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1 American Institute of Aeronautics and Astronautics

Online Predicting the Occurrence of Combustion Instability based on the Computation of Damping Ratios

Tongxun Yi* and Ephraim J. Gutmark† Department of Aerospace Engineering and Engineering Mechanics

University of Cincinnati, Cincinnati, OH 45220-0070

This paper presents a method for online predicting the onset of combustion instability, based on the computation of damping ratios. It is well known that, by Galerkin projection of acoustic eigen-modes onto acoustic equations, combustion instability can be formulated as a set of coupled, second-order, nonlinear oscillators. Under stable combustion and during the initial phase of combustion instability, pressure oscillations are weak, thus heat release perturbations caused by acoustic oscillations can be linearized into functions of pressure and pressure changing rates. These linear terms can be assimilated into the stiff and damping terms of the oscillators, and the broadband, turbulent, background heat release oscillations can be considered as the input signal. In this way, the response of pressure to background heat release oscillations can be characterized by a closed-loop transfer function. By assuming the background heat release oscillations have constant amplitude nearby the resonant frequencies (usually a reasonable assumption), the pressure spectrum may be conceived as a scaled version of the Bode plot. A procedure similar to Discrete-Fourier-Transform, but capable of higher accuracy and more suitable for real time operation, is used for spectrum estimation. The damping ratios are figured out from the spectrum nearby the resonant peaks using a weighted-least-mean-square method. Pressure data measured at two unstable combustors are analyzed, and both show that the damping ratio decreases more than three times and reaches the global minimum before combustion oscillations develop into the nonlinear limit cycle stage.

Nomenclature

ka~ : the kth sample of a real signal;

iβ : scaled amplitude of background heat release rate oscillations;

0θ : initial phase, rad;

φ : equivalence ratio;

iη : mode coefficient of the ith acoustic mode;

iω : the effective or closed-loop resonant frequency of the ith acoustic mode, rad/s; ω : angular frequency, rad/s;

iω~ : resonant frequency of the ith acoustic mode, rad/s;

iω : scaled frequency, ii ωωω = ;

iς~ : damping ratio for the ith acoustic mode;

iς : the effective damping ratio for the ith acoustic mode;

A : signal amplitude; A~ : 0

~ θjAeA = ;

B : column vector for weighted-least-mean-square algorithm; C : matrix for the weighted-least-square algorithm;

ie : background turbulent heat release rate oscillations;

f : frequency, rad/s;

if : forcing terms for the ith acoustic mode;

0F : signal frequency, Hz;

iF : the ith frequency point for spectrum estimation, Hz;

sF : sampling frequency, Hz;

iH : the complex pseudo-amplitude at frequency iF ;

+FH and −FH : normalized components of the pseudo-amplitude; j : complex symbol; L : length of a 1D acoustic field, m; M : number of frequency points for spectrum estimation;

* Graduate assistant, AIAA student member, Dept of Aerospace Engineering and Engineering Mechanics, University of Cincinnati. † AIAA associate fellow, Professor and Ohio eminent scholar, Dept of Aerospace Engineering and Engineering Mechanics, University of Cincinnati.

42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit9 - 12 July 2006, Sacramento, California

AIAA 2006-4734

Copyright © 2006 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


N : sample length; 0p : scaling factor, Pa;

'p : acoustic pressure, Pa;

Xr

: position vector; ( )iip ωˆ : amplitude response of a transfer function;

s : Laplace symbol; s : Laplace symbol, iss ω= ;

t : time, s; T : sample interval, s; x : location in a 1D acoustic field, m; X : solution vector for weighted-least-mean-square

algorithm; W : diagonal weight matrix; RMS: root-mean-square.

( )sWi : scaled transfer function;

I. Introduction odern gas turbines are typically designed to operate in lean premixed or partially premixed mode, where fuel is premixed with a very large percentage of combustion air [1]. Due to the sensitivity of heat release rate to small

variations in equivalence ratios (Ф) and insufficient acoustic damping, combustion instability has become a major technical concern for dry-low-emission gas turbine engines [2]. Accurate theoretical prediction of the onset of combustion instability is challenging, mainly due to the complexities of flame/acoustic wave interactions, flame/vortex interactions, air/fuel mixing, finite chemical kinetics, and complicated engine geometries. However it is desirable for gas turbine operators to know the safety margin to combustion instability, and take corresponding measures before combustion oscillations develop into the detrimental, large-amplitude, nonlinear limit cycle stage. Toward this goal, Lieuwen developed a correlation-function-based procedure for ascertaining safety margin based on pressure measurement, and the damping ratio was obtained by least square fit of the peaks of autocorrelation coefficients [3]. In the case of multiple unstable modes, Lieuwen suggests using bandpass filters. However the unstable frequencies may change with working conditions and may be unknown. In this paper, the safety margin to combustion instability is determined by online computing the damping ratios of acoustic modes. Under stable combustion and during the initial phase of combustion instability, the response of pressure to background heat release oscillations can be characterized by a closed-loop transfer function. By assuming the background heat release oscillations have constant amplitude around the resonant frequencies, the pressure spectrum can be considered as the scaled Bode plot, which is a function of the resonant frequencies and damping ratios. A procedure similar to DFT but suitable for online computation is developed for online spectrum estimation, and a weighted-least-mean-square procedure is used for damping ratio computation. The results show that the damping ratios decrease more than three times and reach the global minima before combustion oscillations develop into the large amplitude limit cycle stage.

II. Methodology 1. Low-order Modeling of Combustion Instability Zinn and Culick show that, by Galerkin projection of the acoustic eigen-modes into the acoustic equations, thermoacoustic instability can be formulated as a set of coupled, nonlinear, second-order oscillator [4][5],

( )∑∞

=

=1

0 )(),('i

ii XtptXprv

ψη , (1)

∞=+⎟⎟⎠

⎞⎜⎜⎝

⎛=++ ,...,1,,...,~~~2 2

2

2

jiedt

df

dtd

dtd

ij

jiiii

iii η

ηηωηωςη . (2)

Here iψ denotes the ith acoustic mode, which should satisfy the boundary conditions. For example, ( )Lxixi πψ sin)( = for

1D constant-area acoustics with open boundaries, and ( )Lxixi πψ cos)( = for 1D acoustics with closed boundaries. L

refers to the tube length. iη is the ith mode coefficient. ( )...if contains the linear and nonlinear “forcing” terms, which are

functions of mode pressure, jη , and pressure changing rates, dtd jη . ( )tei denotes the broadband, turbulent, non-

coherent, background heat release rate oscillations. iς~ refers to the damping ratio of the ith acoustic mode. Under stable

combustion and during the initial phase of combustion instability, pressure oscillations are weak, and ( )...if can be

linearized into functions of jη and dtd jη [3]. By assimilating the linearized ( )...if into the stiff and damping terms in Eq. 2, one obtains,

M


∞==++ ,...,12 22

2

iedt

ddt

diii

iii

i ηωηωςη . (3)

Here iς and iω are referred to as the “effective” or the closed-loop damping ratio and resonant frequency for the ith

acoustic mode. Note that iς is usually much less than 1, i.e. 1<<iς .

2. Laplace Transformation of Eq. 3 Turbulent heat release involves a broadband spectrum, which usually gradually decays with frequency. Here it is assumed that ( )tei has constant amplitude nearby the resonant frequency iω . Thus the pressure spectrum can be conceived as a scaled version of Bode plot. By taking Laplace transform to Eq. 3, one gets,

22 21

)()()(

iiii

ii sssE

ssWωως

η++

== , (4)

which can be further written as,

( ) ⎟⎠⎞⎜

⎝⎛ ==

++==

ii

iiii jjs

sssWsW ω

ωως

ω12

1)( 22 . (5)

Figure 1 shows the magnitude of ( )sWi , i.e. ( ) ( ) 2222 411 iiiii jW ωςωω +−= , as a function of the damping ratio iς . The

resonant peak becomes much sharper for a smaller damping ratio. It is worthwhile to notice that, the largest magnitude of ( )sWi does not occur exactly at 1=iω if the damping ratio is not zero. However for 1<<iς , one can reasonably assume

( ) ( ) iiii jWjW ςω 21max

== . As can be seen from this figure, ( ) 100max

≈ii jW ω for 005.0=iς , and ( ) 50max

≈ii jW ω for

01.0=iς . The sharpness of the resonant peak can be quantified by the bandwidth within which the power density is above 50% of the peak value. This bandwidth can be determined from two frequency points, 2

,+iω and 2,−iω , which should satisfy,

( ) 2/,

2/,

222/, 841 −+−+−+ =+− iiii ξωξω . (6)

One can see that,

( ) ( )( ) ( )11122121221

11122121221

,2222

,

,2222

,

<<−≈→+−−≈+−−=

<<+≈→++−≈++−=

+−

++

iiiiiiiiii

iiiiiiiiii

ςςωςςςςςςω

ςςωςςςςςςω . (7)

So the damping ratio for the ith acoustsic mode can be determined as,

i

i

i

iiiii ω

ωωωωωω

ς222

,,,, Δ=

−=

−= −+−+ . (8)

It is worthwhile to point out that, Eq. 8 may not be directly used for determining the damping ratios, since pressure spectrum usually exhibits multiple smaller peaks around the resonant frequency. To improve accuracy, a weighted-least-mean-square method is used for damping ratio computation.

Fig. 1 Magnitude of ( )sWi at different damping ratios for Eq. 5

3. Pressure Spectrum Estimation

0102030405060708090

100

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

iW

iω

ζ=0.005

ζ=0.01

ζ=0.02

ζ=0.05

ζ=0.1


Discrete-Fourier-Transform (DFT) or Fast-Fourier-Transform (FFT) is commonly used for offline spectrum estimation. FFT has a frequency resolution NFs , and requires NN 2log multiplications. DFT has the same frequency resolution as FFT, but requires ( )2NO multiplications. To improve frequency resolution, one has to use a larger N, and consequently a larger memory and a longer duration. In this paper, a procedure similar to DFT but more suitable for online signal processing and capable of more accurate and faster spectrum estimation is used for spectrum analysis. The procedures are briefly summarized below. Consider a real sinusoid with amplitude A, frequency 0F , and an initial phase 0θ . The kth sample is formulated as,

)1,...,2,0(22

)2cos(~)2()2(

0

0000

−=+=+=++−

NkAeAeTkFAaTkFjTkFj

ok

θπθπ

θπ . (9)

The pseudo-amplitude (PA) at frequency iF is defined as a weighted sum of previous samples,

)(

0

0)(

0

01

0

2 00

)](sin[)](sin[~

)](sin[)](sin[~

~1)( iii FFjTN

i

iFFjTN

i

iN

k

TkFjkii e

FFTFFTN

NAe

FFTFFTN

NAea

NFH +−

−

=

−

++

+−−

== ∑ πππ

ππ

ππ . (10)

Note that 0~ θjAeA = . PA computation can be done recursively. If there are M frequency points where PA is to be

computed, there will be M multiplications with each sample interval. Rewrite (10) as

2;

2

;)2sin()2sin(

;)2sin()2sin(

)(~)(

00

22

ii

F

F

TNFjF

TNFjFii

FFF

FFF

TFNTNF

H

TFNTNF

H

eHeHAFH

+=

−=

=

=

+=

+−

+

+

−

−

+

−

+

+

−

−

πππ

π

ππ

. (11)

The signal frequency is determined as the frequency point with the largest magnitude of PA. It is easy to see that, with 0→−F , 1→

−FH . For accurate frequency estimation, it is necessary to keep TNFjF eHA +

+

π2~ as small as possible, which necessitates a larger N. To show this, consider the modified-Sinc function, which is shown in Fig. 2,

( )222)sin()sin()( πππ ≤=≤−= +TFf

fNNffg . (12)

Without loss of generality, one may ideally assume 1=−FH . Thus g(f) in fact, represents the relative contribution of

+FH to

PA. The side-lobe peaks of +FH occur at frequency points ( ),...2,1== mNmfm π , and the peaks consistently decreases

with mf . A careful examination of Fig. 2 shows that +FH at the 8th and 15th peak is less than 0.04 and 0.02, respectively. A

larger N will reduce the width of the main-lobe and side-lobes, and effectively reduce the magnitude of +FH . Frequency

resolution is jointly determined by the frequency interval and the sample length. For a given frequency interval, say 5 Hz, there is a sample length above which the frequency resolution no longer improves, but the accuracy of amplitude estimation still improves. The computational complexity within each sample interval is proportional to the number of frequency points where the PA is computed. If the ranges of resonant frequencies are known, one may just specify the frequency points within these ranges. If the frequency ranges are not known, one may first identify these ranges using coarser frequency intervals, and then compute PA at frequency points just nearby the resonant frequencies using finer frequency intervals.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4f

g(f)

N=25N=100


Fig. 2. Magnitude of g(f). A larger sample length N is the most effective approach to improve the accuracy of spectrum estimation.

4. Weighted-Least-Mean-Square As mentioned just now, by assuming the background heat release ( )tei has constant amplitude nearby the resonant frequency iω , the pressure spectrum can be considered as a scaled version of the Bode plot. One can express the

magnitude of Eq. 5 at frequency ji,ω as,

( ) ( ) )ˆ(41ˆ 2,0

2,

222,,, ijiijiijiijiji pPP ωηωςωβω =+−= . (13)

Here ji,ω refers to the normalized frequency, ijji F ωπω 2, = . jF is the frequency points used for spectrum estimation.

iβ may be interpreted as the scaled amplitude of background heat release oscillations, which contains the effects of pressure scaling and frequency scaling. Mathematically for each resonant acoustic mode, iς and iβ can be uniquely determined from two points in the pressure spectrum. However, to improve the accuracy, a curve fitting procedure based on weighted-least-mean-square principles is used in this paper. Assume M points nearby the resonant frequency iω are used for damping ratio computation. For each frequency point, one can write down,

( ) ( )MjPP jiijiiii ,...,2,1ˆ1ˆ4 2,

222,

222 =−=− ωωςβ . (14)

Since there are only two unknowns, 2iς and 2

iβ , the problem is over-determined. 2iς and 2

iβ are determined as follows,

( ) WBCWCCX TT

i

i 1

2

2−

=⎟⎟⎠

⎞⎜⎜⎝

⎛=

ςβ , (15)

with

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−

−−

=

2,

22,

21,

41......4141

Mi

i

i

C

ω

ωω

and ( )( )

( ) ⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−

−

−

=

2,

22,

22,

222,

21,

221,

ˆ1...

ˆ1

ˆ1

MiMi

ii

ii

P

PP

B

ω

ωω

. W is the diagonal weight matrix. By taking the square root of the

solution vector X , one obtains iς and iβ .

III. Results Two sets of pressure data measured from unstable combustors are analyzed here. The combustion rig description can be found in Ref. 6. The first data set is obtained from a 0.46-m quartz combustion chamber with interval diameter 0.10 m, which exhibits combustion instability preceding lean blowout. The second data set is obtained from a 0.66-m stainless combustion chamber with interval diameter 0.10 m, which shows the occurrence of combustion instability above lean blowout. In both cases, the unstable mode roughly corresponds to the quarter wave mode of the combustion chamber, with a pressure anti-node nearby the air swirler and fuel injector.

020406080

100120140160180200

0.45 0.5 0.55 0.6 0.65 0.7Equivalence Ratio

RM

S

Chemiluminescence RMS (x100, Volt)Pressure RMS (Pa)

Fig. 3 Pressure and chemiluminescence RMS. This figure shows strong combustion instability before blowout.

Figure 3 shows the RMS of pressure and CH* chemiluminescence for the first data set, where the signal has been filtered using a 4th order Butterworth bandpass filter around the resonant frequency, i.e. around [350 450] Hz. CH* chemiluminescence is measured using a photomultiplier tube at 0.04 m above the dump plane, where heat release is most


intense. Pressure is measured at the chamber exit, 0.2 m away from its center, using a K&J microphone with sensitivity 10 mV/Pa. One can see from this figure that, when reducing equivalence ratio (Ф) from 0.59 to 0.53, although pressure oscillations have been considerably intensified, chemiluminescence RMS changes little. This may be because, within this range, the coherent heat release oscillations caused by thermoacoustic instability has not been well above the background heat release level. PA is computed within [250 575] Hz, with frequency interval 5 Hz. The sampling frequency is 5, 000 Hz, and the sample length is 10, 000. Figure 4 shows pressure spectrum at Ф=0.62. The pressure spectrum exhibits somewhat irregular peaks around the resonant frequency 465 Hz, which is an intrinsic feature of FFT or DFT analysis. Also shown in Fig. 4 is the pressure spectrum by weighted-least-mean-square curve fitting, which is smoother than the original spectrum, and has a model structure exactly the same as a second-order linear oscillator. For results shown in this section, 19 frequency points are used for curve fitting. The diagonal elements of matrix W is ]1,1,1,1,1,2,2,5,20,100,20,5,2,2,1,1,1,1,1[ . Larger weights are imposed for frequency points just around the resonant peak. This is because for a second order linear oscillator, the largest response occurs just nearby the resonant frequency.

Fig. 4 The original pressure spectrum and the curve fitting spectrum

Figure 5 shows iς and iβ at different Ф. When decreasing Ф from 0.67 to 0.53, iς decreases about 3 times, and iβ increases about 3 times, and iς reaches the global minimum before combustion oscillations develop into the large amplitude limit cycle stage. Also shown in Fig. 6 are iς and iβ for Ф<0.53, where pressure oscillations are so strong that Eq. 3~5 are no longer valid. Rigorously speaking, the above procedures for computing iς and iβ should not be applied within this range.

Fig. 5 iς and iβ for the first data set. The procedures for damping ratio analysis should not be used for nonlinear limit cycle stage, where pressure oscillations are intense.

Figure 6 shows RMS of pressure and CH* chemiluminescence for the second data set, where the signal is filtered using a 4th order Butterworth bandpass filter around the resonant frequency, i.e. within [220, 300] Hz. Pressure is measured at 0.08 m above the plenum, using a Kistler pressure transducer with sensitivity 10 kPa/V. CH* chemiluminescence is measured using an optical fiber installed at the fuel injector. Both pressure and chemiluminescence increase sharply when increasing Ф above 0.45. Here it is roughly presumed that the procedures for damping ratio analysis should not be applied for Ф>0.47. Figure 7 shows iς and iβ for the second data set. By increasing Ф from 0.42 to 0.47, iς decreases about five

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.91 0.935 0.96 0.985 1.01 1.035 1.06 1.085

Am

plitu

de

Original Spectrum(x0.2, Pa)Curve Fit (x0.2, Pa)

iω

00.020.040.060.08

0.10.120.140.16

0.46 0.51 0.56 0.61 0.66Equivalence Ratio

Limit Cycle Oscillations

βi

0.0030.0040.0050.0060.0070.0080.009

0.010.0110.0120.013

0.46 0.51 0.56 0.61 0.66Equivalence Ratio

Dam

ping

Rat

io

Limit cycle Oscillations


times, and iβ increases about five times, and iς reaches the global minimum before combustion oscillations develop into the large amplitude limit cycle stage. In this computation, pressure is scaled by a factor of 5.

Fig. 6 Pressure and Chemiluminescence RMS.

Fig. 7 iς and iβ for the second data set. Before combustion oscillations develop into the nonlinear limit cycle stage, iς reaches the global minimum.

IV. Discussion For the present method, it is assumed that the background, non-coherent, turbulent heat release oscillations have constant amplitude around the resonant frequency, and this allows determining damping ratios from the pressure spectrum. It is worthwhile to point out that, the quasi-regular shear layer vortices may continuously perturb the flame front, leading to peaks in heat release spectrum. Such effects may occur even in the absence of acoustic oscillations, and have not incorporated in the forcing term ( )...if . The present method in fact, implicitly assumes that such hydrodynamics-induced heat release peaks are not close to the resonant frequencies or their effects are negligibly small. The computed damping ratios are affected by the weight matrix W, the sample length N, and the number of spectrum points (M) used for weighted-least-mean-square computation. It is found that, although values of iς and iβ may undergo some changes at different W, N and M, the trend is kept the same, i.e. the damping ratio drops to the global minimum before developing into the limit cycle stage. It is further found that good results are still available using fewer points than 19, say 11 points.

V. Conclusion This paper presents a method for online predicting the safety margin of combustion instability in gas turbine engines, based on the computation of damping ratios. It is well known that combustion instability can be described by a set of coupled, second-order, nonlinear oscillators. Under stable combustion and during the initial phase of combustion instability, where pressure oscillations are weak, heat release perturbations caused by pressure oscillations can be linearized into functions of pressure and pressure changing rates. These linear terms can be assimilated into the stiff and damping terms of second-order linear oscillators. Thus the response of pressure to the background heat release oscillations can be characterized by a closed-loop transfer function, which may contain multiple acoustic modes. By assuming the background heat release oscillations have constant amplitude nearby the resonant frequencies, which is usually a reasonable assumption,

0.00E+005.00E-041.00E-031.50E-032.00E-032.50E-033.00E-033.50E-034.00E-03

0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7Equivalence Ratio

βi


0

1

2

3

4

5

6

7

8

9


Pre

ssur

e R

MS

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Che

milu

min

esce

nce

RM

S

Pressure RMS (kPa)

ChemiluminescenceRMS (Volt)

Dam

ping

Rat

io

0.00E+00

5.00E-03

1.00E-02

1.50E-02

2.00E-02

2.50E-02




the damping ratio can be figured out from the pressure spectrum using a weighted-least-mean-square procedure. A procedure similar to Discrete-Fourier-Transform but capable of much faster and more accurate spectrum estimation, and more suitable for real time computation is developed for online spectrum estimation. Pressure data measured from unstable combustors are used for damping ratio analysis, both showing that the damping ratio decreases more than three times and reaches the global minimum before combustion oscillations develop into the nonlinear limit cycle stage.

Reference 1A. H. Lefebvre, “The Role of Fuel Preparation in Low-Emission Combustion,” ASME J. Eng. Gas Turbines Power, Vol.117,

pp.617-65. 2T. Lieuwen and V. Yang, Combustion Instabilities in Gas Turbine Engines: Operational Experience, Fundamental Mechanisms,

and Modeling, Progress in Astronautics and Aeronautics Series, Vol. 210, 2006. 3T. Lieuwen, “Online Combustor Stability Margin Assessment Using Dynamic Pressure Data,” ASME J. Eng. Gas Turbines Power,

Vol.127, pp.478-482. 4B. T. Zinn and E. A. Powell, “Nonlinear Combustion Instabilities in Liquid Propellant Rocket Engines,” Proceedings of the

Combustion Institute (Int.), Vol. 13, 1970. 5F. E. C. Culick, “Nonlinear Growth and Limiting Amplitude of Acoustic Oscillations in Combustion Chambers,” Combust. Sci. and

Technol., Vol. 3, pp. 1-16, 1971. 6T. Yi and E. J. Gutmark, “Combustion Instabilities and Control of a Multi-Swirl Atmospheric Combustor,” ASME J. Eng. Gas

Turbines Power, Vol.127, pp.1-7.

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