Swaminathan N., Bray K.N.C. (eds.) Turbulent Premixed Flames

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3.2 Control Strategies for Combustion Instabilities 183

(t)

Figure 3.24. A Helmholtz resonator connected to the side of a duct.

We consider a Helmholtz resonator connected to a duct of cross-sectional area

. The Helmholtz resonator has a neck of cross-sectional area S

and length  and

a bulb of volume V . There are incoming sound waves in the duct from the left and

right of the Helmholtz resonator of strengths A and D, and reﬂected and transmitted

waves B and C,asshowninFig.3.24.

We denote the pressure perturbation at the open end of the neck of the

Helmholtz resonator by p



(t) and the mass ﬂow into the Helmholtz resonator by

(t). Although there may be a mean ﬂow through the neck of the Helmholtz res-

onator, we assume that any mean ﬂow in the main duct is negligible. p



(t) and ρ



(t)

denote the pressure and density perturbations within the Helmholtz resonator and

are assumed to be independent of position because the bulb of the resonator is

compact (i.e., its size is small in comparison with

c/ω) at the frequencies of interest.

The pressure perturbations match at the neck of the Helmholtz resonator:



(t) = (A +B)e

iωt

= (C + D)e

iωt

. (3.48)

Conservation of mass for the dotted control volume gives

(A − B)e

iωt

= S

(C − D)e

iωt

+ ˆ



(t). (3.49)

As mass ﬂows into the cavity, the density there must rise



= V

dρ



Associated with this density change is a corresponding change in p



(t), the

pressure in the cavity. For isentropic perturbations, p



(t) =



(t), so that



, (3.50)

i.e., for harmonic disturbances of frequency ω,



Viω

. (3.51)

When there is a mean velocity through the neck of the Helmholtz resonator, the

ﬂuctuations in mass ﬂow are related to the pressure difference across the neck by

the conductivity K



, which is independent of amplitude but depends on the length

of the neck and on the Strouhal number of mean ﬂow through the neck. We write

184 Combustion Instabilities

this in the form



= K





− p



). (3.52)

When the neck has negligible length, the entrance to the Helmholtz resonator

just being an circular aperture in a thin combustor wall, we can use the previous

result: K



≈ 2a(η + iδ), where η and δ are the functions plotted in Fig. 3.17 in terms

of the Strouhal number ωa/

u, a is the radius of the aperture, and u is the mean

velocity through it. Then substitution for p



(t)fromEq.(3.51)into(3.52) gives for

disturbances of frequency ω:



2aω(−iη + δ)

− 2a(η + iδ)



. (3.53)

Large mass ﬂow ﬂuctuations are possible near the resonant frequency ω

2a(η + iδ)/V .

We saw in energy balance equation (3.38) that the rate of acoustic energy loss at

a bounding surface S is equal to





u · dS. Hence the rate of absorption of acoustic

energy by a Helmholtz resonator is equal to







2aω(−iη + δ)

− 2a(η + iδ)



. (3.54)

This is large near the resonant frequency of the Helmholtz resonator provided



(t) has a non-negligible amplitude.

Substitution from Eq. (3.53)into(3.48) and (3.49) gives B and C in terms of A

and D. The fraction of incident energy absorbed  then follows from

 = 1 −

. (3.55)

Extension to a longer neck is straightforward [96, 97]: Then the pressure drop

in Eq. (3.52) has an additional term (/S

)dm



/dt. This describes the pressure force

needed to overcome the inertia of the ﬂuid in the neck.

Figure 3.25 shows a comparison between theory and experiment. In general, the

theory works well. The peak absorption near resonance is clear. Enhancement of

this peak absorption can be obtained by ‘tuning’ the mean ﬂow appropriately and,

of course, the Helmholtz resonator should be placed near a pressure antinode, that

is, at a position in the duct near a maximum of p



1rms

If there is no mean ﬂow through the neck of the Helmholtz resonator, the

pressure drop across the neck is related to the instantaneous velocity through the

neck u



(t)[= m



(t)/(ρS

)] by the nonlinear relationship



− p



= 









, (3.56)

where 

is the effective length of the ﬂuid accelerated in the neck and c

is the

discharge coefﬁcient [99]. A numerical integration of Eqs. (3.50) and (3.56) can

be used to determine u



(t) for given neck pressure perturbation p



(t) and hence

to determine the waves in the duct from Eqs. (3.48) and (3.49). The absorption

coefﬁcient then follows from Eq. (3.55). The absorption coefﬁcient depends now on

the amplitude of p



(t), so-called ‘nonlinear absorption’. Sample results are shown

3.2 Control Strategies for Combustion Instabilities 185

0.2

0.3

0.4

0.5

Absorption coefficient

0.6

0.7

0.8

0.9

0.1

0 50 100 150 120

Frequency (Hz)

250 300 350 400 450 500

Figure 3.25. Comparison between theory and experiment for a mean ﬂow of Mach number

0.02 through a perforated plate (from [98]).

in Fig. 3.26. In this case, a pressure node at the neck of the Helmholtz resonator

for a frequency near 280 Hz results in a minimum absorption coefﬁcient there.

Of course, even when there is a mean ﬂow through an aperture, the absorption

coefﬁcient depends on the amplitude of the pressure perturbation at the neck of

the resonator at high-enough sound amplitude. This is illustrated in Fig. 3.27, which

shows absorption from a single aperture. The mean ﬂow velocity through the aperture

is characterised by the mean pressure drop 

p across it. At low sound levels, the

absorption coefﬁcient is ‘linear’, being independent of sound amplitude when there

is a mean ﬂow. At high-enough sound amplitudes, the absorption coefﬁcient becomes

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Absorption coefficient

Frequency (Hz)

0 50 100 150 200 250 300 350 400 450 500

−0.1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Absorption coefficient

Frequency (Hz)

0 50 100 150 200 250 300 350 400 450 500

−0.1

(

)(

)

Figure 3.26. (a) Comparison between the measured and predicted absorption coefﬁcients as

a function of frequency for a sound pressure level (SPL) of 165 dB at the neck of a Helmholtz

resonator (from [82]). (b) Predicted absorption coefﬁcient as a function of frequency for

various SPLs at the neck of a Helmholtz resonator ranging from 120 to 185 dB in increments

of 5 dB. The lowest curve is for 120 dB (from [82]).

186 Combustion Instabilities

0.6

0.5

0.4

0.3

0.2

Absorption coefficient

Pressure amplitude (dB)

0.1

0.0

115 120 125 130 135 140 145

Δp = 500 Pa

Δp = 250 Pa

No mean flow

Figure 3.27. Nonlinearity in the absorption coefﬁcient (from [100]).

nonlinear (implying that the coefﬁcient depends on the amplitude), the sound level

for the onset of nonlinearity increasing with the mean ﬂow velocity.

The sound absorption achieved by a Helmholtz resonator attached to a com-

bustor will depend on its location and on the acoustic mode shape. An example

of this is clear in Fig. 3.26, where the local minimum near 280 Hz is due to the

Helmholtz resonator then being near to a pressure node for this axial acoustic mode.

The annular combustors of aeroengine gas turbines have an axial length that is short

in comparison with their circumference. This means that the lowest resonance fre-

quency of the combustor is often associated with the n = 1 circumferential mode.

When a combustion instability occurs, it is then close to the resonant frequency of

this mode, and to control the instability one needs to understand the effect of one

or more Helmholtz resonators on this mode of oscillation. This can be conveniently

investigated through a network analysis, with the resonator being treated as a side

branch and appropriate conservation conditions applied [101]. The unsteady mass

ﬂow from the Helmholtz resonator is again assumed to be related to the pressure

at its neck through Eq. ( 3.53) for the case with a mean ﬂow. The effectiveness of a

Helmholtz resonator as an acoustic absorber depends on the overall system.

We noted that, for a plane wave, the Helmholtz resonator is ineffective if placed

near a pressure node. For a circumferential travelling wave in a cylindrically sym-

metric geometry, a single Helmholtz resonator does not absorb energy; instead it

just reﬂects an equal and opposite wave, leading to a standing mode with a pressure

node at the neck of the Helmholtz resonator (see Fig. 3.28). There is no absorption of

energy. Indeed, a single Helmholtz resonator may have a detrimental effect because,

at some locations around the annulus, the incident and reﬂected waves will be in

phase, doubling the amplitude of the pressure perturbation from that in the incident

wave.

Multiple resonators are needed to absorb energy in circumferentially travelling

waves. Stow and Dowling [102] investigated the absorption that is due to two or more

Helmholtz resonators located around the circumference of an annular combustor

and give an expression for their optimal locations: If N Helmholtz resonators are to

be used to absorb sound with circumferential wave number n, the best conﬁguration

3.2 Control Strategies for Combustion Instabilities 187

pressure node

reflected

wave

incident

wave

Figure 3.28. A single Helmholtz resonator reﬂects a travelling circumferential wave, giving

a standing wave with a pressure node at the neck of the resonator [102].

is to distribute the Helmholtz resonators evenly over half the wavelength of the

mode. Because of the symmetry, moving the position of any Helmholtz resonator by

multiples of half a wavelength has no effect. This leads to optimum placement of the

jth Helmholtz resonator at



j − 1

+ M



for j = 1,...,N, (3.57)

where M is a positive integer.

The passive techniques described in this subsection can work well at modest

and high frequencies, provided that the Helmholtz resonator or liner is tuned to

give optimal absorption at that frequency. However, the frequency of a combustion

instability varies with the operating condition, and so interest has turned to active

ways of keeping a passive absorption operating at peak absorption.

3.2.3 Tuned Passive Control

Passive control commonly takes the form of adding acoustic dampers to the com-

bustor. The performance of these tends to be highly frequency dependent; effective

damping occurs over only a narrow bandwidth, the frequency of which depends on

the geometry of the acoustic damper. Tuned passive control involves using a control

system and actuator to vary the geometry of acoustic dampers in time so that they

can respond to changes in combustor instability frequency.

For example, Helmholtz resonators (see Subsection 3.2.2) are most effective as

acoustic dampers when their resonant frequency is close to the instability frequency

present in the combustion chamber. Using the ‘thick-neck’ form for the pressure

drop across the neck, we can approximate their resonant frequency ω

by ω

eff

where

c is the speed of sound, S

is the neck area, V is the cavity volume, and l

eff

188 Combustion Instabilities

is the effective neck length. If any of the geometry parameters, S

, V ,orl

eff

,are

altered, a means of responding to frequency changes in the combustion system

is provided. The ability to tune the damping overcomes the main disadvantage

of traditional passive control. Moreover, actuation only needs to be on the time

scale of the changes in operating condition, which is typically much slower than the

time scale of the instability. The required actuator bandwidth is therefore small and

the durability requirements are reduced, representing two major advantages over

active control.

ONLINE IDENTIFICATION OF FREQUENCIES AND AMPLITUDES. To implement tuned pas-

sive control in practice, real-time tracking of the combustor instability frequency and

amplitude level are required. Note that, when the acoustic damping of a passive con-

troller is altered, the acoustic eigenfrequencies and mode shapes of the combustor

change, causing the instability frequency to shift. The consequence of this is that

tuning an acoustic damper needs to be an iterative process, which will typically in-

volve two tuning stages. The ﬁrst implements a ‘guess’ of the optimum geometry,

based on an initial measurement of the instability frequency. The second involves

an online search for the damper geometry that minimises the amplitude level and

indirectly takes account of all the instability frequency changes that occur along the

way [103, 104].

To rapidly track the instability frequency and amplitude in real time, frequency-

domain [fast Fourier transform (FFT)-based] Techniques can be used. However,

these provide fairly slow tracking capabilities, particularly at low frequencies. Time-

domain integration techniques provide a faster alternative and are based on the

premise that integrating the product of a s ignal and a sinusoid extracts the signal

component at the frequency of the sinusoid.

Given a measured pressure signal p(t) with time derivative ˙p(t), the aim is to

ﬁnd the instability frequency ω and the (sine and cosine) amplitude components,

a and b, such that a sin ωt +b cos ωt is a good approximation of p(t). The rules for

updating the online estimates of ω, a and b, are shown below. Derivations can be

found in [104, 105];

n denotes the nth iteration and α is a relaxation coefﬁcient:

n+1

= (1 − α)ω

+ α



p(t) − p(t −2π/ω

)

a(t) cos ω

t +b(t)sinω



, (3.58)

b(t) =

cos ω

t[˙p (t) − ˙p (t −2π/ω

)] + ω

sin ω

t[p(t) − p (t −2π/ω

)]

π(ω

− ω

n+1

)

, (3.59)

a(t) =

sin ω

t[˙p (t) − ˙p (t −2π/ω

)] − ω

cos ω

t[p(t) − p (t −2π/ω

)]

π(ω

− ω

n+1

)

. (3.60)

It is possible to extend these time-domain algorithms to pressure signals contain-

ing multiple frequency components. Such a pressure signal might occur in a combus-

tor exhibiting multiple unstable modes [104, 105]. Algorithms based on time-domain

tracking are used in several of the examples given in the following subsection.

Note that equation 4b in [104] should have the sine and cosine interchanged.

3.2 Control Strategies for Combustion Instabilities 189

combustion

system

sensor

controller

Sensor signal

(feedback signal)

actuator

Controller

signal

Actuator

signal

Figure 3.29. Generic system for feedback control of combustion oscillations.

EXAMPLES OF TUNED PASSIVE CONTROL. Returning to the example of the Helmholtz

resonator, which is the passive damper on which most tuning investigations have

focussed, it is most practical to vary either the cavity volume [103, 106, 107]orthe

neck area [104]. The cavity volume is normally varied by use of a back-piece plunger,

whereas the neck area can be varied by use of an ‘iris’-type valve, which operates

like a camera lens. Both forms of actuation have been demonstrated to effectively

stabilise combustion instabilities over varying operating conditions.

A rather different example of tuned passive control again involves Helmholtz

resonators, but this time considers those whose volume is continually oscillated

with the frequency present in the combustion system (i.e., the instability fre-

quency)

[108, 109]. Whether this volume oscillation enhances or reduces the acoustic

damping depends on the phase of the oscillation. When an online tuning of the oscil-

lation phase is performed such that the amplitude level is minimised, peak damping

performance can be achieved [109]. This approach has the advantage in that the level

of volume oscillation is sufﬁciently small so that it is possible to avoid the sealing

issues that otherwise arise.

Helmholtz resonators are not the only types of passive damper that can be pas-

sively tuned. Perforated liners are another type. In Subsection 3.2.2, it was seen that

the damping of a perforated liner depends on its bias ﬂow rate (i.e., the ﬂow rate of

cooling air through the perforated holes) and its location within the combustor [85].

A two-parameter tuning, in which the bias ﬂow rate and the combustor length were

both successively tuned to maintain maximum damping across varying frequencies,

was successfully performed in cold-ﬂow experiments [110].

3.2.4 Active Control

Active control provides another means of interrupting the coupling between acoustic

waves and unsteady heat release [111–113]. It involves dynamically varying some

input to the combustion system and may be either open loop, for which the input

does not depend on measurements from the system, or closed loop (feedback), for

which the input does depend on system measurements. Closed-loop control offers

better opportunities for suppressing instabilities and so will be the form of active

control assumed from now on.

For closed-loop active control, an actuator modiﬁes some system parameter in

response to a measured signal, as shown in the generic control system in Fig. 3.29.The

This can also be thought of as a type of tuned open-loop active control.

190 Combustion Instabilities

aim is to design the controller – the relationship between the measured signal and the

signal used to drive the actuator – such that the unsteady heat release and acoustic

waves interact differently, leading to decaying rather than to growing oscillations.

Active control offers the possibility for suppressing instabilities over a range

of operating conditions. However, it has the potential to make instabilities worse if

the controller is not well designed, and its practical implementation depends on the

availability of suitable sensors and actuators.

Practical Implementation of Active Control

SENSORS. Sensors are needed to provide measurement of a dynamic quantity related

to the combustion oscillations [114]. Although this is a relatively straightforward task

for an unstable system, it is more difﬁcult for low-amplitude controlled oscillations.

Microphones and pressure transducers are the most commonly used sen-

sors [115–118]. They have the advantage in that, because acoustic waves propagate

throughout the entire combustion system, they do not need to be placed close to

the high temperatures of the heat release zone. They have large bandwidth and are

reasonably robust. Their location within the combustor is important; attention must

be paid to the likely mode shapes so as to avoid placement near a pressure node

and ideally to maximise the signal-to-noise ratio. In annular gas turbines, multiple

sensors are required for capturing the azimuthal modes [119].

The most common alternative is to measure the light emitted by certain progress

chemicals in the ﬂame, most commonly C

, OH, or CH radicals [120, 121]. Optical

ﬁlters in front of photomultipliers ﬁlter out all but the spectral bands related to these

chemicals, with the intensity levels then related to the rate of heat release in the ﬁeld

of view. Such sensors are independent of mode shape, but may be susceptible to

changes in ﬂame location as the stability of the system changes. They also require

optical access to the system; although optical ﬁbres provide a means of achieving

this [121], they have a limited ﬁeld of view and tend to give rather noisy light-intensity

readings.

ACTUATORS. Satisfactory actuation of combustion systems presents a real challenge.

The ﬁrst actuators to be applied to combustion control were loudspeakers [115,

116, 120, 122]. Although these have a good high-frequency response, they are not

sufﬁciently robust for use in industrial systems and their power requirements become

prohibitive at larger scales.

An efﬁcient means of actuation exploits the chemical energy released in com-

bustion by modulating the fuel supply. For ideal actuation, a fuel valve should have

a linear response (to allow linear control theory to be used), large bandwidth, sufﬁ-

ciently large control authority to affect the limit cycle oscillation, a small-amplitude

response that does not exhibit hysteresis, a fast response time, and good robust-

ness and durability. Furthermore, care must be taken to ensure that the acoustics

of the pipework connecting the actuator to the fuel injection position results in a

large response at the likely control frequencies [117, 123]. Obtaining a satisfactory

fuel valve is one of the main challenges facing reliable implementation of feedback

control on practical combustion systems. Solenoid valves, which have the advan-

tage of a linear response, have been used in large and full-scale demonstrations of

control [117, 121, 124]. However, they suffer from limited bandwidth and control

3.2 Control Strategies for Combustion Instabilities 191

authority, exhibit hysteresis at low amplitudes, and are probably not sufﬁciently

durable for practical applications. Magnetorestrictive valves combine fast response

time with increased bandwidth and show signiﬁcant potential, as long as the valve

authority can be maintained at higher frequencies [105].

FEEDBACK CONTROL CHALLENGES. Practical combustors involve turbulence and

combustion; these processes are too complex to use either mathematical or nu-

merical modelling as a basis for the design of controllers that will be implemented

experimentally. This means that controller design must be based on system measure-

ments. The consequences of stability may be so severe that obtaining these is not

straightforward in certain operating regimes. There are likely to be several modes

of instability spanning a wide frequency range, and in such cases it is important that

the controller must be able to control more than one mode. The system will almost

certainly involve substantial time delays caused by factors such as fuel convection

and acoustic-wave propagation; these may be larger than an oscillation period and

provide a real challenge to controller design. Different control parameters may be

needed to cope with different plant-operating conditions, and the presence of tur-

bulent ﬂow will lead to high background-noise levels, which the controller should be

robust to.

HISTORICAL DEVELOPMENT. The ﬁrst experimental demonstration of active control

of a combustion instability was for a Rijke tube with a laminar ﬂame burning on

gauze [120, 125]. An optical ﬁbre and photomultiplier were used to detect the light

emitted by CH free radicals in a Rijke tube ﬂame. This signal was then phase shifted,

ampliﬁed, and used to drive a loudspeaker near one end of the tube. The optimum

gain and phase shift were found by trial and error and gave a sound-pressure-level

(SPL) reduction of 35 dB. Control was then demonstrated on a similar combustor,

but with the sensor signal being a microphone-measured pressure at a location within

the tube [115]; a range of controller phase shifts achieved stability.

These ﬁrst demonstrations both used a loudspeaker to change the boundary

conditions at one end of the tube. In terms of the generalised Rayleigh criterion

discussed in Subsection 3.2.1, this primarily increases the energy lost across the tube

boundaries per cycle, but also alters the acoustic-wave energetics within the tube,

leading to a change in the average energy exchange between the acoustic waves and

combustion. For an appropriate range of phase shifts, inequality (3.39)isreversed

and the system is stabilised. The fact that a range of stabilising phase shifts exists is

consistent with the fact that the inequality only needs to be reversed for instability

to be avoided; unlike in antisound, here we are not relying on an exact cancellation

of signals [125].

Because the power requirements of mechanical actuators such as loudspeakers

become prohibitive at larger scales, attention turned to varying the Rayleigh energy

source term,





dV , directly using unsteady fuel injection. In the ﬁrst experi-

mental demonstration of achieving control through modulation of the fuel supply,

a ducted premixed ﬂame was stabilised by a phase-shift controller, designed by use

of Nyquist methods [126]. An unsteady addition of just 3% excess fuel reduced the

spectral peak corresponding to the main instability mode by 12 dB.

192 Combustion Instabilities

The early demonstrations all used simple gain and phase-shift–time-delay con-

trollers. The controller parameters were obtained for a single operating condition,

usually empirically. Such approaches are unlikely to be adequate for practical com-

bustion systems, in which there may be multiple instability modes. They offer no

guarantees of stability, and a bad choice of controller parameters may increase the

amplitude of the unstable oscillations [117]. Furthermore, controllers involving just

a ﬁxed time delay or phase shift introduce a new oscillation mode, which becomes

unstable as gain is increased and gives rise to a new peak in the pressure spec-

trum [126–128]. A dynamic controller that can introduce suitable gains and phases

at different frequencies is needed, together with a strategy for its design. Some of

the common controller design strategies are described in detail in the following

subsections.

CONTROL STRATEGIES – MODEL-BASED CONTROL. By increasing the sophistication of

the controller structure beyond that of a straightforward gain–time delay, signiﬁcant

improvements in the behaviour of the closed-loop system can be obtained [111].

To design such a controller, some knowledge of the open-loop system is required.

Because mathematical or numerical models can only approximate the open-loop

behaviour of a combustion system, it is currently preferable to design model-based

controllers based on experimental measurements. Model-based controller design

was experimentally implemented on several larger-scale combustion rigs [129–132].

Multisensor multi-actuator control strategies were also recently developed for com-

bustion chambers that are annular in shape and may exhibit unstable longitudinal

and circumferential modes simultaneously [119]. However, we demonstrate the es-

sential concepts of model-based control by applying it to the simplest laboratory-

scale example of a combustion instability with a single unstable mode – the Rijke

tube.

A Rijke tube is a s traight tube, open at both ends, with a heat source inside

and a mean ﬂow passing thorough it. By using a vertical tube, a small amount of

convection ﬂow arises from the mean heat release and there is no need to drive any

additional mean ﬂow through the tube. When the heat source is in the upstream

(bottom) half of the tube and heat addition is in phase with the pressure ﬂuctuation

and in accordance with Rayleigh’s criterion, acoustic waves successively gain energy

and a combustion instability occurs. Rijke tubes were used in the ﬁrst experimental

demonstrations of active control of combustion instabilities [115, 116, 120] and,

because of their simplicity, were used to experimentally demonstrate many advances

in control methodology for combustion instabilities [129, 133, 134].

A vertical quartz Rijke tube, 0.75 m long and with a diameter of 44 mm is con-

sidered, as shown in Fig. 3.30. A propane-fuelled Bunsen burner provides a laminar

ﬂame, which is stabilised on a grid 0.225 m above the bottom of the tube. In the

absence of control, there is an instability at 1533 rad/s (244 Hz). A microphone is

ﬁtted to a tube tapping 0.34 m from the bottom of the tube and actuation is provided

by a 50-W low-frequency loudspeaker situated close to the lower end of the tube.

This has a ﬂat response over the frequency range of interest (500–12 600 rad/s or 80–

2000 Hz); above and below this its dynamics is more complicated. For this reason, a

bandpass ﬁlter with a passband from 500 to 12 600 rad/s is applied to the microphone

signal.