Analysis of different sound source formulations to simulate …web.stanford.edu/group/ihmegroup/cgi-bin/MatthiasIhme/wp-content/paper... · In the flamelet model, [15, 16] a non-premixed

aeroacoustics volume 8 · number 1 & 2 · 2009 – pages 95 – 124 95

Analysis of different sound sourceformulations to simulate combustion

generated noise using a hybrid LES/APE-RF method

T. Ph. Bui,* M. Ihme†, W. Schröder‡ and H. Pitsch§RWTH Aachen University, 52062 Aachen, GermanyUniversity of Michigan, Ann Arbor, MI 48109, USA

Stanford University, Stanford, CA 94305, USA

Received November 12, 2007; revision received April 15, 2008; accepted for publication June 3, 2008

ABSTRACTCombustion noise and sound source mechanisms of the DLR-A flame are investigated. A hybridlarge-eddy simulation/computational aeracoustics (LES/CAA) approach is employed. In the firststep of the hybrid analysis the flamelet/progress variable (FPV) model is employed ascombustion model followed by the acoustic simulation in the second step using the acousticperturbation equations for reacting flows (APE-RF). The flamelet/progress variable database hasbeen extended in terms of acoustic source terms. The analysis of the acoustic field of low MACHnumber reacting flows induced by the thermoacoustic sources such as the unsteady heat releaseleads to a very stiff problem formulation, since the related sources require highly resolved regionsin the source area, which restricts the possible time step. To simulate combustion generated noiseusing such a hybrid approach, a suitable source description has to be used, which preferablysatisfies two requirements, i.e, to efficiently and accurately predict the generated sound field,while the source term can be easily evaluated from the LES. Using the source term, which isexpressed via the scaled partial time derivative of the density, the acoustic field can be reproducedbest up to a maximum STROUHAL number of StD = 2. However, this source formulation requiresa rigorous constraint at the interface of the hybrid approach to avoid spurious noise due toartificial acceleration caused by interpolation. To be more precise, the convection speed ofdensity inhomogeneities has to be preserved during interpolation. A compromise betweenefficiency and accuracy can be achieved using the source formulation expressed via the scaledmaterial derivative of the density, since by definition this formulation does not describe theconvection of density inhomogeneities.

*Research Associate, RWTH Aachen University, ([email protected])†Assistant Professor, University of Michigan, ([email protected]) ‡Professor, RWTH Aachen University, ([email protected])§Assistant Professor, Stanford University, ([email protected])

1. INTRODUCTIONIn the field of aeroacoustics, several methods are available to investigate the soundgenerated from flow effects such as jet noise. For instance, the compressibleconservation equations can be solved numerically, i.e., the noise production mechanismsand the propagation of the resulting acoustic waves are considered simultaneously. Thismethod can also be pursued when combustion noise is to be analyzed. However, to solvethe compressible conservation equations numerically with emphasis on the acousticanalysis high order numerical methods have to be used to preserve the dispersion-relation of the acoustic modes in a wide wave-number range, while several orders oflength and time scales have to be considered, which leads to a stiff problem formulation.This problem is even more pronounced when reacting flows are simulated, since theintroduction of chemical reaction processes in the flow simulation is accompanied byadditional very small time and length scales.

First efforts to quantify combustion noise were mainly based on LIGHTHILL’sacoustic analogy to describe the acoustic pressure response due to aerodynamic effectslike vortex induced sound. Though the dominant noise source in combustion processes,i.e., the unsteady heat release [1–4] differs from the non-reacting case, LIGHTHILL’smain idea of reformulating the conservation equations into a single scalar wave equationwith sources has also been pursued in the combustion noise community [1, 2, 5]. Acousticanalogies, like the very first of LIGHTHILL or derivatives and extensions [6, 7] of thisoriginal formulation, are the basis of hybrid methods in aeroacoustics. Examples ofcombustion noise simulations using a single scalar wave equations have been publishedin the recent past [8–11]. Compared to the acoustic domain of interest, the relevantsource area is usually restricted to a very small region. In terms of combustion noise, forinstance, CHIU & SUMMERFIELD [4] have pointed out that the reactive flow field of anopen turbulent flame can be divided into three parts, namely the outer region, where thespeed of sound is uniform, an intermediate region, in which the mean speed of soundvaries in space, and the reaction zone, where the non-linear reacting processes occur.Since the homogeneous APE-RF system is capable of describing wave propagation in theintermediate and outer region, the necessary source terms have to be evaluated in the innerreaction zone only. According to this analysis, in a hybrid CFD/CAA method tosimulate combustion noise, the usually more expensive CFD simulations can berestricted to the non-linear reaction zone only, provided that the wave operator for theCAA simulation is capable of simulating sound propagation in the intermediate andouter region.

In the present study, the acoustic perturbation equations for reacting flows (APE-RF) areused to simulate the acoustic radiation in a hybrid LES/APE-RF context. The APE-RFsystem, which has been formulated for reacting flows, is based on the acousticperturbation equations (APE) developed by EWERT & SCHRÖDER. [12] To take advantageof the properties of the APE system, the homogeneous APE-RF system does not differfrom the original APE formulation, i.e., the capability of describing wave propagationin a non-uniform irrotational mean flow is also preserved in the homogeneous APE-RFsystem. Additionally, unlike the linearized EULER equations, the system is stable forarbitrary mean flows compared to the linearized EULER equations. Corresponding to theanalysis of CHIU & SUMMERFIELD [4] the APE-RF system becomes homogeneous in the

96 Analysis of different sound source formulations to simulate combustion generated noise using a hybrid LES/APE-RF method

intermediate and outer region such that the reaction process has to be simulated in theinner reaction zone only.

The large-eddy simulation to describe the reacting flow is based on a low MACHnumber variable density approach while the combustion model is applied through theflamelet/progress variable (FPV) model. In a previous work, [13] it has been shown, thatthe radiated acoustic field generated by the scaled substantial time derivative of thedensity as the dominant source term of the APE-RF system is in good agreement withexperimental data in a wide frequency range.

In this study, the major acoustic sub-source of the substantial time derivative of thedensity, i.e., the unsteady heat release rate is extracted from the LES in addition to twosimplified APE-RF source formulations expressed via density fluctuations. This hasalready been done in a hybrid DNS/APE-RF method [14] for a very small reaction zone.However, in the current work, this analysis is performed for a full size non-premixedflame. To evaluate the unsteady heat release rate from the LES, the necessary terms areexpressed through the mixture fraction and the progress variable by extending theflamelet database. The main focus of this study is to investigate different sourceformulations to be used in a hybrid LES/APE-RF approach to efficiently and accuratelypredict the noise emitted by an open turbulent flame. Moreover, the acoustic impact ofthis three formulations provides an insight into the spectral content and the directivitybehavior of different source mechanisms.

The paper is organized as follows. First, the governing equations for the simulationof the flow are presented, followed by the description of the flame configurations. In thesecond part, the APE-RF system with sources is introduced. It is shown how a secondsimplified source formulations of the pressure-density relation besides the formulationusing the material derivative of the density can be achieved to simulate combustionnoise. Special attention is given to the thermoacoustic sources and the method how theyare evaluated from the flow simulation. In the third part, the source term distribution isdescribed and discussed followed by a comparison of the numerical results withexperimental data. Acoustic simulations are performed on two grids with differentresolutions, especially in the source domain. Special emphasis is placed on the analysisof the applicability of each source formulation with respect to the numerical efficiencyand accuracy. The influence of source term interpolation in conjunction with thedifferent APE-RF source formulations is discussed, followed by a presentation of a testcase to evidence the occurence of a spurious effect, namely the “artificial interpolationinduced acceleration”. Subsequently, the results are discussed with respect to theacoustic impact as to the spectral content and the directivity behavior is analyzed.Finally, the results are summarized, followed by some concluding remarks.

2. MATHEMATICAL MODELS2.1. Combustion modelIn the flamelet model, [15, 16] a non-premixed flame is considered as an ensemble oflaminar flamelets. In this model, the thermochemical state of the flame is a function ofthe mixture fraction Z and its dissipation rate

(1) χZ Z Z= 22α ∇ ,

aeroacoustics volume 8 · number 1 & 2 · 2009 97

where αZ denotes the molecular diffusivity. The one-dimensional flame structure isobtained from the solution of the steady laminar flamelet equations, which is denotedby φ

(2)

where ω denotes the thermochemical source term of all species and temperature.Recently, a flamelet/progress variable (FPV) model in which all thermochemical

quantities are parameterized by a reaction progress parameter Λ rather than χZ [17, 18].Using this approach, all flame state can be characterized as

(3)

The reaction progress parameter is independent from Z and is defined through aprogress variable C. The progress variable is a linear combination of some product massfractions and can be obtained from eqn. (3) as

(4)

Assuming that a unique inversion of F exists, the reaction progress parameter in eqn. (3)can be eliminated by inverting the flamelet table, eqn. (4). Then, all chemical species canbe expressed in terms of Z and C only

φ = Gφ (Z, C). (5)

This transformation of eqn. (3) into eqn. (5) is motivated by the fact that the solution ofa transport equation for C is more attractive since it requires less modeling effort thanthat for Λ.

A low MACH number variable density LES is employed in which the FAVRE-filteredform of the continuity and momentum equations are solved

(6a)

(6b)

where u is the velocity vector, ρ is the density, p is the pressure, σ is the stress tensor. The

residual stress tensor, σres =ρu∼u∼ – ρuu∼, is modeled by a dynamic SMAGORINSKY

model [19, 20]. Here, a FAVRE-filtered quantity ψ∼ is defined as

(7)% %&

ψ ψ ( ) ( ) ( ) ( , ; ) ,x x x x x= ′ ′ ′ ′∫1

ρρ G x ∆

∂∂

+ ⋅ + ⋅ + ⋅( ) ∇ ( ) = −∇ ∇ ∇ρ ρ%

% %u

uut

p σ σ res ,

∂ρ∂t

+ ⋅∇ ( ) = 0,ρ %u

C F ZC= ( , ).Λ

φ φ= F Z( , ).Λ

− ∂∂

=χ φZZ2

2

2

ωρ ,


where G is the filter-kernel and ∆ is the filter-width. FAVRE-filtered quantities can be related to REYNOLDS-filtered variables, denoted by an overbar, by the relationρ–ψ∼ = ρψ .

An expression for the FAVRE-filtered thermochemical quantities, required in eqn. (6),can be obtained by integrating eqn. (3) with a presumed joint probability densityfunction (PDF) of mixture fraction and reaction progress parameter. Here, the joint PDFis modeled by a beta-distribution for the mixture fraction and the reaction progressvariable is modeled by a DIRAC distribution. The joint PDF is then parameterized by Z

∼, Λ∼,

and the residual variance of the mixture fraction Z′′2∼

. Then, the FAVRE-averagedthermochemical state relation can be written as

(8)

After expressing Λ∼ in terms of the FAVRE-averaged progress variable C∼, this relation canbe written as

, (9)

and is used as flamelet library in the FPV model. In addition to the solution of theNAVIER-STOKES equations, the FPV model requires the solution of the followingtransport equations for Z

∼and C

∼:

(10a)

(10b)

The residual fluxes, denoted by , are modeled by a dynamicprocedure and the unity LEWIS number assumption is employed. The model,developed by PIERCE & MOIN [21], has been employed for the computation of theresidual mixture fraction variance. In this model, equilibrium between productionand dissipation has been assumed and the model coefficient is determined from adynamic procedure.

The equations of motion (6) and the combustion model (10) are then coupled throughthe density obtained from the flamelet library

(11)

Similarly, the chemical source term, appearing in eqn. (10b), molecular properties, andall acoustic source terms in the APE formulation are also determined from the flameletlibrary using eqn. (9).

ρ = % % %G Z Z Cρρ( , , ).~′′2

% % %ττψ ψ ψ∼res = −ρ ρu u

∂∂

( )( ) ( )

ρ ρ ρα ρω%

% % % % %C

tC CC C+ ∇ ⋅ = ∇ ⋅

∼ ∇ + ∇ ⋅ +u ττ res CC .

∂∂

α( )

( ) ( ) ,ρ ρ ρ

%% % % % %

Z

tZ ZZ Z+ ∇ ⋅ = ∇ ⋅ ∇ + ∇ ⋅u ττ

res

% % % %ψ = ′′G Z Z Cψ ( , , )~2

% % % %ψ = ′′F Z Zψ Λ( , , )~2


2.2. Acoustic perturbation equations for reacting flowsThe acoustic perturbation equations for reacting flows (APE-RF) are based on thehomogeneous APE system [12] to compute trailing edge noise. For combustionnoise simulations this homogeneous system has been chosen to take advantage of itsbenign properties to simulate wave propagation, i.e., its validity for irrotational non-uniform mean flows, while excitations of instabilities are prevented. The APE-RFsystem reads

(12)

(13)

(14)

with the right-hand side sources

(15)

(16)

(17)

where Yn and fn are the mass fraction of species n and the volume force acting on species n,respectively.

The quantity ρe denotes the so-called “excess density”, introduced by CRIGHTONET AL. [22]

(18)ρ ρ ρep p

c= −( ) − −2 .

q cp p

c c

Dp

Dt ce rf p, = − +

−

+

2 2 2

1ρρ ρ

α

+ + + ∇ ⋅ −∂∂

−∇ ⋅

S S Su

xIA IB ICi

jijq

u

τ

ρeeD

Dt

p p

c( )− ⋅ ∇ − −

u ρ 2 ,

qm,rf = ∇ −

− ∇ + −

∇

p p c c

1 1 1 2 2ρ ρ ρ

ρρ

ρ

+ ∇ ⋅ + − ⋅∇( ) − ′ ⋅∇( ) ′ − ′ ×( ) − × ′∑ττ ωω ωωρ Ynn f u u u u un uu( )

qc rf, = −∇ ⋅ ′ ′( )′ρ u

∂∂

∂∂

′ − ′ =pt

ct

qe rf2 ρ

,

∂∂

′ + ∇ ⋅ ′( ) + ∇ ′

=ut

pu u

ρqm, rf

∂∂

′ + ∇ ⋅ ′ + ′( ) =ρ ρ ρt

qc r fu u ,


Perturbation quantities and time averaged mean flow variables are denoted by a primeand an overbar, respectively.

The terms in curly brackets in eqn. (17) can be expressed via the total time derivativeof the density using the energy equation for reacting flows

(19)

Using the species conservation equation and the chain rule, the term SY in eqn. (19) canbe divided into three parts, which will be referred to in the following as thethermoacoustic sources

(20)

where SIA describes the effect due to unsteady heat release, SIB the effect of non-isomolar combustion, and SIC denotes the effect of species diffusion. A detailedderivation of the APE-RF system including an at length discussion of the sources isgiven by BUI ET AL. [23]

2.3. Simplified source term formulations2.3.1. Simplified formulation I: substantial time derivative of the densityComparing the pressure spectra between the reacting and non-reacting case of theradiated field of non-premixed and premixed flames it can be assumed that the effectof chemical reactions dominates over vortex induced noise, which generally is thedominant source term in non-reacting jets. For instance, jet noise simulations of non-reacting jets have been performed by GRÖSCHEL ET AL. [24] for single and coaxialjets using LES and the APE system. The major source term was shown to be theperturbed LAMB vector, which appears on the RHS of the APE-momentum equation.In a previous study [23], an alternative source formulation to simulate combustionnoise, i.e., the scaled material derivative of the density has been proposed. Tosimulate combustion noise, former studies showed the effect of unsteady heat releaseis to be included in the investigation. Since the total time derivative of the densitycontains this major effect and the instantaneous density fluctuation is immediatelyavailable from an LES, this term represents a simplified source expression of theRHS of the APE-RF system, while mean flow effects and the acceleration of density

∂∂

= ∂∂=

∑hYDY

Dt

h

Yn p Y

n

n T p Y

nn

N

S

m m

IA

ρ

ρ ω, , , ,11 244 3344

n

N

p Y n T p Y

nn

N

S

h

Yn m

= =∑ ∑− ∂∂

∂∂

1 1ρρ ω

, , ,

IIB

m

IC

h

Ynn

N

p Y

n

S

1 24444 34444

1 244

− ∂∂

∇ ⋅=

∑1 ρ , ,

,J

44 3444

D

Dt c

Dp

Dt c

h

Y

DY

Dtp n p Y

n

n

N

S

m

Y

ρ α ρρ

= + ⋅ ∂∂=

∑121 , ,1 24444 3444

+ ∇ ⋅ −∂∂

qu

xi

jijτ .


inhomogeneities are neglected. Thus, the RHS of the pressure-density relationreduces to the very convenient form

(21)

2.3.2. Simplified formulation II: partial time derivative of the densityFrom the definition of the “excess density” in eqn. (18) it is evident that neglecting thepressure fluctuations within the source term of the pressure-density relation leads to asource formulation, which is just represented by the partial time derivative of the densitymultiplied by the square of the mean speed of sound. To evidence the sourcemechanisms included in the partial time derivative of the density this source formulationis based on the pressure-density relation in eqn. (17). The acceleration of densityinhomogeneities ∇ ˙ (uρe) plus the effects in the substantial time derivative of the densityare taken into account. If the pressure variation within the source term is neglected(ρe ≈ ρ – ρ

–), eqn. (17) reduces to

(22)

The equation of mass conservation has been used to substitute the divergence of thevelocity field.

2.3.3. Thermoacoustic sourcesUsing the flamelet solutions the acoustic source term contributions, namely the modifiedsource terms

(23)

(24)

(25)

can be determined. However, the acoustic impact of the thermoacoustic source terms SIBand SIC is not directly considered in this study. The influence of SIB and SIC on the

Scp

Sc T

h Y Wc

Y

WIC Mod IC pn n

pn

Nn

nn, = = − ∆ +

∆=

∑α λ λ21 ==

∑1

N

,

S cp S W WIB Mod IBn

nn

N

, ,= ==

∑α ω1

Scp

Sc T

hIA Mod IAp

n nn

N

, ,= ==

∑α ω11

q cD

Dt

c

e r f e, , ∂ρ

ρρ

ρρ ρ

ρρ

= − − ∇ ⋅ ( ) − ⋅ ∇

= −

2

2

u u

DD

Dt

D

Dt

ρρ ρ+ ρ

ρρ

ρρ− ⋅ ∇ − ⋅ ∇ ∇ ⋅ +

− ⋅ ∇

u u u u

= −ct

2 ∂ρ∂

.

q cD

Dte r f D, ,.

ρ

ρρ

ρ= − 2


acoustic field is only estimated. This is addressed in section 4.4. The effect of unsteadyheat release rate (SIA) in comparison with the two simplified source formulations isemphasized in this study. For further investigations and for the sake of completeness, allthree thermoacoustic source terms are tabulated in the flamelet library. The heatcapacity at constant pressure is cp, the sensible enthalpy of the species n is hn, W is themolecular weight, λ is the heat diffusion coefficient, and Yn is the mass fraction ofspecies n. Since the source terms SIA, Mod and SIB, Mod contain only one-point scalarquantities, these terms can be precomputed and stored in the flamelet library. Thecomputation of the term SIC, Mod ’ however, requires information about spatial derivativesof all species mass fractions. Since the complex CH4 mechanism, GRI-Mech 2.11 [25],is employed to characterize the thermochemical system, this would require to store all49 species in the library. The required storage of these species exceeds the availablememory. Therefore, the number of species used for the evaluation of SIC, Mod is reduced.Dimensional reduction was guided by including all major species and species, which arenot in steady state. The steady-state approximation can be applied to species, which areslowly formed by one reaction sequence, but rapidly destroyed in a subsequent reactionstep. Thus, the rate of change of this species concentration approaches zero [26]. Sinceonly acoustic perturbations at medium and high frequencies are of interest, thecontributions of the steady-state species would only be reflected at very low frequenciesand are therefore negligible. Including major species and applying steady-state analysisto the chemical mechanism results in the following set of species, which are included forthe evaluation of the acoustic source term SIC, Mod: N2, O2, CH4, H2O, CO2, CO, H2, O,OH, C2H, C2H2, C2H3, C2H4, C2H5, C2H6, CH2, CH2O, CH2OH, CH3OH.

2.4. Summary of the source term formulationsAs was mentioned before, three different source formulations are examined for theDLR-A flame. These terms include several source mechanisms, whose impact on theradiated acoustic field is analyzed, i.e., the effect of unsteady heat release (qe, rf, SIA), of non-isomolar combustion (qe, rf , SIB), and of species diffusion (qe, rf, SIC). From these mechanismsonly the heat release part is analyzed separately. The overall impact of these three sourcemechanisms including the effect of heat diffusion and viscous terms can be expressedthrough the scaled substantial time derivative of the density (qe, rf, Dρ). If additionally to thelatter source formulation, the acceleration of density inhomogeneities is taken into account,the source formulation results in (qe, rf, ∂ρ), which is given by eqn. (22). Table 1 lists thesource mechanisms inherently contained in the three different source formulations. Inbrief, the various sources read

(26)

(27)

(28)q cte r f, , ∂

= − ∂∂

⋅ρρ2

q cD

Dte r f D, ,.

ρ

ρρ

ρ= − 2

q c Se r f SIA IA Mod, , ,= −2 ρ

ρ


3. NUMERICAL SIMULATION3.1. Experimental configurationThe N2-diluted CH4-H2 /air flame has been experimentally studied by BERGMANN ET AL.[27], MEIER ET AL. [28], and SCHNEIDER ET AL. [29]. The burner configuration for thenon-premixed flame consists of a central fuel nozzle of diameter D surrounded by aco-flow nozzle of square shape. The fuel bulk velocity is Uref . Co-flow air is suppliedat an axial velocity of 7.11 × 10–3Uref . All parameters used in the calculation are givenin Table 2.

3.2. Numerical setup and large-eddy simulationThe FAVRE-filtered conservation equations for mass, momentum, mixture fraction andprogress variable are solved in a cylindrical coordinate system. The residual stresses andscalar fluxes that appear in the transport equations after filtering are modeled by adynamic procedure [17, 30] and the filtered chemical source term is closed using apresumed joint PDF. With the assumption that the mixture fraction and reactionprogress parameter are statistically independent, the marginal PDF of Z is modeled bya β-distribution and a Dirac delta function is used for the PDF of Λ. It has been shown


Table 1. Source mechanisms contained in each simplified formulation of the RHSof the pressure-density relation. The effects due to non-isomolar combustion,

species diffusion, heat diffusion, and viscosity are denoted by ∑ N=5i=2 qe, r f, i. The effect of the acceleration of density inhomogeneities is represented by ∇˙ (uρe).

sources unst. heat rel. pressure fluct.

qe,r f, SIA × – – – –qe,r f, Dρ × × – – –qe,r f, ∂ρ × × × × –

u ⋅∇ ρ∇ ⋅( )uρeqe rf iiN

, ,=

=∑ 25

Table 2. Reference values and characteristicspecifications of the flame configuration.

Parameters DLR Flame A

D 8 × 10–3 mUref 42.15 m/scref 344.33 m/sρref (air) 1.169 kg/m

3

Ma 0.12Re 1.52 × 104

that this is a valid approximation for cases in which extinction and re-ignition effectsare unimportant [31].

Only the upper, stable burning branch of the S-shaped curve is used to parameterizethe flamelet library Gψ (Z, C). This essentially reduces the FPV model to the classicalsteady flamelet model, in which all chemical species are parameterized by the mixturefraction and the scalar dissipation rate. This choice was motivated by the observationthat the simulated flame lifts off of the nozzle and subsequently extinguishes when thecomplete S-shaped curve is used for parameterization. Physically, this occurs because theflame is not stabilized by a pilot and the scalar dissipation rate close to the nozzle is toohigh for stable burning. It can be expected that an unsteady flamelet model, as proposedby PITSCH & IHME [32], would lead to physically more realistic behavior close to thenozzle. However, it was shown that this simplified model leads to excellent results forthe flow field and chemical species distribution and it is likely that the unsteady FPVmodel would only describe the stabilization of the flame more accurately.

The geometry has been non-dimensionalized by the jet nozzle diameter D and thecomputational domain is 120D × 40D × 2π in axial, radial, and circumferentialdirections, respectively. The radial direction is discretized by 160 unevenly spaced gridpoints concentrated in the fuel nozzle. The grid in axial direction uses 320 points and isstretched downstream, beginning at the nozzle exit. The total number of grid points usedfor the simulation is approximately 3.28 million. The minimum and maximum filterwidths in the domain are ∆min = 3.04 × 10

–2 D (shear layer in the nozzle-near region)and ∆max = 1.94 D (outermost grid cell at the outflow plane). The circumferentialdirection is equally spaced and uses 64 points.

The turbulent inlet velocity profile is generated by separately performing a periodicpipe flow simulation by enforcing a constant mass flux.

Convective outflow conditions are used at the outlet and slip-free boundary conditionsare employed at the radial boundaries.

The numerical simulation is run over ten flow-through-times to obtain a statisticallystationary flow field and statistics are collected thereafter over five flow-through-times.

3.3. CAA simulationThe fine CAA domain is discretized using about 20 × 106 grid points covering a physicalextent of 120 D and 70 D in the axial and the radial direction, respectively. The topology ofthe fine grid is chosen such that interpolation errors especially for the thermoacoustic sources(SIA, SIB, SIC) are almost avoided. Since the distribution of the thermoacoustic sources isclosely linked with the surface of stoichiometric mixture, the source region is mainly dividedinto two areas. The grid point distribution in the outer area (r/D ≥ 0.5) coincides in the axialdirection and each fourth grid point in the circumferential direction with the LES mesh, whilethe sources have to be interpolated in the inner area via a trilinear algorithm.

The coarse mesh consists of about 18 × 106 grid points. These points, however, aredistributed over a physical domain of 120 D in the axial and 100 D in the radial direction.The coarse mesh for the CAA simulation is able to resolve STROUHAL numbers up toStD = 1.89 based on 5.4 points per wavelength (PPW) which can be achieved using theDRP scheme. The source terms are sampled with a time increment of ∆tsc∞ /D = 0.3266,


which leads to a representation of the minimum time period of Tmin /∆t = 13.2 points perperiod (PPP) at a maximum STROUHAL number of StD = 1.89. Based on 5.4 PPW, the finegrid is able to resolve far more than StD = 2.0. However, the time increment of thesampled source data limits the maximal resolvable frequency. A temporal resolution of12.5 PPP is assumed to be sufficient at a maximum STROUHAL number of StD = 2.0.

The CAA code is based on the fourth-order dispersion-relation preserving (DRP)scheme of TAM & WEBB [33] for the spatial discretization and the alternating low-dissipation low-dispersion RUNGE-KUTTA (LDDRK) method in the 5/6 mode for thetemporal integration by HU ET AL. [34] to propagate the acoustic waves produced by thesources. A low-pass filtering method, i.e., a 6th-order explicit commutative filter [35, 36] isused at every fifth complete RUNGE-KUTTA time step to remove spurious waves. A dampingzone [37, 38] is introduced on the inner boundaries between the different matching blockscovering the LES and the acoustic domain to supress artificial noise due to a discontinuityin the entropy distribution. Since the APE system does not describe convection of entropyand vorticity modes, the asymptotic radiation boundary condition by TAM & WEBB [33] isused at far field boundaries to lower unphysical reflections into the computational domain.

4. RESULTSThree different sound source formulations have been studied in terms of their numericalapplicability. These formulations differ from each other in their inherently containedsource mechanisms. However, each model includes the effect of unsteady heat releaserate. The impact of each source term will be investigated.

4.1. Source term distributionIn Figure 1 the instantaneous contours of the different source terms are shown in thestreamwise center plane. The source distributions of qe, r f, Dρ and qe, r f, ∂ρ show a veryactive area with high amplitudes in the region of approximately (0 < x/D < 40), whilealso large areas with high amplitudes can be identified in the downstream part of thesource distribution of qe, r f, ∂ρ. However, the thermoacoustic source terms qe, r f, SIA, qe, r f, SIB,and qe, r f, SIC show a distribution which can be characterized by a very high wave-number content in the radial direction, which will be shown in more detail in sec. 4.2,since the zone of heat release is closely related to the surface of stoichiometric conditionand matches the surface of stoichiometric mixture in case of infinitely fast reaction. Thisissue is addressed in more detail in the next subsection.

As has been shown before, the source formulations qe, r f, Dρ and qe, r f, ∂ρ include the effectof unsteady heat release. The influence of the unsteady heat release term on the sourcedistribution of qe, r f, Dρ is also evident from the comparison of Figures 1 (a),(c) especially inthe region (30 < x/D < 70). Figure 1 shows the amplitudes of the source terms qe, r f, Dρ, qe,r f, ∂ρ, and qe, r f, SIA to be of the same order of magnitude. The source distributions differ,since qe, r f, Dρ includes besides the unsteady heat release term the other twothermoacoustic sources and additional effects, which have been pointed out in the previoussection. In terms of combustion noise modeling, however, the source term qe, r f, ∂ρrepresents the most complete formulation of the RHS of the pressure-density relation ofthe APE-RF system. It includes the effect of the acceleration of density inhomogeneities,



Figure 1: (continued)

0

−10

10

−1.50E-04 −9.08E-05 −3.16E-05 2.76E-05 8.67E-05

0

20 40 60x/D

y/D

80 100 120

(d) qe, rf, SIB = −c2 − SIB, Mod ρρ

which apparently dominates the source distribution in the downstream area. Comparingqe, r f, SIB and qe, r f, SIC with the other three source terms, it is evident that the order ofmagnitude of those two thermoacoustic sources are up to three orders of magnitude smaller.The source distribution of qe, r f, SIB , i.e., the effect of non-isomolar combustion, isapparently directly related to the reaction zone, since it includes the reaction terms of thespecies equations ωn. Species diffusion effects occur mainly in the vicinity of the reactionzone, since the production and consumption of species during the combustion process leadsto a sudden change of the species concentration in this area.

4.2. Source term wave number analysisFigure 2 (left) shows the instantaneous source distribution of each formulation at twodifferent axial positions, i.e., at x/D = 10 and x/D = 30 as a function of the radius. Thecorresponding wave number analysis is represented in the right diagramms of Figure 2.It is evident from Figure 2 that the partial time derivative shows the smoothest sourcedistribution, which is characterized by a mainly low wave number content, while the wavenumber analysis of the source formulation qe, r f, SIA, i.e., the unsteady heat release rate,illustrates a very sharp spatial distribution, which is characterized by a wave numbercontent in a wide range. From this analysis it is certain that the interpolated source termspossess the smallest error for qe, r f, ∂ρ, while the interpolation error for SIA should have asignificant impact on the acoustic solution. Additionally, it can be expected from thesefindings that the interpolation error for the source formulation using the substantial timederivative of the density is lower than for SIA. In brief, the formulation with the partialtime derivative of the density is expected to reproduce the acoustic field best. However,it will be evidenced, that this is only true if a special constraint is fulfilled.

4.3. Impact of source term interpolation4.3.1. Coarse grid vs. fine gridIn Figures 3, 4, and 5 experimental and numerical results for different sourceformulations using a coarse and a fine grid are shown. The acoustic spectra arecompared at five axial positions and a radial distance of r/D = 50. A significant deviation


0

−10

10

−8.00E-07 −4.73E-07 −1.47E-07 1.80E-07 5.06E-07

0

20 40 60x/D

y/D

80 100 120

(e) qe, rf, SIC = −c2 − SIC, Mod ρρ

Figure 1: Instantaneous contours in the streamwise center plane of different sourceterms evaluated for the DLR-A flame.


(b) x /D = 30

1.0E.03

8.0E.04

6.0E.04

4.0E.04

2.0E.04

0.0E+00

k/D0.5 1 1.5

SIA

pdrhopdtDrhoDt

0

0.05

0.04

0.03

0.02

0.01

0

Sou

rce

term

s

−0.01

−0.02

SIA

pdrhopdtDrhoDt

0.50 1r/D

x/D = 10

k/D1.5

(a) x /D = 10

2

SIA

pdrhopdtDrhoDt

1.0E.02

1.2E.02

8.0E.03

6.0E.03

4.0E.03

2.0E.03

0.0E+000.50 1 1.5 2 2.5

Sou

rce

term

s

r/D

SIA

pdrhopdtDrhoDt

x/D = 300.006

−0.002

2

0.004

0.002

0

0

64

Figure 2: Instantaneous dimensionless source distribution of different formulationsin the radial direction at different axial positions (left) and associatedwave number content in the radial direction (right). qe, r f, Dρ = “DrhoDt”,qe, r f, ∂ρ = “pdrhopdt”, and qe, SIA = “SIA”

in the higher frequency range starting from StD > 0.65 can be observed between thecoarse and the fine solution for the source formulations qe, r f, Dρ in Figure 4 and qe, rf, SIAin Figure 3. Due to the interpolation onto the coarse grid, this discrepancy has beenexpected and corresponds to a critical maximum wave number of (kD) = 1.54 beingresolved by the interpolation. Furthermore, the acoustic solution in Figure 3 excited byqe, rf, SIA on the coarse mesh also shows an almost constant deviation of ∆Lp = 4dB in thelower frequency range compared to the fine solution, which is in good agreement withthe experimental data in the lower frequency range.


Figure 3: Spectra in the frequency range of 0.2 ≤ StD ≤ 2 are shown. Comparisonof computed sound pressure levels on a coarse and a fine grid at r/D = 50and five different axial x/D locations using qe, rf, SIA. The STROUHAL numberis based on the jet exit velocity and the nozzle diameter D.



Figure 4: Spectra in the frequency range of 0.2 ≤ StD ≤ 2 are shown. Comparison ofcomputed sound pressure levels on a coarse and a fine grid at r/D = 50 andfive different axial x/D locations using qe, r f, Dρ (right). The STROUHALnumber is based on the jet exit velocity and the nozzle diameter D.



Figure 5: Spectra in the frequency range of 0.2 ≤ StD ≤ 2 are shown. Comparisonof computed sound pressure levels on a coarse and a fine grid at r/D = 50and five different axial x/D locations using qe, r f, ∂ρ. The STROUHAL numberis based on the jet exit velocity and the nozzle diameter D.

Analyzing the numerical findings of the partial time derivative of the density revealsan unexpected behavior. It has been shown before that this source term exhibits thesmoothest source distribution, i.e., the impact of the interpolation error on the acousticsolution should be minimal. The results, however, show that besides the aforementioneddeviation in the higher frequency range, which also occurs in the case of qe,r f,Dρ, thereare disparities in the lower frequency range, e.g., in Figures 5(a), (d) (0.3 ≤ StD ≤ 0.65),which can not be explained by ordinary interpolation errors. A possible explanation forthis unexpected behavior is given in the next subsection.

4.3.2. Artificial acceleration due to interpolationWhat exactly is different between qe, r f, Dρ and qe, r f, ∂ρ, which could lead to thisunexpected behavior? To understand this problem, a numerical study is performed inwhich a GAUSSIAN density spot

(29)

with

(30)

is constantly convected with a uniform mean flow (ρ, u−, v−, p−)T = (1, 0.5, 0, 0.7142)T.The difference between the two formulations qe, r f, ∂ρ and qe, r f, Dρ is that the source termqe, r f, ∂ρ incorporates the effect of acceleration of density inhomogeneities. However, inthis case, neither qe, r f, Dρ nor qe, r f, ∂ρ are supposed to excite acoustic responses, since anentropy spot, which is being convected constantly with the mean flow does not produceany noise. Apparently, this directly holds for qe, r f, Dρ, since this source term vanishes bydefinition. Unlike qe, r f, Dρ, the source term qe, r f, ∂ρ is non-zero for a constantlyconvecting entropy spot. Therefore, the influence of the source term interpolation on theacoustic field is investigated in the next step using qe, r f, ∂ρ.

For this test case, the convection of the density distribution in eqn. (29) is analyticallyprescribed with

(31)

on a fine mesh (Figure 6(a)). For the simulation on the coarse grid, the prescribedsources on the fine mesh are interpolated onto the coarser mesh (Figure 6(b)).Numerically, the entropy spot is initialized at x = (–9, 0) without using an embeddedinner damping zone, such that an initial acoustic response occurs. However, in thefollowing, this entropy spot is just being convected at u– and should therefore not produceany additional sound. In the analytically prescribed case in Figure 6(a) no acousticdisturbance is visible due to the convected spot. At this instant illustrated in Figure 6(a)the entropy spot has already been convected 15 units in the positive x-direction.

q cte r f,

= − 2 ∂ρ∂

x t ut0 9 0( ) .= − +

ρ( )=x,( )

.t

x x t y y0.1 exp ln(2)−

−( ) + −( ) 0 2 0 220 5



10

10

5

5

0y y

0x

(a) 401 × 401 grid points

−5

−5−10

−10

−5.84E-04 −4.04E-05 −1.26E-05 1.15E-03

10

10

5

5

0y

0x

(c) qe, rf at t = 30

−5

−5−10

−10

−1.92E-02 −8.22E-03 2.74E-03 1.37E-02

10

10

5

5

0

0x

(b) 112 × 112 grid points

−5

−5−10

−10

−5.84E-04 −4.04E-05 −1.26E-05 1.15E-03

Figure 6: Instantaneous perturbation pressure p′ [–] contours of an entropy spot att = 30 convecting with a mean flow speed of u– = 0.5 in the x-direction, ontwo meshes with different resolutions (a,b). At this moment, the entropyspot has been convected to the position x0 = (6, 0) (c).

In the next step, the source terms of the first prescribed case are sampled with a timeincrement of ∆ t = 0.1 and are interpolated onto the coarser grid, i.e., the same physicaldomain is discretized by (112 × 112) mesh points. Figure 6(c) shows the source positionand the source contours at t = 30. Since the mean flow speed in the x-direction has been

set to be constant at u– = 0.5, it appears that the entropy spot on the coarser grids is notonly been convected, but obviously also been accelerated. The coarse grid has beenchosen such that the mesh points do not coincide with the fine grid points. Duringinterpolation of the GAUSSIAN density distribution it is obvious, that the maximum pointof the GAUSSIAN distribution is permanently dislocated from its analytically prescribedposition. A dipole-like sound field is been produced by this effect, which will be referredto as the “artificial interpolation induced acceleration” effect. The resulting sound fieldis shown in Figure 6(b). It is clearly demonstrated that this effect is able to generatesignificant spurious noise.

The numerical results obtained from the simulation using the fine mesh and thesource formulation qe,r f,∂ρ show the best agreement with the experimental findings.However, as was shown before, the use of this source formulation requires a preservingof the convection speed during source term interpolation.

4.4. Spectral content and directivity behaviorAs mentioned before, each source formulation defined in eqs. (26,27,28) includescertain sound generating effects. A statement about the impact of some sourcemechanisms like SIB or SIC can be given implicitly when the spectral distributions inFigure 7 are compared.

It is evident from Figures 7 (a)-(d) that on the one hand, the acoustic impact in thelower frequency range (StD ≤ 0.45) is in general dominated by the effect of unsteadyheat release rate, except at the location (x/D, r/D) = (50, 50). Overall, this is inagreement with theoretical and experimental findings [1, 39]. On the other hand, thissource effect appears to be insufficient to reproduce the acoustic field in the higherfrequency range (StD > 0.45).

Recall from Table 1 that the source formulation with the substantial time derivative(qe, r f, Dρ) includes additionally to the unsteady heat release rate, the effect of non-isomolar combustion, heat and species diffusion, and a term describing viscousheatinga. Comparing the spectra qe, r f, Dρ with those using qe, rf, SIA shows that theadditional source mechanisms have an impact especially in the higher frequency range.In the downstream direction, i.e., at (x/D, r/D) = (50, 50), the spectrum is overpredictedin the frequency range (StD > 0.8), while at the observer position close to the jet exit(x/D, r/D) = (0, 50), those additional sources are still not able to compensate the lackbetween the acoustic response caused by the unsteady heat release rate and theexperimental data. However, the acoustic field excited by (qe, r f, Dρ) shows a goodagreement in a wide frequency range at almost all observer locations.

During the derivation of the simplified source formulation qe, r f, ∂ρ only one sim-plification has been applied to the RHS of the pressure-density relation, i.e., pressurefluctuations have been neglected. Hence, as also listed in Table 1, this formulation


aIn a hybrid approach, any assumption in the first part, i.e., the equations governing the flow, directly impactsthe subsequent part. Though the APE-RF system is an exact consequence of the conservation equationgoverning multi-component reacting fluids, the flame has been simulated in this case using a flameletcombustion model, in which viscous heating is neglected.


(a) (x/D, r / D) = (0, 50)

(c) (x/D, r / D) = (25, 50)

60

70

50

40

30

200.25 0.5 0.75

St

dB/H

z

1 1.25 1.5 1.75 2

60

70

50

40

30

200.25 0.5 0.75

St

(b) (x/D, r / D) = (12, 50)

dB/H

z

1 1.25 1.5 1.75 2

DrhoDt-fineSIA-fine

pdrhopdt-fineExp.

60

70

50

40

30

200.25 0.5 0.75

St

dB/H

z

1 1.25 1.5 1.75 2

DrhoDt-fineSIA-fine

pdrhopdt-fineExp.

DrhoDt-fineSIA-fine

pdrhopdt-fineExp.


includes the most sound source effects. Compared with the other formulations, theanalysis of Figure 7 shows at each observer location a very good agreement of thenumerical results using qe, r f, ∂ρ with the experimental data. The numerical results,especially in Figure 7(a) suggest the acoustic response from the extra sources, expressedthrough the difference between the two formulations (qe, r f, ∂ρ – qe, r f, Dρ), namely theeffect of the acceleration of density inhomogeneities and the effect of mean densitygradients, to lie in the higher frequency band and to possess a preferred radiationdirection of 90° to the flame axis close to the jet exit. A significant contribution of the

thermoacoustic sources to the acoustic field can be observed mainly in the

higher frequency range (StD > 0.45) at each observer position. Moreover, the spectra inFigures 7(d),(e) reveal an increasing impact of the thermoacoustic sources in the lowerfrequency range at downstream monitor points, for instance at x/D = 50.

qe r f iiN

, ,=

=∑ 25


(d) (x/D, r / D) = (37, 50)

60

70

50

40

30

200.25 0.5 0.75

St

(e) (x/D, r / D) = (50, 50)

dB/H

z

1 1.25 1.5 1.75 2

60

70

50

40

30

200.25 0.5 0.75

St

dB/H

z

1 1.25 1.5 1.75 2

DrhoDt-fineSIA-fine

pdrhopdt-fineExp.

DrhoDt-fineSIA-fine

pdrhopdt-fineExp.

Figure 7: Spectra in the frequency range of 0.2 ≤ StD ≤ 2 are shown. Comparisonof measured and calculated sound pressure level at r/D = 50 and fivedifferent axial x/D locations using different sound source formulations.The STROUHAL number is based on the jet exit velocity and the nozzlediameter D.

5. SUMMARY AND CONCLUSIONSThree different simplified formulations of the RHS of the pressure-density relation havebeen used to simulate the acoustic field generated by the DLR-A flame, a non-premixedturbulent flame. To take advantage of the scale separation in aeracoustics, a hybridsolution approach has been used. That is, the source terms of the APE-RF system havebeen evaluated on the LES mesh and afterwards mapped onto the CAA grid topropagate the sound waves. The flamelet library has been extended in the FPVframework in terms of thermoacoustic source evaluation. The acoustic results for thenon-premixed turbulent flame can be summarized in three categories.

Firstly, the comparison of the results obtained on the coarse grid using the threesource formulations yields:

• Due to interpolation errors the acoustic responses of the sources qe, rf, SIA and qe, r f, Dρshow deviations from the experimental data in a frequency range StD ≥ 0.65.

• The acoustic response due to the interpolated unsteady heat release source on thecoarse grid is characterized by an almost constant drop of ∆Lp = 4 dB.

• Though the source expression qe, r f, ∂ρ exhibits the smoothest source distribution, theacoustic response on the coarse grid also shows deviations in the frequency range(0.3 ≤ StD ≤ 0.65), which cannot be explained by ordinary interpolation errors.

Secondly, the “artificial interpolation induced acceleration” effect has been shown toconsiderably falsify the acoustic solution in conjunction with qe, r f, ∂ρ. The unexpectedaforementioned acoustic response of the source qe, r f, ∂ρ on the coarse grid can at leastbe partially attributed to this effect. The use of this source term requires preserving theconvection speed of density inhomogeneities during interpolation.

Thirdly, the following conclusions about the spectral content and directivity behaviorof the different source mechanisms may be drawn directly or implicitly from the resultsobtained on the fine grid.

• The effect of unsteady heat release dominates the acoustic response in the lowerfrequency range, StD ≤ 0.45, except at (x/D, r/D) = (50, 50). However, the use ofthis source is insufficient to reproduce the acoustic field in the higher frequencyrange StD > 0.45.

• A statement on the impact of the source mechanisms (qe, r f, Dρ – qe, r f, SIA), i.e., theeffect of non-isomolar combustion, species and heat diffusion, can be given implic-itly when the results obtained from qe, r f, SIA and qe, r f, Dρ are compared. These sourcemechanisms have an effect on the higher frequency range. Furthermore, theacoustic impact is even evident in the lower frequency range at the downstreamobserver position.

• The difference between qe, r f, ∂ρ and qe, r f, Dρ represents the effect of acceleration ofdensity inhomogeneities and mean density gradients. These mechanisms producean acoustic output with a preferred radiation direction of 90° to the flame axis.

In terms of numerical applicability of these source formulations it can be concluded thatthe source term qe, r f, SIA is inappropriate to predict the acoustic field of reacting flowsfor two reasons. The evaluation of this source term requires additional effort and the use


of this term leads to a very stiff problem. The most complete formulation qe, r f, ∂ρ shouldbe taken, if the convection speed of density inhomogeneities can be preserved duringinterpolation. When this constraint cannot be maintained, the source expression ofchoice is qe, r f, Dρ, since this formulation is not susceptible to the “artificial interpolationinduced acceleration” effect.

6. ACKNOWLEDGEMENTSThe financial support by the German Research Foundation (DFG) through the ResearchUnit FOR 486 “Combustion Noise” is gratefully acknowledged.

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Analysis of different sound source formulations to simulate …web.stanford.edu/group/ihmegroup/cgi-bin/MatthiasIhme/wp-content/paper... · In the flamelet model, [15, 16] a non-premixed