with peak location atstoichiometric condition. Note that the OH mass fraction is more than an order ofmagnitude smaller than the major species, and given its small concentration andsignificance in the fuel conversion the predictions can considered to be in goodagreement with experimental data.

6. COMBUSTION NOISE RESULTSFollowing the discussion of the flow field and combustion results in the previoussection, acoustic results are presented next. The pressure fluctuations at a far-fieldlocation x were computed as function of frequency using Eq. (34). The three sourceterm contributions, –T=R,

–̂FM, and –̂QR, are obtained from the LES. A total of 4096

instantaneous flow field data sets were collected from the LES. These snapshots areseparated by a constant time increment of at ∆τ = 0.04, spanning a total time intervalof 163.84 non-dimensional time units. For the computation of the acoustic results, thistime interval was subdivided into 41 over-lapping sections, and results were obtained byaveraging over all sub-intervals.

Since sound pressure signals typically cover a wide range of power levels, thedecibel scale is used to measure the sound level. The sound pressure level (SPL) iscomputed as

(43) SPLS

ppp( ) log

ˆ ( ),ω

ω=

10

10

02



0

0.25

0.5

0.75

1

1.25

0

0.05

0.1

0.15

0.2

0

0.025

0.05

0.075

0.1

0

0.01

0.02

0.03

0.04

0.05

0 0.25 0.5 0.75 0.25 0.5 0.75 0.25 0.5 0.75 0.25 0.5 0.750

0.001

0.002

0.003

0.004

0 0 0 1

y = 5 y = 10

Z ~

Z ~

Z ~

Z ~


y = 20 y = 40

〈Y~O

HZ ~

〉〈Y~

CO

Z~〉

〈Y~C

O2

Z~ 〉

〈Y~H

2OZ ~

〉〈Θ~

Z ~〉

Figure 10: Comparison of measured (symbols) and calculated (solid lines)conditional temperature and mass fractions of H2O, CO2, CO, and OH aty = 5, 10, 20, and 40 for DLR flame A. Experimental data are plottedwith estimated uncertainties.

where the reference pressure is p0 = 9.607 × 10–9. The power spectral density is

defined as

(44)

and averaging is performed in time and in azimuthal direction. The overall soundpressure level (OASPL) is a measure of the loudness of the sound, and is defined as

(45)

and can be obtained by integrating the sound pressure level over all ω.

6.1. Source term signalThe source term fluctuations of the streamwise components of –T=R and

–FM, togetherwith

–QR and –ρ are shown in Fig. 11 as function of time for four axial locations alongthe jet lip line. It is interesting to note that the three acoustic source terms –TR,yy,

–FM,y,and –ρ are of the same order of magnitude; however, the source term due to the chemicalreaction rate is approximately two orders of magnitude smaller. The difference in theamplitude of the chemical source term can be attributed to the fact that

–QR is stronglycorrelated with the surface of stoichiometric mixture, and rapidly decays away from Zst.Since the measurement locations are not aligned with the mean location of Zst, the timesignal is of much lower amplitude. Note also that the magnitude of the individual sourceterms alone is not sufficient to determine the acoustic efficiency. This was alreadydiscussed in Sec. 3.3.2. It was shown that the quadrupole and dipole sources scale withM2 and M, respectively, and are therefore less efficient than the monopole source forlow Mach number combustion.

The corresponding frequency spectra for the different source term contributions asfunction of the Strouhal number, St = f, are shown by the solid lines in Fig. 12. Thespectral shape of all source term contributions show a similar behavior for lowfrequencies. With increasing frequency the energy content decreases exponentially;however, at high frequencies the decay rate reduces to an algebraic form with exponentless than –2 for –TR,yy,

–FM,y, and –ρ.The acoustic pressure contribution from these source locations to the far field is

determined by multiplying the source term spectra with their respective directionalcoefficients, Eqs. (35). In the limit when R becomes large compared with the acousticwave length, these coefficients reduce to

(46)

These expressions show that for the computation of the far field pressure the acousticsource terms must also be multiplied with the dependency on the angular frequency ω,

κ ω λ ω µ ω→ → →M M Da2 2 2R R R

, , .

OASPLref

=′ ′

10

10 2log ,

p p

p

ˆ ( ) ( ) ( ) exp{ } ,–

S p t p t ipp ω ζ ωζ ζ= ′ ′ + −∞∞

∫ d


which is a direct consequence of the Fourier transformation of the wave equation. Thedashed curves in Fig. 12 correspond to the spectral contribution of the source terms tothe far-field pressure after multiplying the respective ω-dependence according to Eqs.(46). It can be observed that the spectral contributions at high frequencies saturate forlocations close to the fuel nozzle, but rise further downstream. The reason for thisspurious noise is the algebraic decay rate of the source term spectra. Note that asimilarly unphysical increase in the high frequency part of the spectra has also been


y = 2.5 y = 7.5 y = 15 y = 20

0

0.1

0.2

0.3

0.4

0.5

0.2

0.3

0.4

0.5

0.6

0.7

0.8

− 0.002− 0.0015

− 0.001− 0.0005

0

0.0005

0.001

500 505 510τ τ

515 500 505 510 515 500 505 510 515 500 505 510 515 5200.1

0.2

0.3

0.4

0.5

τ τ

T_R

,yy

F_M

,yQ_

Rρ_

Figure 11: Time series of acoustic source terms for four axial locations along the jetlip line.

observed by Bastin et al. [70] and Rembold [71] for isothermal planar jets. There, thesource of the spurious noise was attributed to limitations due to numerical accuracy andaerodynamic fluctuations which are formed in the nozzle-near shear-layer region of thejet. Spurious noise sources for the particular jet flame configuration have been analyzedin Ref. [46], and the key findings of this analysis are summarized in the next section.


y = 2.5 y = 7.5 y = 15 y = 20

1e− 08

1e− 06

1e− 04

1e− 02

1e+00

| |T_̂

R,yy

ω 2| |T_̂

R,yy

1e− 08

1e− 06

1e− 04

1e− 02

1e+ 00

1e− 08

1e− 07

1e− 06

1e− 05

1e− 04

1e− 03

0.1 1 10 0.1 1 10 0.1 1 10 0.1 1

St St St St

101e− 08

1e− 06

1e− 04

1e+ 02

1e− 00

ω 2|F_̂

M,y|

|F_̂

M,y|

ω|Q_̂

R|

|Q_̂

R|

|ρ_̂

|ω2|ρ

_̂|

Figure 12: Temporal energy spectra of acoustic source terms for four axial locationsalong the jet lip line. The vertical dashed line indicates the location of thecut-off filter.

6.2. Analysis of spurious noise sourcesThe generation of noise from turbulent reacting flows at low Mach-numbers is typicallya rather inefficient process. Because of this inefficiency, the discrepancy betweenacoustic and turbulent length scales, and the small amplitude of acoustic perturbations,it is necessary to provide an accurate characterization of the acoustic source field andutilize numerical schemes for the acoustic propagation that minimize dissipation anddispersion errors.

In the numerical simulation of unsteady turbulent reacting flows and its resultingnoise emission, spurious noise sources can be introduced that pollute the acousticsignal. The following-sources can be identified as primary contributors to the spuriousnoise: (i) spurious noise from LES subgrid scale closure models, (ii) FPV combustionmodel and chemistry tabulation; and Strouhal number limitations arising from (iii)temporal and (iv) spatial resolution. The potential relevance of these different spuriousnoise sources for the unconfined non-premixed jet flame that is here of interest wasanalyzed by Ihme et al. [46]. In this study it was concluded that the chemistry tabulationand spatial resolution of the source region are the major sources of numerical errorspolluting the acoustic pressure field.

The state relation, Eq. (29), in the FPV combustion model is precomputed andtabulated. During the simulation, information from this table is retrieved by tri-linearinterpolation with a truncation error of the order of O(∆Z~

2 , ∆2Z˝2, ∆2C~), and ∆φ denoting

the grid size in φ-direction. It was found that the evaluation of the state relationintroduces spurious noise that leads to a deterioration of the high-frequency spectra ofscalar and flow field quantities. Although these contributions are typically insignificantfor one- and two-point statistics, these noise sources directly affect the acousticpredictions. This high-frequency spurious noise contribution can be reduced byemploying finer table resolution, higher-order interpolation schemes, artificial neuralnetwork techniques [72], or a tetrahedral integration method [73]. In the present work,the tetrahedral integration was used. In this method, a computational cell is subdividedinto tetrahedrons, and the state equation is then integrated for each sub-cell using higher-order quadrature rules. In addition to the apparent reduction of the high-frequency errors,employing this method resulted also in better convergence and numerical stability.

Another relevant source of spurious noise is related to the spatial extent of theacoustic source region and the restrictions on the LES grid resolution. By recognizingthat the acoustic sources are primarily controlled by flow-field advection in ΩF, acritical Strouhal number, associated with this advection process, can be written as

(47)

In the self-similar region of an axisymmetric jet flame, the decay rate of the mean axialvelocity is inversely proportional to the nozzle distance. The grid spacing in axialdirection, shown in Fig. 3, is proportional to y. With this, the axial evolution of thecritical advection Strouhal number can be estimated as Sta ∝ y

–2. Although frequencies

Sta( )( )

( ).y

y

y=

�u

y2∆


larger than Sta can occur in the flow, generated by chemical reactions and turbulence,advective contributions to

–T=R and –FM significantly larger than Sta are not supported by

the computational grid. The critical Strouhal number, evaluated from the local meanflow and grid size, is shown by the vertical dotted lines in Fig. 12. It can be seen thatSta accurately characterizes the onset of spurious noise for

–T=R, –FM , and

–ρ. Note alsothat the acoustic source due to chemical reaction is only marginally affected byadvection effects, and high-frequency contributions at St ≈ 5 are evident for y = 7.5.Based on this analysis, a spectral low-pass filter of the form H(Sta(y) – f ) isintroduced, which eliminates spurious noise beyond the cut-off frequency.

Subsequently it will be shown that the acoustic source term contribution due tochemical reaction extends up to the stoichiometric flame length, Lst. From scalingarguments for diffusion flames follows that Lst ∝ Zst

–1. For the present flame, thenitrogen-diluted methane/hydrogen fuel composition results in a low value for Zst,resulting in an extended flame height. Although such considerations are typically notrelevant for combustion modeling, combustion-noise predictions are critically affectedby extended flame regions, since large computational resources are required tocharacterize and resolve the acoustic source region.

6.3. Acoustic source term correlationsThe space-time evolution of the acoustic source terms TR, FM, and QR, with

(48)

are shown in Fig. 13. The radial location for which the source terms are evaluatedcorresponds to the location of the mean stoichiometric mixture fraction. All threesources exhibit a qualitatively similar behavior. High-frequency dynamic structures areprimarily generated in the shear layer surrounding the potential core-region. Withincreasing downstream distance, small-scale structures rapidly decay; however, largerstructures coalesce to form extended and more coherent structures, which is particularlyevident for y ≥ 10. Associated with the formation of these large-scale coherentstructures is an increase in wavelength and reduction in frequency. Also shown in thesefigures are lines with slopes corresponding to the mean axial velocity evaluated atlocations yst = {7.5, 12.5, 25.0}. From this qualitative comparison of the source termconvection and the mean flow, the following observations can be made. In the regioncontained in the nozzle-near shear layer, the acoustic sources convect at a faster speedthan the local mean flow. With increasing downstream distance, the convection velocityof the acoustic sources approaches the local mean flow, which is particularly evident forFM and QR.

From the source term evolutions, shown in Fig. 13, space-time correlations areevaluated to obtain quantitative information about spatial and temporal length scales ofthe acoustic sources. The space-time correlation coefficient is defined as

(49)Rφ ξ ζ

φ τ φ ξ τ( , , )

( , ) ( ˆ ,y

y y est

st st st=〈 ′′ ′′ + +< <� � ζζ

φ τ φ ξ τ ζ

)

( , ) ( ˆ , ),

〉

〈 ′′ ′′ + + 〉< <� �2 2y y est st st

T F QR R M M R RDa= ∇ ∇ = ∂ ∇ ⋅ = ∂y y y F. . , , ,T Qτ τ


where ê stξ refers to the location along the stoichiometric surface in streamwisedirection. Space-time correlation coefficients are computed for three locations andillustrated in Fig. 14. In these figures, the slope of the solid lines corresponds to the localaxial mean flow velocity 〈~u〉, and the slope of the dashed lines shows the convectivevelocity uc of the source terms.

Compared to the qualitative discussion in Fig. 13, correlation coefficients evaluatedat selected locations provide a detailed understanding about the convective andspatiotemporal properties of the acoustic sources. From the slopes of the solid lines,shown in Fig. 14, it can be seen that the local mean velocity is approximately constantalong the stoichiometric mixture fraction contour with values of 〈u~〉 ≈ 0.17. This is adirect consequence of the similarity solution between conservation equations for axialmomentum and mixture fraction [74]. From the statistical results it is evident that theacoustic sources convect at a faster speed than the local mean velocity. In fact, theconvective speed is approximately three times larger than 〈u~〉 at yst = 5, and uc decayswith increasing downstream distance approaching values of uc ≈ 0.3 at yst = 30. Notethat all three source terms exhibit a very similar convective behavior with only marginaldifferences in the values of uc.


540 0.50.40.30.20.1

− 0.5− 0.4− 0.3− 0.2− 0.1

0

530

520

510

5000 10 20 30

540 0.50.40.30.20.1

− 0.5− 0.4− 0.3− 0.2− 0.1

0

530

520

510

5000 10 20 30

540 0.50.40.30.20.1

− 0.5− 0.4− 0.3− 0.2− 0.1

0

530

520

510

5000 10 20 30

(a) TR = ∇y . ∇y . Τ−=R

yst

(c) R = Da∂τQ_

R

yst

yst(b) FM = ∂τ∇y . F

_M

τ

τ

τ

Figure 13: Space-time evolution of the acoustic source terms evaluated along thelocation of the mean stoichiometric mixture fraction. The slopes of thelines correspond to the mean axial velocity with 〈~u〉 = {0.170, 0.159,0.178}, corresponding to the locations yst = {7.5, 12.5, 25.0},respectively.

The comparison of the correlation coefficients for TR and FM shows that RTR andRFM exhibit a similar evolution for all three axial locations. The Lagrangian integrallength scale, defined as

(50)L R uL i i

cφ φ ξ ξ ξ,

–( ) ( , , / ) ,y y=

∞

∞

∫1

2d


53.752.5

1.250

− 1.25− 2.5

− 3.75− 5

−2.5 0 1.25−1.25 2.5

10.90.80.70.60.50.40.30.20.10−0.1−0.2

T R ζ

yst = 5

53.752.5

1.250

− 1.25− 2.5

− 3.75− 5

−2.5 0 1.25−1.25 2.5

10.90.80.70.60.50.40.30.20.10− 0.1

ζ

yst = 15

53.752.5

1.250

− 1.25− 2.5

− 3.75− 5

−2.5 0 1.25−1.25 2.5

10.90.80.70.60.50.40.30.20.10− 0.1

ζ

53.752.5

1.250

− 1.25− 2.5

− 3.75− 5

−2.5 0 1.25−1.25 2.5

10.90.80.70.60.50.40.30.20.10− 0.1

FM ζ

53.752.5

1.250

− 1.25− 2.5

− 3.75− 5

−2.5 0 1.25−1.25 2.5

10.90.80.70.60.50.40.30.20.10− 0.1

ζ5

3.752.5

1.250

− 1.25− 2.5

− 3.75− 5

−2.5 0 1.25−1.25 2.5

10.90.80.70.60.50.40.30.20.10− 0.1

ζ5

3.752.5

1.250

− 1.25− 2.5

− 3.75− 5

−2.5 0êst ξ êst ξ

1.25−1.25 2.5

10.90.80.70.60.50.40.30.20.10− 0.1

QR ζ

53.752.5

1.250

− 1.25− 2.5

− 3.75− 5

−2.5 0 1.25−1.25 2.5

10.90.80.70.60.50.40.30.20.10− 0.1

ζ

53.752.5

1.250

− 1.25− 2.5

− 3.75− 5

−2.5 0 1.25−1.25 2.5

10.90.80.70.60.50.40.30.20.10− 0.1

ζ

yst = 30

êst ξ

êst ξ êst ξ êst ξ

êst ξ êstξ êst ξ

−0.2 −0.2

−0.2−0.2−0.2

−0.2 −0.2 −0.2

Figure 14: Space-time correlation coefficients for the three acoustic source terms TR,FM, and QR (from top to bottom). The correlation coefficients areevaluated along the stoichiometric mixture fraction surface at threelocations yst = {5, 15, 30}. The slope of the solid lines corresponds to the local mean velocity 〈~u〉, and the slope of the dashed lines representsthe convective velocity uc of the acoustic sources.

increases with increasing yst, which is a consequence of the decaying turbulence inthe jet flame. The correlation coefficients for QR are shown in the last row ofFig. 14, exhibiting a different spatiotemporal structure compared to the other twosource terms. Two notable observations can be made. First, it is apparent that thespatiotemporal correlation decays rapidly with increasing ξ and ζ separation. Thiscan be explained by recognizing that the chemical source term is confined to anarrow region around the stoichiometric surface, which also corresponds to a narrowregion in Z –C composition space. Therefore, the spatiotemporal evolution ofmixture fraction and progress variable results in the intermittency of the chemicalsource term, which is reflected by the localized correlation coefficient. In thisrespect, it is noted that this rapid decay of the correlation coefficient can lead to aninteresting simplification for lower-order modeling approaches of combustion noise.Second, the correlation coefficient along the convection line at yst = 30 shows someevide

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