Non-normality and non-linearity in aero- and thermo-acoustic systems

Transcription

1 Non-normality and non-linearity in aero- and thermo-acoustic systems Wolfgang Polifke Lehrstuhl für Thermodynamik Technische Universität München, Germany November 18th, 2010 Contents 1 Introduction 3 2 Flame dynamics Flame describing function Beyond describing function Analytical models One-way coupling Proper orthogonal decomposition Nonlinear identification Non-normality Mathematical background non-normal Operators Choice of norm Optimal Initial Condition / Disturbance Driving Force Sensitivity of Eigenvalues Numerical Range and Maximum Growth Rate Summary Non-normal thermoacoustics Physics of non-normality in thermoacoustic systems Conclusion 24 5 Acknowledgements 24 polifke@td.mw.tum.de VKI - 1 -

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3 1 INTRODUCTION 1 Introduction and literature review The occurrence of combustion instabilities has been a plaguing problem in the development of combustors for rockets, jet engines and power generating gas turbines. Much of the recent research activity on this problem is due to the introduction of low-emission, leanpremixed combustion technology, which has been shown to be particularly susceptible to combustion driven instability (Keller 1995; Annaswamy et al. 1997; Richards and Janus 1998). Combustion instability arises primarily from an interaction of acoustic waves and fluctuating heat release. Thus, predicting and controlling combustion instability requires an understanding of the interactions between the combustion process and the acoustic waves. Combustion instabilities are self-sustained large amplitude oscillations of the pressure and velocity in combustors with the flame acting as an acoustic actuator and the combustion chamber as an acoustic resonator. If the flame responds to pressure and velocity fluctuations, a feedback loop between combustion and acoustics is established, which can destabilize the system. The occurrence of combustion instability depends on the phase between heat release fluctuations and pressure fluctuations at the flame (Rayleigh 1878). Fluctuating heat addition will amplify the acoustic oscillations, if the maximum and minimum of the heat addition occur during the compression and rarefaction phases of the pressure oscillation, respectively. When pulsations start spontaneously the system is said to be linearly unstable; i.e., the system is unstable with respect to any small amplitude disturbance, as they are always present. This scenario has been modeled by classical linear stability analysis with the assumption of normal modes (Merk 1956; McManus et al. 1993; Dowling 1995; Keller 1995), where the evolution of infinitesimally small perturbations is determined by linearizing the dynamical system about the unperturbed state and examining the eigenvalues (eigenfrequencies) of the system (see also the companion lecture by Polifke (2010b)). If the real parts of all the eigenvalues are negative, i.e. if the amplitudes of all eigenmodes decay exponentially in time, then the system is said to be linearly stable. If at least one of the eigenvalues has a positive real part, the system is linearly unstable with an exponential growth in the amplitude of oscillations. For linear stability analysis, so-called network models are popular, in which each element is modeled using a linear transfer function. In this context, the flame dynamics is described in terms of a flame frequency response. But for the simplest configurations, the frequency response of the premixed flame is determined by experiment or from CFD data and given as input to the network model (Noiray et al. 2008; Kaess et al. 2009; Alemela et al. 2010). Other methods for linear stability analysis have been proposed and developed in recent years, driven by the need to handle complex geometries (see, e.g. the companion lectures by Nicoud (2010) or Polifke (2010b)). Note that even for these advanced system models, information about the flame dynamics is usually given as input in terms of a flame frequency response or generalizations thereof. Linear stability analysis predicts exponential growth of oscillation amplitudes with time. However, exponential growth cannot continue indefinitely. Instead, it is expected that non-linear effects come into play, which reduce the production or increase the dissipation of fluctuation energy. Non-linear effects are represented by additional terms in the governing equations, that are 2nd (or higher) order in fluctuating quantities, e.g. u 2, VKI - 3 -

4 1 INTRODUCTION Production / Dissipation of Fluctuation Energy B A Amplitude Figure 1: Balance of production (magenta, fat line) and dissipation (blue, dashed lines) of fluctuation energy leads to limit cycle in a linearly unstable system. Non-linear effects curb the growth of production with amplitude ( amplitude saturation, case A). Alternatively (or in addition), loss terms may increase in a non-linear fashion (short-dashed blue line), resulting in reduced limit cycle amplitude (case B) or p u, etc. Due to these terms, analytical treatment is in most cases no longer possible. At the same time, interpretation of numerical results can be very challenging, because nonlinear system may exhibit very rich behavior (Sagdeev et al. 1988; Strogatz 2000). In any case, it is expected that eventually oscillation amplitudes will stop growing, and a limit cycle may be established, see Fig. 1. Note that for combustion instabilities, it is also possible that the combustor is destroyed due to excessive mechanical or thermal loads before a limit cycle is reached! Often it is assumed that the flame is the dominant nonlinear element in a combustion system (Margolis 1994; Dowling 1997; Peracchio and Proscia 1999; Hantschk and Vortmeyer 1999; Pankiewitz 2004; Stow and Dowling 2004; Noiray et al. 2008) (In other words, gasdynamic nonlinearities are assumed unimportant). Reduction of the gain of the flame response with increasing oscillation amplitude ( amplitude saturation ) can then lead to a balance between production and dissipation of fluctuation energy. Alternatively, it is also possible that the Rayleigh index is reduced due to a nonlinear change in the phase lag between flucutations of velocity and heat release (Balasubramanian and Sujith 2008a). Finally, is possible that dissipation increases disproportionately, limiting the growth of amplitudes. However, it seems that this point is not discussed in depth in the combustion dynamics community. There is another role that nonlinearity may play in thermoacoustic stability: a linearly stable combustor (i.e. one that does not pulse spontaneously) may be triggered into pulsating operation by the introduction of a finite amplitude disturbance. In practice, such finite amplitude perturbations might be generated by a a sudden shift in operating conditions, or a perturbation of the air flow into a jet engine due to a flight maneuver, etc. A system exhibiting nonlinear instability would be stable with respect to disturbances with amplitudes below a certain threshold value, but transition into pulsating operation will occur when the amplitude of a disturbance exceeds a threshold value. In this case, VKI - 4 -

5 1 INTRODUCTION one speaks of sub-critical transition or sub-critical bifurcation (possibly triggered by transient growth, see below). Furthermore, there are instances of bootstrapping, where a mode that decays initially can grow later and ultimately become unstable due to nonlinear (or non-normal, see below) exchange of oscillation energy with other modes (Yoon et al. 2001). Classical linear stability is not capable of predicting such phenomena. A few words on dynamical systems lingo 1 : a bifurcation is a point in parameter space, where a drastic change in system behavior occurs. The obvious example is the occurrence of an instability, as exemplified in Fig. 2 (images taken from Rocchini (2007)). In these Figure 2: ( Pitchfork )-Bifurcations: Supercritical (left) and subcritical (right). Continuous / dashed lines indicate stable / unstable solutions plots, r is a system parameter, and x denotes the oscillation amplitude. In both cases, there is a stable solution branch with x 0 for parameter values r below a critical value r crit (indicated by the gray vertical lines). As r approaches the critical value, this solution looses its stability. For a supercritical bifurcation (left plot in the figure), new (stable) solution branches are found for parameter values r > r crit. In case of a subcritical bifurcation (right plot in the figure), new (unstable) solution branches exist also for parameter values r < r crit. For triggering, i.e. development of a large amplitude instability (limit cycle) due to a finite amplitude perturbation, the subcritical case is important: for a linearly stable situation with r < r crit, a perturbation can push the system outside the region of stability, i.e. onto or beyond the unstable solution branch (dashed line). Then, as indicated by the arrows pointing up-/downward, amplitudes x will grow even further. The nonlinear response of the flame or, in general, the heat source is believed to play a crucial role also in this context. Despite recent progress based on a describing function for the flame dynamics (Stow and Dowling 2004; Noiray et al. 2008), comprehensive prediction of the conditions for the onset of nonlinear instabilities is a difficult task, which is not yet mastered. In particular, predicting the conditions under which finite-amplitude disturbances destabilize a linearly stable system remains a key challenge, as even in simple laminar flames little is known about key parameters controlling the nonlinear flame dynamics (Zinn and Lieuwen 2005). While considerable research has been performed on nonlinear aspects of thermoacoustic oscillations, the non-normal nature of thermoacoustic interactions has not received much 1 see also VKI - 5 -

6 1 INTRODUCTION attention until the work of Sujith and co-workers (Balasubramanian and Sujith 2008b, a, c; Mariappan and Sujith 2010; Priya and Sujith 2010). Analyzing simplified model problems (Rijke tube, or laminar premix/diffusion flames) they showed that non-normality may have significant consequences for system stability, i.e. it can lead to transient growth of perturbation amplitudes, even if the eigenvalues indicate linear stability, and even if perturbation ampltiudes are initially so small, that nonlinear effects are not yet significant. Transient growth was observed to lead in some instances to high amplitude levels, where nonlinearities become eventually important, resulting in nonlinear driving. The nonlinear driving may then cause a system to become unstable, which would be considered stable according to classical linear stability. Thus contradictory to the general belief about triggering, it is not true that only a high amplitude pulse can cause the system to trigger and finally become unstable. Such behavior can also result from the non-normal, linear growth mechanisms for small initial inputs to the system. Further progress in the analysis and control of thermoacoustic instabilities requires a deeper understanding of nonlinear and non-normal interactions between flame and acoustics. In particular, it must be explored under what circumstances nonlinar or non-normal effects may render classical linear stability analysis invalid. In this context it is also important to formulate adequate models for the flame dynamics as a prerequiste for dependable prediction of system stability in the presence of finite perturbations. With financial support of the FP6 Marie Curie RTN project AETHER (contract nr MRTN- CT ) a summer school / workshop on Non-normality and nonlinearity in Aero- and Thermoacoustics (n3l) took place from May 17th to 20th at TU München. On the first two days, invited lectures by C. Bailly (École Centrale Lyon), M. Juniper (Univ. Cambridge), A. Hirschberg (TU Eindhoven), P. Schmid (École Polytechnique, Paris), T. Schuller (École Centrale Paris) and R.I. Sujith (IIT Madras, Chennai) introduced pertinent fundamental concepts and methods and reviewed the state of the art. On the third and fourth day, recent results covering all aspects of the workshop topic were presented in a series of talks to approximately sixty participants from many European countries (and also outside Europe). Some of the ideas and results concerning non-normality and nonlinearity in thermoacoustis will be discussed in the following. VKI - 6 -

7 2 FLAME DYNAMICS 2 Nonlinear flame dynamics and system models The dynamics of premixed flames under acoustic excitation has been the subject of a large number of investigations, see e.g. the review of premixed flame / acoustic wave interaction by Lieuwen (2003). Existing literature focuses on how flames respond to imposed flow oscillations, i.e. on determination of a flame transfer function F (ω). Theoretical analyses of premixed flame dynamics often involve a kinematic description of the flame front as a hydrodynamic discontinuity, using the so called G-equation. The heat-release oscillations are modeled as being caused by fluctuations in the burning area of the flame, possibly also the burning velocity or the equivalence ratio at the flame surface (Lawn and Polifke 2004). The following papers illustrate the kinematic approach in its application to laminar flames: Boyer and Quinard (1990) developed a linear kinematic model, that was used to study the shape and dynamics of an anchored, laminar premixed flame. Fleifil et al. (1996) developed a one-dimensional kinematic model for an axi-symmetric premixed flame stabilized in a tube on an underlying Poiseuille mean flow profile, subjected to one-dimensional acoustic excitation. Ducruix et al. (2000) developed a similar model and compared the results with experimental measurements of heat-release transfer functions for a conical premixed flame. Good agreement between predicted and observed transfer function magnitude and phase was reported. Schuller et al. (2003) presented a unified model for the flame-transfer functions of conical- and V-flame configurations. A formulation incorporating convective effects provided an improved prediction for gain and phase. However, a satisfactory explanation of the nature and origin of the convective contributions was not given. The influence of the multi-dimensional acoustic near field upon the velocity and pressure distribution just upstream of the flame in a confined configuration was investigated by Lee and Lieuwen (2003). These two-dimensional effects were coupled with the kinematic model to calculate the flame transfer function of a ducted conical flame by Santosh and Sujith (2005). The coupling resulted in significant changes in both the amplitude and phase of the transfer function compared to those predicted by the linear theory of Schuller et al. (2003). 2.1 Flame describing function The nonlinear response of premixed flames to acoustic forcing was investigated, e.g. by Dowling (1999), Balachandran et al. (2005), Hosseini and Lawn (2005), Noiray et al. (2008), Moeck et al. (2009, 2008), Stow and Dowling (2009), Karimi et al. (2009) and Shreekrishna et al. (2010). The nonlinear dynamics of the flame is often described in terms of the (sinusoidal) flame describing function G(ω, A), i.e. a generalized flame transfer function, where both the gain and phase of the response to harmonic forcing may depend on fluctuation amplitude A = u rms. Such a model allows to predict limit cycle amplitudes, nonlinear instabilities ( subcritical bifurcations ) and hysteresis, but neglects non-normal as well as nonlinear modal coupling, as it is assumed that higher harmonics are not important due to the low-pass behaviour of premix flames. Dowling (1999) has explained how limit cycle amplitudes may be determined with a low-order network model : with a linear FTF F (ω), eigenfrequencies are found as roots of the characteristic equation ( dispersion relation ) Det(S(ω r + iω i )) = 0. The eigenfrequencies are in general complex-valued, ω C, and the imaginary part indicates VKI - 7 -

8 2 FLAME DYNAMICS 2.2 Beyond describing function whether an eigenmode grows or decay. With a nonlinear FDF G(ω, A), one identifies limit cycles as roots of Det(S(ω r,a)) = 0 with zero growth rate ω i = 0. Noiray et al. (2008) studied the nonlinear dynamics of laminar, unconfined premixed flames anchored on an orifice plate. Using hot wire probes and photomultiplier to measure the upstream velocity and the intensity of OH emission, respectively, the flame describing function G(ω) was determined over a wide range of frequencies and for amplitudes up to u rms/ū =0.6. Results are shown in Fig. 3 taken from (Noiray et al. 2008). It is evident that for this flame, both gain and phase show nonlinear effects for perturbation amplitudes as low as u rms/ū =0.16. G (a) u rms / u (b) 4π 3π 2π π ϕ f (Hz) f (Hz) 0 Figure 3: Flame describing function of burner stabilized laminar premix flame. Gain (right) and phase (left) vs. oscillation frequency are plotted for a range of excitation amplitudes Noiray et al. (2008) also investigated self-excited oscillations for this type of flame arising due to acoustic interaction with an upstream plenum of adjustable length, which acted as a tunable resonator. Using the describing function, a nonlinear stability map, with plenum length as bifurcation parameter, was generated and successfully compared against experiment, see Fig Beyond describing function Flame describing functions can be measured albeit only with considerable effort and as such provide a realistic description of nonlinear flame dynamics. However, an FDF gives only a simplified, truncated flame dynamics, as it neglects higher harmonics and cannot account for nonlinear interactions between modes. In that sense, simple models of nonlinear heat source dynamics, although highly idealized, exhibit richer behavior, which explains the interest in such models. For example, Moeck et al. (2008) presented a nonlinear flame model that is based on equivalence ratio fluctuations and the associated perturbations in flame speed. With this model, subcritical thermoacoustic instabilities in a premixed combustor test rig with a swirl-stabilized burner were investigated. The model reproduces experimentally observed phenomena, such as triggering of limit cycle oscillations from initially stable states and hysteresis in the variation of the preheat temperature and the air and gas mass flow rates. VKI - 8 -

9 2.2 Beyond describing function 2 FLAME DYNAMICS 0.6 mode 1 mode 2 mode Noiray et al, JFM, 2008 u rms /u ω i > 0 ω i > 0 ω i > 0 f mode 2 mode 1 mode L L 0.6 Figure 4: Stability map of ensemble of unconfined premix flames stabilized on a porous plate with upstream plenum of length L acting as acoustic resonator. Shaded regions indicate modal growth, the continuous lines bordering the shaded regions give the limit cycle amplitudes for given L. Color indicates mode order Analytical models Dowling (1997) introduced a model for a heat source with saturation: the heat release rate Q is proportional to the velocity perturbation u up to a critical level u nl, after that fluctuations of heat release rate no longer increase, see Fig. 5. This model has been used, e.g. by Pankiewitz (2004) to study formation of limit cycles in a multi-burner, annular combustor with a finite-element based solver for a generalized Helmholtz equation. Heckl (1990) study nonlinear acoustic effects in a Rijke tube, and introduced nonlinear behavior into King s law (see e.g. the companion lecture by Polifke (2010b)) as follows: Q u u (t τ) u0 3. (1) This model has been used by Balasubramanian and Sujith (2008c), Juniper et al. (Juniper and Waugh 2010; Waugh et al. 2010) and others to study non-normal and nonlinear effects like transient growth, triggering, and bypass transition in thermoacoustics. Q' u' nl u' Figure 5: Dowling s model of a heat source with saturation VKI - 9 -

10 2 FLAME DYNAMICS 2.2 Beyond describing function One-way coupling The so-called G-Equation, G t + u G = S L G (2) can be interpreted as a front tracking equation for premix flame position Kerstein et al. (1988); Fleifil et al. (1996); Boyer and Quinard (1990). In many cases, this model is formulated for the case of constant density across the flame front, i.e. it is only a oneway-coupling, where the flow affects the flame, but not vice versa. Preetham and Lieuwen (2004) have investigated the nonlinear dynamics of conical and wedge-shaped flames. They discuss the sources of nonlinear behavior in the front tracking equation, and speculate on the type of bifurcations and limit cycle solutions that may occur, depending on characteristics of the nonlinear flame dynamics. Some of the results are presented in terms of describing functions, but the nonlinear G-equation indeed is richer than that. Balasubramanian and Sujith (2008b) formulated a model of the nonlinear dynamics of a laminar diffusion flame. Like the G-equation for premix flames, his model is limited to one-way-coupling, i.e. it assumes constant density. The 2D convection-diffusion equation for the Shvab-Zeldovich variable Z = X Y (where X, Y are mass fractions of fuel and oxidizer, respectively) Z t + u(t) Z x = 1 ( ) 2 Z Pe x + 2 Z (3) 2 y 2 is solved for an imposed, spatially uniform perturbation velocity u(t) with a modal expansion approach 2 ( Galerkin method ) with a few thousand degrees of freedom. When coupled with a numerical model for the acoustics in a duct, Balasubramanian and Sujith (2008a) observed nontrivial phenomena related to the non-normal nature of thermoacoustic interactions (see below) Proper orthogonal decomposition Huang and Baumann (2007) developed a physics-based, reduced-order, nonlinear heat release model for a planar, burner-stabilized, laminar premix flame. Assuming a one-step global reaction, the conservation equations for species and energy are formulated with time t and streamline ψ(x, t) x 0 ρ(x, t) dx. In this formulation, the incoming mass flow rate ρu x=0 becomes an explicit part of the equations. The response to imposed mass flow rate perturbations is simulated with a finite difference method. Using proper orthogonal decomposition (POD) (Berkooz et al. 1993; Chatterjee 2000) and a generalized Galerkin procedure, the infinite-dimensional PDE model was reduced to a set of low-order, nonlinear ordinary differential equations. The issues of model order versus accuracy and the selection of mode shapes to be used in the reduction are discussed. A two-mode linear acoustic model for the combustor is coupled to the unsteady heat release model, qualitative agreement with experimental data is observed. 2 see Culick (2006), or the companion lecture of Polifke (2010b). VKI

11 2.2 Beyond describing function 2 FLAME DYNAMICS Figure 6: Flow chart for limit cycle prediction with POD-based, low order model of local heat source coupled with acoustic system model Selimefendigil and Polifke (2009) also used POD modes to construct a reduced-order model for heat transfer in pulsating flow. When coupled with a model for the system acoustics, it should be possible to predict thermoacoustic limit cycles, see Fig. 6. The conservation equations for 2D laminar, viscous flow and heat transfer along a no-slip wall with a heated section were solved with CFD code. A forcing of the velocity at the inlet was imposed, covering a range of frequencies and amplitudes (up to 200 % of the mean flow velocity). In this way, a multi-variate POD model was constructed by projecting the governing equations on the POD modes. The essential difficulty of the method is related to the inclusion of the forcing term in the model (unlike the model of Huang and Baumann (2007), where the forcing term appears explicitly in the reduced model). POD-based, reduced order model predictions for fluctuating heat transfer rates and the describing function are compared against corresponding CFD results in Figs. 7 and 8. It is seen that the accuracy of POD/LOM results depends on the number of modes retained, especially so for the gain of the describing function. Satisfactory accuracy is observed with 15 modes retained. It is troublesome, however, that a further increase of the number of POD modes retained renders the low-order model unstable. Extending this Ansatz to the case of a turbulent premix swirling flame, say, represents a major challenge Nonlinear identification Strategies for non-parametric, time-series-data-based identification of nonlinear heat source dynamics have been explored by Selimefendigil et al. (Selimefendigil et al. 2010; Selimefendigil and Polifke 2010). The fundamental analogies with linear identification Polifke (2010b) should be obvious, indeed many nonlinear methods like NFIR, NARX, NAR- MAX and NOE are just Nonlinear extensions of corresponding linear methods: Finite Impulse Response, Auto Regressive with exogeneous input, Auto Regressive Moving Average with exogeneous input and Output Error, see Ljung (1999) for a general introduction and Selimefendigil et al. (2010) for specific details. Corresponding nonlinear system models involving higher order transfer functions have also been developed (see below). The procedures and requirements used in the nonlinear system identification are comparable to those of the linear system identification. The input signal covers the relevant range of frequencies, as well as amplitudes. In order to achieve better convergence rates, VKI

12 2 FLAME DYNAMICS 2.2 Beyond describing function CFD 15 modes 13 modes 10 modes Q /Q ss Q /Q ss Nondimensional time Nondimensional time Figure 7: Nondimensionalized (with respect to steady state value) heat transfer rate at the heated section CFD vs. POD/LOM results. Left: forcing amplitude 250%, nondimensional frequency 60, Right: forcing amplitude 350%, nondimensional frequency of CFD 15 modes 13 modes 10 modes G ain P hase [rad] x100 (% ) A/A 0 x100 (% ) A/A 0 Figure 8: Gain and phase of the describing function G(ω, A) for different amplitude ratios at the nondimensional frequency of 60 with CFD and with POD/LOM of different number of modes VKI

13 2.2 Beyond describing function 2 FLAME DYNAMICS it is common to add past outputs in the regressor set and to use a small number of regressors. The actual outputs, or alternatively the computed outputs, may be used in the set of regressors. In the former case, analytical closed form solution to derive the parameters is obtained for some choice of the function like polynomials. It is then easily extended into the frequency domain. In the latter case, the computed outputs from the model (output from the identified model, Q m ) are used in the set of regressors as the past outputs. Even though this strategy appears attractive and is indeed the method of choice in the time domain thermo-acoustic system model (e.g. Galerkin time domain simulation), this identification scheme suffers from stability and nonlinear optimization problems (optimum initial guess, trapping into local minimum, etc.). A schematic representation of the equation error and output error type model structures is shown in Fig. 9. In the figure, z represents the shift operator which simply shifts the one step ahead value of the input or output to the current time. %!## %!## %!#"$ % # %!#"$ % #!!#" $! # '.$/%0)1 '(()( *)+,-!!# #! "!#"$! # #$%&$% '(()( *)+,-! "!##!!#""#! "!#"!#! "!#""# & "" & "" & "" Figure 9: Schematic of equation error and output error model structures As an example, consider second order polynomial identification, which is of type NARX. Past inputs and outputs are included up to second order in the regressor set, such that the model output Q approx is approximated as a second order polynomial, e.g. Q approx (t) = + + N u N Q h u (k)u(t k + 1) + h Q (l)q(t l) k=1 l=1 N u N Q h uq (k, l)u(t k + 1)Q(t l) k=1 l=1 N u N u k=1 m=1 h uu (k, m)u(t k + 1)u(t m + 1). (4) Here the actual output Q is considered as input, so the matrix of inputs U, vector of VKI

14 2 FLAME DYNAMICS 2.2 Beyond describing function unknown parameters P and regressor set Z are defined as, U = [u(t k + 1), Q(t l), u(t k + 1)u(t m + 1), u(t k + 1)Q(t l)],(5) P = [h u (k), h Q (l), h uu (k, m), h uq (k, l)], (6) Z M = [u(t k + 1), Q(t l)], with k, m =1,..., N u, l =1,..., N Q. (7) The error (difference between the approximated and actual outputs) is minimized in a least square sense; V P =0, where V (ZM ; P )= 1 M M (Q PU) 2, (8) i=1 where i is a time index. If the actual outputs Q are used in the regressor set, an analytical closed form solution for the vector of unknown parameters P is computed as the product of the cross correlation between the input matrix U and output vector Q, and the inverse of autocorrelation of the input matrix U : P = 1 M M (QU T ) 1 M i=1 M (UU T ) 1. (9) i=1 Selimefendigil et al. (2010) considered unsteady heat transfer of a cylinder in pulsating cross flow at large fluctuation amplitudes (up to 200 % of mean velocity). For this case, the NARX identification gave excellent agreement between actual and approximated heat transfer rates, see Fig. 10. The sinusoidal describing function could also be reproduced from the NARX model with good accuracy (not shown). 0.2 C FD NL ident 0.1 Q /Q s Time [s] Figure 10: Comparison of CFD data ( actual output Q) vs. nonlinear system identification ( approximated output Q approx ) for model order N u =4,N Q =3 However, if the heat source model is to be used in combination with a system model for acoustics to make possible nonlinear thermoacoustic stability analysis or prediction of limit cycle amplitudes, a time-domain model with actual outputs Q as inputs is not useful. Output error models, which make use of outputs from the model, have been found VKI

15 2.2 Beyond describing function 2 FLAME DYNAMICS to lead to instability and other problems typical for nonlinear optimization (Selimefendigil et al. 2010). Therefore, the approach was extended into the frequency domain with two different approaches, i.e. harmonic balance and harmonic probing approach, which provides the nonlinear transfer function and higher order transfer function of the heat source. In the former case, a system of equations for the coefficients of the harmonic ansatz, and in the latter case, a recursive relation to compute the higher order transfer functions is obtained. The latter approach is computationally inefficient when the computation of the transfer functions with order more than four is considered. But generally, a few higher order transfer functions were found to be sufficient to get convergence in the response (Selimefendigil et al. 2010). VKI

16 3 NON-NORMALITY e 2 R t = 0 e 1 e 2 R t > 0 e 1 Figure 11: Transient growth in a system with two non-orthogonal eigenmodes e 1,e 2 : resultant R at time t>0 is greater than initially (t = 0), although both modes are decaying with time 3 Non-normality In dynamical systems with non-normal eigenmodes, modal interactions can result in transient growth, even if all modes are linearly stable, see Fig. 11 (Schmid and Henningson 2001; Trefethen and Embree 2005; Bale and Govindarajan 2010). Transient growth in non-normal systems is an observable physical characteristic, which has been found in studies related to subcritical transition to turbulence (Schmid and Henningson 2001), magnetohydrodynamics (Krasnov et al. 2004), astrophysics (Mukhopadhyay et al. 2005) and atmospheric flows (Farrell and Ioannou 1996). Non-normal transient growth is a phenomenon that does not require nonlinear interactions between modes. However, if transient growth leads to sufficiently strong fluctuation amplitudes, nonlinear effects will become significant, causing possibly nonlinear driving, resulting in strong growth and eventual saturation of the perturbations in a limit cycle. In that case, the asymptotic stability of the system is changed, as illustrated in Fig. 12, and classical linear stability analysis would fail to indicate long-term system behaviour. Reddy and Henningson (1993) provide a comprehensive analysis on such transient energy growth in the context of hydrodynamic instabilities in a viscous channel flow. The combustion dynamics community has learned in recent years that thermoacoustic systems are in general non-normal (Nicoud et al. 2007; Balasubramanian and Sujith 2008a, c; Nagaraja et al. 2009). Sujith and co-workers have investigated possible consequences of non-normality for the stability of combustion systems. In this context, a variety of comparatively simple configurations a Rijke tube and laminar diffusion or premix flames have been studied, and the scenario of large non-normal transient growth followed by nonlinear instability has been observed (Balasubramanian and Sujith 2008a, c; Nagaraja et al. 2009). VKI

17 3.1 Mathematical background 3 NON-NORMALITY nonlinear triggering nonlinear Energy linear t linear, non-normal transient growth Figure 12: Non-normal transient growth triggers a nonlinear instability, such that asymptotic stability of solutions is not observed 3.1 Mathematical background In this section, which has been adopted from a write-up prepared by Mangesius (2009), some relevant mathematical nomenclature, tools, and concepts are collected. The material is based on lecture notes on Tools for nonmodal stability analysis (P. Schmid, Ladhyx), which were presented also at the n3l workshop. More details are found, e.g. in (Schmid and Henningson 2001; Trefethen and Embree 2005; Schmid 2006) and books on modern linear algebra non-normal Operators Linearization The state vector (e.g. flow velocity) u is decomposed into mean ū and a small amplitude perturbation ɛu of order O(ɛ): u =ū + ɛu + O(ɛ 2 ) (10) Linearization of the governing equations yields an evolution equation for disturbances of 1st order, u t = Lu (11) with the evolution operator L. For the time discrete case consider an evolution: From here on: u u u (t + 1) = Lu (t). (12) Diagonalization A solution to the ODE (11) is represented formally by: u = exp(tl) u 0 with initial condition u 0. For ease of computation and stability insights, L is diagonalized: VKI L = SΛS 1 (13)

18 3 NON-NORMALITY 3.1 Mathematical background with the eigenvalues of L on the diagonal matrix Λ : Λ = diag(λ) and the eigenvectors e in S. If one of the eigenvectors lies in the unstable domain (continuous case: eigenvalue λ> 0, discrete case: λ > 1), then exponential growth will occur. Now the interesting question is: when is the dynamic behavior of exp(tl) essentially captured by exp(tλ)? Bounds on the operator exponential To examine the dynamics, an upper and lower bound of the operator exponential norm is determined. The norm can not decay faster than the least stable mode with eigenvalue λ max, so this yields a lower bound. For the upper bound, consider exp tλmax exp tl = S exp tλ S 1 S S 1 exp tλmax (14) where the term S S 1 κ(s) represents the condition number of S. Distinguish between two cases: 1. κ(s) 1: lower bound is approximately equal to the upper bound, then the system evolution is governed by least stable eigenvalue λ max for all times. 2. κ(s) 1: only the asymptotic behavior is governed by the least stable eigenvalue. non-normality If κ(s) = 1, the linear operator L is said to be normal, the corresponding eigenmodes/-functions/-vectors are orthogonal and thus the diagonalization represents a unitary similarity transformation. If κ(s) 1 the linear operator L is non-normal, the similarity transformation is not unitary and the eigenvectors are not orthogonal. Another definition for non-normal operator L: LL L L, (15) where L is the conjugate transpose (also Hermitian transpose, or adjoint ) of the matrix L. The degree of non-normality can be assessed for instance by using κ(s) or the norm LL L L. Non-orthogonal eigensystem and superposition The similarity transformation exp(tl) =S exp(tλ)s 1 represents a transformation into a non-orthogonal basis for nonnormal L. In consequence, non-orthogonal superposition due to non-orthogonality of the eigenvector-basis may cause transient growth due to different decaying rates of the involved eigenvectors, see Fig. 11. Furthermore, the eigenvalues alone only capture the asymptotic, long-time behavior, not transient effects. Diagonalization of the evolution operator obscures the transient behavior. How else can non-normal operators be analyzed? An alternative decomposition is the Singular Value Decomposition (SVD) exp(tl) =U exp(tσ)v (16) Here, L is diagonalized by using orthogonal vectors. The input basis V and the output basis U are not identical for non-normal L. The transient amplification occurs due to the mapping of one orthogonal basis into the other. So the SVD captures transient amplification. How to analyze the behavior of the operator exponential further? VKI

19 3.1 Mathematical background 3 NON-NORMALITY Maximum amplification and norm of operator exponential The aim is to compute the potential of exp(tl) to amplify a given disturbance over time. The size of the disturbance is measured by an appropriate norm. The maximum amplification G(t) is defined as the ratio of disturbance size to its initial size optimized over all possible initial conditions q G(t) max q 0 q 0 = max exp(tl)q 0 = exp(tl) (17) q 0 q 0 The maximum amplification G(t) over all times is defined as G max Choice of norm The heart of non-normal operator analysis is the non-orthogonality of the eigenvectors e i. The angle between the eigenvectors is computed by the inner product (scalar product), which at the same time is the measure of magnitude of the state variables - the norm, or fluctuation energy. The state vector of a thermoacoustic system ÃŇs given by its states (pressure velocity): q =(p u) (18) The energy is given by the integral over the domain Ω ( ) 1 0 E(t) = q Mq dω = q 2, M =, 0 γ Ma Ma: Mach number. (19) Ω With a decomposition of the weighting matrix M = F F (Cholesky decomposition) and a substitution ξ = Fq, the energy can be formulated as q 2 = ξ ξdω. (20) Now, recall the amplification G(t) reduces to the 2-norm: G(t) = max q 0 Ω q E q 0 E = max q 0 Fq 2 Fq 0 2 max q 0 F exp(tl)f 1 Fq 0 2 Fq 0 2 = F exp(tl)f 1 2 (21) Thus the energy weight is accounted for by the matrices F and F 1. Projection onto eigenvectors The evaluation of the matrix exponential exp(tl) is computationally costly, and prohibitive for real-world problems. To approximate, decompose q into N eigenvectors q n of L (eigenvector expansion) q = N κ n (t) q n (22) with expansion coefficients (weights of the eigenmodes) κ n (t) governed by n=1 VKI d dt κ =Λ κ, Λ= diag(λ i), i=1...n (23)

20 3 NON-NORMALITY 3.1 Mathematical background Substituting this into (20) and (21) yields a convenient way to compute the maximum transient growth... q = Ω q Mq dω = Ω N N ( κ n (t) q n ) M( κ m (t) q m ) dω = n=1 m=1 N n,m=1 κ n (t)m mn κ m (t), (24) where M mn = Ω q nm q m dω represents the so-called Gramian and consists of the energy inner products of the eigenfunctions for all combinations n, m. Finally, the transient amplification is G(t) = F exp(tλ) F 1 2,M mn = F F (25) G(t) represents a maximum growth envelope, i.e. the envelope of individual growth curves that start from different initial conditions. This begs the question: what is the optimal disturbance (init condition κ 0 ) that leads to maximum growth at a specified instant t spec? Optimal Initial Condition / Disturbance Consider that exp(tl) contains full information about evolution of disturbances. Applying a SVD and the results (16) and (25) on the evolution of an initial disturbance κ 0 κ(t spec ), one obtains the result: 1. largest singular value (diagonal element of Σ) is identical to the norm of matrix exponential at any time t spec ) 2. the corresponding optimal initial condition can be extracted from V Driving Force Until now we have considered worst case initial conditions for a homogeneous. What is the maximum possible amplification for a system with external forcing? Considering a driven initial value problem q = Lq + f (26) t with periodic (in time) forcing f = q f exp(iωt). The general solution consists of a homogeneous solution and the particular one, such that q(t) = exp(tl)q 0 +(iωi L) 1 q f exp(iωt) (27) The term (iωi L) 1 is referred to as the resolvent and plays an important role in determining the system behavior for the case of a periodic forcing: 1 dist(λ i,ω) (iωi 1 L) 1 = S diag( dist(λ i,ω) )S 1 κ(s) dist(λ i,ω) (28) 1. κ(s) = 1: response to forcing is determined by the smallest distance of the complex forcing frequency to the eigenvalue of L (e.g. resonance in case dist = 0) VKI

21 3.1 Mathematical background 3 NON-NORMALITY 2. κ(s) 1: large response to forcing even though ω forcing is far away from any natural frequency given by the eigenvalues λ of L In this case one speaks of pseudoresonance. Similar as before the maximum (transient) response to forcing can be obtained by an optimization: R(ω) = max q f q part q f Sensitivity of Eigenvalues = max q f (iωi L) 1 q f q f = (iωi L) 1 (29) To detect non-normality and to estimate the dynamically active eigenmodes, one connects the resolvent norm with the sensitivity of eigenvalues (this results in pseudospectra ). Examine the sensitivity of eigenvalues by perturbing the matrix L: (λi L + E)u = 0 (30) (λi L)u = Eu (31) (λi L) 1 )u ɛ u (32) (λi L) 1 ) ɛ 1 (33) where u is an ɛ- pseudo-eigenvector and λ a ɛ- pseudo-eigenvalue of L with the ɛ-pseudospectrum σ ɛ (L) as the set of all λ C that satisfy (33). An eigenvalue has a high sensitivity, if small perturbations causes large deviations in λ. Modal stability analysis considers the singularities of the resolvent norm. Non-normal operators and their behavior are affected by the sensitivity of the eigenvalue and thus the resolvent norm everywhere in the complex plane. Eigenfunctions associated with sensitive eigenvalues are likely to cause transient growth! Numerical Range and Maximum Growth Rate Consider the growth rate as time derivation of the energy: t q 2 =2Re(Lq, q) (34) where the numerical range is defined as the set of all Rayleigh quotients W (L) =q Lq, q C dim(q), q 2 = 1 (35) By linearizing the expression for the energy ( expansion of exp(tl)) (34) and forming the time derivative one obtains the expression 1 E E t = q 0(L + L)q 0 = W (L + L) λ q 0q max (L + L) (36) 0 for the limit t 0. Thus the initial (max) energy growth rate can easily be determined. VKI

22 3 NON-NORMALITY 3.2 Non-normal thermoacoustics Summary Analyzing and quantifying non-normal operator behavior for systems of the form can be accomplished by applying the following tools / concepts: 1. numerical range governs the behavior in the limit t 0 q = Lq (37) t 2. resolvent allows upper/lower bounds on the max. Transient amplification of a quantity is defined by a norm and is a strong indicator for non-normality, which leads to pseudospectra 3. spectrum determines the behavior in the limit t 3.2 Non-normality and transient growth in flame / acoustic interactions Sujith and co-workers Balasubramanian and Sujith (2008a, c); Nagaraja et al. (2009) have analyzed non-normal (and subsequent nonlinear) behavior in thermoacoustic model systems. Starting point are the acoustic equations in the presence of a heat source, γma u t + p = 0 (Momentum), (38) x p t + γma u x = (γ 1)γ L a Q (Energy). (39) c 0 ρ 0 c 2 0 The heat release rate in the above equation must be calculated from a model for the heat source. In the most general case, the linearized oscillatory heat release rate is written as R (x, ε i ) γ Ma u + S (x, µ i ) p, where R and S can be treated as continuous functions of x (which could even be sharply peaked at the flame location as in the case of a compact heat source), ε i and µ i are parameters which affect heat release rate. The heat release rate could have an explicit dependence on time as well. Equations (38) and (39) can be recast in matrix form as: t x RL a(γ 1) ρ 0 c 3 0 x t SL aγ(γ 1) ρ 0 c 3 0 γmau p = 0 (40) The matrix in Eq. (40) is the thermoacoustic evolution operator L. The above operator does not commute with its adjoint L for non-zero R and S, therefore the thermoacoustic interaction is non-normal 3. In the absence of heat release, the matrix is symmetric and hence normal. It has been shown that the advection-diffusion operator is non-normal (Reddy and Trefethen 1994; Trefethen and Embree 2005), and convection plays a dominant role in 3 The adjoint of a differentiation operator is its negative. VKI

23 3.2 Non-normal thermoacoustics 3 NON-NORMALITY combustors. Priya and Sujith (2010) have shown that the advection-reaction equation is non-normal. Combustion process and the heat release are indeed often modeled (see above) using advection-diffusion equation for the Shvab-Zeldovich variable Z in the case of diffusion burners and advection-reaction equation G ( flame front tracking ) in the case of premixed flames. In a series of papers, see e.g. (Balasubramanian and Sujith 2008b, a, c; Mariappan and Sujith 2010; Priya and Sujith 2010), Sujith and co-workers have analyzed simplified model problems like a Rijke tube with a time-lag model for the heat source according to Heckl (1990), see Eq. (1), a laminar diffusion flame (see Eq. (3)), and a laminar premix flame (see (2)). It was reported that non-normality may have significant consequences for system stability, i.e. it can lead to transient growth of perturbation amplitudes, even if the eigenvalues indicate linear stability, and even if perturbation amplitudes are initially so small, that nonlinear effects are not yet significant. Transient growth was observed to lead in some instances to high amplitude levels, where nonlinearities become eventually important, resulting in nonlinear driving. A discrete-time, state-space model for a generalized Rijke tube showed qualitatively similar behavior (although the modeling ansatz is very different from the modal expansion technique that Sujith and co-workers have used). Details are reported in the companion lecture (Polifke 2010a) and the n3l workshop proceedings (Mangesius and Polifke 2010). These results provide independent corroboration for some of the recently developed ideas on non-normality. Wieczorek et al. (2010) have presented at the n3l workshop results of an eigenvector expansion c.f. Eq. (22). The importance of proper choice of norm was emphasized Physics of non-normality in thermoacoustic systems For trivial acoustic boundary conditions (e.g. open or closed end) and in the absence of heat addition (e.g. combustion), acoustic modes in a resonator are normal. However, the presence of combustion makes a system non-normal. This is formally evident from the thermoacoustic evolution operator (40), but a convincing physics-based ( heuristic ) explanation of the non-normality of thermoacoustic interactions has not yet been established. Sujith and co-workers argue as follows (Balasubramanian and Sujith 2008a): A small disturbance in velocity causes an unsteady response of the combustion process, and thus may act as a source of acoustic oscillations. More precisely, an acoustic mode in a duct is driven by combustion, if combustion and acoustic oscillations are in phase with each other. When such a flame is part of a self-evolving (i.e., self-excited) thermoacoustic system, the phase lag between the two processes itself evolves with time, and thus depends on the system state at earlier times. Whether a given mode at a particular instant in time is excited or damped depends on which mode got excited at earlier times. In general, this leads to a complicated interaction between various modes, which results in non-orthogonal behavior of the eigenmodes. For the special case of a premixed flame, this rather general argument may be refined as follows: In response to forcing by an oscillatory velocity field, an attached premixed flame deforms and develops wrinkles along its length. These wrinkles move along the length of the flame with a convective (not acoustic!) velocity. Due to the finite speed, these wrinkles represent the response to forcing at previous times at upstream positions. Thus, the displacement of the flame at a given point at any VKI

24 REFERENCES instant is the sum total of the response to perturbations at previous times, and at other upstream points. The phase and amplitude of the flame oscillations at a given location are continuously evolving. 4 Conclusion Further progress in the analysis and control of thermoacoustic instabilities requires a deeper understanding of nonlinear and non-normal interactions between flame and acoustics. In particular, it must be explored under what circumstances nonlinar or non-normal effects may render classical linear stability analysis invalid. In this context it is also important to formulate adequate models for the flame dynamics as a prerequiste for dependable prediction of system stability in the presence of finite perturbations. Low order methods are particularly attractive for these studies, because fundamental insight is perhaps most easily gained from models with a small number of degrees of freedom. However, it is equally important to develop tools that are capable of handling real-world problems. Furthermore, it is imperative that experimental confirmation be established for transient growth, bypass transition, triggering, bootstrapping, subcritical bifurcation. As these notes indicate, this is sorely lacking at the time of writing. 5 Acknowledgements The author is indebted to R. I. Sujith, who introduced him to the fascinating ideas of nonnormal effects and non-modal stability analysis. Herbert Mangesius provided his writeup on tools for nonmodal stability analysis, which section 3.1 is based on. Discussions with Peter Schmid, Matthew Juniper and other participants of the n3l and the AIM workshops are much appreciated. References Alemela, P., Fanaca, D., Hirsch, C., Sattelmayer, T., and Schuermans, B. (2010). Determination and scaling of thermo acoustic characteristics of premixed flames. International Journal of Spray and Combustion Dynamics, 2(2): Annaswamy, A., Fleifil, M., Hathout, J., and Ghoniem, A. (1997). Impact of Linear Coupling on the Design of Active Controllers for the Thermoacoustic Instability. Combustion Science and Technology, 128(1): Balachandran, R., Ayoola, B. O., Kaminski, C. F., Dowling, A. P., and Mastorakos, E. (2005). Experimental investigation of the nonlinear response of turbulent remixed flames to imposed inlet velocity oscillations. Combust. and Flame in Press. Balasubramanian, K. and Sujith, R. (2008a). Non-normality and nonlinearity in combustion acoustic interaction in diffusion flames. Journal of Fluid Mechanics, 594: VKI