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Journal of Sound and Vibration (1995) 180(4), 557–581

THE CALCU LATIO N OF THER MOACOU STIC

OSCILLATIONS

A. P. D

Department of Engineering, University of Cambridge, Trumpington Street,Cambridge CB 2 1PZ , England

(Received 1 September 1992, and in final form 23 November 1993)

Thermoacoustic oscillations occur in a wide variety of practical applications in whichheat is supplied to an acoustic resonator. A simple geometry is investigated systematicallyto determine the importance of various flow parameters on the frequency of the oscillations.Detailed consideration of elementary examples shows that the form of the coupling betweenthe heat input and the unsteady flow has a crucial effect on the frequency of oscillation.The same elementary examples are used to compare how well (if at all) different calculationmethods in the literature account for this influence. A mean flow and a distributed region

of heat input significantly complicate the analysis of thermoacoustic oscillations and areoften neglected. Model problems are used to illustrate that mean flow effects can becomesignificant even at modest inlet Mach numbers, and to indicate circumstances under whicha distributed heat input may be treated as concentrated.

1. INTRODUCTION

There is a possibility of thermoacoustic oscillations whenever combustion or heat exchangetake place within an acoustic resonator. They occur because unsteady heating generatessound waves producing pressure and velocity fluctuations. However, within a resonator,these in turn perturb the rate of heat input. Instability is possible if the phase relationshipis suitable because, while the acoustic waves perturb the heat input, the unsteady heat inputgenerates yet more sound.

Rayleigh [1] gave a clear physical description of this phenomenon. Acoustic waves gainenergy when the unsteady rate of heat input is in phase with the pressure perturbations.If this energy gain exceeds that lost on reflection from the boundaries of the resonator,linear acoustic waves grow in strength and the system is unstable. In practice, the

amplitudes of the pressure waves are limited by non-linear effects. Nevertheless, they canbecome so intense that structural damage is done.

Many practical devices are susceptible to destructive thermoacoustic oscillations. Theyoccur in rockets, ramjets, gas boilers and aeroengines. Emphasis has been placed oninvestigating the frequency of the oscillations and the mean heat input level for the onsetof the instability. Two excellent reviews of the literature have been given by Candel andPoinsot [2] and Culick [3].

One particular thermoacoustic oscillation, involving the propagation of low frequencylongitudinal pressure waves, occurs in the afterburners of jet aeroengines. It is commonly

called ‘‘reheat buzz’’. An experimental and theoretical investigation of this problem hasinvolved a series of tests on a laboratory-scale rig, designed to model the essential featuresof an afterburner [4]. The rig geometry involves a flame burning in a duct in the wake of a bluff body. There is a mean flow and the heat release is both distributed and unsteady.

It was found to be necessary to include a proper description of all these effects in the557

0022–460X/95/090557+ 25 $08.00/0 1995 Academic Press Limited


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development of a theory to explain the experimental results [5]. This is at variance withviews and alternative approaches reported in the literature, where one or more of theseeffects are neglected.

The aim of this paper is to investigate systematically, for a very simple geometry, theimportance of various flow effects on the frequency of the thermoacoustic oscillations. Thisis followed by a discussion on how well these effects are described by different calculationmethods.

Section 2 is concerned with the effect of the form of the coupling between the heat inputand the unsteady flow on the frequency of the combustion oscillation. In work on the‘‘buzz’’ oscillation [4, 5], this coupling was found to have a crucial effect on the frequency.However, the complicated details of the experimental geometry and flow conditions

prevented any general conclusions from being drawn from this observation. A very simplygeometry is considered in section 2 to avoid this difficulty.

One-dimensional disturbances are investigated in a duct with no mean flow and oneclosed and one open end. Unsteady heat is added at a single plane, across which the meantemperature changes from T 1 to T 2. Two particular forms for the relationship between therate of heat input and the flow are investigated in detail. Case I is that of no unsteady rateof heat input. This represents a limiting case when the reaction time of the heating processis much longer than the period of oscillation. Then, even though the fluid might enter theheating zone unsteadily, the rate of heat input would be unable to adjust sufficiently rapidlyand would remain constant. In Case II the instantaneous rate of heat input is directlyproportional to the instantaneous mass flow rate into the heating zone: i.e., the heat inputper unit mass is constant. Such a heat input would be appropriate as a limiting case when

the reaction time of the heating process is so much faster than the period of oscillation thatthe rate of heat input responds quasi-steadily to changes of flow rate into the heating zone.

Both forms of heat input are, of course, idealizations. However, it is worth noting thatthe heat released by a non-premixed flame could approximate to Case I, when fuel is forcedinto the combustion region at a constant rate and the combustion is directly proportionalto the fuel injection. Observations [5] of the rate of combustion for a premixed flameburning in a duct are of the form of Case II at low frequencies, but with an additionaltime delay.

The frequencies of the thermoacoustic oscillations are calculated in section 2.1 for these

two forms of the unsteady heat release rate. The relationship between the unsteady heatinput and the flow is found to crucially affect the frequency of oscillation. Indeed, by amean temperature ratio of six, there is nearly a 60% difference between the lowestfrequency predicted for Case II (when the instantaneous rate of heat input is proportional

to the instantaneous mass flow rate) and for Case I (when there is no unsteady rate of heat input).

Some calculation methods commonly used in the literature take no account of the formof the coupling between the unsteady heat input and the flow. In sections 2.2 and 2.3respectively, it is shown that the calculation of the duct acoustic frequency and the Greenfunction technique recommended by Hedge et al . [6] are appropriate only when the rateof heat input is not altered by the unsteady flow. The linearized Galerkin method [3]acknowledges that the form of the coupling between the unsteady heat input and the flowaffects the frequency, but treats the shift in frequency, from that for no unsteady heat

release rate, as linear. This method is tested in section 2.4, by applying it to Case II, withthe instantaneous rate of heat input proportional to the instantaneous mass flow rate. Thepredicted frequency and the true frequencies agree exactly for T 2 = T 1, but divergesignificantly for larger temperature ratios. By a temperature ratio of six, the Galerkin

method predicts a frequency which is less that half the exact value.


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The presence of a mean flow significantly complicates the analysis of a thermoacousticoscillation. An entropy wave, or convected hot spot, becomes coupled to the acousticwaves and a simultaneous solution of equations of conservation of mass, momentum andenergy is required. Many authors choose to neglect the mean flow rather than deal withthis complication, their justification being that the inlet Mach number is low. The errorsintroduced by such an approximation are investigated in section 3. It is striking howsignificant Mach number effects can be. For example, one might be tempted to think that

an inlet Mach number of 0·15 is sufficiently small for the mean flow to be neglected. Butat this Mach number, the frequency of the thermoacoustic oscillation, for Case II,

T 01/T 02 =6, is reduced to half its no flow value.Most practical ways of adding heat also exert a drag force on the flow. In work on the

‘‘buzz’’ oscillation, the drag force exerted by the bluff-body flame holder was found notto have a significant effect [7]. A systematic investigation into the effects of drag is givenin section 4. The drag force exerted by a grid or flame holder with a blockage ratio of 25%or less is found to have a negligible effect on the frequency of thermoacoustic oscillationsfor the range of inlet Mach numbers considered, 0M 1 0·15. A blockage ratio of 50%alters the frequency for inlet Mach numbers greater than 0·10.

When the heat release is distributed over an axial extent d rather than concentrated ata single plane, the strengths of the acoustic and entropy waves vary continuously withposition. It is then easier to determine the unsteady flow directly by integrating theequations of motion. That is done in section 5 to investigate the effects of an extendedregion of heat input.

It is sometimes said that distributed heat input can be considered as concentrated

provided that its axial extent is very much smaller than the acoustic wavelength: i.e.,provided that d /c̄ 1, where is the frequency and c̄ is the speed of sound. However,this is an oversimplification when there is a mean flow. Then there is also an entropy wave,with the much shorter length scale ū/, where ū is the mean flow speed. The entropy waveis affected by even a modest spatial distribution of the heat input, and the frequency of oscillation is altered significantly when this entropy wave is coupled to the acoustic waves.For example, results in section 5 show that for an inlet Mach number of 0·10 and astagnation temperature ratio of nine, distributing the heat input over a length as short as5% of the duct can lead to a 25% change in the lowest frequency of oscillation. To

guarantee, in general, that heat input over a region of length d can be treated asconcentrated requires not only d /c̄1, but also d /ū1. The second inequalityis highly restrictive and circumstances under which it can be relaxed are discussed insection 5.

2. THE EFFECT OF UNSTEADY HEAT INPUT

2.1.

In a region with heat input, the density varies through changes in both the pressure p and specific entropy s. The chain rule of differentiation shows that

D

Dt

= 1

c2

D p

Dt

+

s p

Ds

Dt

. (2.1)

When viscous and heat conduction effects are neglected, T Ds/Dt = q(x, t), where q isthe heat input/unit volume and T is the absolute temperature. Moreover for a perfect gas,/s p = −/c p = −T (−1)/c2, where c is the speed of sound, c p is the specific heat at


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constant pressure and is the ratio of specific heats. Substitution into equation (2.1) leadsto

DDt

= 1c2 DpDt − (− 1)q. (2.2)

Equation (2.2) may be applied to a combusting gas, provided that the reactants andproducts behave as perfect gases, and there is no molecular weight change during thechemical reaction [8].

For linear perturbations in a region where there is no mean heat input or mean velocity,equation (2.2) can be linearized:

DDt

= 1

c̄2 p't − (− 1)q'. (2.3)The overbar denotes a mean value and the prime a perturbation.

A combination of equation (2.3) with the linearized equations of mass and momentumconservation, D/Dt + ¯·u'=0 and u'/t + p'/= 0, leads to

1c̄22 p't2

− ¯·1 ¯ p'= (− 1)c̄2 q't . (2.4)This inhomogeneous wave equation describes the pressure perturbation generated by theunsteady heat input q'(x, t).

The solution of this wave equation can be readily calculated for one-dimensionaldisturbances in the geometry illustrated in Figure 1. A duct of length l has one closed andone open end. The heat input is harmonic of frequency , concentrated at the plane x = b,q'(x, t) = Q'(t)(x − b), where Q'(t) = Q eit and denotes the Dirac -function. In x b,the mean temperature is T 1, the density ¯1 and the speed of sound c̄1. The solution of equation (2.4), that satisfies the rigid end boundary condition at x =0, is

p'(x, t) = A e it(e−ix/c̄1 + eix/c̄1), (2.5)where the complex constant A has yet to be determined. Similarly, in x b, the mean fluidproperties have values T 2, ¯2 and c̄2, and the solution of the wave equation can be writtenin the form

p'(x, t) = B e it(e−i(x − l )/c̄2 − ei(x − l )/c̄2), (2.6)after application of an open end boundary condition, p '(l , t) = 0. Integration of equation(2.4), with q'(x, t) = Q'(t)(x − b), across the region x = b shows that

[ p']x = b+

x = b− = 0 and 1 ¯ p'xx = b+

x = b−

= −(− 1) ¯1c̄21

dQ'dt

, (2.7a, b)

since for a perfect gas, ¯1c̄21 = p̄ = ¯2c̄22. Equation (2.7b) is equivalent to

[u']x = b+

x = b− =(− 1) ¯1c̄21

Q'(t), (2.8)

Figure 1. One-dimensional disturbances in a duct.


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a relationship between the volumetric expansion and the rate of heat input that has beenfrequently used in the literature.

When Q'(t) is specified, the constants A and B may be calculated by substitution for p'(x, t) from equations (2.5) and (2.6) into the jump conditions (2.7). The pressureperturbation within the duct is then determined. However, thermoacoustic instabilitiesoccur when the heat input and the flow perturbations are coupled. As well as the unsteadyheating generating sound as described by equation (2.4), the flow perturbations in turn

affect the rate of heat input. The details of this feedback depend on the precise form of the heating but, in all cases, lead to a relationship between Q and the complex constantsA and B . This relationship, together with the jump conditions (2.7), constitute threehomogeneous equations for the three unknowns A, B and Q , which can be writtensymbolically in the form

X AB Q = 0, (2.9)

where X is a 3 × 3 matrix. Such an equation will have a non-zero solution only if thedeterminant of X vanishes, and this condition determines the frequency of thethermoacoustic oscillation.

In this paper, two particular forms for the unsteady rate of heat input will be consideredas illustrative examples. The first case is that of no unsteady rate of heat input:

Case I, Q'(t) = 0 , no unsteady rate of heat input. (2.10)

Such a heat input would typically occur when the reaction time of the heating process ismuch longer than the period of oscillation. Then, even though the fluid might enter theheating zone unsteadily, the rate of heat input would be unable to adjust sufficiently rapidlyand would remain constant.

In the second case, the instantaneous rate of heat input is taken to be directlyproportional to the instantaneous mass flow rate into the heating zone, the heat input perunit mass being that required to raise the mean temperature from T 1 to T 2: i.e., c p (T 2 − T 1).Since the heat input per unit mass is constant, this case is referred to as that of no unsteadyheat input per unit mass. The rate of heat input is then proportional to the mass flow rate

into the heating zone:Case II, Q'(t) = c p (T 2 − T 1) ¯1u'1 , no unsteady heat input per unit mass, (2.11)

where u '1 (t) denotes u '(b−, t). Such a heat input would be appropriate when the reactiontime of the heating process is so much faster than the period of oscillation that the rateof heat input responds quasi-steadily to changes of flow rate into the heating zone.

Of course, the two forms for the rate of heat input in (2.10) and (2.11) are idealizations.However, it is worth noting that the heat released by a non- premixed flame couldapproximate to Case I (expression (2.10)), if the fuel was forced into the combustion regionat a constant rate and the rate of combustion was directly proportional to the rate of fuelinjection. From a series of experiments on a confined premixed flame, Bloxsidge et al . [5]postulated a rate of heat input which, for harmonic disturbances, can be expressed in theform Q'(t)= kc p (T 2 − T 1) ¯1u'1 (t − ); k and the time delay depend on the frequency.

Over a range of measured frequencies, k varied between 0·5 and 1·0. There are, therefore,also circumstances under which the Case II form of the heat input in expression (2.11) isachievable in practice. The aim of this section is not to obtain detailed predictions for aparticular geometry of flame or heated grid, but rather to demonstrate that the form of

the unsteady heat input can have a significant effect on the frequency of a thermoacoustic


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oscillation. This leads on to an investigation of how this effect is accounted for (if at all)in some of the methods of solution in the literature.

For the rate of heat input in Case I, described by equation (2.10), substitution for p'(x, t)from equations (2.5) and (2.6) into the jump conditions (2.7) leads to an equation for .After some algebra the frequency of oscillation is found to be given implicitly by

tan () tan () = ¯1c̄1/ ¯2c̄2 for Case I, (2.12)

where =b/c̄1 and =(l − b)/c̄2. It is a straightforward matter to solve equation (2.12)numerically. The roots are real and the lowest frequency solution is shown in Figure 2 forvarious values of T 2/T 1 and for the particular geometry b = l /2.

When the rate of heat input in Case II described by equation (2.11) is used, a different

equation for the frequency is obtained. It is evident from equation (2.5) that

u'1 (t) = A e it(e−ib/c̄1 − eib/c̄1)/ ¯1c̄1. (2.13)

When this is used in equation (2.11), substitution into equation (2.7) leads to

tan () tan () = c̄1/c̄2 for Case II. (2.14)

Again the roots are real, and the lowest frequency solution is plotted in Figure 2 forcomparison with the solution of equation (2.12).

When the temperature is uniform along the duct (T 2 = T 1), the predicted frequencies justcorrespond to that for which the duct length is a quarter-wavelength. For othertemperature ratios, the relationship between the unsteady heat input and the flow cruciallyaffects the frequency of oscillation. Indeed by a temperature ratio of six, there is nearly

a 60% difference between the frequency predicted for no unsteady heat input per unit massand no unsteady heat input per unit volume.

Of course, it is well known that the unsteady heat input affects the frequency. Rayleigh[1] commented that ‘‘the pitch is raised if heat be communicated to the air a quarter periodbefore the phase of greatest condensation: and the pitch is lowered if the heat becommunicated a quarter period after the phase of greatest condensation’’. In the geometryconsidered here, the velocity fluctuation u '1 lags the pressure perturbation and so the rateof heat input in equation (2.11) lags the greatest condensation and leads to a lowerfrequency than that for Q' = 0. The numerical calculations show that this shift in frequency

Figure 2. The lowest frequency of oscillation as a function of temperature ratio T 2/T 1 for b = l /2. ——, CaseI, no unsteady rate of heat input (roots of equation (2.12)); – – – – , Case II, no unsteady heat input per unitmass (roots of equation (2.14)).


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can be significant, and it is appropriate to discuss how this effect is accounted for in someof the methods of solution in the literature.

2.2.

Many authors just assume that thermoacoustic oscillations occur at the ‘‘acousticfrequency’’ of a duct. By this, they invariably mean the frequency at which a solution of the homogeneous wave equation

1c̄22 p't2

− ¯·1 ¯ p'= 0 (2.15)satisfies the boundary conditions. This frequency takes no account of the relationshipbetween the unsteady heat release rate and the flow. For example, for the particulargeometry in Figure 1 it leads to

tan () tan () = ¯1c̄1/ ¯2c̄2, (2.16)

for any form of the unsteady heat release rate. A comparison of equations (2.4) and (2.15)shows why this is so. Equation (2.4), which was derived from the equations of motion,reduces to the homogeneous form (2.15) only when there is no unsteady rate of heat input

per unit volume. When the rate of heat input is unsteady, the pressure perturbations satisfy

the inhomogeneous wave equation (2.4) and the form of the heat input must be consideredin a calculation of the frequency of oscillations in the duct.

2.3.

Hedge et al . [9] used a Green function technique to determine the one-dimensionalpressure perturbation generated by unsteady combustion within a duct. They essentiallysolved the inhomogeneous wave equation (2.4) for q'(x, t) = q̂(x) e it, by introducing aGreen function G(xx0) which satisfies

¯c̄2 d

dx 1 ¯ dGdx+2G = −c p (− 1)(x − x0) (2.17)and the boundary conditions. The solution of equation (2.4) is then given by

p'(x

, t) = p̂(x) eit

, where

p̂(x) =ic p G(xx0)q̂(x0) dx0. (2.18)

In particular, when the heat input is concentrated on the plane x = b, q̂(x0) = Q (x0 − b),and

p̂(0)=iQG(0b)/c p . (2.19)

A general form for the Green function was given by Hedge et al . [9, equation 14]. Forthe geometry illustrated in Figure 1, their expression simplifies to

G(0b) = −c̄1c̄2

sin ()T 1c̄2 sin () sin () − T 2c̄1 cos () cos ()

. (2.20)

Hedge et al . found the resonance frequencies of the duct by minimizing the magnitude of the denominator of the Green function. For the particular form (2.20), the denominatorvanishes at frequencies that satisfy

tan () tan () = T 2c̄1/T 1c̄2 = ¯1c̄1/ ¯2c̄2, (2.21)


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a condition identical to that in equation (2.12). If the unsteady heat input is a specifiedbroadband source, equation (2.19) shows that large pressure perturbations are generatedat this frequency. However, it is misleading to interpret this as the frequency of athermoacoustic instability, as suggested in reference [6].

A thermoacoustic instability involves coupling between the heat input and the flow. Theunsteady heat input occurs as a response to fluctuations in flow. This can be expressedin the form

Q = Z () p̂(0), (2.22)

for some function Z (). Substitution for Q in equation (2.19) leads to

p̂(0)(1−iZ ()G(0b)/c p ) = 0. (2.23)

The frequency of the thermoacoustic oscillation is the frequency at which self-sustainingoscillations can occur. It is clear from equation (2.23) that p̂(0) can be non-zero only if

{1/G(0b)} − iZ ()/c p = 0. (2.24)

This reduces to 1/G(0b) = 0 only if Z () = 0. This explains why the zeros of 1/G(0b) werefound in equation (2.21) to be identical to frequencies calculated for the particular caseof no unsteady rate of heat input. When the heat input responds to the flow, Z () isnon-zero and the frequency is shifted.

As an example, consider Case II with no unsteady heat input per unit mass. The modeshape in x b, as described by equations (2.5) and (2.13), shows that

u'1 (t) = −i sin () p̂(0) eit/ ¯1c̄1. (2.25)

Hence the unsteady heat input described by equation (2.11) can be written in the formQ = −ic p (T 2 − T 1) sin () p̂(0)/c̄1, leading to

Z () = − ic p (T 2 − T 1) sin ()/c̄1. (2.26)

Substitution of the Green function and Z (), from equations (2.20) and (2.26),respectively, into equation (2.24) leads, after some algebra, to the same equation for asthat given in equation (2.14).

In summary, a Green function can be a useful tool in the investigation of thermoacoustic

oscillations. However, it is not sufficient simply to inspect the Green function alone. Theform of the coupling between the heat input and the flow must also be considered. In thenotation of this section, this coupling is described by the function Z (). The frequencyof the thermoacoustic oscillation is then a zero of {1/G(0b)} − iZ ()/c p . The form of the relationship between the unsteady heat input and the flow affects the frequency.

2.4.

Culick and his colleagues used a Galerkin method to solve the inhomogeneous waveequation (2.4). This method acknowledges that the form of the coupling between theunsteady heat input and the flow affects the frequency, but linearizes the change infrequency and mode shape. Reference [3] clearly describes the method and gives a reviewof earlier work.

For the one-dimensional problem illustrated in Figure 1, the method involves expanding

the pressure perturbation as a Galerkin series:

p'(x, t) =

m = 1

m (t)m (x). (2.27)


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The functions m (x) are eigensolutions of the homogeneous wave equation,

2mc̄2 m + ¯

ddx 1 ¯ dmdx = 0, m = 1, 2, . . . , (2.28)

that satisfy the boundary conditions

dm /dx = 0 at x = 0 and m = 0 at x = l . (2.29)

It is a straightforward matter to show from this definition that the functions m areorthogonal,

l

0

m (x)n (x) dx = 0 for m n. (2.30)

For the geometry in Figure 1, the mean temperature is uniform in the two regions x band x b. The homogeneous wave equation in (2.28) then simplifies to

d2mdx2

+2mc̄21

m = 0 in 0x b, d2m

dx2 +

2mc̄22

m = 0 in bx l , (2.31a, b)

together with two jump conditions across x = b:

[m ]x = b+x = b− = 0 and

1 ¯

dmdx

x = b+

x = b−

= 0. (2.32)

It is a matter of straightforward algebra to check that

m (x) = cos (mx/c̄1) in x b(cos m /sin m ) sin (m (l − x)/c̄2) in x b (2.33)in a solution of equations (2.31) and (2.32) that satisfies the boundary conditions (2.29).m is the mth root of tan m tan m = ( ¯1c̄1)/( ¯2c̄2) and m =mb/c̄1 and m =m (l − b)/c̄2.

Substitution for p'(x, t) from equation (2.27) into equation (2.4) leads to

m = 1 d2m

dt2 +2mmm (x) = (− 1) q't . (2.34)

After multiplication by n (x) and integration with respect to x, the orthogonality

condition (2.30) shows that equation (2.34) simplifies to

d2ndt2

+ 2nn =(− 1)

E n l

0

q't n (x) dx, (2.35)

where

E n = l

0

2n dx = 12(b + (l − b) cos2 m /sin2m ), (2.36)

after substitution for n from equation (2.33).No approximation has yet been made, but in the next stage in the analysis it is assumed

that q'/t is small in magnitude. It is then argued that since q'/t is small, it need onlybe evaluated approximately. The acoustic approximations n (t)n (x) and (̊ n (t)/ ¯2n ) dn /

dx for the pressure and velocity perturbation are then used when calculating q'/t. If second derivatives of the amplitudes arise, they are replaced by the zeroth orderapproximation ¨ n (t)−2nn (t). The errors induced by these approximations can bechecked by applying the method to Case II to find the lowest frequency of combustion

oscillation when there is no unsteady heat input per unit mass.


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Then the rate of heat input has the form given by equation (2.11) andq'(x, t) = c p (T 2 − T 1) ¯1u'1(x − b). Rewriting the velocity perturbation in the acousticform ( ¨ 1(t)/ ¯21) d1/dx leads to

q'(x, t) = c p (T 2 − T 1) ̊ 1(t)21

d1dx

(b−). (2.37)

The right side of equation (2.35) therefore involves the second derivative, ¨ 1(t). This isreplaced by −211(t), following Culick’s rules, to give

¨ 1 +211 = −1c pE 1

(−1)(T 2 − T 1) d1

dx (b−)1(b)

= 1c p1(−1)(T 2 − T 1) sin 1 cos 1/E 1c̄1, (2.38)

after substitution for 1 from equation (2.33). Equation (2.38) is a straightforward secondorder differential equation for 1(t). The frequency of oscillation of 1(t), , can be foundby substituting 1(t) = C e it into equation (2.38) to give

2 =21 − c p1(−1)(T 2 − T 1) sin 1 cos 1/E 1c̄1. (2.39)

As in this method it is assumed that the difference −1 is small, Culick [3] thereforerecommended that the square root in equations such as (2.39) be evaluated approximately

by using the binomial theorem. Equation (2.39) then simplifies to

=1 − c p (−1)(T 2 − T 1) sin 1 cos 1/2E 1c̄1. (2.40)

This approximation to the lowest frequency of combustion oscillation is plotted in Figure

3 for comparison with the exact value.The linearized Galerkin method calculates the change in frequency due to the unsteady

heat input, but treats the shift as small. The form of heat input in equation (2.11), i.e.,no unsteady heat input per unit mass, vanishes identically for T 2 = T 1 and is infinitesimallysmall for small T 2 − T 1. Hence, the frequency predicted from the linearized Galerkinmethod and the true frequency agree exactly for T 2 = T 1. The two curves of frequencyagainst temperature ratio also have the same slope at this point, but they divergesignificantly for larger temperature ratios, as shown in Figure 3. By a temperature ratioof six, the Galerkin method predicts a frequency which is less than half the exact value.

Figure 3. The lowest frequency of oscillation for Case II with no unsteady heat input per unit mass for b = l /2. ——, Correct solution; ····, acoustic resonance frequency and the value predicted by investigating zeros of 1/G(0b); – – – – , frequency predicted by linearized Galerkin theory; · · · · , frequency predicted by ray theory.


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2.5.

The phase change of a high frequency sound wave, as it travels through a region of

gradually varying sound speed c̄(x), can be described by ray theory. According to raytheory, negligible sound energy is reflected as the ray propagates through such a region,and a sound wave of frequency , propagating in the x-direction undergoes a phase change l 0 dx/c̄(x) over a distance l [10].

Ray theory is appropriate for short wavelength waves travelling through a region inwhich the sound speed varies gradually and there is no unsteady heat input per unitvolume. It is therefore entirely the wrong physical limit for low frequency thermoacousticoscillations, where the wavelength is longer than, or comparable with, the length of theregion of heat input and the oscillations are driven by the unsteady heating. Nevertheless,ray theory is used in industry to estimate the frequency of thermoacoustic oscillations andso it is worthwhile to compare its predictions with the exact frequency in some modelcalculations.

According to the ray theory, the phase of a ray is changed by (b/c̄1 + (l − b)/c̄2), aftertravelling along the duct in Figure 1. The pressure wave undergoes phase changes at 0 and on reflection at the closed and open ends of the duct respectively. An estimate of theresonance frequency then follows from a requirement that the total phase change afterpropagation up and down the duct be an integer multiple of 2: i.e.,

2(b/c1 + (l − b)/c̄2) += 2n, n integer. (2.41)

This clearly gives frequencies which are independent of the form of the rate of heat release.The lowest frequency solution of equation (2.41) is plotted in Figure 3. It agrees with

the true frequency of oscillation only for the trivial case with no mean temperaturegradient, T 2 = T 1. When the mean temperature varies along the duct over a length scalewhich is much shorter than the wavelength, ray theory is inadequate. This is because itdoes not take reflections from the region of heat input into account and so is unable to

describe the flow perturbations correctly.

2.6.

Thermoacoustic oscillations have been investigated for the geometry illustrated in Figure1. Results for the frequency of oscillation for two particular forms of the unsteady heat

release rate are summarized in Figure 2. They show that the form of the relationshipbetween the unsteady heat input and the flow crucially affects the frequency of oscillation.Many authors have neglected this effect or treated it as small.

It has been said [3] that good approximations to the frequency of a thermoacoustic

oscillation can often be obtained by calculating the acoustic frequencies of the geometry.That is equivalent to solving the case of no unsteady heat release rate per unit volume.The results in Figure 3 serve as a warning that this procedure is not universally appropriate.

Hedge et al . [6] recommended the use of a Green function G to investigateone-dimensional perturbations in a duct. They then interpreted the roots of 1/G = 0 as thefrequency of the thermoacoustic oscillation. In Figure 3 it is shown that this leads to resultswhich can be significantly in error for the particular case of no unsteady heat input perunit mass. The analysis in section 2.3 explains why it is not sufficient simply to inspectthe Green function alone. The frequency of a thermoacoustic oscillation is a zero of 1/G

only if the rate of heat input per unit volume does not vary significantly in response tothe flow perturbations. If it does, the form of the coupling between the heat input and theflow affects the frequency and must be considered.

Culick and his colleagues have developed a Galerkin method to investigate thermoa-

coustic oscillations. This method acknowledges that the form of the coupling between the


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. . 568

unsteady heat input and the flow affects the frequency, but treats the shift in frequency,from that for no unsteady heat release rate per unit volume, as linear. The methodtherefore works exactly when there is no unsteady heat release rate per unit volume. Insection 2.4 the method was applied to the case of no unsteady heat input per unit massand once again the results are plotted in Figure 3. The predicted frequency and truefrequencies agree exactly for T 2 = T 1. The two curves of frequency against temperatureratio also have the same slope at this point, but they diverge significantly for larger

temperature ratios. By a temperature ratio of six, the Galerkin method predicts a frequencywhich is less than half the exact value.

In section 2.5 the frequency ‘‘predicted’’ by ray theory was compared with the truefrequency. There is no theoretical basis for believing that ray theory can accurately describe

low frequency thermoacoustic oscillations. Indeed, the plots in Figure 3 show that raytheory is unable to describe correctly the variation of the frequency of oscillation with thetemperature ratio T 2/T 1.

3. THE EFFECT OF A MEAN FLOW

Most practical thermoacoustic oscillations involve a mean flow. However, often theMach number of the oncoming flow is so small that it is tempting to neglect the mean

velocity entirely. The errors introduced by such an approximation are investigated in thissection.

A mean flow has two main consequences. Trivially, it affects the speed of thepropagation of the acoustic waves, which then travel downstream with speed c̄ + ū andupstream at c̄ − ū. In addition, a mean flow admits a second type of linear waves. Theseare convected with the mean velocity and involve convected entropy or vorticity. As aconsequence, the equations of conservation of mass, momentum and energy becomecoupled.

These effects may be illustrated by considering one-dimensional thermoacousticoscillations in the geometry in Figure 4. The inlet is choked so that the mass flow rate isconstant there:

('/ ¯1) + (u'/ū1) = 0 at x = 0. (3.1)

The boundary condition of a downstream open end can again [10] be taken as

p' = 0 at x = l . (3.2)

The heat input is concentrated at the fixed plane x = b, the rate of heat input per unitcross-sectional area being denoted by Q'(t).

Consider disturbances with time dependence eit. Upstream of the zone of heat input,there are acoustic waves propagating in both directions and the flow is isentropic. The

Figure 4. One-dimensional disturbances in a duct with a mean flow and mean rate of heat release.


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571

with

e1 = e−ib/c̄1(1 + M 1), e2 = eib/c̄1(1 − M 1), e3 = e−ib/c̄2(1 + M 2),

e4 = eib/c̄2(1 − M 2), e5 = e−il /c2(1 + M 2), e6 = eil /c̄2(1 − M 2),

a1 = M 1 + 12M 21 + (− 1)

−1, a2 = M 1 − 12M 21 − (− 1)

−1,

a3 = M 2 + 12M 22 + (− 1)

−1 and a4 = M 2 − 12M 22 − (− 1)

−1.

For Case II, with no unsteady heat input per unit mass, the last row in the matrixdefining X is to be replaced by

−c p (T 02 − T 01)(1+ M 1)e1/c̄21 c p (T 02 − T 01)(1− M 1)e2/c̄21 0 0 0 1. (3.16)

Equation (3.14) has a unique trivial solution A = B = C = D = S = Q =0 unless thedeterminant of X vanishes. Therefore only disturbances for which det X = 0 canpropagate. For given mean flow properties, this is an equation for and, in general, ithas complex roots. Since the time dependence is of the form e it, the sign of Imaginary determines whether disturbances grow or decay. Real gives the frequency of the mode.

It is a straightforward matter to determine the zeros of det X numerically, and someresults are shown in Figure 5. Inspection of the form of the matrix X in equation (3.15)shows that the leading order effect of the mean flow is O(M ). However, it is apparent fromthe numerical results that the importance of the the mean flow depends not only on the

inlet Mach number and stagnation temperature ratio (which affect the value of M 2), butalso on the form of the coupling between the heat input and the unsteady flow. It is strikinghow significant Mach number effects can be. For example, one might be tempted to thinkthat an inlet Mach number of 0·15 is sufficiently small for the mean flow to beneglected. But at this Mach number, the frequency of the thermoacoustic oscillation forCase II, T 02/T 01 = 6, is reduced to half its no flow value. Care needs to be taken beforesimply concluding on the basis of a low inlet Mach number that the mean flow may beneglected.

Figure 5. The lowest frequency of oscillation as a function of inlet Mach number for b = l /2. T 01 = 288 K, p̄2 =1 bar and T 02/T 01 = 6. ——, Case I (no unsteady rate of heat input); – – – –, Case II (no unsteady heat inputper unit mass); , Case I with no mean flow (root of equation (2.12)); , Case II with no mean flow (root of equation (2.14)).


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. . 572

4. THE EFFECTS OF DRAG

Most practical ways of adding heat also exert a drag force on the flow. In this section,the effect of this drag on the frequency of thermoacoustic oscillations is investigated.

If the instantaneous drag can be modelled quasi-steadily, the drag force per unit ductarea may be written in the form 12C D1u

21(t), where C D is a drag coefflcient. In particular,

when a bluff body or grid is used for heat transfer or to stabilize a flame, its blockage

reduces the duct area available for the flow. If the expansion from this reduced area isabrupt, an elementary one-dimensional quasi-steady analysis can be used to relate the dragcoefficient to the blockage ratio r. This gives [12]

C D = ( 1 − ( 1 − r)−1)2. (4.1)

For linear perturbations the momentum equation (3.6) becomes

p'1 + '1 ū21 + 2 ¯1ū1u'1 = p'2 + '2 ū22 + 2 ¯2ū2u'2 + 12C D'1 ū21 + C D ¯1ū1u'1 , (4.2)

and the third row of the matrix X in equation (3.15) is replaced by

( 1 + 2M 1s + M 21s)e1 ( 1 − 2M 1s + M 21s)e2 −(1+ M 2)2e3 −(1− M 2)2e4 M 22 0, (4.3)

where

s = 1 −1

2C D =1

2 + ( 1 − r)−1

−

1

2(1 − r)−2

, (4.4)after substitution for C D from equation (4.1).

Results for various inlet Mach numbers and blockage ratios are shown in Figure 6. Thedrag force exerted by a grid or flame holder with a blockage ratio of 25% or less is foundto have a negligible effect on the frequency of thermoacoustic oscillations for the rangeof inlet Mach numbers considered. A blockage ratio of 50% alters the frequency for Machnumbers greater than 0·1.

5. THE EFFECTS OF DISTRIBUTED HEAT INPUT

So far, the heat input has been considered as concentrated at a single plane, but in manypractical applications it is distributed over a significant length of the resonator. For

Figure 6. The lowest frequency of oscillation as a function of inlet Mach number for b = l /2. T 01 = 288 K, p̄2 = 1 bar and T 02/T 01 = 6 and various values of the blockage ratio. ——, No drag; – – – – , 25% blockage ratio;····, 50% blockage ratio.


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573

example, when a flame is stabilized in a duct in the wake of a bluff body, the combustionis initiated as the flow passes the flame holder, but the fluid continues to burn throughoutthe downstream portion of the duct [4]. The conditions under which a distributed heatinput can be considered as concentrated are discussed in this section.

It is sometimes said that heat input with axial extent d can be treated as lumped at asingle plane provided d is small in comparison with the acoustic wavelength (or moreprecisely provided d c̄/). However, this is an oversimplification when there is a meanflow. Then there is also an entropy wave, with the much shorter length scale ū/ (seeequation (3.4)).

In a region of distributed heat input’ the strengths of the acoustic and entropy waveschange continuously with position. It is then easier to determine the flow by working

directly from the equations of motion. The one-dimensional equations of mass, momentumand energy conservation for the mean flow are respectively:

d

dx ( ¯ū) = 0 ,

d

dx ( p̄ + ¯ū2) = 0 ,

d

dx (c pT + 12ū

2) =q̄(x) ¯ū

, (5.1–5.3)

where q(x, t) is the rate of heat input per unit volume. When q̄(x) is specified as a functionof x , and the flow at one axial position is known, equations (5.1)–(5.3) can be integratedwith respect to x in a straightforward way to determine the mean flow throughout the duct.

Linear perturbations can be calculated in a similar way. For oscillations proportional

to eit, the unsteady one-dimensional equations of mass, momentum and energy may bewritten in the form

(d/dx)( ¯u' +'ū) = − i', (5.4)

(d/dx)( p' +'ū2 + 2 ¯ūu')=−i( ¯u' +'ū), (5.5)

(d/dx)[( ¯u' +'ū)(c pT + 12ū2) + ¯ū(c pT ' + ūu')]=q' − i['(cvT + 12ū

2) + ¯(cvT ' + ūu')].

(5.6)

The upstream boundary conditions determine the inlet flow perturbations. Then, once themean flow, the frequency and the relationship between q' and the unsteady flow has beenspecified, equations (5.4)–(5.6) can be integrated along the duct, thus determining the flowperturbations at all positions. The details are given in reference [5].

At a general value of , the exit boundary condition is not satisfied. It is therefore

necessary to iterate in , at each state calculating the flow in the duct, until the complexvalues of , for which the exit boundary condition is met, are determined. These are thefrequencies of the thermoacoustic oscillations. Only disturbances with these particularfrequencies satisfy all the boundary conditions and can exist as free modes of the heatedduct.

As an illustrative sample, consider the duct geometry of section 3, where the inlet flowis isentropic and choked, and the exit is open. The inlet boundary condition is describedby equation (3.1), together with the isentropic relationships ' = p'/c̄21 and c pT ' = p'/ ¯1.Equation (3.2) shows the exit boundary condition to be p '(l , t) = 0. The mean heat inputis uniformly distributed over a length d , centred on x = b:

q̄(x) =

c p ¯1ū1(T 02 − T 01)/d

0

for x − b 12d for x − b 12d .

. (5.7)

The lowest frequency of oscillation has been calculated for this distributed heat input byintegrating equations (5.1)–(5.6), for Case I:

q'(x, t) = 0, no unsteady rate of heat input per unit volume. (5.8)


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. . 574

Figure 7. The lowest frequency of oscillation as a function of stagnation temperature ratio for a uniformdistribution of mean heat input over a distance d . Case I, no unsteady rate of heat input, q ' =0, with M 1 =0·1,b = l /2, T 01 = 288 K, p̄2 = 1 bar. ——, Concentrated mean heat input; – – – –, d =0·05l ; ····, d = 0·25l .

The frequencies predicted for two values of d /l are shown in Figure 7. Also plotted forcomparison are results for the concentrated mean heat input

q̄(x) = c p ¯1ū1(T 02 − T 01)(x − b). These are the roots of det X = 0, where X is the 6×6matrix in equation (3.15). It is apparent from these plots that there are significantdifferences between the predicted frequencies for concentrated and distributed mean heatinput. For example, distributing the mean heat over a length as short as 5% of the ductcan lead to a 25% change in the frequency of the oscillation at the higher temperatureratios, even though d /c̄1 is only 0·1.

The detailed form of the axial distribution of mean heat input is not so important. InFigure 8, results are given for a triangular distribution in the rate of mean input:

q̄(x) =4c p ¯1ū1(T 02 − T 01)(x + 12d − b)/d 4c p ¯1ū1(T 02 − T 01)(b + 12d − x)/d

0

for b − 12d x bfor bx b + 12d for x − b 12d . . (5.9)

The plots are virtually indistinguishable from those in Figure 7, where the mean input isdistributed uniformly over the same axial extent d and with the same centroid b.

Figure 8. As Figure 7, but with a triangular distribution of mean heat input.


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575

Hence, even if the mean heat has only a modest spatial distribution over a lengthd 0·1c̄/, it can still lead to appreciably different frequencies of oscillation from thatwhen the heat input is concentrated at a single axial plane.

These numerical results can be interpreted and explained analytically. In the Appendix,the conservation equations (5.4)–(5.6) are integrated across a region of heat input. Thereit is shown (see equation (A13)) that the pressure and velocity fluctuations on either sideof a region of heat input are unaffected by its spatial distribution over a length d provided

d c̄1 1 and

c p 2

1

¯(ū − ū2)s' dx p'. (5.10)

Here s ' denotes the perturbation in entropy and 1 and 2 are the positions x = b − 12d andb + 12d on either side of the region of heat input. Entropy fluctuations, with their shorterlength scale ū/ are more strongly influenced by a distribution in heat input. If the entropywaves propagating downstream of a region of heat input of axial length d are to have thesame strength as those produced by concentrated heat addition, it is evident from equation(A15) that d must be small in comparison with ū /. For a low Mach number mean flowthis is a strong constraint.

In the examples considered in this paper, the pipe exit is open and the boundarycondition of zero pressure perturbation does not involve the entropy waves. Hence, the

two conditions (5.10) are sufficient to ensure that the frequency of oscillation is unaffectedby a spatial distribution of heat input. The entropy fluctuations in the region of heat inputmust be determined before the integral in conditions (5.10) can be estimated.

For a mean flow and linear perturbations, the entropy equation T Ds/Dt = q(x, t) maybe written in the form

s't

+ ū s'x

=q̄R p̄ q'q̄ − u'ū − p' p̄. (5.11)

Consider Case I, in which q'(x, t) = 0. For disturbances of frequency and low mean flowMach numbers, equation (5.11) simplifies to

is' + ū s'/x = −q̄Ru'/ p̄ū. (5.12)

When d /ū is very small in comparison with unity, ū s'/x is the largest term on the leftside of equation (5.12). After substitution for q̄ (x) from equation (5.7), integration leadsto

ūs'(x)Oc pT 2T 1 − 1 x − b +12d

d u'1 for d ū 1, (5.13)

where the initial condition s '1 = 0 has been used. Note that the estimate of ū2s'2 obtainedby putting x = b + 12d in expression (5.13) agrees with the value calculated in equation (3.9)for d = 0. Expression (5.13) may be used to estimate the integral in conditions (5.10) andgives

c p 2

1

¯(ū − ū2)s' dxO

d

c̄1 T 2

T 1− 1

2

p'

for

d

ū 1. (5.14)

Hence the two conditions (d /c̄1)(T 2/T 1 − 1)2 1 and d /ū 1 are sufficient to ensurethat the frequency of oscillation is not altered by the distribution of heat input over a small

length d .


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577

Figure 10. As Figure 7, but for q '/q̄ = (u'/ū) + ( p'/ p̄).

equations (5.1)–(5.6), with the unsteady heat input (5.18). Results for the concentratedmean heat input were found by replacing the last row in the matrix X in equation (3.15)by

−c p (T 02 − T 01)(1+ M 1)e1/c̄21 c p (T 02 − T 01)(1− M 1)e2/c̄21 0 0 0 1, (5.19)

and finding the roots of det X = 0. As predicted, for the form of unsteady heat input (5.18),the frequency of oscillation is less affected by the distribution of heat input than that forCase I with q' = 0 shown in Figure 7.

The results for Case II, in which the rate of release is proportional to the mass flow rate,show similar trends. For a concentrated heat input, the frequency of oscillation is a rootof det X =0, where X is described by equation (3.16). When the heat input is distributed,the unsteady heat input generalises to

Case II, q'

q̄ =

u'ū

+' ¯

, no unsteady heat input per unit mass. (5.20)

This is close in form to equation (5.18) and only weak entropy waves are generated. Onceagain, the frequency of oscillation is altered little by the distribution in heat input.

In summary, entropy waves have a short length scale, ū/, and are more affected byan axial distribution in the heat input than the acoustic waves with their longer length scale

Figure 11. As Figure 7, but for Case II with q '/q̄ = (u'/ū) + ('/ ¯).


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. . 578

c̄/. Acoustic and entropy waves are coupled through a region of heat input. The twoconditions d /c̄11 and ¯(ū − ū2)s' dx/c p p' must be satisfied, if the acoustic wavesare to be unaffected by the distribution of heat input over an axial distance d . Results inFigure 7, for M 1 = 0·1 and Case I with no unsteady rate of heat input per unit volume,show that distributing the heat input over a length as short as 5% of the duct can leadto a 25% change in the frequency of oscillation at the higher temperature ratios. Toguarantee in general that heat input distributed over a length d can be treated asconcentrated, one requires both d /ū1 1 and d /c̄1 1. This is very restrictive.

However, flow conditions that reduce the volume term ¯(ū − ū2)s' dx, that accountsfor the coupling between the acoustic and the entropy waves, reduce the sensitivity of thethermoacoustic oscillations to the axial extent of the heat input. The results in Figures 7–11

show that this can be achieved in several ways:(i) A reduction in the mean temperature ratio. The results for small (T 02 − T 01)/T 01 in

Figure 7 show that then the frequency of oscillation is only slightly affected by a modestdistribution in heat input.

(ii) A reduction in mean inlet Mach number . A comparison of Figures 7 and 9 showsthat this reduces the effect of the axial extent of the heat input.

(iii) Particular forms for the unsteady heat input . The unsteady heat input in equation(5.18) generates no entropy waves, while only weak waves are produced when the heatinput per unit mass is constant. The results in Figures 10 and 11 show that then thefrequencies of the thermoacoustic oscillations are insensitive to a spatial distribution inheat input, when the downstream boundary condition involves only the acoustic waves.

6. CONCLUSIONS

Model problems with very simple geometries have been considered to investigate theinfluence of various flow effects on the frequency of thermoacoustic oscillations.

The form of the coupling between the heat input and the unsteady flow has beendemonstrated to have a crucial effect on the frequency of oscillation. Indeed, for theparticular case of heating at a single axial plane in a duct with one open and one closedend, there is nearly a 60% difference between the frequency predicted for no unsteady heatinput per unit mass and no unsteady rate of heat input. A number of calculation methods

recommended in the literature have been tested by applying them to model problems. Thishas shown that they do not account fully for this effect.The presence of a mean flow significantly complicates the analysis of a thermoacoustic

oscillation, since an entropy wave becomes coupled to the acoustic field. It is, therefore,

tempting to neglect the mean flow, whenever the inlet Mach number is low. However, greatcaution must be exercised before making such an assumption. Mean flow effects are foundto be surprisingly significant. For example, with a mean stagnation temperature ratio of six, the frequency of a thermoacoustic oscillation for an inlet Mach number of 0·15 canbe reduced to half its no flow value.

Many practical ways of adding heat also present a blockage to the flow and exert a dragforce on the fluid. A drag force exerted by a grid or flame holder with a blockage ratioof 25% or less is found to have a negligible effect on the frequency of thermoacousticoscillations for inlet Mach numbers in the range 0M 10·15. A blockage ratio of 50%

alters the frequency for inlet Mach numbers greater than 0·1.Even a modest distribution of the heat input over an axial distance d can lead to a

significantly different frequency of oscillation from that when the heat input is concen-trated. For example, even an axial extent of heat input as short as 5% of the duct length

can lead to a 25% change in the frequency of oscillation. Entropy waves have a short


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579

length scale ū / and are more affected by an axial distribution in the heat input than theacoustic waves with their longer length scale c̄/.

Hence, to guarantee, in general, the heat input distributed over a length d can be treatedas concentrated, one requires both d /ū1 and d /c̄1. However, these conditions canbe relaxed when the downstream boundary condition involves only acoustic waves, andthe acoustic waves (and hence the frequency of oscillation) are not affected by the entropywaves. This is the case at low mean flow Mach numbers or for mean temperature ratios

near unity. It also occurs for certain forms of the unsteady heat input, which lead to noor only very weak entropy waves. In all these cases heat input, distributed over a distance

d , can be considered as concentrated provided that d is compact.

REFERENCES

1. L R 1896 The Theory of Sound , London: Macmillan.2. S. M. C and T. J. P 1988 Proceedings of the Institute of Acoustics 10, 103–153.

Interactions between acoustics and combustion.3. F. E. C. C 1988 AGARD-CP -450. Combustion instabilities in liquid-fuelled propulsion

systems—an overview.4. P. J. L 1988 Journal of Fluid Mechanics 193, 417–443. Reheat buzz—an acoustically

coupled combustion instability, part I: experiment.5. G. J. B, A. P. D and P. J. L 1988 Journal of Fluid Mechanics 193,

445–473. Reheat buzz: an acoustically coupled combustion instability, part 2: theory.6. U. G. H, D. R, B. T. Z and B. R. D 1987 AIAA-87-0216. Fluid mechanicallycoupled combustion-instabilities in ramjet combustors.

7. A. P. D 1988 AGARD-CP -450. Reheat buzz—an acoustically coupled combustioninstability.

8. F. A. W 1965 Combustion Theory. Reading, Massachusetts: Addison-Wesley.9. U. G. H, D. R and B. T. Z 1988 American Institute of Aeronautics and Astronautics

Journal 26, 532–537. Sound generation by ducted flames.10. A. P. D and J. E. F W 1983 Sound and Sources of Sound . Chichester: Ellis

Horwood.11. A. M. C 1982 Journal of Fluid Mechanics 121, 59–105. Low frequency sound radiation

and generation due to the interaction of unsteady flow with a jet pipe.12. A. P. D and G. J. B 1984 AIAA-84-2321. Reheat buzz—an acoustically driven

combustion instability.

APPENDIX

The aim of this Appendix is to determine conditions under which equations (5.4)–(5.6),describing changes through a distributed region of heat input, integrate to give the jump

conditions (3.5)–(3.7) for concentrated heat input.Integration of equations (5.4)–(5.6) across the region of heat input, from position 1,

x = b = 12d , to 2, x = b +12d , leads to

[ ¯u' +'ū]21 = −i 2

1

' dx, (A1)

[ p' +'ū2 + 2 ¯ūu']21 = −i 2

1

( ¯u' +'ū) dx, (A2)

[( ¯u' +'ū)c pT 0 + ¯ū(c pT ' + ūu')]21 − Q'=−i

2

1

('(cvT + 12

ū2) + ¯(cvT ' + ūu')) dx,

(A3)

where Q ' = 21 q '(x) dx. It is evident that equations (A1)–(A3) reduce to the concentratedheat input jump conditions when their right sides are negligible.


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. . 580

In a perfect gas, the entropy s = cv ln p − c p ln + constant. For linear perturbations,this may be rearranged to show that

' = ( p'/c̄2) − ( ¯/c p )s'. (A4)Similarly,

c pT ' = ( p'/ ¯ ) + T s'. (A5)The inlet boundary conditions ensure that there are no incoming entropy waves and sos'1 = 0. Equations (A1)–(A3) will now be rearranged so that they relate p'2 , u'2 and s'2 tothe inlet flow with additional integrals over the region of heat input.

First note that '2 can be eliminated by subtracting the product of equation (A1) andū2 from equation (A2) to give

p'2 + ¯2ū2u'2 = p'1 + '1 ū1(ū1 − ū2) + ¯1(2ū1 − ū2)u'1−i 2

1

('(ū − ū2) + ¯u') dx. (A6)

A second equation for p'2 and u'2 is derived by noting that the energy equation,

[u(c pT + 12u2)]21 = Q −

2

1

t ((cvT + 12u

2)) dx, (A7)

may be rewritten in the form

− 1 pu +12u3

2

1

= Q − 2

1

t p− 1 +12u2 dx, (A8)

since c pT = p/(− 1). Equation (A8) is equivalent to

− 1 pu2

1

+1u1[12u2]21 + [u]21 12u

22 = Q − t p− 1 + 12u2 dx. (A9)

For linear perturbations of frequency this leads to

− 1 (u'2 p̄2 + ū2 p'2 ) + ¯1ū1ū2u'2

=Q' +

− 1 (u'1 p̄1 + ū1 p'1 ) + ¯1ū21u'1 + ( ¯1u'1 + '1 ū1)

12(ū

21 − ū22)

−i 2

1 p'− 1 + ¯ūu' +'12(ū2 − ū22 ) dx, (A10)

after substitution for [ ¯u' +'ū]21 from equation (A1).When ' is expanded in terms of pressure and entropy from equation (A4), equations

(A6) and (A10) become, respectively,

p'2 + ¯2ū2u'2 = p'1 +'1 ū1(ū1 − ū2) + ¯1(2ū1 − ū2)u'1

−i 2

1 p'c̄2

− ¯s'c p (ū − ū2) + ¯u' dx, (A11)

and

− 1 (u'2 p̄2 + ū2 p'2 ) + ¯1ū1ū2u'2

=Q' + − 1 (u'1 p̄1 + ū1 p'1 ) + ¯1ū21u'1 + ( ¯1u'1 + '1 ū1)12(ū21 − ū22)

−i 2

1 p'− 1

+ ¯ūu' + p'c̄2 − ¯c p s'12(ū2 − ū22) dx. (A12)


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581

Equations (A11) and (A12) describe the changes in p' and u' across a region of heatdistributed over a length d . It is evident that these are equal to the changes for aconcentrated heat input provided that the contributions from the integrals are negligible:i.e., provided that

d c̄1 1 and

c p

2

1

¯(ū − ū2)s' dx p', (A13)

and the flow is subsonic. These conditions are sufficient to ensure that the acoustic waveson either side of a region of heat input are unaffected by its distribution over a finite length.

To determine s '2 , the strength of the entropy wave, note that equations (A1) and (A3)may be combined to give a modified energy equation:

¯1ū1(c pT '2 + ū2u'2 − c pT '1 − ū1u'1 ) + ( ¯1u'1 + '1 ū1)c p (T 02 − T 01) − Q'

=i 2

1 '(c pT 02 − cvT − 12ū2) − ¯(cvT ' + ūu') dx. (A14)Substitution for ' and T ' from equations (A4) and (A5) into equation (A14) leads to

¯1ū1T 2s'2 + p'2

¯2+ ū2u'2= Q' + ¯1ū1 p'

1

¯1+ ū1u'1− ( ¯1u'1 +'1 ū1)c p (T 02 − T 01)

+i 2

1 −s' ¯c p

(c pT 02 − 12ū2) +

p'c̄2

c p (T 02 − T 0) − ¯ūu' dx. (A.15)Estimation of the integral in equation (A15) shows that

d /ū1 (A16)

is required if the strength of an entropy wave produced by heat addition is to be unaffectedwhen the heat input is distributed over a length d .

The implications of the conditions (A13) and (A16) are discussed in section 5.

Dowling 1995fsdfasdfasdf

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