May 11, 2000 - 1.1 Flow-flame-acoustic interactions in combustion dynamics ..... of the coefficients, is not fixed a priori, so we may hope to find one or indeed ...
Six Lectures on Thermoacoustic Combustion Instability
21st CISM-IUTAM International Summer School on Measurement, Analysis and Passive Control of Thermoacoustic Oscillations
Wolfgang Polifke Professur für Thermofluiddynamik Technische Universität München, Germany June 29 – Jul 3 2015, Udine, Italy
Copyright © 2015 by Wolfgang Polifke Permission granted to reproduce for personal and educational use only. Commercial copying, hiring, lending is prohibited. Corrected printing (September 2015)
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Contents 1 Introduction 1.1 Flow-flame-acoustic interactions in combustion dynamics . . . . . . . . . . . . . 1.2 Rayleigh’s stability criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Outline of these notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 5 6 9
2 Stability Analysis 2.1 Unsteady simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Determination of eigenmodes and eigenfrequencies . . . . . . . . . . . . . . . . 2.2.1 Eigenfrequencies of a Rijke tube . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Further remarks on dynamic stability analysis . . . . . . . . . . . . . . . . 2.3 The Nyquist Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Nyquist plots in control theory . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Application of the Nyquist criterion to thermoacoustic systems . . . . . 2.3.3 Identification of eigenfrequencies and growth rates from a Nyquist plot . 2.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Overview of System Models 3.1 Finite-Volume / Finite-Element models . . . . . . . . . . . . 3.1.1 Computational fluid dynamics . . . . . . . . . . . . . 3.1.2 Computational acoustics . . . . . . . . . . . . . . . . 3.2 Galerkin method . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Network Models . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Review of previous work . . . . . . . . . . . . . . . . . 3.3.2 Compact element . . . . . . . . . . . . . . . . . . . . . 3.3.3 Duct elements . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Compact flame . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Example network calculations . . . . . . . . . . . . . 3.4 State-space models . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Acoustic eigenmodes of a resonator tube . . . . . . . 3.4.2 A state-space “network model” for a resonator tube . 3.4.3 A state-space model for the n-τ system . . . . . . . .
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4 Modelling premixed flames as distributed time lags 4.1 A simplistic model for the response of a lean premixed flame to equivalence ratio fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Results obtained with Gaussian time delay distribution . . . . . . . . . . . 4.2 Kinematic response of a perfectly premixed flame . . . . . . . . . . . . . . . . . . . 4.2.1 Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Determining the Impulse Response Function . . . . . . . . . . . . . . . . . 4.2.3 IR to Uniform Velocity Perturbation . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 IR to Convective Velocity Perturbation . . . . . . . . . . . . . . . . . . . . . 3
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4.2.5 Interpretation of Impulse Response Functions . 4.2.6 Cutoff Behavior . . . . . . . . . . . . . . . . . . . 4.2.7 Excess Gain Behavior . . . . . . . . . . . . . . . . 4.3 Practical premixed flame with staged fuel injection . . 4.4 Turbulent premixed swirl flame . . . . . . . . . . . . . .
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Chapter 1
Introduction In science and technology, the term instability characterizes situations where small (”input”) perturbations of initial or boundary conditions result in a violent response, such that system (”output”) variables change drastically or grow to very large amplitudes. After a transition period, a new system state is established, which is usually quite different from the initial state. Indeed, a complete breakdown of the initial state is often observed. Many instabilities involve large-amplitude spatio-temporal oscillations of the system variables; in this case a so-called limit cycle may be established as a final state. A wide variety of instabilities are known in mechanics, hydrodynamics, plasma physics, geophysics, control theory, aeronautics, or even economics. Well-known examples are acoustic feedback in public address systems (electro-mechanical feedback), the collapse of the Tacoma Narrows bridge (aeroelastic flutter), the oscillations of the London Millennium bridge (an instability involving ”positive biofeedback”), the formation of large scale vortices in shear layers (Kelvin-Helmholtz instability; Karman vortex street), blade flutter or surge and stall in the compressor of gas turbines (aeroelastic flutter or pressure rise / mass flow instability), etc. Thermoacoustic instabilities arise from an interaction of acoustic waves and unsteady heat release. In general such instabilities are undesirable, because they can produce excessive noise, limit operational flexibility and may even result in massive structural damage. This is particularly true for thermoacoustic combustion instabilities (the term combustion dynamics is also popular) which are a cause for concern in applications as diverse as small household burners, process heaters, gas turbines, or rocket engines.
1.1 Flow-flame-acoustic interactions in combustion dynamics Unlike other flow instabilities, thermoacoustic instabilities are not a local phenomenon in the sense that stability properties are not determined solely by the dynamics of the heat source and the flow field in its immediate vicinity. Instead, acoustic waves travel back and forth across the extent of the system, and as a consequence acoustic boundary conditions far away from the heat source – at the top of the chimney, say – may strongly influence stability properties. In (premixed) flames, convective transport of fuel inhomogeneities from the fuel injector to the flame, or of entropy inhomogeneities (”hot spots”) from the flame to the exhaust, are other non-local phenomena which influence system stability. All these effects are linked together in feedback loops, as illustrated in Fig. 1.1 for a premix combustor. It should be obvious that the complete and accurate description of a thermoacoustic instability can be a daunting task, since the system under consideration may involve a variety of fluid-dynamic and physico-chemical phenomena, covering a wide range of space and time scales. Whether the interaction of the fluctuations of heat release rate, pressure, velocity, fuel concentration, etc., actually gives rise to an instability depends in an essential manner on the rela5
Flame
Swirl / Vorticity
Velocity
Heat Release Fluctuations
Front Kinematics
Burner Pressure Loss Volume Flow
Equivalence Ratio Entropy waves
Pressure Air / Fuel Supply
Acoustic Waves
Combustion Chamber
Figure 1.1: Interactions between flow, acoustics and heat release in a combustor as an example of a thermoacoustic system which may exhibit instability. tive phases, and in particular on the phase between fluctuations of heat release and pressure at the heat source. This has been known since Rayleigh (1878); the stability criterion named after him is discussed in the next section. It is also shown why the Rayleigh criterion represents a necessary, but not sufficient criterion for instability.
1.2 Rayleigh’s stability criterion Consider flow of a gas past a heat source, e.g. a premix or diffusion flame, or a surface with convective heat transfer to the gas (the so-called ”stack” in a thermoacoustic machine, or the wire in a Rijke tube, see below.). Mass conservation in steady state requires that (ρu)c = (ρu)h .
(1.1)
With the density ρ h on the downstream side (index h for ”hot”) being lower than the density of the upstream side (index c for ”cold”), the velocity u, i.e. the volume flux, must increase across the heat source. Now, if the rate of heat released by the source fluctuates, the volume ”produced” will also fluctuate, thereby generating sound1 – just like a loudspeaker box with its oscillating membrane. The heat release rate in a flame, say, may be perturbed by turbulent fluctuations of the velocity field upstream of the flame front. This gives rise to combustion noise, e.g. a camping burner or a blow torch which ”hisses” or ”roars”. Combustion noise often exhibits a broad band frequency distribution, which derives from the size distribution of the turbulent eddies perturbing the flame. Combustion noise may also be generated by fairly large scale, vortical coherent structures, originating from hydrodynamic instability of the base flow (e.g. a shear layer or swirling flow). In any case, if one speaks of combustion noise, it is usually implied that there is no significant feedback from the sound emitted back to the flow fluctuations which perturbed the heat release in the first place. However, if the heat source is enclosed in a chamber that acts as an acoustic resonator, sound will in general be reflected back to the flame such that a feedback loop is established (see again Fig. 1.1). If the phase between the sound field established in the chamber and the fluctuations of heat release is just right, a self-excited feedback instability may occur. Then small 1 Note that sound production by heat release with its monopole character is a very strong sound source and – if
present – usually dominates other source processes.
6
p,u
p
t
2' 3'
2"
Q
2' 3'
1 1
4'
4'
t
3" 4"
Q 1
4" 2"
v
3"
t
Figure 1.2: Thermodynamic interpretation of Rayleigh’s stability criterion. Left: p-v diagram of isentropic compression / expansion ( —— ), with heat addition in-phase with pressure fluctuations (1-2’-3’-4’) and out-of phase (1-2”-3”-4”). The dashed gray lines represent lines of constant entropy. Right-top: Fluctuations of pressure ( —— ) and velocity ( - - - ) in a standing wave, Right-middle/bottom: in-phase / out-of-phase fluctuations of heat release rate. (infinitesimal) perturbations are amplified ever more, until eventually some kind of saturation mechanism kicks in and a limit cycle is established. For saturated thermoacoustic combustion instabilities, limit cycle velocity fluctuations often exceed the mean flow velocities, amplitudes of pressure fluctuations can reach more than 120 dB in atmospheric flames, and several MPa in rocket engines. Damage to the combustion equipment can then result very quickly due to excessive mechanical or heat loads. If such thermoacoustic instability occurs, the frequency spectrum of the resulting pressure and velocity oscillations typically exhibit one or several distinct peaks, with frequencies often (but not necessarily) close to the acoustic eigenfrequencies of the enclosure (or the complete combustion system) without unsteady heat release. Rayleigh (1878) has proposed a criterion which tells us when the phases between the sound field and the heat release fluctuations are ”just right”: For instability to occur, heat must be released at the moment of greatest compression. A more general formulation of the Rayleigh criterion states that a positive correlation between the fluctuations of heat release and pressure, respectively, is required for thermoacoustic instability to be possible: I p 0 Q˙ 0 d t > 0. (1.2) Here the integration runs over one period of the oscillation, p 0 denotes fluctuations of pressure at the heat source, Q˙ 0 the fluctuations of the heat transfer (or release) rate. There is also a straightforward generalization of this ”Rayleigh integral” for spatially distributed heat release R rate, which includes a spatial integration . . . dV over the volume where heat is released. The Rayleigh criterion (1.2) may be made plausible by analogy with a thermodynamic cycle. Consider a small volume of gas, which is compressed and expanded by a standing acoustic wave. Sound waves are isentropic, so in a p-v diagram the volume moves back and forth on an isentrope (see the line in the diagram on the left side of Fig. 1.2). What happens if heat is added and extracted periodically to the gas? Well, addition of heat will result in a comparative increase of the specific volume v of the gas, and if the heat addition is in-phase with pressure fluctuations, the state of the gas volume moves clockwise around a thermodynamic cycle (curve 1-2’3’-4’ in the diagram). So this is a ”thermoacoustic heat engine”, which feeds mechanical energy into the sound wave – and a self-excited instability may occur, as Rayleigh suggested. If fluctuations of heat addition are not perfectly in phase with pressure fluctuations, then the area of the cycle 1-2’-3’-4’ will be smaller, and the efficiency of the engine will decrease. In the extreme case where fluctuations of heat release rate Q˙ 0 are opposite in phase with p 0 , the system moves 7
counterclockwise through the cycle 1-2”-3”-4” – and mechanical energy is extracted from the acoustic wave. The mechanical work performed by the thermodynamic cycle resulting from acoustic perturbations with heat release illustrated in Fig. 1.2 may be formulated as follows: I
v p dv = − γp 0
0
I
0
0
p dp +
I
0
p dv
0(Q)
I
= 0+
d v 0(Q) dt ∼ p dt 0
I
p 0 Q˙ 0 d t .
(1.3)
Here the changes v 0 in specific volume have been split into an isentropic part, for which v 0 = −v p 0 /γp (where γ is the ratio of specific heats), and a part v 0(Q) which is due to heat addition (or removal). The rate of change of this term with time is proportional to fluctuations of the rate of heat addition. One may conclude that the work done by the ”thermoacoustic engine” is positive (energy is added to the acoustic field), if the integral of p 0 Q˙ 0 over one period of oscillation is positive – just as Rayleigh has argued. Rayleigh’s criterion is ”necessary, but not sufficient” for instability to occur2 . This means that a system is guaranteed to be thermoacoustically stable, if the stability is not fulfilled – but if the criterion holds, it is not guaranteed that an instability will indeed develop. The reason for this limitation is the following: if the criterion is fulfilled, Rayleigh’s thermoacoustic engine feeds mechanical energy into the acoustic field. However, oscillation amplitudes will only grow, if losses of acoustic energy, which may occur elsewhere in the system, do not exceed the rate of energy generated by the fluctuating heat source. More complete stability criteria have been developed, they are discussed in chapter 2, Stability Analysis. Let’s conclude this section with an important observation: Rayleigh’s criterion is stated in terms of fluctuations of heat release rate and pressure at the heat source. Nevertheless, acoustic boundary conditions far away from the heat source can and in general will determine stability properties in a decisive manner. Why is that? To answer this question, one must consider that convective heat transfer rates as well as the heat release rate in a premix flame usually respond much stronger to changes in velocity u 0 than in pressure p 0 . The phase between fluctuations of pressure p 0 and u 0 at the flame is determined by the acoustics of the system, e.g. the acoustic impedance at the boundaries, and the wave pattern – perhaps with partial reflections and transmissions – inside the enclosure. As a very simple example, consider the fundamental λ/2mode in a straight duct of length L with two open ends and a compact heat source at position x = L/4 as shown in Fig. 1.3. Such a system has been known to exhibit thermoacoustic instability for a long time and is known as a”Rijke tube” Rayleigh (1878); McManus et al. (1993); Heckl (1988). Acoustic fluctuations of velocity in the left half of the tube are opposite to those in the right half of the tube (velocity u 0 is ahead of pressure p 0 by π/2 in the left half and lags behind by −π/2 in the right half). If we assume that a heat source placed in the tube responds to changes H in velocity, then clearly the sign of the Rayleigh Integral p 0Q˙ 0 d t will depend on the position of the heat source: a heat source, which gives rise to self-excited instability when placed in the left half of the tube, will not go unstable when placed in the right half, and vice versa. This is explained in more detail in the following. It follows that it is usually the combination of heat release dynamics and system acoustics that controls stability. System acoustics determines largely both the impedance at the heat source (the phase between p 0 and u 0 ) as well as the total losses of acoustic energy (dissipation inside the system, also radiation to the environment at the boundaries). For this reason, thermoacoustic stability analysis is in general not possible without a model for the acoustics of the enclosure and for the interactions between acoustics and the heat source. Such a model is called a thermoacoustic system model, various ways of constructing such a model will be 2 Other stability criteria found in the literature, which are based in a similar manner only on phase relationships, suffer also from this limitation.
8
. Q
p =0
i
c
p =0
h
x
Figure 1.3: Rijke tube of length L with two ”open end” boundary conditions (inlet ”i”, exit ”x”) and a compact heat source Q˙ at x c = L/4 with x h − x c ¿ L. Distribution of fluctuating velocity and pressure / density at one particular instant during the oscillation are indicated by arrows and shading, respectively. described in chapter 3. For noteable exceptions, see the recent papers on intrinsic thermoacoustic feedback and the corresponding intrinisc instabilities of premix flames by Hoeijmakers et al. (2014); Bomberg et al. (2015); Courtine et al. (2014) and Emmert et al. (2015).
1.3 Outline of these notes In chapter 2, an overview of various methods for stability analysis is given. Since researchers in thermoacoustics often come from different backgrounds, it should not come as a surprise that a wide variety of methods have been proposed. Various approaches for modelling ”the system” are introduced in chapter 3 ranging from use of full-scale computational fluid dynamics (CFD) to simple, low-order network models, as they are popular in duct acoustics. The low-order models are limited in geometrical flexibility and may seem simplistic – nevertheless they often produce non-trivial results and are most helpful to develop insight into instability mechanisms. It is shown for a selected simple geometries how a mathematical description of network model elements may be derived from fundamental principles. A few example computations obtained with simple network models are presented and discussed. The concept of a state-space model is introduced in simplistic form. Chapter 4 argues that a wide variety of premixed flames may be modelled in terms of distributed time lags (DTL). Such a representation reflects the fact that convective processes are important in flame dynamics. DTL models may be derived by paper & pencil, or identified from CFD data, and they may be formulated in time as well as frequency domain. It is straightforward to combine DTL models with a variety of system models in order to carry out linear stability analysis. In these notes, it has not been attempted to give reference to, let alone discuss in adequate detail all publications which have contributed to the development of the field. Suggestions for further reading are given in the short review sections at the end of some chapters – but the selection is admittedly not free from personal bias and features the work of the combustion dynamics groups at ABB Corporate Research and TU München more prominently than would be adequate for a proper review of the subject.
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Chapter 2
Stability Analysis A system (or a system state) may be called stable against a perturbation, if some time after the perturbation is imposed, the initial system state is re-established. It follows quite naturally from this definition that in order to investigate the stability properties of a system, one must determine the response of the system to perturbations. Often it is fairly easy to compute the response to perturbations with infinitesimally small amplitudes. In this case the governing equations may be simplified by neglecting the so-called non-linear terms, i.e. terms which are higher order in perturbation amplitudes. A system is accordingly called linearly stable, if it returns to its initial state after a slight perturbation. Note that it is not specified what kind of perturbation is imposed, except that the amplitude be small in some sense. In linearly unstable systems, a dominant or most unstable mode – i.e. a distinct pattern of vibration with a particular frequency ω – develops in the very early stages of the instability. This most unstable mode has – as the term implies – a growth rate that is larger than that of any other unstable mode. After a while, the most unstable mode dominates the behaviour of the system such that the whole system oscillates at the frequency ω of the mode. In this stage, the frequency spectrum typically shows a single peak at the frequency ω. The oscillation amplitudes keep growing exponentially, until saturation of amplitude sets in and eventually a limit cycle, i.e. oscillation with large, but finite amplitudes, is established. Significant growth of amplitudes is associated with linear instability, correspondingly the departure from exponential growth is due to non-linear terms, i.e. those terms of higher order in oscillation amplitudes, which were neglected in the linearized analysis. During the non-linear phase of the evolution of the instability, the frequency spectrum typically shows more than one peak – often a dominant peak at the frequency ω and lower-amplitude, but still rather distinct peaks at integer multiples n ω of the fundamental frequency (harmonics). This scenario of linear instability → growth of dominant mode → non-linear saturation → limit cycle is a well established paradigm in instability theory (Drazin, 2002; Keller, 1995; Dowling, 1995). Experimental results for a Rijke tube obtained by Lumens (2006) are shown in Fig. 2.1. Nevertheless, there is evidence that this scenario does not always apply. For example, some linearly stable thermoacoustic systems operate stably for a long time, but exhibit strong instability after suffering a strong external perturbation. This kind of non-linear instability behaviour is called triggering. It has been observed in thermoacoustic systems ranging from Rijke tubes to rocket engines (Culick, 1989, 1994; Flandro et al., 2007). More recently, the relevance of non-normal effects for thermoacoustic instabilities has been brought to attention by Balasubramanian and Sujith (2007a,b). Furthermore, many combustion systems, while not exhibiting a strong thermoacoustic instability, show significant fluctuation levels (of pressure, velocity, heat release rate) over a band of frequencies (often in the vicinities of combustor eigenfrequencies). Such a state should per-
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Q = 385 ± 7 W; vmean = 0.0218 ± 0.0002 m/s
Figure 2.1: Time trace of pressure in an electrically heated Rijke tube. After a limit cycle is established, a plug is pulled from the tube, such that the instability collapses. haps not be described as the limit cycle of an instability, but rather as resonant amplification of combustion noise by the combustion chamber.
2.1 Unsteady simulation Probably the most straightforward method to assess the stability of a thermoacoustic system makes use of unsteady simulation in the presence of small (initial) perturbations: If the perturbations grow in amplitude with time, the system is understood to be unstable, and stable otherwise. Provided that non-linear effects are taken into account properly by the simulation model – which is in principle possible for many models in use – a limit cycle is established once the simulation is continued for a sufficiently long time, As an example of this strategy, Fig. 2.2 shows the temporal development of the velocity just upstream of the heat source in a generalized Rijke tube, i.e. a resonator with closed-open boundary and heat source with time lag placed in the center. The computational model used to generate the data shown in Fig. 2.2 was based on simplified 1-D conservation equations for mass, energy and momentum in laminar compressible flow. ”Lumped parameter” source terms for energy and momentum were used to model the heat source, see Polifke et al. (2001c). Obviously, the simulation exhibits instability: the oscillation amplitudes grow until a limit cycle with negative velocities, i.e. reverse flow past the heat source during part of the oscillation cycle, is established1 . The use of unsteady simulation for stability analysis is conceptually a straightforward approach, and has been used by a number of authors (Janus and Richards, 1996; Dowling, 1997; Veynante and Poinsot, 1997; Peracchio and Proscia, 1998; Polifke et al., 2001c; Pankiewitz and Sattelmayer, 2003b; Sung et al., 2000). This does not imply, however, that this approach is easy to implement or generally superior to alternative strategies. The following drawbacks may be 1 It is straightforward to interpret this result: if hot gases are swept back to the heat source, the temperature
difference and therefore also the heat transfer between the hot wire and the gas decreases, thereby reducing with increasing fluctuation amplitude the gain of the transfer function and the Rayleigh integral until the instability saturates.
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(
30
vc [m/s]
20
10
0
-10 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
t [s]
Figure 2.2: Temporal development of velocity v c upstream of the heat source in a generalized Rijke tube. Time lag τ of the heat source was adjusted to fulfill the Rayleigh criterion. identified: • Unsteady simulation is a ”brute force” approach which can be computationally very expensive. This is in particular the case if a computational fluid dynamics simulation model for (turbulent, reacting), incompressible flow is used, i.e. the unsteady Reynoldsaveraged Navier-Stokes (URANS) or Large Eddy Simulation (LES). • In general, only the most unstable mode is identified. Stable modes, or modes that are unstable, but have smaller growth rates than the most unstable mode, cannot be observed. In this sense, the stability assessment based on unsteady simulation is not comprehensive. • In some instances, it is not easy to discern a spurious numerical instability from a genuine physical instability. A related problem is that it can be difficult to numerically generate the initial state, from which the instabilities are supposed to develop. • Results can depend on initial conditions, i.e. details of the initial perturbation (typically a random perturbation of very small amplitude) can influence which mode develops. For example Evesque et al. (2003), a Finite-Element based model of linear acoustics, showed that for non-plane acoustic waves in an annular combustor, the initial conditions can determine whether a standing or a rotating mode develops. Further analysis showed that rotating modes with clockwise and counterclockwise sense of rotation are degenerate eigenmodes for this problem. This explains the sensitivity against initial conditions in this particular case. • In general, the acoustic impedance at the boundary of the computational domain is a complex-valued function Z (ω) which depends on frequency ω. The implementation of appropriate acoustic boundary conditions in the time domain is a non-trivial problem. See Schuermans et al. (2005); Huber et al. (2008) for recent advances on this problem in the context of CFD-based models of thermoacoustics in the presence of mean flow.
2.2 Determination of eigenmodes and eigenfrequencies The stability of a system can be determined by identifying the eigenmodes and in particular the eigenfrequencies of the system. This classical method, known also as dynamic stability analysis, is very popular and can be applied to a wide variety of problems.
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What is an eigenmode and an eigenfrequency? The German prefix Eigen can be translated as own, peculiar to, characteristic or individual. An eigenfrequency is then a frequency that is easily excited in a system. Once excited, the system, left to itself, will continue to oscillate for some time at that frequency. The corresponding pattern of vibration, e.g. the spatial distribution of nodes and anti-nodes, is identified as the shape of the eigenmode. Typically, a dynamic system has more than one – even infinitely many – eigenmodes and corresponding eigenfrequencies. Trying for a bit more mathematical rigor, the eigenvectors ~ x m of an operator A are those vectors which are left unchanged under the operator up to factors χm , the eigenvalues of the operator: A~ x m = χm ~ xm . (2.1) This should be familiar from linear algebra. The notions of ”eigenvector” (or ”eigenfunction” or ”eigenmode”) and ”operator” may be interpreted in a rather general sense. For example, consider acoustic oscillations of pressure p 0 in a duct of length L with two open end boundary conditions p 0 (0) = p1(L) = 0, obeying the 1D Helmholtz-equation, 1 ∂2 p 0 ∂2 p 0 − = 0. c 2 ∂t 2 ∂x 2
(2.2)
The eigenmodes are the well-known standing waves p 0 (x, t ) ∼ exp(i ωm t ) sin(k m x). The wave numbers k m , which are restrained to the values k m = nπ/L in order to fulfill the boundary conditions, determine the shape of each eigenmode. The Helmholtz-equation then is re-written as ∂2 p 0 = −ω2m p 0 , (2.3) ∂t 2 for the eigenfrequencies ωm = k m c. When formulated in this way, the analogy with (2.1) should be obvious: The eigenmodes correspond to particular oscillations of the gas column in the duct, such that the second derivative of pressure with respect to time is simply the shape of the pressure distribution scaled by a factor. This factor, the eigenvalue, turns out to be −ω2m , where ωm is the angular frequency of the oscillation. The fact that the time derivative operator is Hermitian2 leads to several important properties, such as that the standing wave patterns are orthogonal functions. For ideal systems without dissipative effects, an eigenmode oscillates without decay, corresponding to a purely real eigenfrequency ω ∈ R. If damping (due to viscous losses or radiation losses at the system boundary) is non-negligible, oscillation amplitudes decrease in time. If amplitude levels are small, non-linear terms may be neglected, then an exponential decrease of amplitudes is typical. Assuming harmonic time dependence ∼ exp(i ωt ) an eigenfrequency ωm with positive imaginary part ℑ(ωm ) > 0 describes this situation3 . Unstable modes in systems which can exhibit self-excited instability – this is of course the case most interesting in the present context – are characterized by eigenfrequencies with negative imaginary part ℑ(ωm ) < 0. 2 In mathematics, on a finite-dimensional inner product space, a self-adjoint operator is one that is its own ad-
joint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. By the finite-dimensional spectral theorem such operators have an orthonormal basis in which the operator can be represented as a diagonal matrix with entries in the real numbers (from Wikipedia). More details on ”Sturm-Liouville theory” (the theory of linear differential equations, differential operators and their eigenvalues / eigenfunctions) are found in many mathematics text books. 3 Note that often the opposite convention ∼ exp(−i ωt ) for the time dependence is chosen. In this case, a negative imaginary part indicates a decaying eigenmode.
13
A growth rate or cycle increment of an eigenmode m can be derived from its eigenfrequency: ½ ¾ ℑ(ωm ) Γ ≡ exp −2π − 1. (2.4) ℜ(ωm ) The growth rate indicates by which amplitude ratio a mode increases or decreases per cycle. For example, a growth rate of Γ = 0.2 indicates an increment of the amplitude of 20% per cycle. To illustrate this method of stability analysis, it is shown in the next subsection how the eigenfrequencies of a Rijke tube can be determined from a low-order thermoacoustic model.
2.2.1 Eigenfrequencies of a Rijke tube The Rijke tube is a simple, yet interesting example of a system that can exhibit self-excited thermoacoustic instability. It has been discussed previously by many authors for its ”pedagogical” value (McManus et al., 1993; Dowling, 1995; Polifke et al., 2001c; Polifke, 2004). Here it is shown how the eigenfrequencies and thereby the stability characteristics of a Rijke tube can be determined from the characteristic equation (also dispersion relation), which in turn is derived from an approximate low-order model of the Rijke tube. We consider a system as shown in Fig. 1.3, i.e. a heat source placed in a duct. It is assumed in the following that there is an acoustically closed end (u 0 = 0) at the left boundary. This little trick simplifies the analysis a bit. Also, the position of the heat source is not at position x c = L/4, as it is for a ”real” Rijke tube. Instead, the length of the duct to the left of the heat source is denoted as l c , while to the right we have l h = L − l c . It is assumed that the reader is familiar with basic acoustics, so instead of starting ”from scratch”, we just introduce some notation and go into medias res quickly: Riemann Invariants f and g (also called p + and p − by some authors) are related to the primitive acoustic variables as follows, µ ¶ µ ¶ 1 p0 1 p0 0 0 f = +u , g = −u . (2.5) 2 ρc 2 ρc One may think of these ”Riemann invariants” (also called ”characteristic wave amplitudes”) simply as acoustic waves propagating in the down- and upstream direction, respectively4 . Now consider the length of duct to the left of the heat source, i.e. the ”cold” part of the Rijke tube. A downstream traveling wave f undergoes a change in phase exp{i ωl c /c c } as it travels with the speed of sound c across the distance l c = x c − x i from the inlet to the cold side of the heat source. Similar for the wave g traveling in the upstream direction. Along the length of the duct, there is no interaction between the waves f and g . From these coupling relations the transfer matrix of the duct is readily obtained: µ
fc gc
¶
µ
=
e −i kc l c 0
0 e i kc l c
¶µ
fi gi
¶
.
(2.6)
with a wave number k c ≡ ω/c c for plane waves without mean flow and dissipative effects. For the ”hot” duct to the right of the heat source, the transfer matrix is of the same form, but of p course c = γRT and l may be different, so the coefficients of the transfer matrix may be different, too. For the heat source (which is a wire mesh or ”gauze” in a Rijke tube), one observes that the pressure drop across the heat source is negligible for sufficiently small mean-flow Mach number, p c0 = p h0 , (2.7) 4 A note on nomenclature: we make in these notes no explicit distinction between acoustic variables in the time
domain and in the frequency domain. Most of the time we are in frequency space, and the f ’s and g ’s are to be understood as complex-valued Fourier transforms of time series.
14
while for the velocity we assume that u h0 (t ) = u c0 (t ) + nu c0 (t − τ).
(2.8)
This is the famous n −τ model for a heat source with time lag. This simple model has played an important role in the development of the theory of combustion instabilities in rocket engines, is used frequently as a pedagogical example (as in these notes), and is even relevant as a building block for models of ”real-world” turbulent premix flames. One can show (see e.g. Polifke (2004); Poinsot and Veynante (2005)) that the second term on the r.h.s. appears if the heat release of a flame (or some other source of heat) responds to a change in velocity u with a certain time lag τ. The important thing about the time-lag is that it may allow some phase-alignment between fluctuations of pressure and heat release, respectively, even with standing acoustic waves (where the phase between p 0 and u 0 is ±π/2). So, for given impedance Z = p 0 /u 0 at the heat source, the time lag τ in combination with the frequency ω controls the sign of the Rayleigh integral and therefore system stability. For the interaction index n, one finds under certain assumptions – which hold true for the gauze of a Rijke tube, but not necessarily for a premix flame – that n = Th /Tc − 1, i.e. n is determined by the increase in mean temperature T across the heat source. In terms of the Riemann invariants, the coupling relations across the heat source are ρ h ch ( f h + g h ) = ( f c + g c ), ρ c cc ´ ³ f h − g h = 1 + n e −i ωτ ( f c − g c ).
(2.9) (2.10)
At the left boundary u 0 = 0 and therefore f i − g i = 0, while at the right boundary, p 0 = 0 and correspondingly f x + g x = 0. This completes the construction of a low-order model for the Rijke tube: there are eight equations (2 boundary conditions, 2×2 equations from the ducts on both sides of the heat source, and 2 equations from the heat source) for the eight unknowns f i , g i , . . . g x . In matrix & vector notation, this reads
Matrix of Coefficients
fi .. . = gx
0 .. . . 0
(2.11)
As indicated, the system is homogeneous (the right hand side is a vector with all 0’s), so a nontrivial solution exists only if the determinant of the system matrix S vanishes, i.e. if Det(S) = 0.
(2.12)
How can this condition be satisfied? Inspection of the coupling relations (2.6) - (2.10) shows, that the coefficients of the system matrix depend on geometrical and physical parameters (length l , speed of sound c, density ρ in the ducts plus the interaction index n and the time lag τ), which are fixed for a given system. However, the frequency ω, which appears in several of the coefficients, is not fixed a priori, so we may hope to find one or indeed several eigenfrequencies ωm of the system such that Det(S)|ω=ωm = 0. In this sense Eq. (2.12) is the equivalent to the characteristic equation Det(A − χI) = 0 in linear algebra. For the Rijke tube with ξ ≡ ρ h c h /ρ c c c = c c /c h , ³ ´ Det(S) = 4{cos(k c l c ) cos(k h l h ) − ξ sin(k c l c ) sin(k h l h ) 1 + n e −i ωτ }, (2.13) see McManus et al. (1993) for the derivation.
15
The characteristic equation Det(S) = 0 is a transcendental equation, i.e. it cannot be solved explicitly for eigenfunctions ωm . In general one has to resort to numerical root finding to determine the eigenfrequencies (see the Appendix for a simple implementation in Matlab). An approximate analytical solution for the special case l c = l h , ρ h = ρ c and c c = c h = c has been derived by McManus et al. (1993). Introducing non-dimensional variables with L and L/c as characteristic lenght- and time-scales, respectively, the characteristic equation (2.13) reduces for this configuration to ³ω´ cos(ω) − n e −i ωτ sin2 = 0. (2.14) 2 For n = 0, i.e. in the absence of thermoacoustic coupling between velocity and heat release at the gauze, the solutions to this equation are the familiar quarter-wave duct eigenmodes with wave lengths λ = L/4, 3L/4, . . . and frequencies ω(0) m =
π 3π 5π π , , , . . . = (2m + 1) ; m = 0, 1, 2, . . . . 2 2 2 2
(2.15)
These eigenfrequencies are real-valued, so there is no amplification or damping of the eigenmodes. For non-zero, but small interaction index n ¿ 1, the eigenfrequencies can be determined (0) 0 0 approximately as ωm ≈ ω(0) m + ωm , where the deviations ωm from the eigenfrequencies ωm are (0) assumed to be small, ω0m ¿ ωm . To first order in ω0m , cos(ω(0) + ω0m ) ≈ (−1)m+1 ω0m , Ã m ! (0) 0 ´ ω + ω 1³ m m sin2 ≈ 1 + (−1)m ω0m , 2 2 e −i ωm τ
(0)
≈ e −i ωm τ (1 − i ω0m τ).
Retaining only terms that are 1st order in the small quantities n or ω0m , one obtains from Eq. (2.14) ω0m
≈ (−1)m+1
n −i ω(0) m τ . e 2
(2.16)
The coupling between acoustics and heat release, which results from the non-zero interaction index n, leads to complex-valued eigenfrequencies: m+1 ℜ(ωm ) ≈ ω(0) m + (−1)
ℑ(ωm ) ≈ (−1)m
n cos(ω(0) m τ), 2
n sin(ω(0) m τ), 2
(2.17) (2.18)
where ω(0) m = (2m + 1)π/2, see Eq. (2.15). The result for the real part of ωm shows that the frequency of an unstable mode will in general differ slightly from the eigenfrequency of the corresponding acoustic mode with n = 0. This is due to the phase change introduced by the time-lagged response of the heat source. The imaginary part ℑ(ωm ) determines the stability of the mode. With the convention ∼ exp (i ωt ) for the time dependence, an eigenmode is unstable if the imaginary part of its eigenfrequency is negative, ℑ(ωm ) < 0, because in this case the amplitude grows with time as exp (−ℑ(ωm )). The stability limits for the time lag τ resulting from this criterion for eigenfrequencies according to Eq (2.16) are shown in Fig. 2.3 for modes 0 to 3. For almost any value of the time lag τ, at least one of the first four eigenmodes is unstable. In a more realistic model, viscous losses and especially radiation of acoustic energy to the environment at the open end should be taken into account. Both effects tend to increase stability, and they both tend to increase with frequency, so that the width of the stable regions in Fig. 2.3 would increase, and 16
m=3 m=2 m=1 m=0 0
1
2
3
τ
Figure 2.3: Stability map of the first four modes in a ”Rijke” tube with one closed end and weak coupling n → 0. Gray regions indicate stability, i.e. positive imaginary part of the eigenfrequency ωm . Re ω 1.75 1.7 1.65 1.6 0.5 1.55 1.5 1.45
Im ω 0.2 0.15 0.1 0.05 1
1.5
2
2.5
3
-0.05 -0.1 -0.15
0.5
1
1.5
2
2.5
3
Figure 2.4: Comparison of eigenfrequencies determined with numerical (——) and approximate analytical solutions (dashed lines) of the dispersion relation (2.14) with interaction index n = 0.3 for the fundamental mode m = 0. With increasing n, m the differences between numerical and analytical solutions will increase (not shown). especially so for the higher modes (Indeed, in a real Rijke tube, one usually observes only the fundamental mode m = 0). Note that the Rayleigh criterion as stated in Eq. 1.2) does not take into account the beneficial effect of losses on system stability. It is thus overly pessimistic. It is not difficult to solve the dispersion relation numerically for larger values of n (and indeed for arbitrary values of the parameters l , c, n, τ as well as arbitrary reflection coefficients at the duct ends). A simple Mathematica script is given in the Appendix, results are shown in Fig. 2.4.
2.2.2 Further remarks on dynamic stability analysis • What has been done here ”by hand” for the simplistic model of a Rijke tube can be formalized and cast into a numerical algorithm – a network model. With this approach, the thermoacoustic system is modelled as a collection of network elements. Each network element is represented mathematically by its transfer matrix (or scattering matrix), derived from coupling relations akin to Eq. (2.6). With given network structure and a library of network elements, the system of equations (2.11) (or the system matrix) representing the complete network may then be generated automatically by the computer without any ”paper and pencil” work. This can be done for networks of arbitrary topology, representing combustion systems with fuel and air supply ducts, cooling channels, etc. as suggested in Fig. 2.5. Of course, the search for eigenfrequencies is then also performed numerically. • The computation and inspection of eigenfrequencies as presented above for the Rijke tube is compatible with Rayleigh’s criterion: If one determined from a network model 17
Fuel Supply Air Supply
Burner
Flame
Combustor
Figure 2.5: Network model of a combustion system. the relative phases between fluctuations of pressure p 0 and heat release Q˙ 0 at the heat source and evaluated the Rayleigh integral (1.2), one would obtain results in complete agreement with Eq. (2.18). However, this is only so because we neglected losses and assumed ideal boundary conditions with acoustic reflection coefficients |r | = 1. Because loss mechanisms can easily be taken into account with a low-order model, the dynamic stability criterion is more powerful than Rayleigh’s criterion. • The dynamic stability criterion is in principle applicable to any homogeneous linear system. However, for complicated systems, the roots of the characteristic equation (2.12), i.e. the eigenfrequencies ωm , can in general not be determined analytically, because the system of equations (2.11) is simply too large for paper-and-pencil work. Instead, iterative numerical algorithms for root-finding are required. The iterative search for the roots gives rise to a number of problems: In order to assure that a system is stable with respect to self-excited thermoacoustic oscillations, it is necessary (and sufficient) to verify that all eigenfrequencies ωm are located in the upper half of the complex plane. However, numerical root-finding always starts from an initial guess and moves iteratively by ”trial and error” towards a root. The roots may be located anywhere in the complex plane; the basins of attraction of the various roots may be quite different in size, and some roots may indeed be very hard to find. So one can never be sure that one has really found all eigenfrequencies – but one unstable root that goes unnoticed may render an otherwise stable system unstable! • Sometimes the coefficients of the system matrix are known only for real-valued frequencies (e.g. if transfer matrices are determined by experiment or from computational fluid dynamics). In this situation, one cannot solve for complex-valued roots of the characteristic equation, because the determinant of the system matrix away from the real axis is not known. In the next section, an alternative approach making use of Nyquist diagrams as they are familiar from control theory is presented, which does not suffer from some of the shortcomings mentioned. In particular, iterative searches for eigenfrequencies in the complex plane are not required with that approach.
2.3 The Nyquist Criterion The Nyquist stability criterion is a well-known tool in control theory. It allows to test for stability of a closed-loop system by inspection of the Nyquist plot of the open-loop transfer function (OLTF) (Jacobs, 1993). In this section, a brief review of the method is given, then the application to thermoacoustic systems is discussed.
18
x0
x1
G(s)
1 Figure 2.6: Open loop system with unit negative feedback.
2.3.1 Nyquist plots in control theory Consider a system as shown in Fig. 2.6 with open loop transfer function G(s) and unit negative feedback H (s) = 1, such that x 1 = G(s) (x 0 − x 1 ), or x 1 =
G(s) x0 . G(s) + 1
(2.19)
The characteristic equation, from which stability can be deduced, is then G(s) + 1 = 0.
(2.20)
The system is stable, if all the roots s n of this equation are located in the left half of the complex plane. In that case, all perturbations of the system will decay exponentially ∼ e st with time. To locate all the roots can be tedious, therefore alternative methods to assess the stability have been developed. For example, in control theory the transfer function is usually a fraction of polynomials, P N (s) an s n + . . . + a0 G(s) = = . (2.21) P D (s) b m s m + . . . + b 0 The Routh-Hurwitz stability criterion exploits this fact and allows to determine system stability from the polynomial coefficients (Jacobs, 1993). Alternatively, one can deduce stability from the Nyquist plot, i.e. a polar plot of the imaginary axis5 mapped through the open loop transfer function. The associated Nyquist stability criterion is based on Cauchy’s argument principle, which states the following: Consider an analytical function f (z) with a number Z of zeros f (z) = 0 and a number P of poles f (z) → ∞ within a simple closed contour C in the complex plane. Then the winding number of the image curve of the contour C mapped f (z) around the origin 0 + i 0 is equal to Z − P . For the characteristic equation (2.20) and the Nyquist plot, Cauchy’s argument principle implies that the number Z of zeros of the OLTF G(s) in the right half of the complex plane is related to the number N of positive encirclements (i.e. in clockwise direction) of the ”critical point” (−1 + 0i ) and the number of poles P of G(s) as follows: N = Z − P.
(2.22)
For a stable system, no roots of the characteristic equation (2.20) should be on the right side of the s-plane, i.e. Z = 0. Nyquist’s criterion follows with Eq. (2.22): for stability, the number N of anticlockwise encirclements about the critical point (−1 + 0i ) must be equal to P , the number of open loop poles in the right half plane. 5 Strictly speaking, the Nyquist plot is the image of the Nyquist contour, which is a closed contour encompassing
the right half of the complex plane. Starting from −i ∞ one moves along the imaginary towards +i ∞ and then in a clockwise circular arc with radius r → ∞ back to the point −i ∞.
19
fˆu
1
ˆu g
fˆd ˆd g
Figure 2.7: Diagnostic dummy
2.3.2 Application of the Nyquist criterion to thermoacoustic systems It is by no means obvious how the Nyquist criterion can be employed to analyze the stability of thermoacoustic systems. A network-model-based approach for generation of a Nyquist plot has been proposed by Polifke et al. (Polifke et al., 1997; Sattelmayer and Polifke, 2003a): To establish an analogy to control systems and to define the equivalent to the OLTF, the network model must be ”cut open”. This can be done for networks of arbitrary topology by introducing a ”diagnostic dummy” element in the network. As indicated in Fig. 2.7, two of the four variables of this network element are simply ”shortened out”, e.g. gˆd = gˆu , while the second pair of variables is not connected at all. Instead, a fixed value is assigned to one of those ports, e.g. f d = 1. Note that by inserting the diagnostic dummy, the homogeneous system of equations (2.11) changes into an inhomogeneous system S0 ~ x 0 = b with r.h.s. b = (0, . . . , 1, . . . , 0). Now one can define the OLTF of the thermoacoustic system with diagnostic dummy as fˆu G(ω) = − . (2.23) fˆd The minus sign is introduced to maintain close analogy with negative feedback control systems. In general, i.e. for arbitrary frequency ω, the solution ~ x 0 (ω) of the inhomogeneous system with diagnostic dummy will not be equal to any of the eigenmodes of the original system Eq. (2.11). In this case the values of the unconnected variables across the diagnostic dummy will not be equal, i.e. fˆd 6= fˆu . However, for every eigenfrequency ωn , the solutions ~ x 0 (ωn ) of the system with diagnostic dummy will be identical to the corresponding eigenvector ~ x n of the original system (up to an arbitrary scaling factor), and the acoustic variables will match across the ”cut”, fˆd = fˆu . It follows that Eq. (2.23) defines a mapping, which maps every eigenmode of the homogeneous system Eq. (2.11) to the critical point (-1 + 0i), i.e. G(ωn ) = −1 for every eigenfrequency ωn . This important property of the OLTF will be exploited in the following. Although the equivalent to the open loop transfer function is now defined, the classical Nyquist criterion, as it was formulated in the previous section, is not directly applicable to a thermoacoustic system: As already mentioned, low-order models for thermoacoustic stability analysis are commonly formulated with harmonic time dependence ∼ e i ωt , and the imaginary part of the angular frequency ω determines stability. For a stable system, no eigenfrequencies ωn must be located in the lower half of the complex plane. This implies for the Nyquist plot, that the real axis of the ω-plane is mapped by the open loop transfer function to the G(ω)-plane. Once these modifications are taken into account, one could in principle proceed with application of the criterion in complete analogy to control theory. However, transfer functions in thermoacoustics are in general not polynomials in ω or fractions thereof, but involve harmonic or exponential functions6 . The identification of poles is in this case no easier than the determination of eigenfrequencies by iterative numerical solution of the characteristic equation (2.20). Moreover, the coefficients of acoustic transfer matrices – the building blocks of network models – are not always given in analytical form. In this case, the OLTF cannot be evaluated for 6 Incidentally, for this reason application of the Routh-Hurwitz criterion to thermoacoustic systems is not possi-
ble.
20
complex-valued frequencies with imaginary part ℑ(ω) 6= 0. These difficulties are also discussed by Polifke et al. (1997); Sattelmayer and Polifke (2003a). Therefore, a modified rule for the interpretation of Nyquist plots has been put forward, which is more suitable for application to thermoacoustic systems (Polifke et al., 1997; Sattelmayer and Polifke, 2003a). The proposed rule is based on a property of analytical functions: An analytic function is conformal, i.e. it preserves local angles or ”handedness”, at any point where it has a nonzero derivative (Jänich, 1983). Consider now the open loop transfer function G(ω) as a conformal mapping from the ω-plane onto the G-plane, see Fig. 2.8. The real axis ℑ(ω) = 0 in the ω-plane (left graph) and its image in the G-plane (right graph) are indicated by the thick dashed line with arrow head. According to Eq. (2.23), the eigenfrequencies (roots of the characteristic equation (2.20)) ωn are mapped to the critical point (-1 + 0i). Because the conformal mapping ω → G(ω) preserves handedness, the critical point will lie to the left (right) of the image curve of the real axis if the corresponding root ωn lies in the upper (lower) half of the complex ω-plane. The situation shown in the Fig. 2.8 corresponds to an unstable mode. These deliberations suggest the following modified Nyquist criterion: Consider the image curve of the positive half of the real axis ω = 0 → ∞ under the OLTF mapping in the G(ω)plane, as shown in Fig. 2.8. As one moves along the image curve in the direction of increasing frequency ω, an eigenmode with eigenfrequency ωn is encountered each time the image curve passes the critical point (−1+0i ) (In Fig. 2.8, only one sweep past an eigenfrequency is shown). If the critical point lies to the right of the image curve, the eigenmode is unstable, because then its frequency ωn is located below the real axis in the ω-plane. On the other hand, if the critical point is located to the left, the eigenmode is stable. If the image curve passes through the critical point, the mode is neutrally stable. In comparison to the classical Nyquist stability criterion, it is perhaps easier to foster an intuitive understanding of the modified Nyquist rule. However, it is admitted that no strict mathematical proof for the modified criterion is known. It has been validated successfully for a number of cases, where eigenfrequencies and growth rates can be determined by solution of the characteristic equation (Polifke et al., 1997; Sattelmayer and Polifke, 2003a). Nevertheless, one must concede that erroneous predictions may be obtained if the derivative of the OLTF vanishes for some real-valued frequency ω ∈ R (the mapping is then not conformal). Furthermore, if an eigenfrequency has a large imaginary part such that the OLTF curve passes the critical point at a large distance, the proposed criterion may fail because conformality is a local property, i.e. it holds only in a finite-size neighborhood of the point considered. Fortunately, very large growth rates with ℑ(ω) ¿ 0 are not observed in realistic network models, while very large damping rates with ℑ(ω) À 0 correspond to strongly damped modes, which are of no concern for the overall stability of a combustion system.
2.3.3 Identification of eigenfrequencies and growth rates from a Nyquist plot Conformality, i.e. the local preservation of angles under a mapping f : z → f (z), implies that an orthogonal grid of lines with constant real or imaginary part, respectively, is mapped to an orthogonal grid of lines in the image plane (with the exception of points where the derivative of f is zero). In other words, under a conformal mapping, the neighborhood of any point is rotated and stretched or shrunk, as illustrated in Fig. 2.8 (Jänich, 1983). This interpretation of conformality implies that it is possible to estimate the frequency of an eigenmode as well as its rate of growth or decay from the image curve of the OLTF: 1. the real part of the eigenfrequency is approximately equal to the frequency ω for which the distance between the image curve and the critical point attains a local minimum. 2. the imaginary part ℑ(ωn ) is approximately equal to the minimum distance from the critical point to the OLTF image curve divided by the scaling factor of the mapping. 21
ωn
Figure 2.8: Conformal mapping ω → G(ω) of the real axis under the OLTF for an unstable eigenfrequency. The scaling factor can be roughly determined by evaluating how an interval (ℜ(ωn ) − ∆ω; ℜ(ωn ) + ∆ω), is mapped to a segment G(ℜ(ωn ) − ∆ω) → G(ℜ(ωn ) + ∆ω) of the OLTF image curve. The scaling factor is then estimated as the arc length divided by 2∆ω. A more precise way of deducing the growth rate from the image curve has been proposed by Sattelmayer and Polifke, based on the identity theorem of functional analysis (Sattelmayer and Polifke, 2003b). The theorem assures that a polynomial fit for the OLTF, which approximates the transfer function G(ω) with good accuracy for a range of purely real frequencies ω1 ≤ ω ≤ ω2 , with ω, ω1 , ω2 ∈ R will locally approximate the OLTF also for complex-valued frequencies ω ∈ C. This consideration suggests to determine the growth rate of an eigenmode as follows: generate a polynomial fit PG (ω) = g m ωm + . . . + g 0 which approximates the OLTF curve close to the critical point, i.e. in the vicinity of an eigenmode ωn , see Fig. 2.8. Then determine with a numerical root finding algorithm the frequency ω∗ for which the approximating polynomial PG (ω∗ ) = −1. If the eigenmode ωn is not too far away from the real axis, then ωn ≈ ω∗ . It is remarkable that in this way complex-valued eigenfrequencies of an acoustical system can be determined from the OLTF image curve, which is computed for purely real frequencies ω ∈ R. This is very convenient when analytical expressions for transfer matrix coefficients are not known, which is usually the case for transfer matrices or response functions determined from experiment. In concluding this section we remark that with the proposed rule for interpretation of Nyquist plots, it is obvious that the frequency at which the OLTF curve crosses the real axis should not be identified with an eigenfrequency. Indeed, it has been shown by example that this popular, but incorrect ”heuristic” version of the Nyquist criterion does lead to erroneous predictions (Lundberg, 2002; Sattelmayer and Polifke, 2003a).
2.4 Suggestions for further reading Stability criteria which are like the Rayleigh criterion based on relative phases between fluctuations of velocity, pressure, fuel concentration, heat release rate, entropy, etc., have been developed and used by Richards and Janus (1998); Lawn (2000); Polifke et al. (2001b). Such criteria are certainly helpful for qualitative analysis of relevant feedback mechanisms, but one must be aware of their limitations! As discussed above, phase-based criteria can be overly pessimistic regarding the stability of a combustion system, because they cannot include the stabilizing effects of loss of acoustic energy. Furthermore, Polifke et al. (2001b) have shown that it is not always possible to determine the relative phases involved with sufficient accuracy without a
22
full network model of the combustion system. In Fig. 2.3 a simple stability map is shown, i.e. the stability limits vs. the value of the time lag τ. Similar maps can be constructed in many ways to illustrate how the stability properties of a system depend on one or more system parameters. Of course, one can do more than merely indicate parameter regions of stable/unstable behavior. For example, one may plot frequency of limit cycle oscillation, instability growth rates, ”mode switching”, etc. as a function of time lags, spread of time lags, combustor residence times, or Helmholtz resonator location, to name but a few possible arrangements (Richards and Janus, 1998; Lieuwen and Zinn, 1998; Polifke et al., 2001b; Flohr et al., 2001; Stow and Dowling, 2001, 2003). A comparison of the dynamic and "diagrammatic" approaches for system stability analysis has been carried out by Sattelmayer and Polifke (2003a). It is found that the approach proposed by Polifke et al. (1997), which has been presented above, produces the same results as the search for eigenmodes and eigenfrequencies, while the earlier rules for interpretation of the Bode diagram (Deuker, 1995; Krüger et al., 2000) are often in error.
23
Chapter 3
Overview of System Models Miscellaneous methods of stability analysis for thermoacoustic systems have been discussed in the previous section. It should have become apparent that stability analysis in general requires an underlying system model, which represents the relevant thermo-fluid- dynamic and acoustic phenomena. A wide variety of system models have been proposed for thermoacoustic stability analysis, ranging from simple estimates for phase lags based on convective time scales to highresolution CFD simulation – notably Large Eddy Simulation (LES). Some system models – notably CFD-based approaches – include ”by design” sub-models for all relevant processes which one encounters in (turbulent, reacting) compressible flow in complicated geometries. However, due to the high numerical effort associated with such a ”brute force” approach, a strategy of divide and conquer is often preferred, where the propagation (and dissipation, and reflection) of acoustic waves is modelled explicitly, while information on the dynamics of the heat source is supplied as input to the model. Such input data are obtained from experiment or CFD models, and supplied to the system model in terms of a flame transfer function, or a burner matrix, say. This chapter gives an overview of the most important types of system models. A categorization of the various methods is attempted in Fig. 3.1.
Mode Based
Time Domain
Frequency Domain
Modal Expansion Methods
Network Models!
"Galerkin Methods", system of ODEs
(nonlinear) algebraic equations
FD, FE, FVol!
State Space Models ! continuous / discrete time system of ODEs! algebraic equations
CFD!
computational (Aero-)Acoustics! linearized PDEs ! APEs, LEE, LNSE, LRF, Helmholtz
(reacting) compressible ! Navier Stokes
Nonlinear Effects
linearized Equations
Figure 3.1: Overview of models for thermoacoustic systems.
24
3.1 Finite-Volume / Finite-Element models 3.1.1 Computational fluid dynamics Modelling of unsteady (turbulent, reacting) compressible flow is a powerful tool, particularly attractive for thermoacoustic instabilities because advanced CFD tools can capture in principle all relevant processes. An introduction to unsteady flow modelling is given in companion lectures of this lecture series by Nicoud (2007), the book by Poinsot and Veynante (2005) is recommended for further reading. An obvious drawback of this approach is the high computational cost, which is certainly one of the reasons why it has been used only rarely until now (Hantschk and Vortmeyer, 1999; Murota and Ohtsuka, Indianapolis, 1999; Wall, 2005; Schmitt et al., 2007). If CFD is used for thermoacoustic stability analysis, a significant technical difficulty is the implementation of boundary conditions which enforce not only the required mean and turbulent flow conditions, but also the appropriate acoustic boundary conditions (impedance or reflection coefficient). See Selle et al. (2004); Polifke et al. (2006); Schuermans et al. (2005); Huber et al. (2008) for recent publications on this problem. A third shortcoming of the CFD approach is a point that is not always appreciated by practitioners of that art: interpretation of CFD results in the thermoacoustic context is usually not straightforward! Insight into a thermoacoustic instability mechanism is often gained only after sophisticated post-processing of CFD data, usually supplemented with acoustic analysis. See the publication of Martin et al. (2006) for an example along these lines. Note that ”brute force” simulation of transient system behaviour in the presence of small perturbations is not the only possible way of applying CFD to the study of thermoacoustic instabilities. CFD can also be used as an element in a divide and conquer strategy. In such a framework, flow simulation is often used to determine the flame transfer function or a transfer matrix (Sklyarov and Furletov, 1975; Deuker, 1995; Krüger et al., 1998; Bohn et al., 1998; Polifke et al., 2001c; Polifke and Gentemann, 2004; Gentemann et al., 2004; Armitage et al., 2004; Gentemann and Polifke, 2007; Zhu et al., 2005; Giauque et al., 2008; Huber and Polifke, 2008). It is also possible to use flow simulation to compute the OLTF of a (sub-)system, which is then used to generate a Nyquist plot (see section 2.3). Explicit determination of a flame transfer function or burner transfer matrix is not required, which can be an advantage for some system configurations (Kopitz and Polifke, 2005, 2008; Neunert et al., 2007).
3.1.2 Computational acoustics In this section, system models for thermoacoustic analysis that make use of a finite-elementor finite-volume-formulation of linearized perturbation equations are discussed. From the equations for conservation of mass and momentum in flow of an ideal gas, the linearized Euler equations for small fluctuations of velocity ~ u 0 , pressure p 0 , and density ρ 0 can be derived ∂u 0 ∂ρ 0 ∂ρ 0 ∂ρ ∂u i + ui + u i0 + ρ i + ρ0 ∂t ∂x i ∂x i ∂x i ∂x i 0 0 0 0 ∂u i ∂u ∂u i 1 ∂p ρ ∂p + u j i + u 0j + − 2 ∂t ∂x j ∂x j ρ ∂x i ρ ∂x i ∂u 0 ∂p 0 ∂p 0 ∂p ∂u i + ui + u i0 + γp i + γp 0 ∂t ∂x i ∂x i ∂x i ∂x i
= 0,
(3.1)
= 0,
(3.2)
= (γ − 1) q˙ 0 .
(3.3)
Here q˙ denotes a volumetric rate of heat release, γ is the ratio of specific heats. The variables without apostrophe 0 represent a mean flow state, and may depend on position ~ x , but not on time. 25
One can show – see e.g. Chu and Kovasznay (1958); Pierce (1981) – that every perturbation of the mean flow state described by this system of equations can be interpreted as a superposition of three different modes: an acoustic mode, a vorticity mode, and an entropy mode. If the mean flow is homogeneous and heat sources are absent, then the three modes propagate independently from each other. In that case, the acoustic mode represents propagation of pressure waves (with the speed of sound), the vorticity mode represents production and transport of fluctuations of vorticity, the entropy modes describes production and transport of ”hot spots” (regions of temperature and density different from the mean value). The latter two modes are transported convectively. If the mean fields are not spatially homogeneous, coupling between the three modes is to be expected. This can give rise, e.g., to low-frequency oscillations in combustors, where socalled entropy waves are an important element of the thermoacoustic feedback loop (Keller et al., 1985; Polifke et al., 2001b; Eckstein, 2004). The linearized Euler equations Eqs. (3.1) – (3.3) represent a very general description of the generation and propagation or transport of small disturbances in compressible flow. This set of coupled, partial differential equations is – although linearized – notoriously difficult to handle numerically. Therefore, the linearized Euler equations are not often used for computational acoustics in the form shown above . Ewert and Schröder (2003) have shown how a set of acoustic perturbations equations (APEs) can be derived from from Eqs. (3.1) – (3.3), where after suitable variable decomposition, each equation represents one of the perturbation modes. This formulation is a good basis to introduce simplifications and isolate the phenomena of interest in a specific problem setup. In FE/FV-based system models for thermoacoustic analysis, even simpler formulations are typically employed. Often the vorticity mode is neglected completely, while entropy waves are considered only in a rudimentary manner (if at all). For example, Pankiewitz and Sattelmayer (2003a,b) have developed a finite-element-based model of acoustic wave propagation in 3D geometries in the presence of heat release with non-constant speed of sound and density. The model is based on a generalized wave equation for pressure fluctuations µ ¶ 1 D2p0 ∂ 1 ∂p 0 γ − 1 D q˙ 0 − ρ . = c2 Dt 2 ∂x i ρ ∂x i c2 Dt
(3.4)
In low-Mach-number flow, the substantial derivative operator D . . . /D t may be replaced by the standard partial derivative w.r.t. time ∂ . . . /∂t . Then it is apparent that Eq. (3.4) is a generalized Helmholtz equation, with spatial dependence of speed of sound c and density ρ taken into account, and a source term due to fluctuations of the heat release rate q˙ 0 . For the heat source, models based on time lags between the acoustic velocity u b0 (computed from the gradient of pressure p 0 ) at a reference position and the momentary heat release rate in a certain region of the combustion chamber are employed. In the simplest case, a formulation equivalent to the famous n − τ model is used u (t − τ(~ x )) q˙ 0 (~ x, t ) =n b . ˙ x) q(~ ub 0
(3.5)
More advanced formulations are discussed by Pankiewitz (2004). The interaction index n – which is a measure of the strength of the thermoacoustic coupling – and the distribution of time lags τ(~ x ) cannot by computed by the FE-model, but must be determined by other means (experiment, CFD, . . . ) In this sense, the approach of Pankiewitz and Sattelmayer (2003a,b) is one example of the ”divide and conquer strategy” mentioned above. With a time-domain, finite-element based solver for Eq. (3.4), the investigation of system stability is now in principle a straightforward task, if one follows the strategy outlined in Section 2.1: first a reference solution for Eq. (3.4) without the source term is computed, then the 26
(1,0,0)
0.8
(1,1,0)
cies and lead to a not only qualitative agreement.
(1,0,0)
0.7
CONCLUSIONS We have presented two novel te coustic analysis of annular combusto especially attractive for its speed of ex simulation has the advantage that it c bitrary geometries and does not rely the propagation and coupling of the ac indicate that both approaches are suit and have the potential to be developed
f
0.6 0.5 0.4 0.3 0.2 0.8
1
1.2
1.4
1.6
τ
1.8
2
2.2
2.4
REFERENCES Figure 5. of FREQUENCIES AND CORRESPONDING TYPES computed with an Figure 3.2: Eigenfrequency dominant modes in an annular MODE combustor [1] S. Hubbard and A. P. Dowling. A FOR DIFFERENT DELAY TIMES. (!,") LOW ORDER MODEL, (•,◦) FE (triangles) and a low-order model (circles), respectively. Filled symbols indicatedmix unstable Burners. AIAA Paper AIAA-9 TIME DOMAIN SIMULATION. (!,•) UNSTABLE MODE, (◦,") STABLE [2] T. Sattelmayer. Influence of the modes. As the frequency increases, the non-plane (1,1,0) mode becomes dominant. MODE. on Combustion Instabilities from ations. ASME Paper 2000-GT-00 response to small initial perturbations is simulated. Results obtained for the eigenfrequency of [3] F. E. C. Culick. Some Recent Re domain simulation, from the time evolution of the acoustic quanthe dominant mode in an annular combustor are shown in Fig. 3.2. This figure is taken from tics in Combustion Chambers. tities the frequency and the cycle increment (and thus the growth Pankiewitz et al. (2001). It isbeseen that with varying time lag τ of the heat release model, differ-1994. 146–169, rate) can determined. [4] Murota and M. Ohtsuka. Large ent plane (mode (1,0,0))For and non-planeof (mode (1,1,0)) eigenmodes As T. expected, a comparison the two techniques we carriedare outdominant. calbustion Oxcillation in the Premix culations for depends a model combustor with lag. a geometry similar to fig. 1 the stability of the modes also on the time per 99-GT-274, 1999. twelve burners. Weresponse employed of a simple heatsource release to model Note that local with nonlinearities in the the heat large amplitude oscil[5] U. Kr¨uger, J. H¨uren, S. Hoffman adopted from Dowling [8]. At one burner j this model reads lations can be implemented in the formulation of Pankiewitz et al.. If dissipative effects are D. Bohn. Prediction and Meas also included – e.g. by imposing non-ideal acoustic boundary conditions – a limit tic cycle can Improvements in Gas Turbin " 1 d q˙ j state then be observed as an asymptotic simulation. Of course, the formulation of physi" of the " tion Systems. Journal of Enginee (7) + q˙ j = ηuB, j (t − τ) , j = 1, . . . , 12 , ω0 dt models 123, pp. 557–566, 2001. cally reasonable, let alone accurate for non-linear effects and loss mechanismPower, remains [6] S. R. Stow and A. P. Dowling. a formidable challenge. ¯˙ u¯B is the ratio of the mean heat release rate to the in an Annular Combustor. A where η = q/ Pankiewitz (2004) has also implemented a frequency-domain solver for Eq. (3.4) intions the pres2001. mean burner exit velocity and ω 0 is the cut-off frequency of the ence of a harmonic forcing term. This allows to compute the response of a thermoacoustic res[7] W. Krebs, G. Walz, and S. Hoffma low-pass filter. For the calculations with the low order model onator to an external From distributions transfer ysis of Annular Combustor. AI eq.excitation. (7) is transfered to pressure the frequency domain. obtained in this way, the matrix of geometricallyFigure non-trivial elements can be computed. 1999. 4 shows the frequencies (given in a non-dimensional A frequency-domain that stabil[8] the A. P. Dowling. Nonlinear self form) formulation as a function of thedetermines delay time τ eigenfrequencies for a given value of ωωm0 .to assess flame. Journal of Fluid the type of the (unstable stable, resp.) ity – see Section 2.2Additionally – of a thermoacoustic system has and beenleast proposed by Benoit et al.ducted (Benoit, 1997. mode is2005; given.Martin Eitheretthe axial mode (1,0,0) or the first equation290, 2005; Benoit and Nicoud, al.,pure 2004; Selle et al., 2006). A wave for small combined axial-circumferential mode (1,1,0) (see fig. 4) is obpressure perturbations (Poinsot and Veynante, 2005) is considered, served in distinct regions for the value of τ. µ 2 0 0¶ p ∂model q˙ 0 time domain Both the ∂low-order (model A) and∂the 2 ∂p , (3.6) − c = (γ − 1) simulation predict same type ∂t 2 the∂x ∂x i of mode for∂tthe same values i of τ. The corresponding frequencies are similar in both models, which is – for smallthough Machinnumbers and homogeneous mean are pressure p – equivalent to Eq. the time domain simulation the values somewhat (3.4). In the simplest case,However, the flame is modelled as a purely acoustic element. Neglecting the lower. in the low-order model the calculated level of effects of local turbulence, chemistry orhigher. heat loss, the heat release responds to fluctuations instability is generally Therefore some modesrate which are stableposition, in the timesee domain simulation, predicted to be unstable of velocity at a reference again Eq. (3.5).are For the boundary conditions, the acoustic in the low-order model. impedance must be expressed in the form The differences in frequency and stability can be attributed 1 α1of temperature from the plenum/burner– to the sudden change = + α2 + α3 ω with ω ∈ C. (3.7) Z ω chamber. This discontinuity at the mosection to the combustion is treated differently the two to approaches. Using a classicalment Galerkin finite elementinmethod discretizeSo Eq.already (3.6) on a finite element the purely acoustical eigenfrequencies (not shown here) differ, mesh with m nodes, one obtains an algebraic system of equations which affects the stability analysis. We are confident that a more elaborate treatment this remove the]. discrepan[A][P ] +of ω[B ][Pproblem ] + ω2 [Pwill ] = [D(ω)][P (3.8)
27
4
[11] is potentially high whereas in the case with swirler, the amplitude of the mode is far less important in this zone.
Z Y X
Figure 3.3: Mixed axial/azimuthal acoustic mode in an annular combustor. No unsteady heat release (Courtesy of L. Benoit).
Figure 2.6: 1st circumferential mode of the combustor in the configuration without swirler, f 836hz, acoustic Here [P ] is the vector of unknowns p 0 , the matrix [A] represents the spatial-derivative operator pressure modulus. on the r.h.s. of Eq. (3.6), [B ] represents the boundary terms. If the source term due to unsteady heat release is neglected, [D] = 0, a suitable variable transformation yields a problem of dimension 2m that is linear in the frequency ω. This problem can be solved by a direct method, e.g. QR-based, or an Arnoldi method. The latter is preferable, because only the first few modes are usually of interest. If the source term on the r.h.s. of Eq. (3.6) is included, the resulting problem cannot be solved by standard methods of linear algebra. Benoit and co-workers propose two methods to find eigenfrequencies 1. an expansion for ”weak thermoacoustic interaction” with an expansion parameter Z 1 ²≡ n(~ x ) dV. V V 0(0) 0 Eigenmodes (ωm , p m ) are sought as first-order expansions around the modes (ω(0) m , pm ) without heat release fluctuations,
Z Y
ωm 0 pm
(1) 2 = ω(0) m + ²ωm + O (² ),
=
0(0) 0(1) pm + ²p m + O (²2 ).
(3.9) (3.10)
0(0) After some algebra, an explicit expression for ²ω(1) in terms of the pressure field p m (~ x ), the heat release fluctuations q˙ 0 (~ x ) and the boundary conditions is obtained. The imaginary part of ω(1) then indicates whether the thermoacoustic coupling destabilizes the see Benoit (2005); Benoit and Nicoud for details. with swirler, f Figure 2.7: system, 1st circumferential mode of the combustor in (2005) the configuration 753hz, acoustic X
pressure modulus. A validation of this approach against a resonator with local, time-delayed heat source – again a generalized Rijke tube – gives excellent agreement for very weak thermoacoustic coupling (interaction index n = 0.01 and expansion parameter ² = 0.003). However, with coupling strength representative of combustion, i.e. interaction index n = 5 and expansion parameter ² = 1.6 significant deviations in the computed growth rates are observed. 2. The non-linear eigenvalue problem Eq. (3.8) can be solved iteratively for a sequence of eigenfrequencies ω(k) quadratic problem m , k = 1, 2, . . . from the14 ³ ´ [A] − [D(ω(k−1) )] [P ] + ω(k) [B ][P ] + (ω(k) )2 [P ] = 0. (3.11) 28
Martin et al. (2004) report that usually less than 5 iterations are enough to achieve convergence (starting from the homogeneous problem [D] = 0 for the first step of the iteration. Using this system model in combination with LES results, Martin et al. (2004) discuss acoustic energy and modes in a turbulent swirled combustor. The computational acoustics models based on linearized PDEs for perturbations in compressible flow with heat release presented in this section have not yet been validated in a comprehensive, quantitative manner against experimental data. A significant difficulty is the implementation of realistic models for the fluctuating heat release q˙ 0 (~ x ).
3.2 Galerkin method This section introduces a system modelling approach that relies on a modal expansion of the acoustic field is introduced: the Galerkin method, which is – for obvious reasons – also called modal expansion method (Culick, 2006). Another noteworthy approach that makes use of a modal expansion is the state-space approach developed by Schuermans and co-workers (Schuermans et al., 2002, 2003; Schuermans, 2003). Although that approach is quite flexible, powerful and efficient, it is not discussed further in these introductory notes. The Galerkin method is a classical method, that can be interpreted as a particular variant of a weighted residual method (Fletcher, 1991). The application of Galerkin methods is by no means restricted to thermoacoustic problems – on the contrary the method is applicable to a wide variety of differential equations. The presentation here follows the discussion of Culick (1989) and is restricted to a one-dimensional situation for ease of presentation. Note, however, that the Galerkin approach may very well be applied to geometries of applied interest, e.g. annular gas turbine combustion chambers (Krebs et al., 1999). Other recent applications are those of Ananthkrishnan et al. (2005); Tyagi et al. (2007); Huang and Baumann (2007). Consider wave propagation in the absence of a fluctuating heat source, represented by the Helmholtz equation (like Eq. (3.4), but with zero right-hand side). In the one-dimensional case with 0 < x < L and with constant speed of sound, eigenmodes ψm (x) of the pressure perturbations satisfy the equation ∂2 ψm 2 + km ψm = 0, (3.12) ∂x 2 with wave number k m = ωm /c = mπ/L for a chamber open at both ends, p 0 = 0 (a chamber closed at both ends, u 0 = 0, has the same set of possible wave numbers). The solutions to this homogeneous problem are the well-known orthogonal modes, i.e. ψm (x) = sin(k m x),
(3.13)
where orthogonality implies L
Z 0
2 ψm ψn d x = E m δmn .
(3.14)
The modes are orthogonal, because – simply speaking – the Laplace operator in the Helmholtzequation is Hermitian. More mathematical details on ”Sturm-Liouville theory” (the theory of linear differential equations, differential operators and their eigenvalues / eigenfunctions) are found in many text books. The essential feature of the Galerkin method is that the acoustic field in the presence of the heat source is expressed as superposition of these ”homogeneous eigenmodes” ψm , X p 0 (x, t ) = η m (t )ψm (x), (3.15) m
X η˙ m (t ) d u 0 (x, t ) = ψm (x). 2 m γk m d x
29
(3.16)
Substituting this expansion into the governing equations (3.4), multiplying with a basis funcRL tion ψn , integrating over the domain 0 , and exploiting the orthogonality property (3.14), a system of ordinary differential equations for the amplitudes η m (t ) is obtained: d 2 ηm + ω2m η m = F m , dt2
(3.17)
with a forcing term Fm =
γ−1 2 Em
L
Z 0
q˙ 0 (x) ψm (x) d x.
(3.18)
Note that there may be an additional term that stems from the boundary conditions, see (Culick, 1989). Also, non-linear terms, which may arise from local non-linearities in the response of the heat source to large amplitude flow perturbations, or from non-linear acoustics, can be integrated in the formulation. In that case nonlinear terms, which may take the form P l a ml η m η l or similar, appear in Eq. (3.17). From the Galerkin expansion, a system of ordinary differential equations for the amplitudes η m (t ) is obtained. Naturally then, stability analysis proceeds by ”unsteady simulation” as outlined in Section 2.1. Note that the shape and the frequency of the dominant mode, which develops in the course of the simulation, need not be equal (not even similar) to one of the eigenmodes ψm of the homogeneous problem. In general, an eigenmode of the inhomogeneous problem (with heat source) is a collective phenomenon, to which many normal modes ψm contribute. Especially in the case of nonlinear interactions, very complex behaviour with transfer of energy between normal modes during the time evolution of the ODE (3.17) is to be expected. When complex geometries of technical interest – an annular gas turbine combustor – are to be considered, the normal modes ψm can no longer be determined analytically. Instead, they are determined by computational acoustics, usually based on an FE formulation (and of course without the heat source term). The projection of the partial differential equation to generate the ODEs for the amplitudes η(t ), which requires integration over the computational domain with the acoustic variables and the ψm ’s as integrands, is then also performed numerically (Krebs et al., 1999). One must make certain that from such a procedure indeed a normal set of base functions ψm is produced. This property is often taken for granted – but as Benoit (2005) has shown, non-trivial boundary conditions are enough to lead to a non-normal set of eigenmodes for the homogeneous problem!
3.3 Network Models of Acoustic Systems In section 2.2.1, a low order model of the Rijke tube has been constructed. This model represents the Rijke tube as an assembly of three elements – the upstream duct, the gauze, the downstream duct – plus the boundary conditions. Assuming linear acoustics and harmonic time dependence ∼ exp{i ωt }, a mathematical model of the relevant thermoacoustic processes is obtained in terms of a linear system of equations, see Eq. (2.11). The unknowns are the (Fourier-transformed) acoustic variables1 of pressure p 0 and velocity u 0 at the junctions – the ports – of the elements. Many thermoacoustic systems can be represented in an approximate manner as such a network of acoustic elements, with each element corresponding to a particular component of the system, e.g. a duct, a nozzle or orifice, a burner, a flame, some kind of acoustic termination, etc., see Fig. 2.5. The coupling relations for the unknowns across an element are combined into 1 Instead of the primitive acoustic variables p 0 and u 0 , the so-called Riemann invariants f , g are often used, but
this is a matter of convention and convenience.
30
the transfer or scattering matrix of the element. The transfer matrix coefficients of all network elements are combined into the system matrix S of the network. What has been done ”by hand” for the Rijke tube, can also be done ”by software”, which is particularly useful if an network comprises many elements. The numerical tools that build up a representation of the system as an assembly of interconnected elements, construct the corresponding system matrix from this network, and then solve the algebraic problem, are often called ”network models” or ”loworder network tools” in the acoustics community. In the EU Research and Training Network AETHER, TU München is responsible for ”System Modelling and Stability Analysis”. The network tool ”taX” will be made available to network members. A structured training event is planned in March 2008, where AETHER researchers will have the opportunity to learn how to use the software. In the following, previous work on network models is reviewed and suggestions for further reading are given. Then, a few simple examples of network elements and results obtained with network models are discussed.
3.3.1 Review of previous work An abundant literature exists on low-order models of acoustic wave propagation in ducts, good introductions are found in Munjal (1986) and Poinsot and Veynante (2005) (the latter with consideration also of combustion instabilities). The use of low order models for thermoacoustic stability analysis has been pioneered by Merk (1957). The low-order method has been applied to study combustion instabilities in afterburners (Bloxsidge et al., 1988), Rijke tubes (Heckl, 1988) and lean premixed stationary gas turbines (Keller, 1995). Validations of 1D low-order models against experimental data from a single burner test rig have been published by Bloxsidge et al. (1988) and Schuermans et al. (2000). For the particular case of annular gas turbine combustors, the low-order method has been extended to quasi 2D or 3D geometries. Under the guidance of J. J. Keller, Curlier (1996) developed a numerical tool to study the propagation of acoustic waves in annular geometries in the presence of a mean flow. The underlying approach to represent the acoustic wave field goes back to Tyler and Sofrin (1962). Krüger et al. (Indianapolis, USA 1999) used a network of four-pole acoustic elements to investigate the stability limits of azimuthal modes in a heavyduty annular gas turbine combustor. Their particular approach, however, seems to be limited inasmuch as it is not capable of handling mixed modes – see Fig. 3.5 below – in a reasonable manner. The interaction of entropy waves with acoustics in an annular combustor with a choked exit has been investigated by Polifke et al. (1999), details of the method and additional results have been published by Polifke et al. (2001b); Polifke (2004). An extension of the method, capable of describing the effect of burner-to-burner variability (”broken symmetry”) and mode coupling has been proposed and validated by Evesque and Polifke (2002). This is important, because Berenbrink and Hoffmann (2000) showed experimentally that judiciously introduced differences between burners could be exploited as a means of passive control. The difference and significance of spinning vs. standing modes in annular combustors has been discussed by Evesque et al. (2003). Bohn and Deuker (1993) were perhaps the first to propose the implementation of low-order methods as a network tool, where a library of acoustic elements is generated, each of them represented by its respective transfer matrix, and combined in a flexible and to some extent automated manner to represent the dynamic properties of complicated systems. Since then, network tools have been implemented by the research groups at RWTH Aachen for Siemens Power Generation (Krüger et al., Indianapolis, USA 1999), at ABB Corporate Research in Dättwil (Polifke et al., 1997, 2001b) and more recently also at the University of Cambridge (Stow and Dowling, 2001). Stow and Dowling (2001, 2003) also implemented a low-order model for annu31
l Au Ad xd
xu
Figure 3.4: Acoustically compact contraction. lar combustors. Particular attention was paid to the boundary conditions appropriate for a gas turbine combustor, otherwise the formulation is in essence identical to the method proposed by Keller and co-workers above Keller (1995); Curlier (1996); Polifke et al. (2001b). A new, qualitatively different and seemingly quite powerful variant of the network approach has been proposed recently by Schuermans et al. (2003); Schuermans (2003), which makes use of a state space approach.
3.3.2 Compact element with inertia and loss Consider the propagation of acoustic perturbations through a contraction as shown in Fig. 3.4 with length l = (x d − x u ) such that kl ¿ 1 (acoustically compact). A transfer matrix for such an element may be derived in an approximate manner from momentum and mass conservation. Conservation of momentum is exploited to derive a coupling relation between acoustic perturbations p 0 , u 0 on both sides of the contraction by integrating the unsteady Bernoulli equation (Hirschberg, 2001): ¶ µ ∂ ∂ϕ u 2 γ p = 0. (3.19) + + ∂x i ∂t 2 γ−1 ρ along a streamline of the mean flow from x u to x d . First consider the term with the velocity potential ϕ: Z Z Z xd ∂ xd ∂ϕ ∂ xd ∂ Au d xi = ui d xi ≈ uu d x. (3.20) ∂t xu ∂x i ∂t xu ∂t A(x) xu The approximation u(x) ≈ u c A(x) is valid because it was assumed that the element is acoustically compact – then the flow through the element is effectively incompressible. Introducing an extended length Z xd Au l ext ≡ d x, (3.21) x u A(x) and assuming harmonic time-dependence, one obtains: ∂ ∂t
Z
xd xu
∂ϕ d x i = i ω l ext u u0 . ∂x i
(3.22)
The remaining two terms in (3.19) are simply evaluated at x u and x d . Linearization and dividing by c yields · ¸d p0 0 0 i k l ext u u + M u + = O (kl )2 , (3.23) ρc u with [v]du ≡ v d −v u for any variable v. This result is exact to first order in the Helmholtz number k l , because the time derivative in (3.20) introduces a factor k l , so that any deviations of u(x)
32
from u c A(x) appear as second order terms in (3.23). Neglecting terms of first order in Mach number, this result may be simplified ¯ ¯ p 0 ¯¯ p 0 ¯¯ = − i k l ext u u0 + O (kl )2 + O (M ). ρ c ¯d ρ c ¯u
(3.24)
Note that l ext may be significantly larger than the geometrical length l of the element if the area A(x) < A u , A d between the up- and downstream terminations. Mass conservation between upstream (”u”) and downstream (”d ”) demands that d dt
Z
ρ dV +
Z
ρu i d A i = 0.
(3.25)
For the geometry considered (see Fig. 3.4) and assuming quasi-one-dimensional flow, this can be simplified as follows Z xd d ρ(x) A(x) d x + [ρu A]du = 0. (3.26) d t xu For the density, ρ 0 (x) = ρ 0u + O (kl ) and with the assumed harmonic time dependence i ωρ 0u
Z
xd xu
A(x) d x + [(ρ 0 u + ρu 0 ) A]du = O (kl )2 .
Diving by the mean density ρ and defining a reduced length Z xd A(x) dx l red ≡ Ad xu one obtains:
(3.27)
(3.28)
¯ ·µ ¶ ¸d p0 p 0 ¯¯ 0 + u +M i k l red A d A ≈ 0. ρ c ¯u ρc u
(3.29)
Neglecting terms of first order in M , this simplifies further to u d0 A d
= u u0 A u − i k l red
¯ p 0 ¯¯ Ad + O (kl )2 + O (M ). ρ c ¯u
Finally, the transfer matrix for a compact acoustic element follows: 0 µ ¶ p 1 −i k l eff − ζM ρc = −i k l α red u0 d
(3.30)
with loss is approximated as p0 ρc u0 u
(3.31)
This is a combination of (3.24) and (3.30), with an area ratio α ≡ A u /A d and a pressure loss term ζM . The loss term ζM has been introduced to account in an approximate manner for loss of acoustic energy due to, e.g., flow coupling. With this term included, a change in volume flow rate u 0 A through the element results in a change in pressure drop p u0 − p d0 across the element. • The loss coefficient ζ for the acoustic pressure that appears in Eq. (3.31) is in general not equal to the hydrodynamic loss coefficient ζd for the total head. It is fortunately not difficult to establish a relation between the two loss coefficients, which may be expected to hold with reasonable accuracy a) in the quasi-stationary limit and b) if the relevant loss mechanisms flow and acoustics are the same. Consider Bernoulli’s equation from position ”u” upstream to position ”d” downstream with a loss term: ρ ρ ρ (3.32) p d + u d2 = p u + u u2 − ζd u d2 . 2 2 2 33
By convention, the loss is expressed in terms of the downstream velocity u d . Continuity implies that u d = α u u with α = A u /A d , and the downstream pressure p d can be expressed in terms of upstream pressure p u and velocity u u as follows: ρ 2 u (1 − α2 (1 + ζd )). 2 u
(3.33)
− [α2 (1 + ζd ) − 1] M u u d0 .
(3.34)
pd = pu + Linearization and division by ρc yields p d0 ρc
=
p u0 ρc
The term in angular[. . .] brackets appears as the acoustic loss coefficient ζ in Eq. (3.31). For a sudden change in flow cross-sectional area, conservation of momentum implies for the loss coefficient ζd = (1 − 1/α)2 (neglecting the vena contracta effect for contractions). Then ζ = 2α(α − 1), which is small and negative for 0 < α < 1 and increases rapidly for area ratios α > 1 (contractions). • The effective length introduced in (3.31) combines the extended length – see the definition (3.21) – with end corrections l ec , which may appear at sharp corners, l eff ≡ l ext + l ec ,u + l ec ,d
(3.35)
For an element with significant ”intermediate contraction”, A(x) < A u , A d over some length, the following inequalities should hold: l red < l < l ext < l eff . Physically, the effective length accounts for the inertia effects, i.e. a change in pressure difference p u0 − p d0 will accelerate the fluid between the two reference positions ”u” and ”d”, resulting in a gradual change of velocity u 0 through the element, which leads to a phase difference between p u0 − p d0 and u 0 . An expression similar to (3.31) has been introduced by Paschereit and Polifke (1998) to approximate the transfer matrix of a premix burner (without flame). It turns out that this form is indeed quite general and applicable to all sorts of compact elements, e.g. sudden changes in cross sectional area, an orifice, etc. (Gentemann et al., 2003). Computational fluid dynamics may be used to determine the values of the coefficients appearing in (3.31), see e.g. Flohr et al. (2003)
3.3.3 Duct elements In this section, we present transfer matrices for a few non-trivial duct elements. Thin annular duct The equation (2.6) given above for wave propagation in a duct with wave number k = ω/c is valid only for plane waves without mean flow. In general, non-plane waves (”higher-order modes”) may propagate in a duct at sufficiently high frequency (above the cut-off frequency). Of particular interest for gas turbine combustion are thin annular ducts, with the height of the annulus significantly smaller than the radius R or the length L, such that the amplitude of acoustic waves depends on the axial coordinate x and the circumferential position φ, but not on the radius r , ¡ ¢ p 0 ∼ exp i ωt − i k x x − i k ⊥ Rφ , (3.36) 34
(a) Axial mode (1,0)
(b) Azimuthal mode (0,1)
(c) Mixed mode (1,1)
Figure 3.5: Plane and higher order eigenmodes in a thin annular duct. and similarly for velocity u 0 or the Riemann invariants (but see the discussion of standing vs. rotating modes by Evesque et al. (2003)). Different mode shapes in such a thin annulus are shown in Fig. 3.5. For the azimuthal or perpendicular component of the wave vector, k ⊥ ≡ ±m/R with mode index m, because the wave field must be periodic in the azimuthal direction. If one plugs the expression (3.36) into the 2-D form of the convective wave equation, (
∂ +~ u · ∇)2 p 0 − c 2 ∇2 p 0 = 0, ∂t
(3.37)
one obtains the following condition for the axial component of the wave vector s µ ¶ k⊥ 2 ω/c −M ± 1 − (1 − M 2 ) . k x± = 1− M2 ω/c
(3.38)
Here M ≡ U /c is the mean flow Mach number for uniform mean flow in the axial direction with velocity U . The transfer matrix for non-plane wave propagation in a thin annular duct with mean flow is similar in form to the previous result (2.6): µ
fd gd
¶
µ
=
e −i k x+ l 0
¶µ
0 e −i k x− l
fu gu
¶
.
(3.39)
Note, however, that the relation between Riemann invariants and the primitive acoustic variables is a bit more complicated, p0 = f +g, ρc with a coefficient κ± ≡
u 0 = κ+ f + κ− g .
k x± k − M k x±
.
(3.40)
(3.41)
For zero Mach number and plane waves m = M = 0, this reduces to the previous results (2.6) with k x± = ±ω/c. With this formalism, it is possible to describe efficiently the acoustic wave field in a modern gas turbine with an annular combustor from the compressor exit to the turbine inlet, say. Ducts with varying cross-sectional area A duct with varying cross sectional area – also an annular duct with curved walls, like a gas turbine combustion chamber – can be effectively modelled by ”stair-stepping” the contour of 35
the duct, i.e. as an alternating succession of short straight duct elements and small discontinuous changes in cross sectional area. The transfer matrices of these ”building blocks” have been introduced above (a change in cross-sectional area can be described as a compact element as in 3.3.2), the transfer matrix of the complete duct is obtained as the matrix-product of the individual transfer matrices. Note that in general for each sub-element, the cross-sectional area A and the radius R may be different, therefore the wave vector (k x , k ⊥ ), the Mach number and the area ratio will not be constant. Experience shows that geometries of applied interest can be represented with sufficient accuracy with about one dozen of sub-elements. Joints and forks In networks with non-trivial topology, c.f. Fig. 2.5, elements are needed to describe a situation where two flow paths are joined into one, or one flow path splits into two. In principle, this requires a 3 × 3 transfer matrix to interconnect the six Riemann invariants at the three ports of such an element. In the simplest case, one may assume equal pressures, i.e. p i0 = p 0j = p k0 and conservation of mass, i.e. u i0 A i + u 0j A j + u k0 A k = 0 to construct the transfer matrix. Of course, if end corrections, loss coefficients, etc. must be considered, the coupling relations are more complicated, and one has to take into account mean flow direction – but there is no essential difficulty in constructing transfer matrices for such elements.
3.3.4 Transfer matrix of a compact flame In this section a closure formulation for the transfer matrix of a compact flame (more generally speaking: a compact source of heat) is presented. ”Compact” means that the heat release is concentrated in a region much smaller than acoustic wave lengths. In this approximation, a premix flame front is considered as a discontinuity of negligible thickness, which adds a certain amount of heat q per unit mass (units J/kg = m2 /s2 ) to the flow of an ideal gas. For steady flow of gas, Chu (1953) has shown that conservation of energy, mass and momentum across a thin source of heat leads to the following Rankine-Hugoniot relations: c h2
ξ
µ
uh ¶
p ρc
h
= c c2 + (γ − 1) q + O (M 2 ), q = u c + (γ − 1) M c + O (M 2 ), cc µ ¶ p q = − (γ − 1) M c + O (M 2 ), ρc c cc
(3.42) (3.43) (3.44)
with ξ ≡ ρ h c h /ρ c c c . Here the index ”c” stands for the cold (upstream) side of the discontinuity, ”h” stands for the hot (downstream) side A detailed derivation of these relations (using the present notation) is given in (Polifke et al., 2001b). A useful result follows from Eq. (3.42): Th q − 1 = (γ − 1) 2 . Tc cc
(3.45)
The linearization of the Rankine-Hugoniot relations (3.42) - (3.44) in the presence of (small) acoustic fluctuations shows how fluctuations of pressure and velocity up- and downstream of the flame are related to fluctuations of the heat release rate: ¶ µ 0 µ 0¶ µ 0¶ µ ¶ u c Q˙ 0 p p Th ξ = − − 1 uc Mc + , (3.46) ρc h ρc c Tc u c Q˙ µ ¶ µ ˙0 ¶ p0 Th Q u h0 = u c0 + − 1 uc − c . (3.47) Tc Q˙ p c where Q˙ ≡ ρu q˙ is defined as rate of heat addition per unit area, and terms of order M 2 or higher are neglected. Again, a detailed derivation of these results is found in (Polifke et al., 2001b). 36
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