Transient energy growth of acoustic disturbances in triggering self-sustained thermoacoustic oscillations

Energy xxx (2015) 1e12

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Transient energy growth of acoustic disturbances in triggering self-sustained thermoacoustic oscillations Zhiguo Zhang a, Dan Zhao b, *, S.H. Li b, C.Z. Ji b, X.Y. Li b, J.W. Li c a

School of Mechanical and Automotive Engineering, Zhejiang University of Science and Technology, Hangzhou City, Liuhe Road, Zhejiang, 310023, China School of Mechanical and Aerospace Engineering, College of Engineering, Nanyang Technological University, Singapore, 639798, Republic of Singapore c School of Aerospace Engineering, Beijing Institute of Technology, 100081, Beijing, China b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 29 April 2014 Received in revised form 13 January 2015 Accepted 14 January 2015 Available online xxx

Thermoacoustic instability occurs in many modern combustion systems. It most often arises due to the coupling between unsteady heat release and acoustic waves. Transient energy growth of acoustic disturbances could trigger thermoacoustic instability in a non-normal combustion system. In this work, transient energy growth analysis of a modelled choked combustor with a gutter confined is conducted. The non-normal interaction between acoustic disturbances and the anchored V-shaped flame is studied first. The thermoacoustic system is shown to be non-normal and characterized by non-orthogonal eigenmodes. Transient energy growth analysis is then performed to gain insights on its finite-time stability behaviour, which cannot be predicted by classical linear theory. To characterize the non-normality, two different energy measures are defined and estimated. One involves with acoustic travelling waves. The other is concerned with not only the travelling waves but the monopole-like flame. Comparison is then made between the two measures. It is found that the maximum transient energy growth of combustionexcited oscillations is about 102
104 times greater than that of acoustic disturbances. Furthermore, the ‘critical’ time taken to reach the maximum transient growth rate is about half of period of the fundamental mode, which is about 90% shorter than that when only acoustic disturbances are considered. In addition, the most ‘dangerous’ location at which the flame is more susceptible to thermoacoustic instability is estimated. Finally, experiments are conducted on an open-ended thermoacoustic system. It is found that transient growth of flow disturbances can trigger nonlinear limit cycle oscillations. © 2015 Elsevier Ltd. All rights reserved.

Keywords: Thermoacoustic instability Premixed flame Non-normality Transient energy growth Monopole Non-orthogonality

1. Introduction Thermoacoustics (i.e. heat-to-sound or vice versa conversion mechanism) has fascinated scientists for many decades now, due to its practical energy applications [1]. Large-amplitude heat-driven sound is wanted in thermoacoustic heat engines [2e6]. However, such self-sustained pressure oscillations (also known as thermoacoustic instability) are undesirable in aeroengines and gas turbines, since they can cause structural damage [7e10]. Over last decades, thermoacoustic instability has been one of the subjects of intense research activity, aiming to better understand and predict them [11e15]. Linear stability analysis [16,17] is generally conducted via

* Corresponding author. E-mail address: [email protected] (D. Zhao).

calculating thermoacoustic system's eigenfrequencies. However, when the eigenmodes are nonorthogonal, the combustion system is non-normal, and acoustic disturbances undergo transient growth [18,19]. If the transient growth is large enough, thermoacoustic instability might be triggered [18e21]. To gain insight on the finite-time stability behaviours, nonnormality analysis of thermoacoustic systems receives attention recently [22e25]. Experimental measurement of the transient growth in a lean-premixed gas turbine combustor was performed in Cambridge [26] with two swirlers mounted upstream of the flame. The inlet of the combustion system is neither acoustically open nor closed. To theoretically investigate the non-normality and nonlinearity, a 1D theoretical model of a Rijke tube was developed [18]. By neglecting the mean temperature gradient, the thermoacoustic system [27] was shown to be non-normal, and the maximum transient growth was about 400% larger than that of initial acoustic disturbances. To extend the model by including the

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Fig. 1. Time evolution of (a) orthogonal eigenvectors (b) nonorthogonal eigenvectors.

mean flow and entropy effects, non-normality analysis of a thermoacoustic combustor with a choked exit was performed [11]. Much of the preceding work conclude that if a thermoacoustic system is associated with non-orthogonal eigenmodes as shown in Fig. 1(b), it has the potential to trigger thermoacoustic instabilities due to the fact that small initial disturbances may undergo transient growth [28]. Thus understanding of the transient energy growth of acoustic disturbances and its prediction are important for a given system. So far non-normality analysis has tended to focus on combustors with Dirichlet boundary conditions. However, in practice, there are many combustion systems, which involve choked end (corresponding to Robin-type boundary condition) [22,26,29,31,32]. Thus there is a need to characterize the non-normality in a choked combustor [31]. It is also needed to identify a measure to quantify the resulting transient growth and to understand the non-normal interaction between the flame and acoustic disturbances. This partially motivated the present study. In this work, non-orthogonality and transient energy growth analysis of thermoacoustic oscillations is conducted. A thermoacoustic system with a choked inlet and an open exit is considered first. In Sect. 2, the governing equations are discussed. In Sect. 3 nonorthogonality analysis of the thermoacoustic eigenmodes is conducted. In Sect. 4, transient energy growths of two different types of flow disturbances are defined and estimated. In Sect. 5, experiments are conducted to illustrate that flow disturbances transient growth can trigger limit cycle thermoacoustic oscillations.

Fig. 2. Schematic of a cylindrical duct with a premixed V-shaped flame anchored on the flame-holder/gutter.

temperature T, the speed of sound c and the gas density r undergoing a sudden jump. Thus, we use subscript u and d to denote the parameters in the pre-burn region and after-flame region respectively. The length of the combustor is L and the flame is being axially placed at zf. Across the flame, the conservation of mass, momentum and energy hold as

2 v6 6pR2 vt 4 b

2 v6 6pR2 vt 4 b

d

d

7 zþ f 2 rdz7 5 ¼ pRb ½ruz
f ¼ 0

(1)

z
f

zþ

Zf

3

 þ  7 zf zþ f 2
þ ru½u
¼ pR ½p ¼0 rudz7 z z b 5 f f

(2)

z
f

3 zþ    Zf  2 7 zþ v6 ru gp g 1 h 2 izþf f 6pR2 7¼
þ dz ½pu ru u þ zf 5 z
vt 4 b g
1 g
1 2 2 f z
f

¼ Q ðtÞ (3) where Rb is the radius of the combustor. p, u and r are the instantaneous pressure, velocity and density. g ¼ 1:4 is the ratio of specific heats. Q ðtÞ ¼ Q þ Q 0 ðtÞ is the instantaneous heat release rate. The flow parameters consist of a mean value denoted by an overbar and a fluctuating part denoted by a prime. If the flow disturbances in different combustion regions are modelled as right- and left travelling waves pþ(z,t) and p
(z,t), as shown in Fig. 2, then the instantaneous pressure and velocity fluctuations can be described as

  1   13 2 0 0 8 > z
zf z
zf > > @ @ > p þ 4pþ  A þ p
 A5 zu  z  zf > u t
u tþ > > cu 1 þ M u cu 1
M u < pðz; tÞ ¼   1   13 2 0 0 > > > z
zf z
zf > > þ
A A5 zf < z  zd 4 @ @ > > : p þ pd t
c 1 þ M  þ pd t þ c 1
M  d

3

2

2. Thermoacoustic model of a premixed choked combustor A 1D thermoacoustic system with a confined premixed V-shaped flame burning in the recirculation zone of a bluff body (flameholder) is considered. The combustor is associated with a choked inlet and an open exit. The choked end corresponds to a Robin-type boundary condition [31]. The flame with constant fuel-air ratio burning in the combustor is shown in Fig. 2. The geometry of the combustor and flow conditions are same as that in the experiment conducted by Langhorne [32] and summarized in Table A.1. The flame is assumed to be acoustically compact and it is modelled as a thin sheet. The presence of the flame results in the mean

zþ

Zf

(4)

d

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Z. Zhang et al. / Energy xxx (2015) 1e12

3

  1   13 2 0 0 8 > z
zf z
zf > 1 > þ
4p @t
 > uu þ A
pu @t þ  A5 zu  z  zf > > > ru cu u cu 1 þ Mu cu 1
M u < uðz; tÞ ¼   1   13 2 0 0 > > > z
zf z
zf > 1 > u þ
@ 4 þ@ A A5 zf < z  zd > > : d r c pd t
c 1 þ M 
pd t þ c 1
M  d d d d d d

is the unburned gas velocity, and nf points downstream as shown in Fig. 3. Mathematically this statement is equivalent to dG=dt ¼ vG=vt þ ðuf
U f nf Þ$VG. After substituting nf ¼ VG=jVGj and Gðz; r; tÞ ¼ z
xðr; tÞ, the familiar G-equation [29] is obtained

where M u and M d denote the Mach numbers of upstream and downstream of the flame. It is worth noting that the mean flow effect is considered in the travelling-wave expansion method as suggested in Refs. [16,29]. When Mu and M d are vanishingly small, fluctuating parts of Eqs. (4) or (5) can be shown to be the solution of conventional acoustic wave equation. The combustor boundaries are modelled using pressure reflection coefficients [30,31]. At the open downstream end, the reflecc d ðuÞ is used to couple the reflected wave p
ðzd ; tÞ tion coefficient Z

vxðr; tÞ ¼ uf ðtÞ
U vt

the

cd ð1
M d Þ

c u ðuÞ is used to couple pþ ðzd ; tÞ to p
ðzd ; tÞ, as stream end, Z u u   þ
þ
2juðzf
zu Þ c u ðuÞ b b p u exp
p ðuÞ . Here b p ðuÞ and b p u ð
zu ; uÞ ¼ Z 2

b

cu ð1
Mu Þ

denote Fourier transform of p (t) and p
(t) respectively. Since the mass flow rate at the choked inlet is constant, it gives that the c u ¼ ð1
M u Þ=ð1 þ M u Þ. The downstream reflection coefficient Z

Q ðtÞ=Q =fAf ðt
tf Þ=Af . To capture the flame dynamic responses, the classical G-equation is used for flame-front tracking [29] to

A f ðtÞ ¼ A f þ A 0

¼

dq 0

Ra

vxðr; tÞ vr

2 #1=2 (6)

u u

a

(7)

Here H ðtÞ is the Heaviside step function and ɸ0 is the initial phase. Fig. 3 shows the evolution of the flame front in one period. It is interesting to observe that the flame moves upstream of the flame-holder during part of the oscillations as 2p=4  ut  6p=4, since the fluctuating part of the oncoming flow velocity is larger than it's mean value. Note that xðr; tÞ is evaluated at 300 locations across the combustor radius and a 4th order Runge-Kutta algorithm is used for the numerical time integration. Once xðr; tÞ is known, the instantaneous flame surface area A f ðtÞ can be calculated by

solved if Q ðtÞ ¼ Q þ Q 0 ðtÞ is known. Following the previous works [9,29], the heat release Q ðtÞ from the premixed flame imitates the evolution of its surface area ration such that

ZRb



uf ðtÞ ¼ 1 þ B sinðut þ f0 ÞH ðtÞ uf

c d is set to
1.0 for simplicity. It is worth reflection coefficient Z c u can also be set to
1.0 to characterize an upstream noting that Z open end. In this way, the developed thermoacoustic model describes a Rijke-type combustor. The system governing equations as given in Eqs. (1)e(3) can be

Z2p

f

1þ

radius of the flame-holder and the combustor respectively. In order to illustrate the flame dynamic responses, an unsteady inlet flow uf(t) as given as Eq. (7) is applied to the flame.

þ

0 f ðtÞ

"

Eq. (6) describes the location of the flame surface. And the flame front evolution is found to be dynamically coupled with the unsteady velocity uf(t) at the flame-holder, which is related to the oncoming acoustic disturbance velocity uu as i h R2b 1 þ
uf ðtÞ ¼ ðR2
R2 Þ uu þ r c ðpu ðtÞ
pu ðtÞÞ . Here Ra and Rb are the

d

incident one pþ ðz ; tÞ as d d  
þ 2juðzd
zf Þ c d ðuÞ b b p d ðzd ; uÞ ¼ Z p d exp
. And at the choked up2

with

(5)

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  0 2  2 vx vx0 vx vx þ dr r 1þ þ2 vr vr vr vr

0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  ZRb ZRb r vx ðrÞ vxðrÞ B vxðrÞ vr vr ffi dr þ O sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2pB dr þ r 1þ @ 2  vr vxðrÞ Ra Ra 1þ vr

study the non-normal interaction between the flame and flow disturbances. The main features of the nonlinear flame model [29] are reproduced here for completeness and integrity. The flow is assumed to be axisymmetric and combustion is assumed to occur on a thin surface whose axial position at radius r is given by Gðz; r; tÞ ¼ z
xðr; tÞ ¼ 0. The flame surface is assumed to move in the direction of its normal nf with speed uf $nf
U f , where uf(uf,vf)

1  0 2 ! C vx þ /C A vr

(8)

Further simplification and analysis leads to

A

f

¼

  p R2b
R2a uf U

(9)

f

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Fig. 3. Time evolution of the premixed V-shaped flame surface anchored to the tip of the gutter during one period of the oncoming acoustic disturbance, as Ra/Rb ¼ 0.25, B ¼ 1:2, u=2p ¼ 59 Hz, U f ¼ 3:6 m/s, uf ¼ 37 m/s, and f0 ¼ 2:7.

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R u b ZRb u U 2f Z dx0 ðr; tÞ 0 dr ¼ c A f ðtÞzt1
2 2pr x0 ðr; tÞdr dr uf Ra

(10)

Ra

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where c ¼
2p 1
U 2f =u2f . To simplified Eq. (10), integration by parts and the boundary conditions are applied. If the flame front is discretized into K flame elements equal of radial length dr, then A 0f ðtÞ can be expressed as

A 0f ðtÞz

K X k¼1

A

0 fk

¼c

K X

lk x0k ðtÞdr

(11)

k¼1

where lk is the weight factors corresponding to the trapezoidal integration formula as given as lk ¼ 0:5, as pþ ðtÞk ¼ 1 or K. Otherd wise lk ¼ 1. Substituting Eq. (1) into Eq. (3) and linearising the momentum and energy conservation Eqs. (2) and (3) give the time evolution of 0 the outgoing travelling waves p
u ðtÞ and generated from Q (t). By using these flow conservation equations across the flame and applying the end boundary conditions, the acoustic waves present in the combustion system can be predicted by solving the resulting matrix equation as

0

1  2 zf
zu C B p
B A B u @t
 2 B ! cu 1
M u B p
u ðtÞ B A ¼BB 0 þ  1  B pd ðtÞ B 2 z
z d f B þB C @ p @t
 A d 2 cd 1
M d 0

1 C C 0 1 C 0 C C B Q 0 ðtÞ C Cþ@ A C 2 C c pR u b C A

(12)

where A and B are coefficient matrices as given in Appendix B. The time evolution of the flow perturbations from specified conditions as given in Table A.1 can now be determined by numerically solving Eqs. (2), (6) and (12). Fig. 4(a) shows the time evolution of the pressure fluctuation p0 ðtÞ=p. It can be seen that initial disturbances grow into a nonlinear limit cycle at approximately t ¼ 0.5 s. The nonlinearity is indicated by the presence of the harmonics ½u2 ; u3 … as shown in the spectra of the pressure fluctuations in Fig. 4(c). Rayleigh index S is generally used as a critical indicator of the combustion system stability [8]. It describes the energy exchange rate between unsteady heat release and the pressure oscillations. If S > 0, the acoustical energy in the system is increased and tends to drive the unstable process into a limit cycle. However, a negative S indicates a ‘destruction’ interaction between unsteady heat release and pressure waves and so thermoacoustic oscillations are decaying. Following the previous works [8,33], we define a normalized Rayleigh index S(t) as given as

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Z. Zhang et al. / Energy xxx (2015) 1e12



〈b pm; b pn〉 ¼

Zzd

5

 b pmb p n dz ¼

zu


gp u2n þ

Zzd

b pm

zu

  b n v 1 vp dz vz r vz

jðg
1Þ un

Zzd

  b  d z
z dz b pm Q f

(15)

zu

b ðuÞ needs to be estimated. If To determine the inner product, Q we assume the flame Strouhal number St is small in the present system [34], then the classical time-lag N
t formulation can be used [29,35]. It has been demonstrated in Sect. 2 that the flame model can be linearized into the N
t formulation as

Q 0 ðtÞ ¼

Fig. 4. Time evolution of (a) pressure fluctuation p0 (t)/p, (b) Rayleigh index S(t) and (c) frequency spectra of the limit cycle.

Zzd   Zt p0 ðu; jÞQ 0 ðu; jÞdj sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dj SðtÞ ¼ d z
zf dz sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z Z zu

2p=u

0

2p=u

p02 ðu; jÞ

0

gpN u0f ðt
tv Þ

where N denotes the interaction index describing the intensity of the heat-to-sound interaction, tv is the time lag between the velocity perturbation and heat release oscillation, which is assumed to be equal to 0:4zd =uu . If we approximate u0f ðt
tv Þ in Eq. (16) to the first order i.e. u0f ðtÞ
tv vu0f ðtÞ=vt, taking Fourier transform to b ðuÞ into Eq. (15), then the both sides of Eq. (16) and substituting Q inner product can be shown as

 〈b pm; b pn


gp 〉¼ 2 un

Q 02 ðu; jÞdj

0

3. Non-orthogonality analysis Now non-orthogonality analysis of eigenmodes is performed to gain insight on the thermoacoustic system [22,31]. The nonorthogonality is characterized by the inner product of two eigenmodes [22]. If the inner product is zero, then the ei2genmodes are orthogonal. The system with decaying orthogonal eigenmodes will be stable. However, when the inner product is non-zero, the system is non-normal and characterized with non-orthogonal eigenmodes. In such non-normal system, finite-time growth of acoustic disturbances might trigger thermoacoustic instability. Let's first consider the system with no damping involved. The flame results in the mean temperature undergoing a jump, i.e. T d > T u . And it produces the travelling acoustic waves as given as   2 0 0 0 ðg
1Þ v p vp vQ 1 v 1
vz ¼ gp vt . Taking Fourier transform to it and r vz gp vt 2

Zzf

þ

0

Zzd zf

ð


jum z
zf

1

  b n v 1 vp dz vz r vz

1
jum ðz
zf Þ jum ðz
zf Þ   1 vb pn þ
cd ð1þM d Þ cd ð1
M d Þ A v @b dz p d;m e þb p d;m e vz r vz

  jN cu ejun tv R2b  þ
þ
   b p u;m b p u;n p u;n
b p u;m þ b un R2b
R2a

where um and un are the mth and nth modes frequencies. They can be estimated by solving Eq. (12) in frequency domain as given by the resulting block matrix equation as

10 1 0 1 W1 V1 0 0 0 … 0 0 B B V2 C B C 0 C W2 0 … 0 T B 0 TB 0 C C B C ¼c @ A c @ « « « « 1 « « A@ « A 0 VN 0 0 0 … 0 WN |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 0

(18)

W


þ

where Vm ¼ ½ b p u;m ðum Þ; b p d;m ðum ÞT , cT¼[1,1…,1]. Subscript m denotes the mth mode with frequency of um and Wm is a 2  2 matrix as given as

0  B
jum tu B Wm ¼ BA
B e @ 0 

0

Þ

0

0

0




jum td

e 0

B
jum tv b u e
jum tu B 1
Z þBN e   @ p R2b
R2a ðg
1Þ

(14)

b denotes Fourier transform of Q (t). By manipulating Eq. where Q  (14), the inner product between the eigenmodes b p m and b p n can be shown as

ð

jum z
zf


þb p u;m ecu ð1
Mu Þ A

(17)

taking the complex conjugate yields

   b n
u2n  v 1 vp g
1b  b pn
¼ Q d z
zf ðjun Þ vz r vz gp gp

Þ

þ @b p u;m e cu ð1þMu Þ

zu


gp þ 2 un

(13) Fig. 4(b) shows the variation of the Rayleigh index S(t) with time. It can be seen that when the initial perturbation grows into a limit cycle, S is positive and keeps increased and finally ‘saturated’ at approximately 0.90. This indicates the ‘construction’ interaction between heat and sound.

(16)

ðg
1Þ

where

the

time

delay



0

11

CC CC CC 0 AA

(19)

2

tu ¼ 2ðzf
zu Þ=cu ð1
Mu Þ

and

2 Md Þ

and the matrices A and B are given in td ¼ 2ðzd
zf Þ=cd ð1
Eq. (12). Theoretically, the acoustic disturbances consist of infinite

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Z. Zhang et al. / Energy xxx (2015) 1e12

modes. For simplicity, we assume that first N modes are involved PN b
b
and in the flow fluctuations, i.e. p u ¼ m¼1 p u;m ðum Þ, PN þ þ b p d ¼ m¼1 b p d;m ðum Þ. Eq. (18) has solutions only if the determinant of the matrix Wm vanishes, i.e. N Y

detfWg ¼

detfWm g ¼ 0

(20)

m¼1

The root satisfying det{Wm}¼0, i.e. um ¼ Um þ jsm is the complex eigenfrequencies of the present system. Physically, sm denotes the oscillation grow rate. When sm > 0, the system is unstable. However, if sm < 0, the long-term evolution of eigenmodes are decaying. However, if the eigenmodes are non-orthogonal, they may interact and transient energy growth might occur. um is the oscillation mode frequency. They can be found by Newton's iteration method [17]. Fig. 5(a) shows the variation of estimated u1 with the flame location zf/L. Compared with the experimental measurements [32], qualitatively good agreement is observed. It is worth noting that the measured frequency at fixed flame location is changed with varied fuel-air ratio. When the upstream inlet is set to be acoustically open, the dominant eigenfrequency is predicted as denoted by the dash dot line in Fig. 5(a). It can be seen that when the boundary conditions are varied, the thermoacoustic eigenfrequency is changed dramatically.  With um estimated, the inner product 〈 b pm; b p n 〉 characterizing eigenmodes non-orthogonality can be calculated, by assuming that c b
bþ b
bþ b
p u ¼ 1:0 þ j0, p u ¼ Z u p u expf
jutu g, p d ¼
p u W11 =W12 , and
þ cdb b p expf
jutd g. Fig. 5(b) illustrates the variation of the p ¼Z d

d



inner product 〈 b pm; b p n 〉 with zf/L. It is apparent that the thermoa



p n 〉 > 0. Furthercoustic modes are non-orthogonal, since 〈 b pm; b more, the non-orthogonality depends on zf/L. It is also observed that the non-orthogonality is increased with increased flameacoustic interaction index N , as shown in Fig. 5(b). This indicates that the system non-normality becomes more pronounced with the

interaction intensity between the flame and oncoming acoustic disturbances. This finding is consistent with the transient energy growth analysis as discussed in the following Section. Note that in c u ¼ ð1
M u Þ=ð1 þ M u Þ is used to characour previous analysis, Z c u is set to
1.0, terize the upstream choked inlet. However, if Z then it characterizes an acoustically open inlet. This physical configuration is corresponding to a Rijke-type thermoacoustic system, which is the same to our experimental setup as discussed later. Eigenfreqeuncy and non-orthogonality analysis of such Rijketype combustor can then be conducted by following the same procedure as described above. 4. Transient energy growth analysis In order to validate the findings obtained from the nonorthogonality analysis, transient energy growth analysis of flow disturbances is conducted. To characterize the disturbances transient energy growth, two different energy measures are defined. One is concerned with the acoustic travelling waves. Its energy E a per unit cross-sectional area consists of both kinetic E k and potential energy E p. For 1D plane wave travelling to the right, i.e. pþ it has been shown [36] that u ðz; tÞ, 2

þ

ðz;tÞ ; E k ¼ E p ¼ p 2gp

and

Ea ¼EkþEp ¼p

þ

ðz;tÞ2 . gp

Thus the total

acoustical energy in the combustor is given as

Z2p Ea ðtÞ ¼

ZRb dq

rdr 0

0

Zzd

2

zu

i 1 h þ p ðz; tÞ2 þ p
ðz; tÞ2 dz gp

2  2 1 0 2sðz
zf Þ  2sðz
zf Þ   
  þ 
c 1þM c 1
M u u u u ð Þ þ b ð Þ  Adz @ b ¼  p u ðuÞe  p u ðuÞe       zu 3 2  2 1 0 2sðz
zf Þ  2sðz
zf Þ   Zzd 

 þ   7 c 1þM c 1
M ð Þ ð Þ d  d  A  p ðuÞe d p d ðuÞe d þ @ b d  þ b  dz5     e2st pR2b gp

6 4

Zzf

zf

(21) If we define a state variable vector x consisting of the travelling waves as given in Appendix C, then the acoustical energy Ea can be Z zd b ejut ÞH U x b ejut Þdz ¼ < xðz; tÞH ; xðz; tÞ > U. ðð x rewritten as Ea ðtÞ ¼ zu

H

b ejUt est is the conjugate transpose of x(z,t) and U Here xðz; tÞH ¼ x is the weighted diagonal matrix as given as in Appendix C. The maximum transient growth factor of the acoustical energy ðtÞ ðz;tÞ;xðz;tÞ > U is then defined as ga ðtÞ ¼ max EEaað0Þ ¼ maxxðz;0Þs0 < H

U

estimate ga from the complex signal x(z,t), a procedure as reported in Refs. [11,37] is used. Such complex signal can be linearly expanded in terms of its complex eigenmodes as b ðzÞexpf
jFtgk. Here vector k contains the coefficients of xðz; tÞ ¼ x the linear combination of eigenmodes and F ¼ diagðu1 ; u2 /uN Þ stores the complex frequencies for the expansion. The acoustical energy can then be rewritten as

Ea ðtÞ ¼

Zzd 

H b ðzÞe
jFt kdz b H ðzÞU x e
jFt k x

zu

H    ¼ e
jFt k Q a e
jFt k Fig. 5. (a) comparison between the predicted mode frequency u1 and the experimentally measured ones, as L ¼ 1.92 m, M u ¼ 0:08, tv ¼ 0:01 s, (b) variation of



 

= < b p1 ; b p 1 > with the flame location zf/L, as tv ¼ 0:01 s, L ¼ 1.92 m, and M u ¼ 0.08, cu ¼ 340 m/s and Tu ¼ 288 K.

where matrix Qa is given as Q a ¼

(22) R zd zu

b H ðzÞU x b ðzÞdz. It can be seen x

that Qa is a positive Hermitian matrix. And so the Cholesky

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Z. Zhang et al. / Energy xxx (2015) 1e12

7

decomposition can be applied and yields the square matrix Sa of size N such that SH a Sa ¼ Q a . Substituting Qa into the acoustic energy of Eq. (22) leads to

Ea ðtÞ ¼ ðSa expf
jFtgkÞH ðSa expf
jFtgkÞ

(26) (23)

Since the Cholesky factor Sa is not singular, the maximum transient growth factor at time t is given as

  2 Ea ðtÞ ¼ max kSa expf
jFtgS
1 ga zf ; t ¼ max a k2 Ea ð0Þ Sa

(24)

This indicates that ga(zf,t) at time t is determined by the largest singular value of ðSa e
jFt S
1 a Þ, which can be calculated by using singular decomposition methods [22]. The optimal initial perturbation is given by the corresponding right singular vector of ðSa e
jFt S
1 a Þ. The transient growth factor ga(t) obtained at time t consists of all perturbations for the first N eigenmodes. By maximizing ga(t), the global maximum transient growth factor is estia ¼ maxt/∞ ga ðzf ; tÞ. mated as gmax The alternative energy measure takes all the disturbances present in the thermoacoustic system into account. It has been shown in Sect. 2 that the flame present behaves like a monopole sound source. Unsteady heat input from the flame to the surrounding gas at constant pressure causes a corresponding unsteady expansion of the gas. Thus the disturbances present involve not only the travelling waves but also the unsteady volumetric flow rate due to the gas expansion(see Eq. (2) [18]). The disturbance energy due to dilatation at the flame is then given as

Z2p Ef ðtÞ ¼

ZRb dq

0

0

  Ea ðtÞ þ Ef ðtÞ < yðz; tÞH ; yðz; tÞ > F ¼ max gf zf ; t ¼ max Ea ð0Þ þ Ef ð0Þ yðz;0Þs0 < yðz; 0ÞH ; yðz; 0Þ > F

K  2 1 c2 ru ðg
1Þ Q X ru du02 rdr ¼ lm x0m ðtÞDr f 2 2gp A f k¼1

(25) Thus the transient growth factor of both flow and flame disturbances energy is given as

where y is the new state variable vector and F is a weighted diagonal matrix as given in Appendix C. Similarly, the maximum tranf f sient growth factor gmax is given as gmax ¼ maxt/∞ gf ðzf ; tÞ. To select a proper eigenmode number N, a convergence check is conducted first to calculate the relative change in ga(t) with increasing number of acoustic eigenmode. Fig. 6(a) shows the relative change of the transient growth factor with time, as N is set to 3, 5 and 6. It can be seen that the change between ga(N ¼ 6) and ga(N ¼ 5) is less than 3%. Thus the number of total eigenmodes is chosen to be N ¼ 6 in our analysis. In addition, the number of discretized flame front elements is needed to select. In this work, the flame front is discretized into 300 elements. This is to ensure that the relative change in the transient growth factor gf(t) with increase in number of the flame elements is less than 5%. To check the convergence of the flame elements number, the relative change f f between gmax ðK ¼ 250Þ and gmax ðK ¼ 300Þ is estimated, as shown in Fig. 6(b). It can be seen that the relative change    f  f ;K¼300 f is small. This justifies Dgmax ðK ¼ 250Þ
gmax ðK ¼ 300Þ=gmax

using 300 discretized flame front elements, i.e. K ¼ 300 to estimate the transient growth factor gf(t). As the numbers of the eigenmodes and discretized flame front element are properly chosen, ga(t) and gf(t) are calculated respectively. Fig. 7 illustrates the variation of ga(zf,t) with time, when first 6 eigenmodes are involved in the acoustic disturbances. It can be seen that when the interaction index N is small i.e. a N ¼ 1:0  10
3 , gmax z5:72 > 1:0. This indicates that the acoustical disturbances undergo a transient growth, which is about 572% as large as the initial disturbances. Further increasing N (i.e. N ¼ 0:3) a / þ ∞ and the combustion system becomes linearly results in gmax a unstable. The finding of increased gmax with N is consistent with our non-orthogonality analysis.

Fig. 6. Convergence check (a) the relative change of ga(zf,t) normalized with its maximum value with time, as zf/L ¼ 0.614, tv ¼ 0:01 s and M u ¼ 0:08 and the mode number N is set to 3 different values, (b) The relative change of Dgf(zf,t) normalized with its maximum value with time, as zf/L ¼ 0.614, tv ¼ 0:01 s and M u ¼ 0:08 and flame element number K is set to 250 and 300.

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Z. Zhang et al. / Energy xxx (2015) 1e12

Fig. 9. Variation of gf(zf,t) with normalized time u1 t=2p as N ¼ 6, K ¼ 300, tv ¼ 0:0212 s, M u ¼ 0:08 and zf/L ¼ 0.25. Fig. 7. The variation of ga(zf,t) with time, as N ¼ 6, zf/L ¼ 0.614, tv ¼ 0:01 s and M u ¼ 0:08: d N ¼ 3:0  10
1 , - - - N ¼ 1:0  10
3 .

Fig. 8 illustrates the transient growth factor of the total acoustic and flame disturbances gf(zf,t) with time t/T1, as the flame is place at zf/L ¼ 0.614. Comparison is then made between gf(zf,t) and ga(zf,t). It f can be seen that gmax is approximately 3 orders of magnitude as f

a large as that of gmax . The increased order of magnitude of gmax is due to the increased degrees of freedom of the monopole-like

f f gmax . However, as the flame is placed at 0.063 < zf/L < 0.3, gmax becomes infinite large (corresponding to an unstable system). Such predicted ‘dangerous locations’ via calculating the maximum f transient growth factor gmax is different from the conventional linear modal analysis, which calculates the mode growth factor and predicts the long-term stability behaviours only.

5. Measurement of thermoacoustic instability triggered by disturbances transient growth

f

flame. Furthermore, gmax is found to occur at u1 t ¼ p as shown in the embedded graph. That is to say, the maximum possible amplification of total acoustic and flame disturbances is reached after half of a period of the fundamental mode with frequency u1. The transient energy growth of both flame and acoustic disturbances has been shown to depend on the flame location, as indif cated in Eq. (26). Fig. 9 shows the estimated gmax varied with t/T1. It f can be seen that gmax /∞, as t/∞. This indicates the system becomes unstable. In order to predict the ‘dangerous’ regions at which the flame is f more susceptible to thermoacoustic instabilities, gmax is estimated as the flame location is varied, as shown in Fig. 10. It can be seen that when the flame is placed at 0 < zf/L < 0.063 or zf/L > 0.3, the system is linearly stable but with a finite transient growth factor

Fig. 8. Comparison between gf(zf,t) and af(zf,t) with normalized time u1 t=2p as K ¼ 300, N ¼ 6, tv ¼ 0:01 s, M u ¼ 0:08 and zf/L ¼ 0.614.

In order to validate our theoretical findings, experiments need to be conducted on a choked combustor. Recently, Kim & Hochgreb [26] experimentally measured the transient growth in a choked combustor. And they confirmed that transient growth of flow disturbances can trigger thermoacoustic oscillations. To be different from the previous work and for simplicity, we conduct a transient growth experiment on a Rijke-type combustion system, as shown in Fig. 11(a). The length and diameter of the tube are 110 and 5 cm respectively. A premixed laminar flame is anchored on a Bunsen burner, which is placed at 15 cm away from the bottom open end. A PC fan with power 0.9 Watt is used to generate impulse disturbances to trigger thermoacoustic instability. A pressure sensor

f f Fig. 10. Variation of gmax ðzf Þ=gmax ðzf =L ¼ 0:063Þ with the normalized flame location zf/L.

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Z. Zhang et al. / Energy xxx (2015) 1e12

9

Fig. 11. (a) Schematic of the experimental setup, (b) measured temperature contour of the Rijke-type thermoacoustic system.

(B&K 4957 microphone) is paced at 55 cm away from the bottom open end. An infrared thermal imaging camera is used to measure the temperature contour of the Rijke tube, as shown in Fig. 11(b). As the fan is turned on at t ¼ 5 s for 0.1 s, the measured pressure fluctuations are observed to undergo transient growth, as shown in

the shaded area of Fig. 12(a). The flow disturbances first decay, then are amplified and finally grow into limit cycle oscillations (see t > 7.5 s). This reveals that thermoacoustic instability could be triggered by the transient growth of acoustic disturbances occurred in the system associated with nonorthogonal eigenmodes [22]. It is

Fig. 12. (a) Time evolution of the measured pressure fluctuations, (b) frequency spectrum of the measured pressure fluctuations, (c) comparison between measured and predicted dominant thermoacoustic frequency u1 =2p from the Rijke tube.

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10

Z. Zhang et al. / Energy xxx (2015) 1e12

worth noting that the present experiment involves using an openeopen combustion system, which is different from the choked combustor as discussed in the theoretical analysis section. However, the transient growth effect on triggering thermoacoustic instability has been successfully observed not only in the choked combustor [26] but also the open-ended combustor. However, for comparison the dominant eigenfrequency varied with the flame position is measured and compared with that predicted by using c u as shown in Fig. 12(c). It can be seen the developed model with Z that good agreement is obtained. This confirms that the developed thermoacoustic model can be used to characterize a choked or an open-ended Rijke-type thermoacoustic system, depending on the c u. setting of the inlet boundary condition, i.e. Z

6. Conclusions In this work, transient energy growth of acoustic disturbances in triggering thermoacoustic instability is considered. Transient growth analysis of a choked combustor is conducted first. It is shown that the thermoacoustic system is non-normal and characterized by nonorthogonal eigenmodes. Such non-orthogonality results in the disturbances transient energy growth. To quantify the transient growth, two different energy measures are defined and calculated. Comparison is then made between these two measures. It is found that the maximum transient energy growth f factor gmax of both acoustic and flame disturbances is 2 orders of a magnitude larger than that of only acoustic disturbances gmax .

0

  u 2
1 þ Mu 2
d
M u uu

B B ! A¼B 2 B  u2d Mu  @ 1
gM u 2 þ Mu

1 1
Mu g
1 2 u2u 0

    1
Mu u 2 1 þ Mu 2
d þ Mu uu 1 þ Mu

B B B ! B¼B 2 B 1 þ gM  u2d M  2 u @ þ Mu
u 1 þ Mu
1 g
1 2 u2u

1 þ Mu

ru cu rd cd

the National Natural Science Foundation of China (51206148 and 51176172). As a visiting Professor, Prof. Zhang ZG would like to acknowledge the support provided by Nanyang Technological University. Appendix A. Geometry and flow parameters of the setup Table A.1 Geometry and flow parameters of the combustion system. Geometry

Value

Combustor length, L (m) Radius of the combustor,Rb (m) Radius of the flame-holder/gutter, Ra (m)     Upstream length of the combustor, zf
zu  ðmÞ     Downstream length of the combustor zd
zf , (m)

1.92 0.035 0.0175

Combustor inlet Mach number, M u Combustor inlet temperature, Tu (K) Combustor inlet and outlet pressure, pu ¼ pd ðPaÞ Fuel, Mean heat release rate Q , (J) Fuel-air ratio normalized on stochiometric fuel-air ratio, Flame speed (assumed constant) (m/s) Combustion efficiency

0.08 288 1.01325  105 ethylene 3.16  106 0.7 3.6 0.8

1.18 0.74

Appendix B. Coefficient matrices The coefficient matrices A and B are given as

1

C C C C ru cd 1 þ gM d A Mu Md þ rd cu g
1 r cu 1
Mu u rd cd

(B.1)

1

C C C C cd 1
gM d ru C
Mu Md A
cu g
1 rd

Furthermore, such distributed monopole-like flame front is shown to play an important role in decreasing the ‘critical’ time taken to f reach gmax . It is found that the ‘critical’ time is about half of period of the fundamental mode, which is about 90% shorter than that a taken to reach gmax . This reveals that in order to characterize the non-normality of a thermoacoustic system, the flow fluctuation energy should account for contributions from all the constituent phenomena, i.e. both flame and acoustics. In addition, the ‘unstable’ regions of the combustor at which the flame is more susceptible to f thermoacoustic instabilities are predicted in terms of gmax . Finally, experiments are conducted on a Rijke-type combustion system. It is found that acoustic disturbances undergo transient growth and nonlinear limit cycle oscillations are triggered. Acknowledgements This work is financially supported by the Ministry of Education e Singapore, under the grant AcR-Tier1-RG91/13-M4011228 and by

(B.2)

Appendix C. State variable vector x and y and coefficient matrices U and F

0

1 ð Þ B C
B bþ C B p u ðuÞe cu ð1þMu Þ C B C B C juðz
zf Þ B C B
C c 1
M uð uÞ C B b ðuÞe p B u C xðzÞ ¼ B C B juðz
zf Þ C B C
þ Bb C c 1þM B p d ðuÞe d ð d Þ C B C B C juðz
zf Þ B C @
cd ð1
Md Þ A b p d ðuÞe

0

ju z
zf

and

1 B gp B B B B 0 B B U¼B B B 0 B B B @ 0

0

0

1 gp

0

0

1 gp

0

0

1 0 C C C C 0 C C C C C 0 C C C C 1 A gp (C.1)

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Z. Zhang et al. / Energy xxx (2015) 1e12

1 Þ ÞC C C B C B juðz
zf Þ C B B
cu ð1
Mu Þ C C B b p e C B u C B juðz
zf Þ C B B þ
c 1þM C Bb dð dÞ C C   B pd e C B y b x ¼B pu; b C juðz
zf Þ C B B b
cd ð1
Md Þ C C B pd e C B C B b B x 1 ðDrÞ C C B C B C B b ð2DrÞ x 2 C B C B C B « A @ b x K ðKDrÞ 0

þ Bb B pu e




0

ð ð

ju z
zf

1 B gp B B B B 0 B B B B B 0 B B B F¼B B 0 B B B B 0 B B B 0 B B B 0 @

cu 1þM u

and

0

0

0

0

0

0

1 gp

0

0

0

0

0

0

1 gp

0

0

0

0

0

0

1 gp

0

0

0

0

0

0

hl1

0

0

0

0

0

0

hl2

0

0

0

0

0

0

1

0

0

0

0

0

0

0

1 0 C C C C 0 C C C C C 0 C C C C C 0 C C C C 0 C C C 0 C C C 0 C A

11

(C.2)

hlK

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where h ¼

c2 ru ðg
1Þ Q Dr. 2gp Af

It is easy to show that the combustion-driven acoustic waves and the flame surface element disturbances are satisfying the following system equation as

0 B B1 B B B B B B B B0 B B B0 B B B B0 B B B B0 B B B0 B B B B0 B @ 0

0

Mu
1 e Mu þ 1

ð

1

Þ


2ju zf
zu 2 cu 1
M u

0 ð

0

0

/

0

0

/

Þ


2ju zd
zf

e cd

2 1
M d

1

0

0

W11

W12

0

0

/

0

W21

W22

0

0

/

0

z1

0

1

0

/

0

z2

0

0

1

/

0

«

0

0

0

1

0

zK

0

0

0

/

C 0 1 0 0C C C0 C 1 B B 0 C C þ B C b C CB p u ðuÞ C B C B CB
C C CB b p d ðuÞ C B 0 C C 0 CB
C B B CB b C BQ CB p u ðuÞ C b C ðuÞ C B C C B C 0 CB b þ B C B cu pR2 C CB p d ðuÞ C C ¼ b B CB C C C b 0C x 1 ðuÞ C B CB C C B 0 B CB C B C B CB b C ðuÞ x C B C 0 CB 2 C B 0 C CB B C@ « C C A B C 0C B C b C B 0 C C x K ðuÞ @ C A 0C C 0 A 0

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