Numerical study on intrinsic thermoacoustic instability of a laminar premixed flame

Numerical study on intrinsic thermoacoustic instability

of a laminar premixed flame

Camilo F. Silva1, Thomas Emmert2, Stefan Jaensch3, Wolfgang Polifke4

Fachgebiet fur Thermodynamik. Technische Universitat Munchen D-85747 Garching,Germany

Abstract

A study on the velocity sensitivity and intrinsic thermoacoustic stabil-ity of a laminar, premixed, Bunsen-type flame is carried out. Direct nu-merical simulation (DNS) of the flame, placed in an acoustically anechoicenvironment and subjected to broad-band, low-amplitude acoustic forcing,generates time series of fluctuating heat release rate, velocities and pressure.The time series data is post-processed with system identification to estimatethe impulse response and transfer function of the flame. The associated fre-quency response is validated against experiment with good accuracy. DNSresults obtained with acoustic excitation from the inlet or outlet boundary,respectively, confirm that the flame responds predominantly to perturbationsof velocity. The stability of eigenmodes related to intrinsic thermoacousticfeedback is investigated with a network model. Both stable and unstableintrinsic thermoacoustic modes are predicted, depending on details of theconfiguration. The predicted modes are directly observed in direct numericalsimulations, with good agreement in frequencies and stability.

Keywords: Intrinsic thermoacoustic feedback, Finite Impulse Response(FIR), Flame Transfer Function (FTF), combustion instability

Email address: [email protected] (Camilo F. Silva)1Corresponding author. Post-doc, Technische Universitat Munchen2PhD Student, Technische Universitat Munchen, [email protected] Student, Technische Universitat Munchen, [email protected], Technische Universitat Munchen, [email protected]

Preprint submitted to Combustion and Flame May 25, 2015

1. Introduction

Thermoacoustic combustion instabilities have been an important subjectof combustion research for many decades, originating from the enormousproblems encountered during the design and test of rocket engines [1, 2]. Theincreased interest in recent years is due to the occurrence of thermoacousticinstabilities in gas turbine power-plants, aero engines and other low emissioncombustion systems, which often operate under lean premixed conditions[3, 4].

The established understanding of self-excited thermoacoustic instabilitiesinvolves feedback between unsteady heat release by the flame and acousticwaves in fuel and air supply and the combustion chamber: The unsteadyflame acts as a monopole source of sound and generates acoustic waves, whichare reflected by the combustion system back towards the flame, where theyperturb in turn the heat release rate. This feedback loop may go unstableprovided phase lags are favourable [5] and the loss of perturbation energynot excessive.

In the simplest case, premix flames are described as velocity sensitive: theglobal heat release rate Q′ responds to perturbations u′ of velocity upstreamof the flame. The link between these two quantities is commonly given interms of the flame frequency response or flame transfer function F(ω), suchthat

ˆQ(ω)¯Q

= F(ω)u(ω)

u, (1)

or the corresponding flame impulse response h(t)

Q′(t)¯Q

=1

u ∫T

0h(τ) u′(t − τ) dτ. (2)

In these equations, (′) denotes the deviation of a quantity from its mean

value () and the () stands for the Fourier transform. The symbols ω andτ represent angular frequency and time delay, respectively. T accounts herefor the duration of the impulse response h(t).

The flame transfer function (FTF) as defined in Eq. (1) has been usedextensively in the study of thermoacoustic instabilities of laminar and tur-bulent flames. Experimentally, the FTF is typically evaluated by imposing asinusoidal excitation of the flow upstream of the flame and by measuring the

corresponding response in heat release ˆQ(ω) [6, 7, 8, 9, 10, 11]. Sinusoidal

2

excitation also gives satisfactory results in numerical studies of flame dynam-ics [12, 13], but only at considerable computational cost. The identificationof the flame response through the corresponding impulse response h(τ) – seeEq. (2) – offers an alternative, which is computationally more efficient. Thismethod was introduced in the work of Polifke et al. [14], Zhu et al. [15] andGentemann et al. [16], and subsequently used and refined in a number ofstudies [17, 18, 19, 20]. Once the flame transfer function is known, it can beimplemented in analytical dispersion relations [21, 22, 23], network models[21, 24, 25] or Helmholtz solvers [26, 27, 28] to study the thermoacousticstability of combustion systems.

The present paper originated from an attempt to estimate the dynamicsof a laminar premix flame by direct numerical simulation (DNS) combinedwith system identification, (see the Ref. [29] for a description of the approachand previous applications in aero- and thermoacoustics). A computationalmodel (case B in section 4.2) with acoustically non-reflecting boundary con-ditions was set up, because it is understood that low reflection coefficientsfoster accurate identification [30]. However, it was observed that this compu-tational model responded to low amplitude broad-band excitation with verystrong, resonant, low-frequency oscillations in flame shape and heat releaserate, such that identification of the flame dynamics in the limit of linearbehaviour was not possible. This was not expected, since acoustically non-reflecting boundaries imply a significant loss of acoustic energy. Furthermore,the low frequencies of the exhibited thermoacoustic instability could not beexplained. Due to the small size of the domain, any acoustic cavity reso-nances would occur at frequencies much higher than that of the combustiondynamics.

Hoeijmakers et al. [31] observed in an independent study that low-orderacoustic network models of a combustion system with zero acoustic reflec-tion coefficients can indeed possess unstable eigenmodes. This might appearunphysical at first sight, because non-reflecting boundaries should break thefeedback between the flame and the system acoustics introduced above. How-ever, Hoeijmakers et al. [31] demonstrated analytically with a simple n-τmodel for the flame dynamics that under some conditions the flame scatter-ing matrix may indeed be an “intrinsically unstable element”. An interpre-tation of the physics of the instability was not developed, but instabilitiesobserved in a combustion test rig with low acoustic reflection coefficients[32] lend some support to the argument that flame-acoustic coupling may beintrinsically unstable.

3

A physics-based explanation of Intrinsic Thermo-Acoustic (ITA) modesin terms of flow – flame – acoustic interactions was developed by Bomberget al. [33]. The interactions were analyzed in a representation that respectscausality, i.e a formulation that allows to distinguish input from output sig-nals and thus cause from effect. In this framework a thermoacoustic feedbackloop intrinsic to the thermo-acoustic coupling was identified, which may gounstable, or exhibit resonant amplification. As a result, the generation ofacoustic energy by a perturbed flame may exhibit very strong peaks, whichcan be associated with poles of the flame scattering matrix ([34, 31]).

The ITA feedback loop was detailed by Bomberg et al. [33] and Emmertet al. [34] as follows: a flow perturbation u′ just upstream of the flame causesperturbations that are convected along the length of the flame and causein turn a fluctuation in heat release rate Q′, see Eq. (1). The fluctuationQ′ then generates acoustic waves that propagate away from the flame. Theacoustic wave propagating in the upstream direction modulates the upstreamvelocity u′, thus closing the feedback loop. A more detailed analysis of the“physics” of ITA feedback and the structure of the corresponding modes isgiven by Courtine et al. [35].

In the light of these findings, it was understood that the unexpectedbehaviour of the laminar premix flame DNS described above should be in-terpreted as a result of resonance of the external excitation with the ITAfeedback loop. Without persistent excitation, the DNS develops a stronginstability, which can be identified as an unstable ITA mode (see section4.2). Further analysis of the case with low-order models helped to identify aconfiguration where the intrinsic mode is stable, such that identification offlame dynamics is possible.

Before completion of the present study, several questions, which indeedare strongly related to each other, were pending. It was questioned whetherthe thermoacoustic intrinsic instability is a real physical phenomenon ormerely a spurious result of simplistic network models. Furthermore, it wasdebated whether the response of a compact premixed flame is actually veloc-ity sensitive, or whether flame dynamics should be described as a responsealso to pressure. In addition, more direct evidence for the validity of theITA feedback as described by Bomberg et al. [33] was sought. Accordingly,the present study has the following objectives: First, to demonstrate, byDirect Numerical Simulation, that the ITA feedback is an authentic physi-cal phenomenon that is present in premixed combustion systems. Second,to give evidence, by an adequate identification of the FTF, that compact

4

laminar premixed flames indeed respond predominantly to perturbations ofupstream velocity and thus may be treated as velocity sensitive elements.Furthermore, we confirm that thermoacoustic network models in conjunc-tion with the identified FTF are indeed capable of analysing the stability ofITA modes.

This article is organized as follows. In the next section, a brief overviewof system identification is given. In the third section the flame transfer func-tion of the laminar flame under study is estimated with DNS/SI, by imposingupstream as well as downstream broadband acoustic forcing, and validatedagainst experimental and computational results published previously [36, 12].Results are analyzed under the description of a laminar premix flame consid-ered as a velocity sensitive element. In the fourth section, low-order modelsof a burner-stabilized laminar flame are developed to compute the frequenciesand growth rates of acoustic modes for various acoustic reflection coefficientsof the inlet-outlet boundaries, including the limiting cases of zero reflectioncoefficients. Stable as well as unstable intrinsic modes are found, dependingon the details of the configuration. Corresponding results from DNS of lam-inar premix flame corroborate the results of the low order models, and someproperties of the ITA mode are studied.

In concluding this study, the reader is cautioned that intrinsic thermoa-coustic instabilities are not to be mistaken for intrinsic thermo-diffusive orhydrodynamic instabilities of premix flames, as reviewed e.g. by Clavin [37].

2. Overview of system identification

In order to infer a model of the relation between inputs and outputs of asystem, system identification uses the information contained in correspond-ing time series data [38]. If a Single Input Single Output (SISO) system,considered as a black box, is assumed to be Linear Time Invariant (LTI), it isthen completely characterized by its Impulse Response (IR). Be aware thatthe IR is a property of a such a system, not a model. When the system doesnot exhibit internal feedback, i.e. when the outputs of the system can beassumed to be related to the input by merely time delays, a model for thesystem can be written as

rt = b0st + b1st−1 +⋯ + bnbst−nb

, (3)

where rt and st denote the output and the input of the system at discretetime t, respectively. In this case, the IR of the system can be modeled as a

5

polynomial

G(q) = b0 + b1q−1 + ⋅ ⋅ ⋅ + bnb

q−nb , (4)

where the shift operator is defined by the property q−kst = st−k. Thecoefficients bk describe the response of the system to a unit impulse st = δt0. Inthis situation we consider G(q) as a Finite Impulse Response (FIR), becauseafter a certain (discrete) time the contribution of the coefficients bk, for k > nb,is small and therefore negligible. The FIR model has been used in severalstudies that combine numerical simulations and system identification (seereview of Ref. [29]) when studying the dynamic response of flames.

In addition to the discrete time representation of LTI systems, the corre-sponding response in the frequency domain is very useful, since it character-izes the associated filter behavior of the system. The flame transfer function(FTF) is derived as the z-transform of the impulse response G(q). It reads:

F(ω) =ng

∑k=0bke

−iωk∆t. (5)

where i denotes the imaginary unit and ∆t is the sample interval in thediscrete time domain. Note that the angular frequency ω is considered hereas a complex-valued variable. Identifying or “estimating” a FIR model meansfinding the optimal number and values of the corresponding coefficients. Thecoefficients nb and bk in a FIR model are commonly obtained by correlationanalysis in terms of the Wiener-Hopf equation [38]. In the present study, thesystem identification tool box of Matlab 2013b was used.

3. Numerical assessment of the flame impulse response and flametransfer function

This section describes the methodology to obtain by direct numericalsimulation (DNS) the flame impulse response and the corresponding flametransfer function of the laminar flame studied experimentally by [36]. Twodifferent cases are evaluated with acoustic excitation from the inlet or outletboundary, respectively. This procedure is based on System Identification (SI)and is similar to the methods used by Kaess et al. [17], Foller and Polifke [39]and Tay-Wo-Chong et al. [19]. Discussion on the velocity sensitivity of theflame under study is presented.

6

Ta

Td Tw

l0

l1

l2

2 mm3 mm

slit

plenum

combustionchamber

uin

Figure 1: Test rig configuration

3.1. Experimental set-up

The test rig used by Kornilov et al. [36] consists of a vessel with 8 laminarBunsen-type flames stabilized at a flat perforated plate. This plate is of 1mm thickness and perforated by 8 rectangular slits of dimensions 12×2 mm,separated from each other by 3 mm. The premixture is composed of air andmethane at an equivalence ratio equal to 0.8. The operating and thermalconditions are found in table 1 and illustrated in Fig. 1. Following the studyof Duchaine et al. [12], the temperatures Td and Tw are assumed to be around373 K, even though it is known by experiments [36] that this temperaturelays between 373 K ± 50 K during steady combustion.

Table 1: Operating and thermal conditions

Equivalence Inlet Premixture Duct Combustor Wallratio velocity uin temperature Ta temperature Td temperature Tw0.8 0.4 m.s−1 293 K 373 K 373 K

7

3.2. Numerical solver

The DNS/LES (Large Eddy Simulation) numerical tool AVBP5 devel-oped by CERFACS and IFP is used in this work to solve the compressibleset of Navier-Stokes equations. In the present study, AVBP discretizes thespatial terms of the equations considering a cell-vertex finite volume formu-lation based on the second-order accurate Lax-Wendroff scheme. Temporalevolution is computed through a second order accurate Runge-Kutta ap-proach. The Navier Stokes Characteristic Boundary Conditions (NSCBC)[40] are implemented in AVBP to treat waves crossing the boundaries. To-tally non reflecting acoustic boundary conditions are imposed based on atechnique known as plane wave masking [41]. Since this approach needs asan input the plane acoustic waves that are leaving the computational domain,a Characteristic Based Filter [42] is also considered. A two-step chemistry(2S-CM2) [43] is used since full chemistry schemes, for the flame exposed inthe present work, do not present considerable differences in results. Detailedexplanations on the subject can be found in the work of Duchaine et al. [12].The flame front is resolved by 8 grid points on a 3D mesh, which leads to atetrahedral computational grid containing around 100000 nodes.

3.3. Numerical identification of flame dynamics

In the time domain, Eq. (2) relates the response Q′(t) to incoming velocityperturbations u′(t). Let us define the input as s = u′u/uu and the output as

r = Q′/ ¯Q, where u′u = uu − uu and Q′ = Q − ¯Q. The index ‘u’ denotes the flowat a location immediately upstream of the flame (see Fig. 2). Equation (2) iswritten in discrete time as an FIR model (see Eq. (3)) and the correspondingFTF is computed as the z-transform of the impulse response based on Eq. (5).When determined in this way, the FTF is well-defined not only for purely realfrequencies, but throughout the complex frequency plane. This is importantfor the accurate determination of growth rates of thermoacoustic instabilities[44].

The velocity perturbation u′u just upstream of the flame – see location (u)in Fig. 2 (right) – is regarded as the input signal of the FIR model. This sig-nal cannot be imposed directly in the numerical simulations, but results fromacoustic perturbations imposed at the boundaries of the computational do-main. In this context, recall the definition of characteristic wave amplitudes

5http://www.cerfacs.fr/4-26334-The-AVBP-code.php

8

l1

l2

A1

ginfin

fout

DNSu

A2

(d)(u)

gout

gin

(in)

(out)

DNSd

fout

Figure 2: DNS set-ups for identification of flame dynamics with excitation signal fin atthe upstream boundary (left), and signal gout at the downstream boundary (right). Platethickness l1 = 1 mm, length l2 = 8 mm

f, g

f ≡1

2(p′

ρc+ u′) , g ≡

1

2(p′

ρc− u′) , (6)

where ρ denotes density and c the speed of sound. The characteristic wavef is propagating in the downstream direction, while g is traveling upstream.

In this study, two different set-ups are used to identify the impulse re-sponse and subsequently the corresponding FTF. In the first case, calledDNSu and illustrated in Fig. 2 (left), the acoustic excitation is imposed atthe inlet boundary of the computational domain as a wave fin traveling inthe downstream direction. Due to the non-reflecting boundary conditions,any upstream traveling signal gin leaves the domain without reflection at theinlet, while gout is equal to zero by design. In the second case, called DNSd

and depicted in Fig. 2 (right), the excitation signal is imposed as gout at theoutlet boundary. The anechoic conditions at the boundaries assure that fin

is zero and fout leaves the domain without being reflected at the outlet.In order to obtain information suitable for identification of flame dynamics

from the DNS, the signals fin and gout are designed to exhibit the followingproperties: First, the spectral energy should be uniformly distributed over thefrequency range of interest. This assures that all the frequencies of interestare excited. Second, it should demonstrate fast decorrelation with itself, sothat the length of the time series, a key parameter for a reliable estimationof the coefficients bk under correlation analysis, remains reasonably short.

9

Third, the signal should exhibit a crest factor close to one. This is donewith the purpose of imposing high energy signals with the lowest amplitudepossible so that non-linearities are avoided [38]. In contrast to the worksof Kaess et al. [17], Foller and Polifke [39] and Tay-Wo-Chong et al. [19],the signal preferred in the present study is not a Discrete Random BinarySignal (DRBS), but instead is constructed based on Daubechies Wavelets[45]. Although this type of signal has a slightly worse (larger) crest factor,the spectral energy distribution is closer to the desired one.

The velocity perturbation signal u′u is evaluated in the post-processingpart of the simulations from the time series of volume flow rate at location(u). The relation of the perturbation signal to the acoustic waves at thislocation is simply

u′u = fu − gu, (7)

which follows immediately from Eq. (6).In order to evaluate a satisfactory impulse response from data using cor-

relation analysis, as in the present study, experience suggests that the lengthof data series should extend at least over ten times the length of the im-pulse response [38, 46]. In our case, data captured over 100 ms is consideredenough to evaluate the impulse response and subsequently the FTF of theflame, since the length of the IR is around 10 ms (see Fig. 3(a)). Note thatonly one simulation is needed to retrieve the frequency response for the fre-quency band of interest. If, in contrast, harmonic excitation is performed,the frequency response for only one frequency can be computed in one simu-lation. Moreover, if data during 100 ms is captured with such an input signal,good results in the frequency response would be expected only for frequen-cies equal or higher than 10 Hz (in the most favorable situation when dataover one cycle is considered sufficient). Contrary to broadband excitation,information at lower frequencies would not be well captured.

The impulse response of the flame, modeled by a FIR structure and ob-tained after solving Eq. (3) (with na = 0), is shown in Fig. 3(a). It exhibitsa maximum at τ1 = 3.9 ms and a minimum τ2 = 6.4 ms, which is indeed adynamics that commonly characterizes laminar flames. The maximum (mini-mum) impulse response corresponds to the moment of maximum (minimum)flame elongation and surface area. Figure 3(b) shows the flame frequencyresponse computed from Eq. (5) and compares it with the numerical re-sults of Duchaine et al. [12] and the experimental measurements of Kornilovet al. [36], respectively. Good agreement is observed for upstream as well

10

0 0.002 0.004 0.006 0.008 0.01400

200

0

200

400

600

Time (s)

Am

plitu

de

0 100 200 300 400 500 600 7000

0.5

1

1.5

Gain

0 100 200 300 400 500 600 70020

10

0

Frequency (Hz)

Phas

e (r

ad) π

107

147

(a) (b)

Figure 3: Characterization of laminar premix flame dynamics. (a) Impulse Responseestimated from DNSu (dark circle) and DNSd (gray circle). (b) Flame frequency responsefrom DNSu (dark line) and DNSd (gray line), Experiments of Kornilov et al. [36] (dashedline) and DNS of Duchaine et al. [12] (triangles).

as downstream forcing. The overshoot of the FTF at 107 Hz results fromconstructive superposition of contributions of the positive and negative co-efficients bk of the impulse response. If all coefficients bk were of the samesign, an overshoot could not exist [47]. It should be pointed out that in thestudy of Duchaine et al. [12], which was based on harmonic excitation, sevenDNS runs were required to obtain the values of the flame frequency responseat seven discrete points in the frequency spectrum. In the present study, onthe other hand, a single DNS run was sufficient to obtain the FTF over theentire frequency band of interest.

Now we turn to the question whether the flame is indeed velocity sen-sitive, which has been questioned in the context of ITA instabilities. Letus recall that a flame is considered velocity sensitive when the correspond-ing fluctuations of heat release rate are related only to upstream velocityperturbations. Note, however, that a flame in general can also respond vig-orously to pressure fluctuations. Examples are partially premixed flames ornon-compact flames, i.e. flames whose length is of the order of the acous-tic wavelength. In what follows we will show, based on our results, thatthe flame under investigation, which is compact and perfectly premixed, ispredominantly velocity sensitive.

For the sake of our argument, we will perform a proof by contradiction.Let us assume that the flame response to upstream velocity perturbations

11

0 100 200 300 400 500 600 7000

0.5

1

1.5

Frequency (Hz)

Gai

n

0 100 200 300 400 500 600 700−8

−6

−4

−2

0

Frequency (Hz)

Phas

e (r

ad)

Figure 4: Impedance Zu for the upstream excitation case (DNSu).

is different depending whether the flame is acoustically excited from the

upstream or downstream side. Subsequently we assume that ˆQ is linked toboth uu and pu as

ˆQ¯Q= Fvel(ω)

uu

uu

+Fpres(ω)pu

ρucuuu

(8)

and suppose that pressure fluctuations are related to velocity fluctuationsthrough an acoustic impedance Z

Z =p

ρcu=f + g

f − g. (9)

It follows

ˆQ¯Q= (Fvel(ω) +ZuFpres(ω))´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

F(ω)

uu

uu

. (10)

Recalling that both upstream and downstream boundary conditions areacoustically non reflective, we consider now the same two cases previouslydiscussed. In the first case, when the flame is acoustically excited from down-stream, the impedance Zu is equal to -1 for all frequencies since fu is zerofor all frequencies. In the second case, when the flame is acoustically excitedfrom upstream, the impedance Zu is not anymore -1, as illustrated in Fig. 4.

Following that idea, it is clear that if ˆQ is related to both uu and pu as sug-gested in Eqs. (8) and (10), then F(ω) must be different for these two casesif Fpres(ω) ≠ 0. The fact that F(ω) is independent of the excitation position(upstream or downstream as shown in Fig. 3(b)) indicates that Fpres(ω) = 0.We conclude therefore that the premix flame investigated in this work is ve-locity sensitive, while it does not (or only very weakly) respond to pressure

12

perturbations. In other words, the SISO model structure implied by Eq. (2)with u′u as input signal indeed respects the “physics” of the flame dynam-ics and in particular the causality of acoustic – flow – flame interactions:For the case DNSu with excitation from the inlet boundary, the resultingacoustic signal fu at location (u) generates a velocity perturbation accord-ing to Eq. (7), which then causes a modulation of heat release rate Q′. Onthe other hand, if the acoustic excitation is imposed at the outlet boundaryfor the case DNSd, the resulting characteristic wave g travels towards andthrough the flame, such that it eventually causes an acoustic perturbationgu in the slit. In the absence of a downstream traveling wave fu (see Fig. 2,right), the velocity signal is reduced to u′u = −gu according to Eq. (7). Forthis case, which is slightly more involved than DNSu, the causality of acous-tic – flow – flame interactions can be represented as gout → gu → u′u → Q′.Note that in the two cases discussed the fluctuations of heat release Q′ willgenerate acoustic perturbations that propagate away from the flame in bothdirections. As explained in the introduction, the upstream traveling per-turbations will contribute to gu at the upstream location, resulting in ITAfeedback. More detailed explanations of the physics involved can be foundin the works of Courtine et al. [35], Bomberg et al. [33], Emmert et al. [34]and Silva et al. [48].

4. DNS of intrinsic thermoacoustic instability

This section presents results on ITA instability observed directly in DNSof a laminar premix flame. Two configurations are considered, both witha flame and flame holder that corresponds to the experimental set-up ofKornilov et al. [36]. The first case, here denoted “A”, is the one alreadystudied in section 3. It consists of two cavities of different cross sections asillustrated in Fig. 5(a). One cavity represents the slit of the flame-holdingplate, while the adjacent cavity represents the combustion chamber. Thesecond case “B” considered includes an additional plenum cavity upstreamof the plate, see Fig. 5(b).

Before presenting DNS results for these two configurations, results of lin-ear stability analysis obtained with a network model will be introduced. Thisnetwork model uses as input the Flame Transfer Function F(ω) obtained insection 3. Note that F(ω) is defined throughout the complex frequencyplane (not only for purely real frequencies), since it was computed by thez-transform of the impulse response (Eq. (5)).

13

l1

l2

A1

A2

(d)(u)

(out)

(in)

(d)(u)

A1

A2

l0 = 6 mm

l1 = 1 mm

l2 = 8 mm

(out)

(in)

(a) Case A (b) Case B

Figure 5: Computational domains of DNS of laminar premix flame stabilized on a flat platwith and without upstream plenum cavity.

The eigenmodes identified with the network model are useful for the inter-pretation of the simulation results. DNS results are presented subsequently,with acoustically non-reflecting boundary conditions imposed at inlet andoutlet [41]. Both stable as well as unstable modes are observed, which can-not be related to the acoustic modes of the configuration. These acousticmodes, called AC modes in this work, are considered as directly related tothe geometry, boundary conditions and mean flow of the system under study,but being independent of the flame dynamics.

4.1. Stability analysis with low order network model

Network models are useful tools to determine and interpret eigenmodesof (thermo-)acoustic systems. The models can be conveniently formulatedin terms of characteristic wave amplitudes f and g at the nodes of networkelements that comprise the acoustic system. Each element is described by itstransfer matrix. In this section, conservation principles are combined withflame transfer function obtained numerically to produce a semi-analyticalformulation. Following Polifke et al. [25] or Paschereit et al. [49], the acoustictransfer matrix T for the characteristic waves f , g between an upstream plane(in) and a downstream plane (out) is defined as

[fout

gout] = [

T11 T12

T21 T22] [fin

gin] (11)

Let us consider now the locations (u) and (d) immediately up- and down-stream of an infinitesimally thin, planar zone of heat release, as illustrated

14

in Fig. 5. Linearization of the quasi-1D Euler equations in the low Machnumber regime yields [22]:

pd = pu (momentum equation), (12)

A2ud = A1uu +(γ − 1)

γpˆQ (mass and entropy equations) (13)

where A denotes the cross-sectional area. The perturbations of the global

heat release rate ˆQ is now described by the Flame Transfer Function:

F(ω) ≡ˆQ(ω)/ ¯Q

uu(ω)/uu

(14)

Defining now the area change α = A1/A2 and relative temperature incre-ment θ = (Td/Tu − 1), where T denotes here temperature, we obtain a simpleexpression of Eq. (13):

ud = uuα(1 + θF(ω)), (15)

Recalling the relation (6) between Riemann invariants and primitive acousticvariables

u = f − g and p = (f + g)ρc, (16)

and by considering Eqs. (12) and Eq. (15), we obtain the coefficients of theflame transfer matrix T (F ) between planes (u) and (d)

T(F )11 (ω) = T

(F )22 (ω) = 1

2(ξ + α + αθF(ω)),

T(F )12 (ω) = T

(F )21 (ω) = 1

2(ξ − α − αθF(ω)),

(17)

where ξ = ρucu/ρdud denotes the specific acoustic impedance. In order toderive an expression of the transfer matrix T (A) (see Fig. 5(a)) with respectto the inlet and outlet planes (in) and (out), the phase change due to wavepropagation in the adjacent ducts must be included:

T (A) = [e−ωl2/cd 0

0 eωl2/cd] [T(F )11 T

(F )12

T(F )21 T

(F )22

] [e−ωl1/cu 0

0 eωl1/cu] . (18)

The transfer matrix of configuration B (see Fig. 5(b)) is computed as:

T (B) = T (A)1

2[(1 + 1/α) e−ωl0/cu (1 − 1/α) eωl0/cu(1 − 1/α) e−ωl0/cu (1 + 1/α) eωl0/cu

] (19)

15

An acoustic network model representation of both configurations A, B resultin a 4 × 4 linear system of equations

⎡⎢⎢⎢⎢⎢⎢⎢⎣

1 −Rin 0 00 0 −Rout 1T11 T12 −1 0T21 T22 0 −1

⎤⎥⎥⎥⎥⎥⎥⎥⎦

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶M

⎡⎢⎢⎢⎢⎢⎢⎢⎣

fin

gin

fout

gout

⎤⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

0000

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(20)

where T may be either T (A) or T (B), and Rin and Rout stand for the acousticreflection coefficients of inlet and outlet, respectively. In order to find a non-trivial solution of the linear system, it is necessary to assure that the solutionsatisfies det(M) = 0. For the systems described in Eq. (20), the determinantof M is easily found, and consequently the characteristic equation to besolved reads

det(M) = 0 ⇒ T22 −RoutT12 +RinT21 −RinRoutT11 = 0. (21)

A case similar to configuration A was studied by Emmert et al. [34]. Thatstudy describes the acoustics of the system by an open loop scattering matrixS(bc)S(F ), where S(bc) = [Rin, 0 ; 0, Rout] is a 2×2 matrix containing theinformation on the boundary conditions. The corresponding eigenfrequenciesare found by solving the open loop characteristic equation det(S(bc)S(F ) −I) = 0. Identical results are obtained for cases Rin,Rout ≠ 0. However, theformulation used in the present study based on the transfer matrix – whichis also used by Hoeijmakers et al. [31] – is more convenient in the limit ofvanishing reflection coefficients Rin,Rout = 0. In that case the characteristicequation resulting from defined by Eq. (21) reduces for case A simply to

T(A)22 = 0 ⇔ eωl1/cueωl2/cd(ξ + α + αθF(ω)) = 0. (22)

The first two terms cannot satisfy eωl1/cu = 0 or eωl2/cd = 0 and therefore theydo not need to be considered in the analysis. The solution of ξ+α+αθF(ω) = 0can be realized and is related to the ITA eigenmodes in complete agreementwith the results of Hoeijmakers et al. [31], Emmert et al. [34] and Silva etal. [48].

The characteristic equation for case B with non-reflecting boundary con-ditions reads

T(B)22 = 0 ⇔ [e−ωl1/cu(1 − 1/α)(ξ − α − αθF) + eωl1/cu(1 + 1/α)(ξ + α + αθF)] = 0.

(23)

16

In the general case, solutions of Eq. (23) are not easily separable into ACand ITA modes. The AC modes (see definition of AC at the beginning ofthis section) of the system can be found by setting the flame insensitive toacoustic perturbations (F = 0). Eq. (23) then reduces to

e−ωl1/cu(1 − 1/α)(ξ − α) + eωl1/cu(1 + 1/α)(ξ + α) = 0. (24)

For cu = 324 l1 = 1 mm, α = 0.4 and ξ = 2.81, the first AC eigenfrequencyis around 162000 Hz, which is close to the frequency corresponding to thehalf-wavelength of the slit. This frequency is very high if compared withthe solutions once the flame is present (F ≠ 0), whereas the correspondinggrowth rate is highly negative. It should be emphasized that if reflectingacoustic boundary conditions are considered, solutions of Eq. (21) for caseB, while considering F = 0, still lie above 8000 Hz.

Let us consider now Eq. (21) for both A and B systems with an activeflame, where F(ω) is the one identified by DNSu (see section 3). An instabil-ity map based on different values of Rin and Rout is constructed and shownin Fig. 6. The eigenmodes shown are linked to the ITA feedback, since onlyfrequencies around 150 Hz are present, which cannot be related to any ACmode. For both cases A and B, it is observed that a positive growth rate ispredicted for several combinations of Rin and Rout.

Indeed, the growth rate of the ITA mode is enhanced when an open-end(Rin = −1) is imposed at the inlet and a closed-end (Rout = 1) is imposed atthe outlet. On the contrary, the growth rate is damped when a closed-endand an open-end are considered at the inlet and outlet, respectively. Theseresults exhibit a different trend than the stability analysis of Hoeijmakerset al. [31] for a simple duct-flame-duct configuration. In that study, thestability map shows maximum instability when a closed inlet is combinedwith an open outlet. The description of the heat source by a simple n-τmodel may account for the differences in stability prediction.

One particular case is of great interest. It is observed that when (Rin,Rout =

0) a positive growth rate is predicted for Case B. Although these results areexpected in the light of the results of Hoeijmakers et al. [31] and Emmertet al. [34], it is still counter intuitive that instability occurs in this particu-lar situation when a maximum of acoustic energy is evacuated through theboundaries and, accordingly, acoustic losses are at the highest level.

17

1000 500 0 500120

140

160

180

Freq

uenc

y (H

z)

1000 500 0 500120

140

160

180

Freq

uenc

y (H

z)

1000 500 0 500120

140

160

180

Freq

uenc

y (H

z)

1000 500 0 500120

140

160

180

Freq

uenc

y (H

z)

1000 500 0 500120

140

160

180

!"#$%&'"(%)'*+ ,-

.")/0)123'*45-

1000 500 0 500120

140

160

180

Freq

uenc

y (H

z)

1000 500 0 500120

140

160

180

Freq

uenc

y (H

z)

1000 500 0 500120

140

160

180

Freq

uenc

y (H

z)

1000 500 0 500120

140

160

180

Freq

uenc

y (H

z)

1000 500 0 500120

140

160

180

!"#$%&'"(%)'*+ ,-

.")/0)123'*45-

Rin = −1

Rin = −0.5

Rin = 0

Rin = 0.5

Rin = 1

−→Rout −→Rout

(152, 170)

Case A Case B

UnstableStableUnstableStable

(147,−12)

Figure 6: Eigenfrequencies derived from Eq. (21). Rout varies from -1 (black) to 1 (lightgray). Symbols in red specify results for Rin, Rout = 0. The dashed line indicates thefrequency (147 Hz) which is the solution of ξ + α + αθF(ω) = 0.

0 0.01 0.02 0.03 0.04 0.05 0.060.2

0

0.2

time (s)

Am

plitu

de

0 0.01 0.02 0.03 0.04 0.05 0.06

1

0

1

time (s)

Am

plitu

de

(a) (b)

Figure 7: Time series of input signal fin (black) and output signal fout (gray). (a) CaseA. (b) Case B.

18

0 500 1000 1500 2000

10 5

Frequency (Hz)

PSD

151 Hz

0 500 1000 1500 2000

10 5

Frequency (Hz)

PSD

164 Hz

(a) (b)

Figure 8: Power Spectral Density of Input signal fin (black) and output signal fout (gray).(a) Case A. (b) Case B.

4.2. ITA modes in direct numerical simulation

In order to validate the low order stability predictions for cases A andB, DNS are performed. Configuration A is considered first. Since the ITAmode of this case is predicted to be stable, one way to observe this mode inDNS is by exciting the system, which is seen as a stable acoustic oscillator,with a broadband signal of uniform spectral energy. If an acoustic mode ofthe system lies between the frequency range of forcing, it will be excited anda resonance peak will be identified. The resulting resonance peak indicatesthe eigenfrequency of the stable mode.

Figure 7(a) illustrates both the input signal fin and output signal fout

during a period of 60 ms. It is observed that the output signal is oscillatingwith a period of roughly 6 ms. The power spectral density (PSD) of bothsignals is shown in Fig. 8(a): whereas the energy of the signal is uniformlydistributed for the input, the spectrum of the output signal shows a resonancepeak at 151 Hz, in good agreement with the mode predicted at 147 Hz bythe network model analysis.

Figure 7(b) illustrates both input and output signals for case B. Here theinput signal fin contains very low energy and is only applied during 0.005s at the beginning of the simulation in order to trigger the instability. Theoutput signal fout exhibits clearly a high amplitude oscillation at a frequencyof 164 Hz, which is in fair agreement with the ITA flame eigenfrequencyof 152 Hz predicted by the network model analysis. Figure 8(b) shows thePSD for input and output signals of case B. It is interesting to remark thatnetwork model predictions are more accurate for Case A. One possible ex-planation relies on the linearity assumption in which the network model isbased. Whereas acoustics in Case A still can be described by a linear model,an accurate prediction of the limit cycle frequency of Case B might need theinclusion of non-linearities in the description of the flame dynamics.

19

0 0.01 0.02 0.03 0.04 0.05 0.06

1

0

1

time (s)

Am

plitu

de

Figure 9: Signals of interest. (black) u′in, (gray) Q′/ ¯Q, (dashed) p′out/(ρoutcout). Notethat since fin = 0 and gout = 0 (totally non-reflecting acoustic boundary conditions), thenuin = −gin and p′out/(ρoutcout) = fout.

In order to verify that the thermal source of acoustic energy productionis positive, according to the Rayleigh criterion [5], we superpose now thenormalized signals of Q′ and p′out in Fig. 9. Note that p′out is indeed very closeto the fluctuating pressure throughout the domain due to the large value ofthe wavelength with respect to the length of the system. In addition, thesignal u′in, which exhibits the same phase of u′u, is also present in order tovisualize the phase shift between u′in and Q′. It is observed that Q′ and p′out

are perfectly in phase and therefore the Rayleigh criterion for thermoacousticenergy production is verified. Furthermore, it is seen that u′in and Q′ arecompletely out of phase (φ = π) as expected from the analysis in the previoussection. These results agree with the observations of Courtine et al. [35],Emmert et al. [34], Bomberg et al. [33] and Hoeijmakers et al. [31] andconfirm that the instability is thermo-acoustic in nature.

4.3. Conclusions

We have demonstrated, by Direct Numerical Simulation, that intrinsicthermoacoustic feedback is an authentic physical phenomenon that is presentin premixed combustion systems. The corresponding ITA modes are in thepresent case independent of acoustic (AC) modes. Both stable and unstablemodes have been observed.

We have given evidence, by an adequate identification of the FTF, thatcompact laminar premixed flames are velocity sensitive. This corroboratesthe structure of the flow-flame-acoustic interaction with ITA feedback sug-gested by Bomberg et al. [33].

Finally, we have shown that thermoacoustic network models in conjunc-tion with an FTF identified from DNS are capable of predicting stabil-ity/instability of this kind of acoustic mode. This notion might seem ob-

20

vious, but has been questioned in the context of intrinsic thermoacousticinstabilities.

In summary, the present study gives strong evidence that intrinsic ther-moacoustic instabilities are not an artifact of simplistic models, but a typeof combustion instability that until recently has escaped the attention of thecombustion dynamics research community and warrants further study.

Acknowledgments

The authors gratefully acknowledge CERFACS for making available theCFD solver AVBP, and the Research Association for Combustion Engines(Forschungsvereinigung Verbrennungskraftmaschinen e.V. FVV) for the Fi-nancial support to Stefan Jaensch. We are also indebted to Leibniz Rechen-zentrum LRZ for providing access to HPC resources (SuperMUC).

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