## Abstract

This paper presents an asymptotic approach to combustion instability in premixed flames under the assumptions of large activation energy and small Mach number. The entire flow consists of four distinct yet fully interactive sub-regions, which accommodate the chemical reaction, heat transport, hydrodynamics and acoustics, respectively. A reduced system was derived to describe the intricate coupling between the flame and acoustics that underlies the combustion instability. The asymptotically reduced system was employed to study the weakly nonlinear interaction between the Darrieus–Landau instability and the longitudinal acoustic mode of the combustion chamber. The general asymptotic formulation includes the influence of enthalpy fluctuation in the oncoming mixture. It is shown that one-dimensional enthalpy fluctuation, through its interaction with flame, produces sound waves, and may cause parametric instability of the flame. The mutual coupling between the sound wave and parametric instability is analysed at the instability thresholds.

## 1. Introduction

Combustion instability generally refers to large-amplitude pressure fluctuations of acoustic nature in combustion chambers. It has a number of adverse effects, including undesired unsteady thrust, structural fatigue, interruption to the feeding lines to cause ‘flash-back’ or flame blow-off.

Combustion instability is essentially a self-excited oscillation, arising from a two-way coupling between the unsteady heat release and the acoustic fluctuation in the chamber (e.g. Yu *et al*. 1991). According to Rayleigh's criterion, unsteady heat release leads to amplification of acoustic pressure when the two are ‘in phase’. The acoustic pressure, on the other hand, may affect heat release via a number of distinct physical mechanisms:

sound waves affect burning rate directly;

sound waves affect heat release by modulating the flame surface area (Markstein 1970);

sound waves may modulate the inlet feeding rate of fuel, causing equivalence ratio oscillations (e.g. Lieuwen & Zinn 1998);

sound waves excite Kelvin–Helmholtz instability modes at the inlet, which subsequently break down into small-scale turbulence to affect unsteady heat release (Poinsot

*et al*. 1987).

Establishing the relations between unsteady heat release and sound from a theoretical standpoint requires detailed analyses of these processes, which are problematic. A semi-empirical approach has often been taken to extrapolate suitable relations from experimental data. For example, in their work on ‘reheat buzz’, Bloxsidge *et al*. (1988) took the unsteady heat release to be proportional to the flow velocity at the flame front with a suitable time delay, while Dowling (1997) employed a nonlinear relation to describe the self-sustained oscillations. A somewhat different model, which relates heat release to the flame surface area, was proposed by Fleifil *et al*. (1996) and was further extended by Dowling (1999).

In the semi-empirical approach, the hydrodynamic and chemical processes are completely by-passed. To understand the acoustic-flame coupling on a first-principles basis, one has to look into the structure of the flame as well as its associated hydrodynamic field. A theoretical framework for this is the large-activation-energy asymptotics (AEA; Williams 1992). The key physical basis that leads to mathematical simplification is the scale disparity, that is, chemical reaction takes place in an inner layer much thinner than the flame thickness *d*, so that matched-asymptotic-expansion technique can be used to analyse systematically each of these regions. Excellent and extended reviews of the subject have been given by Clavin (1985, 1994). Here, we highlight some major steps in its development so as to put the present work in an appropriate context.

The early theoretical work employed a diffusion-thermal type of model (e.g. Barenblatt *et al*. 1962), in which the flame was assumed to propagate in a prescribed hydrodynamic field, that is, the effect of flame on the flow field was neglected. Later, Matkowsky & Sivashinsky (1979) demonstrated that the diffusion-thermal model can be derived in the large-activation-energy limit from the fundamental equations governing chemically reacting flows.

The second phase of AEA was represented by the work of Matalon & Matkowsky (1982) and Pelce & Clavin (1982), who considered the effect of flame on hydrodynamics through gas expansion. By analysing the internal diffusive structure of flame, they derived a set of jump conditions relating to the flow fields on the burnt and unburnt sides of the flame front. These results provide an asymptotic justification of the earlier phenomenological modelling of flame as a discontinuity (Landau 1944), and clarify the link between that approach and the diffusion-thermal model.

The formulations of Matalon & Matkowsky (1982) and Pelce & Clavin (1982) exclude acoustics. A formal formulation of acoustic–flame coupling was given by Harten *et al*. (1984) for what may be called the ‘high-frequency’ regime, where the acoustic time-scale is comparable to the transit time of the flame, while the hydrodynamics have a length-scale of O(*d*). The theory was formulated primarily to address how a flame responds to an externally imposed acoustic wave. The inverse process of flame influencing sound, and as such, the nature of the two-way coupling, remain somewhat implicit.

The two-way flame–acoustic coupling was studied by Clavin *et al*. (1990). Using AEA, they calculated the modification to the burning rate by the acoustic pressure, and thereby estimated the amplification rate of sound waves. McIntosh (1991) analysed the interaction of flame with pressure fluctuations of various time-scales. These analyses provide a complete understanding of mechanism (i) listed above, albeit for a flat flame, for which the hydrodynamics is absent. For a curved flame, there is an intrinsic hydrodynamic field and mechanism (ii) operates. A detailed analysis of this mechanism based on first-principles has only recently begun.

In order to provide a unified framework for analysing the flame–acoustic interaction that leads to combustion instability, Wu *et al*. (2003) developed a completely self-consistent theory using AEA in the low-Mach-number limit. The theory shows that an unsteady flame has an intrinsic acoustic field, which is coupled to Euler's equations describing the hydrodynamics via the equation governing the flame front.

In this paper, the formulation of Wu *et al*. (2003) will be extended to include the effect of enthalpy (i.e. temperature and/or reactant mass fraction) fluctuation in the oncoming mixture. This is an issue of practical importance, because enthalpy fluctuation can be present owing to ‘unmixedness’ of the fresh mixture, or as a response of the inlet to the combustion-generated acoustic pressure. In controlled situations, they may be generated by actuators, e.g. by secondary fuel injection. The present work is motivated by potential application to this type of active control of combustion instability (McManus *et al*. 1993; Sivasegaram *et al*. 1995).

## 2. Formulation: governing equations and scalings

Consider the combustion of a homogeneous premixed combustible mixture in a duct of width *h** (see figure 1). A one-step irreversible exothermic chemical reaction is assumed, and the mixture is taken to be a perfect gas. The fresh mixture has a density *ρ*_{−∞} and temperature *Θ*_{−∞}. The mean temperature (density) behind the flame increases (decreases) to *Θ*_{∞}(*ρ*_{∞}). An important non-dimensional parameter is(2.1)where *E* is the activation energy and is the universal gas constant. The flame propagates into the fresh mixture at the mean laminar flame speed *U*_{L}, and it has an intrinsic thickness , where is the thermal diffusivity. For late reference, we define the length ratio *δ*=*d*/*h** and Mach number *M*=*U*_{L}/*a**, where is the speed of sound.

On assuming that density *ρ* and temperature *θ* are non-dimensionalized by *ρ*_{−∞} and *Θ*_{−∞}, respectively, the Arrhenius law for the reaction rate *Ω* can be written as(2.2)where *Y* is the mass fraction, *Θ*_{+}=1+*q* is the adiabatic flame temperature, and the factor *Ω*_{0} is chosen so that the non-dimensional speed of a flat flame is unity.

A key simplifying assumption is that of large activation energy, corresponding to *β*≫1. In AEA, the Lewis number *Le* is required to be close to unity, or more precisely *Le*=1+*β*^{−1}*l* with *l*=O(1). Instead of the reactant concentration *Y*, it is more convenient to work with enthalpy *H*,

Let (*x*, *y*, *z*) and *t* be the space and time variables normalized by *h** and *h**/*U*_{L}, respectively. Assume that the flame front is given by *x*=*f*(*y*,*z*,*t*). It is convenient to introduce a coordinate system attached to the front,and to split the velocity (normalized by *U*_{L}) as * u*=

*u*

*+*

**i***, where*

**v***is the unit vector along the duct. Then the governing equations can be written as (Matalon & Matkowsky 1982)(2.3)(2.4)(2.5)(2.6)(2.7)supplemented by the state equation 1+*

**i***γM*

^{2}

*p*=ρθ, where

*Pr*and

*γ*denote the Prandtl number and the ratio of specific heats, respectively, and is the gravity force. We have definedwith the operators ∇ and ∇

^{2}being defined with respect to the transverse variables

*η*and

*ζ*.

To make analytical progress, we assume, in addition to *β*≫1, that(2.8)

Four distinct asymptotic regions then emerge (see figure 1). In addition to the thin reaction and pre-heated zones, there are also hydrodynamic and acoustic regions, which scale on *h** and *λ**=O(*h**/*M*), respectively. Within the reaction zone of O(*d*/*β*) width, the heat release (species variation) owing to the reaction balances the thermal (mass) diffusion (Matkowsky & Sivashinsky 1979). In the pre-heated zone, the dominant balance is between the advection and diffusion. All four regions are fully interactive in the sense that the final complete solution relies on the investigation of all these regions.

The direct interaction between sound wave and flame is through the hydrodynamic region, where the solution expands as(2.9)

The mean density *R*_{0}=1 for *ξ*<0 and *R*_{0}=1/(1+*q*) for *ξ*>0. In the following, the subscript ‘0’ will be omitted. Substitution of (2.9) into (2.3)–(2.5) leads to the equations governing (*u*_{0}, **v**_{0}, *p*_{0}):(2.10)where .

The hydrodynamic field exhibits jumps across the flame front, which can be derived by analysing the internal structure of the flame. Our analysis (details of which will be published separately), shows that at leading order, the jumps across the pre-heated zone, denoted by 〚.〛, are(2.11)and 〚*h*_{0}〛=0. The front evolution is governed by(2.12)where , with , which must be determined by solving the equation governing the transport of enthalpy fluctuation,(2.13)

When *h*_{−∞}=0, (2.11) and (2.12) reduce to those of Matalon & Matkowsky (1982).

The leading-order system, consisting of (2.10)–(2.13), suffices for most of the present work, but improved results may be obtained if the jumps at O(*δ*) and O(*βM*) are used.

## 3. Acoustic–flame–enthalpy interaction

### (a) Acoustic zone

The appropriate variable describing the acoustic motion in this region is(3.1)

Owing to the transverse length being much smaller than the longitudinal length, the motion is a longitudinal oscillation about the uniform mean background, and the solution can be written as(3.2)where *U*_{±} are the mean velocities behind and ahead the flames, respectively, with *U*_{+}−*U*_{−}=*q*. The unsteady field is governed by the linearized equations(3.3)

As ,

As will be shown in §3*b*, *p*_{a} is continuous across the flame, but the flame induces a jump in *u*_{a}, i.e.(3.4)where stands for the space average of *ϕ* in the (*η*, *ζ*) plane, and *F*_{0} is defined in (3.5) below. The result indicates that an unsteady flame is deemed to generate an acoustic field. As will be shown in §3*b*, the sound wave in turn exerts a leading-order back effect on the flame and hydrodynamics, unlike typical aeroacoustic problems, where sound is a passive by-product.

### (b) Hydrodynamic region

In order to facilitate the matching with the solution in the acoustic regions, we subtract from the total field the acoustic components as well as the mean background flow by writing(3.5)where *P*_{±} is the mean pressure (with *P*_{+}−*P*_{−}=−*q*), and . Let **v**_{0}=**V**_{0}. It follows from (2.10) that the leading-order hydrodynamic field satisfies(3.6)(3.7)(3.8)where *h*(*ξ*) is the Heaviside step function, =[*u*_{a}], and . Matching with the outer acoustic solution requires that as *ξ*→±∞. The pressure and transverse velocity jumps in (2.11) become(3.9)where . The equation governing the advection of *H*_{0} becomes(3.10)

Since (*U*_{0}, **V**_{0})→0 as *ξ*→−∞, *H*_{0} satisfies the upstream condition(3.11)where function *H*_{−∞} represents the enthalpy fluctuation approaching the hydrodynamic zone.

The hydrodynamic motion affects the ambient acoustic regions by inducing a longitudinal velocity jump. To derive this key result, we take the spatial average of (3.6) in the (*η*, *ζ*) plane, and then integrate with respect to *ξ* to obtain . Inserting the first equation in (3.5) into the first jump relation in (2.11) and taking the spatial average, we findwhich on using (3.9), simplifies to (3.4). The jump condition across the flame for *U*_{0} becomes(3.12)

The hydrodynamic equations (3.6)–(3.8) and the enthalpy advection equation (3.10) are coupled to the acoustic equation (3.3) via (3.4) and the flame front equation(3.13)to form an overall interactive system. Upstream enthalpy is advected by the hydrodynamic field associated with the flame. The advection, however, is not passive, because enthalpy also influences hydrodynamics by modifying the jumps through its interaction with the thermal field of the flame. The flame motion and enthalpy also radiate sound, whose acceleration creates in essence a Taylor–Rayleigh effect to act on the hydrodynamics and ultimately on the flame.

In the rest of the paper, the general asymptotic framework will be applied to two limiting cases, for which further analytical progress can be made.

## 4. Interaction between acoustic mode and Darrieus–Landau (D–L) instability

An important manifestation of mechanism (ii) of large-scale combustion instability is the coupling of acoustic modes with small-scale D–L instability, which causes flame wrinkling and thus modulates heat release. This mechanism was considered by Pelce & Rochwerger (1992) in connection with the experiments of Searby (1992), who observed that sound was generated when a wrinkling flame was propagating downwards in a tube. However, their theoretical treatment relied on some *ad hoc* assumptions, which cannot be justified mathematically. Recently, Wu *et al*. (2003) applied the general framework of §3 (with *h*_{−∞}=0) to derive a self-consistent weakly nonlinear theory for a flame stabilized by gravity. The main result is now briefly summarized to demonstrate the advantage of the present systematic asymptotic formulation.

For simplicity, a two-dimensional flame was considered. A D–L mode with wavenumber *k*=*π* is nearly neutral when *U*_{L} is close to . Suppose that the magnitude of such a mode is of O(*ϵ*). The weakly nonlinear interaction takes place on the time-scale of O(*ϵ*^{−2}) (Stuart 1960), so that the appropriate slow variable is *τ*=*ϵ*^{2}*t*. In keeping with this, *U*_{L} is assumed to deviate from *U*_{c} by O(*ϵ*^{2}). The velocity and pressure in the hydrodynamic region expand as(4.1)while the pressure and velocity of the acoustic fluctuation expands as(4.2)where *A* and *B* are the amplitude functions of the D–L and acoustic modes respectively, and , are constants to be determined by imposing the continuity conditions across the flame, [*u*_{a,1}]=0 and [*p*_{a}]=0, and the end conditions: *u*_{a,1}=0 at and *p*_{a}=0 at , where *L* is related to the dimensional length of the duct *l** via *L*=*Ml**/*h**, and *σ* is a parameter characterizing the mean position of the flame front. This leads to the dispersion relation for acoustic modes(4.3)

The analysis of the hydrodynamics (4.1) and acoustics (4.2) can be carried out to O(*ϵ*^{3}) to give the coupled amplitude equations(4.4)with all the constants being evaluated explicitly. For a flat flame (*A*=0), (4.4) reduces to *B*′=*m*_{p}*B*, which is the result of Clavin *et al*. (1990), describing mechanism (i). If the flame amplitude *A* is artificially taken to be constant, (4.4) reduces to the result of Pelce & Rochwerger (1992), which is one-way coupling, as the back effect of sound wave on the flame is ignored. The present two-way coupled theory unifies mechanisms (i) and (ii). The system (4.4) was solved for the parameters close to those in Searby's (1992) experiments to predict the sound pressure *p*_{e}, the flame amplitude and position. The result is shown in figure 2, while Searby's measurement is reproduced in figure 3. The theory faithfully captures sound amplification and subsequent flattening of the flame. It may be noted that there is even a good degree of quantitative agreement.

## 5. Acoustic radiation of enthalpy and parametric instability

Assume that the upstream enthalpy fluctuation is one-dimensional, independent of *η* and *ζ*. Then the system admits a flat-flame solution,

For simplicity we take . The enthalpy fluctuation produces sound through the jump (3.4), which can be written as a Fourier series . Assuming that none of the harmonics in the enthalpy fluctuation has a frequency coinciding with any eigenfrequency of the duct modes, we find that the acoustic acceleration(5.1)where(5.2)with Δ_{s}(*ω*;*σ*) being given by (4.3).

The enthalpy fluctuation and the sound wave exert a periodic forcing on the flame, leading to a parametric instability. To study this, we perturb the flat flame so that(5.3)

Since (*u*_{0},**V**_{0})≪1, and equations (3.6)–(3.8) can be linearized to yield(5.4)

Then we have the solution(5.5)

Substituting (5.5) into (5.4), solving the resulting equations and making use of the linearized jump conditions and the linearized front equation *F*_{0,t}=*U*_{0}(0^{−},*η*,*ζ*), we obtain(5.6)where . When *h*_{−∞}=0, (5.6) reduces to the familiar one governing D–L instability.

Equation (5.6) admits solution of the form *α*=e^{μt}*ϕ*(*t*), with *ϕ* being a periodic solution, and *μ* the Floquet multiplier, whose real part *μ*_{r} represents the growth rate. We solved (5.6) for the same parameters as those in figure 2, except that *U*_{L}=30 cm s^{−1}. The variation of *μ*_{r} with is shown in figure 3. At 100 Hz, moderate level of enthalpy fluctuation (regime I) reduces the growth rate, and may completely stabilize the flame. However, rapid parametric instability (regime II) is induced when the enthalpy intensity exceeds a critical value. It is found that the instability in regimes I and II correspond to fundamental and subharmonic resonance, respectively. The instability is very sensitive to frequency, e.g. fluctuation at 75 Hz significantly destabilizes the flame.

As the parametric instability develops, it will produce additional sound by altering the flame surface area, and the sound wave in turn affects the parametric instability. The mutual interaction may be studied analytically when the parametric instability is close to its threshold, i.e. when is close to the boundaries of the ‘stable window’ in figure 3. Suppose that the magnitude of the parametric instability wave is O(*ϵ*). It alters the flame surface area by O(*ϵ*^{2}) to generate extra sound waves of the same magnitude, which in turn reacts back on the parametric instability. This suggests the introduction of the slow time variable *τ*=ϵ^{2}*t*. We also let so that linear and nonlinear effects are comparable. The solution for *α*(*t*) can be written as(5.7)with *α*_{0} satisfying (5.6). The parametric instability contributes an additional source , which is expressed as a Fourier series , so that the extra sound

By a standard weakly nonlinear instability analysis, we obtain at O(*ϵ*^{3}) the amplitude equation(5.8)

Figure 3 shows the evolution of *A* at the fundamental (*h*_{c}≈0.8) and subharmonic (*h*_{c}≈2.5) thresholds. For the former, the nonlinear effect has a stabilizing effect so that parametric instability does not produce substantial noise. For the latter, nonlinearity is destabilizing.

## 6. Concluding remarks

Combustion instability is a complex phenomenon involving several important processes (chemical reaction, heat transport, hydrodynamics and acoustics) that take place on distinct spatial/time-scales. By using AEA, the chemical reaction and heat transport are dealt with analytically, allowing the flame to be represented by a single order parameter, front position *F*. This leads to an asymptotic theory for combustion instability, in which the acoustic–flame interaction is described by Euler's equations coupled to the acoustics equation and the front equation (Wu *et al*. 2003). When enthalpy fluctuation is present in the oncoming mixture, that system must be further coupled with the equation governing the advection of enthalpy.

Application of the general asymptotic formalism to special cases has provided useful insights into some key aspects of combustion instability, but a thorough investigation of the fully nonlinear system is yet to be undertaken. Specifically, by solving the system numerically, we may be able to predict the self-sustained large-amplitude pressure oscillations, which occur due to a strong interaction between acoustic modes and the parametric instability of the flame.

The present asymptotic theory, which was derived for the ‘open loop’ case, may be easily adjusted to address the mechanism (iii) of combustion instability, where enthalpy fluctuation arises owing to the pressure interfering with the feeding line. A further necessary ingredient is a suitable model describing the feeding line response to the acoustic pressure.

The most important application is active control of combustion instability by using secondary fuel injection. The present asymptotic theory, which describes the underlying cause–effect relation, could be employed to guide the design of appropriate controllers, as well as to test their performance *a priori*.

## Footnotes

One contribution of 19 to a Theme ‘New developments and applications in rapid fluid flows’.

- © 2005 The Royal Society