analysis and control of an unstable mode in a combustor with tuneable end condition

Analysis and control of an unstable modein a combustor with tuneable end

conditionMaria Heckl* and Bla Kosztin

Department of Mathematics, Keele University, Staffordshire ST5 5BG, U.K.

(Submission date: April 13, 2013; Revised Submission date: July 13, 2013; Accepted date: July 14, 2013)

ABSTRACTA major problem in the development of low-pollution combustion systems are thermo-acousticinstabilities, i.e. large-amplitude oscillations generated by a feedback between the unsteady heatrelease and acoustic waves. In order to develop robust control strategies, it is necessary to havea predictive model that captures the physics of the phenomenon. The aim of this paper is topresent such a model for a dump combustor with a generic heat release law, and fitted at the inletend with a perforated plate backed by a tuneable cavity. Our model leads to a simple governingequation for one acoustic mode in the combustor, and from this equation stability predictions canbe made with a minimum of numerical effort. We will use it to examine the effect of varioussystem parameters.

1. INTRODUCTIONThermo-acoustic instabilities have been a major problem in the development of low-pollutant combustion systems [1]. Such instabilities occur in systems involvingcombustion in an acoustic resonator: heat release fluctuations act as a source ofacoustic fluctuations and vice versa, and the coupled system can reach large-amplitude self-sustained oscillations. In such a case, the acoustic energy gained by thecombustion-acoustic interaction exceeds the acoustic energy lost due to soundradiation, friction, etc.

It is important to be able to prevent thermo-acoustic instabilities because they canlead to catastrophic hardware damage. In order to develop control strategies, it isnecessary to understand those physical effects that play a key role in such instabilities.A control strategy that has received much attention in recent years is the use ofperforated plates (or liners), backed by a tuneable cavity (see e.g. [25]). By fitting

International journal of spray and combustion dynamics Volume .5 Number . 3 .2013 pages 243 272 243

*Corresponding author: [email protected]

such a device to the boundary of a thermo-acoustic system, it is possible to manipulatethe acoustic energy loss at that boundary in such a way that it exceeds the acousticenergy gain, and the system then becomes stable.

The aim of this paper is to present a simple, yet accurate, analytical model for alaboratory burner controlled by a cavity-backed perforated plate at the inlet boundary,and to use this model for stability predictions under different operating conditions.We consider the burner as a combined system of acoustic resonator and flame, withlongitudinal acoustic modes. Based on a Greens function approach, we will derive agoverning equation for one such mode; this will turn out to be the equation of aharmonic oscillator, forced by the rate of heat release, which in turn depends on theacoustic field. It includes all relevant properties of the acoustic resonator, in particularthe inlet boundary condition, and the properties of an idealised flame, such as time-lag and power. Stability predictions for any operating condition can be made easilyfrom the sign of the damping term in this governing equation.

Our analytical model can be seen as a generalisation of the Galerkin approach, whichis widely used for thermoacoustic stability analyses (see e.g. [6] or [7]). The Galerkinapproach requires explicit knowledge of the eigenfunctions of the system (see [1], chapter13), but this is available only in a few special cases, such as a one-dimensional combustorwith simple end conditions and a uniform temperature distribution. Also, the method islimited by the requirement of weak forcing. Another limitation relates to damping: thisneeds to be estimated from other sources and then incorporated into the equation ofmotion in retrospect (see e.g. [8]). Our approach overcomes these limitations.

Moreover, our analytical model adds an extra dimension to the work by Tran et al.[2], who investigated a tuneable tube termination comprising of a perforated platebacked by a hard-walled cavity of variable length. This group studied an unstable dumpcombustor with a swirl flame and applied such a termination at the inlet section of theirburner. They demonstrated experimentally that by suitable choice of the cavity length,the reflection coefficient could be minimised, and that the instability could then besuppressed. The reflective behaviour of a cavity-backed perforated plate wassubsequently investigated in detail by the same group [9, 10]. They found that maximumabsorption is reached for the quarter-wave modes of the back-cavity. Their theoreticalstudies considered such a termination in isolation, i.e. the features of the acousticresonator and those of the flame were not included. Although it is physically intuitivethat maximising the energy losses at one end of the burner has a stabilising effect, it isnot obvious how the complete thermo-acoustic system reacts to such a measure. Forexample, the acoustic energy gain, which depends very strongly on the phase betweenthe acoustic pressure and the rate of heat release (see e.g. [11]), is likely to be affectedby a change in reflection coefficient. The present paper builds on the work by Tranet al. in that the complete thermoacoustic system, i.e. acoustic resonator and flame, ismodelled analytically.

Thermo-acoustic systems are nonlinear in general. For control purposes, however,knowledge about linear stability/instability is paramount. We restrict our considerationsto purely linear situations. Our method can in principle be extended to nonlinearsituations, but this is beyond the scope of the present paper.

244 Analysis and control of an unstable mode in a combustor with tuneable end condition

Our paper is structured as follows. The derivation of the governing equation of asingle mode is shown in section 2. We will then proceed to use this equation in order topredict the stability behaviour of a specific thermo-acoustic system with tuneable inletboundary condition, and this is shown in section 3. Our predictions will turn out to befully in line with observations on how tuning the cavity length can stabilise the burner.Over and above that we will make predictions about the influence of parameters thatcharacterise the flame. Conclusions will be drawn in section 4.

2. ANALYTICAL DESCRIPTION FOR A GENERIC DUMP COMBUSTOR2.1. The modelled configurationWe consider a configuration, which simulates the premixed laboratory combustorstudied experimentally by Tran et al. [2]. This is shown schematically in Figure 2.1.

The outer shell is a tube with two sections: an inlet section with uniform cross-sectional area A1 and a combustion chamber with a larger, and also uniform, cross-sectional area A2. Cold premixed gas enters the combustion chamber, where it is burntin a compact flame and turned into hot gas. The mean temperature in the two sectionsis uniform: T

1 (cold) in the inlet section, and T

2 (hot) in the combustion chamber; the

corresponding mean densities are 1 and

2.We treat the acoustic field in the tube as one-dimensional with upstream and

downstream travelling waves in each section. This is justified, given that the diameter ofeach section is of the order of 0.1 m, whereas the acoustic wavelength is of the order of1 m. We ignore the geometrical complications due to the two-stage injection system. Thetube is orientated along the x -axis, as shown in Figure 2.1. The upstream end at x = 0 andthe downstream end at x = L are described by pressure reflection coefficients R0 and RL,respectively. The interface, where the cross-sectional area and mean temperature jump, issituated at x = l. The speed of sound is uniform on either side of the interface, where itjumps from c1 to c2.

Our theoretical study is based on a Greens function approach and will involve thefollowing steps:(1) Calculate the tailored Greens function for the acoustic waves in the tube.(2) Derive the governing equation for a single mode driven by an oscillating flame.(3) Calculate the complex eigenfrequency of such a mode and predict its stability

behaviour.

International journal of spray and combustion dynamics Volume .5 Number . 3 .2013 245

R0

Inlet section

Flame Acoustic wavesAcoustic waves

Combustion chamberA2, T2A1, T1

_ _

RL

x =x = 0 x = L

Figure 2.1: Cross-section of the modelled configuration.

2.2. The tailored Greens functionThe Greens function G(x, x, t t) is the response observed at position x and time t bya point source at position x firing an impulse at time t. Its governing equation is

(2.1)

where c is the speed of sound (equal to c1 and c2 in the cold and hot region,respectively). The tailored Greens function is the solution of (2.1), which satisfies the same conditions at the boundaries and interface as the acoustic field. We choose theGreens function to be a velocity potential and calculate it for the case, where theimpulsive source is situated in the combustion chamber, i.e. between x = l and x = L.

The frequency-domain equivalent of (1) is

(2.2)

G (x, x, ) is the Fourier transform of G(x, x, t t), and physically it represents theresponse to a harmonic point source at x with time dependence eit. The frequency-domain Greens function has the following form,

(2.3)

This is a superposition of upstream and downstream travelling waves (see Figure2.2) with wave numbers k1 = /c1 and k2 = /c2, respectively in the inlet region andcombustion chamber.

a+, a, b+, b, c+, c are (complex) velocity potential amplitudes, which need to bedetermined. Two equations for these amplitudes come from the boundary conditions:

at x = 0: (2.4)a R ak k+

=e ei i1 1

0l l

( , , )

( ) ( )

G x x

a a xk x k x

=+ <

at x = L: (2.5)

Another two equations come from continuity conditions across the interface at x = l.

pressure: (2.6)

volume flow rate: (2.7)

The remaining two equations come from the requirement that G satisfies thegoverning equation (2.2). Extracting the required equations from (2.2) involvesmathematical manipulations (shown in Appendix A), and the result is

(2.8)

(2.9)

With equations (2.4) to (2.9), we have six linear equations for the amplitudes. Theycan be solved with Cramers rule to give results for a+, a, b+, b, c+, c. Substitutioninto (2.3) then leads to

with

(2.11)

(2.12a)A x R x c x c( , ) / / = 0 1 1e ei i+

i

1

2 20

11 1 2 2

cR RL( )( )e e e ei i i i+

Fc

R RL( ) ( )( ) = + iAA

1

2 10

11 1 2 2e e e ei i i i

(2.10) ( , , )

( , ) ( , )

( )

( ,G x x

A x C x

Fx

D x =

< 0) in grey.

The period of the 272 Hz oscillation is 0.0037 s, so the time-lag values coveredin Figures 3.3 to 3.5 (up to 0.004 s) include the whole range from zero to one period.Beyond one period, the pattern repeats along the axis; this is just discernible in thefigures.

For Q

= 0, there is stability for all points in the plane. The stability map for10 kW is shown in Figure 3.3. It is dominated by a stable region, but also has a region


0.0 0.10.000

0.001

Stable

Unstable

Tim

ela

g (s)

0.002

0.003

0.004

0.2Cavity length (m)

0.3 0.4 0.5

Figure 3.3: Stability map for Q

= 10 kW.

of instability. As Q

increases, this unstable region grows, and a second unstable regionappears on the right edge of the map. A typical case is shown in Figure 3.4, whereQ

= 20 kW. As Q

increases further, the two unstable regions grow towards each other,and eventually, they join up as shown in Figure 3.5, where Q

= 30 kW. Beyond 30 kW,

the picture barely changes; the map in Figure 3.5 is representative for heater powerslarger than 30 kW.

We will now discuss these results in the light of our finding of section 3.1, whichsuggests that one can control an instability by back-cavity tuning. The optimal cavitylength is a quarter wavelength, which is 0.31 m for the case considered here (frequencyof unstable mode: 272 Hz, speed of sound: 342 m/s). At this cavity length, stability isachievable for powers below about 25 kW. Above that power, there are time-lags forwhich stability cannot be achieved. Such a case can be seen in Figure 3.5, where theinstability region spans the full range of cavity lengths.

Our findings can also be explained in terms of the acoustic energy balance: Duringa thermoacoustic instability, the acoustic energy gained from the flame exceeds theenergy lost at the combustor ends. By adding a cavity-backed perforated plate at onecombustor end, the acoustic energy loss at that end can be manipulated and greatlyenhanced by choosing a cavity length that makes the end non-reflective. In such acase, acoustic energy is transferred from the combustor to the back-cavity, and thereit is no longer able to contribute to the thermoacoustic feedback, which originallydrove the instability. As a consequence, the combined system (combustor and back-cavity) is stable.


0.0 0.10.000

0.001

Stable

Unstable

Tim

ela

g (s)

0.002

0.003

0.004


0.3 0.4 0.5


= 20 kW.

3.4. Comparison with earlier resultsThe robustness of this technique was examined by Scarpato et al. [9], who studied acavity-backed perforated plate in isolation. They found that the absorption bandwidtharound the peak frequency is very large. This is in line with our predictions, whichindicate that (unless the power of the flame is quite large) stability is achieved for awide range of cavity length values centred around the optimal cavity length.

Also, our predictions are in line with the experimental observations reported by Tranet al. [2]. In their laboratory burner the fundamental mode was unstable at 272 Hz, andthe instability was controlled if the cavity length was = 0.28 m. This is quite similarto our optimal value of = 0.31 m; the slight discrepancy is probably due to thesimplifications we made to the geometry of the combustor.

Unfortunately, there is no information in [2] about the power or the FTF of Transset-up, so our comparison between theoretical and experimental results cannot be takenfurther. More experimental data is required for a credible validation of our model.

4. CONCLUSIONSWe derived the governing equation for a single mode in a dump combustor, which couldthen be analysed with a minimum of numerical effort to predict the stability of thatmode. Included in our model are- the acoustic properties of the resonant chamber- the coupling between chamber acoustics and heat release rate- a tuneable termination of the inlet chamber.


0.0 0.10.000

0.001

Stable

Unstable

Tim

ela

g (s)

0.002

0.003

0.004


0.3 0.4 0.5


= 30 kW.

We applied our model to a specific thermo-acoustic system, which was fitted with atuneable termination of the inlet chamber in order to control the single (unstable) modein this system. Our predictions agree with the experimental observation that stability isachieved for a wide range of tuning parameters.

The predictive capability of our model goes beyond the back-cavity tuning. In alsoincludes parameters that characterise the flame. This is an improvement on earlieranalytical studies as these ignore the flame. We found that the flame does have an effecton the robustness of the control system. While deviations from the optimal cavity lengthdo not matter much if the burner power is low, the robustness deteriorates as the burnerpower increases. Also, the time-lag in the heat release law determines if control can beachieved.

REFERENCES[1] T.C. Lieuwen and V. Yang, Combustion instabilities in gas turbine engines.

American Institute of Aeronautics and Astronautics, 2005.

[2] N. Tran, S. Ducruix and T. Schuller, Damping combustion instabilities withperforates at the premixer inlet of a swirled burner, Proceedings of theCombustion Institute, 2009, 32, 29172924.

[3] L. Li, Z. Guo, C. Zhang, C. and X. Sun, A passive method to control combustioninstabilities with perforated liner, Chinese Journal of Aeronautics, 2010, 23,623630.

[4] D. Zhao, A.S. Morgans and A.P. Dowling, Tuned passive control of acousticdamping of perforated liners, AIAA Journal, 2011, 49, 725734.

[5] X.F. Sun, Vortex sound interaction and its application in aerospace propulsionsystem. Proceedings of the 19th International Congress on Sound and Vibration,Vilnius, Lithuania, 812 July 2012.

[6] B. Tulsyan, K. Balasubramanian and R.I. Sujith, Revisiting a model forcombustion instability involving vortex shedding, Combustion Science andTechnology, 2009, 181, 457482.

[7] D. Zhao, Transient growth of flow disturbances in triggering a Rijke tubecombustion instability, Combustion and Flame, 2012, 159, 21262137.

[8] K.I. Matveev and F.E.C. Culick, A model for combustion instability involvingvortex shedding, Combustion Science and Technology, 2003, 175, 10591083.

[9] A. Scarpato, N. Tran, S. Ducruix and T. Schuller, Modeling the dampingproperties of perforated screens traversed by a bias flow and backed by a cavityat low Strouhal number, Journal of Sound and Vibration, 2012, 331, 276290.

[10] A. Scarpato, S. Ducruix and T. Schuller, A comparison of the damping propertiesof perforated plates backed by a cavity operating at low and high Strouhalnumbers, Comptes Rendus Mecanique, 2013, 341, 161170.

[11] A.P. Dowling and J.E. Ffowcs Williams, Sound and sources of sound. EllisHorwood, Chichester, 1983.

[12] Maria A. Heckl, Analytical model of nonlinear thermo-acoustic effects in a matrixburner, Journal of Sound and Vibration, 2013, 332, 40214036.


[13] N. Noiray, Linear and nonlinear analysis of combustion instabilities, applicationto multipoint injection systems and control strategies. PhD thesis, Ecole CentraleParis, Laboratory EM2C, 2007.

[14] Maria A. Heckl, Nonlinear thermo-acoustic effects in a matrix burner amplitudedependent time lag. Internal report, Department of Mathematics, KeeleUniversity, 24 August 2012.

[15] I.J. Hughes and A.P. Dowling, The absorption of sound by perforated linings,Journal of Fluid Mechanics, 1990, 218, 299335.

[16] M.S. Howe, On the theory of unsteady high Reynolds number flow through acircular aperture, Proceedings of the Royal Society London Series A, 1979, 366,205223.

[17] T. Komarek and W. Polifke, Impact of swirl fluctuations on the flame response ofa perfectly premixed swirl burner, Journal of Engineering for Gas Turbines andPower - Transactions of the ASME, 2010, 132, paper no 061503.

[18] A.D. Polyanin and A.V. Manzhirov, Handbook of integral equations. CRC Press,Boca Raton, 1998.

[19] I. Bronstein and K. Semendjajew, Taschenbuch der Mathematik, 14th edition.Verlag Harry Deutsch, Zrich and Frankfurt/Main, 1974.

[20] A.H. Nayfeh, Introduction to perturbation techniques. John Wiley & Sons, 1981.

APPENDIX A: DERIVATION OF EQUATIONS (2.8) AND (2.9)We assume that both the observer and the source are located in the combustionchamber, i.e. the expressions involving the amplitudes b+, b and c+, c in (2.3) apply.The two expressions can be combined with the Heaviside function,

(A.1)

This is now treated as a trial solution, which will be differentiated with respect to xand substituted into (2.2). When differentiating, it is worth making use of the identities

(A.2)

and

(A.3)

Then

= ++ 2

2 22 2 2

H( )( ) ( ) (

G

xx x k b bk x k xe ei il ll)

e ei i = k x k xx x x x2 2( ) ( )( ) ( ).l l

d

d

H( )( ),

x x

xx x

=

( , , ) H( ) ( ) ( )G x x x x b bk x k x = + + e ei i2 2l l + + +

H( ) ( ) ( )x x c ck x k xe ei i2 2l l


(A.4)

When substituting this into (2.2), several terms cancel, and the remaining terms canbe grouped into those with factor (xx) and those with factor (xx),

(A.5)

Equating the coefficients of (xx) and (xx) on either side of (A.5) gives twoequations; by adding and then subtracting these, two equivalent, simpler equations areobtained, which are identical to (2.8) and (2.9) in the main text.

APPENDIX B: DERIVATION OF THE GOVERNING EQUATIONSB.1. Integral equation for the acoustic velocityOur aim is to derive an integral equation from the governing PDE for the Greensfunction G(x, x, t t), equ. (2.1), and from the acoustic analogy equation (2.20) for thevelocity potential (x, t).

We assume that the initial conditions are zero,

(B.1a,b)

and note that boundary conditions described by a reflection coefficient, as in (2.4) and(2.5), are homogeneous, i.e.

(B.2a,b)

(B.2c,d)

=

=

x

G

xG

xx

xx2 2

, ,

=

==

==

=

x

G

xG

xx

xx

01 0

01 0

, ,

( ) , ,ttt t= =

=

=0

0

0 0

+ ++ + ( ) )( ) ( )x x b b ck x k x ke e ei i i2 2l l 22 2( ) ( ) ) ( ).x k xc x x + = l le i

( ) ( ) ( )x x k b b ck x k x k + ++ +i e e ei i i2 2 2l l 22 2( ) ( )x k xc +l le i

+ + +

( ) .( ) ( )x x c ck x k xe ei i2 2l l

+ ++

( ) ( ) ( )x x k c ck x k xi e ei i2 2 2l l

+ + ++

H( )( ) ( ) ( )x x k c ck x k x2

2 2 2e ei il l

+ ++

( ) ( ) ( )x x b bk x k xe ei i2 2l l

+

( ) ( ) ( )x x k b bk x k xi e ei i2 2 2l l


where 1 and 2 are constants.(2.1) and (2.20) are now written in terms of the source coordinates x and t (noting

that G satisfies the reciprocity principle, G(x, x, tt) = G(x, x, tt), and that

(B.3a)

(B.3b)

We multiply by G and as indicated, and then subtract the resulting equations fromone another, to get

(B.4)

This equation is now integrated with respect to source position (over the length of thecombustor, x = 0...) and with respect to source time (up to the observer time, t = 0...t)

The individual terms simplify as indicated, and (B.5) becomes,

(B.6)

For a compact flame with heat release rate (2.19), this simplifies to

(B.7)( , ) ( , , ) ( ) .x t B G x x t t q t tt

t

q= =

0

d

( , ) ( , , ) ( , )x t B G x x t t q x t xt

t

x

= = =

0 0

d ddt.

(B.5)

t

t

x

x t x x t t x t

x t = =

=

0 0

( , ) ( ) ( )

( , )

d d

=

t

t

0 xx

x t

t

B G q x t x t

c

G

t

=

=

=

=

=

0

20 0

21

( , ) d d

222

2

0

=

Gt

t x

d d

= =

t

t

x

G

xG

x0 0

2

2

2

2

=

d dx t

0

..

(zero initial conditions) (homogeneous boundary connditions)

( , ) ( ) ( ) ( , )x t x x t t B G q x tc

G

t =

12

2

2

2

2

2

2

2

2

Gt

G

xG

x

.

12

2

2

2

2c

G

t

G

xx x t t x t

= ( ) ( ) ( , ).

12

2

2

2

2c t xB q x t G x x t t

= ( , ) ( , , ) ,

=

2

2

2

2

G

t

G

t,


Equation (B.7) can be written for the velocity u at the flame by differentiating with

respect to x, evaluating at x = xq and using

(B.8)

The Greens function is given by (2.17), with H(t t) = 1 throughout the integrationrange, and its derivative is

with the velocity amplitudes Gn as given by (2.24) in the main text.

B.2. Differential equation for the acoustic velocity (single mode)Our aim is to convert the integral equation (B.8) to a differential equation for theacoustic velocity of a single mode, say mode n. Disregarding all the other modes, (B.8)reduces to (after replacing Re[...] by 1/2([...] + [...]*), where * denotes the complexconjugate),

(B.10)

By abbreviating the integral and its complex conjugate,

(B.11)

(B.10) can be written more compactly as

(B.12)

In order to convert this into a differential equation, we follow the method describedon p. 127 in [18]. This involves differentiation with respect to t, which appears not onlyin the integrand, but also in the integration limit in (2.27) (see e.g. p. 349 in [19]).

uB

n n= +{ }2 I I * .

Int

t

nt tG q t tn=

=

0

e di ( ) ( ) ,

u tB

G Gt

t

nt t

nt tn n( ) ( ) * ( )*= +

=

2 0

e ei i q t t( ) .d

(B.9)

=

=

=

=

G x x t tx x xx x

Fq

q

n n n

( , , )Re

( )

1

i

gg x x

xx x

x xG

nq

q

n

( , , ) =

==

1 2444444 3444444

e in t t( ) ,

u t BG x x t t

xx x

x x

q t tt

t

q

q

( )( , , )

( )=

= =

=

0

d .

u tx t

x x xq( )

( , ),=

=


With

(B.13)

the first and second derivative of (B.12) become

(B.14a)

(B.14b)

The general idea now is to find an explicit expression for In, which can then besubstituted into (B.14b) to give a second-order differential equation for u(t). To this end,we multiply (B.14a) by (in) and (B.12) by (nn*), and add the resulting equations to get

(B.15)

This gives the expression

(B.16)

and with this, the last two terms in (B.14b) become after a few straightforwardmanipulations

(B.17)

(B.14b) itself then becomes after a few further manipulations

(B.18)

Several terms in this equation can be simplified,

(B.19a)i( ) Im( ),* n n n = 2

= + + + +B G G q t B G G G G Gn n n n n n n n n n2 2( ) ( ) i ( )* * * * nn n n nG q t* * * ( ).

u u un n n n+ + =i( )* *

n n n n n n n n nu Bu

Bq t G2 2

2 2I I+ = + + +( ) i ( ) ( )(* * * * GGn n n* *) i( ). +

Inn n

n n nBu u G G q t=

++( ) +

1 2

** *i i( ) ( ) ,

i i( ) ( ) ( ) .* * *u uB

G G q tn n n n n n+ = + + +{ } 2 I

uB

G G q t G G q tn n n n n n n n= + + + 22( ) ( ) ( i i ) ( )* * * I (( ) .* *n n2 I{ }

uB

G G q tn n n n n n= + +{ }2 ( ) ( ) i i ,* * * I I

= +I In n n ntG q ti ( ),


(B.19b)

(B.19c)

(B.19d)

and this leads to

(B.20)

APPENDIX C: DETAILS OF THE STABILITY ANALYSISOur starting point is the governing equation (2.29),

(C.1)

We introduce a nondimensional time t , based on the resonance frequency of moden,

(C.2)

With

(C.3a,b)

equation (C.1) becomes

(C.4)

d

d

d

d

2

22

02u

t

u

tB n Gn n n n (Re ) Re( ) Im( ) Re( )[ ] +

uu t B n G B n G u tn n n n n( ) | | Im( ) Im( ) ([ ]* * 2 0 1 =

)

Re( )Re( )( )

,

+

B n G

u t

tn n1d

d

d

d

d

d

d

d

d

dand

d

d

d

d

u

t

u

t

t

t

u

t

u

t

u

tn= = =

%

%

% %Re( )

2

2

2

222(Re ) ,n

%t t n= Re( ).

= + B n G u t B n G u tn n n 1 1Im( ) ( ) Re( ) ( ).* &

&& &u B n G u B nn n n n + [ Im( ) Re( )] [| | Im(2 02

0 GG un* )] =

&& &u u u B G q t B Gn n n n n + = +2 2Im( ) | | Im ) ( )* ( Re( )) ( ).&q t

BG G G G G Gn n n n n n n n n n n n2

i ( )* * * * * * + + = BB Gn nIm( ),*

BG G B Gn n n2

( ) Re( ),*+ =

n n n* | | ,= 2


where

(C.5)

is a nondimensional time-lag. We make no assumption that the time-lag is small comparedwith the period of the oscillation, i.e. may have order of magnitude 1 (or less).

Dividing (C.4) by 2r gives

(C.6)

We denote the coefficients in this equation as indicated. , , 1 and 2 arenondimensional parameters, defined by

(C.7)

(C.8)

(C.9)

(C.10)

is a frequency and is of order 1. is the ratio between the imaginary and real partof n, i.e. the ratio between damping and frequency of mode n in the passive resonator.We assume that this is a small parameter. The two parameters 1 and 2 are small aswell, as can be shown by the following estimate.

From (2.24), with (2.11), (2.12) and (2.18), we can see that Gn = O(1/); the time lag is of the same order as the period of the oscillation, hence Gn n/(2). The constant, which is given by (2.28), can be estimated by assuming parameter values, which are

2 0 2

= B nGn n

r

Im( ).

*

1 0

= B nGn

r

Re( ),

=Im( )

Re( ),n

n

%

=| |

Re( )n

n

=

=

+B n G

nn

u t Bn n

r

1 2

1

02

Im( )( )

*

1 244 344

% % nnG

nn

u t

tn

r1

11

0

Re( ) ( )

=

1 244 344

% %

%d

d

d

d

d

d

2

2 02u

t

u

tB n

Gn

r

=

Im( ) Re(

nnr

n

r

u t)

( )| |

=

+

=1

2

2

2

=

B nGn n

r

0 2

2

Im( )*

=

% = Re( )n


typical for a premixed laboratory burner: Q 10 kW, u 10 ms1, 1 kg m3,

A 103 m2. The result is 106 m2s2. The estimate for B = (1)/c2 comes from c 500 ms1 and = 1.4, and turns out to be B 1.6.106 m2s2. This gives 1 0.25 101; 2 is also of the order 101.

Equation (C.6), rewritten into a more compact form with (C.7) to (C.10),

(C.11)

is well suited for the application of the method of multiple scales (see e.g. [20], section5.4), because we have two different time scales: A fast time that relates to the oscillationfrequency Re(n), and a slow time that relates to the growth of the oscillation amplitudeIm(n). We put

(C.12a)

(C.12b)

and expand u(t ) in powers of of the form

(C.13)

terms of order O(2) and higher will be ignored. With the chain rule,

(C.14)

we can convert the derivatives of u with respect to variable t to variables t0 and t1,

(C.15a,b)

The time-lag term u(t ) can also be expanded in powers of . From (C.13) we get

(C.16)

and expansion about t1 gives with

(C.17b)

This turns (C.16) into

(C.18)

The derivative of this can be obtained from (C.15a) and (C.16),

(C.19)

Then the governing equation (C.11) becomes

(C.20)

Equating coefficients of like powers of to zero yields two equations,

(C.21)

(C.22)

which can be solved sequentially. (C.21) has solution

(C.23)u t t A t t0 0 1 1 0( , ) ( ) .=e i

=

+

+ +2 22

0

0 11

0

02 0

1

02 0

u

t t

u

tu

n

nu( ) (tt t

n

n

u t t

t0 11

01

0 0 1

0

+

%%

, ) ( , ) ,

12

1

02

21:

+

=

u

tu%

02

0

02

20 0: ,

+ =u

tu%

= +

n

nu t t

n

n

u t t

t1

02 0 0 1

1

01

0 0 1

0

( , ) ( , ) .%%

+

+

2

0

02

20

0 1

21

02

0

0

2 2u

t

u

t t

u

t

u

t ( 11

20 1

22 0) + + =% % u u u

d

d

u t

t

u t t

t

u t( ) ( , ) (% %%

%%

%

0 0 10

20 0 ,, ) ( , )

( ,

t

t t

u t t

t

u t t

1

0 1

1 0 1

0

0 0 1

+

+

%

% )).

t1

u t u t tu t t

t( , ) ( , )

( , )% % % %

%

0 0 1 0 0 1

1

++ u t t1 0 1( , ).%

u t t u t tt

u t1 0 1 1 0 11

1 0( , ) ( , ) (

% % % % %% , ).t1


With that, and with

(C.24a)

(C.24b)

(C.24c)

(C.24d)

equation (C.22) becomes

(C.25)

The right hand side of this ODE represents forcing at the resonance frequency. Inorder to avoid resonant solutions, this forcing term needs to be zero. This conditiongives a 1st-order ODE for A(t1),

(C.26)

with

(C.27a)

(C.27b)a1 2= i .

an

n

n

n0 1 21

02

1

012= +i( ) i ,i i

e e

+ =a A t a At0 1 1 1

0( ) ,d

d

=e i + + t At

A t A tn

n0 2 2

11 1 2 1

1

0

i ( ) i ( ) ( )d

d 2 1 1

01 1A t

n

nA t( ) ( ) .ie

i e

i

+

=

21

02

21

u

tu%

= u t tt

A t t0 0 1

01

0( , )

i ( ) ,i i

e e

u t t A t t0 0 1 10( , ) ( ) , = % % % % e ei i

= 2

0

0 1 1

0u

t t

A

tti ,i d

de

= ut

A t t0

01

0i ( ) ,i e


The solution of (C.26) is

(C.28)

where

(C.29)

The real part of this expression represents the growth rate of the amplitude A(t); weabbreviate it by ,

(C.30)

is an indicator for the stability behaviour:

(C.31)

We convert the expression in (C.29) into standard form so that the real and imaginarypart are explicit,

(C.32)

and switch back to dimensional quantities to obtain

(C.33)

This gives the growth rate in terms of parameters that describe the burner (n andBGn) and parameters that describe the heat source (, , n1 and n0). is periodic in .

=

Im( )

Re( ) Re( )

Im( )Re( )

Re( )n

n n

n n

n

Bn

G

2 0

+ n G n

Gn n

n n

nn11 Re( )cos | |

Im( )

| |sin | |

*

.

+ + ( )

i

2 21

02 1%

% % % % %

nn

cos sin ,,

aa

n

n0

11

1

01 2

1

22= + +

%% % % %( ) cos sin %% %( )

+

stable

unstableif

0

0

=

Rea

a0

1

aa

n

n0

11 2

1

02 1

1

22=

+ + ( )i

i( ) ii

e

.

A t A Aa t a t( ) ,/ ) / )1 0 00 1 1 0 1= =e e(a (a %


analysis and control of an unstable mode in a combustor with tuneable end condition

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