This section previews the structure and content of this book and provides suggestions for how readers of different backgrounds can use it most profitably. The bulk of Chapter 1 is dedicated to reviewing the basic equations to be used in this text. The remainder of the book is divided into three main sections: Chapters 2–6, 7–9, and 10–12. The first section, Chapters 2–6, discusses flow disturbances in combustors. Chapter 2 details how different types of disturbances arise and propagate in inhomogeneous, reacting combustor environments. By introducing the decomposition of flow disturbances into acoustic, vortical, and entropy disturbances, this chapter sets the stage for Chapters 3–6, which delve into the dynamics of disturbances in inhomogeneous environments in more detail. Specifically, Chapters 3 and 4 focus on the evolution of vortical disturbances in combustor environments. Chapter 3 provides a general overview of hydrodynamic stability theory and details some general features controlling the conditions under which flows are unstable. Chapter 4 then details specific canonical flow configurations that are particularly relevant to combustor environments, such as shear layers, wakes, and swirling jets. This chapter also discusses effects of flow inhomogeneity and acoustic forcing effects on flow instabilities.

Chapters 5 and 6 treat acoustic wave propagation in combustor environments. Chapter 5 provides a general introduction to acoustic wave propagation, boundary conditions, and natural acoustic modes. Chapter 6 then provides additional treatment of the effects of heat release, mean flow, and complex geometries on sound waves. This chapter also includes an extensive discussion of thermoacoustic instabilities.

Overview

This chapter describes the processes associated with spontaneous ignition (or auto-ignition) and forced ignition. The forced ignition problem is of significant interest in most combustors, as an external ignition source is almost always needed to initiate reaction. Two examples in which the autoignition problem is relevant for flowing systems are illustrated in Figure 8–1 [1–12]. Figure 8–1(a) depicts the autoignition of high-temperature premixed reactants in a premixing duct. This is generally undesirable and is an important design consideration in premixer design. Figure 8–1(b) depicts the ignition of a jet of premixed reactants by recirculating hot products. In this case, autoignition plays an important role in flame stabilization and must be understood in order to predict the operational space over which combustion can be sustained. Although not shown, autoignition can also occur during the injection of a fuel, air, or premixed reactants jet into a stream of hot fuel, air, or products. For example, a vitiated H2/CO stream reacts with a cross-flow air jet in RQL combustors [13].

Figure 8–2 shows several canonical configurations used to study ignition that are referred to in this chapter. These are (a) the ignition of premixed reactants by hot gases, (b) the ignition of a non-premixed flame by either a hot fuel or air stream or an external spark, and (c) stagnating flow of fuel or premixed reactants into a hot gas stream [14].

This chapter continues the treatment initiated in Chapter 3 on the evolution of vorticity in flows. We now focus on specific flow fields and include the effects of heat release and external forcing. Hydrodynamic flow stability is a large, rich field; this chapter can provide only a brief introduction to the many fascinating instabilities that arise [1]. For these reasons, attention is specifically focused on high Reynolds number flows and several specific flow configurations of particular significance in combustor systems, including shear layers, wakes, jets, and backward-facing steps.

The vorticity that controls the hydrodynamic stability features of the flow originates largely from the boundary layers in approach flow passages or other walls. The separating boundary layer characteristics serve, then, as an important initial condition for the flows of interest to this chapter. Discussion of the stability and coherent structures present in boundary layers is outside the scope of this book, but several important characteristics of boundary layers are summarized in Aside 4.1.

This chapter starts with a discussion of free shear layers in Section 4.1, the most fundamental hydrodynamic instability of interest to practical combustors. It then considersmore complex flows that largely involve interactions of multiple free shear layers or of shear layers with walls. For example, two-dimensional wakes (Section 4.2) and jets (Section 4.3) are equivalent to two free shear layers separated by some distance, a, of oppositely signed vorticity. It is recommended that readers using this book as a text focus on Section 4.1, and then use the remaining sections as references for other specific flow configurations as required.

A key focus of this text is to relate the manner in which fluctuations in flow or thermodynamic variables propagate and interact in combustion systems. In this chapter, we demonstrate that combustor disturbances can be decomposed into three canonical types of fluctuations, referred to here as acoustic, entropy, and vorticity disturbances. This decomposition is highly illustrative in understanding the spatial/temporal dynamics of combustor disturbances [1]. For example, we show that unsteady flow motions can be decomposed into acoustic fluctuations, which propagate as waves at the speed of sound, and vorticity fluctuations, which are advected by the flow. This decomposition is important because, as shown in Chapters 11 and 12, two velocity disturbances of the same magnitude can lead to very different influences on the flame, depending on their phase speeds and space–time correlation. Aside 2.2 further emphasizes how this decomposition provides insight into behavior measured in a harmonically oscillating flow field.

This chapter is organized in the following manner. Section 2.1 introduces the basic approach for analyzing disturbances, and illustrates the formal process of perturbation expansions used throughout the text. Section 2.2 then considers small-amplitude disturbance propagation in homogeneous flows. This limit is helpful for understanding key aspects of the problem, as the disturbance modes do not interact and are not excited. Section 2.3 closely follows this material by treating the effects of boundary conditions, finite amplitude disturbances, and inhomogeneities, and shows how these effects cause interaction and/or excitation of these modes. Sec-tion 2.4 then considers the energy density and energy flux associated with these fluctuations.

Chapter 8 considered ignition and the processes associated with autoignition and forced ignition of a nonreactive mixture. In this chapter, we focus on premixed and non-premixed flames and the key physics controlling burning rates and extinction processes. Section 9.1 summarizes basic issues associated with the structure and burning rate of steady, premixed flames in one-dimensional flow fields. This includes discussions of the effects of pressure, temperature, and stoichiometry on burning rates. Section 9.2 discusses how these one-dimensional characteristics are altered by stretch; that is, fluid mechanic shear or flame curvature. We then discuss how these lead to changes in burning rate and, for large enough levels of stretch, cause the flame to extinguish. Section 9.3 treats the effects of unsteadiness in pressure, fuel/air ratio, and stretch rate. Specifically, we discuss how the flame acts as a low pass filter to disturbances in most cases, and that its sensitivity to disturbances diminishes with increasing frequency. These results have important implications for many combustion instability phenomena, in which the flame is perturbed by time varying flow and composition variations.

We then move to non-premixed flames. Section 9.4 reviews non-premixed flames in the fast chemistry limit, and Section 9.5 discusses finite rate kinetic effects. This section shows that large gradients in fuel and oxidizer concentrations can lead to flame extinction.

Chapter 11 described the dynamics of flamelets forced by velocity or burning rate oscillations and illustrated the key physics controlling the spatiotemporal dynamics of the flame position. This chapter focuses on the impacts of these disturbances on the mass burning rate and/or heat release rate itself. For example, a key quantity of interest for the thermoacoustic instability problem is the heat release fluctuations that are induced by the flame wrinkling processes described in Chapter 11. Section 12.1 overviews basic mechanisms through which flow disturbances lead to heat release oscillations, and differentiates among velocity coupling, fuel/air ratio coupling, pressure coupling, and acceleration coupling. These are quantitatively analyzed in the linear regime in Section 12.2. Key questions addressed in this section are the gain and phase responses of the unsteady heat release in response to different types of disturbances. For example, given a disturbance velocity fluctuation of magnitude ɛ, what are the magnitude and phase shift of the resulting unsteady heat release, Q̇(t)? This phase shift has profound implications on thermoacoustic instability limits in particular. We also detail how these gain and phase shifts are functions of the flame configuration, such as its length and spreading angle, as well as the frequency. Nonlinear effects are discussed in Section 12.3. As detailed in Section 6.7.2.2, the amplitude dependence of the flame response is critically important in controlling the limit cycle oscillations in self-excited instabilities.

Section 12.4 then treats broadband flame excitation and the generation of sound by turbulent flames. Section 12.4.1 discusses the influence of broadband fluctuations on the time-averaged burning rate, a key problem in turbulent combustion. Section 12.4.2 treats the spectrum of heat release fluctuations induced by broadband flow disturbances, an important problem for combustion noise applications. Finally, Section 12.4.3 treats the sound generated by unsteady heat release fluctuations.

Introduction

This book is about unsteady combusting flows, with a particular emphasis on the system dynamics that occur at the intersection of the combustion, fluid mechanics, and acoustic disciplines – that is, on combustor physics. In other words, this is not a combustion book – rather, it treats the interactions of flames with unsteady flow processes that control the behavior of combustor systems. Whereas numerous topics in reactive flow dynamics are “unsteady” (e.g., internal combustion engines, detonations, flame flickering in buoyancy dominated flows, and thermoacoustic instabilities), this text focuses specifically on unsteady combustor issues in high Reynolds number, gas phase, subsonic flows. This book is written for individuals with a background in fluid mechanics and combustion (it does not presuppose a background in acoustics) and is organized to synthesize these fields into a coherent understanding of the intrinsically unsteady processes in combustors.

Unsteady combustor processes define many of the most important considerations associated with modern combustor design. These unsteady processes include transient, time harmonic, and statistically stationary, stochastic processes. For example, ignition, flame blowoff, and flashback are transient combustor issues that often define the range of fuel/air ratios or velocities over which a combustor can operate. As we discuss in this book, these transient processes involve the coupling of chemical kinetics, mass and energy transport, flame propagation in high shear flow regions, hydrodynamic flow stability, and interaction of flame-induced dilatation on the flow field – much more than a simple balance of flame speed and flow velocity.

The final two chapters of this book treat the response of flames to forced disturbances, both time-harmonic and random. This chapter focuses on local flame dynamics; that is, on characterizing the local space-time fluctuations in position of the flame. Chapter 12 treats the resulting heat release induced by disturbances, as well as sound generation by heat release fluctuations. These two chapters particularly stress the time-harmonic problem with more limited coverage of flames excited by stochastic disturbances. This latter problem is essentially the focus of turbulent combustion studies, a topic that is the focus of dedicated treatments [1–3].

These unsteady flame–flow interactions involve kinetic, fluid mechanic, and acoustic processes over a large range of scales. Fundamentally different physical processes may dominate in different regions of the relevant parameter space, depending on the relative magnitudes of various temporal/spatial scales. Section 11.1 starts the chapter by reviewing the key length and time scales involved with flame–flow interactions. Then, Sections 11.2 and 11.3 analyze premixed and non-premixed flame dynamics, respectively.

Chapters 2 through 6 focused on disturbances in combustor environments and how they evolve in space and time. This chapter initiates the second section of this book, Chapters 7 through 9, which focus on reactive processes and their interactions with the flow. This particular chapter treats the hydrodynamic influence of the flame on the flow field in the thin flame limit. In this limit, the internal flame structure does not need to be considered. The flame acts as a volume/energy source that leads to discontinuities in flow properties or their derivatives, such as velocity, vorticity, or entropy. Wrinkling on the flame also leads to modification of the approach flow velocity field. Kinetically controlled phenomena are treated in Chapter 8, which treats ignition processes, and in Chapter 9 which treats premixed and non-premixed flames. This chapter focuses almost exclusively on premixed flames where the flame–flow coupling must be explicitly accounted for to describe many important phenomena. In contrast, the gas expansion induced by non-premixed combustion modifies the flow field, but its impact is more quantitative than qualitatative.

Section 7.1 works out the jump conditions across a thin, premixed flame and shows how flames modify flow vorticity and velocity. There is no specific section on non-premixed flame jump conditions, so we briefly note here that such jump conditions, based on one-step kinetics, stipulate that the diffusive fluxes of fuel and oxidizer into the reaction sheet occur in stoichiometric proportions (see Section 9.4 and Eq. (9.27) specifically) and that the jump in sensible enthalpy gradient on the fuel and oxidizer side is directly proportional to the fuel/oxidizer diffusive flux (i.e., the mass burning rate). The reader is referred to Section 5.5.1 in Law [1] or Section 3.1.5 in Williams [2] for these non-premixed flame derivations.

This chapter discusses acoustic wave propagation in combustor environments. As noted in Chapter 2, acoustic waves propagate energy and information through the medium without requiring bulk advection of the actual flow particles. For this reason, and as detailed further in this chapter, the details of the time-averaged flow has relatively minor influences on the acoustic wave field (except in higher Mach number flows). In contrast, vortical disturbances, which propagate with the local flow field, are highly sensitive to the flow details. For these reasons, there is no analogue in the acoustic problem to the myriad different ways in which vorticity can organize and reorganize itself as in the hydrodynamic stability problem. Rather, the acoustic field is insensitive to these details and is largely controlled by the boundaries and sound speed field.

The acoustic problem, however, has its own unique, distinctive set of rich physics. In particular, sound waves reflect off of boundaries and refract around bends or other obstacles. In contrast, vortical and entropy disturbances advect out of the domain in which they are excited – the only way in which they can further influence the disturbance field in the system is if they excite backward-propagating sound waves, a topic discussed in Section 6.4.2. The wave propagation nature of sound waves also implies that an acoustic disturbance in any part of the system will make itself felt in every other region of the flow. For example, an acoustic disturbance in the combustor propagates upstream and causes oscillations throughout the air flow passages, into fuel supply systems, and all other locations downstream of a sonic point.