GT GT-38168

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1 Proceedings of ASME Turbo Expo 23 Power for Land, Sea, and Air June 16 19, 23, Atlanta, Georgia, USA Proceedings of ASME TURBO EXPO 23 Power for Land, Sea, and Air June 16 19, 23, Atlanta, Georgia, USA GT GT MODELLING OF CIRCUMFERENTIAL MODAL COUPLING DUE TO HELMHOLTZ RESONATORS Simon R. Stow and Ann P. Dowling Department of Engineering, University of Cambridge Cambridge, CB2 1PZ, United Kingdom Tel: , Fax: , ABSTRACT Lean premixed prevaporised (LPP) combustion can reduce NO x emissions from gas turbines, but often leads to combustion instability. Acoustic waves produce fluctuations in heat release, for instance by perturbing the fuel-air ratio or flame shape. These heat fluctuations will in turn generate more acoustic waves and in some situations self-sustained oscillations can result. A linear model for thermoacoustic oscillations in LPP combustors is described. A thin annular geometry is assumed and so circumferential modes are included but radial dependence is ignored. The formulation is in terms of a network of modules such as straight ducts and area changes. At certain operating conditions, the flow is predicted to be unstable, with linear waves growing in amplitude. Helmholtz resonators can be used to absorb acoustic energy and, when carefully designed and installed at appropriate locations, can stabilise the flow. Helmholtz resonators are included in the model. Connecting a Helmholtz resonator to an annular duct destroys the axisymmetry of the geometry. This results in coupling of the circumferential modes which must be calculated. The model is used to investigate the best arrangement of resonators around the circumference of an annular duct to achieve maximum damping of a circumferential oscillation. c speed of sound e energy flux H specific stagnation enthalpy k constant in flame model M Mach number m mass flux N largest circumferential wavenumber considered n circumferential wavenumber p pressure q axial momentum flux R (mean) radius S entropy T temperature u axial velocity w circumferential velocity θ circumferential coordinate κ Rayleigh conductivity of Helmholtz resonator neck λ n parameter for mode n at main inlet µ n boundary condition on mode n at main outlet ρ density ω complex frequency NOMENCLATURE A cross-sectional area A E complex amplitude of entropy wave A V complex amplitude of vorticity wave complex amplitudes of acoustic waves A ± Superscripts () mean value () perturbation () ˆ complex perturbation, assuming factor of e iωt 1 Copyright 23 by ASME

2 1 INTRODUCTION LPP combustion is a method of reducing NO x emissions from gas turbines by mixing the fuel and air more evenly before combustion. However LPP combustion is highly susceptible to thermoacoustic oscillations. The interaction between sound waves and combustion can lead to large pressure fluctuations, resulting in structural damage. Active control (such as by modulating the fuel flow, see [1]) could potentially remove such instabilities. However a simpler method may be to use passive damping devices such as Helmholtz resonators. A combustion instability model that includes the effects of Helmholtz resonators is required in order to find their best design and placement. We present a linear theory to predict combustion instabilities in thin annular LPP combustors, typical of aeroengines. As a thin annular geometry is assumed, radial variation is ignored. The formulation is in terms of a network of modules describing the feature of the geometry. Helmholtz resonators are included in the model. These destroy the axisymmetry of the geometry and hence introduce coupling of circumferential modes which must be considered. The use of Helmholtz resonators to damp combustion instabilities has been considered by Bellucci et al. [2]. Their results were calculated using a numerical acoustic package meaning that modal coupling did not have to be explicitly considered. Evesque and Polifke [3] considered coupling of circumferential modes in a thin annular geometry due to non-identical premix ducts. Krüger et al. [4, 5] have used networks of modules to prediction combustion instabilities in annular combustors. Their modules were one dimensional with the axial and circumferential variation being given by linking modules in both directions. Oscillations in annular combustors have also been computed by Hsiao et al. [6, 7] using a hybrid CFD/acoustics-based model on a three-dimensional grid of cells. In Section 2 we describe the various components of the model and the method of solution, including the calculation of the circumferential modal coupling. In Section 3 we apply the model to some test cases. In particular we investigate the best placement of Helmholtz resonators to damp a circumferential oscillation. Our conclusions are presented in Section 4. 2 DESCRIPTION OF THE MODEL We now give a description of the model. The geometry consists of string of modules starting with a specified acoustic boundary condition at the inlet and finishing with another boundary condition at the outlet. The modules describe features such as straight ducts, area changes and combustion zones. Modelling for Helmholtz resonators is also included. The model described by the authors in [8] treated the circumferential modes as independent. The introduction of Helmholtz resonators means that circumferential modes become coupled. We use cylindrical polar coordinates x, r and θ and write the pressure, density, temperature and velocity as p, ρ, T and (u,v,w), respectively. We will assume a perfect gas, p = R gas ρt. Since a thin annular geometry is assumed, we will ignore variation in the radial direction and take v to be zero. The flow is taken to be composed of a steady axial mean flow (denoted by bars) and a small perturbation (denoted by dashes), e.g. p = p(x) + p (x,θ,t). We will consider perturbations with complex frequency ω and hence write p = Re[ ˆp(x,θ)e iωt ], with ˆp(x,θ) = n= ˆp n (x)e inθ, and similarly for the other perturbation variables. Here ˆp n represents the component with circumferential wavenumber n. For n large ( n > N, say) the component will be highly cutoff, i.e. instead of propagating it will decay rapidly with x, and hence can be ignored. We can therefore approximate by taking ˆp = N n= N ˆpn e inθ, etc. The key part of the model is the straight ducts, described in Section 2.1. Other components of the model are described in Sections The method of solution to find resonate modes of the geometry is given in Section Straight ducts The straight ducts are assumed to be the only components of the geometry with significant axial length, i.e. all other features are treated as acoustically compact. The ducts are assumed to be thin and annular, with cross-sectional area A and (mean) radius R. The mean flow along the duct is constant and hence is given by the values at the start of the duct. Each circumferential component of the perturbations can be written the sum of four waves, where [ ˆp n,û n, ˆρ n,ŵ n] T = F w(x) (1a) w(x) = [ A + e ik + x,a e ik x,a E e ik x,a V e ik x] T, 1 1 F = k + /( ρα + ) k /( ρα ) nū/( ρ c 2 ) 1/ c 2 1/ c 2 1/ c 2, n/(r ρα + ) n/(r ρα ) ωr/( ρ c 2 ) (1b) (1c) k ± = Mω (ω 2 ωc 2 ) 1/2 c(1 M 2, ) α ± = ω + ūk ±, (1d) ω c = n c R (1 M 2 ) 1/2, k = ω/ū, (1e) c is the speed of sound and M is the Mach number (see [8, 9]). Note that the flow is assumed to be subsonic. The cutoff frequency of the duct is ω c /(2π). A + and A represent two acoustic waves, whereas A E and A V represent convected entropy and vorticity waves, respectively. 2 Copyright 23 by ASME

3 Given ˆp n, û n, ˆρ n and ŵ n at the start of the duct we can find w there by inverting Eq. (1a). The wave amplitudes at the end of the duct are then given by Eq. (1b). In the model we often consider perturbations in fluxes. The mass flux is m = Aρu, the axialmomentum flux is q = Ap + mu, the angular-momentum flux is mrw and the energy flux is e = mh, where H = c p T u2 is the (specific) stagnation enthalpy. We will also require the perturbation in entropy, Ŝ n = c v ˆp n / p c p ˆρ n / ρ. Using matrices we can easily convert between wave amplitudes, flow perturbations and flux (and entropy) perturbations. 2.2 Area changes and combustion At regions of area change and zones of combustion, conservation laws are used to relate the fluxes before and after the change. These regions are assumed to be axisymmetric and so each circumferential mode can be treated independently. For combustion, the energy flux is increased by the rate of heat input, Q. Typically, Q is found by specifying the value of the mean temperature after combustion. ˆQ is given by a flame transfer function which may come from an analytical model (see [1]), from CFD calculations (see [11, 12]) or from experiments (see [13]). A simple model is to take ˆQ ˆmn = k n Q m e iωτ (2) where ˆm and m are the values at the point of fuel input, τ is the convection time from fuel input to combustion and k is a constant, typically taken to be unity. More details on the modelling of area changes and combustion are given in [8]. 2.3 Inlet At the inlet, the mean flow is calculated by specifying the mass flux, pressure and temperature. For the perturbation, an acoustic boundary condition is specified. This may be an open end ( ˆp = A E = A V = ), a closed end (û = A E = A V = ), a semiinfinite pipe (no downstream acoustic waves and A E = A V = ) or a choked inlet ( ˆm = Ĥ = Rŵ =, see [9]). These boundary conditions must apply to all the circumferential mode independently. Hence the perturbations for each circumferential mode, n, can be found except for an unknown parameter λ n, say. For instance, for a choked inlet we may take ˆp n = λ n, û n = λ n (ū/ p)c p T /(c p T +ū 2 ), ˆρ n = λ n ( ρ/ p)c p T /(c p T +ū 2 ), ŵ n =. If the circumferential modes are coupled, the relative magnitudes and phases of the λ n must be calculated. 2.4 Outlet At the outlet, an acoustic boundary condition is specified. This may be an open end ( ˆp = ), a closed end (û = ), a semiinfinite pipe (no upstream acoustic waves) or a choked outlet r 1 2 Figure 1. θ centreline x 1,2 θ = θh Schematic diagram of flow near a Helmholtz resonator. ( ˆM =, see [9]). This boundary condition must apply to all the circumferential modes. We define µ n to be the error in the boundary condition for circumferential mode n. For instance, for a choked inlet we may take µ n = ρ c(2û ū ˆT / T ). The method to find solutions satisfying these boundary conditions is described in Section Helmholtz resonator A Helmholtz resonator is a damping device consisting of a large volume connected via a short neck to a duct in which we wish oscillations to be reduced. Connecting a Helmholtz resonator to an annular duct will destroy the axisymmetry of the geometry. The effect of this asymmetry on the mean will be small and is ignored. However for the perturbations, the circumferential components will become coupled. Helmholtz resonators have an increased damping effect if there is a net flow through the neck into the duct. Typically the Helmholtz resonator would be attached to the combustion chamber of the gas turbine, with the through flow supplied by connecting the resonator volume (or neck) to part of the cooling air system. The flow would provide part of the cooling as well as increasing the damping from the resonator. A schematic diagram of flow near a Helmholtz resonator is shown in Fig. 1. In the figure the flow runs along an annular duct from regions 1 to 2. The Helmholtz resonator consists of a large volume (region 4) connected to the duct between the two regions via a neck (region 3) at circumferential location θ = θ H. Even with a net flow through the neck, the effect of the resonator on the mean flow will be small and hence in the present model the mean flow at region 2 is taken to be the same as at region 1. For the perturbations, conservation of mass, energy, axial momentum and angular momentum give ˆm 2 (θ) = ˆm 1 (θ)+ ˆm 3,θ (θ), ê 2 (θ) = ê 1 (θ) + ê 3,θ (θ), ˆq 2 (θ) = ˆq 1 (θ) and Rŵ 2 (θ) = Rŵ 1 (θ). Here ˆm 3,θ (θ) is the mass flux into the annular duct from the neck of the resonator and clearly should only be nonzero for θ = θ H. Writing the mass flux in the neck itself as ˆm 3, we should take ˆm 3,θ (θ) = 2πδ(θ θ H ) ˆm 3. Similarly, ê 3,θ (θ) = 2πδ(θ θ H )ê 3 is the energy flux from the neck into the annular duct. (Since the flow in the neck is assumed to be one-dimensional and perpendicular to the annular duct, its contribution to the axial and θ r 3 Copyright 23 by ASME

4 angular momentum is zero.) The mass flow, ˆm 3, is driven by the difference between the pressure at the resonator-volume end of the neck, ˆp 4, and the pressure at the annular-duct end of the neck, ˆp 1 (θ H ). The Rayleigh conductivity, κ, of the neck is defined by κ = iω ˆm 3 ˆp 4 ˆp 1 (θ H ). (3) The rate of change of mass in the resonator volume is related to the mass flux out of it: m 3 = (V 4 ρ 4 )/ t, ˆm 3 = iωv 4 ˆρ 4 = iωv 4 ˆp 4 / c 2 4 where V 4 is the volume of the resonator. Here we have assumed that the perturbations in the resonator volume are isentropic and that any perturbations in the cooling supply have a negligible influence. This leads to κ ˆp ˆm 3 = 1 (θ H ) iω + c 2 4 κ/(iωv (4) 4 ). We assume that the flow shearing past the neck, u 1, is negligible compared with the through it, u 3. Hence for a circular neck with radius r H (= A 3 /π) and negligible length, the result by Howe [14] for κ/(2r H ) as a function of Strouhal number, ωr H /(ū 3 /2), is applicable. We denote the Rayleigh conductivity for that case as κ. For a neck of finite length, l 3, it has been shown that κ can be modelled as κ = κ /(1 + κ l 3 /A 3 ) (see [15]). Therefore ˆm 3 can be found from ˆp 1 (θ H ) = N n= N ˆpn 1 einθ H. Writing the mass flux equation in terms of the circumferential components we have N ˆm n n = N 2 ein θ N = n = N ˆm n 1 ein θ 2πδ(θ θ H ) ˆm 3. (5) Multiplying this equation by e inθ and integrating between and 2π gives ˆm n 2 = ˆmn 1 + ˆm 3 e inθ H, for all values of n. Similarly, ê n 2 = ên 1 + ê 3 e inθ H, ˆq n 2 = ˆqn 1 and Rŵn 2 = Rŵn 1. To calculate ê 3, we take Ĥ 3 = Ĥ 4 = [(γ 1)/γ]c p T ˆp 4 / p. Hence given all the circumferential components of the perturbations at 1 we can find all the components at 2. The modes are coupled because ˆp 1 (θ H ) depends on all the components. 2.6 Method of solution In this section we describe the calculation of the mean flow and perturbations. For the mean flow, we start at the inlet where the flow is known and step through the modules to find the flow in the rest of the geometry. For the perturbations, given ω and λ = [ λ N,...,λ N] T, all the circumferential components at the main inlet are known. We step through the geometry, as for the mean flow, calculating all the circumferential modes at each module before continuing to the next. For the solution thus found, each mode will have an error at the outlet, µ n. We must find ω and λ to satisfying µ n =, thus giving a resonant mode of the geometry. For a given ω, we define the matrix M to be such that M n,m is the value of µ n for the solution with λ = e m, where e j = δ i i, j. For a general λ, µ n = M n,m λ m since the perturbations are linear. Hence for the correct values of ω and λ, Mλ =. Since for a solution to exist λ, this implies that detm =. Thus the procedure to find a complex resonant frequency, ω, is to first guess the value of ω and calculate the matrix M (which involves calculating 2N +1 perturbation solutions), and then iterate the value of ω to achieve detm =. For this value of ω, a λ will exist giving Mλ =. Finally, this correct λ is calculated using an inverse iteration method as follows: We start with a random guess of λ which is normalised by replacing λ by λ/λ n, where n is such that λ n = max λ n. This is then updated by solving Mλ new = λ old, and re-normalised. This process is then iterated until the change in λ is small. M will have an eigenvalue that is very small (though not exactly zero due to numerical errors) so these iterations will converge to the corresponding eigenvector, which is the value of λ required. The modeshape for the resonant mode can then be calculated using this λ. The resonant frequency of the mode is Reω/(2π) and its growth rate is defined as Imω. The growth rate indicates the rate at which linear perturbations will grow or decay with time, in particular modes with positive growth rate are unstable whereas modes with negative growth rate are stable. In the present work, Helmholtz resonators are the only source of modal coupling as we will assumed all other components of the geometry are axisymmetric. (Annular LPP combustors are typically axisymmetric except for the presence of premix ducts. In fact, it can be shown that a ring of identical premix ducts does not introduce coupling of circumferential modes provided that N is less than half the number of ducts [16]; see also [17] where premix ducts lead to coupling of radial modes only, and [3] where circumferential modal coupling only occurred for non-identical premix ducts). 3 EXAMPLE RESULTS We now apply the model to some examples to investigate the effect of Helmholtz resonators. The geometry for the following examples is a uniform annular duct with length.2 m, radius.3 m and cross-sectional area.3 m 2. This simple geometry means that it will be easier to identify the effect of resonators themselves. For the mean flow, the temperature is 2 K, the pressure is 4 MPa and the mass flow is 5 kg s 1. The inlet and outlet of the duct are taken to be acoustically closed ends. All the resonators are taken to have neck radius 7 mm, neck length 3 mm and a temperature of 1 K inside the volume, with speed of the flow through the neck set to be 1 m s 1 (see [15] for an investigation of these parameters). When no Helmholtz resonators are present, the circumferen- 4 Copyright 23 by ASME

5 Table 1. n 1 st axial mode 2 nd axial mode ± ± ± Resonant frequencies of duct with no Helmholtz resonators. tial modes are not coupled and their lowest resonant frequencies are shown in Table 1. For n, the first frequency corresponds to a mode independent of axial position and is the cutoff frequency for that circumferential wave. The growth rates for all of these modes is zero, i.e. they are neutrally stable. We now add Helmholtz resonators to the geometry to try make the growth rate of modes become large and negative, indicating a damping of the mode. Before concentrating on the circumferential modes, we briefly investigate the plane mode at 213 Hz. The modeshape of this mode is a half wave, with a pressure node halfway along the duct and anti-nodes at the inlet and outlet. If a Helmholtz resonator is added, this mode is replaced by two modes, one at a slightly lower frequency than the original and one at a slightly higher frequency. However, in general one of these modes has a very large negative growth rate. For both modes the circumferential components are coupled, although the main component is n =. Figure 2 shows the variation of frequency and growth rate of one of the modes with resonator position (relative to the duct inlet) for a resonator volume of m 3. (The other mode is not of much interest as it has highly damped, having a growth rate of less than 8.) The solid, dashed, dash-dotted and dotted lines denote N =, 1, 4 and 1, respectively. The effect of including more circumferential components by increasing N is noticeable but small, and even treating the problem as uncoupled (i.e. taking N = ) is a fair approximation. As we would expect the resonator has no effect when placed halfway along the duct, since this is a pressure node and so there is no unsteady mass flow into the resonator neck (see Eq. (4)). Furthermore, we would expect the resonator to provide more damping when placed at a position where the pressure amplitude of the original mode is higher, in particular the best positions should be the inlet and outlet. This is approximately the case. (The fact that the correspondence is not exact appears to due to the resonator volume not being the ideal value, which is m 3.) In Fig. 3, the circles show the amplitude of circumferential components of pressure perturbation at the resonator neck for this mode with the resonator at x = and taking N = 1. We see that by far the main component is n =, with the components being very small position of resonator (m) Figure position of resonator (m) Variation of frequency and growth rate with resonator position for less stable mode originally at 213 Hz (n = ). The solid, dashed, dash-dotted and dotted lines denote N =, 1, 4 and 1, respectively. amplitude of pressure component n Figure 3. Amplitude of circumferential components of pressure perturbation at resonator neck for various modes. (Note that the scale is arbitrary.) The circles, crosses, squares and pluses denote modes originally at 213 Hz (n = ), 447 Hz (n = ±1), 893 Hz (n = ±2) and 134 Hz (n = ±3), respectively. for n 2. We now consider the n = ±1 mode at 447 Hz. This mode does not vary axially. Note that n = 1 and n = 1 represent circumferential waves spinning in opposite directions. Placing a single Helmholtz resonator on the duct (say at θ = ) has no damping effect. The n = 1 and n = 1 waves become coupled to produce a standing wave with a pressure node along θ = (and, by symmetry, along θ = 18 also). In order to have damping, at least two resonator must be used. Figure 4 shows the variation of frequency and growth rate with N for the case of two resonators with volume m 3. The first resonator is at θ = and the second at θ = 6. Here, and throughout the rest of the examples, the axial location of all resonators is halfway along the duct. The variation with N is seen to be small. As shown in the figure, the original two modes (n = 1 and n = 1) have been replaced by four coupled modes. The crosses in Fig. 3 show the amplitude of 5 Copyright 23 by ASME

6 N N Figure 4. Variation of frequency and growth rate with N for modes originally at 447 Hz (n = ±1) volume of both resonators (m 3 ) x volume of both resonators (m 3 ) x 1 4 Figure 6. Variation of frequency and growth rate with resonator volume for modes originally at 447 Hz (n = ±1). x (m) θ (degrees) Figure 5. Amplitude of pressure perturbation for 459 Hz mode. (Note that the scale is arbitrary.) The circles indicate the positions of the two resonators. circumferential components of pressure perturbation at the resonator neck for the modes (taking N = 1). Both figures suggest that N = 4 is more than sufficient; unless otherwise stated this is the value used from now on. Two of the modes (at 451 Hz and 417 Hz) have a pressure node between the resonators, whereas the other two (at 459 Hz and 412 Hz) have a pressure anti-node there. The modeshape of the 459 Hz mode (for N = 4) is shown in Fig. 5. As is the case for all four modes, we see that there is little axial variation. Figure 6 shows the variation of the four modes as the volume of the resonators is changed. The best configuration is when the largest growth rate is smallest, i.e. there is maximum overall damping. Here this occurs at a volume of m 3. Sim amplitude of pressure perturbation Figure 7. Variation of frequency and growth rate with circumferential location of second resonator for modes originally at 447 Hz (n = ±1). (The first resonator is fixed at θ =.) The resonator volume is m 3. ple theory suggests that the resonator volume should be taken to be c 2 4 A 3 /[l 3 (2π f )2 ], where f is the frequency of the (original) oscillation, c 4 is the speed of sound in the resonator volume, and A 3 and l 3 are the cross-section area and length of the neck, respectively (see [18]). Here this corresponds to m 3. Equation (4) suggests a better estimate is c 2 4 Re(κ)/(2π f )2, with κ being the Rayleigh conductivity at the original frequency, giving m 3. In Fig. 7 we show the effect of varying the circumferential location of second resonator (the first being fixed at θ = ) for a resonator volume of m 3. The best angle here is 94. This is roughly 9, which is perhaps what we 6 Copyright 23 by ASME

7 Figure 8. Variation of frequency and growth rate with circumferential location of second resonator for modes originally at 447 Hz (n = ±1). (The first resonator is fixed at θ =.) The resonator volume is m Figure 9. Variation of growth rate with circumferential location of second resonator for two least stable modes originally at 893 Hz (n = ±2). (The first resonator is fixed at θ =.) might expect for a predominantly n = ±1 mode. Figure 8 shows equivalent results for a resonator volume of m 3. The best angle is now 117, however there is not much variation in the growth rates between around 5 and 14. This shows that if the volume of the resonators is well chosen, the precise angle is less important. Next we consider the n = ±2 mode at 893 Hz and the n = ±3 mode at 134 Hz. As before, in both cases adding two Helmholtz resonators replaced the original uncoupled modes by four coupled modes. Figure 9 show the variation in growth rate for two of the modes that were originally the 893 Hz mode for a resonator volume of m 3 (the other two modes have large negative growth rates and are not shown). We find that the best angles are 47 and 135. Figure 1 shows corresponding results for the 134 Hz mode, taking the resonator volumes to be m 3. The best angles are now 31, 91 and 15. The amplitude of circumferential components of pressure perturbation at the resonator neck for these four modes are denoted in circumferential location of third resonator (degrees) Figure 1. Variation of growth rate with circumferential location of second resonator for two least stable modes originally at 134 Hz (n = ±3). (The first resonator is fixed at θ =.) Figure 11. Variation of growth rate of least stable mode originally at 447 Hz (n = ±1) with circumferential location of second and third resonators. (The first resonator is fixed at θ =.) Fig. 3 by squares for the two modes originally at 893 Hz and by pluses for the two modes originally at 134 Hz for an angle of 6 between the resonators (taking N = 1). The results in Figs. 7 1 suggest that when placing two resonators to damp a circumferential disturbance the best angles to use between them correspond to odd multiples of half the circumferential wavelength of the (original) oscillation. We now investigate the placement of three Helmholtz resonators on the circumference. In Fig. 11 we show the variation in the growth rate of the least stable of the modes originally at 447 Hz (n = ±1) with the circumferential location of the second and third resonators, the first being fixed at θ =. The volume of the resonators is taken to be m 3. The best configurations are found to when the resonators are around 6 apart, and when they are 12 apart (i.e. equally distributed around the circumference). Poor configurations exist that give no damping, for example when two resonators are at the same location with the third is directly opposite. For these configurations it is possible to have a circumferential standing wave with pressure nodes at all the resonator necks. Figure 12 shows corresponding results Copyright 23 by ASME

8 π/ n * circumferential location of third resonator (degrees) Figure 12. Variation of growth rate of least stable mode originally at 893 Hz (n = ±2) with circumferential location of second and third resonators. (The first resonator is fixed at θ =.) circumferential location of third resonator (degrees) Figure 13. Variation of growth rate of least stable mode originally at 134 Hz (n = ±3) with circumferential location of second and third resonators. (The first resonator is fixed at θ =.) Figure 14. for the modes originally at 893 Hz (n = ±2) with the volume of the resonators taken to be m 3. Again there are poor configurations, for example if the three resonators are separated by 9, i.e. they are at, 9 and -9, then it is possible to have pressure nodes at all three resonators. There are eight best configurations shown in the figure (although some are equivalent to others under rotations and reflections of the geometry). One of these is when the resonators are close to 3 apart, and the others are equivalent to rotating any of the three resonators by multiples of 9. Results for the modes originally at 134 Hz (n = ±3) are shown in Fig. 13 for a resonator volume of m 3. There are several best configurations, one is where the resonators are close to 2 apart and the others are where any of these are roπ/ n * π /(h n * ) Explanatory diagram of best positions of resonators on the circumference of a duct. tated by multiples of 6. These results suggest that for three resonators one best configuration is (approximately) equivalent to the resonators being at angles of, π/(3n ) and 2π/(3n ) and the others are equivalent to rotating any of the three resonators by multiples of π/n (where ±n are the circumferential wavenumbers of the original oscillation). From the symmetry of the ±n modes, it is not surprising that moving a resonator around the annulus by π/n (i.e. half the wavelength of the dominant components) has little effect. Combining this with the findings for two resonators suggests a general rule that a best configuration is to distribute the resonators evenly along half a wavelength of the mode (i.e. along an sector of angle of π/n ), and that rotating any of these resonators by multiples of half the wavelength is also a best configuration. Figure 14 gives an explanatory diagram. The solid circles indicate the resonators being distributed evenly along half a wavelength, and the dashed-outline circle indicates one of the resonators being rotated by half the wavelength. Finally, we consider an example including unsteady combustion. The same geometry is used except that there is now a combustion zone at x =.25 m raising the mean temperature from 1 K to 2 K. The unsteady heat release is given by the simple flame model in Eq. (2). The time delay, τ, is taken to be.5 ms as this gives a strongly unstable mode for n = ±1 (at around 425 Hz) when k = 1. The growth rate of this mode as k is varied between.5 and 1.5 is shown as a solid line in Fig. 15. We now add two resonators with volume m 3, 9 degrees apart, at x =.5 m to try and damp this mode. As before, four modes are then found to be present. The growth rate of most unstable mode is shown as a dashed line in Fig. 15. We see that the growth rate of the mode has been reduced, in particular the mode is now only unstable for k > 1, as opposed to k >.61 without the resonators. 4 CONCLUSIONS A linear model for thermoacoustic oscillations in thin annular LPP combustors has been described. This model could be used when designing LPP combustors to predict combustion instabilities and to investigate the effect of varying parameters such as combustor temperature or duct lengths. Its modular form 8 Copyright 23 by ASME

9 flame model constant, k flame model constant, k Figure 15. Variation of growth rate of mode without resonators (solid line) and of the most unstable mode with resonators (dashed line) with combustion model constant, k. means that it is very flexible and new components could easily be incorporated. Helmholtz resonators introduce circumferential model coupling which is calculated. The model has been used to investigate the best placement of Helmholtz resonators around the circumference of an annular duct (for instance, an aeroengine combustion chamber) to damp a circumferential oscillation. A general guide appears to apply. Firstly distributing the resonators evenly along a sector corresponding to half the circumferential wavelength gives very good damping. Secondly, rotating any of these resonators by multiples of half the wavelength also gives very good damping. An example including unsteady combustion has demonstrated that the damping can be sufficient to stabilise unstable modes. ACKNOWLEDGMENT This work was funded by the European Commission whose support is gratefully acknowledged. It is part of the GROWTH programme, research project ICLEAC: Instability Control of Low Emission Aero Engine Combustors (G4RD-CT2-215). REFERENCES [1] A. J. Riley, A. P. Dowling, A. Annaswamy, S. Park and S. Evesque, 23, Adaptive closed-loop control of an atmospheric gaseous lean-premixed combustor, ASME Paper GT [2] V. Bellucci, C. O. Paschereit, P. Fohr and F. Magni, 21, On the use of helmholtz resonators for damping acoustic pulsations in industrial gas turbines, ASME Paper 21- GT-39. [3] S. Evesque and W. Polifke, 22, Low-order acoustic modelling for annular combustors: validation and inclusion of modal coupling, ASME Paper GT [4] U. Krüger, J. Hürens, S. Hoffmann, W. Krebs and D. Bohn, 1999, Prediction of thermoacoustic instabilities with focus on the dynamic flame behavior for the 3A-series gas turbine of Siemens KWU, ASME Paper 99-GT-111. [5] U. Krüger, J. Hürens, S. Hoffmann, W. Krebs, P. Flohr and D. Bohn, 2, Prediction and measurement of thermoacoustic improvements in gas turbines with annular combustion systems, ASME Paper 2-GT-95. [6] G. C. Hsiao, R. P. Pandalai, H. S. Hura and H. C. Mongia, 1998, Combustion dynamic modeling for gas turbine engines, AIAA Paper [7] G. C. Hsiao, H. C. Mongia, R. P. Pandalai and H. S. Hura, 1998, Investigation of combustion dynamics in dry low emission engines, AIAA Paper [8] S. R. Stow and A. P. Dowling, 21, Thermoacoustic oscillations in an annular combustor, ASME Paper 21-GT- 37. [9] S. R. Stow, A. P. Dowling and T. P. Hynes, 22, Reflection of circumferential modes in a choked nozzle, J. Fluid Mech., 467, pp [1] A. P. Dowling, 1999, Thermoacoustic instability, 6 th International Congress on Sound and Vibration. [11] M. Zhu, A. P. Dowling and K. N. C. Bray, 23, Studies of flame transfer function with three dimensional calculations, ASME Paper GT [12] C. A. Armitage, R. S. Cant, A. P. Dowling and T. P. Hynes, 23, Linearised theory for LPP combustion dynamics, ASME Paper GT [13] W. S. Cheung, G. J. M. Sims, R. W. Copplestone, J. R. Tilston, C. W. Wilson, S. R. Stow and A. P. Dowling, 23, Measurement and analysis of flame transfer function in a sector combustor under high pressure conditions, ASME Paper GT [14] M. S. Howe, 1979, On the theory of unsteady high Reynolds number flow through a circular aperture, Proc. R. Soc. Lond. A, 366, pp [15] I. D. J. Dupère and A. P. Dowling, 23, The use of Helmholtz resonators in a practical combustor, ASME Paper GT [16] S. R. Stow and A. P. Dowling, 22, Report on incorporation of acoustic dampers into low-order models, Deliverable D4.1, EU Project No. GRD (Unpublished). [17] S. Akamatsu and A. P. Dowling, 21, Three dimensional thermoacoustic oscillation in an premix combustor, ASME Paper 21-GT-34. [18] A. P. Dowling and J. E. Ffowcs Williams. Sound and sources of sound. Ellis Horwood, Copyright 23 by ASME