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.

The two- and three-dimensional truss examples presented in this chapter demonstrate the complex and often unexpected load deflection behavior exhibited when, in particular, geometrical nonlinearity is included in structural analysis. Each point on the various graphs shown below represents an equilibrium configuration; however, these configurations may be structurally stable or unstable. For a chosen load it can be observed that the structure can be in a variety of equilibrium configurations. For most structures subjected to “in service” loadings, this is clearly unacceptable (not to say alarming), nevertheless such analysis can indicate possible collapse scenarios. While the points on a load deflection graph refer to equilibrium configurations, it must not be assumed that connecting adjacent points necessarily represents smooth continuity of the motion of the structure as the loading changes. However, such smooth motion is likely to be the case if a large number of load increments are employed in the solution, but it cannot be guaranteed.

A situation where a small change in load leads to a dramatic change in configuration is known as “snap-through” behavior. There are “structures” that rely on snap-through behavior to fulfill a useful function. Indeed such structures are vastly more numerous than everyday structures; for example, a shampoo container cap when opened carefully will suddenly “flick” into a fully opened position. A child's hair clip often employs snap-through behavior to lock into position, while perhaps the most common item is the simple light switch.

This worked examples text is intended primarily as a companion to the second edition of the textbook Nonlinear Continuum Mechanics for Finite Element Analysis by Javier Bonet and Richard D. Wood. However, to be reasonably self-contained, where necessary key equations from the textbook are replicated in each chapter.

Textbook equation numbers given at the beginning of each chapter are indicated in square brackets.

Exercises are presented in a mix of direct (tensor), matrix, or indicial notation, whichever provides the greater clarity. Indicial notation is used only when strictly necessary and with summations clearly indicated.

The textbook is augmented by a website, www.flagshyp.com, which contains corrections, software, and sample input data. Updates to this worked examples text will also be included on the website as necessary.