Unity maximum transient energy growth of heat-driven acoustic oscillations

Energy Conversion and Management 116 (2016) 1–10

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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman

Unity maximum transient energy growth of heat-driven acoustic oscillations Xinyan Li a, Dan Zhao a,b,⇑, Xinglin Yang b, Shuhui Wang c a

School of Mechanical and Aerospace Engineering, College of Engineering, Nanyang Technological University, Singapore 639798, Singapore School of Energy and Power Engineering, Jiangsu University of Science and Technology, Zhenjiang City, Mengxi Road 2, Jiangsu Province 212003, China c College of Mechanical and Vehicle Engineering, Hunan University, 410082 Changsa City, Hunan Province, China b

a r t i c l e

i n f o

Article history: Received 2 January 2016 Accepted 19 February 2016 Available online 5 March 2016 Keywords: Energy conversion Acoustical energy Thermal energy Transient energy growth Heat-to-sound Thermoacoustic oscillations

a b s t r a c t Transient energy growth of acoustic disturbances may trigger thermoacoustic instability in a non-normal thermoacoustic system. In this work, minimizing transient energy growth of heat-driven acoustic oscillations in an open-ended thermoacoustic system is considered. For this, a state-space thermoacoustic model with an acoustically compact heat source and distributed monopole-like actuators is developed. The heat source gives rise to the mean temperature jump, as experimentally measured. It is modeled with a modified King’s Law. Coupling the unsteady heat release model with a Galerkin series expansion of the acoustic waves present enables the time evolution of flow disturbances and acoustical energy to be calculated, thus providing a platform on which to gain insight on the system’s transient stability behaviors and the non-normal response of the system to the dynamic actuators. It is first shown that implementing a linear-quadratic regulator (LQR) leads to the system being asymptotically stabilized. However, the LQR optimization strategy fails in eliminating the transient growth. This finding is consistent with Pseudospectra analysis of the present system. In order to achieve unity maximum transient growth, a Lyapunov-based optimization strategy is systematically designed. It is found that this optimization strategy achieves both exponential decay of the acoustical energy and unity maximum transient growth. Furthermore, the sound pressure level is reduced by approximately 25 dB. In addition, the number of the actuators K is shown to be related to the mode number N as K ¼ N. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Self-sustained heat-driven acoustic oscillations are wanted in thermoacoustic heating or cooling systems [1–6,47]. However, such oscillations are undesirable in many combustion systems, such as aero-engine afterburners, rocket motors, ramjets, boilers and furnaces [9,10]. The oscillations are also known as thermoacoustic instability. It is generated by a dynamic interaction between unsteady heat release and flow disturbances present [11–15]. When unsteady heat is added in phase with the pressure oscillations [16], the energy of acoustic disturbances increases. Unsteady heat release is a monopole-like sound source to produce acoustic waves [17]. The sound waves [1,2,18] propagate within the system and partially reflect from boundaries to arrive back at the heat source due to impedance change [47]. And more unsteady heat release may be caused under certain conditions. This interac⇑ Corresponding author at: N3-02c-72, Division of Aerospace Engineering, 50 Nanyang Avenue, Singapore 639798, Singapore. E-mail address: [email protected] (D. Zhao). http://dx.doi.org/10.1016/j.enconman.2016.02.062 0196-8904/Ó 2016 Elsevier Ltd. All rights reserved.

tion between unsteady heat release and acoustic disturbances may result in large-amplitude and damaging self-sustained pressure oscillations [19] (also known as thermoacoustic instability). Such oscillations can become so intense that they cause overheating, flame flashback or blow-off, structural vibration and costly mission failure [11,20]. Thermoacoustic instability is currently one of the major challenges for land-based gas turbine and aero-engine manufacturers [21–23]. To mitigate thermoacoustic instability, there are two typical approaches, which break the coupling between the unsteady heat release and acoustic waves. One is passive and the other is active/feedback control [18]. Passive control [24] involves redesigning the ignition system or changing operating conditions or applying Helmholtz resonators [25] and acoustic liners to increase acoustic damping/loss. Applying such passive approach in practical engine systems is well-reviewed [24]. Passive approach is low-cost and simple. However, it cannot respond to changes in operating conditions due to the lack of dynamic actuators. However, feedback control approach can be implemented to various types engine systems and applied to a wide range of operation.

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X. Li et al. / Energy Conversion and Management 116 (2016) 1–10

Nomenclature C1 ; C2 c cp ; cv dw E G Gmax k K Lw L M N
p p0 Q_ a Q_ s Q Rk ; S k S Sak t T Tw um
; u0 u V

damping coefficients in Eq. (16) speed of sound in air, m=s heat capacity ratio at constant pressure and volume, kJ=kg K the diameter of the heated wire, m acoustical energy, J transient growth maximum transient growth kth eigenmode, heat conductivity, W=mK total length of the heated wire, m total length of the combustor, m mean flow Mach number number of eigenmodes, mean pressure, Pa pressure fluctuation, Pa actuation signal in Eq. (3), J/m3 unsteady heat release rate, KJ=s matrix defined in Eq. (26) actuation parameters in Eq. (5) cross-sectional area of the combustor, m2 cross-sectional area of the kth actuator, m2 time, s Temperature, K Heated wire temperature, K control input in Eq. (15) mean flow velocity and fluctuating part, m/s Lyapunov function defined in Eq. (31)

Typically, active control [19,26,27] is implemented in closedloop configuration. The active controller drives a dynamic actuator in response to a sensor measurement. One of the general actuation actions is to modulate the acoustic field by using a monopole-like sound source such as a loudspeaker [28]. Experimental investigation of using loudspeakers to mitigate thermoacoustic instabilities was conducted by Campos-Delgado et al. [29] on a non-premixed spray combustor via implementing linear quadratic Gaussian (LQG)/loop transfer recovery (LTR) and H1 loop-shaping. Comparison was then made by evaluating the performance of these controllers and the conventional phase-shifted controller. It is found that the phase-shifted controller resulted in worse attenuation of the pressure oscillations than LQG/LTR or H1 . Another general actuation is to modulate the unsteady heat release rate by using a secondary fuel injector [30,31]. Bernier et al. [31] applied a secondary fuel injector to stabilize a lean premixed pre-vaporized swirl-involved combustor. The combined transfer function of the burner and the actuation system were measured by using two methods. Insightful reviews of thermoacosutic instability and its feedback control were reported by McManus et al. [19]. The objective of traditional linear controllers applied on a thermoacoustic system is to make all the eigenmodes decay exponentially, i.e. to make the system stable under classical linear stability. However, when the thermoacoustic eigenmodes are nonorthogonal as typically found in a practical system, controlling the dominant eigenmode alone may cause other modes being excited due to the coupling effect [32]. Non-orthogonality of thermoacoustic eigenmodes (i.e., nonnormality) has received more attentions recently [33–35]. It has been shown that in a linearly stable but non-normal system (characterized with non-orthogonal eigenmodes) [8,34,35] there can be significant transient energy growth of small-amplitude acoustic

x xf

axial location along the combustor, m axial distance of the heated wires, m

Greek symbols an ; bn coefficients defined in Eq. (11) aak cross-sectional area ratio d Kronecker delta function g time-varying function as defined in Eq. (6) c the ratio of specific heat jn coefficient defined in Eq. (6) k conduction coefficient, W=mK l viscosity, kg=m s x oscillation frequency, rad=s w basis function as defined in Eq. (6) q air density, kg=m3 s time delay, s tak the kth actuation signal f damping coefficient as defined in Eq. (16) Subscript 1, pre-heating 2, after-heating a, actuator Superscript instantaneous value . time derivative  mean value 

perturbations. It has also been shown [33] that the nonorthogonality results from the presence of unsteady heat release or complex impedance boundary conditions. When the transient growth of acoustic disturbances is large enough, thermoacoustic instability might be triggered. Experimental measurement of the transient growth was performed on a lean-premixed gas turbine combustor in Cambridge [22]. Transient growth of acoustic disturbances cannot be predicted by such classical linear stability theory [28], since it provides information only about the long-term evolution of the eigenmodes [35]. Similarly, implementing conventional linear controllers might be associated with transient growth. Kulkarni et al. [35] might be the first group of researchers studying the effect of non-normality on the performance of such linear controllers on stabilizing a Rijke-type thermoacoustic system [36–40]. Theoretical analysis was conducted by expanding the acoustic perturbation via Galerkin technique. It was claimed by Kulkarni et al. [35] that the eigenfunction wðxÞ ¼ F ðjpxÞ representing the mode-shape is the same as that when there is no mean temperature. This finding is different from those obtained from the previous works [41,44]. The implementation of conventional linear controller such as pole-placement [35] or LQR [44] were shown to lead to the thermoacoustic system being non-normal. And these controllers failed in preventing the ‘nonlinear driving’ of low amplitude acoustic disturbances by controlling their transient growth. For this, so-call ‘transient growth controllers’ were developed [35,44]. It is based on the critical condition derived by Whidborne and McKernan [42]. Kulkarni et al. [35] showed that the pole-placement controller needs 6 actuators to minimize transient growth in a system with 3 modes. However, neither the ‘transient growth controller’ nor the relationship between the actuators and eigenmodes was systematically designed. Furthermore, the values of the distributed

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X. Li et al. / Energy Conversion and Management 116 (2016) 1–10

actuators response functions were set by the authors without justification. Zhao & Reyhanoglu [44] attempted to conduct a systematic study on minimizing transient growth in a thermoacoustic system with multiple modes present. It was shown that the number of actuators K is related to the mode number N as

2. Description of the thermoacoustic model

Heckl Approximation Matveev

800 700

Temperature [K]

K ¼ N 2 . However, for simplicity, the system they considered is in the absence of mean temperature gradient. The effect of mean temperature gradient on designing transient growth controller is not examined. Neither is minimizing the number of actuators. Lack of such investigations partially motivated the present work. In this work, a simplified Rijke-type thermoacoustic system with a mean temperature gradient and multiple distributed actuators present is considered. In Section 2, the nonlinear model equations are developed. And the acoustic perturbations are expanded by using Galerkin series. By linearizing the unsteady heat source term, system equations are formulated in state-space form for controller evaluation and pseudospectra analysis of the system’s nonnormality. In Section 3, the total energy of acoustic disturbances is defined as a measure to characterize the transient growth and to study non-normal behavior. To achieve strict dissipativity (unity maximum transient growth), a Lyapunov-based transient growth controller is systematically designed to minimize the maximum transient growth. This is described in Sect. 4. The relationship between the actuator number and the eigenmodes to be controlled is identified. The dynamic response of the thermoacoustic system before and after control is evaluated in Section 5. For comparison, a linear-quadratic-regulator (LQR) is also tested in the system with two eigenmodes. Finally, key findings are summarized in Section 6.

900

600 500 400 300

→ Heater

200 100

0

0.2

0.4

0.6

0.8

1

Axial position [m] Fig. 2. Measured mean temperature
¼ 1:01325  105 Pa and u
1 ¼ 0:5 m/s. p

varied

with

axial

position,

as

to downstream region, i.e. T 1 ¼ T 2 , the predicted mode frequencies and mode shapes are dramatically different. This may lead to wrong prediction of the characteristics of unstable modes. Note that we use subscript 1 and 2 to denote the regions upstream and downstream of the heat wires. The instantaneous flow variables in the thermoacoustic system consist of a mean and a fluctuating part as


ðxÞ þ u0 ðx; tÞ; qðx; tÞ ¼ q
þ p0 ðx; tÞ
ðxÞ þ q0 ðx; tÞ; pðx; tÞ ¼ p uðx; tÞ ¼ u ð1Þ

Following the previous works [33,34], we develop a simplified mathematical model of a thermoacoustic system with a mean temperature jump and multiple monopole-like actuators attached, as shown in Fig. 1. The length of the tube is L. Heated wires are axially placed at x ¼ xf and work as an acoustically compact heat source. It is modeled by a modified form of King’s law [34,40]. The presence
of the heat source causes the mean temperature T, the mean flow u
undergoing a variation from the region and gas density q 0 6 x 6 xf to xf < x 6 L. This has been experimental measured by Matveev [36,37], as shown in Fig. 2. The mean temperature variation is approximated by a sudden jump from upstream to downstream in the present work, as denoted by the dash line in Fig. 2. The approximation is good enough to characterize the thermoacoustic system in terms of eigenfrequencies and mode shapes. Fig. 3 illustrates the mode shapes at x1 and x2 . Comparing the predicted and measured mode shapes reveals that good agreement is observed. However, when the mean temperature is assumed to be unchanged from upstream

where u is the velocity, p is the pressure, q is the density. The overbar denotes a mean value and the prime describes a perturbation. The Mach number of the mean flow is assumed to be negligible. The working gas is assumed to be perfect, inviscid and non heatconducting. The thermoacoustic system with K distributed actuators [44–46] are governed by momentum and energy conservation equations as given as @u0 1 @p0 þ ¼0
@x @t q

ð2Þ

 cp 
@
XK 1 @ 2 p0 @ 2 p0 c  1 @
_ Q s dðx  xf Þ þ 2  2 ¼ 2 a t dðx  xak Þ k¼1 ak ak
c2 @t 2
c @t
c @t @x |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð3Þ

Q_ a

where c and
c denote the ratio of specific heat and sound speed. dðxÞ is the Dirac delta function which is used to define the location of the heat source and the actuators. Here Eq. (3) describes an ‘ideal’ energy conservation, since it does not involve with any damping/ loss. However, acoustic damping is generally involved in practical system. For this, the equation needs to be modified by introducing an additional ‘damping’ term as discussed later. The unsteady heat release rate Q_ s ðx; tÞ depends on the acoustic field. Modified King’s law [35,40] can be used to describe the unsteady heat transfer from a cylindrical hot wire in a horizontal flow as

Lw ðT w  TÞ pffiffiffi Q_ s ¼ S 3

Fig. 1. Schematic of closed-loop thermoacosutic system with multiple distributed actuators implemented.

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffi# qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" 

u1 þ u0 ðx; t  sÞ  u1
1 dw 2pkC m q 3  3

ð4Þ

where S is the duct cross-sectional area. T w , dw and Lw denote the temperature, diameter and length of the heated wires respectively.
1 is the oncoming mean C v is the specific heat at constant volume, u flow velocity. s is the time lag between the moment when the oncoming velocity perturbation acts and that when the corresponding heat release is felt. The time delay results from the processes of

4

X. Li et al. / Energy Conversion and Management 116 (2016) 1–10

Heckl → Heater

1

ω1=215

0.5

Matveev

→ Heater

1

ω1=193

|u′(ω1 , x)|

1

|p′(ω , x)|

Approximation

0.5

ω1=202 0

0

2

ω =427 2

0

0.5

0.5

1

→ Heater

1

ω2=430

0.5

0

0

1

→ Heater

1

|p′(ω , x)|

0.5

|u′(ω2, x)|

0

ω =384 2

0.5

0

1

0

0.5

Axial position (m)

1

Axial position (m)


¼ 1:01325  105 Pa and u
1 ¼ 0:5 m/s. Fig. 3. Comparison of normalized mode shapes of acoustic pressure p0 ðxf ; tÞ and velocity u0 ðxf ; tÞ, as L = 1.0 m, xf ¼ 0:25 m, p

surface heat transfer and subsequent thermal diffusion between the heated wires and the ambient fluid. In Eq. (3), aak denotes the ratio of the cross-sectional area Sak of the kth actuator [35,43] to the cross-sectional area S of the duct, i.e. aak ¼ Sak =S. tak is the diaphragm velocity of the kth actuator [32,35]. It is given as

tak ¼ Rk

0

0

u ðxak ; tÞ p ðxak ; tÞ þ Sk
1 u cM1 p


ð5Þ

By optimizing actuator parameters Rk and Sk via a properly designed controller, the acoustic field can be modified to lead the Rayleigh integral [16] to be negative. This is how an unstable thermoacoustic system being stabilized via feedback control approaches. The acoustic parameters such as pressure p0 and velocity u0 in Eqs. (2) and (3) can be expanded by using Galerkin series [33– 35] as a superposition of the duct natural modes as

p0 ðx; tÞ ¼

N X 1

jx n¼1 n n

g_ n ðtÞwn ðxÞ

ð6Þ

RL where j2n ¼ 0 w2n ðxÞ dx. xn denotes the eigenfrequency of thermoacoustic oscillations. The basis function wn ðxÞ is the eigen-solution of the classical acoustic wave equation without considering the heat source and the damping effect as given as

1 @ 2 p0 @ 2 p0  2 ¼0
c2 @t 2 @x

2 n 2
c2

d wn x þ wn ¼ 0; dx2

for x 2 ½0; xf ;

ð8Þ

for x 2 ½xþf ; L

ð9Þ

The open-ended boundary conditions and the pressure continuity and velocity jump conditions [33] across the heat source at x ¼ xf are given as

wn ð0Þ ¼ wn ðLÞ ¼ 0;

0 6 x 6 xf

ð11Þ

xþf 6 x 6 L

where an ¼ xn xf =
c1 and bn ¼ xn ðL  xf Þ=
c2 . It can be further shown that the basis function wn satisfies the condition of orthogonality, i.e.

Z



L

0

wn ðxÞwm ðxÞ dx ¼

0

m–n

j2n m ¼ n

The eigenfrequency xn is determined by the characteristic equation, i.e.


c2 sin am cos bn ¼ 
c1 sin bn cos an : By replacing the pressure oscillation p0 ðx; tÞ in the moment Eq. (2), the acoustic velocity perturbation u0 ðx; tÞ can be obtained as a superposition of the duct natural modes as

0

u ðx; tÞ ¼

8 N
 X g ðtÞ > xn x > n > > q
1
c1 jn cos
c1 <

for 0 6 x 6 xf

n¼1

N
 > X > gn ðtÞ xn ðLxÞ sin an > > :
c2 q
2
c2 jn cos sin bn

ð12Þ for

xþf

6x6L

n¼1

2

2

8
 > < sin x
cn x 1
 wn ðxÞ ¼ > sin an :  sin xn ðxLÞ
c2 sin bn

ð7Þ

Note that acoustic waves produced by the heat source will propagate along the combustor. They needs to satisfy with the homogenous acoustic wave equation, i.e. Eq. (7). Since the presence of the heat source leads to the mean temperature undergoing a jump from the upstream region and downstream of the heated wires, substituting Eq. (6) into Eq. (7) yields

d wn x2n þ 2 wn ¼ 0;
c1 dx2

With these conditions applied, the basis function wn ðxÞ can be shown as

x¼xþ

wn ðxÞjx¼xff ¼ 0;

x¼xþ 1 dwn  f ¼0 q
ðxÞ dx x¼x f

ð10Þ

Substituting Eq. (6) into Eq. (3) leads to N X 1 n¼1

2 d g_ n

jn xn

dt

2

!

þ x g_ n wn ðxÞ 2 n

 @
_ Q s ðx; tÞdðx  xf Þ @t ! K X @t
aak ak dðx  xak Þ þ cp @t k¼1

¼ ðc  1Þ

ð13Þ

Multiplying Eq. (13) with wm ðxÞ, and then integrating over the whole spatial domain and using the orthogonality property, gives rise to

 2 x m xf d gm xm _ 2 Q þ x g ¼ ð1  c Þ ðx ; tÞ sin s f m m
c1 jm dt 2  K X x x x
m aak tak ðxak ; tÞ sin m
ak  cp c1 jm k¼1

ð14Þ

5

X. Li et al. / Energy Conversion and Management 116 (2016) 1–10

When acoustic damping or boundary loss is considered, Eq. (14) can be modified as

 2 x m xf d gm dgm xm _ 2 þ um þ 2f x x g ¼ ð1  c Þ ðx ; tÞ sin þ Q m m m m
c1 dt jm s f dt2 ð15Þ where the damping coefficient fm is frequency-dependant and it is given as

fm ¼

rffiffiffiffiffiffiffiffi  1 xm x1 C1 þ C2 2p x1 xm

ð16Þ

where C1 and C2 are constant. um is defined as the control input and it is written as



K x X x x
m um ¼ cp a t ðx ; tÞ sin m
ak jm k¼1 ak ak ak c1

ð17Þ

Substituting Eqs. (6) and (12) into Eq. (5), the kth actuation signal can be shown as





N N R X gn ðtÞ xn xak Sk X g_ n ðtÞ  tak ¼
k cos


u n¼1 q1 c1 jn c1 cM1 p
n¼1 jn xn  xn xak

 sin


c1

um ¼

b k xm R

jm

k¼1

N N X xm X Anmk gn þ b Sk Bnmk g_ n n¼1

jm

ð19Þ

!

ð20Þ

n¼1

4. Design of Lyapunov-based transient growth controller To stabilize an unstable thermoacoustic system and to minimize transient growth of acoustical energy, a Lyapunov-based feedback control algorithm is designed. According to Eq. (15), the control input is proposed as

um ¼ ð2fm xm  Bm2 Þg_ m þ ðx2m  Bm1 Þgm þ ðc  1Þ  x m xf xm _  Q s ðxf ; tÞ sin
c1 jm ¼

g€ 1 þ 2f1 x1 g_ 1 þ x21 g1

Eðxf ; tÞ Eðxf ; 0Þ

t!1

2

3

ð1  cÞ xj11 Q_ s ðxf ; tÞ sin




x 1 xf

3


c1

Q

ð26Þ Matrix Q is proposed to be constructed as

q1s ¼

x1 ; for m ¼ 1; j1

Q NN ð21Þ

ð22Þ

qms ¼

xm Anms jm An1s

for m – 1

And it is given as x1 j1

x1 j1

6 6 x2 sinðx2
c1xa1 Þ 6 6 j2 sinðx1
cxa1 Þ 1 6 ¼6 .. 6 6 . 6 4 xN sinðxN
c1xa1 Þ jN sinð

x1 xa1
c 1



x2 xa2

x2 sinð


c1

x1 xa2

j2 sinð


c1

.. . jN sinð



Þ

.. xN xa2

xm sinð

Þ

Þ


c1

x1 xa2
c 1

Þ

.



Þ

3

x1 j1

7 7 7 j2 sinð
c Þ 7 1 7 7 .. 7 7 . xN xaN 7 xN sinð
c Þ 5 1 x2 sinð

ð27Þ

x2 xaN
c1

Þ

x1 xaN

jN sinð

x1 xaN
c 1

ð28Þ

Þ

Note that Q is invertible, if the locations of the actuators xak is properly chosen. The actuated thermoacoustic system Eq. (26) can be further simplified to

g€ m þ Bm2 g_ m þ Bm1 gm ¼ 0;

m ¼ 1; . . . ; N

ð29Þ

The analytical solution to Eq. (29) can be shown as

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

And the maximum transient growth is given as

Gmax  maxfGðxf ; tÞg

ð25Þ

6
7 7 6 7 6 € 2 þ 2f2 x2 g_ 2 þ x22 g2 7 6 ð1  cÞ x2 Q_ s ðxf ; tÞ sin x
2 xf 7 6 g j2 c1 7 6 7 6 7¼6 7 6 .. 7 6 7 6 . . .. 5 6 7 4 5 4
 x N xf g€ N þ 2fN xN g_ N þ x2N gN xN _ ð1  cÞ jN Q s ðxf ; tÞ sin
c1 2 32 3 q11 q12    q1N I1 6q 76I 7 q    q 22 2N 7 6 2 7 6 21 76 7 þ6 .. .. .. 7 6 .. 7 6 .. 4 . . . . 54 . 5

2

The transient acoustical energy growth is defined as the ratio of acoustical energy at instant moment to the initial status, i.e.

tP0

m ¼ 1;    ; N

for

s¼1

xf

Gðxf ; tÞ ¼ max

N X qms I s

IN qN1 qN2    qNN |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

When the ‘triggering’ and transient stability behavior of a given thermoacoustic system are interested, the effect of non-normality must be considered and combined in the design of a feedback controller. To characterize the transient growth of the acoustic disturbances in an 1D thermoacoustic system, an energy measure is needed. Here we choose the total acoustic energy Eðxf ; tÞ per unit cross-sectional area as the measure. The definition of the total acoustical energy is given as:

 02
1 u02 p1 ðx; tÞ q 1 ðx; tÞ dx Eðxf ; tÞ ¼ þ
2cp 2 0 Z L  02
2 u02 p2 ðx; tÞ q 2 ðx; tÞ þ dx þ
2 2 c p xf N  1 X g_ 2m 2 ¼ g þ m
2cp x2m m¼1

It is worth noting that the designed Lyapunov function is based on Eq. (24) to achieve unity maximum transient growth. This is one of the main differences between the present and previous works.

2

3. Definition of transient acoustical energy growth

Z

ð24Þ

The actuated system equation can then be expressed in a matrix form as

then the control input as involed in Eq. (17) can be rewritten as K X

dE 60 dt

ð18Þ

bk; b If we define R S k , Anmk and Bnmk as

 

b k ¼  aak Rk ; Anmk ¼ cos xn xak sin xm xak R jn ;
c1
c1 M1  

aak Sk xn xak xm xak b sin ; Bnmk ¼ sin jn xn Sk ¼

c1 c M1 1

If Gmax ¼ 1, then there is no transient growth of the acoustical energy. This is also known as strict dissipativity or unity maximum transient growth. In physics, it describes that the energy of acoustic disturbances is decaying, i.e.

ð23Þ



gm ¼ lm e

Bm2 þ

B

m2 2

þ4Bm1

t

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

þ #m e



Bm2 

B

m2 2

þ4Bm1

t

;

for m ¼ 1; . . . ; N ð30Þ

X. Li et al. / Energy Conversion and Management 116 (2016) 1–10

lm ; #m are coefficients depending on the initial conditions. The control parameters Bm1 and Bm2 can be optimized to ensure the monotonic decrease of the acoustic transient energy growth. This is achievable by designing a proper Lyapunov function V as given as

N  1 X g_ 2 2 g þ m m
2cp Bm1

Differentiating the proposed Lyapunov function with respect to time leads to

b k xm R

k¼1

jm

N N X xm X Anmk gn þ b Sk Bnmk g_ n

jm

n¼1

!

n¼1

¼

N X qms I s

k ¼ 1; . . . ; N

ð34Þ

n¼1

b k and b where R S k can be obtained as:

bk ¼
R PN

PN

n¼1 An1k

n¼1 An1k

b Sk ¼


PN

n¼1 An1k

gn

2

þ

PN

gn


P N

n¼1 Bn1k

g_ n
P N

g_ n

n¼1 Bn1k

gn

2

þ

n¼1 Bn1k

g_ n

0

−0.01 −0.02 −0.03 −0.04 −30 −200

−100

0

p′(t)

100

200

−0.05 −0.3 −0.2 −0.1

0

0.1 0.2 0.3

u′(t)

Fig. 4. Phase diagram of acoustic pressure p0 ðxf ; tÞ and velocity u0 ðxf ; tÞ, as the actuators are not actuated (i.e. open-loop), N = 2, L = 1.0 m, xf ¼ 0:65 m, s ¼ 4 ms, Lw ¼ 1 m, T w ¼ 1680 K, dw ¼ 5  104 m, S ¼ 1:56  103 m2, k ¼ 0:0328 W/m K,
¼ 1:01325  105 Pa and u
1 ¼ 0:5 m/s. C v ¼ 719, c ¼ 1:4, c1 ¼ 0:3, c2 ¼ 0:03, p

ð33Þ

s¼1

N N X X An1k gn þ b S k Bn1k g_ n ¼ I k ; n¼1

0.01 0

−20

By replacing qmk with Eq. (27), Eq. (33) can be simplified to

bk R

0.02

10

ð32Þ

The non-positive sign of Eq. (32) guarantees that the system is asymptotically stable by Lasalle’s principle. If we set Bm1 ¼ x2m , the proposed Lyapunov function V can be converted into the acoustical energy expression as defined in Eq. (21). The nonpositive gradient of V ensures strict dissipativity (unity maximum transient growth) to be achieved. Substituting Eq. (25) into Eq. (17) leads to

(b)

0.03

−10

N dV 1 X Bm2 2 ¼ g_  0
dt cp m¼1 Bm1 m

K X

20

ð31Þ

m¼1

0.05

(a)

0.04

p˙ (t)

V¼

30

u˙ (t)

6

2 I k ;

2 I k

ð35Þ

Therefore to stabilize N unstable modes, only K ¼ N actuators are needed. The number of actuators is at least 50% less than that required in the previous works of Kulkarni et al. [35] and Zhao & Reyhanoglu [44]. This is another key contribution of the present work in comparison with the previous ones [35,44].

dospectra analysis are performed. The geometry and flow conditions of the thermoacousitc system is summarized in table below (see Table 1). Fig. 4 shows phase plots of acoustic pressure and velocity fluctuations, when the actuators are not actuated. It can be seen that initial small-amplitude flow disturbances grow into large-amplitude limit cycle oscillations, as denoted by the outer circle. This means that the open-loop system is unstable and there are unstable modes with real part of eigenfrequencies protruding in the right half plane. This is confirmed by our pseudospectra analysis, as shown in Fig. 5(a). It is apparent that there is one unstable mode. The dash line describes the numerical abscissa. Part of it protrudes into the right half plane. This indicates that the system is non-normal and transient growth [47,48] occurs. Fig. 6 shows the transient growth behavior the system. It can be seen that the maximum transient growth Gmax ! 1, as t ! 1. The finding from the transient growth analysis is consistent with our Pseudospectra analysis, i.e. the open-loop system is unstable.

5. Results and discussion 5.2. Closed-loop results 5.1. Open-loop results Before the designed transient growth controller is evaluated to stabilize the thermoacoustic system, its characteristics in openloop configuration needs to be studied. For this, modal and Pseu-

Table 1 The geometry and flow conditions of the thermoacoustic system. Parameter

Value

Mean temperature in upstream region T 1 Mean temperature in downstream region T 2 Heated wire diameter dw

300 K 400 K

Heated wire temperature T w Cross-sectional area of the duct S Specific heat at constant volume C t Specific heat ratio c Conductivity coefficient k Total length of the duct L
1 Mean density in upstream region q
Mean pressure p
1 Inlet mean velocity u

5  104 m 1680 K 1:56  103 m 719 J/Kg K 1.4 0.0328 W/m K 1m 1.17 kg/m3 1:01  105 Pa 0.5 m/s

Now the designed Lyapunov-based controller is applied to minimize the transient growth and stabilize the system with two modes. For comparison, a conventional linear controller, i.e. linear quadratic regulator (LQR) is also implemented. The performances of these controllers are evaluated and compared in terms of minimizing transient growth and sound level reduction. It is worth noting that the performance of the Lyapunov-based controller is related to the system modeling. In this work, we implement it on an unstable thermoacoustic system with a temperature jump configuration. Such configuration agrees well with the experimental measurements. Thus the modeling is more accurate and the control performance is better as expected. However, the Lyapunovbased controller can be easily modified to be implemented on a thermoacoustic system with an invariant mean temperature configuration by following the same procedure. Fig. 7 shows phase plots of acoustic pressure and velocity, when LQR and Lyapunov-based controllers are implemented. It can be seen that the limit cycle oscillations are quickly dampened. Furthermore, LQR controller works more rapidly and effectively in minimizing thermoacoustic oscillations. This might be due to the

7

X. Li et al. / Energy Conversion and Management 116 (2016) 1–10

(b)

(a) 10

5

5

0

0

−5

−5

−10

−10

ℑ{z}

10

−10

−5

0

5

10

−10

−5

0

5

10

ℜ{z}

10

(c)

log ξ 10 −1

ℑ{z}

5

−2

0

−5

−3

−10 −10

−5

0

ℜ{z}

5

10

−4

Fig. 5. Pseudospectra of the thermoacoustic system in open- and closed-loop configurations. The contour values are calculated by using log10 n ¼ log10 kðzI  AÞ1 k and the
1 ¼ 1:01325  105 Pa, u
1 ¼ 0:5 m/s, B12 ¼ 100, B22 ¼ 200. (a) open-loop, (b) maximum value corresponds to eigenvalues, as N = 2, xf ¼ 0:65 m, s ¼ 4 ms, c1 ¼ 0:3, c2 ¼ 0:03, p closed-loop with LQR controller implemented. (c) closed-loop with Lyapunov-based controller applied.

10

Lyapunov−based control

Open−loop Unity Transient Growth

6

20

LQR

0.04 limit cycle

→∞

max

p˙ (t)

10

2

G(t)=1

10

0

0.03

10

0.02

5

0.01

u˙ (t)

G

4

G (t)

10

15

0

0

−5

−0.01

−10

−0.02

−15

−0.03 limit cycle

0

0.005

0.01

0.015

0.02

0.025

Time (s)

−20 −200

−100

0

p′(t) Fig. 6. Time evolution of transient growth of acoustical energy in the open-loop thermoacoustic system, as N = 2, L = 1.0 m, xf ¼ 0:65 m, s ¼ 4 ms, c1 ¼ 0:3,
¼ 1:01325  105 Pa and u
1 ¼ 0:5 m/s. c2 ¼ 0:03, p

assumption that all the state vectors are known to the controller. However, implementing the LQR controller is associated with transient growth of acoustical energy, as denoted by the dash line in Fig. 8. The Lyapunov-based controller is found to be able to achieve unity maximum transient growth, as denoted by the cross line in Fig. 8. Pseudospectra analysis of the actuated system, as shown in Fig. 5(b) and (c) reveals that both controllers lead to all eigenmode being stable (negative real parts). Furthermore, implementing Lyapunov-based controller is associated with no part of numerical abscissa protruding in the right half plane. This explain how strict dissipativity is achieved. However, part of numerical abscissa line associated with LQR controller application is protruding in the right half plane. The performance of these controllers are further evaluated by assessing the sound pressure level before and after control. Fig. 9

100

200

−0.04 −0.4

−0.2

0

0.2

0.4

u′ (t)

Fig. 7. Phase diagrams of pressure p0 ðtÞ and velocity oscillation u0 ðtÞ in an actuated thermoacoustic system, as N = 2, L = 1.0 m, xf ¼ 0:65 m, s ¼ 4 ms, Lw ¼ 1 m, T w ¼ 1680 K, dw ¼ 5  104 m, S ¼ 1:56  103 m2, k ¼ 0:0328 W/m K, C v ¼ 719,
¼ 1:01325  105 Pa
1 ¼ 0:5 m/s, c ¼ 1:4, c1 ¼ 0:3, c2 ¼ 0:03, p and u B12 ¼ 100; B22 ¼ 200.

shows that the sound pressure level is reduced by approximately 25 dB, as the Lyapunov-based controller is applied. This is comparable with that of LQR controller implementation. In order to gain insight on the eigenmodes interaction, acoustical energy analysis is conducted. Fig. 10 illustrates time evolution of acoustical energy at x1 and x2 . It can be seen the acoustical energy of both modes (Fig. 10(a)) are ‘saturated’, when there are no controllers implemented. However, the magnitude of E_ 1 is comparable with E_ s ðtÞ (see Fig. 10(a) and (c)) but much greater than E_ 2 . This indicates that the generated limit cycle oscillations are dominated by the lower frequency mode. When the actuators are turned on, the acoustical energy

X. Li et al. / Energy Conversion and Management 116 (2016) 1–10

Lyapunov−based controller LQR

1

E˙ 1 (t)

1.2 1

G(t)=1.0 G

>1.0

(a)

0

G

=1.0

2

max

E˙ 2 (t)

0.2

0 0

0.005

0.01

0.015

0.02

x 10

0.02

Open−loop

Lyapunov−based control

2

5

0

5

x 10

−3

1 (e)

0

0

−1

−1

SPL (dB) 90

6

0

(a)

0.02

0.04

0.06

0

0.02

400

600

6

Control on →

0.06

0

800

−6 0

0

0.2

0.4

0.6

−6

0

0.2

R˙ 1

2

0.4

0. 6

Time (S)

Time (s)

Fig. 9. Variation of sound pressure level before and after the LQR and Lyapunovbased controllers are implemented.

Control on→

−3

−3

Frequency (Hz)

(d)

3

5

(b) S˙ 1

200

0.04

Es(t)

va2

v

a1

3

0

x 10

2

Fig. 10. Phase diagrams of acoustical energy E1 ðx1 Þ and E2 ðx2 Þ, as N = 2, L = 1.0 m, xf ¼ 0:65 m, s ¼ 4 ms, Lw ¼ 1 m, T w ¼ 1680 K, dw ¼ 5  104 m, S ¼ 1:56  103 m2,
¼ 1:01325  105 Pa and k ¼ 0:0328 W/m K, C v ¼ 719, c ¼ 1:4, c1 ¼ 0:3, c2 ¼ 0:03, p
1 ¼ 0:5 m/s, B12 ¼ 100, B22 ¼ 200. Note that Es ¼ E1 þ E2 denotes the total acousu tical energy. (a)–(c) for open-loop system. (d)–(f) for closed-loop system with Lyapunov-based controller implemented.

Δ≈ 31.8 dB

120

10 −4

E (t)

Es(t) Δ≈ 24.5 dB

0.06

1

−4

x 10

E˙ s (t)

E˙ s (t)

x 10 (c)

0.04

E (t)

−4

0

10

E2(t)

−3

0.02

x 10 (f)

−2 0

150

60

0

0.06

0

1

LQR

0.04

E1(t)

−4

Time Fig. 8. Comparison of the transient growth of acoustical energy in the actuated thermoacoustic system with LQR and Lyapunov-based controller implemented, as
1 ¼ 1:01325  105 Pa, u
1 ¼ 0:5 m/s, N = 2, xf ¼ 0:65 m, s ¼ 4 ms, c1 ¼ 0:3, c2 ¼ 0:03, p B12 ¼ 100, B22 ¼ 200.

0

(b)

−2

0.025

−3

−1 0

0.6 0.4

x 10 1 (d)

−1

max

−3

E˙ 2 (t)

G (t)

0.8

x 10

E˙ 1 (t)

8

0

(e)

0

at both eigenfrequencies are simultaneously minimized, as shown in Fig. 10(d)–(f). To minimize transient growth of acoustical energy in the thermoacoustic system with two modes, two actuators are driven by the proposed Lyapunov-based controller. Fig. 11(a) and (d) shows time evolution of the actuation signals to obtain the desired system responses as shown in Fig. 7. It can be seen that initial large-amplitude actuation signals are needed to stabilize the system. However, actuation signals decay quickly. More detailed information about the control parameters are summarized in phase plots of R1 ; R2 ; S1 and S2 as shown in Fig. 11(b), (c), (e) and (f) respectively. These plots indicate that both modes are stabilized simultaneously. It is worth noting that transient growth control of acoustical energy is achieved via properly designing a Lyapunov-based function in the present work. This is different from the previous works

Fig. 11. (a) and (d) time evolution of actuation signal ta1 and ta2 respectively; (b), (c), (e) and (f) Phase diagrams of actuator parameters R1 ; R2 , S1 and S2 as xa1 ¼ 0:25 m, xa2 ¼ 0:35 m. aa1 ¼ aa2 ¼ 0:01.

[35,44], of which a critical condition L þ LT derived by Whidborne and McKernan [42] needs to be satisfied. Here, L is the operator of the linearized governing equation. In addition, the number of actuators needed is greatly reduced in comparison with that suggested in the previous works [35,44]. Finally, the developed thermoacoustic model simulates a real thermoacoustic system by considering the mean temperature effect. And the dominant two modes are

simultaneously stabilized. When unstable modes to be controlled are increased, the number of actuators will linearly increased. This means that it will be challenging to ensure that all actuators are not placed at pressure nodes. Thus there is a compromise between the modes to be controlled [49,50] and the actuators to be

−2 −10

0

R1

−5 −10

5

(c)

0

−5 −10

0

R

2

0

10

20

10

20

S1

S˙ 2

R˙ 2

5

10

10

(f)

0

−5 −10

0

S

2

X. Li et al. / Energy Conversion and Management 116 (2016) 1–10

implemented, especially to achieve unity maximum transient growth, i.e. strict dissipativity. Lyapunov-based control strategy has great potential to be applied in a practical system. However, its performance depends strongly on the system modeling. Large modeling error and uncertainty can lead to a control failure. In order to minimize the modeling error and uncertainty and to achieve better control performance, a robust system state estimator/observer may be needed.

6. Conclusions Transient energy growth may occur in a non-normal thermoacoustic system. When the transient growth is large enough, it might trigger the system to become unstable. Such transient energy growth is unwanted in gas turbines and aeroengines. In this work, Lyapunov-based minimizing transient growth of heat-driven acoustic oscillations is studied in an open-ended thermoacoustic system with multiple modes and a mean temperature jump across the heat source. An acoustically compact heated wire is confined in the system and works as an unsteady heat source. It is described by using a modified form of King’s law. Coupling the unsteady heat release model with a Galerkin series expansion of acoustic fluctuations provides a platform on which to gain insights on the onset of limit cycle and evaluating the performance of feedback controllers. To study the non-normality, Pseudospectra and transient energy growth analysis are conducted to the thermoacoustic system with and without actuators actuated. To mitigate thermoacoustic oscillations produced in the system with two eigenmodes, a linear-quadratic regulator (LQR) is implemented first to tune the actuators. The system becomes asymptotically stable. However, it fails in eliminating acoustical energy transient growth, which is potentially dangerous in triggering thermoacoustic instability. In order to achieve strict dissipativity (i.e. unity maximum transient energy growth, i.e., Gmax ¼ 1:0), a Lyapunov-based optimization strategy is systematically designed and tested. The number K of actuators is shown to be equivalent to the number N of unstable modes to be stabilized. Approximately 50% less actuators than that suggested in recent works are needed to stabilize an unstable system with N P 2 modes. Comparison is then made between the performance of the LQR and that of the Lyapunov-based transient growth optimization strategy. It is shown that the Lyapunov-based strategy can lead to the acoustical energy being decreased exponentially. Sound pressure level is reduced by approximately 25 dB. This finding is consistent with our Pseudospectra analysis.

Acknowledgements The work is supported by Singapore Ministry of Education AcRF-Tier1 Grant RG91/13-M4011228 and the National Natural Science Foundation of China under Grant No. 51506079 and Hunan Province Natural Science Foundation under Grant No. 12JJB008. This financial support is gratefully acknowledged. We would like to thank Prof. Sebastien Candel, Ecole Cnetrale Paris and Prof. Ann Karagozian UCLA, USA for helpful discussion.

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