Review
Transverse combustion instabilities: Acoustic, fluid mechanic, and flame processes

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Abstract

Thermoacoustic oscillations associated with transverse acoustic modes are routinely encountered in combustion chambers. While a large literature on this topic exists for rockets, no systematic reviews of transverse oscillations are available for air-breathing systems, such as in boilers, aircraft engines, jet engine augmentors, or power generating gas turbines. This paper reviews work on the problem for air-breathing systems, summarizing experimental, modeling, and active control studies of transverse oscillations. It then details the key physical processes controlling these oscillations by describing transverse acoustic wave motions, the effect of transverse acoustic waves on hydrodynamic instabilities, and the influence of acoustic and hydrodynamic fluid motions on the unsteady heat release. This paper particularly emphasizes the distinctions between the direct and indirect effect of transverse wave motions, by arguing that the dominant effect of the transverse acoustics is to act as the “clock” that controls the frequency and modal structure of the disturbance field. However, in many instances, it is the indirect axial flow disturbances at the nozzles (driven by pressure oscillations from the transverse mode), and the vortices that they excite, that cause the dominant heat release rate oscillations. Throughout the review, we discuss issues associated with simulating or scaling instabilities, either in subscale experimental geometries or by attempting to understand instability physics using identical nozzle hardware during axial oscillations of the same frequency as the transverse mode of interest. This review closes with a model problem that integrates many of these controlling elements, as well as recommendations for future research needs.

Introduction

This paper reviews the current state of understanding of transverse combustion instabilities, which are problematic in a variety of systems. For example, approaches for meeting increasingly stringent emissions regulations and efficiency demands have led to thermoacoustic oscillation problems in gas turbines used for power generation and propulsion. In particular, combustors running in a lean, premixed mode are highly susceptible to self-excited, combustion driven oscillations. This is in part due to the loss in acoustic damping associated with the secondary air inlet holes that are present in conventional diffusion flame combustors. For example, transverse oscillations in annular aircraft or aeroderivative combustors have been described by General Electric, Pratt & Whitney, Solar Turbines, and Rolls Royce [1], [2], [3], [4]. Similarly, transverse acoustic oscillations in annular frame-engine combustors have been described by Siemens and Alstom [5], [6]. Transverse acoustic oscillations in can-type combustors are generally higher frequency than in annular systems (the exception being radial modes in annular combustors) and have been described by Sewell and Sobieski [7]. Finally, transverse oscillations have long been an issue in jet engine augmentors [8], [9], [10], [11], [12], [13]. Referred to as “screech” because of its high-pitched tone, the transverse mode in augmentors can be excited in addition to longitudinal acoustic oscillations, referred to as “rumble” or “buzz” [14]. Transverse oscillations have also been problematic in numerous rocket systems. While we note analogies and commonalities in physics throughout, the focus of this review is on air-breathing combustors and does not attempt to comprehensively address the large literature on transverse oscillations in rockets.

Combustion instability, a coupling between resonant combustor acoustics and heat release rate fluctuations, is one of the leading challenges in developing and operating both aircraft and power-generation gas turbines [1]. Combustion instability is manifested by acoustic and heat release rate oscillations inside the combustor chamber. This feedback cycle between combustor acoustics and heat release rate oscillations typically involves the three steps outlined in Fig. 1. Heat release rate fluctuations add energy to the acoustic field, leading to acoustic pressure and velocity fluctuations that propagate throughout the combustor. These acoustic fluctuations then excite vortical structures and fuel/air ratio oscillations that, in turn, lead to further heat release fluctuations that close the feedback loop.

A necessary condition for the oscillations to be self-excited is that net energy must be added to the acoustic disturbance field by the heat release. It can be shown that this occurs when the phase of the heat release rate and pressure oscillations is within ±90°, i.e., the “Rayleigh criterion” [15]. This criterion is stated in a more general time domain formulation in Eq. (1), which indicates that the product of the heat release rate oscillation and the pressure oscillation integrated over an acoustic cycle must be greater than zero for the heat release to add energy to the acoustic field.3Vtp1(x,t)q˙1(x,t)dtdV>0

Physically, the criterion states that the oscillating heat release, which results in oscillations in local gas dilatation, transfers energy to the acoustic field by periodically doing work on the gas when the oscillations in heat release and pressure are in phase.

The conditions under which a system is linearly unstable, and the resultant amplitude of the acoustic oscillations, are dependent on the balance between disturbance growth and damping in the combustor system. In an undamped system, the value of the integral in Eq. (1) must exceed zero for oscillations to grow. However, all real systems possess some acoustic damping, and so the value of the integral must be greater than the volume-integrated system damping. Combustor damping sources include losses at inflow and outflow boundaries, and flow oscillations through cooling holes in the combustor liner excite vortices, transferring acoustic energy to vortical flow motion [19]. Similarly, narrowband acoustic damping at known problematic frequencies is often added with quarter-wave tubes and Helmholtz resonators. These damping methods are discussed further in Sec. 2.3.

When small amplitude oscillations are self-excited in linearly unstable systems, the amplitude initially grows exponentially in time. However, nonlinear effects modify the energy balance at higher amplitudes, causing the amplitude to saturate into a limit cycle, or some more complex orbit in the phase plane. For example, the unsteady heat release oscillations often do not grow linearly with acoustic disturbance amplitude, such as induced by saturation in flame area fluctuations [20].

Even if the oscillations are not self-excited (i.e., if the heat release rate oscillations do not add energy to the acoustic field, or if the rate of energy addition does not exceed system damping), significant oscillations can occur at resonant frequencies in damped systems. These noise-driven oscillations can be understood by considering the excitation of a slightly damped resonator with broadband noise, which can achieve very large amplitudes for small damping rates [17], [21].

Combustion driven oscillations can be associated with any of the natural acoustic modes of the combustor system, including the longitudinal or the transverse modes – both radial and azimuthal. By focusing on transverse acoustic disturbances, this review concentrates on the physics associated with acoustic oscillations perpendicular to the mean-flow direction in the combustor. This is in contrast to longitudinal oscillations, where acoustic fluctuations oscillate in the direction of mean flow. Longitudinal oscillations have been the focus of a number of studies; see reviews by Lieuwen and Yang [1], McManus et al. [22], Ducruix et al. [23], Huang and Yang [24], and Candel et al. [25]. While the dominant coupling mechanisms between the acoustics and the flame response, reviewed later in this section, are nominally the same between longitudinal and transverse oscillations, the symmetry of the acoustic disturbance is significantly different in these two cases. Longitudinal modes excite symmetric disturbances along the flame, meaning that the acoustic disturbance amplitude is roughly constant at all azimuthal locations around the flame. However, this is not true during transverse oscillations, where the incident acoustic oscillations are non-axisymmetric with respect to the downstream axis of the flame. This introduces new coupling pathways and additional considerations. Since combustion driven oscillations are associated with an unstable combustor system, another important distinction between longitudinal and transverse oscillations is the ability to accurately simulate them in sub-scale (laboratory) environments. Such approaches are routinely used for longitudinal mode studies, such as by studying the dynamics of a single or small number of nozzles, and can be done with reasonable replication of the acoustic mode structure. However, such sub-scale arrangements are generally more challenging for transverse oscillations.

The mechanisms by which acoustic fluctuations in the combustor excite flame heat release rate fluctuations have been reviewed by several authors [17], [23], [26], [27]. In general, coupling between the acoustic field and the flame takes place through one or more pathways where disturbances in the flow field, including fluctuations in velocity, pressure, or mixture composition, drive fluctuations in the flame heat release rate. We discuss these coupling pathways with regards to transverse instabilities in Section 7, but provide a basic overview here. “Velocity-coupled response” refers to the sensitivity of the heat release rate to velocity disturbances. The velocity disturbances are associated with both acoustic and vortical velocity fluctuations [20], [28], [29], [30], [31], [32], [33], [34], where the vortical velocity fluctuations are excited by acoustic fluctuations, as discussed in Section 6. Velocity fluctuations drive flame heat release rate fluctuations through multiple pathways, including flame area fluctuations, mass burning rate fluctuations (induced by oscillatory flame stretch and/or scalar dissipation rate), and oscillatory atomization/breakup processes. Typical characteristic delay times between the excitation of the flow disturbance and the heat release are convective in nature, usually involving the time required for a vortex to convect from its point of initiation to the midpoint of the flame (see also Secs. 4.2 and 5)

Fluctuations in mixture composition, or equivalence ratio, are common in air-breathing systems as the fuel and air flow rates are typically sensitive to acoustic disturbances. Thus, acoustic fluctuations in the combustion chamber drive fluctuations in fuel flow rate into the combustor, resulting in oscillations in local equivalence ratio [1], [35], [36], [37], [38], [39], [40]. These variations in equivalence ratio drive heat release rate fluctuations through a number of pathways, including oscillations in the mixture heat of reaction and burning rate. Typical characteristic delay times between the excitation of the fuel/air disturbance and the heat release are also convective in nature, associated with the convection time from the fuel injection point to the midpoint of the flame (e.g., see Eq. (7)).

In addition, acoustic oscillations drive heat release rate oscillations through fluctuations in pressure, temperature, and density, which are isentropically related in the acoustic field. In air-breathing systems, the effect of pressure fluctuations is often of O (M) relative to velocity and fuel/air ratio coupling and, thus, is weak in low Mach number flows [17], [41], [42], [43], [44], [45], [46]. In liquid-fueled systems, fluctuations in pressure or velocity may also drive variation in fuel injection, atomization, and vaporization rates, further leading to fluctuations in local equivalence ratio; much of the literature on this liquid-driven coupling mechanism focuses on liquid-rocket applications, although a smaller general literature focuses on the response of sprays and droplets [47], [48], [49].

Users and operators of air-breathing systems often categorize combustion instabilities by their frequency. For example, “high frequency” instabilities are often termed “screech,” and “low frequency” ones are referred to as “rumble” or “growl”. While this distinction is useful in identifying different ranges in practice, it does not necessarily imply that key coupling physics are different. A more physics-based distinction is based upon relative time and length scales; in particular, the flame response time scale relative time to the acoustic period (the flame Strouhal number, St, in Eq. (39)) and the acoustic wavelength relative to the nozzle dimension or flame length - a flame that is small relative to a wavelength is referred to as “acoustically compact”. For acoustically compact flames, the distribution of the unsteady heat release is usually not important; rather, the spatially integrated unsteady heat release controls stability limits. In contrast, the spatial distribution of the unsteady heat release is very important in non-compact flames, as it is the relative phase of the local heat release and pressure that controls the value of the Rayleigh integral in Eq. (1).

Transverse oscillations have been the focus of significant investigation in liquid rockets. They are addressed in a number of books and monographs, including Yang and Anderson [50], Crocco et al. [51], Harrje and Reardon [52], and Dranovsky [53]. Initially, large-scale rocket testing revealed several issues, including mid-to high-frequency oscillations driven by a coupling between acoustic disturbances and flow and/or combustion processes. These oscillations were often abated with the use of baffles, Helmholtz resonators, and alterations to spray and impingement patterns. Initially, design of these abatement systems was largely empirical, but better understanding of the structure of the acoustic mode, particularly the transverse acoustic modes, reduced the number of design iterations required. Although the baffle and resonator solutions proved useful, questions regarding the underlying coupling mechanisms were often unanswered.

Several laboratory experiments have been designed to replicate the dominant transverse mode measured in the rocket combustion chambers, enabling a more detailed understanding of instability mechanisms [50], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66]. Test configurations include both single- and multiple-injector configurations in a variety of high-pressure chambers. A number of the laboratory combustors are rectangular; in this configuration, the injectors are located on one side of the chamber and the port for acoustic forcing is on a perpendicular face. Some chambers, however, are cylindrical, including those for both high- and low-pressure testing. The flame is typically forced with a siren [63]. This configuration allows for a high back-pressure on the forcing apparatus in order to achieve significant excitation amplitudes, as well as frequency selection by variation of the siren's rotation speed.

In addition, Reardon and coworkers [67], [68] developed a “sector motor” in which only a fixed sector angle of the engine was tested by using copper inserts that blocked portions of the chamber. For example, a combustor was fabricated that consisted of a 180° chamber (half the chamber), that displayed only the standing form of an azimuthal mode. They also tested angles of 20, 60, 90, 120, 210, and 270°, and found that changing the angle of their variable angle sector motor caused transition from stable operation at small angles, to a self-excited “1T” mode, then to a self-excited “2T” mode for angles larger than 180°.

These facilities enabled insight into physical mechanisms responsible for self-excited oscillations. “Injector coupling,” meaning the acoustic coupling between the pressure oscillations at the injector face associated with the acoustic mode in the chamber and oscillations in reactant mass flow rates from the injector elements, was identified as a significant mechanism [69]. For example, low pressure-drop injectors were shown to be highly susceptible to thermoacoustic oscillations; increases in pressure drop, decreasing the acoustic sensitivity of the nozzle flow rate, reduced the oscillation magnitude [70].

However, other mechanisms are also important. For example, one set of tests showed that increasing the pressure drop did not eliminate the oscillations. Rather, in that case, disturbance amplitudes decreased by increasing the velocity ratio between the fuel and oxidizer streams in a co-axial injector configuration [70]. This, along with positive results stemming from recessing one stream, increasing the hydrogen temperature, and increasing the thickness of the oxidizer tube, all suggest the additional role of coupling of acoustic fluctuations and jet hydrodynamic instabilities. Finally, jet breakup, atomization, and subsequent mixing of liquid fuel and oxidizer streams also profoundly influence thermoacoustic instability limits by affecting time delays between injection and ignition [71], [72], [73].

While much research in the area of combustion instability in rockets is on-going, many lessons from this research can be applied to air-breathing architectures. For example, an analog of injector coupling is similarly a dominant mechanism of transverse oscillations in annular combustors [74], [75], and is the focus of Sec. 3.2.1. Additionally, variations in liquid jet breakup, atomization, and mixing are similarly important coupling mechanism in liquid-fueled combustors [76].

As described above, there is a large literature on transverse oscillations in rocket environments. However, we are not aware of a focused review of the problem in air-breathing systems. The objective of this review is to summarize, synthesize, and suggest areas for needed work for air-breathing systems, particularly gas turbines. While this review addresses the general issues associated with instabilities in both can and annular systems, it has a disproportionally larger treatment of annular combustors, where azimuthal disturbances are often the dominant mode of combustion instability. The simple reason for this focus is that annular systems are the subject of the majority of the open literature on the topic.

Fig. 2 shows a conceptual model of an annular combustor, outlining the major geometric, acoustic, and flame features. We start with the common upstream compressor discharge plenum (not shown), which discharges into a ring of fuel nozzles. Here, only a single ring of nozzles is shown, although multiple rings do exist in fielded configurations [77]. Nozzle ‘s’ has a cross sectional area, Acs,s, a length h, and a nozzle spacing ds. While the plenum has its own acoustic characteristics that may also support azimuthal disturbances (e.g., see Bauerheim et al. [78]), for simplicity we parameterize everything upstream of the nozzle by the impedance boundary condition, Zˆo, which is a function of operating condition, frequency, and acoustics of the upstream plenum. The flow-field created by these nozzles is often highly complex and hydrodynamically unstable, and the acoustic flow oscillations generally excite large scale vortical disturbances that also enter the problem. The flames, shown in blue, are located downstream of the nozzle outlet.

The bolded black line in the figure indicates a transverse distance. An important simplification that we will utilize in several places is that the cross sectional area of each inlet nozzle is much smaller than that of an axially oriented cut through the combustor, i.e., Acs,s/Acs,A ≪ 1. In this case, the unsteady volume flow rate associated with transverse flow oscillations is generally much larger than the volume flow oscillations in the nozzles, enabling us to decouple their acoustics in many instances. In this way, the azimuthal acoustic fluctuations act as a “clock,” setting the overall frequency of the disturbance. These disturbances excite axial motions in the nozzle, which are often the dominant source of flame response.

The acoustic boundary condition at the exit of the combustor is denoted as Zˆout. Parameterizing the outflow by this local boundary condition is an approximation that is accurate in cases where the nozzle dimensions are small relative to the acoustic wavelength [79].

Although no corresponding figure is included, many similar considerations apply for a can-combustion system. A key difference however, is that the cross sectional area of the nozzles is on the same order of magnitude as that of the combustor can, so that the area ratio argument described above does not apply. In this case, the acoustics of the region up and downstream of the flame are integrally linked into an overall system mode, with a mode structure that cannot be decoupled. This renders the can-combustor problem significantly more complicated.

The rest of this paper is organized as follows. Sec. 2 provides a review of the experimental, analytical, and modeling studies of transverse combustion instabilities in air-breathing combustor systems. In addition to experimental and computational studies, we discuss investigations into the control of transverse combustion instabilities. Then, subsequent sections describe the key physical processes during transverse combustion instabilities, with a discussion of acoustic field structure of transverse oscillations (Sec. 3), excitation of hydrodynamic instabilities by transverse forcing (Sec. 4), flame response to transverse modes (Sec. 5), and an illustrative model problem for transverse thermoacoustic instability (Sec. 6). We conclude by identifying key gaps in understanding, and recommendations for future research topics.

Section snippets

Experimental, modeling, and combustion control efforts

This section summarizes experimental, modeling, and instability control efforts in the area of transverse combustion instability for gas turbine applications, which are listed in Table 1.

Transverse modes in circular geometries

This section summarizes transverse acoustic mode shapes and frequencies of circular combustion chambers. We will quickly summarize classical duct acoustic results, and refer the readers to more detailed references for further discussion [17], [187]. Transverse modes in rectangular geometries are a straightforward generalization of one-dimensional axial modes (axial modes are denoted with the index, n) to the transverse direction, and are not discussed further. In circular geometries, the

Hydrodynamic instabilities

Although combustion driven oscillations are usually associated with natural acoustic frequencies that serve as the “clock” for the system frequency, the dominant source of disturbances that actually lead to heat release rate oscillations are often vortical flow disturbances. These large scale vortical disturbances themselves arise from concentration of the vorticity that largely originates from boundary layers in approach flow passages or other walls. For example, the separating free shear

Flame response

This section discusses more fully the sensitivity of the flame to transverse instabilities. Starting with transverse flow and pressure disturbances as the input, and heat release oscillations as the output, there are multiple pathways through which heat release oscillations are excited, as shown in Fig. 30. The flame is dominantly sensitive to flow disturbances (referred to as “velocity coupling”), as well as any fuel/air ratio oscillations excited by these disturbances (“fuel/air ratio

System thermoacoustic instability model problem

Having considered different elements of the problem, we now present results from a simplified, quasi one-dimensional model that combines many of the problem's key physical elements into an overall thermoacoustic stability analysis of azimuthal modes. This analysis follows the presentations by Stow et al. [79] and Parmentier et al. [122], and will consider azimuthal oscillations in a thin annular combustor, ignoring plenum acoustics (see, e.g., Bauerheim et al. [78], for inclusion of plenum

Concluding remarks and future work

Transverse modes are a dominant mode of combustion instability in a range of combustion devices, and the challenges associated with studying and controlling these modes are arguably greater than those associated with longitudinal instabilities. Prominent observations from this review are organized next under the headings of acoustics, hydrodynamics, and flame response:

  • Acoustics:

    • In annular combustion systems with large area ratios between nozzles and the combustor cross sectional area, Acs,s/Acs,

Acknowledgments

Vishal Acharya and Timothy Lieuwen acknowledge support from the US Department of Energy contract DE-NT0005054 (contract monitor Mark Freeman) and from the National Science Foundation contract CBET-1235779 (contract monitor Professor Ruey-Hung Chen). Jacqueline O'Connor gratefully acknowledges the support of the College of Engineering and the Department of Mechanical and Nuclear Engineering at the Pennsylvania State University. The authors also gratefully acknowledge many helpful comments and

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