20th Australasian Fluid Mechanics ConferencePerth, Australia5-8 December 2016

Proper Orthogonal Decomposition Analysis of Sound Generationin a Turbulent Premixed Flame

D. Brouzet1, A. Haghiri1, T. Colonius2, M. Talei1 and M. J. Brear1

1Department of Mechanical EngineeringUniversity of Melbourne, Victoria 3010, Australia

2Division of Engineering and Applied ScienceCalifornia Institute of Technology, Pasadena, CA 91125, USA

Abstract

This paper presents proper orthogonal decomposition (POD)method of a Direct Numerical Simulation (DNS) dataset of aturbulent premixed flame. The POD results show that the com-bustion process is the dominant source of noise at low frequen-cies, whereas the inlet noise becomes an increasingly importantsource at higher frequencies. It is also shown that the first threePOD modes, that are mainly associated with the noise producedby the flame, can provide a reasonable estimate of the DNSspectra at low frequencies.

Introduction

Lean premixed combustion is a promising way of decreasingregulated pollutant emission in modern gas turbines. However,combustors operating with lean flames are susceptible tothermo-acoustic instabilities [8]. This phenomenon involvesa strong coupling between the flame dynamics and acousticwaves and results in large pressure fluctuations leading tothe combustor failure in extreme cases [7, 10]. As a result,achieving a better understanding of how flames produce soundis of great importance for environmental and safety reasons.

A number of different frameworks have been used toanalyse combustion-generated noise. One of these involves em-ploying a form of acoustic analogy, obtained by rearranging theequations of motion into different forms of the inhomogeneouswave equation. The source terms in an acoustic analogy may bethen interpreted as the main sources of sound in a given flow.For instance, in an early work by Strahle [9], the fluctuationsof the heat release rate was identified as the dominate sourceof noise in combustion by using Lighthill’s acoustic analogy[6]. Much of the subsequent works started from that basis andprovided useful insight [11].

Proper orthogonal decomposition (POD) is a widely knowntool that decomposes a dataset into a series of independent,orthogonal modes. In many mechanical systems, this techniqueis used to identify the most energetic behaviours of physicalvariables of a system. POD is also a potential tool to analysesound generated by flames as it can show how different flowevents that occur with regular frequency and are mutuallycorrelated over large spatial extent contribute to the generatedsound. While acoustic analogies provide some informationabout the distribution of sound sources in a flow, POD can pro-vide more details about the contribution of coherent structuresto the produced sound. For example, Kabiraj et al. performeda POD analysis on flame images in an experimental study of aturbulent premixed flame [3]. They further established a linkbetween the coherent structures observed in the flame and pres-sure fluctuations at a given location in the far field. However,the POD analysis in their study as well as others reported inthe literature, in the context of turbulent combustion, have been

limited to the flame region.

This paper therefore applies POD to the pressure fluctua-tions over the entire computational domain of a DNS datasetthat captures a turbulent, premixed flame producing sound [2].This enables unambiguous determination of any correlationbetween the far-field sound and structures in the flame.

Numerical methods

DNS dataset

A DNS dataset of a turbulent premixed stoichiometric flamefeaturing sound generation was used in this study [2]. The DNSwas performed using a modified version of S3D [1], referred toas S3D-SC [4]. It uses an 8th order central differencing schemefor spatial derivatives, combined with a 6-stage, 4th order ex-plicit Runge-Kutta time integrator.

Jet diameter DDomain size (Lx×Ly×Lz) 25D×16D×16DGrid resolution (nx×ny×nz) 2628×1040×1040Mean inlet Mach number (M) 0.35Co-flow Mach number 0.0035Heat release parameter (α) 2.08Jet Reynolds number (Re) 5300Inlet turbulent intensity (u′/U j) 3.7% (at centreline)Thermal flame thickness (δth/D) 0.07Sound speed (are f ) 1.755Laminar flame speed (SL/are f ) 0.00422Zeldovich number (β) 7.9Damkholer number (Da) 8.6Prandtl number (Pr) 0.72Lewis number (Le) 1

Table 1: DNS parameters.

The DNS data features a round jet of unburned premixed mix-ture (reactant) at stoichiometric condition issuing into an openenvironment of combustion products at the adiabatic flame tem-perature. The jet Reynolds number was equal to 5300 and a ho-mogeneous turbulence field with a turbulence intensity of 3.7%at the jet centreline was fed into the mean velocity field usingthe Taylor hypothesis. Four jet flow-through times were usedfor post-processing. Table 1 summarizes the DNS parameters.A representation of the flame at the jet centreline with the iso-surface of maximum reaction rate can be seen in figure 1.

POD formulation

To study the flame acoustics, the POD analysis was performedon the pressure fluctuations field p′ of a 2D plane passingthrough the flame centreline. Both temporal and spectral

X

x/D=14

Y

y/D=±3.5

1..1 . . . . .. . . . . . . 1

. . . . .. . . . . . . 1..

1..1 . . . . .. . . . . . . 1

. . . . .. . . . . . . 1..

0

Figure 1: Weighting matrix used in the POD formulation.

POD approaches were used to analyse the data. In temporalPOD, the data matrix Q used in the formulation is com-posed of spatial vectors of p′ at different instants such that:Q = [P′(t1) . . . P′(tN)]. The vectors P′(ti) represent the 2Dp′ field at instant ti, i.e. P′(ti) = [p′(x1,y1) . . . p′(xm,yn)]

Tt=ti

where m and n are the number of grid points in the streamwiseand spanwise directions, respectively.

In spectral POD, spectra of the pressure fluctuations P ′are first computed at each point in space using a number ofrealizations N f from the pressure field. They are then usedto construct the data matrix Q for a particular frequency fp,such that: Q = [PPP′1( fp) . . . PPP′N f

( fp)]. Vectors PPP′i( fp) represent

the power spectral density value at frequency fp for the ith

realization, i.e. PPP′i( fp) =[P ′(x1,y1, fp) . . . P ′(xm,yn, fp)

]Ti .

In that case, the modes found are representative of the pressurefield behaviour at a given frequency fp.

In classical POD, the most energetic recurring behavioursof a system are found by solving the following eigenvalueproblem [5]:

KΨΨΨ = QT QΨΨΨ =ΨλΨλΨλ. (1)

In equation 1, K is the covariance matrix, ΨΨΨ is the eigenvectors’matrix and λλλ is the eigenvalues’ diagonal matrix.

In this paper, a particular focus was on pressure fluctua-tions outside the flame. For this purpose, a weighting matrixH was defined as equal to 1 if x/D < 14 and | y/D |> 3.5, andequal to 0 otherwise (see figure 1). Then the diagonal weightingmatrix Hs was constructed from H and the covariance matrixK was changed to:

K = QT HsQ. (2)

This can be seen as a correlation between HsQ and Q, the firstmatrix being composed of p′ values in the unity weightingregion. In this way the POD analysis shows how the sound inthe specific unity weighting region correlates with the pressurein the whole computational domain.

Then, the ith proper orthogonal modes Φi was found bymultiplying Q to the ith eigenvector and normalizing it:

Φi =QΨi

‖ (QΨi)T HsQΨi ‖1/2

. (3)

The relative importance of the ith mode was given by the ratioof its corresponding eigenvalue λi by the sum of all eigenvalues.

Strouhal number

0 0.5 1 1.5 2

Psd

×10−9

0

2

4

6

Figure 2: Pressure spectrum at location [14D;6D].

Using both equations 1 and 3, it can be shown that:

ΦΦΦT HsΦΦΦ = I. (4)

With this weighted formulation, the modes are not orthogonalto each other in the entire domain, as in classical POD. Thisrelation can be seen as the modes being independent to eachother only outside the jet wake, where Hi j = 1 (see figure 1).

Finally, the POD coefficients a = [a1 a2 . . . aN ]T , where

vectors ai represent the time coefficients for mode Φi, werecomputed. Taking the ansatz Q = ΦΦΦa, it can be shown that thetime coefficients ai for the corresponding mode Φi is given bythe following equation:

ai = ΦTi HsQ. (5)

If equation 5 is used to rearrange the ansatz, the following rela-tion is obtained:

ΦΦΦa =ΦΦΦΦΦΦT HsQ. (6)

So the ansatz and equation 5 will be valid if and only ifΦΦΦΦΦΦT Hs = I, which will be the case in the unity weightingregion.

A reconstruction matrix Qr with M modes may be obtainedusing the following equation:

Qr =M

∑i=1

Φiai. (7)

Note that summing over all modes would give a perfect recon-struction so that Qr = Q in the unity weighting region.

Results and discussion

A typical spectrum of the acoustic pressure fluctuations outsidethe flame, at location [14D;6D], is shown in figure 2. This wascomputed by averaging Fast Fourier Transform (FFT) of 14realizations including 512 timesteps each. A hanning windowwas used to avoid any spurious high frequency content and anoverlap of 75 % between consecutive time intervals was used.With the choice of these parameters, both a good spectrumconvergence and spectral resolution were achieved. Thedimensionless Strouhal number used throughout this paper wasdefined based on the inlet diameter D and the Mach number Msuch that:

St =f DM

(8)

St = 0.11 (52.6%) St = 0.45 (87.0%) St = 1.00 (67.5%)

St = 2.01 (60.6%) St = 3.01 (62.9%) St = 3.96 (80.7%)

Figure 3: Most energetic spectral POD modes at various Strouhal numbers.

Mode 1 (40.4%) Mode 2 (22.8%) Mode 3 (8.1%)

Figure 4: Three most energetic temporal POD modes.

The most energetic mode for Strouhal numbers equal to 0.11,0.45, 1.00, 2.01, 3.01 and 3.96 can be seen in figure 3, showingflame acoustics behaviour for a wide range of frequencies. Therelative importance of the first mode compared with the othermodes at a given frequency is displayed as a percentage on topof each subplot. The weighting region is also shown on the topleft subplot. The two modes corresponding to lower frequenciesshow pressure waves coming from the flame, indicating that thecombustion process is mainly responsible for the sound pro-duced at those frequencies. It can be especially noted that themost energetic mode at St = 0.45 is accountable for 87% of theproduced noise at that frequency, illustrating how important thesound coming from the flame is at low frequencies. However, atSt = 1.00 and St = 2.01, the inlet noise appears to increasinglycontribute to the sound generation process. At St = 2.01,the inlet and the combustion noise seem to be of comparablestrength (orange and blue areas respectively at inlet and flametip). The inlet noise then becomes stronger at St = 3.01 andfinally is the dominant source of noise at St = 3.96. Here, the

fact that the most energetic mode at St = 3.96 is responsiblefor 80% of the produced sound shows that the inlet noisedominates the combustion-generated noise at high frequencies.Overall, figure 3 shows that the two phenomena responsiblefor the produced noise contribute differently relative to eachother at different frequencies. However, the spectral POD doesnot shed light on the strength of these sources to the overallproduced sound.

Temporal POD will provide an answer to this questionby showing the most energetic modes, without breaking downthe dominant structures based on their frequency. The resultsof temporal POD with the relative importance of each modeare displayed in figure 4. Mode 1 is accountable for 40.4%of the noise coming from the central region of the flame. Thesecond mode is almost twice less important and displays soundgeneration from the tip and inlet. The third mode features apure inlet noise behaviour but is only accountable for 8.1%of the overall sound, confirming that the inlet noise is far less

important than the combustion noise in the stoichiometricturbulent premixed flame studied here.

The combined contribution of the three most energeticmodes is equal to 71.3%, suggesting that the reconstructionof the data with these three modes might give a reasonableestimation of the DNS data. The average spectrum of thereconstructed signal at the point [14D;6D] was computed andis plotted in figure 5, with the DNS data spectrum shown as areference. It can be seen that there is a good agreement betweenthe reconstructed data and DNS for frequencies less that 0.7.However, the reconstructed signal is not representative forhigher frequencies. This was expected, as the most energeticmodes mainly represent the noise produced by the combustionprocess which was shown to be associated to lower frequencies(see figure 5).

The individual spectra of modes 1 to 3, displayed in fig-ure 6, show different peak frequencies. The main peak atSt = 0.11 is mainly represented by mode 1 while the secondpeak at St = 0.61 is almost due to mode 3.

Strouhal Number

0 1 2 3 4 5

Psd

10−10

Sum DNS

Figure 5: Time signal reconstruction with 3 most energeticmodes of temporal POD compared with DNS data in the spec-tral domain.

Strouhal Number

0 0.5 1 1.5

Psd

10−10

Mode 1 Mode 2 Mode 3 Sum DNS

Figure 6: Individual mode spectra compared to time signal re-construction and DNS spectra.

Conclusion

Spectral and temporal proper orthogonal decomposition (POD)was used to study noise production by a turbulent, premixedflame. These decompositions were performed on a recentlydeveloped DNS dataset that resolved both the turbulent flamemotions and its radiated sound.

This analysis revealed that the dominant acoustic modesat low frequencies were mainly associated with combustion,whereas the inlet forcing played an important role at higherfrequencies. Both sources were of comparable importanceat intermediate frequencies. Reconstructed acoustic spectra

from the 3 most energetic POD modes also gave reasonableagreement with the DNS data at low frequencies.

Further investigations will consider sound production bylean flames, which are important when considering realdevices. They will also attempt to relate this POD approach toa more physical understanding of the relationships between theradiated sound and the flame structure.

Acknowledgements

The research was supported by computational resources on theAustralian NCI National Facility through the National Compu-tational Merit Allocation Scheme and Intersect Australia part-ner share and by resources at the Pawsey Supercomputing Cen-tre.

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